Magnettorheological flud technology  applications in vehicle systems

Physical.Properties

The discovery and development of magnetorheological (MR) fluids began with Jacob Rabinow at the U.S National Bureau of Standards in the late 1940s Notably, although MR fluids were introduced around the same time as electrorheological (ER) fluids, they garnered more patents and publications in the late 1940s and early 1950s However, the lack of high-quality MR fluids has led to limited relevant literature, aside from a brief surge of publications following their initial discovery Fortunately, there has been a recent resurgence of interest in MR fluids, signaling a renewed focus in this area of research.

MR.fluids.belong.to.a.family.of.rheological.materials.that.undergo.rheological phase-change under the application of magnetic fields Typically,.

Magnetorheological (MR) fluids consist of soft ferromagnetic or paramagnetic particles, ranging from 0.03 to 10 micrometers, dispersed in a carrier fluid These magnetizable particles, typically pure iron, carbonyl iron, or cobalt powder, are combined with a non-magnetic carrier fluid, often silicone or mineral oil In the absence of a magnetic field, MR particles are randomly distributed; however, when exposed to a magnetic field, they align and form chains, creating a reversible yield stress in the fluid This yield stress can be rapidly adjusted in response to changes in the magnetic field intensity, providing MR fluid-based devices with advantages such as a continuously variable dynamic range and quick response times.

From a fluid mechanics perspective, magnetorheological (MR) fluid behaves as a Newtonian fluid without a magnetic field, but demonstrates unique Bingham plastic characteristics when exposed to such a field Consequently, MR fluid is typically modeled as a Bingham fluid, with its constitutive equation defined accordingly.

The dynamic yield stress (τ_y) of magnetorheological (MR) fluid is influenced by the applied magnetic field, which can be represented by magnetic flux density (B) or magnetic field strength (H) As the magnetic field increases, the dynamic yield shear stress also rises, indicating a transition from Newtonian to Bingham behavior The total shear stress in MR fluid comprises two components: viscous-induced stress, proportional to the shear rate, and field-dependent yield shear stress, which has an exponential relationship with the electric field MR fluid exhibits distinct rheological behaviors across different regions: it behaves as a linear viscoelastic material in the pre-yield region, a non-linear viscoelastic material in the yield region, and a plastic material in the post-yield region Understanding the rheological properties in these regions is crucial for designing MR application devices such as dampers, mounts, valves, and clutches, where the field-dependent dynamic yield shear stress and complex modulus (G*) are key properties to consider.

The microstructure of magnetorheological (MR) fluids plays a crucial role in the pre-yield region, which is essential for designing smart structures aimed at noise control and shock wave isolation In smart structures, the wave motion within the MR fluid domain generates minimal strain in the pre-yield region, contributing to their effectiveness in various applications.

A rotational shear-mode viscometer is typically utilized to assess the yield shear stress and current density of magnetorheological (MR) fluids As illustrated in Figure 1.4, a parallel disk rheometer is specifically designed to evaluate the properties of MR fluids effectively.

Bingham.behavior.of.MR.fluids.

Increasing Magnetic Field Pre-Yield Post-Yield

Shear.stress–shear.rate.relationship.of.MR.fluids.

The MR cell features a magnetic core that generates a magnetic field, with a gap between a rotating disk and a stationary plate filled with MR fluid A DC servomotor regulates the disk's rotational speed to control the shear rate, while a torque transducer measures the torque produced This data is converted through an analog-to-digital converter to a personal computer, which calculates the corresponding shear stress To assess the field-dependent viscoelastic properties of the MR fluid, a rotary oscillation test is conducted using a dynamic spectrometer The instrument utilizes a parallel-plate fixture where transducers on the bottom plate measure torque and normal force, while the servo drives the top plate MR fluid is placed in the gap between the circular plates, and a magnetic field is applied, allowing for the measurement of the complex shear modulus at varying oscillatory frequencies under constant magnetic fields.

Magnetorheological (MR) fluids are magnetic analogues of electrorheological (ER) fluids, sharing similar rheological characteristics However, MR fluids exhibit several distinct features, including a significantly higher achievable yield stress, ranging from 50 to 100 kPa, compared to 2 to 10 kPa for ER fluids Additionally, MR fluids demonstrate resilience against impurities and contaminants encountered during manufacturing and operation Devices utilizing MR fluids typically require power below 50 Watts, operating at 12 to 24 volts and 1 to 2 amps, which can be easily supplied by conventional batteries.

Experimental.apparatus.for.Bingham.characteristics.measurement.

The synthesis of high-performance magnetorheological (MR) fluids involves several critical factors These fluids should exhibit high yield stress under maximum magnetic fields while maintaining low viscosity in the absence of a field Additionally, the MR effect must remain stable across a broad temperature range Research should also focus on sedimentation, incompressibility, specific heat transfer properties, wear characteristics, fatigue properties, and lubrication to support the commercial development of advanced MR fluids A key aspect from a control perspective is the fluid's response time to the magnetic field, which must be quantitatively assessed in relation to particle size, shape, base oil viscosity, and particle conductivity Understanding the fluid time constant is essential for creating a dynamic model for fluid-based actuators and implementing real-time feedback control for MR devices Notably, commercially available MR fluids have a response time of less than 3 milliseconds, making them suitable for dynamic systems in vehicles, such as suspension and engine mounts.

Potential.Applications

When MR fluid is utilized as a controllable actuator in application devices, its working behavior is categorized into three distinct modes, as illustrated in Figure 1.6 In the shear mode, it is typically assumed that

Quartz (6.5 OD) Window Fiber Optic

The parallel-plate fixture designed for a rheometric mechanical spectrometer features one of the two field activation parts that can freely translate or rotate around the other Devices that operate in shear mode, such as clutches, brakes, and vibration isolation mounts for small magnitude excitation, benefit from this design Additionally, rotational type MR clutches, as illustrated in Figure 1.7, are effective for torque transmission and tension control in various dynamic systems.

Operating.modes.of.MR.fluids.

The transmitted torque in MR fluid systems can be effectively controlled by adjusting the field intensity applied to the fluid domain In flow mode, the two field activation components are fixed, and the actuating force is managed by varying the pressure difference between two control volumes Applications of flow mode devices include shock absorbers in vehicle suspension systems, seat dampers, recoil dampers, landing gear, large stroke dampers for bridges and buildings, seismic dampers, and vibration isolation mounts For instance, MR dampers in passenger vehicles adjust the pressure difference between damper chambers to provide optimal damping forces under various road conditions, enhancing passenger comfort and driving stability Additionally, flow mode dampers are utilized in medical devices, such as prosthetics While most MR dampers in flow mode operate semi-actively to dissipate energy, hydraulic valve systems can leverage MR fluid as an active actuator, allowing control over pressure and flow rates through field intensity adjustments This capability enables precise control of position and velocity in integrated cylinder systems, positioning MR hydraulic valve systems as viable replacements for conventional servo-valve systems across various applications.

The MR clutch operates in various modes, with the squeeze mode allowing for the adjustment of the activation gap in the vertical direction In this mode, the MR fluid is compressed by a normal force, enabling control over both tensile and compressive forces through field intensity Applications for devices in squeeze mode include vibration isolation systems designed for small magnitude excitations, such as the mount illustrated in Figure 1.11 Additionally, MR fluids can be integrated into traditional host structures to create smart structures.

Photograph.of.MR.damper.for.a.leg.

The complex modulus of smart structures can be controlled by adjusting the field intensity, as illustrated by the MR shock absorber in Figure 1.12 This capability allows for the modulation of the modal characteristics, including natural frequencies and mode shapes, effectively managing unwanted vibrations and noises caused by resonance or external excitations Applications of smart structures utilizing MR fluids span various fields, including dash panels, aircraft wings, helicopters, and flexible robotic arms.

Photograph.of.the.MR.valve.

Photograph.of.the.MR.mount.

Most devices utilizing MR fluid technology require integration with semi-active or active control systems to achieve commercial viability Essential elements for reliable MR device design include a straightforward methodology, precise dynamic modeling, and effective control algorithms A reliable design should prioritize simplicity, compactness, and durability, alongside ease of operation Simplifying design can be accomplished by minimizing components and eliminating moving parts when feasible The mechanism's size is crucial for real-world applications, and optimization should consider MR effects related to design parameters like field-activated gap and length Durability encompasses both mechanical aspects, such as fatigue and wear resistance, and magnetic durability, which reflects the mechanism's operational longevity in a magnetic field This longevity is influenced by gap uniformity during motion and material selection User-friendliness is vital for both operation and maintenance, including the process of refilling MR fluid Achieving these design objectives necessitates accurate analysis and dynamic modeling of the fluid's behavior to ensure reliable specifications.

MR fluid-based control systems comprise sensors, signal converters, a microprocessor, a voltage isolator, a current amplifier, and a control algorithm Current sensors, like accelerometers, can be modified to assess the dynamic response of devices The microprocessor, which features A/D and D/A signal converters, is crucial for effective closed-loop control timing.

Smart structures utilizing magnetorheological (MR) fluid require a microprocessor with a minimum of 12 bits for effective control software and the ability to handle high sampling frequencies of up to 10 kHz The current amplifier's response time must be sufficiently rapid to support the feedback control system's operation Most existing control algorithms for MR fluid-based vibration attenuation are semi-active, with performance improvements achievable in active mode without necessitating large power sources The skyhook control algorithm is a widely adopted approach due to its straightforward formulation and implementation Other semi-active control schemes include groundhook controllers, sky-groundhook controllers, and neuro-fuzzy skyhook controllers Potential active controller candidates for these systems encompass PID, LQG, sliding mode, H-infinity, and Lyapunov-based state feedback controllers It is crucial to consider that semi-active actuators cannot enhance the mechanical energy of the control system, necessitating careful adaptation of active control strategies Conversely, an active control system can be developed by integrating a hydraulic MR valve-cylinder system in a closed-loop configuration, allowing the application of existing active control strategies without modification.

[1].Rabinow,.J 1948 The.magnetic.fluid.clutch AIEE Transactions.67:.1308-1315. [2].Rabinow,.J 1948 Magnetic.fluid.clutch National Bureau of Standards Technical

[3].Rabinow,.J 1951 Magnetic.fluid.torque.and.force.transmitting.device United. States.Patent.2.

[4].Jolly,.M R.,.Jonathan,.W B.,.and.Carlson,.J D 1999 Properties.and.applications. of.commercial.magnetorheological.fluids Journal of Intelligent Material Systems and Structures.10:.5–13.

[5].Carlson, J D., Catanzarite, D M., and St Clair, K A 1996 Commercial. magneto-rheological devices International Journal of Modern Physics B 10:. 2857–2865.

[6].Li,.W H.,.Chen,.G.,.and.Yeo,.S H 1999 Viscoelastic.properties.of.MR.fluids

[7].Claracq,.J.,.Sarrazin,.J.,.and.Montfort,.J P 2003 Viscoelastic.properties.of.magnetorheological.fluids Rheologica ACTA.43:.38–49.

[8].Wang,.Y 2006 Modeling.of.polymer.melt/nanoparticle.composites.and.magneto-rheological.fluids,.Ph D Dissertation,.Ohio.State.University,.Columbus.

[9].Wei,.B.,.Cong,.X.,.and.Jiang,.W 2010 Influence.of.polyurethane.properties.on. mechanical.performances.of.magnetorheological.elastomers Journal of Applied

[10].Waigh,.T A 2005 Microrheology.of.complex.fluids Reports on Progress in Physics. 68:.685–743.

[11].Wang, X and Gordaninejad, F 2007 Flow analysis and modeling of field- controllable,.electro-and.magneto-rheological.fluid.dampers Journal of Applied

[12].Li,.W H.,.Chen,.G.,.Yeo,.S H.,.and.Du,.H 2002 Dynamic.properties.of.magnetorheological.materials Key Engineering Materials 227:.119–124.

[13].Or,.S W.,.Duan,.Y F.,.Ni,.Y Q.,.Chen,.Z H.,.and.Lam,.K H 2008 Development. of.magnetorheological.dampers.with.embedded.piezoelectric.force.sensors.for. structural.vibration.control Journal of Intelligent Material Systems and Structures. 19:.1327–1338.

[14].Carlson,.J D and.Weiss,.K D 1994 A.growing.attraction.to.magnetic.fluids

[15].Dyke,.S J.,.Spencer.Jr.,.B F.,.Sain,.M K.,.and.Carlson,.J D 1996 Experimental. Verification of Semi-active Structural Control Strategies Using Acceleration. Feedback Third International Conference on Motion and Vibration Control, Chiba,. pp 1–6.

[16].Yang,.G.,.Spencer.Jr.,.B F.,.Carlson,.J D.,.and.Sain,.M K 2002 Large-scale.MR. fluid.damper:.modeling.and.dynamic.performance.considerations Engineering

[17].Boelter,.R and.Janocha H 1997 Design.rules.for.MR.fluid.actuators.in.different. working.modes Proceedings of the SPIEs 1997 Symposium on Smart Structures and

[18].Wereley,.N M.,.Cho,.J U.,.Choi,.Y T.,.and.Choi,.S B 2008 IJ-209.Magnetorheo- logical.dampers.in.shear.mode Smart Materials and Structures.17:.1–11.

[19].Hongsheng,.H.,.Hongsheng,.H.,.Juan,.W.,.Liang,.C.,.Jiong,.W.,.and.Xuezheng.J 2009 Proceedings of the IEEE International Conference on Automation and Logistics,. Shenyang,.China,.pp 1248–1253.

[20].Choi,.S B.,.Hong,.S R.,.Cheong,.C C.,.and.Park,.Y K 1999 Comparison.of.field- controlled characteristics between ER and MR clutches Journal of Intelligent

In recent studies, Saito and Ikeda (2007) explored the development of a normally closed type magnetorheological clutch, emphasizing its application in safe torque control systems for human-collaborative robots, as published in the Journal of Intelligent Materials Systems and Structures Additionally, Jin, Lin-qing, and Chang-hua (2006) conducted a design analysis of cylindrical magnetorheological clutches, which was featured in the Journal of Functional Materials.

[23].Lee,.U.,.Kim,.D.,.Hur,.N G.,.and.Jeon,.D Y 1999 Design.analysis.and.experimental.evaluation.of.an.MR.fluid.clutch Journal of Intelligent Material Systems and Structures.10:.701–707.

Recent studies have focused on the innovative applications of magnetorheological (MR) fluids in automotive technologies Kavlicoglu et al (2008) explored the heating mechanisms in high-torque MR fluid limited slip differential clutches, highlighting their effectiveness in enhancing performance Additionally, Neelakantan and Washington (2005) contributed to the understanding of MR transmission clutches by modeling and reducing centrifuging effects, which is crucial for improving automotive efficiency These advancements underscore the potential of MR fluids in optimizing vehicle dynamics and performance.

[26].Jedryczka,.C.,.Piotr,.S.,.and.Wojchiech,.S 2009 The.influence.of.magnetic.hysteresis on magnetorheological fluid clutch operation International Journal for

Computation and Mathematics in Electrical and Electronic Engineering.28:.711–721.

[27].Kavlicoglu,.N C.,.Kablicoglu,.B M.,.Liu,.Y.,.Evrensel.C A.,.Fuchs,.A Korol,. G.,.and.Gordaninejad.F 2007 Response.time.and.performance.of.a.high-torque. magneto-rheological fluid limited slip differential clutch Journal of Smart

[28].Hakogi,.H.,.Ohaba,.M.,.Kuramochi,.N.,.and.Yano,.H 2005 Torque.control.of. a.rehabilitation.teaching.robot.using.magneto-rheological.fluid.clutches JSME

[29].Li,.W H and.Du,.H 2003 Design.and.experimental.evaluation.of.a.magnetorheological.brake International Journal of Advanced Manufacturing Technology.21:.508–515. [30].Karakoc, K., Park, E J., and Suleman,.A 2008 Design considerations for an. automotive.magnetorheological.brake Mechatronics.18:.434–447.

[31].Hong,.S R and.Choi,.S B 2005 Vibration.control.of.a.structural.system.using. magneto-rheological.fluid.mount Journal of Smart Material Systems and Structures. 16:.931–936.

[32].Choi, S B., Hong, S R., Sung, K G., and Sohn, H W 2008 Optimal control. of.structural.vibrations.using.a.mixed-mode.magnetorheological.fluid.mount

International Journal of Mechanical Sciences.50:.559–568.

[33].Ahmadian,.M and.Ahn,.Y K 1999 Performance.analysis.of.magneto-rheological.mounts Journal of Smart Material Systems and Structures.10:.248–256.

[34].Ahn,.Y K.,.Ahmadian,.M.,.and.Morushita,.S 1999 On.the.design.and.development.of.a.magneto-rheological.mount Vehicle System Dynamics: International

Journal of Vehicle Mechanics and Mobility.32:.199–216.

[35].Yao,.G Z.,.Yap,.F F.,.Chen,.G.,.Li,.W H.,.and.Yeo,.S H 2002 MR.damper.and.its. application.for.semi-active.control.of.vehicle.suspension.system Mechatronics. 12:.963–973.

[36].Yu,.M.,.Dong,.X M.,.Choi,.S B.,.and.Liao,.C R 2009 Human.simulated.intelligent.control.of.vehicle.suspension.system.with.MR.dampers Journal of Sound and Vibration.319:.753–767.

Research by Du et al (2005) explores semi-active H∞ control methods for vehicle suspension systems utilizing magneto-rheological dampers, published in the Journal of Sound and Vibration Additionally, Choi, Lee, and Park (2002) examine the H∞ control performance of a complete vehicle suspension system that incorporates magneto-rheological dampers, as detailed in Vehicle System Dynamics.

International Journal of Vehicle Mechanics and Mobility.38:.341–360.

Research has explored innovative approaches to vibration control in suspension systems, particularly through the use of magnetorheological (MR) fluid dampers Lai and Liao (2002) demonstrated the effectiveness of these dampers in enhancing suspension performance, as published in the Journal of Vibration and Control Additionally, Yokoyama, Hedrick, and Toyama (2001) developed a model-following sliding mode controller specifically designed for semi-active suspension systems equipped with MR dampers, highlighting the potential for improved vehicle dynamics and ride comfort.

Proceedings of the American Control Conference,.Arlington,.VA,.4:.2652–2657.

[41].Choi,.S B and.Han,.Y M 2003 MR.seat.suspension.for.vibration.control.of.a. commercial.vehicle International Journal of Vehicle Design 31:.202–215.

[42].Yu,.M.,.Liao,.C R.,.Chen,.W M.,.and.Huang,.S L 2006 Study.on.MR.semi-active. suspension.system.and.its.road.testing Journal of Intelligent Material Systems and

[43].Guo,.D L.,.Hu,.H Y.,.and.Yi,.J Q 2004 Neural.network.control.for.a.semi-active. vehicle.suspension.with.a.magnetorheological.damper Journal of Vibration and

[44].Lee,.H S and.Choi,.S B 2000 Control.and.response.characteristics.of.a.magneto-rheological fluid damper for passenger vehicles Journal of Intelligent

[45].McManus,.S J.,.St Clair,.K A.,.Boileau,.P É.,.Boutin,.J.,.and.Rakheja,.S 2002 Evaluation of vibration and shock attenuation performance of a suspension. seat.with.a.semi-active.magnetorheological.fluid.damper Journal of Sound and

Recent studies have explored advanced vehicle suspension systems utilizing magnetorheological fluid dampers Liu et al (2004) conducted an experimental study on fuzzy skyhook control for such systems, highlighting its effectiveness in enhancing vehicle performance Additionally, Wereley and Pang (1998) performed a nondimensional analysis of semi-active electrorheological and magnetorheological dampers, employing approximate parallel plate models to better understand their dynamics These findings contribute significantly to the development of smart materials and structures in automotive engineering.

[48].Liao, W H and Wang, D H 2003 Semiactive vibration control of train suspension.systems.via.magnetorheological.dampers Journal of Intelligent Material

[49].Lam,.A H F and.Liao,.W H 2003 Semi-active.control.of.automotive.suspension.systems.with.magneto-rheological.dampers International Journal of Vehicle

[50].Song,.X.,.Ahmadian,.M.,.Southward,.S.,.and.Miller,.L R 2005 An.adaptive.semi- active.control.algorithm.for.magnetorheological.suspension.systems Journal of

[51].Batterbee,.D C and.Sims,.N D 2007 Hardware-in-the-loop.simulation.of.magnetorheological.dampers.for.vehicle.suspension.systems Journal of Systems and

[52].Shen, Y., Golnaraghi, M F., and Heppler, G R 2006 Semi-active vibration. control schemes for suspension systems using magnetorheological dampers

Journal of Vibration and Control.12:.3–24.

[53].Choi,.S B.,.Lee.S K.,.and.Park,.Y P 2001 A.hysteresis.model.for.field-dependent. damping.force.of.a.magnetorheological.damper Journal of Sound and Vibration. 245:.375–383.

Recent advancements in automotive suspension systems have highlighted the effectiveness of magnetorheological dampers Sassi et al (2005) developed an innovative design that underwent thorough experimental characterization, demonstrating significant improvements in performance Additionally, Prabakar et al (2009) explored optimal semi-active preview control responses in a half-car vehicle model equipped with magnetorheological dampers, further emphasizing their potential to enhance vehicle dynamics and ride comfort These studies collectively underscore the importance of magnetorheological technology in modern automotive engineering.

[56].Choi,.S B.,.Lee,.H S.,.and.Park,.Y P 2002 H-infinity.control.performance.of.a. full-vehicle.suspension.featuring.magnetorheological.dampers Vehicle System

[57].Zhang, C W., Ou, J P., and Zhang, J Q 2006 Parameter optimization and. analysis.of.a.vehicle.suspension.system.controlled.by.magnetorheological.fluid. dampers Structural Control and Health Monitoring.13:.885–896.

[58].Guo, S., Li, S., and Yang, S 2006 Semi-active vehicle suspension systems. with magnetorheological dampers IEEE International Conference on Vehicular

[59].Cao,.J.,.Liu,.H.,.Li,.P.,.and.Brown,.D J 2008 State.of.the.art.in.vehicle.active. suspension.adaptive.control.systems.based.on.intelligent.methodologies IEEE

Transactions on Intelligent Transportation Systems.9:.392–405.

[60].Ahmadian,.D S M 2001 Vehicle.evaluation.of.the.performance.of.magneto.rheological.dampers.for.heavy.truck.suspensions Journal of Vibration and Acoustics 123:.365–375.

[61].Dogruer,.U.,.Gordaninejad,.F.,.and.Evrensel,.C A 2008 A.new.magneto-rheological.fluid.damper.for.high-mobility.multi-purpose.wheeled.vehicle.(HMMWV)

Journal of Intelligent Material Systems and Structures.19:.641–650.

[62].McManus,.S J and.St Clair,.K A 2002 Evaluation.of.vibration.and.shock.attenuation.performance.of.a.suspension.seat.with.a.semi-active.magnetorheological. fluid.damper Journal of Sound and Vibration 253:.313–327

[63].Choi, S B., Nam, M H., and Lee, B K 2000 Vibration control of a MR seat. damper for commercial vehicles Journal of Intelligent Material Systems and

[64].Han,.Y M.,.Nam,.M H.,.Han,.S S.,.Lee,.H G.,.and.Choi,.S B 2002 Vibration. control.evaluation.of.a.commercial.vehicle.featuring.MR.seat.damper Journal of

Smart Material Systems and Structures.13:.575–579.

[65].Lee,.Y R and.Jeon,.D Y 2002 A.study.on.the.vibration.attenuation.of.a.driver. seat.using.an.MR.fluid.damper Journal of Smart Material Systems and Structures. 13:.437–441.

[66].Choi,.S B and.Han,.Y M 2003 MR.seat.suspension.for.vibration.control.of.a. commercial.vehicle International Journal of Vehicle Design.31:.201–215.

Ahmadian and Poynor (2001) conducted an evaluation of magneto-rheological dampers for effectively controlling gun recoil dynamics, as published in Shock and Vibration In a subsequent study, Ahmadian, Appleton, and Norris (2002) performed an analytical investigation into fire out of battery scenarios utilizing magneto-rheological dampers, also featured in Shock and Vibration.

[69].Facey,.W.,.Rosenfeld,.N.,.Choi,.Y T.,.Wereley,.N.,.Choi,.S B.,.and.Chen,.P.C 2005 Design and testing of a compact magnetorheological impact damper. for high impulsive loads International Journal of Modern Physics Part B. 19: 1549–1555.

[70].Batterbee, D C., Sims, N D., Stanway, R., and Wolejsza, Z 2007 Magnet- orheological.landing.gear:.A.design.methodology Smart Materials and Structures. 16:.2429–2440.

[71].Choi,.Y T and.Wereley,.N M 2003 Vibration.control.of.a.landing.gear.system. featuring.electrorheological/magnetorheological.fluids AIAA Journal of Aircraft. 40:.423–439.

[72].Spencer.Jr.,.B F.,.Dyke,.S J.,.Sain,.M K.,.and.Carlson,.J D 1997 Phenomenological. model.of.a.magnetorheological.damper ASCE Journal of Engineering Mechanics. 123:.230–238.

[73].Yang, G 2001 Large-scale magnetorheological fluid damper for vibration. mitigation:.modeling,.testing.and.control Ph.D Thesis,.University.of.Notre. Dame.

[74].Liu,.Y.,.Gordaninejad,.F.,.Evrensel,.C A.,.Wang,.X.,.and.Hitchhock,.G 2005 Comparative study on vibration control of a scaled bridge using fail-safe. magneto-rheological fluid dampers ASCE Journal of Structural Engineering. 131:.743–751.

[75].Hiemenz, G., Choi, Y T., and Wereley, N M 2003 Seismic control of civil. structures.utilizing.semi-active.MR.braces Journal of Computer-Aided Civil and

[76].Dyke,.S J.,.Spencer.Jr.,.B F.,.Sain,.M K.,.and.Carlson,.J D 1996 Modeling.and. control.of.magnetorheological.dampers.for.seismic.response.reduction Smart

[77].Choi,.Y T and.Wereley,.N M 2005 Semi-active.vibration.isolation.using.magnetorheological.isolators AIAA Journal of Aircraft.42:.1244–1251.

[78].An, J and Kwon, D S 1997 Modeling of a magnetorheological fluid (MRF). actuator.using.a.magnetic.hysteresis-based.model Proc of 14th Korean Automatic and Control Conference,.Yongin,.South.Korea,.pp 219–222.

[79].Kim,.K H.,.Nam,.Y J.,.Yamane,.R.,.and.Park,.M K 2009 Smart.Mouse:.5-DOF. haptic.hand.master.using.magneto-rheological.fluid.actuators Journal of Physics:

[80].Yoo,.J H.,.Siroshi,.J.,.and.Wereley.N M 2005 A.magnetorheological.piezo- hydraulic actuator Journal of Intelligent Material Systems and Structures. 16: 945–953.

[81].Yoo, J H and Wereley, N M 2004 Performance of a magnetorheological. hydraulic power actuation system Journal of Intelligent Material Systems and

[82].Nosse,.D T and.Dapino,.M J 2007 Magnetorheological.valve.for.hybrid.elec- trohydrostatic.actuation Journal of Intelligent Material Systems and Structures.18:. 1121–1136.

[83].Nguyen,.Q H.,.Choi,.S B.,.and.Welerey,.N M 2008 Optimal.design.of.magnetorheological.valves.via.a.finite.element.method.considering.control.energy.and. a.time.constant Smart Materials and Structures.17:.1–12.

[84].Choi,.S B.,.Song,.H J.,.Lee,.H H.,.Lim,.S C.,.and.Kim,.J H 2003 Vibration. control of a passenger vehicle featuring magnetorheological engine mount

International Journal of Vehicle Design.33:.2–16.

[85].Schweiger,.J 2002 Aircraft.Control,.Applications.of.Smart.Materials,.Encyclopedia of Smart Materials,.Wiley.Online.Library.

[86].Chen,.L and.Hansen,.C H 2005 Active.vibration.control.of.a.magnetorheological.sandwich.beam Proceedings of ACOUSTICS,.Busselton,.Western.Australia,. pp.93–98.

[87].Marathe,.S.,.Gandhi,.F.,.and.Wang,.K W 1998 Helicopter.blade.response.and. aeromechanical stability with a magnetorheological fluid based lag damper

Journal of Intelligent Material Systems and Structures.9:.241–282.

[88].Wereley,.N M.,.Kamath,.G M.,.and.Madhavan,.V 1999 Hysteresis.modeling. of.semi-active.magnetorheological.helicopter.lag.damper Journal of Intelligent

[89].Zhao,.Y.,.Choi,.Y T.,.and.Wereley,.N M 2004 Semi-active.damping.of.ground. resonance in helicopters using magnetorheological dampers Journal of the

Introduction

An automatic controller compares the actual output of a plant with the desired reference input, generating a control signal to minimize the deviation between these values A typical control system, as illustrated in Figure 2.1, integrates with magnetorheological (MR) devices The controller identifies the actuating error signal, which is typically a low power level, and amplifies it to a sufficient level using a current amplifier for MR devices The actuator then generates the plant input based on the control signal received.

Conventional actuators like electric motors are typically utilized in active control systems, while magnetorheological (MR) fluid-based actuators are more commonly found in semi-active control systems Active actuators can both supply and dissipate control energy, whereas semi-active actuators are limited to energy dissipation MR devices can function as active actuators when integrated with a hydraulic servovalve mechanism and a pump Additionally, widely used sensors, such as accelerometers, can be adapted to measure the dynamic responses of control systems that incorporate MR devices This chapter explores effective control methodologies for vehicle application systems that utilize MR technology.

Semi-Active.Control

Control energy in systems can be managed through active control devices, which allow for energy to be added or removed In contrast, semi-active devices, such as MR (Magnetorheological) devices, can only dissipate control energy, enhancing the stability of the control system By adjusting the intensity of the applied magnetic field, the damping of the system can be continuously regulated, achieving the desired damping force across various velocities This capability is a key advantage of semi-active control systems utilizing MR devices To attain the desired damping force within the controllable domain, three semi-active control methods can be employed: sky-hook, ground-hook, and sky-groundhook controllers.

The skyhook controller, introduced by Karnopp et al., is recognized for its straightforward logic and ease of implementation in real-world applications A schematic configuration of the skyhook controller is illustrated in Figure 2.3, showcasing how the desired damping force, F_d, can be adjusted accordingly.

Velocity Zero Field Non-zero Field

Semi-active.control.characteristics.with.MR.devices.

A typical control system featuring MR devices includes a block diagram where the control gain, represented as C sky, physically indicates the damping This control gain can be carefully adjusted using the magnetic field to achieve the desired damping force In practical applications, the control gain can also be selected through on-off logic.

The on-off control logic of the skyhook controller is commonly utilized in vehicle suspension systems to optimize ride quality and stability In this system, the shock absorber generates high damping force during rebound motion and low damping force during jounce motion, allowing for effective control of hard and soft modes by manipulating the magnetic field without the need for directional check valves The skyhook method isolates the vibrations of the sprung mass connected to the ground, while a groundhook controller addresses the vibrations of the unsprung mass By employing skyhook control, ride comfort is enhanced by minimizing the body motion, whereas groundhook control improves steering stability by reducing tire motion To achieve a balance between ride comfort and steering stability, a sky-groundhook controller can be implemented, featuring two ideal dampers—one anchored to the ceiling to manage the vehicle body vibrations and the other fixed to the ground to control damping forces.

A semi-active system utilizing a skyhook controller effectively manages the vibration of the wheel, specifically the unsprung mass (m_us) By properly adjusting the damping forces associated with magnetorheological (MR) dampers, we can enhance both ride quality and steering stability, ensuring optimal performance in various driving conditions The desired damping force can be precisely calibrated to achieve these improvements.

The equation F_d = σC_sky (1 - σ)C_ground describes the control dynamics of a system, where C_sky represents the skyhook control gain, C_ground signifies the groundhook control gain, and σ (ranging from 0 to 1) is a crucial weighting parameter that influences control performance When σ is set to zero, the damping force focuses solely on steering stability, highlighting the importance of accurately determining the weighting parameter based on the dynamic motion of vibrating structures or systems Utilizing a fuzzy algorithm is one effective approach to achieve this optimal determination.

The skyhook controller serves as an effective application for managing the damping force of a magnetorheological (MR) shock absorber, as illustrated in Figure 2.5 The damping force generated by the shock absorber can be accurately derived through this control method.

The equation \( F_k x = g p + c_e p + F_{MR}(H) + F_f \) describes the dynamics of a gas chamber, where \( k_g \) represents the stiffness, \( c_e \) indicates the equivalent viscous damping coefficient, \( F_{MR} \) denotes the controllable damping force dependent on the magnetic field \( H \), \( F_f \) is the frictional force, and \( x_p \) signifies the piston velocity By substituting the actual damping force \( F \) with the desired damping force \( F_d \), we can determine the necessary control magnetic field required to achieve the desired performance.

A semi-active system utilizing a sky-groundhook controller effectively achieves desired performance values in vehicle suspension systems For optimal implementation of the magnetorheological (MR) shock absorber, it is essential to have a high damping force during rebound (extension) motion, while a lower damping force is necessary during jounce (compression) motion.

To achieve optimal performance in shock absorption, traditional passive or semi-active dampers utilize a directional check valve that permits fluid flow between upper and lower chambers only during jounce motion, thereby reducing damping force In contrast, Magnetorheological (MR) shock absorbers can attain the same performance objectives through a different mechanism, allowing for precise control of damping levels—both high and low—by simply adjusting the magnetic field, eliminating the need for a directional check valve This innovative approach highlights the controllability of damping forces in MR shock absorbers.

If 0 (jounce), at zero field x F Cx C x F F p d p p d

The desired performance of the MR shock absorber entails generating a high damping force during rebound motion and a low damping force during jounce motion Figure 2.6 illustrates the controlled responses that align with these damping trajectories The tracking response demonstrates that effective damping force tracking is accomplished through magnetic field control.

The Skyhook damping force control of a magnetorheological (MR) shock absorber involves applying the required control input determined by the Skyhook controller during the rebound motion In contrast, no magnetic field is activated during the jounce motion, optimizing the shock absorber's performance for varying road conditions.

PID.Control

An effective method for achieving the desired damping force in a magnetorheological (MR) damper system is through the use of a proportional-integral-derivative (PID) controller The PID controller is well-regarded for its straightforward implementation and robust performance in the face of system uncertainties, making it a popular choice in practical applications Each component of the PID controller plays a crucial role in optimizing control actions to enhance system stability and responsiveness.

Damping.force.control.response.using.a.skyhook.controller (From.Choi,.S.B et.al.,.Mechatronics,.8,.2,.1998 With.permission.)

I, and D is shown in Figure 2.7 [6] From the block diagram, the input is. expressed.by

( ) ( ), for action ( ) ( ), for action ( ) ( ), for action u s k E s P u s k s E s I u s k sE s D p i d

The Laplace variable is represented as \( s \), while \( k_p \), \( k_i \), and \( k_d \) denote the control gains for the Proportional (P), Integral (I), and Derivative (D) components of the PID controller The feedback error signal, \( E(s) \), reflects the difference between the desired value and the actual output value Therefore, the structure of the PID controller can be defined accordingly.

Control.action.of.P,.I,.and.D.components.

The P-controller functions as an amplifier with an adjustable gain, denoted as k_p, which, when increased, enhances the response time of the control system However, excessively high feedback gains can lead to instability within the system The I-controller modifies the control value, u(t), at a rate proportional to the actuating error signal, e(t), ensuring that for zero error, u(t) remains constant This action effectively reduces or eliminates steady-state errors, a crucial aspect in tracking control problems To enhance system stability, the D-controller can be employed, although it may amplify noise and cause saturation in the MR actuator It's important to note that the D-controller should not be used in isolation, as its effectiveness is limited to transient periods Achieving optimal control performance requires careful determination of control gains k_p, k_i, and k_d, which can be accomplished through methods such as Ziegler-Nichols, adaptive, and optimal techniques.

To illustrate the effectiveness of the PID controller, this study utilizes the MR damper for passenger vehicles, focusing on controlling its damping force A block diagram is presented to achieve the desired damping force of the MR damper through the PID controller, highlighting the importance of precise control in enhancing vehicle performance.

The microprocessor stores the tracked damping force (F d) and compares it with the actual damping force (F a) produced by the MR damper The resulting actuating error signal is then fed back to the PID controller, which adjusts the control input accordingly.

PID.control.scheme.for.damping.force.control.of.an.MR.damper.

The control input, which involves the current or magnetic field, is amplified through the voltage/current (v/c) converter As illustrated in Figure 2.9, the results achieved by implementing the PID controller show that the actual damping force from the MR damper closely follows the desired force by adjusting the control input current set by the PID controller This demonstrates that the damping force of the MR damper can be continuously regulated by the input current linked to the PID controller.

LQ.Control

Linear Quadratic (LQ) control is a widely used control technique applicable to various systems, including MR actuator-based control systems This method assumes the plant operates as a linear system in state space form, with a performance index represented as a quadratic function of the plant states and control inputs A significant advantage of LQ control is its ability to produce linear control laws that are straightforward to implement and analyze In the context of Linear Quadratic Regulator (LQR) optimal control, a specific state equation is utilized.

Damping force control response using a PID controller (From Lee, H.S et al., Journal of

Intelligent Material Systems and Structures,.11,.80,.2000 With.permission.) where.x.is.the.state.vector,.u.is.the.input.vector,.A.is.the.system.matrix,.and.

B.is.the.input.matrix The.impending.problem.is.to.determine.the.optimal. control.vector u(t).=.−Kx(t) (2.10) so.as.to.minimize.the.performance.index

The equation J = ∫ 0 ∞ x Q x u R u dt T + T (2.11) defines a cost function where Q is a positive-semidefinite state weighting matrix and R is a positive-definite input weighting matrix These matrices, Q and R, play a crucial role in determining the relative importance of error minimization and control energy expenditure When the system represented by (A, B) is controllable, the feedback control gain can be effectively derived.

K R B P= − 1 T (2.12) where.P.is.the.solution.of.the.following.algebraic.Riccati.equation:

If.the.performance.index.is.given.in.terms.of.the.output.vector.rather.than. the.state.vector,.that.is,

J = ∫ ∞ y Q y u R u dt T + T (2.14) then.the.index.can.be.modified.by.using.the.output.equation y =.Cx (2.15) to

The process of determining the feedback gain K, which minimizes the index in Equation (2.16), mirrors the approach used for minimizing the index in Equation (2.11) Additionally, the LQ optimal control framework can be seamlessly adapted to address the linear quadratic Gaussian (LQG) problem when the control system and performance index are influenced by white Gaussian noise, as referenced in [8].

The equation (2.18) describes a system influenced by random noise disturbances (ω) and random measurement noise (ν), both of which are white Gaussian zero-mean stationary processes Given that both the states and control inputs are random, the performance index will also exhibit randomness The objective is to determine the optimal control strategy that minimizes the average cost By employing a procedure analogous to that used in the Linear Quadratic Regulator (LQR) problem, the solution can be effectively derived.

In the previous equations, Q1 is identified as a positive semi-definite matrix, while R1 is a positive definite matrix Notably, the problem can be addressed in two distinct stages: determining the controller gain K and calculating the estimator gain Ke.

Sliding.Mode.Control

Despite the numerous advantages of feedback control systems, MR devices face certain system perturbations and uncertainties For example, the field-dependent yield shear stress of MR fluid can change with temperature variations Additionally, the dynamic behavior of an MR device is influenced by the magnetic field, which may lead to non-linear hysteresis in the damping force of an MR damper system To ensure the control robustness of systems featuring MR devices, it is essential to implement a robust controller that addresses these system uncertainties.

A sliding mode controller (SMC), also known as a variable structure controller, is highly regarded for its robustness against system uncertainties and external disturbances Originating from the literature of the former Soviet Union, SMC research and development are now applied globally across various engineering systems The principal operation modes in variable structure systems are achieved through appropriate discontinuous control laws, which ensure invariance to parameter variations and external disturbances during sliding mode motion To illustrate this invariance property under sliding mode motion, one can consider a second-order system.

= − > < (2.21) when k < 0, the eigenvalues of the system become.

Therefore,the.phase.portrait.of.the.system.is.a.saddle.showing.unstable.motion. except.stable.eigenvalue.linear.(refer.to.Figure 2.10a) When.k >.0,.the.eigenvalues.become c c

The system's phase portrait exhibits a spiral source, indicating unstable motion, as illustrated in Figure 2.10b When switching occurs along the line defined by \( s_g = cx_1 + x_2 \) at \( x = 0 \), it follows specific switching logic.

The system achieves asymptotic stability for any arbitrary initial conditions, as illustrated in Figure 2.10c Two subsystems converge to a sliding line or surface, known as the switching line Upon reaching this sliding line, the system's behavior can be effectively characterized.

The original system response remains unaffected by system parameters during sliding mode motion, ensuring the system's robustness against uncertainties and external disturbances Achieving sliding mode motion typically requires meeting specific sliding mode conditions.

The aforementioned condition can be understood as a criterion for Lyapunov stability To outline the design steps for Sliding Mode Control (SMC), we will examine a control system that is affected by external disturbances.

Invariance.property.of.the.SMC. where d is external disturbance and a is parameter variation These are. bounded.by

As.a.first.step,.we.choose.a.stable.sliding.line.as

Then,.the.sliding.mode.dynamics.becomes

Thus,.if.we.design.the.SMC,.u,.by

The.sliding.mode.condition.in.Equation.(2.33).can.be.satisfied.as:

In the controller defined by Equation (2.38), the variable k represents the discontinuous control gain, while sgn(⋅) denotes the signum function This design approach can be readily applied to higher-order control systems.

[1].Leitmann, G 1994 Semiactive control for vibration attenuation Journal of

Intelligent Material Systems and Structures.5:.841–846.

[2].Karnopp,.D 1974 Vibration.control.using.semi-active.force.generators ASME

Journal of Engineering for Industry 96:.619–626.

[3].Guclu,.R 2005 Fuzzy.logic.control.of.seat.vibrations.of.a.non-linear.full.vehicle. model Nonlinear Dynamics.40:.21–34.

[4].Terano,.T.,.Asai,.K.,.and.Sugeno,.M 1987 Fuzzy System Theory and Its Applications,. Boston:.Harcourt-Brace.

[5].Ogata,.K 1990 Modern Control Engineering,.Englewood.Cliffs,.NJ:.Prentice-Hall. [6].Han,.Y M.,.Nam,.M H.,.Han,.S S.,.Lee,.H G.,.and.Choi,.S B 2002 Vibration. control.evaluation.of.a.commercial.vehicle.featuring.MR.seat.damper Journal of

[7].Lee,.H S and.Choi,.S B 2000 Control.and.response.characteristics.of.a.magnetorheological.fluid.damper.for.passenger.vehicles Journal of Intelligent Material

[8].Chen,.C T 1999 Linear System Theory and Design,.New.York:.Oxford.University. Press.

[9].Zinober, A S I 1990 Deterministic Control of Uncertain Systems, London:. Peter.Peregrinus.Press.

[10].Slotine,.J J E and.Sastry,.S S 1983 Tracking.control.of.nonlinear.systems.using. sliding.surfaces.with.application.to.robot.manipulators,.International Journal of

[11].Park,.D W and.Choi,.S B 1999 Moving.sliding.surfaces.for.high-order.variable.structure.systems International Journal of Control.72:.960–970.

Introduction

Hysteretic behavior is commonly observed in various devices involving magnetic, ferroelectric, and mechanical systems Smart materials, including magnetorheological (MR) fluids, electrorheological (ER) fluids, piezoceramics (PZTs), and shape memory alloys (SMAs), are increasingly utilized as actuators in active and semi-active control systems However, these materials exhibit nonlinear hysteretic responses that negatively impact actuator performance In response to the growing demand for enhanced control performance, researchers have extensively explored nonlinear hysteresis models and robust feedback control schemes to improve or compensate for the hysteretic behavior of smart material actuators.

Magnetorheological (MR) fluid is a key focus in research for applications in valve systems, shock absorbers, and engine mounts due to its ability to change properties instantaneously when exposed to a magnetic field Typically modeled as Bingham fluid, MR fluid demonstrates Newtonian behavior post-yield but also exhibits hysteretic behavior influenced by dynamic conditions and control inputs This shear rate hysteresis has garnered significant attention, leading to various investigations into MR device performance Researchers like Stanway et al and Spencer et al have proposed mechanical models, including a Coulomb friction element and the Bouc-Wen model, to describe the behavior of MR dampers, while Kamath and Wereley introduced a nonlinear viscoelastic-plastic model to capture the fluid's behavior across pre-yield and post-yield regimes Additionally, a nonlinear biviscous model for MR/ER dampers has been suggested by Wereley et al.

This chapter explores new hysteresis modeling techniques for magnetorheological (MR) fluid that can be seamlessly integrated into control systems It highlights a critical aspect of MR device control by establishing a general hysteresis model of MR fluid in relation to an applied magnetic field While conventional hysteresis models provide a decent approximation of the post-yield behavior of MR devices, there is a need for more precise models to accurately capture the complex hysteretic behavior of MR fluids The magnetic field serves as the control input for MR fluid applications, making hysteretic behavior crucial for device control performance The classical Preisach independent domain model, recognized in the ferromagnetic field, is introduced to represent these characteristics of MR fluids effectively This model has previously been utilized for piezoelectric transducers (PZT) and shape memory alloys (SMA) Through rheometer tests on a commercial MR product (MRF-132LD; Lord Corporation, U.S.), key properties such as the minor loop property and wiping-out property are examined, demonstrating the Preisach model's alignment with the physical hysteresis phenomena of MR fluid A Preisach model for MR fluid is then proposed and numerically identified, with several first-order descending (FOD) curves derived to establish the relationship between shear stress and applied magnetic field The identified model is validated by comparing predicted shear stress with measured values, and its predictions are also contrasted with those of the conventional Bingham model.

Section 3.3 introduces a polynomial hysteresis model for magnetorheological (MR) devices, featuring a variable control input coefficient for simplified implementation Recent advancements in semi-active MR dampers have been proposed to effectively reduce vibrations in various dynamic systems, including vehicle suspensions Experimental results indicate that MR dampers can selectively control unwanted vibrations when paired with suitable control strategies The accuracy of the damping force model, which captures the inherent hysteretic behavior of MR dampers, is crucial for achieving optimal control performance Specifically, a more precise damper model is essential for open-loop control, which is easier and more cost-effective to implement compared to closed-loop control.

Several damper models have been developed to predict the field-dependent hysteretic behavior of MR dampers, including the Bouc-Wen model, the non-linear hysteretic biviscous model, and a modified Bingham plastic model The effectiveness of these models in accurately predicting hysteretic behavior has been positively validated through comparisons with experimental results.

Achieving effective tracking control performance in a control system, whether open-loop or closed-loop, is challenging due to the variability of experimental parameters in the models concerning the applied field intensity.

This section focuses on a cylindrical MR damper suitable for mid-sized passenger vehicles, examining its hysteretic behavior through experimental evaluation in the damping force versus piston velocity domain The study compares the measured hysteretic characteristics of field-dependent damping forces with predictions from previously established models Additionally, the accuracy of damping force control using the proposed model is experimentally validated through an open-loop control scheme.

Preisach.Hysteresis.Model.Identification

Hysteresis.Phenomenon

In the previous chapter, we explored MR (magnetorheological) fluids, which are suspensions of magnetizable particles in a low permeability base fluid These fluids exhibit an instantaneous change in rheological properties due to the polarization of suspended particles when exposed to a magnetic field This phenomenon leads to a yield stress in the suspension, allowing MR fluids to be classified as Bingham fluids, characterized by a specific constitutive equation.

The shear stress (τ) in magnetorheological (MR) fluids is determined by the equation τ = ηγ + τy(H), where η represents dynamic viscosity, γ is the shear rate, and τy(H) is the dynamic yield stress influenced by the magnetic field (H) The Bingham model effectively controls MR devices by providing a favorable approximation for describing their post-yield behavior However, it falls short in capturing the hysteretic behavior of MR fluids.

The experimental apparatus designed to measure shear stress produced by MR fluid is depicted in Figure 3.1 Shear stress, dependent on the applied field, was quantified using a rheometer (MCR 300, Physica) equipped with an MR cell (TEK 70MR, Physica) To reduce hysteresis effects from mechanical friction, an air bearing was implemented, and the system operated at a controlled temperature of 25°C The setup features a rotating disk with a diameter of 20 mm, and a measuring gap of 1 mm is maintained between the rotating and stationary disks, which is filled with the MR fluid A personal computer, along with A/D and D/A converters, manages both the shear rate and the input field.

Figure 3.2 illustrates the measured hysteretic behaviors of MR fluid (MRF-132LD, Lord Corporation) used in this test Two distinct hysteretic behaviors are observed, as shear stress depends on both dynamic conditions (shear rate) and control input (magnetic field) In Figure 3.2(a), shear stress is recorded while varying the shear rate under a constant magnetic field, whereas Figure 3.2(b) shows shear stress changes with varying magnetic field strength The hysteresis loop is traced counterclockwise, revealing that hysteresis loops are influenced by both shear rate and magnetic field intensity, with the magnetic field having a significantly greater impact on the hysteretic behavior.

The schematic configurations of the experimental apparatus highlight the influence of shear rate on performance In application devices, the magnetic field serves as a crucial control input Consequently, the significant hysteretic behavior related to this control input can greatly impact the overall control performance of these devices.

This section focuses on the measured shear stress at a shear rate of 1 s⁻¹, which is used as an approximate yield stress to establish a hysteresis model linking yield stress (output) to magnetic field (input) By measuring shear stress at this constant rate, it is assumed to be close to the dynamic yield stress, as illustrated in Figure 3.3(a) Additionally, the field-dependent viscosity remains below 50 Pa·s, as shown in Figure 3.3(b) It is important to note that the shear stress values depicted in the figures correspond to the dynamic yield stress.

Shear Rate (1/s) (a) Shear rate vs shear stress

Magnetic Field (kA/m) (b) Magnetic field vs shear stress

Hysteretic.behavior.of.the.MR.fluid (From.Han,.Y.M et.al.,.Journal of Intelligent Material Systems and Structures,.19,.9,.2007 With.permission.)

The Preisach model is a valuable tool for modeling hysteresis in ferromagnetic materials and shows promise in capturing the hysteretic behavior of magnetorheological (MR) fluids, as the chain clusters in these fluids are formed by induced magnetic dipoles through electromagnetization This phenomenological model is defined by two key properties: the minor loop property and the wiping-out property The minor loop property indicates that two comparable minor loops, created by transitioning between identical pairs of input maxima and minima, will align perfectly if one is shifted appropriately in the output parameter Meanwhile, the wiping-out property determines which values from the previous input trajectory influence the current output, specifically identifying which dominant maxima and minima can negate the effects of prior inputs.

Magnetic Field (kA/m) (b) Plastic viscosity

Approximated.yield.stress.of.the.MR.fluid (From.Han,.Y.M et.al.,.Journal of Intelligent Material

To assess the alignment of a physical hysteresis phenomenon with the Preisach model, it is essential to analyze both the minor loop and the wiping-out properties This examination helps in understanding the behavior of dominant maxima and minima within the system's structures.

The applied magnetic field trajectory for the minor loop experiment is illustrated in Figure 3.4(a), featuring triangular input signals The resulting shear stress, depicted in Figure 3.4(b), demonstrates a favorable congruency Meanwhile, Figure 3.5(a) presents the magnetic field trajectory for the wiping-out experiment, which consists of two sets of dominant maxima and minima Figure 3.5(b) indicates a strong agreement between the first and second sets of dominant maxima.

Magnetic Field (kA/m) (b) Minor loop response

Minor loop property of the MR fluid (From Han, Y.M et al., Journal of Intelligent Material

Systems and Structures,.19,.9,.2007 With.permission.)

The effect of the previous inputs (a1, a2, a3) has been overshadowed by the larger maxima (a1) of the latter inputs (a1, a2, a3) Consequently, the preceding input values do not influence the current outputs This demonstrates that the wiping-out property is adequately fulfilled for the MR fluid utilized in this test.

The results illustrated in Figures 3.4 and 3.5 indicate a favorable congruency among the comparable minor loops and demonstrate satisfactory wiping-out properties for the MR fluid This suggests that the Preisach model is suitable for describing the hysteresis behavior of the MR fluid Consequently, the hysteresis identification of the MR fluid was conducted using the Preisach model.

Time (sec) (b) Wiping-out response a 1 a 1 a 2 a 3 a 2 a 3

Sh ea r S tr es s ( kP a)

Wiping-out property of the MR fluid (From Han, Y.M et al., Journal of Intelligent Material

Preisach.Model

The.Preisach.model.that.we.adopt.to.describe.the.hysteretic.behavior.of.the. MR.fluid.can.be.expressed.as.[14]:

The Preisach plane serves as a foundational model for understanding hysteresis in magnetorheological (MR) fluids, where each hysteresis relay is defined by a pair of switching values (α, β) with α ≥ β As the magnetic field, denoted as H(t), fluctuates over time, individual relays adjust their outputs, contributing to the overall system output through a weighted sum, as illustrated in Figure 3.6(a) The simplest hysteresis relay for MR fluids, depicted in Figure 3.6(b), operates in two states, -1 and 1, reflecting the polarization of ferromagnetic materials, resulting in an output that varies between zero and a maximum yield stress This behavior creates a hysteresis loop within the first quadrant of the magnetic field-yield stress plane The Preisach plane can be geometrically interpreted as a one-to-one mapping between relays and their switching values, bounded within a triangle defined by the maximum yield stress, with α0 and β0 marking the limits of the magnetic field input at 0 kA/m and 257.25 kA/m, respectively This framework allows for the analysis of the state of individual relays, which are categorized into two time-varying regions.

The.two.regions.represent.that.relays.are.on.0.and.1.positions,.respectively Therefore,.Equation.(3.2).can.be.reduced.to

The numerical technique for identifying the Preisach model has proven to be an effective approach for smart materials In this context, the Preisach model for magnetorheological (MR) fluid is implemented numerically by utilizing a numerical function for each mesh value of (α, β) within the Preisach plane These mesh values correspond directly to the data found in the frequency of operation (FOD) curves.

(b) Hysteresis relay for the MR fluid β 0 α 0 α β α = β γ αβ τ y à α 1 β 1 à α 2 β 2 γ α 1 β 1 γ α 2 β 2 à α n β n γ α n β n

The configuration of the Preisach model is based on experimental data, as illustrated in Figure 3.7(a), which presents one of the mesh values (α1, β1) and its corresponding FOD curve This curve exhibits a monotonic increase to the value α1, followed by a monotonic decrease to β1 After the input peaks at α1, the decrease sweeps out area Ω, creating the descending branch within the major loop A numerical function T(α1, β1) is subsequently defined to represent the output change along this descending branch.

Numerical.identification.and.implementation.of.the.Preisach.model.

From.Equation.(3.4).and.Equation.(3.5),.we.can.determine.an.explicit.formula.for.the.hysteresis.in.terms.of.experimental.data.

Figure 3.7(b) illustrates the variations in an input magnetic field, which can be divided into n trapezoids, denoted as Qk The area of each trapezoid, Qk, is calculated as the difference between the areas of two triangles associated with T(αk, βk-1) and T(αk, βk) Consequently, the output for each trapezoid Qk is derived from this geometric representation.

By.summing.the.area.of.trapezoids.Q k over.the.entire.area.Γ+,.the.output.is. expressed.by

Consequently,.for.the.cases.of.the.increasing.and.decreasing.input,.the.output.τ y ( )t of.the.Preisach.model.is.expressed.by.the.experimentally.defined.

The numerical implementation of the Preisach model necessitates the experimental determination of T(α, β, k) at a limited number of grid points within the Preisach plane The finite grid points and their corresponding measured output values present challenges, as some magnetic field input values may not align with these grid points and may lack measured output values To address this issue, this work utilizes specific interpolation functions to ascertain the corresponding T(α, β, k) values when the magnetic field input does not coincide with the grid points.

, for square cells , for triangular cells

Hysteresis.Identification.and.Compensation

To calculate the numerical function of the Preisach model, FOD datasets are constructed as an initial step As illustrated in Figure 3.8(a), the magnetic field input is utilized to gather FOD curves, ranging from 0 kA/m to 257.25 kV/m, which is divided into 10 sub-ranges The magnetic field is applied incrementally, with each step sustained for 20 seconds to ensure accuracy in the measurements.

Time (sec) (a) Magnetic field input

Measured FOD curves (From Han, Y.M et al., Journal of Intelligent Material Systems and

In this study, we analyzed the steady-state condition of a magnetorheological (MR) fluid by measuring shear stress with a rheometer The resulting flow curve data, illustrated in Figure 3.8(b), was utilized to numerically identify the Preisach model for the MR fluid, providing insights into its behavior under varying conditions.

A.hysteresis.prediction.using.the.Preisach.model.was.tested.under.three. different types of input trajectories of the magnetic field: step, triangular,. and arbitrary signals The predicted results were compared to the results. predicted by the Bingham model in Equation (3.1) Figure 3.9 shows the. Bingham.characteristics.of.the.MR.fluid.adopted.in.this.test,.from.which.the. field-dependent.yield.stress.is.obtained.by.τ y ( ) 7.5H = H 1.4 Pa.

The actual and predicted hysteresis responses under a step input are illustrated in Figure 3.10(b), with the input shown in Figure 3.10(a) During the increasing step, both the Preisach model and the Bingham model yield similar results; however, the hysteresis effect becomes more pronounced in the subsequent decreasing step The Preisach model demonstrates a high level of accuracy in predicting actual responses, outperforming the Bingham model significantly This is further validated by the prediction errors depicted in Figure 3.10(c), where maximum prediction errors for the Preisach model and the Bingham model are calculated at 0.72 kPa and 2.34 kPa, respectively, under steady-state conditions.

The hysteresis response under a triangular input trajectory is illustrated in Figure 3.11, demonstrating that the proposed Preisach model effectively captures the hysteresis nonlinearity with minimal prediction error, as indicated in Figure 3.11(c) The maximum prediction errors for the Preisach and Bingham models are 0.43 kPa and 2.49 kPa, respectively Additionally, Figure 3.12 presents a comparison of the field-dependent hysteresis loops, highlighting the alignment between actual measurements and those identified by the proposed Preisach model.

Bingham.characteristics.of.the.MR.fluid (From.Han,.Y.M et.al.,.Journal of Intelligent Material

Time (sec) (a) Step input trajectory

Actual Preisach model Bingham model

Actual.and.predicted.hysteresis.responses.under.a.step.input (From.Han,.Y.M et.al.,.Journal of

Intelligent Material Systems and Structures,.19,.9,.2007 With.permission.)

Time (sec) (a) Triangular input trajectory

Actual.and.predicted.hysteresis.responses.under.a.triangular.input (From.Han,.Y.M et.al.,.

The Preisach model effectively captures the field-dependent hysteretic behavior of shear stress in magnetorheological (MR) fluids, as demonstrated in the Journal of Intelligent Material Systems and Structures.

Figure 3.13 illustrates the prediction responses based on an arbitrary input trajectory, which includes various slopes of ramp and constant inputs The Preisach model effectively captures the hysteresis output, demonstrating its capability to accurately represent the hysteresis behavior of the MR fluid in relation to the output (yield stress) and input (magnetic field).

The control of magnetorheological (MR) devices involves adjusting the magnetic field to achieve the desired shear or yield stress, which is influenced by the actuating force A highly effective strategy for managing hysteresis nonlinearity is inverse model control, allowing for open-loop compensation through Preisach model inversion This approach enables the prediction of shear stress generated by MR fluid based on a known shear rate and applied magnetic field Conversely, the magnetic field required to produce a specific shear stress can be calculated using the inverse model The performance of this control strategy is significantly dependent on the accuracy of the formulated model The proposed control algorithm, illustrated in a flow chart, involves predicting and linearizing hysteresis nonlinearity using the Preisach model After defining desired shear stress trajectories, the corresponding magnetic field is computed using the nominal relationship derived from the Bingham characteristic yield curve of the MR fluid The predicted shear stress is then compared with the desired shear stress to facilitate linearization within the algorithm.

Hysteresis.loops.of.the.MR.fluid (From.Han,.Y.M et.al.,.Journal of Intelligent Material Systems and Structures,.19,.9,.2007 With.permission.)

Time (sec) (a) Arbitrary input trajectory

Actual.and.predicted.hysteresis.responses.under.an.arbitrary.input (From.Han,.Y.M et.al.,.

The Journal of Intelligent Material Systems and Structures published in 2007 discusses the iterative process of updating control inputs until the error is minimized Consequently, the final control input, denoted as \( u(k) \), can be expressed mathematically as a function of various parameters and feedback mechanisms.

The error bound, denoted as ε, and the number of updating times, m, are essential parameters in the control process The values of α and β are derived from the Bingham characteristic yield curve of the MR fluid Once the kth control input is established, the desired shear stress for the next k+1th step is introduced, and this iterative process continues for the entire desired shear stress set The proposed hysteresis model for MR fluids demonstrates seamless integration into control systems.

Polynomial.Hysteresis.Model.Identification

Hysteresis.Phenomenon

The schematic configuration of the automotive MR damper used in the test is depicted in Figure 3.15 This MR damper features a piston that separates the upper and lower chambers and is completely filled with the Lord product MRF132-LD MR fluid Key design parameters include an outer radius of the inner cylinder measuring 30.1 mm and the length of the magnetic pole.

10 mm; the gap between the magnetic poles: 1.0 mm; the number of coil. turns:.150;.and.the.diameter.of.the.copper.coil:.0.8.mm The.gas.chamber.is. fully.charged.by.nitrogen.and.its.initial.pressure.at.the.maximum.extension. (up.motion.of.the.piston).is.set.at.25.bar Figure 3.16.presents.the.measured. damping force versus piston velocity at various input currents (magnetic. fields) The.result.is.obtained.by.exciting.the.MR.damper.with.the.excitation. frequency.of.1.to.4.Hz.and.the.exciting.magnitude.of.±20.mm The.details.

The configuration of the MR damper for measurement procedures is thoroughly detailed in Reference [20] Figure 3.16 illustrates that the damping force magnitude rises with increasing piston velocity and input current Additionally, the hysteresis loop expands as the input current increases, indicating a significant relationship between current input and damper performance.

Polynomial.Model

To predict the damping force characteristics of conventional damper models, we begin with the simple Bingham model, illustrated in Figure 3.17(a) This model highlights the field-dependent damping force characteristics of the MR damper, as shown in Figure 3.16 In the Bingham model, the yield stress (τy) of the MR fluid plays a crucial role in determining the damping performance.

The damping force of the magnetorheological (MR) damper can be expressed by the equation τ = α y H β, where H represents the magnetic field and α and β are intrinsic values specific to the MR fluid, which must be identified experimentally.

In Equation (3.12), \( x_p \) represents the piston displacement, while \( \upsilon \) denotes the piston velocity The stiffness constant attributed to gas compliance is indicated by \( k_e \), and \( c_e \) refers to the damping constant resulting from the viscosity of the MR fluid Additionally, \( \alpha_1 \) is a geometrical constant.

Measured.damping.force.characteristics (From.Choi,.S.B et.al.,.Journal of Sound and Vibration, 245,.2,.2001 With.permission.)

The controllable damping force is influenced by the input magnetic field or current, as illustrated in Figure 3.17(b) This figure depicts a fundamental mechanism of the Bouc-Wen model, commonly used to analyze nonlinear hysteresis behavior The damping force of the magnetorheological (MR) damper can be expressed as referenced in [7].

Models.for.damping.force.prediction. where.x 0.is.the.initial.displacement.due.to.the.gas,.γ.is.the.pressure.drop.due. to.the.MR.effect.(yield.stress),.and.z.is.obtained.by

The parameters ε, δ, and A in the Bouc-Wen model significantly influence the hysteresis behavior in the pre-yield region These experimental parameters vary depending on the intensity of the input field, making it challenging to implement an open-loop control system that achieves the desired damping force.

Figure 3.17(c) illustrates the schematic configuration of the third model discussed in this section The hysteresis loop depicted in Figure 3.16 can be divided into two distinct regions: positive acceleration (lower loop) and negative acceleration (upper loop) Each of these loops can be fitted using a polynomial based on piston velocity Consequently, the damping force of the MR damper can be represented mathematically.

The experimental coefficient \( a_i \) is determined through curve fitting, with the polynomial order selected via trial and error It was found that polynomials up to the fifth order failed to accurately represent the measured hysteresis behavior, whereas sixth and higher order polynomials effectively captured this behavior with minimal differences Consequently, a sixth order polynomial was chosen to balance accuracy and computational time, which is crucial for real-time control of the damping force The coefficient \( a_i \) in Equation (3.15) can be expressed in relation to the intensity of the input current, as illustrated in Figure 3.18 The plots show the measured values indicated by dark squares and the solid curve representing the linear fit of the coefficient \( a_i \) Additionally, the coefficients \( a_1, a_3, a_4, a_5, \) and \( a_6 \) are detailed in Reference [21], demonstrating that \( a_i \) can be linearized concerning the input current.

As.a.result,.the.damping.force.can.be.expressed.by

The coefficients \( b_i \) and \( c_i \) are derived from the intercept and slope of the plots illustrated in Figure 3.18 The specific values of \( b_i \) and \( c_i \) utilized in this test are detailed in Table 3.1 It is important to note that the coefficients \( a_i \), \( b_i \), and \( c_i \) remain unaffected by the magnitude of the input current.

Thus,.we.can.easily.realize.an.open-loop.control.system.to.achieve.a.desirable.damping.force This.is.presented.in.section.3.3.3.

Hysteresis.Identification.and.Compensation

The measured damping force is compared with the predicted damping forces from the Bingham model, the Bouc-Wen model, and the proposed polynomial model, as illustrated in Figure 3.19 The excitation frequency is set at 1.4 Hz with a magnitude of ±20 mm, while the input current to the MR damper is maintained at 1.2 A Notably, the Bingham model fails to accurately capture the non-linear hysteresis behavior, although it does reasonably predict the magnitude of the damping force at a specific piston position.

Input Current (A) (a) Positive acceleration (a 0 ) (b) Positive acceleration (a 2 )

Input Current (A) (c) Negative acceleration (a 0 ) (d) Negative acceleration (a 2 )

The relationship between input current and velocity is explored through the Bouc-Wen model and polynomial model, which effectively predict the measured hysteresis behavior To validate the model's general effectiveness, various excitation conditions and input currents are tested, with comparative results illustrated in Figure 3.20, showing the model's accurate predictions without altering the experimental coefficients The accuracy of the damping force control in the MR damper is contingent upon the chosen damper model An open-loop control system is established, as depicted in Figure 3.21, to achieve the desired damping force The control input current necessary to attain this force is derived from the damper model and applied accordingly, as outlined in Equation (3.17).

= (3.18) where.F d is.the.desirable.damping.force.to.be.tracked The.desirable.damping. force.is.normally.set.by.F d =c v sky The.coefficient.c sky is.control.gain,.and.it.is.

Coefficients.b i and.c i of.the.Polynomial.Model

The comparison of damping forces between measurement and prediction reveals that the proposed model demonstrates significantly better control accuracy than the Bingham model As illustrated in Figure 3.22, the open-loop control system effectively realizes damping force controllability Notably, the Bingham model shows poor tracking accuracy at peak values, indicating its limitations in capturing the dynamics of the system.

Damping.force.characteristics.at.various.operating.conditions (From.Choi,.S.B et.al.,.Journal of

Sound and Vibration,.245,.2,.2001 With.permission.)

The block diagram for damping force control illustrates the behavior of damping force at zero and near-zero piston velocities, as depicted in Figure 3.19(a) In contrast, the polynomial model effectively aligns with the desired outcomes across the entire range of piston velocities.

Some.Final.Thoughts

In.this.chapter,.two.hysteresis.modeling.techniques.are.discussed.as.potential.candidates,.which.can.be.easily.integrated.into.a.control.system.adopting. MR.fluids.

The first section presents a comprehensive hysteresis model of magnetorheological (MR) fluid in relation to the applied magnetic field, which serves as a control input for MR devices The field-dependent hysteretic behavior was analyzed using the Preisach model, demonstrating its relevance to MR fluid through minor loop tests and wiping-out tests Subsequently, the hysteresis model was developed based on first-order descending curves.

Tracking.control.responses.of.the.damping.force (From.Choi,.S.B et.al.,.Journal of Sound and

The effectiveness of the Preisach model in predicting field-dependent shear stress has been validated through comparisons with measured data This model significantly reduces prediction errors across various input trajectories when compared to the traditional Bingham model.

The second section explores a hysteresis model that employs a polynomial expression to represent the field-dependent damping force of the MR damper It compares the hysteretic damping force predicted by this polynomial model with those from the Bingham and Bouc-Wen models The findings indicate that the polynomial model effectively captures the non-linear hysteresis behavior of the MR damper Furthermore, the superior control accuracy of the proposed model over the Bingham model was confirmed through the implementation of an open-loop control system designed to track a desired damping force.

The newly introduced hysteresis compensation methods can be seamlessly integrated into control systems that utilize magnetorheological (MR) devices, such as electronic control suspensions with MR shock absorbers The hysteretic characteristics of MR fluid depend on both dynamic conditions, like shear rate, and control inputs, such as the magnetic field To improve the control accuracy of MR applications, it is essential to account for this complex and extensive hysteretic behavior during controller synthesis Consequently, the hysteresis model of MR fluid gains significant importance in practical applications that require straightforward implementation.

[1].Mittal, S and Menq, C H 2000 Hysteresis compensation in electromagnetic.actuators.through.Preisach.model.inversion IEEE/ASME Transactions on

[2].Song,.D and.Li,.J C 1999 Modeling.of.piezoactuator’s.nonlinear.and.frequency. dependent.dynamics Mechatronics 9:.391–410.

[3].Choi,.S B and.Lee,.C H 1997 Force.tracking.control.of.a.flexible.gripper.driven. by.a.piezoceramic.actuator ASME Journal of Dynamic Systems, Measurement and

[4].Shames, I H and Cozzarelli, F A 1992 Elastic and Inelastic Stress Analysis,. Upper.Saddle.River,.NJ:.Prentice.Hall.

[5].Choi, S B., Nam, M H., and Lee, B K 2000 Vibration control of a MR seat. damper for commercial vehicles Journal of Intelligent Material Systems and

In the 1987 study by Stanway, Sproston, and Stevens, a non-linear model for an electro-rheological vibration damper was presented, highlighting its effectiveness in controlling vibrations (Journal of Electrostatics, 20: 167–184) Additionally, the 1997 research by Spencer Jr., Dyke, Sain, and Carlson introduced a phenomenological model for a magnetorheological damper, emphasizing its application in engineering mechanics (Journal of Engineering Mechanics).

American Society of Civil Engineers.230:.3–11.

[8].Kamath,.G M and.Wereley,.N M 1997 Nonlinear.viscoelastic-plastic.mechanisms-based.model.of.an.electrorheological.damper Journal of Guidance, Control and Dynamics.20:.1125–1132.

[9].Wereley,.N M.,.Pang,.L.,.and.Kamath,.G M 1998 Idealized.hysteresis.modeling.of.electrorheological.and.magnetorheological.dampers,.Journal of Intelligent

[10].Han,.Y M.,.Choi,.S B.,.and.Wereley,.N M 2007 Hysteretic.behavior.of.magnetorheological.fluid.and.identification.using.Preisach.model Journal of Intelligent

[11].Han,.Y M.,.Lim,.S C.,.Lee,.H G.,.Choi,.S B.,.and.Choi,.H J 2003 Hysteresis. identification of polymethylaniline-based ER fluid using Preisach model

[12].Ge,.P and.Jouaneh,.M 1997 Generalized.Preisach.model.for.hysteresis.nonlinearity.of.piezoceramic.actuators,.Precision Engineering.20:.99–111.

[13].Gorbet,.R B.,.Wang,.D W L.,.and.Morris,.K A.,.1998 Preisach.model.identification.of.a.two-wire.SMA.actuator Proceedings of the IEEE International Conference on Robotics & Automation,.Leuven,.Belgium,.pp 2161–2167.

[14].Mayergoyz,.I D 1991 Mathematical Models of Hysteresis New.York:.Springer-Verlag [15].Hughes,.D and.Wen,.J T 1997 Preisach.modeling.of.piezoceramic.and.shape. memory.alloy.hysteresis Smart Materials and Structures 6:.287–300.

[16].Choi, S B and Lee, S K 2001 A hysteresis model for the field-dependent. damping.force.of.a.magnetorheological.damper Journal of Sound and Vibration. 245: 375–383.

[17].Choi,.S B.,.Choi,.Y T.,.and.Park,.D W 2000 A.sliding.mode.control.of.a.full-car. ER.suspension.via.hardware-in.the-loop-simulation Journal of Dynamic Systems and.Measurement and Control.122:.114–121.

[18].Carlson,.J D and.Sproston.J L 2000 Controllable.fluid.in.2000-status.of.ER.and. MR.fluid.technology Seventh International Conference on New Actuators,.Bremen,. Germany,.pp 126–130.

[19].Sims,.N D.,.Peel,.D J.,.Stanway,.R.,.Johnson,.A R.,.and.Bullough,.W A 2000 The electrorheological long-stroke damper: a new modeling technique with. experimental.validation Journal of Sound and Vibration.229:.207–227.

Choi et al (1998) explored the control characteristics of a continuously variable electrorheological (ER) damper, highlighting its potential applications in mechatronics Additionally, Lee (2000) developed a hysteresis model for the damping forces of a magnetorheological (MR) damper specifically designed for passenger cars in his master's thesis at Inha University, Korea These studies contribute valuable insights into the advancements in damping technologies for automotive applications.

Introduction

Vehicle vibration caused by different road conditions necessitates effective attenuation Successfully suppressing vibration enhances the lifespan of vehicle components, improves ride comfort, and ensures better steering stability This is typically achieved through a vehicle's suspension system, which primarily includes springs and shock absorbers (or dampers) Various suspension methods have been developed to manage vehicle vibration, and these systems are classified into three types—passive, active, and semi-active—based on the level of external power they require.

Passive suspension systems with conventional dampers are simple and cost-effective but suffer from performance limitations due to uncontrollable damping forces, making them inadequate for superior vibration isolation on varying road conditions In contrast, active suspension systems enhance performance by incorporating additional active forces and control algorithms, allowing for improved responsiveness to disturbances across a wide frequency range However, these systems necessitate large power sources and complex components like sensors and servo valves A semi-active suspension system, also known as adaptive-passive, mitigates these issues by integrating a control scheme with tunable devices, replacing active force generators with modulated variable components such as variable rate dampers This approach enables semi-active suspensions to deliver desirable performance without the need for extensive power sources or costly hardware.

Recent advancements in semi-active shock absorbers utilizing magnetorheological (MR) fluid have shown promising applications in vehicle suspension systems Carlson et al introduced a commercially available MR damper suitable for both on- and off-highway vehicles, demonstrating enhanced damping force and control through the application of a magnetic field Spencer Jr et al developed a dynamic model to predict the damping force of MR dampers, validating their predictions against experimental measurements Similarly, Kamath et al proposed a semi-active MR lag mode damper model, confirming its accuracy through comparative analysis of predicted and measured forces Yu et al assessed the performance of MR suspension systems through road tests, while Guo and Hu presented a nonlinear stiffness MR damper model, which they validated through simulations and experiments.

Du, Sze, and Lam proposed an H-infinity control algorithm for a vehicle MR damper, demonstrating its effectiveness through simulation Shen, Golnaraghi, and Heppler introduced a load-leveling suspension system utilizing an MR damper Pranoto and Nagaya developed a 2DOF-type rotary MR damper and validated its efficiency Ok et al applied MR dampers in cable-stayed bridges, confirming their effectiveness with a semi-active fuzzy control algorithm Choi et al manufactured an MR damper for passenger vehicles, detailing its damping force control characteristics, and further extended their research to assess the control performance of the proposed MR damper using hardware-in-the-loop simulation (HILS).

This chapter explores a semi-active suspension system utilizing magnetorheological (MR) shock absorbers, focusing on optimal design, damping force control, and vibration control Section 4.1 details the optimal design of MR fluid-based shock absorbers and presents a case study on damping force controls using a Preisach hysteresis compensator, followed by a discussion on full vehicle suspension with an MR damper Section 4.2 addresses the geometric optimization of MR dampers, incorporating advanced objective functions such as damping force, dynamic range, and inductive time constant Recent research by Rosenfeld and Wereley introduced an analytical optimization design method for MR valves and dampers, based on the assumption of constant magnetic flux density to prevent premature saturation in the magnetic circuit However, this assumption may lead to suboptimal results, as valve performance is influenced by both the magnetic circuit and the geometry of the ducts through which the MR fluid flows.

Nguyen et al proposed an optimal finite element method (FEM)-based design for magnetorheological (MR) valves constrained within a specified volume, focusing on minimizing the valve ratio derived from finite element analysis while considering all geometric variables However, this research did not address the control energy and time response of the MR valves In a subsequent study, Nguyen et al explored an optimal MR valve design that meets specific operational requirements, including pressure drop and minimum control energy, while also accounting for the inductive time constant as a state variable.

This study focuses on optimizing the power consumption of a valve while treating pressure drop and inductive time constants as state variables To enhance computation efficiency and avoid low-quality design parameters, the optimization problem is reformulated from constrained to unconstrained The cylindrical MR damper for vehicle suspension, proposed by Lee and Choi, is examined, with its damping force and dynamic range derived from a quasi-static model based on the Bingham model of MR fluid The control energy and inductive time constant are calculated, and initial geometric dimensions are established under the assumption of constant magnetic flux density in the damper’s magnetic circuit The optimization objective is defined based on the initial damper solution, and an optimization procedure utilizing a golden-section algorithm and local quadratic fitting technique is implemented through ANSYS Parametric Design Language (APDL) The developed optimization tool identifies optimal MR damper solutions constrained within a specific cylindrical volume defined by radius and height.

Section 4.3 discusses the hysteretic behavior of MR dampers influenced by magnetic fields and introduces a new control strategy for precise damping force management MR fluids inherently display hysteretic nonlinear responses, prompting various efforts to address these behaviors in smart material actuators One method involves implementing robust control schemes to mitigate the negative impacts of hysteretic nonlinearity, treating model parameter variations as actuator uncertainties managed through feedback control Alternatively, a nonlinear actuator-driving model can be utilized to estimate and compensate for hysteretic effects in a feed-forward loop The Preisach model emerges as a promising candidate for hysteresis modeling and compensation in MR fluids.

This section explores the hysteretic behavior induced by applied magnetic fields, specifically input currents, in the damping force control of MR dampers The study utilizes Delphi Corporation's Magneride™ MR damper, examining its damping force characteristics through experimental evaluation A Preisach hysteresis model is established for the MR damper, with its first-order descending curves identified experimentally A feed-forward hysteretic compensator is formulated for effective damping force control and subsequently verified through experiments Additionally, a quarter-vehicle suspension model is developed, incorporating a sky-hook controller alongside the hysteretic compensator for enhanced vibration control The performance of vibration control is experimentally assessed in a quarter-vehicle test facility, comparing results with and without the hysteretic compensator.

Section 4.4 explores a full-vehicle suspension system equipped with MR dampers, highlighting that most research has traditionally focused on quarter-car models, while some studies have examined vibration isolation in half- or full-vehicle systems The robustness of MR suspension control systems is often challenged by parameter uncertainties and external disturbances This section's key contributions include the development of a mathematical model for a full-vehicle MR suspension system and demonstrating its effectiveness in attenuating unwanted vibrations in passenger vehicles A dynamic model is derived, and a controller is designed to minimize vibration levels from external road excitations Control responses for vibration isolation are evaluated using the HILS methodology and presented in both time and frequency domains.

Optimal.Design

Configuration.and.Modeling

The cylindrical MR damper for vehicle suspension, proposed by Lee and Choi, consists of upper and lower chambers divided by a piston and filled with MR fluid As the piston moves, the MR fluid flows between the chambers through orifices and an annular duct An external gas chamber serves as an accumulator for the MR fluid flow generated by piston motion By disregarding frictional forces between oil seals and assuming quasi-static behavior, the damping force of the MR damper can be expressed mathematically.

The force exerted by the damper can be calculated using the formula F_d = P_A2 * p - P_A1 * (p - A_s), where A_p and A_s represent the areas of the piston and piston shaft, respectively In this equation, P_1 and P_2 denote the pressures in the upper and lower chambers of the damper Additionally, the relationship between P_1, P_2, and the pressure in the gas chamber, P_a, is critical for understanding the damper's performance.

The equations P2 = +ΔPaP2 and P1 = -Δ - ΔPaP1P3 describe the pressure drops (ΔP1, ΔP2, and ΔP3) associated with the flow of MR through the lower and upper valve orifices, as well as the annular duct between the outer and inner cylinders These pressure drops are crucial for calculating the pressure within the gas chamber.

The initial pressure (P₀) and volume (V₀) of the accumulator are critical parameters in determining its performance The coefficient of thermal expansion (γ) varies between 1.4 and 1.7 during adiabatic expansion, influencing the behavior of the system Piston displacement (xₚ) plays a significant role in the operation, and by neglecting minor losses, the pressure drops (ΔP₁, ΔP₂, and ΔP₃) can be accurately calculated.

The configuration of the MR damper involves several key parameters: τy represents the yield stress of the MR fluid influenced by the applied magnetic field, while η denotes the field-independent plastic viscosity, also known as base viscosity The inner cylinder's length is indicated as L, and tg refers to the gap within the annular duct between the inner and outer cylinders R1 and R2 are the average radii of the intermediate pole and the annular duct, respectively Additionally, Lm signifies the length of the magnetic pole, and tm indicates the gap of the orifice in the MR valve structure The coefficient c, which varies based on the flow velocity profile, ranges from a minimum of 2.07 to a maximum of 3.07, and can be estimated using established methods.

= + η η+ π τ (4.5) where.Q.is.the.flow.rate.of.MR.fluid.flow.through.the.valve.structures From. Equation.(4.1),.Equation.(4.2),.and.Equation.(4.4),.the.damping.force.of.the.MR. damper.can.be.calculated.by

The first term in Equation (4.6) signifies the elastic force generated by gas compliance, while the second term indicates the damping force attributable to the viscosity of magnetorheological (MR) fluid The third term represents the force resulting from the yield stress of the MR fluid, which can be precisely controlled by adjusting the intensity of the magnetic field within the MR fluid ducts This term is crucial as it constitutes the primary component of the damping force, anticipated to be substantial in MR damper designs The dynamic range of the damper, defined as the ratio of peak force under maximum current input to that under zero current input, can be approximated as follows:

The dynamic range is a crucial factor in assessing the overall performance of the MR damper, as a higher dynamic range is anticipated to offer an extensive control range for the damper's functionality.

In this test, the commercial MR fluid (MRF132-LD) from Lord Corporation is utilized Figure 4.2 illustrates the relationship between the induced yield stress of the MR fluid and the applied magnetic field intensity (H mr) Through the least square curve fitting method, the yield stress of the MR fluid can be accurately approximated.

In Equation (4.8), the yield stress is measured in kPa, while the magnetic field intensity is expressed in kA/m The coefficients C0, C1, C2, and C3 are determined to be 0.3, 0.42, -0.00116, and 1.0513E–6, respectively To compute the damping force of the MR damper, it is essential to analyze the magnetic circuit of the damper The solution of this magnetic circuit allows for the determination of the yield stress of the MR fluid within the active volume, where the magnetic flux interacts, using Equation (4.8) Subsequently, the damping force can be calculated from Equation (4.6).

The.magnetic.circuit.can.be.analyzed.using.the.magnetic.Kirchoff’s.law.as:

The magnetic field intensity in the kth link of the circuit is represented by H_k, while l_k denotes the overall effective length of that link The number of turns in the valve coil is indicated by N_c, and I represents the applied current in the coil wire Additionally, the circuit adheres to the magnetic flux conservation rule.

Yield.stress.of.the.MR.fluid.as.a.function.of.magnetic.field.intensity (From.Nguyen,.Q.H et.al.,.

Smart Materials and Structures discuss the relationship between magnetic flux and magnetic properties in circuits, where magnetic flux density (B_k) increases proportionally with magnetic intensity (H_k) at low magnetic fields, described by the equation B_k = μ_0 μ_k H_k Here, μ_0 represents the magnetic permeability of free space, and μ_k is the relative permeability of the kth link material As the magnetic field intensifies, the material's ability to polarize decreases, leading to magnetic saturation The magnetic properties of materials are typically represented by a nonlinear B-H curve Due to the complexity of finding an exact solution for magnetic circuits, approximate solutions are commonly employed The proposed MR damper's approximate magnetic circuit is illustrated in Figure 4.3, with relevant equations provided for further clarification.

Due.to.small.gap.size.of.the.MR.ducts.and.small.thickness.of.the.intermediate.pole,.Equation.(4.11).and.Equation.(4.12).can.be.approximately.expressed.as:

Approximate.magnetic.circuit.of.the.MR.damper.

H_mr and B_mr represent the effective magnetic field intensity and flux density of the MR fluid link Additionally, A_mr and l_mr denote the effective cross-sectional area and length of the MR link, respectively.

The magnetic circuit of the valve can be analyzed using the B-H curves of the magnetorheological (MR) fluid and the material of the valve structure, as indicated by Equations (4.13) and (4.14) In low magnetic fields, the magnetic field intensity across the MR fluid link can be approximated effectively.

A mr c m mr mr mr mr mr mr

= + à à + à à + à à (4.16) where.μ mr and.μ.are.the.relative.permeability.of.MR.fluid.and.the.valve.core. material,.respectively.

To enhance the accuracy of the magnetic circuit solution, Finite Element Method (FEM) has been utilized, specifically employing commercial software ANSYS Given the axisymmetric geometry of the valve structure, a 2D-axisymmetric coupled element (PLANE13) is utilized for electromagnetic analysis It is important to highlight that a highly accurate solution is anticipated through the use of FEM with a fine mesh, as this approach accounts for a significant number of links and the variation of flux density within the active MR volume.

Design.Optimization

This section employs a finite element method integrated with an optimization tool to determine the optimal geometric dimensions of the MR damper, aiming to minimize a specific objective function In vehicle suspension design, two conflicting performance indices—ride comfort and suspension travel (rattle space)—must be addressed Achieving a high damping force is essential for reducing suspension travel, while low damping force is preferred to enhance ride comfort, necessitating a large dynamic range Additionally, a fast time response from the MR damper is crucial for improving the controllability of the suspension system To meet these requirements, a comprehensive objective function is proposed.

The equation α + α λ λ + α represents the relationship between the yield stress force (F MR), dynamic range (λ d), and inductive time constant (T) of the damper, which are derived from the finite element (FE) solution of the damper's magnetic circuit The reference values for damping force (F MR,r), dynamic range (λ d,r), and inductive time constant (T r) are also specified The weight factors α F, α d, and α T correspond to the damping force, dynamic range, and inductive time constant, respectively, with the constraint that their sum equals one (α F + α d + α T = 1).

The selection of weighting factors in suspension system design is crucial and varies based on the intended use For systems intended for uneven or unpaved roads, a high damping force is essential, leading to the selection of larger values for αF and αT Conversely, when designing suspension systems for flat roads, a larger value of αd is typically utilized.

The.inductive.time.constant.and.control.energy.of.the.MR.damper.can.be. calculated.as:

N I R= 2 w (4.19) where.L in is.the.inductance.of.the.valve.coil.given.by.L in =N c Φ/I,.and.R w is. the.resistance.of.the.coil.wire.that.can.be.approximately.calculated.as:

In Equation (4.20), key parameters include Lw, the length of the coil wire; rw, the resistance per unit length; dc, the average diameter; Aw, the cross-sectional area of the coil wire; and r, the resistivity of the coil wire, specifically 0.01726E-6 Ωm for copper The number of coil turns, Nc, can be approximated by Nc = AA c / w, where Ac is the cross-sectional area of the coil The reference damping force, dynamic range, and inductive time constant of the damper are derived from the initial design parameters of the MR damper These initial geometric dimensions, including the core radius Rc, coil width wc, and valve housing thickness th, are established under the assumption of constant magnetic flux density throughout the magnetic circuit of the damper.

In Equation (4.21), Rv represents the outer radius of the valve structure The initial pole length, L m, is set to ensure that the magnetic flux density remains below the saturation level of the valve core material, which is 1.5 T for silicon steel in this test It is important to note that the permeability of the MR fluid is significantly lower than that of the valve core material Consequently, the magnetic field intensity of the MR fluid link can be approximated using the formula Hmr = Nc I/2tm To comply with the saturation constraints of the valve core material, specific conditions must be met.

To optimize the MR damper using Finite Element Method (FEM), an analysis file is created with ANSYS Parametric Design Language (APDL) to solve the damper's magnetic circuit and calculate the objective function Key design variables, including coil width, pole length, MR orifice gap, and core radius, are coded and assigned initial values During the optimization process, the valve structure's geometric dimensions are adjusted, and meshing size is defined by the number of elements per line Given that the magnetic field intensity varies along the pole length, it is essential to establish paths within the MR active volume for magnetic flux The average magnetic field intensity across the MR ducts is computed by integrating the field intensity along the defined path and normalizing it by the path length, leading to the determination of magnetic flux and magnetic field intensity.

∫ ∫ Φ = π = (4.24) where B mr (s) and H mr (s) are the magnetic flux density and magnetic field. intensity.at.each.nodal.point.on.the.defined.path.

The first-order method implemented in the ANSYS optimization tool is utilized to determine the optimal design parameters for the MR damper The process begins with initial design variable values, and by executing the analysis file, key metrics such as magnetic flux, field intensity, damping force, dynamic range, and inductive time constant are calculated The ANSYS optimization tool then reformulates the optimization problem by converting constrained design variables into unconstrained ones using penalty functions The dimensionless, unconstrained objective function is defined as f.

In the initial iteration (j = 0), the search direction for design variables (DVs) is determined by the negative gradient of the unconstrained objective function The reference objective function value, OBJ 0, is selected from the current group of design sets, while P xi serves as the exterior penalty function for the design variable x i The direction vector is then calculated accordingly.

The.values.of.DVs.in.the.next.iteration.(j + 1).is.obtained.from.the.following. equation.

Initial Values of Design Parameters Run Analysis File

Calculate New Values of DV (Golden-Section Algorithm)

Calculate Equivalent Unconstrained Objective Function (f )

Calculate Equivalent Unconstrained Objective Function f

Find DV Direction Vector (Polak-Ribiere Recursion Formula)

Find DV Direction Vector (Negative of the Gradient of f )

Magnetic flux Magnetic field intensity Objective function

To achieve optimal design parameters for the MR damper, a flowchart outlines the process where the line search parameter \( s_j \) is determined using a combination of the golden-section algorithm and local quadratic fitting techniques The analysis file is executed with the updated design variable (DV) values, and the convergence of the objective function is monitored If convergence is achieved, the current DV values are considered optimal; if not, additional iterations are conducted These subsequent iterations follow a similar procedure to the initial iteration, with direction vectors calculated using the Polak-Ribiere recursion formula.

Thus,.each.iteration.is.composed.of.a.number.of.sub-iterations.that.include.search.direction.and.gradient.computations.

Optimization.Results

This section computes the optimal solution for the MR damper using the previously established optimization procedure The valve core is made from silicon steel, whose magnetic properties are depicted in Figure 4.5(a), while the magnetic properties of the MR fluid are shown in Figure 4.5(b) The coil wire, constructed from copper, has a relative permeability equivalent to that of free space (μc = 1) The base viscosity of the MR fluid is assumed to be constant at η = 0.092 Pa·s, and the piston velocity for the optimization problem is set at vp = 0.4 m/s Significant dimensions of the MR damper, including the coil width (wc), valve housing thickness (th), valve orifice gap (tm), and pole length (Lm), are designated as design variables, as illustrated in Figure 4.6 The coil wires are specified as 24-gauge (diameter = 0.5106 mm), with a maximum allowable current of 3 A, while the maximum applied current to the coil is I = 2 A The current density applied to the coils can be approximately calculated using J = I/Aw.

The effectiveness of a simulation improves with a smaller mesh size; however, this also increases computational costs To achieve convergence in the solution, the mesh size should be defined by the number of elements per line instead of the element size This is quantified by a parameter known as the basic meshing number, which, when set to 10, has been proven sufficient for ensuring convergence in finite element (FE) solutions In this study, the FE model of the valve structure is illustrated, utilizing a basic meshing number of 10 For MR dampers, an initial MR orifice value of 0.75 mm is selected, leading to design variable initial values of Lm = 7.5 mm, wc = 2.4 mm, and th = 5.6 mm The magnetic flux density and magnetic flux derived from the FE method show flux density values of approximately 1.43 to 1.52 T at critical sections, aligning closely with the saturation flux density of 1.5 T from prior analytical calculations, indicating strong agreement between analytical and finite element results.

Magnetic Field Intensity H (kA/m) (a) B–H curve of silicon steel.

Magnetic Field Intensity H (kA/m) (b) B–H curve of the MR fluid

Magnetic.properties.of.silicon.steel.and.MR.fluid (From.Nguyen,.Q.H et.al.,.Smart Materials and Structures, 18,.1,.2009 With.permission.)

The optimal solution for the MR damper is achieved with equal weighting factors (αF = αd = αT = 1/3), where design variable limits are set as 5 mm ≤ Lm ≤ 13 mm, 2 mm ≤ wc ≤ 8 mm, 0.5 mm ≤ tm ≤ 1.5 mm, and 3 mm ≤ th ≤ 6 mm Initial values yield a stress force of 2065 N, a dynamic range of 28.85, and an inductive time constant of 50.5 ms, with an objective function value of 1 After nine iterations, the solution converges to a minimal objective function value of 0.943, resulting in optimal values of yield stress force at 1715 N, dynamic range at 35.64, and an inductive time constant of 40.6 ms The optimal design variables are Lm = 11.7 mm, wc = 3 mm, tm = 1.1 mm, and th = 4.05 mm, demonstrating significant improvements in dynamic range and inductive time constant, although the yield stress force is slightly reduced compared to initial values Additionally, power consumption at the optimum is calculated to be 6.21 W, lower than the initial 8 W, which can be easily supported by a commercial power unit for vehicles, thus power consumption was excluded from the objective function.

For suspension systems tailored for uneven or unpaved roads, a significant damping force is essential, necessitating a high value of α F The optimal solution for the magnetorheological (MR) system is illustrated in Figure 4.10.

Geometric.dimensions.of.the.MR.damper.

Finite element solution of the damper magnetic circuit at initial design variables (From. Nguyen,.Q.H et.al.,.Smart Materials and Structures, 18,.1,.2009 With.permission.)

Finite.element.model.for.solving.magnetic.circuit.of.the.damper (From.Nguyen,.Q.H et.al.,.

Smart Materials and Structures, 18,.1,.2009 With.permission.)

Iteration (c) Yield stress force, time constant, and dynamic range

Optimized solution of the MR damper when the weighting factors are selected as follows:. αF = αd.=.αT.=.1/3 (From.Nguyen,.Q.H et.al.,.Smart Materials and Structures, 18,.1,.2009 With.permission.)

(c) Yield stress force, time constant, and dynamic range

The optimized solution of the damper is achieved with selected weighting factors: αF = 0.5, αd = 0.2, and αT = 0.3 After eight iterations, the solution converges, yielding a damping force of 2630 N, a dynamic range of 20, an inductive time constant of 42.3 ms, and an objective function value of 0.937 The optimal design variable values are Lm = 12.26 mm, wc = 2.72 mm, tm = 0.81 mm, and th = 4.17 mm Results indicate significant improvements in yield stress force and conductive time constant, although the dynamic range is reduced compared to initial design variables Adjusting the weighting factors allows for a new optimal damper solution, where the term with the largest weighting factor becomes dominant The power consumption at this optimum is 5.16 W.

To enhance ride comfort, it is essential to minimize the damping force resulting from MR fluid viscosity when no current is applied, necessitating a high dynamic range The optimal solution for the MR damper, determined with weighting factors of αF = 0.2, αd = 0.5, and αT = 0.3, converges after 10 iterations At this optimum, the damping force is 1170 N, dynamic range is 63, power consumption is 6.6 W, inductive time constant is 40.3 ms, and the objective function value is 0.826 The design variables' optimal values are Lm = 11.5 mm, wc = 3.3 mm, tm = 1.47 mm, and th = 4.1 mm These results demonstrate the successful achievement of improved dynamic range for enhanced ride comfort.

To validate the accuracy of the finite element (FE) model, this section examines how the optimal solution depends on meshing size It is widely recognized that smaller meshing sizes yield better results; however, they also increase computation time A convergence of the solution is anticipated when the meshing size is reduced to a certain threshold, as illustrated in Figure 4.12, which shows that convergence occurs when the basic meshing number reaches 10 or more, with an error of only 0.15% when increasing from 10 to 12 Consequently, the results depicted in Figures 4.9 to 4.11 are deemed highly accurate Additionally, Figure 4.13 presents the performance evaluation of optimized MR dampers across varying applied current ranges, showing the damping force and inductive time constant for initial and optimized dampers The optimized cases, labeled as “Opt.1,” “Opt.2,” and “Opt.3,” demonstrate that yield stress force approaches saturation when the applied current exceeds 1.5 A Furthermore, at lower applied current levels, the inductive time constant remains nearly constant due to the small induced magnetic flux density, which increases proportionally with magnetic field intensity at higher current levels.

(c) Yield stress force, time constant, and dynamic range

Optimized.solution.of.the.damper.when.the.weighting.factors.are.selected.as.follows:.αF.=.0.2;.αd.=.0.5;.αT.=.0.3 (From.Nguyen,.Q.H et.al.,.Smart Materials and Structures, 18,.1,.2009 With.permission.)

Dependence of the optimal solution on the meshing size (From Nguyen, Q.H et al., Smart

Materials and Structures, 18,.1,.2009 With.permission.)

The yield stress force and inductive time constant of optimized and initial MR dampers vary with different applied values As the density approaches the magnetic saturation of the material, there is a corresponding saturation of yield stress force and a reduction in the inductive time constant Notably, the decrease in the inductive time constant is nearly proportional to the applied current.

Damping.Force.Control

MR.Damper

A.commercial.MR.damper,.Delphi.Corporation’s.Magneride TM that.is.used. for.high-class.passenger.vehicles,.is.adopted Figure 4.14.shows.the.photograph of the MR damper featuring continuous controllability Figure 4.15. presents the measured field-dependent damping force of the MR damper. with.respect.to.the.piston.velocity This.plot.is.obtained.by.calculating.the. maximum damping force at each velocity The piston velocity is changed.

The study investigates the performance of a commercial MR damper by varying the excitation frequency from 0.25 Hz to 3.5 Hz while keeping the excitation amplitude constant at ±20 mm It is observed that as the magnetic field strength increases, the damping force also rises, demonstrating the damper's effectiveness Notably, the MR damper achieves a damping force range comparable to that of a conventional hydraulic passive damper, with a maximum force around 1000 N This damping force behavior is modeled using a biviscous model, as detailed in the following section Additionally, Figure 4.16 illustrates the measured dynamic bandwidth of the system.

The.field-dependent.damping.force.of.MR.damper (From.Seong,.M.S et.al.,.Smart Materials and

The dynamic bandwidth of the MR damper, as reported by Seong et al in "Smart Materials and Structures," is determined by analyzing the damping force in the frequency domain while sweeping input current frequencies at 2 A The findings reveal that the MR damper has a dynamic bandwidth of approximately 38 Hz at -3 dB This capability allows for effective control of both the vehicle body mode (1 to 2 Hz) and the wheel mode (10 to 15 Hz) in passenger vehicles, highlighting the potential of MR dampers in enhancing vehicle performance.

The MR damper exhibits two types of damping force hysteretic behavior: velocity-dependent hysteresis and field-dependent hysteresis The magnetic field, linked to the control input, significantly influences the system's control characteristics Figure 4.17 illustrates the measured hysteresis response of the MR damper relative to input current, with the hysteresis loop exhibiting a counterclockwise direction To analyze hysteresis, we utilize the Preisach model, initially designed to represent hysteresis in magnetic materials, characterized by two key properties: the minor loop property and the wiping-out property The minor loop property indicates that two similar minor loops generated by identical input extremes will overlap after a shift in the output parameter, while the wiping-out property identifies which prior input values affect the current output, determining dominant maxima and minima that can negate smaller values To verify the applicability of the Preisach model to physical hysteresis phenomena, one can assess these properties In this study, we will evaluate the field-dependent hysteretic behavior of the damping force using the Preisach model to enhance the control accuracy of the MR damper.

Hysteretic behavior of MR damper with respect to input current (From Seong, M.S et al.,.

Smart Materials and Structures,.18,.7,.2009 With.permission.)

Preisach.Model

The.Preisach.model.that.we.adopt.to.describe.the.hysteretic.behavior.of.an. MR.damper.can.be.expressed.as:

The equation F MR (t) = ∫∫ Γ à α β γ( , ) αβ [ ( )]i t d d α β represents the damping force due to the magnetic field in a Preisach model, where Γ is the Preisach plane and γ ⋅αβ[ ] denotes the hysteresis relay The current i(t) indicates the magnetic field input, while the weighting function μ(α, β) describes the contribution of each relay to the overall hysteresis Each relay is defined by a pair of switching values (α, β) with α ≥ β, and as the input changes over time, the relays adjust their outputs based on the magnetic field The system output is determined by the weighted sum of all relay outputs, with the simplest hysteretic relay configuration for the MR damper illustrated in Figure 4.18(a), representing a modification of a classical relay design featuring two states.

The configuration of the Preisach model involves states -1 and 1, reflecting the opposite polarization of ferromagnetic materials The output of the relay is binary, either 0 or 1, making it suitable for modeling the hysteresis of an MR damper, where the damping force fluctuates between zero and a maximum value In this context, a hysteresis loop is situated in the first quadrant of the input-output plane, specifically the current-damping force plane The Preisach plane serves as a one-to-one mapping between relays and their switching values (α, β), as illustrated in Figure 4.18(b) The maximum damping force confines the plane within a triangular area, where α0 and β0 denote the upper and lower limits of the magnetic field input, set at 2A and 0A, respectively This model delineates the state of each individual relay, resulting in the division of the Preisach plane into two time-varying regions.

{( , ) | output of at is 0 {( , ) | output of at is 1 Γ = α β ∈Γ γ Γ = α β ∈Γ γ

The.two.regions.represent.that.relays.are.on.the.0.and.1.positions,.respectively Therefore,.the.first.equation.can.be.reduced.to

The numerical technique for identifying the Preisach model has proven to be effective for smart materials In this study, the Preisach model for the MR damper is numerically implemented by utilizing a numerical function for each mesh value of (α, β) within the Preisach plane The mesh values correspond to the data obtained from the experimentally derived FOD curves One example of a mesh value (α1, β1) is shown in Figure 4.19(a), where its corresponding FOD curve exhibits a monotonic increase to α1, followed by a monotonic decrease to β1 After the input peaks at α1, the decrease sweeps out area Ω, creating the descending branch within the major loop A numerical function T(α1, β1) is then defined to represent the output change along this descending branch.

From.Equation.(4.32).and.Equation.(4.33),.we.can.determine.an.explicit.formula.for.the.hysteresis.in.terms.of.experimental.data.

Figure 4.19(b) illustrates the fluctuating series of an input magnetic field, where Γ+ can be divided into n trapezoids, denoted as Qk Geometrically, the area of each trapeoid Qk is calculated as the difference between two triangle areas associated with T(α, βk−1) and T(α, βk) Consequently, the output for each trapezoid Qk is derived accordingly.

Numerical.identification.and.implementation.of.the.Preisach.model.

By.summing.the.area.of.trapezoids.Q k over.the.entire.area.Γ+,.the.output.is. expressed.by

Consequently, for the case of the increasing and decreasing input, the. output.F MR ( ) of the Preisach model is expressed by the experimentally.t defined.T( , )α β k k :

The numerical implementation of the Preisach model involves determining the parameters T(α, β, k) at a finite number of grid points within the Preisach plane However, the limited number of grid points and the measured output values can lead to challenges, as some magnetic field input values may not correspond directly to these grid points Additionally, there may be instances where the necessary output values are not available To address this issue, interpolation functions are utilized to estimate the corresponding T(α, β, k) values when the magnetic field input does not align with the grid points.

To identify hysteresis, FOD data sets are created to compute the numerical function of the Preisach model The input current, which represents the magnetic field, varies from 0 A to 2 A and is divided into 10 sub-ranges This current is applied in a stepwise fashion, with each magnetic field step maintained for 5 seconds to ensure steady-state conditions The damping force is measured using a load cell, allowing for the determination of the FOD curves Subsequently, the Preisach model for the MR damper is numerically identified using the collected FOD data sets.

Hysteresis.prediction.using.the.Preisach.model.is.tested.under.two.different.types.of.input.trajectories.of.the.current:.triangular.and.arbitrary.signals

Predictions.using.the.Preisach.model.are.compared.to.the.results.predicted. by.the.Bingham.model.

Bingham.characteristics.of.the.MR.damper.are.experimentally.measured. and.fitted.to.the.function:.F MR B ( ) 486i = i B 1.16

The analysis of the triangular input depicted in Figure 4.21(a) reveals the actual and predicted damping force responses, as shown in Figures 4.21(b), (c), and (d) During the increasing input phase, both the Preisach and Bingham models yield similar predicted responses However, the hysteresis effect becomes prominently evident in the subsequent decreasing step, as illustrated in Figure 4.21(d) Notably, the results indicate that the Preisach model accurately predicts the measured responses with high precision.

The damping force behavior of the MR damper in response to a sinusoidal input trajectory is illustrated in Figure 4.22(b), with detailed magnifications provided in Figures 4.22(c) and (d) These findings indicate that the Preisach model effectively captures the hysteretic behavior of the damping force, demonstrating its accuracy in reconstruction.

Controller.Formulation

This section establishes a control algorithm to assess the performance of the MR damper's damping force control Figure 4.23 illustrates the block diagram of the MR damper control system, which incorporates a hysteretic compensator Initially, the desired damping force (F_t) is defined within the system The controllable damping force (F_MR) is calculated using a biviscous model To determine the current (i_B) needed for generating the controllable damping force, the inverse Bingham model is employed Additionally, to address hysteresis, the feed-forward hysteretic compensator adjusts the current (i_P) Ultimately, the control input (i) is derived by combining i_B and i_P to achieve the desired damping performance.

Experimental result Preisach model Bingham model (B) (A)

(a) Damping force responses (b) Damping force responses

Experimental result Preisach model Bingham model

Actual and predicted damping force responses under triangular input (From Seong, M.S et.al.,.Smart Materials and Structures,.18,.7,.2009 With.permission.)

Experimental result Preisach model Bingham model (A)

(a) Current input trajectory (b) Damping force responses

Experimental result Preisach model Bingham model

Time (sec) (c) Magnification of (A) (d) Magnification of (B)

Actual and predicted damping force responses under sinusoidal input (From Seong, M.S et.al., Smart Materials and Structures,.18,.7,.2009 With.permission.)

The block diagram illustrates the damping force control of a magnetorheological (MR) damper It outlines how the desired damping force is applied to the MR damper to achieve precise tracking of the specified force The implementation of the control algorithm is detailed in three key steps.

The MR damper exhibits distinct damping coefficients for pre-yield and post-yield stress velocities, as illustrated in Figure 4.15 To model this behavior, the biviscous model is utilized, as depicted in Figure 4.24(a) In this model, c_pr represents the pre-yield stress damping coefficient, c_po denotes the post-yield stress damping coefficient, and x_y indicates the yield stress velocity Notably, the yield stress velocity, x_y, is influenced by the input current, allowing it to be expressed accordingly.

Biviscous.model. where.a,.b,.and.c.are.obtained.from.the.experiment.as:.a =.−0.0315,.b =.0.18557,. and.c =.0.00773 The.total.damping.force.of.an.MR.damper.F t can.be.expressed. using.the.biviscous.model:

F c x x x x c x f F x x c x f F x x t pr y y po y MR y po ( y MR ) y

In Equation (4.40), the parameters c_pr and c_po are derived from experimental results, with values of c_pr = 816 Ns/m and c_po = 0 Ns/m The variable f_y can be calculated using the equation f_y = c_x * pr_y, where i_B = 0 To simplify the damping force model in Equation (4.40), we can separate the total damping force, F_t.

Letting.c s be.c po ,.the.simplified.total.damping.force.has.no.damping.coefficient.term.after.yield.stress.velocity.as.shown.in.Figure 4.24(b).and.is.mathematically.represented.as:

(4.42) where.c pr *.is.c pr −c po and.f y *

.is.c x pr y * , (i b =0) From.Equation.(4.42),.the.damping.force.F MR due.to.the.magnetic.field.is.obtained.as:

For.calculating.the.current.i B ,.the.inverse.Bingham.model.is.adopted.using. Equation.(4.38).as:

The implementation flowchart for a Preisach hysteretic compensator is illustrated in Figure 4.25 This compensation algorithm involves the estimation and linearization of nonlinear hysteretic behavior through a Preisach hysteresis model Comprehensive algorithms are provided for further clarity.

I Desired.damping.force.of.kth.order.F MR d , ( )k.is.determined.

II Calculate.current.i k B ( )using.inverse.Bingham.model. i P (k) = i p (k) + Δi i(k) = i B (k) + i P (k) Δi = ((F MR,d − F MR,r )/α B ) 1/

Compute F MR,r (k) Using Preisach Model

Flow.chart.for.implementation.of.Preisach.hysteretic.compensator.

III Determine.the.control.current.i(k).by.adding.current.i k B ( ).and.i k p ( ). IV Determine.actual.damping.force.F MR r , ( )k using.a.Preisach.model.and. compare.with.desired.damping.force.F MR d , ( )k.

V Compare.the.difference.of.actual.damping.force.F MR r , ( )k and.desired. damping.force.F MR d , ( )k.with.error.limit.ε.

VI If.the.difference.is.larger.than.error.limit.ε,.calculate.the.Δi.and.re- calculate.control.current.i(k).

To ensure optimal performance, repeat the algorithm until the difference falls below the error limit, ε Once the difference is within this acceptable range, implement the control current, i(k), to the MR damper.

In.this.work,.the.error.limit.ε.is.set.to.5N,.and.to.prevent.the.unlimited.repeat,.cycle.limitation.is.set.to.100.

Control.Results

An experiment was conducted to implement the control system depicted in Figures 4.23 and 4.25 The experimental setup for the damping force control of a magnetorheological (MR) damper is illustrated in Figure 4.26 The MR damper was subjected to an excitation amplitude of ±20 mm and a frequency of 2.4 Hz using a hydraulic exciter The damping force of the MR damper was measured using a load cell, while the movement of the piston was also recorded.

The experimental configurations for damping force control of a magnetorheological (MR) damper are measured using a linear variable differential transformer (LVDT) A control signal is generated by a computer data acquisition (DAQ) system, which is then fed back to the current amplifier and applied to the MR damper.

Figure 4.27 illustrates the results of damping force control under sinusoidal conditions, with Figure 4.27(b) displaying the magnification graph of (A) from Figure 4.27(a) The control results indicate that the proposed control algorithm utilizing a hysteretic compensator achieves more accurate damping force control performance compared to scenarios without the compensator Additionally, as shown in Figure 4.27(c), the control current with the hysteretic compensator is lower than that without it Figure 4.28 presents damping force control results under an arbitrary desired trajectory, revealing similar control characteristics as observed in Figure 4.27 This enhanced accuracy in damping force control can lead to improved vibration control performance in suspension systems, which will be further explored in the following section.

This study examines the impact of damping force controllability on the vibration control of a magnetorheological (MR) suspension system, utilizing a quarter-vehicle suspension model The model, depicted in Figure 4.29, features two degrees of freedom and assumes a linear spring for the suspension, with the tire represented as a linear spring component A state space equation is formulated based on the mechanical model for the quarter-vehicle MR suspension system.

In.the.above,.m s and.m u are.the.sprung.mass.and.unsprung.mass,.respectively k s is the stiffness coefficient of the suspension, c s is the damping.

Control with compensator Control w/o compensator

Damping.force.control.results.for.sinusoidal.trajectory (From.Seong,.M.S et.al.,.Smart Materials and Structures, 18,.7,.2009 With.permission.)

Desired force Control with compensator Control w/o compensator

The damping force control results for arbitrary trajectories, as discussed by Seong et al in Smart Materials and Structures, highlight the significance of the suspension coefficient and the tire stiffness coefficient The vertical displacements of the sprung mass, unsprung mass, and excitation are represented by z_s, z_u, and z_r, respectively.

This article discusses the implementation of a straightforward yet highly effective skyhook controller The skyhook controller's logic is recognized for its ease of application in real-world scenarios, allowing for the desired damping force to be effectively established.

In this study, the control gain, denoted as C_sky, is set at 7000 through a trial-and-error method, representing the physical damping coefficient The damping force, represented by u in Equation (4.45), plays a crucial role in the suspension system To effectively manage the damping of the suspension system, it is essential to adjust it based on the suspension travel motion, leading to the implementation of a semi-active actuating condition.

Once.the.control.input.u.is.determined,.the.input.current.to.be.applied.to.the. MR.damper.is.obtained.by.damping.force.control.algorithm.

Vibration.control.characteristics.of.the.quarter-vehicle.MR.suspension.system.are.evaluated.under.two.types.of.excitation.(road).conditions The.first. m u m s z s z u z r k s c s

Mechanical.model.of.the.quarter-vehicle.MR.suspension.system. excitation,.normally.used.to.reveal.the.transient.response.characteristic,.is.a. bump.described.by

The equation \( z_r = A_m [1 - \cos(\omega t)] \) describes the vertical displacement of a vehicle traveling over a bump, where \( \omega = 2\pi f \), \( f = 1/T \), and \( T \) is the period of the excitation In this scenario, the bump has a height \( A_m = 0.07 \, m \) and a width \( D = 0.8 \, m \) The vehicle moves over the bump at a constant speed of 3.08 km/h, equivalent to 0.856 m/s Additionally, the second excitation is represented as a sinusoidal function.

z r =A m sinωt (4.49) where.ω.(0.5.∼.15.Hz).is.the.excitation.frequency.and.A m (1.∼.10.mm).is.the. excitation.amplitude.

Figure 4.30 illustrates an experimental setup for a quarter-vehicle magnetorheological (MR) suspension system designed to assess the effectiveness of a vibration isolation control algorithm This system comprises an MR damper, spring, sprung mass, and tire, all integrated into a hydraulic framework The displacement of the sprung mass and suspension travel are monitored using two Linear Variable Differential Transformers (LVDTs), while a wire sensor captures the excitation signal The hydraulic system simulates road profiles to test the MR suspension system, with a current amplifier regulating the input.

The experimental setup for the quarter-vehicle magnetorheological (MR) suspension system involves controlling the current based on a specific control algorithm applied to the MR damper The system parameters for this quarter-vehicle MR suspension are selected according to the design specifications of a conventional suspension system intended for high-class passenger vehicles, as detailed in Table 4.1.

Figure 4.31(a) presents displacement versus time responses of the MR. suspension system for the bump excitation and Figure 4.31(b) presents.

System.Parameters.of.the.Quarter-Vehicle.Suspension.System

Uncontrolled Control with compensator Control w/o compensator

Time (sec) (a) Displacement vs time (b) Magnification of (A)

Time (sec) (c) Acceleration vs time (d) Magnification of (B)

Control performances under bump excitation (From Seong, M.S et al., Smart Materials and

The article discusses the magnification of acceleration versus time responses, illustrated in Figure 4.31(c), with a focus on region "B." Additionally, tire deflection versus time responses are presented in Figure 4.31(e), with the magnification for region "C" shown in Figure 4.31(f).

Control performances under bump excitation (From Seong, M.S et al., Smart Materials and

The implementation of the Preisach hysteretic compensator in the MR suspension system effectively suppresses unwanted vibrations caused by bump excitation As demonstrated, the input current required with the compensator is lower compared to when it is not used The transmissibility of the MR suspension system shows a significant reduction during suspension travel, particularly around the body resonance frequency of 1 to 2 Hz, when the controller is activated Overall, the performance of vibration control is enhanced further with the use of the hysteretic compensator.

Full-Vehicle.Test

MR.Damper

The MR damper's schematic configuration and photograph are illustrated in Figure 4.33 It consists of upper and lower chambers separated by a piston, which is completely filled with MR fluid As the piston moves, the MR fluid circulates through orifices at both ends, transferring between the chambers An external gas chamber serves as an accumulator, effectively absorbing sudden pressure variations in the lower chamber caused by the rapid motion of the piston.

Transmissibility.for.suspension.travel.under.sine.excitation (From.Seong,.M.S et.al.,.Smart

Materials and Structures,.18,.7,.2009 With.permission.)

(a) Configuration of the MR damper

(b) Photograph of the MR damper

The configuration and analysis of the proposed MR damper assume that the MR fluid is incompressible and that pressure within one chamber is uniformly distributed Additionally, it is considered that the frictional force between the oil seals and fluid inertia is negligible Consequently, the damping force of the MR damper can be expressed mathematically.

In.the.above,.x p and.x p are.the.piston.displacement.and.velocity,.respectively The.magnetic.field.is.related.by.H NI h= /2 m

The first term in Equation (4.50) denotes the spring force from gas compliance, while the second term represents the damping force due to the viscosity of the MR fluid The third term arises from the yield stress of the MR fluid, which can be controlled by the intensity of the magnetic field For this study, the MR fluid used is the commercial product MRF132-LD from Lord Corporation, with experimental parameters α and β identified as 0.083 and 1.25 using a Couette-type viscometer Utilizing the Bingham model, the required damping force for the MR damper was determined for application in a mid-sized passenger vehicle, with design parameters including an outer radius of the inner cylinder at 30.1 mm, a magnetic pole length of 10 mm, a gap between poles of 1 mm, 150 coil turns, and a copper coil diameter of 0.8 mm The damping force relative to piston velocity at various magnetic fields is illustrated in Figure 4.34, showing that increased magnetic fields result in higher damping forces For instance, at a piston velocity of 0.38 m/s, the damping force increases from 452 N to 2189 N with a current input of 2.0 A The simulated damping force aligns closely with the measured data, validating the MR damper model from Equation (4.50) Figure 4.35 presents the measured dynamic response characteristics of the MR damper, obtained by applying a constant current input of 1.6 A across various frequencies from 0.5 Hz to 100 Hz, indicating an approximate dynamic response bandwidth for the MR damper.

The MR damper effectively controls both the vehicle body mode (1 to 2 Hz) and the wheel mode (10 to 13 Hz) in passenger vehicles, achieving optimal performance at 28 Hz with a -3 dB response As illustrated in Figure 4.36, the power consumption of the MR damper is crucial for generating the necessary damping force, with approximately 8 W required to produce a damping force of 2000 N This power demand can be adequately met by the commercial batteries typically found in passenger vehicles, ensuring efficient energy conversion from electrical to mechanical energy.

Dynamic response characteristic of the MR damper (From Choi, S.B et al., Vehicle System

Comparison of the field-dependent damping force between the simulated and measured.results (From.Choi,.S.B et.al.,.Vehicle System Dynamics,.18,.7,.2002 With.permission.)

Full-Vehicle.Suspension

A mathematical model can be developed for a full-vehicle magnetorheological (MR) suspension system that features four MR dampers, as illustrated in Figure 4.37 The vehicle body is considered rigid and possesses degrees of freedom in its motion.

Mechanical Power Generation/ Electrical Power Consumption

Mechanical.power.generation.to.electrical.power.consumption (From.Choi,.S.B et.al.,.Vehicle

The mechanical model of the full-vehicle magnetorheological (MR) suspension system accounts for vertical, pitch, and roll movements This model is connected to four rigid bodies that represent the wheel's unsprung masses, each possessing a vertical degree of freedom To derive the governing equations of motion, the bond graph method is employed, which is particularly effective for modeling hydraulic-mechanical systems Through this approach, the governing equations for the full-vehicle MR suspension system are established.

J af af bf bf aF aF bF bF

J cf df cf df cF dF cF dF m z f f F m z f f F m z f f F m z f f F g s s s s MR MR MR MR s s s s MR MR MR MR s s s s MR MR MR MR us s t MR us s t MR us s t MR us s t MR

The state space equations can be defined as follows: the state vector is represented as \( x = [z_{us1}, z_{us2}, z_{us3}, z_{us4}, g, \theta_1, \theta_2, \phi_1, \phi_2]^T \), while the control input vector is \( u = [F_{MR1}, F_{MR2}, F_{MR3}, F_{MR4}]^T \) and the disturbance vector is \( w = [z_1, z_2, z_3, z_4]^T \) The system dynamics are captured in the equations \( f_{si} = k_z si (si - z_{usi}) + c_z si (si - z_{usi}) \) and \( f_{ti} = k_z ti (usi - z_i) \) for \( i = 1, 2, 3, 4 \).

= (4.52) where.A∈ℜ 14 14 × ,B∈ℜ 14 4 × ,C∈ℜ 4 14 × ,.and.L∈ℜ 14 4 × are.the.system.matrix,.the.control.input.matrix,.the.output.matrix,.and.the.disturbance.matrix,.respectively These.matrices.are.given.by

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Controller.Design

In practical applications, the sprung mass is influenced by loading conditions, including the number of passengers and payload, which in turn affects the pitch and roll moment of inertia To address these structural system uncertainties, we utilize the Loop Shaping Design Procedure (LSDP).

H ∞ robust.stabilization.proposed.by.McFarlane.and.Glover [29] As.a.first. step,.we.establish.nominal.and.perturbed.plants.described.by.normalized. left.co-prime.factorization.as:

− (4.53) where.M.and.N.satisfy.the.following.condition:

The symbol (*) represents a complex conjugate transpose In Equation (4.53), the parameters ∆m and ∆n must satisfy the condition ||[∆ ∆m, n]||₁/∞ ≤ γ, where 1/γ indicates the maximum allowable uncertainty bound Based on the system parameters outlined in Table 4.2, we implement a 30% variation in the sprung mass and the corresponding moment of inertia Figure 4.38(a) illustrates the singular value of the nominal plant, showing a small magnitude at low frequencies, which indicates low gain and bandwidth To effectively reject disturbances and enhance the closed-loop bandwidth, a specific weighting function is designed.

We.formulate.now.the.shaped.plant.as:

W diag e W eW eW eW se c c c c

The singular value plot of the shaped plant, illustrated in Figure 4.38(b), demonstrates a significant increase in magnitude within the low-frequency range, indicating enhanced robustness against external disturbances.

As a second step, the optimal solution of.γ min for robust stabilization is. computed.for.the.shaped.plant.as:

Parameters.of.a.Full-Vehicle.MR.Suspension

The Hankel norm, denoted as L 0.22.m h m 0.001.m η 0.2.Paãs, serves as a crucial design indicator in control systems When loop shaping is executed effectively, a sufficiently small value for γ min is achieved By selecting a suitable γ slightly larger than γ min, we determine γ = 3.22 for the proposed system Utilizing this γ value, we can formulate the suboptimal controller K ∞ as outlined in reference [1].

B X D se c h se se se h h se se se

The article discusses the formulation of state-space system matrices for a shaped plant, represented as A_se, B_se, C_se, and D_se It introduces the equations governing the dynamics, including the relationship between the input and output through the generalized control algebraic Riccati equation (GCARE) and the generalized filtering algebraic equation The variables G_se, X, and Z are highlighted as positive definite solutions within this framework, emphasizing their significance in the control and filtering processes of the system.

Frequency (rad/s) (c) Sensitivity function (d) Complementary sensitivity function

Singular.value.plots.of.the.H ∞ controller (From.Choi,.S.B et.al.,.Vehicle System Dynamics,.18,.7,.2002 With.permission.)

Riccati equation (GFARE), respectively Finally, combining it with pre- designed.weighting.functions,.the.final.H ∞.controllers.are.obtained.as:

Figures 4.38(c) and (d) illustrate the sensitivity and complementary sensitivity functions, revealing a small magnitude in the low-frequency range and a steady 0 dB in the high-frequency range for the sensitivity function This indicates that the designed controller for each MR damper effectively suppresses unwanted vehicle vibrations caused by road disturbances Additionally, the complementary sensitivity function demonstrates robust sensor noise suppression and resilience to system uncertainties, as evidenced by its low magnitude in the high-frequency range and 0 dB in the low-frequency range Figure 4.39 showcases the block diagram of the H ∞ control system, where the control input derived from the H ∞ controller is applied to the MR damper based on suspension travel motion, leading to the imposition of specific actuating conditions.

. u u for u z z for u z z i i i si usi i si usi

The.above.condition.implies.physically.that.the.activating.of.the.controller. u i only.assures.the.increment.of.energy.dissipation.of.the.stable.system.[30]

Block.diagram.of.the.LSDP.H ∞ control.system.

Once.the.control.input.u i is.determined,.the.input.current.to.be.applied.to.the. MR.damper.is.obtained.by:

Performance.Evaluation

Developing new components for complete systems is often time-consuming and costly To mitigate these challenges, theoretical analysis through computer simulation is commonly employed However, this approach can overlook complex real-world scenarios that are difficult to model, leading to inaccuracies in predicting system performance To address these limitations, Hardware-in-the-Loop Simulation (HILS) has recently emerged as a viable alternative The HILS method offers significant advantages, including easy modification of system parameters and cost-effective testing facilities Additionally, it allows for the exploration of a wide range of operating conditions that can emulate practical situations in a laboratory setting This study evaluates the performance of a full-vehicle Magnetorheological (MR) suspension system using the proposed HILS configuration.

Full-vehicle MR suspension model

Schematic.diagram.of.the.hardware-in-the-loop-simulation.(HILS).for.the.full.vehicle.suspension.system.

The Hardware-in-the-Loop Simulation (HILS) consists of three main components: interface, hardware, and software The interface includes a computer equipped with a mounted DSP board, while the hardware comprises the MR damper, current amplifier, and hydraulic damper tester The software features a theoretical model of the full-vehicle MR suspension system along with a control algorithm Initially, a computer simulation of the full-vehicle MR suspension system is conducted using an initial value, integrating with the hydraulic damper tester to apply displacement based on demand signals from the computer The current amplifier then applies control currents to the MR damper as determined by the control algorithm The damping force of the MR damper is measured by the hydraulic damper tester and fed back into the computer simulation, which runs both the tester and the current amplifier in a simulation loop For this test, the front left MR damper is selected based on the hydraulic damper tester's capacity, and the measured damping force is used to calculate the damping forces of the other three MR dampers in the model, assuming their field-dependent dynamic characteristics are similar to one of the MR dampers used in hardware implementation.

The control characteristics for vibration suppression in a full-vehicle Magnetorheological (MR) suspension system are assessed under two types of road excitations using Hardware-in-the-Loop Simulation (HILS) The initial excitation typically employed to analyze the transient response characteristics is a bump.

The equation for the bump excitation is given by ω = π r² V D Z/b, where r represents the half of the bump height (0.035 m), D is the bump width (0.8 m), and D car denotes the wheelbase (2.4 m), which is the distance between the front and rear wheels During the bump excitation, the vehicle moves over the bump at a constant speed of 3.08 km/h (0.856 m/s) Additionally, the second type of road excitation used for evaluating the frequency response is characterized as a stationary random process with a zero mean.

In the analysis of a vehicle's performance on a paved road, the equation \( z_i + \rho_r Z_i V = VW_{ni} \) (where \( i = 1, 2, 3, 4 \)) is utilized, with \( W_{ni} \) representing white noise characterized by an intensity of \( 2\sigma \rho_r^2 V \) The road irregularity values are selected under the assumption that the vehicle maintains a constant speed of 72 km/h (or 20 m/s) For this analysis, parameters such as \( \rho_r = 0.45 \, m^{-1} \) and \( \sigma^2 = 300 \, mm^2 \) are adopted to reflect the conditions of the paved road Additionally, the system parameters for the magnetorheological (MR) suspension system are derived from those of a conventional suspension system tailored for a mid-sized passenger vehicle, as detailed in Table 4.2.

Figure 4.41 illustrates the time responses of the MR suspension system under bump excitation, highlighting the importance of displacement and acceleration of the sprung mass, along with tire deflection, in assessing ride comfort.

(c) Pitch angular displacement (d) Pitch angular acceleration

Bump.responses.of.the.MR.suspension.system.via.HILS (From.Choi,.S.B et.al.,.Vehicle System

The implementation of the H ∞ controller significantly reduces the vertical displacement and acceleration of the sprung mass in vehicles, enhancing road holding Notably, pitch angular displacement, acceleration, and tire deflection are also improved with the control input Controlled frequency responses under random excitation reveal that applying the H ∞ controller leads to a reduction of approximately 20% in vertical acceleration and 40% in tire deflection around body resonance (1 to 2 Hz) Additionally, a reduction of about 30% in vertical acceleration at wheel resonance (10 Hz) is observed These results demonstrate that utilizing a semi-active MR suspension system can effectively enhance both ride comfort and steering stability in vehicles.

Some.Final.Thoughts

This chapter explores a semi-active suspension system that utilizes magnetorheological (MR) shock absorbers to reduce vehicle vibrations caused by different road conditions, enhancing both ride comfort and steering stability.

In Section 4.2, an optimal design procedure for the MR shock absorber was developed using ANSYS parametric design language The optimization aimed to identify the geometric dimensions of the valve structure in the MR damper to minimize an objective function Based on the Bingham model of MR fluid, the damping force and dynamic range were assessed, leading to the formulation of an objective function that combines the ratios of damping force, dynamic range, and inductive time constant against their reference values, weighted accordingly Reference values were established under the assumption of constant magnetic flux density, ensuring it remained below the saturation level of the valve material Optimal solutions for the damper were derived with varying weighting factor sets tailored to specific design objectives, demonstrating that different weighting factors yield distinct optimal results, with the dominant term being influenced by the larger weighting factor The convergence of the optimal solutions was verified to confirm the results, and the performance of the optimized MR dampers was evaluated across various applied current levels, highlighting the optimal trends in yield stress force and inductive time constant.

Random.road.responses.of.the.MR.suspension.system.via.HILS (From.Choi,.S.B et.al.,.Vehicle

Section.4.3.introduced.a.new.control.strategy.for.damping.force.control.of.

The effectiveness of MR dampers was experimentally verified, leading to the formulation of a feed-forward hysteretic compensator based on the Preisach hysteresis model After assessing the damping force characteristics of a commercial MR damper (Delphi Magneride TM), the control scheme was implemented, demonstrating enhanced damping force controllability and reduced power consumption Quarter-car vehicle tests revealed that the MR suspension system, utilizing a skyhook controller integrated with the proposed hysteretic compensator, significantly outperformed the skyhook controller alone in vibration control The hysteretic behavior of MR dampers, influenced by the magnetic field, is crucial for precise damping force control, necessitating consideration of factors such as temperature variation and fluid sedimentation, which may impact overall control performance.

A full-vehicle suspension system utilizing MR dampers was proposed and evaluated through Hardware-in-the-Loop Simulation (HILS) A cylindrical MR damper was designed based on the Bingham model of MR fluid, and its field-dependent damping characteristics were assessed Subsequently, a complete suspension system featuring four independent MR dampers was constructed, and the governing equations of motion were derived To enhance control performance amidst parameter uncertainties and external disturbances, a robust H ∞ controller was designed The control characteristics for vibration suppression under various road conditions were thoroughly evaluated using HILS Results showed that for bump excitation, the implementation of the H ∞ controller significantly reduced vibration levels, as indicated by the acceleration of the sprung mass and tire deflection Additionally, for random excitation, the control characteristics were notably improved, resulting in reduced vertical acceleration of the sprung mass and tire deflection.

The.results.presented.in.this.chapter.are.self-explanatory,.justifying.that. the.MR.suspension.system.is.very.effective.for.vibration.isolation.of.a.passenger.vehicle.

In their 1996 study, Carlson et al explored the applications of commercial magnetorheological fluid devices, highlighting their significance in modern physics and engineering Building on this foundation, Spencer Jr et al (1997) developed a phenomenological model for a magnetorheological damper, contributing to the understanding of these advanced materials in engineering mechanics.

[3].Kamath,.G M.,.Wereley,.N M.,.and.Jolly,.M R 1998 Characterization.of.semi- active magnetorheological fluid lag mode damper SPIE Conference on Smart

Structures and Integrated Systems,.pp.356–377.

[4].Yu,.M.,.Liao,.C R.,.Chen,.W M.,.and.Huang,.S L 2006 Study.on.MR.semi-active. suspension.system.and.its.road.testing Journal of Intelligent Material Systems and

[5].Guo,.D and.Hu,.H 2005 Nonlinear.stiffness.of.a.magnetorheological.damper

[6].Du,.H.,.Sze,.K Y.,.and.Lam,.J 2005 Semi-active.H-infinity.control.of.vehicle.suspension.with.magnetorheological.dampers Journal of Sound and Vibration.238:. 981–996.

[7].Shen, Y., Golnaraghi, M F., and Heppler, G R 2007 Load-leveling suspension system with a magnetorheological damper Vehicle System Dynamics. 45: 297–312.

[8].Pranoto,.T and.Nagaya,.K 2005 Development.on.2DOF-type.and.rotary-type. shock absorber damper using MRF and their efficiencies Journal of Materials

[9].Ok,.S Y.,.Kim,.D S.,.Park,.K S.,.and.Koh,.H M 2007 Semi-active.fuzzy.control of cable-stayed bridges using magnetorheological dampers Engineering

[10].Choi, S B., Lee, S K., and Park,.Y P 2001 A hysteresis model for the field- dependent.damping.force.of.a.magnetorheological.damper Journal of Sound and

[11].Choi,.S B.,.Lee,.H S.,.and.Park,.Y P 2002 H-infinity.control.performance.of.a. full-vehicle.suspension.featuring.magnetorheological.dampers Vehicle System

[12].Nguyen, Q H and Choi, S B 2009 Optimal design of a vehicle magnetorheological.damper.considering.the.damping.force.and.dynamic.range Smart

[13].Rosenfield,.N C and.Wereley,.N M 2004 Volume-constrained.optimization.of. magnetorheological.and.electrorheological.valves.and.dampers Smart Materials and Structures 13: 1303–1313.

[14].Nguyen,.Q H.,.Han,.Y M.,.Choi,.S B.,.and.Wereley,.N M 2007 Geometry.optimization.of.MR.valves.constrained.in.a.specific.volume.using.the.finite.element. method Smart Materials and Structures 16: 2242–2252.

[15].Nguyen,.Q H.,.Choi,.S B.,.and.Wereley,.N M 2008 Optimal.design.of.magnetorheological.valves.via.a.finite.element.method.considering.control.energy.and. a.time.constant Smart Materials and Structures 17:.1–12.

[16].Lee,.H S and.Choi,.S B 2000 Control.and.response.characteristics.of.a.magnetorheological.fluid.damper.for.passenger.vehicles Journal of Intelligent Material

[17].Seong,.M S.,.Choi,.S B.,.and.Han,.Y M 2009 Damping.force.control.of.a.vehicle MR damper using a Preisach hysteretic compensator Smart Materials and

[18].Han,.Y M.,.Choi,.S B.,.and.Wereley,.N M 2007 Hysteretic.behavior.of.magnetorheological.fluid.and.identification.using.Preisach.model Journal of Intelligent

[19].Delivorias,.P P 2004 Application.of.ER.and.MR.fluid.in.an.automotive.crash.energy.absorber Report.No MT04.18.

[20].Walid, H E A 2002 Finite element analysis based modeling of magnetorheological dampers M.S Thesis, Virginia Polytechnic Institute and State. University,.Blacksburg.

[21].Li, W H., Du, H and Guo, N Q 2003 Finite element analysis and simulation.evaluation.of.a.magnetorheological.valve International Journal of Advances

[22].Mayergoyz, I D 1991 Mathematical Models of Hysteresis, New York:. Springer-Verlag.

[23].Hughes,.D and.When,.J T 1994 Preisach.modeling.of.piezoceramic.and.shape. memory.alloy.hysteresis Smart Materials and Structures 6:.287–300.

[24].Ge,.P and.Jouaneh,.M 1997 Generalized.Preisach.model.for.hysteresis.nonlinearity.of.piezoceramic.actuators Precision Engineering.20:.99–111.

[25].Wereley,.N M.,.Pang,.L.,.and.Kamath,.G M 1998 Idealized.hysteresis.modeling.of.electrorheological.and.magnetorheological.dampers Journal of Intelligent

[26].Ma, X Q., Rakheja, S., and Su, C Y 2007 Development and relative assess- ments.of.models.for.characterizing.the.current.dependent.hysteresis.properties. of.magnetorheological.fluid.dampers Journal of Intelligent Material Systems and

[27].Choi,.S B.,.Choi,.Y T.,.Chang,.E G.,.Han,.S J.,.and.Kim,.C S 1998 Control.characteristics.of.a.continuously.variable.ER.damper Mechatronics.8:.143–161. [28].Karnopp,.D C.,.Margolis,.D L.,.and.Rosenberg,.R C 1991 System Dynamics: A

Unified Approach,.New.York:.John.Wiley.&.Sons.

[29].McFarlane,.D C and.Glover,.K 1989 Robust Controller Design Using Normalized

Coprime Factor Plant Description,.Berlin:.Springer-Verlag.

[30].Leitmann, G 1994 Semiactive control for vibration attenuation Journal of

System for Tracked and Railway Vehicles

Introduction

Vehicles can be classified into two main types: wheeled vehicles and tracked vehicles Wheeled vehicles encompass both passenger and railway vehicles As discussed in Chapter 4, the ride comfort and driving stability of passenger vehicles can be significantly improved by implementing MR suspension systems, which effectively reduce vibrations caused by road conditions Next, we will explore the suspension systems used in tracked vehicles.

Tracked vehicles operate on continuous tracks rather than wheels, making them ideal for various terrains, including soft ground and off-road conditions These vehicles encompass construction machinery, military armored units, and unmanned ground vehicles, all designed for high mobility Military tanks and armored vehicles, in particular, require excellent off-road capabilities However, as the speed of these high-mobility tracked systems increases, so does the vibration from rough terrain, which can lead to fatigue for crewmembers and affect the delicate instruments within the vehicle Additionally, vibration in a gun barrel can diminish shooting accuracy.

Excessive vibration in high mobility tracked vehicles restricts their maximum speed, negatively impacting survivability and operational efficiency in combat scenarios While some studies have explored vibration control in tracked vehicle systems using active and semi-active suspension, most research has primarily focused on passenger vehicles This article addresses the lack of research on the application of magnetorheological (MR) fluid in tracked vehicles by proposing a novel suspension system that incorporates an MR fluid-based valve A new double rod-type MR suspension unit (MRSU) is introduced, along with a two-coil annular MR valve design Dynamic modeling of the proposed suspension system is conducted, and optimal design parameters for the MR valve are determined using ANSYS The study also evaluates the vibration control responses of the tracked vehicle equipped with the MRSU through computer simulations.

Section 5.3 discusses the importance of MR suspension systems in railway vehicles, particularly for high-speed trains that enhance transportation efficiency for passengers and freight However, the operation of these high-speed vehicles can lead to car body vibrations, which may result in issues such as reduced ride stability, compromised ride quality, and increased track abrasion Therefore, effective vibration control is essential to improve ride quality and maintain car body stability To address the limitations of passive suspension systems, researchers have increasingly advocated for the implementation of active suspension technologies, including oil valves and pneumatic actuators.

A semi-active system utilizing a magnetorheological (MR) fluid damper has been proposed as a viable option for railway vehicle suspensions This article discusses a feasibility study of the MR damper as a secondary suspension system Initially, the MR damper is designed and integrated with the governing motion equations of the railway vehicle, which account for secondary suspension Following this, the acceleration responses of the car body are analytically assessed using the skyhook control law under various track irregularities, demonstrating the effectiveness of the MR damper system in enhancing railway vehicle suspension performance.

Tracked.Vehicles

System.Modeling

One of the key passive suspension systems for tracked vehicles is the in-arm suspension unit (ISU), which comes in two configurations: single rod and double rod The double rod design offers enhanced load leveling capabilities and improved weight control This section discusses modifications made to the passive double rod ISU to create the modified reactive suspension unit (MRSU) As illustrated in Figure 5.1, the proposed MRSU includes a magnetorheological (MR) valve connected to the orifice of the passive ISU When the pressure difference between the upper and lower chambers exceeds a specific threshold, the MR valve is activated by a magnetic field, allowing for the adjustment of damping forces.

To derive the piston velocity from external disturbances and the moment arm of the MRSU, a coordinate system is introduced as shown in Figure 5.2 This fixed coordinate system, denoted as x_i - y_i, rotates in relation to the component The coordinates (x_ij, y_ij) indicate point i in relation to the coordinate system (x_j - y_j) Point 1 represents the center of rotation of the MRSU, while point 2 indicates the center of the wheel Additionally, point 6 marks the center of the actuating piston, and point 4 signifies the end of the connecting rod.

Points 3 and 7 indicate the ends of the actuating cylinder, while Point 5 represents the intersection of a perpendicular line drawn from Point 1 to the connecting rod To determine the position of this point, we can calculate the coordinates x = 43 and y = 43.

The equation is represented as \( x = -\theta + \theta \), where \( x_{41} = l_{fx} \), \( y_{41} = l_{fy} \), and \( \theta = \theta + \theta_p c \), as illustrated in Figure 5.2 The value 0 is assigned to \( y_{63} \) since point 6 lies on the coordinate system's axis defined by \( x^3 - y^3 \) Consequently, we can derive the following equation.

Hence, 1 sin ( / ).1 y l 43 d θ = − The.piston.position.x 63.with.respect.to.the.coordinate.system.x 3 − y 3.can.be.obtained.as.follows:

To.drive.the.piston.velocity,.x 43,y 43.can.be.obtained.as.follows:

In.addition,.the.angular.speed.θ1.is.defined.by

Configuration.of.the.proposed.MRSU.

Thus,.the.piston.velocity.can.be.calculated.by

The.moment.can.be.derived.by.the.piston.force.and.the.distance.between. point.1.and.point.5 The.moment.arm.l m is.given.by

l m = −y 15= −[x 41sin(θ + θ − p 1) y 41cos(θ + θ p 1)] (5.7) Therefore,.the.moment.acting.on.point.1.can.be.obtained.by

T g =l F m s / cos( )θ1 (5.8) where.F s is.the.spring.force.acting.on.the.position. l fy l fx l v l m l d l n l a

Rotation Center of MRSU Position 1

Edge of Crank Shaft Position 4 Actuating

Piston Position 6 y 3 y 1 x 1 x 3 x 1 x 5 y 1 y 5 ex) relative coordinate x 3 y 3 x 43 , y 43 Position 4

Coordinate.system.of.an.in-arm.MRSU.

To calculate the spring and damping torques of the proposed MRSU, refer to the motion diagram illustrated in Figure 5.3 It is assumed that the gas typically does not exchange significant heat with its surroundings, allowing the gas pressure to be expressed accordingly.

P P V V= st ( / ) st 1.4 (5.9) where P st and.V st = πr l g g 2 are the pressure and volume, respectively, in the. static.equilibrium.state The.volume.V of.the.gas.chamber.is.given.by

V V= st + πr x p 2 ( p −x st ) (5.10) where.x p ,.x st is.the.moving.piston.position.and.static.piston.position.of.each. case r p is.the.radius.of.piston.

On.the.other.hand,.when.the.MR.fluid.flows.through.the.MR.valve,.the. pressure.drop.can.be.expressed.by

The pressure drop in a magnetorheological (MR) fluid can be expressed as the sum of two components: ∆P MR, which accounts for the field-dependent yield stress, and ∆P v, which is attributed to the fluid's viscosity These pressure drops are determined by the geometry of the MR valve, with the equation for ∆P MR given by ∆P MR = 2ct h caτ t / + τ a /h Understanding these relationships is crucial for optimizing the performance of MR fluids in various applications.

∆ = ηP v 6 LQ h R/π 3 1 (5.13) where.c.is.a.coefficient.that.depends.on.the.flow.velocity.profile,.η.is.the.base. viscosity.of.MR.fluid,.Q.is.the.flow.rate.through.the.MR.valve,.and.R 1 is.the. x p

Due to pressure drop between upper chamber and lower chamber

Dynamic.motion.diagram.of.MRSU. average.radius.of.the.annular.duct Thus,.the.damping.torque.at.the.rotational.center.of.the.wheel.arm.is.obtained.as

∆ = ∆P m P MR + ∆P v (5.14) where.F d is.the.damping.force.at.the.actuating.piston.expressed.by

Now,.the.damping.torque.at.the.rotational.center.of.the.wheel.arm.is.obtained. as.follows:

In deriving the pressure equations, two key assumptions are made: the piston velocity is relatively slow, and the compressibility of the MR fluid is negligible.

Optimal.Design.of.the.MR.Valve

The passive double rod ISU has been modified to create the MRSU, which features an MR valve integrated into the manifold of the passive ISU The manifold of the MRSU includes both the MR valve and a two-coil annular valve A simplified structure of this two-coil annular MR valve is illustrated in Figure 5.4, highlighting its constrained specific volume Key geometric characteristics of the valve include valve height (L), valve radius (R), valve housing thickness (d_h), MR channel gap (h), iron flange thickness (t), and the relationships between these dimensions.

Magnetic.circuit.of.the.MR.valve. distance.between.the.upper.coil.and.lower.coil.a When.electric.current.is. applied.to.the.coil,.a.magnetic.field.appears.as.shown.in.Figure 5.4.

To optimize the MR valve, ANSYS is integrated with an optimization tool to determine the optimal geometric dimensions of the MRSU valve The primary objective is to minimize the valve ratio, which is defined as the ratio of the viscous pressure drop to the field-dependent pressure drop of the MR valve A lower valve ratio significantly impacts the performance characteristics of the MR valve, making it desirable to achieve a small value The valve ratios for the two-coil MR valve are calculated using Equations (5.12) and (5.13).

The yield stress of the MR fluid, influenced by the magnetic circuit, has been measured, resulting in a polynomial curve illustrated in Figure 5.5 This study utilizes a commercially available MR fluid, specifically MRF-132DG manufactured by Lord Company, as detailed in Table 5.1 The polynomial curve represented in Figure 5.5 can be described by the following equation.

τ(kPa) 52.962= B 4 −176.51B 3 +158.79B 2 +13.708B+0.1442 (5.18) where.τ is.the.yield.stress.caused.by.the.applied.magnetic.field.and.B.is.the. magnetic.flux.density.of.the.applied.magnetic.field.

The geometry and dimensions of MR valves, including coil width, flange thickness, and valve housing thickness, significantly influence valve performance and are classified as design variables (DVs) Initially, a log file is created to analyze the magnetic circuit of the valve and calculate the pressure drop and valve ratio using ANSYS parametric design language.

In the optimization process of MR fluid valves, it is essential to input variables and assign initial values The geometry dimensions of the valves change during optimization, necessitating the specification of mesh size by the number of elements per line rather than by element size After solving the magnetic circuit of the valve, the average magnetic flux density (B) through MR flows is determined from the finite element (FE) solution by integrating the flux density along a defined path and dividing it by the path length These paths are established along the ducts traversed by the magnetic circuit A flowchart detailing the optimization procedures is illustrated in Figure 5.6.

To achieve optimal design parameters for the MR valve, the first-order method of the ANSYS optimization tool is employed This tool converts the constrained optimization problem into an unconstrained one using penalty functions Subsequently, the dimensionless, unconstrained objective function is formulated accordingly.

(5.19) where.λ1.is.the.objective.function,.λ0.is.the.reference.objective.function,.q.is. the.response.surface.parameter,.P x is.the.exterior.penalty.function,.and.P g is. the.extended.interior.penalty.function.

In the initial iteration, the search direction for the design variable is assumed to be the negative gradient of the unconstrained objective function, with the direction vector calculated accordingly.

The.values.of.design.variables.in.the.next.iteration.are.calculated.by.line. search.parameter.s j

y ( 1) j + =y ( ) j +s d j ( ) j (5.21) where the line search parameter s j is calculated using a combination of a. golden-section.algorithm.and.a.local.quadratic.fitting.technique The.log.file.

Magnetic.Properties.of.the.Valve.Components

MR.fluid MRF-132DG B-H.Curve B-H.Curve

The execution of cap/bobbin steel is guided by the new values of decision variables (DVs) and the convergence of the objective function The convergence is assessed through the parameter λ; if convergence is achieved, the DVs at the jth iteration represent the optimal values If convergence does not occur, further iterations will be conducted according to a specified formula.

Thus,.each.iteration.is.composed.of.a.number.of.sub-iterations.that.include. search.direction.and.gradient.computations.

Calculate equivalent unconstrained objective function

Find DV direction vector (Negative of the gradient)

Calculate new values of design variable (Golden-section algorithm)

Calculate equivalent unconstrained objective function

Calculate new value of DV y n

Flowchart.to.achieve.optimal.design.parameter.

The optimal solution for a Magnetorheological (MR) valve is derived from the optimization procedure outlined in previous equations, utilizing the magnetic properties of the MR valve as detailed in Table 5.1 These valves are housed in a cylinder with a radius of 0.23 mm and a height of 120 mm It is assumed that the base viscosity of the MR fluid remains constant at 0.092 Pas, with a flow rate of 3 × 10⁻⁴ m³/s For this test, a valve gap of 1 mm is selected, and the current density applied to the coils can be estimated using the appropriate equation.

In this analysis, the current (I) applied to the coil is defined by the equation J I A = / w (5.24), where A w represents the cross-section of the wire The wire is specified as 24-gauge, with a resistance of 0.18 mΩ per unit length and a maximum allowable current of 3 A To assess the power consumption of the valve coil, the power consumption (N) is expressed in the relevant formula.

N I R= 2 w (5.25) where.R w is.the.resistance.of.the.coil.wire,.which.can.be.calculated.approximately.as.follows:

The equation Rw = Lrw = Vc / w (5.26) defines the relationship between the length of the coil wire (Lw), the resistance per unit length of the coil wires (rw), and the total volume of all coils in the MR valve (Vc) Additionally, the calculations for the valve core radius (Rc), the thickness of the iron flange (t), and the thickness of the valve housing (dh) are derived from specific equations.

Figure 5.7 illustrates the optimal solution for a two-coil annular MR valve, constrained within a specific volume when a current of 2.0 A is applied to the valve coils At these initial conditions, the valve ratio is λ = 0.087, the pressure drop is ∆P = 1.47 bar, and the power consumption is N = 0.51 W, with coil width W = 0.5 mm, valve housing thickness d = 0.4 mm, and iron flange thickness t = 0.20 mm The optimal design variables are determined to be t_opt = 0.88 mm, W_opt = 1.5 mm, and d_h_opt = 5.53 mm Under these optimal conditions, the power consumption reduces to N_opt = 0.32 W, and the pressure drop decreases to ∆P_i_opt = 0.23 bar Additionally, the difference in magnetic flux density between the outer and inner ducts is 0.0398 T, which is considered small and acceptable according to the FEM analysis presented in Figure 5.8.

Vibration.Control.Results

The vehicle model discussed in this section features a single-axis military vehicle with a sprung mass and independent suspension The vehicle body is considered rigid, allowing for vertical movement (X direction) Figure 5.9 illustrates a mechanical model that includes both the wheel and the body The model incorporates the gas spring torque (Tg), the MR damping torque (Td), and the vertical force (Fy) in its analysis The governing equation for this model can be derived from these components.

Optimization.results.of.an.MR.valve.

Magnetic.flux.density.of.an.MR.valve.

Mechanical.model.of.a.wheel.and.the.body.

In Equation (5.30), the variables represent key components of a vehicle's dynamics: \(M_s\) is the sprung mass, \(M_u\) is the unsprung mass, and \(F_y\) denotes the vertical force exerted by the track Additionally, \(k_s\) and \(c_s\) are essential coefficients for spring and damping, influenced by the initial pressure within the gas chamber and the viscosity of the magnetorheological (MR) fluid.

The comparison of spring torque and damping torque in the MRSU is illustrated in Figures 5.10 and 5.11, highlighting the differences between the initial and optimized configurations The parameters utilized in the computer simulation are detailed in Table 5.2 Notably, the spring torque demonstrates a nonlinear behavior in both scenarios, attributed to the compressive effect of the gas chamber, which is proportional to the product of the piston area and the distance moved Additionally, the spring characteristics are influenced by the gas pressure within the chamber, while an increase in damping torque is also observed.

D am pi ng T orque (kNm)

The characteristics of the optimized Magnetic Resonance Stimulation Unit (MRSU) reveal an increase in the magnetic field compared to its initial design Additionally, the damping torque of the optimized MRSU is greater than that of the initial version under the same control input This indicates that the optimized MRSU demonstrates enhanced energy efficiency and overall performance.

The proposed semi-active Magnetic Resonance Suspension Unit (MRSU) may experience reduced vibration isolation performance when subjected to excessive magnetic fields To ensure effective isolation under varying road conditions, it is essential to implement an appropriate control scheme This article adopts the semi-active skyhook controller as a suitable control algorithm, known for its straightforward logic and ease of use.

Sp ring To rque (kNm)

D am ping To rque (kNm)

Characteristics.of.the.MRSU:.optimal.design. to.implement.in.a.practical.field The.desired.damping.force.for.the.MR. damper.is.set.by

The equation \( C_{sky} = u \cdot C_{s} \) (5.31) represents the gain of the skyhook controller, which serves as a physical indicator of the damping coefficient It is essential to control the damping force of the suspension system based on the motion of the suspension travel, necessitating specific actuating conditions.

In the computer simulation, the gas spring's initial pressure is set at 82 bars based on its characteristics Figure 5.12 illustrates the bump responses of a tracked vehicle as it traverses a 16-inch bump at a speed of 20 km/h, demonstrating a significant reduction in vertical motion when control current is applied to the MRSU compared to the uncontrolled scenario Furthermore, the optimized MRSU shows greater vibration reduction than the initial model, even with lower control energy, as depicted in Figure 5.12(b) Additionally, Figures 5.10(b) and 5.11(b) reveal that the optimized MRSU has a larger damping torque control range under the same control input, indicating superior vibration isolation performance due to its enhanced dynamic control range Overall, the results confirm the effectiveness of the MR suspension system.

Radius.of.actuating.piston r p 76.mm

Flow.flux.of.MR.fluid Q 0 3.×.10 −4 m 3 s −1

Viscosity.of.MR.fluid η 0.092.Pas

Initial.pressure.of.gas.spring P st 1.01225e 5 Pa

Initial.volume.of.gas.spring V st 1.0053096e −3 m 3

Vertical.force F y 80,000.N system.can.be.successfully.installed.not.only.to.passenger.vehicles.but.also.to.tracked.vehicles.to.attenuate.unwanted.vibrations.

Railway.Vehicles

System.Modeling

The governing equations of motion for railway vehicles with suspension systems can be derived from Newton's laws A 15-degree-of-freedom passenger railway vehicle model is utilized to analyze the lateral response on a tangent track in relation to random track irregularities.

Bump.response.of.the.tracked.vehicle.

Table 5.3 presents the model degrees of freedom for the railway vehicle's wheel-set, which is assumed to follow the track vertically while the wheel-set motion is influenced by creep force input The roll of the bogie frame is disregarded as a degree of freedom Creep forces from wheel-rail interaction are calculated using Johnson and Vermeulen’s creep theory The equations of motion for the vehicle are outlined, incorporating track input terms, and the lateral response of the car body is analyzed for representative alignment conditions.

Wheel-set h 3 k pz a k py body Car

Wheel-set y x l k sy c sy k py k px b x y z

Mechanical.model.of.a.railway.vehicle. and.creep.force.input.[8] The.governing.equations.for.the.wheel-set.can.be. expressed.as.follows:

The.governing.equations.for.the.bogie.frame.can.be.expressed.as.follows:

I b k b b k b k d k d f py py sy sy MR fy py py px px δ − δ − δ − δ − δ − δ + δ

The.governing.equations.for.the.car.body.can.be.expressed.as.follows:

I lk h lc h lF lk h lc h lF

I h k h h c h h F h k h h c h h F c sy sy MR sy sy MR cy sy sy MR sy sy MR cr sy sy MR sy sy MR δ + δ − δ + δ + δ − δ + δ +

(5.35) where.m c,f,w and.I c,f,w are.the.mass.and.inertia.moment.of.car.body,.bogie.frame,. and.wheel-set,.respectively K p,s and.c s are.the.stiffness.and.damping.ratio.of.

The.Degree.of.Freedom.of.a.Railway.

The proposed MR suspension system, illustrated in Figure 5.14, consists of distinct damper and spring components, with the damper divided into upper and lower chambers by a piston head Both the MR damper and safety damper are completely filled with MR fluid, which flows through orifices during piston motion For analytical simplicity, it is assumed that the MR fluid behaves as incompressible and that pressure is uniformly distributed within each chamber Additionally, frictional forces between oil seals and fluid inertia are considered negligible, allowing the damping force to be expressed under the assumption of quasi-static behavior.

The equation F_d = P_A2p - P_A1(p - A_s) describes the force exerted by a piston in a MR damper, where A_p and A_s represent the effective cross-sectional areas of the piston and piston shaft, respectively The pressures in the upper and lower chambers of the damper are denoted as P1 and P2, and the relationship between these pressures and the accumulator pressure, P_a, can be further articulated.

Configuration.of.an.MR.damper.

A typical single-coil MR valve, as illustrated in Figure 5.15, consists of a valve coil, core, and housing MR fluid flows from the inlet through annular ducts between the core and housing to the outlet When the coil is powered, a magnetic field is generated, affecting the MR fluid in the ducts and causing it to transition into a semi-liquid or solid state, effectively halting the flow This functionality allows the MR valve to serve as a relief valve, pressure control valve, or flow rate control valve.

Assuming Bingham fluid behavior for magnetorheological (MR) fluid and neglecting the unsteady effects of its flow in an annular duct, the pressure drop across a single MR valve can be calculated.

The equation ∆ = ∆ + ∆ = η π + τ (5.38) describes the relationship between the viscous and field-dependent pressure drops (∆P vis and ∆P y) in a single annular MR valve In this context, Q represents the flow rate through the MR valve, while τ y indicates the induced yield stress and η denotes the post-yield viscosity of the MR fluid.

The average radius of the annular duct is represented by R_d, while L denotes the overall effective length The annular duct gap is indicated by t_d, and L_p refers to the length of the magnetic pole The coefficient c is influenced by the velocity profile of the MR flow, which can be mathematically expressed.

The.yield.stress.of.the.MR.fluid,.τ y ,.is.a.function.of.the.applied.magnetic.field. intensity,.which.can.be.approximately.expressed.by

The schematic diagram of an MR damper illustrates the applied magnetic field (H_mr), measured in A/mm, which is defined by the equation H_mr = NI_t/2d, where N represents the number of coil turns in the MR valve and I denotes the current Additionally, the coefficient C is an intrinsic value of the MR fluid that must be determined experimentally The pressure in the accumulator can be calculated using established formulas.

The initial pressure and volume of the accumulator are represented as P0 and V0, respectively The coefficient of thermal expansion, γ, varies between 1.4 and 1.7 during adiabatic expansion The piston displacement, denoted as δp, refers to the relative movement between the car body and the bogie frame of a railway vehicle The damping force can be determined using Equations (5.36), (5.37), and (5.38) as referenced in [12].

The first term in Equation (5.42) signifies the elastic force generated by the accumulator, while the second term denotes the damping force resulting from the viscosity of the MR fluid The third term represents the force attributed to the yield stress of the MR fluid, which can be continuously regulated by the magnetic field.

In this study, a commercial magnetorheological fluid (MRF 132 DG) from Lord Corporation is utilized The coefficients C0, C1, C2, and C3 are derived through experimentation and the least squares curve fitting method, yielding values of C0 = 0.3, C1 = 0.42, C2 = 1.16E3, and C3 = 1.05E−6 Using the Bingham model for the MR fluid, the necessary size and level of damping force are established to ensure the MR damper's suitability for military vehicles Key design parameters for the MR damper include a valve radius (Rd) of 28 mm, a magnetic pole length (Lp) of 34 mm, a magnetic pole gap (td) of 1 mm, and 280 coil turns (N).

Vibration.Control.Results

To evaluate the performance of the proposed MR damper, this study adopts the semi-active skyhook controller as a control algorithm The skyhook logic is recognized for its simplicity and ease of practical implementation The desired damping force for the MR damper is established by this effective control strategy.

The control gain of the skyhook controller, represented as C_sky, serves as a physical indicator of the damping coefficient It is crucial to manage the damping force of the MR damper in relation to the piston movement, necessitating specific actuating conditions to be established.

0, ( ) 0 u C for for sky carbody bogieframe carbody bogieframe

The power spectral density (PSD) of lateral, yaw, and roll acceleration of a railway vehicle's car body under random track irregularities is depicted in Figure 5.16 The uncontrolled scenario utilizes a conventional passive system with viscous dampers, while the controlled scenario employs magnetorheological (MR) dampers operated in semi-active control mode The results clearly demonstrate that the secondary suspension system with MR dampers significantly reduces lateral, yaw, and roll vibrations compared to the uncontrolled case, achieving an impressive 24% to 30% improvement in vibration reduction.

The dynamic performance of railway vehicles is crucial for safety and is assessed using specific performance indices that quantify ride quality and vehicle stability Ride quality refers to the vehicle's suspension ability to maintain motion within comfortable limits for passengers while preventing loading damage Factors influencing ride quality include displacement, acceleration, the rate of change of acceleration, and environmental conditions such as noise, dust, humidity, and temperature The ride quality of a railway vehicle is typically evaluated using the ride index method, where the ride index \( V_r \) is derived from octave-band accelerations and their corresponding center frequencies.

The octave-band acceleration, represented as a_rms, and the octave-band center frequency, denoted as f_c, are essential components in calculating ride indices for each frequency band The overall ride index is derived from these calculations, providing a comprehensive assessment of ride quality across different frequency bands.

Power.spectrum.densities.of.the.car.body.acceleration.of.the.railway.vehicle.

Using subjective ratings, the ride indices were correlated with ride quality, as demonstrated in Table 5.4 The findings indicate that railway vehicle suspension systems equipped with Magnetorheological (MR) dampers significantly enhance ride quality in lateral, yaw, and roll directions when compared to uncontrolled systems.

Some.Final.Thoughts

This chapter discusses the successful application of MR suspension systems in both passenger and tracked vehicles to mitigate vibrations caused by road conditions A double-rod type MR suspension unit (MRSU) was proposed for tracked vehicles, with an analysis of its damping and spring characteristics An optimization procedure utilizing finite element analysis was developed to determine the optimal geometry of the MR valve, a crucial component of the MRSU The results showed improved performance metrics, including damping torque Computer simulations of a tracked vehicle equipped with the MRSU demonstrated a significant reduction in vertical motion during bump tests when paired with a skyhook controller Furthermore, the optimized MRSU exhibited superior vibration control performance compared to the initial design, requiring less control energy.

In Section 5.3, a semi-active secondary suspension system utilizing MR dampers was analyzed for railway vehicles A 15-degree-of-freedom model was developed, incorporating three vibration motions: lateral, yaw, and roll of the wheel-set, bogie frame, and car body The governing equations of vehicle motion were integrated with MR dampers, and computer simulations were conducted using a skyhook control algorithm to assess performance under random irregularities with creep force The results confirm that the ride quality of the railway vehicle can be significantly enhanced by implementing the proposed MR damper system.

[1].Wang,.W.,.Chen,.B.,.Yan,.X.,.and.Gu,.L 2011 Design.and.dynamic.simulation. on adjustable stiffness hydro-pneumatic suspension vibration characters of. tracked vehicle Electric Information and Control Engineering (ICEICE), Beijing,. China,.pp 4056–4059.

[2].Kamath,.G M and.Wereley,.N M 1997 Nonlinear.viscoelastic.plastic.mechanism based model of an electrorheological damper Smart Materials and

[3].Gavin,.H P.,.Hanson,.R D.,.and.Filisko,.F E 1996 Electrorheological.dampers,. part.2:.testing.modelling Journal of Applied Mechanics 63:.676–682.

In their 1998 study, Choi et al explored the control characteristics of a continuously variable electrorheological (ER) damper, highlighting its potential applications in mechatronics (Mechatronics, 8, 570–576) Later, in 2009, Ha et al focused on the optimal design of a magnetorheological fluid suspension specifically for tracked vehicles, contributing valuable insights to the field of physics (Journal of Physics).

[6].Iwnicki,.S 2006 Handbook of Railway Vehicle Dynamics Boca.Raton:.CRC/Taylor.

[7].Ha, S H., Choi, S B., Lee, K S., and Cho, M W 2011 Ride quality evaluation.of.railway.vehicle.suspension.system.featured.by.magnetorhological.fluid. damper 2011 International Conference on Mechatronics and Materials Processing

[8].Ha,.S H.,.Seong,.M S.,.Kim,.H S.,.and.Choi,.S B 2011 Performance.evaluation of railway secondary suspension utilizing magnetorhological fluid. damper The 12th International Conference on Electrorheological (ER) Fluids and

Magnetorheological (MR) Suspensions,.Philadelphia,.PA,.pp 142–148.

[9].Coodall, R M., Williams, R A., Lawton, A and Harborough, P R Railway. Vehicle Active Suspensions in Theory and Practice 1981 Vehicle System

Dynamics: International Journal of Vehicle Mechanics and Mobility.10:.210–215.

[10].Choi,.S B.,.Shu,.M S.,.Park,.D W.,.and.Shin.M J 2001 Neuro-fuzzy.control. of.a.tracked.vehicle.featuring.semi-active.electro-rheological.suspension.units

[11].Liao,.W H and.Wang,.D H Semiactive.vibration.control.of.train.suspension. systems via magnetorheological dampers 2003 Journal of Intelligent Material

[12].Nguyen,.Q H.,.Han,.Y M.,.Choi,.S B.,.and.Wereley,.N M 2007 Geometry.optimization of MR valves constrained in a specific volume using finite element. method Smart Materials and Structures 16:.2242–2252.

[13].Coxon,.H E and.McHaughton,.L D 1971 The.elements.of.bogie.design.for.Australian conditions, NSW Railways–Institution of Mechanical Engineers.(Australia).

MR Applications for Vibration and Impact Control

Introduction

Chapters 4 and 5 explored the rheological properties of magnetorheological (MR) fluids, which can be controlled by varying magnetic field intensity These fluids are highly effective in vibration control systems, particularly in semi-active suspension systems that utilize MR shock absorbers Vehicles experience multiple sources of vibration or impact, including engine excitation and external collisions, with road excitation being a significant contributor to vehicle body vibration.

Engine excitation is a significant source of vibration in passenger vehicles, prompting the development of various passive engine mounts to reduce unwanted noise and vibration Recent advancements in controllable engine mounts have shown improved vibration control performance compared to traditional passive mounts A promising solution involves semi-active engine mounts utilizing magnetorheological (MR) fluids Additionally, enhancing vehicle safety during collisions is crucial, leading to research focused on advanced braking systems that preemptively apply braking force This research aims to protect drivers and vehicles from collisions, with studies highlighting the effectiveness of airbags and safety belts in reducing injury rates Investigations into vehicle collision characteristics have been conducted, including analyses of pedestrian accidents Furthermore, innovative solutions like friction dampers have been proposed to minimize impact forces and enhance vehicle security, validated through finite element method (FEM) analysis Lastly, studies have indicated that changes in vehicle stiffness can improve stability under various conditions, utilizing computer simulations to model these effects.

Jawad developed a smart bumper collision controller that adjusts stiffness before impact using a hydraulic system Recent research has focused on shock reduction through smart fluids with reversible properties under an applied field Lee, Choi, and Wereley proposed the use of a magnetorheological (MR) damper to minimize shock transmission in helicopters, designing and validating their controller through mathematical modeling and real-system application Additionally, Ahmadian, Appleton, and Norris introduced a shock reduction mechanism utilizing an MR damper in scenarios involving bomb detonations Further studies by Lee and Choi, as well as Lee, Choi, and Lee, confirmed the effectiveness of an MR damper in vehicle suspension systems Song et al also suggested a shock damper to mitigate impact, demonstrating its performance through acceleration decrements in a one-degree-of-freedom system.

Section 6.2 introduces a mixed-mode MR engine mount designed to effectively reduce unwanted vibrations in passenger vehicles This innovative mount combines shear and flow modes, utilizing MR fluids for optimal performance After confirming that the damping force can be adjusted through magnetic field intensity, the MR engine mount is integrated into a full-vehicle model The governing equations of motion are derived, taking into account the engine's excitation force A semi-active skyhook controller is then formulated and implemented through hardware-in-the-loop simulation (HILS), allowing for the evaluation of vibration control responses at the driver's position in both frequency and time domains.

Section 6.3 presents a magnetorheological (MR) impact damper designed to minimize the force transmitted to a vehicle's chassis during frontal collisions The governing equations of motion for the MR impact damper are derived from a hydraulic model, incorporating a field-dependent force based on a Bingham model of the MR fluid The effectiveness of the collision mitigation performance is assessed through computer simulations utilizing a vehicle model that includes an occupant.

MR.Engine.Mount

Configuration.and.Modeling

The schematic configuration of the mixed-mode MR engine mount is illustrated in Figure 6.1(a), featuring an upper section made of main rubber to ensure optimal stiffness and damping properties The magnetic pole, fixed to the upper plate, moves vertically with the rubber in response to external excitation Motion of the MR fluid occurs through a gap between the housing and the magnetic pole, regulated by the intensity of the magnetic field or current applied to the coil, allowing for controlled damping force Assuming incompressibility of the MR fluid and uniform pressure in the chamber, the dynamic equation of the MR engine mount is derived from the hydraulic model presented in Figure 6.1(b).

+ − τ − (6.1) where.m.is.the.engine.mass,.k.is.the.stiffness.of.the.rubber,.b.is.the.damping. constant.of.the.rubber,.A.is.the.flow.area,.h.is.the.gap.of.the.magnetic.pole,

The.proposed.mixed-mode.MR.engine.mount.

The piston area of the upper chamber is denoted as A_p, while η represents the viscosity of the magnetorheological (MR) fluid Additionally, τ_ys indicates the yield stress of the MR fluid in shear mode The pressure drop between the upper and lower chambers is derived from the findings in reference [20].

In the context of fluid dynamics, Q_f represents the flow of fluid driven by pressure differences, while τ_yf indicates the yield stress of the magnetorheological (MR) fluid in flow mode The flow behavior is characterized by the index n, according to the Herschel-Bulkley model Additionally, L denotes the length of the magnetic pole and w signifies its width The pressures P_1 and P_2 in Equation (6.2) can be derived from the relevant continuity equations.

(6.3) where.Q.is.the.total.fluid.flow,.and.C 1 and.C 2 are.the.compliance.of.the.upper. and.lower.chambers,.respectively.

From.Equation.(6.2).and.Equation.(6.3),.we.can.obtain.the.following.equation:

In the absence of a magnetic field, the flow resistance, denoted as R, plays a crucial role Given that the ratio (C1 + C2) / (C * C * R1 * R2) is significantly greater than one, we can simplify our analysis by assuming that Qf equals zero Moreover, the pressure drop response induced by the magnetic field occurs at a much faster rate than the exciting frequency, allowing us to disregard the derivative term in Equation (6.4) Consequently, we arrive at the governing equation for the MR engine mount.

Based on the governing model from Equation (6.5), a properly sized MR engine mount has been designed and manufactured, as illustrated in Figure 6.2, specifically for a mid-sized passenger vehicle To explore the field-dependent dynamic performance of the MR engine mount, an experimental setup, depicted in Figure 6.3, was established with a 60 kg engine mass The study evaluates the displacement transmissibility between input excitation and output displacement across various magnetic fields, measured by two sets of proximitors, with different current inputs applied to the current amplifier Results, shown in Figure 6.4, demonstrate that the MR engine mount effectively suppresses vibrations near resonance, indicating that the damping force can be controlled by adjusting the current intensity To validate the controllability of the damping force, a semi-active skyhook controller was designed and experimentally implemented, with Figure 6.5 presenting the measured damping force controllability of the MR engine mount, confirming that the damping force arises solely from the viscosity of the MR fluid.

The MR engine mount fluid is effectively monitored before any control action is taken, ensuring that the desired damping force is accurately imposed by the microprocessor This system is enhanced by the activation of the MR engine mount in conjunction with the skyhook controller The remarkable controllability of the damping force leads to efficient vibration control across the entire vehicle system.

Full-Vehicle.Model

The 13 degrees-of-freedom full-vehicle model, depicted in Figure 6.6, represents a front-engine-front-drive (FF) vehicle supported by three engine mounts The first and second MR engine mounts are located at the front and back left corners, while the third MR engine mount is positioned at the right edge of the engine within the (x,z) plane The vehicle body is considered rigid, whereas the engine has degrees of freedom in pitch, roll, and yaw directions Additionally, the vehicle body is connected to four rigid bodies that represent the wheel unsprung mass, each possessing vertical degrees of freedom.

Shaker Current Amplifier Micro-processor

An.experimental.setup.for.the.MR.engine.mount.test. freedom The.governing.equations.of.motion.of.the.full-vehicle.model.are. derived.as.[20]:

1 Vertical.motion(z b ).of.the.vehicle.body:

3 3 3 3 3 3 1 2 3 m z k z z c z z k z z c z z k z z c z z k z z c z z k Z Z c Z Z k Z Z c Z Z k Z Z c Z Z F F F b b s b u s b u s b u s b u s b u s b u s b u s b u e Z e eb e Z e eb e Z e eb e Z e eb e Z e eb e Z e eb MR Z MR Z MR Z

Displacement.transmissibility.of.the.MR.engine.mount (From.Choi,.S.B et.al.,.International

Journal of Vehicle Design,.33,.1-3,.2003 With.permission.)

2 Roll.motion(ϕ).of.the.vehicle.body:

J t k z z c z z t k z z c z z t k z z c z z t k z z c z z t k Z Z c Z Z t k Z Z c Z Z t k Z Z c Z Z t F t F t F br s b u s b u br s b u s b u bl s b u s b u bl s b u s b u bel e Z e eb e Z e eb bel e Z e eb e Z e eb ber e Z e eb e Z e eb bel MR Z bel MR Z ber MR Z

Damping.force.controllability.of.the.MR.engine.mount (From.Choi,.S.B et.al.,.International

3 Pitch.motion(θ).of.the.vehicle.body:

J l k z z c z z l k z z c z z l k z z c z z l k z z c z z l k Z Z c Z Z l k Z Z c Z Z l k Z Z c Z Z l F l F l F bf s b u s b u br s b u s b u br s b u s b u bf s b u s b u bef e Z e eb e Z e eb ber e Z e eb e Z e eb bem e Z e eb e Z e eb bef MR Z ber MR Z bem MR Z θ = − + − − − + −

F F F F e e Z e eb e Z e eb e Z e eb e Z e eb e Z e eb e Z e eb

Full-vehicle.model.with.the.MR.engine.mounts.

2 2 2 3 3 3 3 3 3 m X k X X c X X k X X c X X k X X c X X F e e X e eb e X e eb e X e eb e X e eb e X e eb e X e eb eX

2 2 2 3 3 3 3 3 3 m Y k Y Y c Y Y k Y Y c Y Y k Y Y c Y Y F e e Y e eb e Y e eb e Y e eb e Y e eb e Y e eb e Y e eb eY

8 Rolling.motion.(α).of.the.engine:

J l k Z Z c Z Z h k Y Y c Y Y l k Z Z c Z Z h k Y Y c Y Y l F l F M ef e Z e eb e Z e eb e e Y e eb e Y e eb er e Z e eb e Z e eb e e Y e eb e Y e eb ef MR Z er MR Z eX α = + − + −

9 Yawing.motion.(β).of.the.engine:

J l k X X c X X t k Y Y c Y Y l k X X c X X t k Y Y c Y Y t k Y Y c Y Y M ef e X e eb e X e eb el e Y e eb e Y e eb er e X e eb e X e eb el e Y e eb e Y e eb er e Y e eb e Y e eb eZ β = − − + −

10 Pitching.motion(γ).of.the.engine:

J h k X X c X X t k Z Z c Z Z h k X X c X X t k Z Z c Z Z t k Z Z c Z Z t F t F t F M e e X e eb e X e eb el e Z e eb e Z e eb e e X e eb e X e eb el e Z e eb e Z e eb er e Z e eb e Z e eb el MR Z el MR Z er MR Z eY γ = − − + −

The.variables.used.from.Equation.(6.6).to.Equation.(6.15).are.well.defined. with.specific.values.in.Table 6.1.

Now,.in.order.to.solve.the.equations.of.the.motion,.we.have.to.determine. engine.excitation.force These.are.given.by.[24,.25]:

M eX eY eZ p eX cn c gn gn n p eZ c eY p r eZ

In this article, we analyze the mechanics of an in-line, 4-cylinder, 4-stroke engine, focusing on key parameters such as piston mass (m p), the rotational radius of the crank arm (r), angular velocity of the crankshaft (ω), and the length of the connecting rod (l) We also consider the gas pressure in each cylinder (P cn), the piston area of the engine cylinder (A c), and the gas explosion phase between each cylinder (ϕ gn) to understand engine performance.

Specifications.of.the.Vehicle.Parameters

Variable Value Unit Variable Value Unit m b 868 kg k e1 ,.k e2 ,.k e3 133–240 N/m m e 244 kg t el 0.25 m m u1 ,.m u4 29.5 kg t er 0.52 m m u2 ,.m u3 27.5 kg l ef 0.19 m c s1 ,.c s4 3200 N200e l er 0.21 m c s2 ,.c s3 1700 N700e h e 0.16 m k u1 ,.k u2 ,.k u3 ,.k u4 200,000 N/m l bef 1.3 m k s1 ,.k s4 20,580 N/m l bem 1.11 m k s2 ,.k s3 19,600 N/m l ber 0.9 m

Control.Responses

The HILS configuration, depicted in Figure 6.7, is designed to assess the vibration control performance of the MR engine mount This system comprises a software component, which is the full-vehicle model, a hardware component featuring a shaker, load cell, proximitor, and MR engine mount, and an interface component made up of a microprocessor with a digital signal processing (DSP) board, along with A/D and D/A converters The damping force of the MR engine mount is experimentally measured and applied to the corresponding part of the full-vehicle model Subsequently, the vehicle's vibration control response is evaluated, and the shaker re-excites the engine mount based on the controlled motion This iterative process continues until a satisfactory control response is achieved.

In this test, the magnetic field or current applied to the MR engine mount is determined using a semi-active skyhook controller The desired damping force at each position of the MR engine mount is defined by the equation U_t(i) = C_sky(i)(Z_te(i) - Z_te(bi)) = F_MRi(t) for i = 1, 2, 3.

Interface Part (AD/DA Board) Excitation

Y X β φ zu2 ks2 cs2 mb mu2 zb2 ku2 ku3 mu1 mu4 ku1 zu1 ku4 ks1 cs1 ks4 cs4 zb1 zb4 zu4 zb3

The schematic configuration of the HILS for vibration control involves the control gain, denoted as C skyi, which represents the damping coefficient To effectively manage vibrations, the damping force defined by Equation (6.17) must be applied to the MR engine mount, taking into account the relative motion between components This leads to specific actuating conditions that must be adhered to for optimal performance.

U t U t for Z t Z t Z t for Z t Z t Z t i i ei ei ebi ei ei ebi

The physical condition indicates that the control input \( U_i \) solely ensures the increase of energy dissipation in a stable system Once the control input \( U_i \) is established, the necessary input magnetic field for the MR engine mount can be determined.

−α + α + α + α (6.19) where.α1.and.α2.are.experimental.constants.of.the.field-dependent.yield. stress.of.the.MR.fluid.given.by ( ) 1 2

In this study, the MR fluid product MRF-132LD from Lord Corp is utilized, with experimental evaluations yielding values of α1 and α2 at 3.16 and 224.25, respectively It is important to note that the control magnetic field, derived from Equation (6.19), can be easily converted to the control input current by accounting for the number of coil turns Additionally, Equation (6.19) is based on the assumption that the yield stress of the MR fluid in flow mode significantly exceeds that in shear mode.

Before assessing vibration control performance, the forces transmitted from the engine to the vehicle body are analyzed, as shown in Figure 6.8 The results indicate a significant reduction in transmitted forces through the damping control of the MR engine mount, leading to decreased engine and vehicle body vibrations Notably, uncontrolled responses occur without the control current Figures 6.9 and 6.10 illustrate the frequency responses of displacement and acceleration at the engine's center of gravity, demonstrating effective vibration suppression within the analyzed frequency range when the MR engine mount is activated To further evaluate vertical acceleration, the time history of acceleration at the driver's position is analyzed and presented in Figure 6.11, with responses recorded at an engine idle speed of 750 rpm The findings clearly show that the acceleration magnitude is significantly reduced by applying the control input current, as depicted in Figure 6.11(b).

Force transmitted from the engine to the vehicle body (From Choi, S.B et al., International

Root.mean.square.(RMS).displacement.at.engine.C.G point (From.Choi,.S.B et.al.,.International

Journal of Vehicle Design,.33,.1–3,.2003 With.permission.)

RMS Displacement (mm) z Direction y Direction

RMS.acceleration.at.engine.C.G point (From.Choi,.S.B et.al.,.International Journal of Vehicle

MR.Impact.Damper

Dynamic.Modeling

The configuration of an impact damper, designed for installation in a vehicle's bumper structure, is illustrated in Figure 6.12 This section discusses the use of magnetorheological (MR) fluid to generate damping force during external impacts, such as frontal collisions The MR impact damper features an upper chamber filled with MR fluid, a magnetic circuit that produces a magnetic field, a lower chamber situated beneath the magnetic circuit, and a diaphragm that contains and accumulates the fluid A flow channel connects the upper and lower chambers, and the magnetic circuit is integrated within the inner cylinder, enhancing the damper's effectiveness upon impact.

Acceleration.at.driver’s.position.(750.rpm) (From.Choi,.S.B et.al.,.International Journal of Vehicle

The MR impact damper operates through the deformation of its upper chamber, which features a bellows structure that allows for significant fluid volume changes and large deformations This design provides excellent resistance to oil, moisture, pressure, and ensures airtight functionality When subjected to a magnetic field during impact, the damper generates a damping force at the magnetic pole, which can be continuously adjusted by varying the magnetic field's intensity To analyze the magnetic flux density within the flow channel, a conventional finite element analysis program, ANSYS, is utilized, demonstrating that the flow channel maintains a uniform magnetic field.

The configuration of the MR impact damper is determined by analyzing the magnetic flux density, which dictates the dimensions of the flow channel and magnetic pole For this analysis, it is assumed that the MR fluid is incompressible and that the motion of the MR impact damper occurs in a single direction.

Figure 6.14.shows.a.hydraulic.model.of.the.MR.impact.damper The.transmitted.force.into.the.chassis.body.is.obtained.as:

12:14:17 NODAL SOLUTION STEP = 1 SUB = 1 TIME = 1 HSUM RSYS = 0 PowerGraphics EFACET = 1 AVRES = Mat SMX = 233811

FEM.result.of.the.magnetic.flux (From.Woo,.D et.al.,.Journal of Intelligent Material Systems and

The hydraulic model of the MR damper involves key parameters such as the stiffness (k_b) and damping constant (c_b) of the bellows, along with the pressure (P_1) in the upper chamber and the effective piston area (A_p) The damping force generated is a result of the pressure drop, which encompasses both fluid resistance (ΔP_fp) and the yield stress (ΔP_MR) of the MR fluid Fluid resistance occurs during flow in the absence of a magnetic field, while yield stress arises from the intensity of the magnetic field between poles The overall pressure drop can be expressed as a combination of these factors.

The pressure difference between two points, represented by the equation P_t2 - P_t1 = ΔP_fp + ΔP_MR, consists of two components: ΔP_fp and ΔP_MR The term ΔP_fp is calculated as R_fp Q_fp(t), where R_fp denotes fluid resistance and Q_fp(t) is the flow rate, defined by the product of A_fp, the flow path area, and x_fp(t), the flow velocity Meanwhile, ΔP_MR is expressed as (2/L) * h * τ_y(H), where L represents the length of the magnetic pole (50 mm), while h and H correspond to the gap size (1 mm) and the magnetic field, respectively.

When.the.flow.is.generated.by.external.excitation.x(t),.continuity.equations. at.the.upper.and.lower.chambers.are.obtained.as:

= − (6.22) where.C 1 and.C 2 are.the.compliances.of.the.upper.and.lower.chambers From. Equation (6.20) to Equation (6.22), the transmitted force to body is represented.as:

F k x t c x t k x t x t k x t x t k x t c x t x t F cb b b i fp i fp d fp fp fp fp MR

In.the.above.equation,.F MR is.the.field-dependent.damping.force.due.to.yield. stress.of.the.MR.fluid It.can.be.expressed.by

The dynamic performance of the proposed damper is assessed through computer simulations based on the dynamic model outlined in Equation (6.23) Figure 6.15 illustrates the damping force simulated under harmonic excitation, considering three different input displacement magnitudes of ±20 mm, ±50 mm, and ±100 mm The results are presented for various magnetic fields in both frequency and time domains, revealing that the damping force increases with the intensity of the applied magnetic field.

Collision.Mitigation

A.vehicle.system.is.simplified.to.chassis,.engine.room,.and.bumper Its.motion. is.assumed.to.be.in.one.direction Figure 6.16.shows.the.vehicle.model.with. the.MR.impact.damper.that.has.3.degrees-of-freedom.[28,.29] The.vehicle. model.includes.an.occupant.model.that.has.2.degrees-of-freedom.considering.the.upper.and.lower.part.of.the.body.[30] Therefore,.the.dynamic.model. of.the.vehicle.system.including.occupant.is.derived.as:

The total mass of the vehicle components includes the mass of the bumper (M1), the engine room (M2), the frame (M3), the lower part of the body (M4), and the head mass (M5).

Dynamic.characteristics.of.the.MR.impact.damper (From.Woo,.D et.al.,.Journal of Intelligent

Material Systems and Structures, 18,.12,.2007 With.permission.)

In.this.section,.the.collision.mitigation.performance.is.evaluated.by.computer.simulation Initial.conditions.of.the.vehicle.are.imposed.as:

In a vehicle crash, the collision process consists of three critical phases: crash initiation, airbag deployment, and occupant contact Each phase necessitates varying levels of damping force, prompting the formulation of a control algorithm for the MR damper tailored to the specific timing of the crash.

F kN if t m kN if t m kN if t m cb < ≤

For.performance.evaluation,.a.vehicle.crash.severity.index.(VCSI).is.introduced.as.a.performance.criterion VCSI.is.calculated.by.deceleration.data.as:

Figure 6.17 illustrates the collision mitigation performance of vehicles at three distinct speeds: 32 km/h, 48 km/h, and 56 km/h Notably, at a crash speed of 48 km/h, the occupant deceleration in conventional vehicles is highlighted.

The mathematical model of the vehicle system shows a reduction in gravitational force from 34 G to 31 G when an MR impact damper is installed Concurrently, the Vehicle Crash Severity Index (VCSI) decreases from 459 to 421 Analysis of vehicle body deformation at a crash speed of 56 km/h reveals a decrease in deformation from 65 cm to 58 cm with the use of the MR impact damper These findings indicate that vehicle collision impacts are effectively mitigated by applying varying damping forces during different crash phases Additionally, energy dissipation during the crash is illustrated, with the area under the solid line representing energy dissipated by the MR damper, confirming that the impact energy is efficiently managed through the activation of the MR damper.

Standard vehicle With MR damper

Time (msec) (b) Deceleration of the occupant (48 km/h)

Comparison of the vehicle crash severity index (From Woo, D et al., Journal of Intelligent

Material Systems and Structures, 18,.12,.2007 With.permission.)

Some.Final.Thoughts

This chapter explores two notable applications of magnetorheological (MR) technology in vibration attenuation and collision mitigation for vehicles Section 6.2 focuses on a semi-active engine mount designed to reduce vibrations, evaluated through a comprehensive full-vehicle model The engine serves as a major source of vibration in vehicles, and traditional passive mounts exhibit performance limitations To address this issue, a semi-active mount utilizing MR fluids was introduced, demonstrating enhanced vibration control capabilities within a Hardware-in-the-Loop Simulation (HILS) framework that incorporates the full vehicle model.

100 Standard vehicle With MR damper

Standard vehicle With MR damper

Response.comparison.with.and.without.MR.damper (From.Woo,.D et.al.,.Journal of Intelligent

The study published in Material Systems and Structures demonstrates that both displacement and acceleration at the engine's center of gravity can be effectively minimized by activating three MR engine mounts Furthermore, the research indicates that utilizing control input current for these MR mounts significantly reduces the acceleration levels experienced at the driver's position.

Section 6.3 discusses the implementation of an impact damper designed for vehicle bumpers to minimize driver injuries and reduce vehicle body damage Collision mitigation is a crucial area of research in vehicle engineering, with MR (magnetorheological) fluid emerging as a promising solution The vehicle collision process involves three distinct phases—crash initiation, airbag deployment, and occupant contact—each requiring varying levels of damping force This section presents an analytical crash model that incorporates the MR impact damper, occupants, and vehicle body structure Simulations indicate that utilizing the MR impact damper with a straightforward step-wise control strategy can significantly decrease Vehicle Crash Severity Index (VCSI) and frame deformation, tailored to each phase of the collision.

[1].Barber,.D E and.Carlson,.J D 2010 Performance.characteristics.of.prototype. MR.engine mounts containing glycol MR.fluids Journal of Intelligent Material

[2].Mansour,.H.,.Arzanpour,.S.,.Golnaraghi,.M F.,.and.Parameswaran,.A M 2010 Semi-active engine mount design using auxiliary magnetorheological fluid. compliance.chamber Vehicle System Dynamics.49:.449–462.

Energy.dissipation.with.and.without.MR.damper (From.Woo,.D et.al.,.Journal of Intelligent

Material Systems and Structures,.18,.12,.2007 With.permission.)

[3].Duclos,.T G 1987 An.externally.tunable.hydraulic.mount.which.uses.electrorheological.fluid SAE Technical Paper Series 870963.

[4].Morishita, S and Mitsui, J 1992 An electronically controlled engine mount. using.electro-rheological.fluid SAE Technical Paper Series 922290

[5].Williams, E W., Rigby, S G., Sproston, J L., and Stanway, R 1993 Electro- rheological fluids applied to an automotive engine mount Journal of Non-

[6].Choi,.S B and.Choi,.Y T 1999 Sliding.mode.control.of.a.shear-mode.type.ER. engine.mount KSME International Journal 13:.26–33.

Choi and Song (2002) explored the vibration control of passenger vehicles using a semi-active electrorheological (ER) engine mount, highlighting its effectiveness in enhancing vehicle dynamics Witteman (1999) focused on improving vehicle crashworthiness design by controlling energy absorption in various collision scenarios, presenting significant findings in his Ph.D dissertation at Eindhoven University of Technology These studies contribute to advancements in automotive engineering, emphasizing the importance of innovative technologies in vehicle safety and performance.

[9].Mizuno, K and Kajzer, J., 1999 Compatibility problems in frontal, side, single car collisions and car-to-pedestrian accidents in Japan Accident Analysis

[10].Witteman,.W 2005 Adaptive.frontal.structure.design.to.achieve.optimal.deceleration pulses Proceedings of the 19th International Technical Conference on the

Enhanced Safety of Vehicles,.Washington,.D.C.,.pp 1–8.

[11].Wồgstrửm, L., Thomson, R., and Pipkorn, B 2004 Structural adaptivity for. acceleration level reduction in passenger car frontal collisions International

[12].Jawad, S A W 1996 Intelligent hydraulic bumper for frontal collision mitigation Crashworthiness and Occupant Protection in Transportation Systems ASME 218:.181–189.

[13].Lee,.D Y.,.Choi,.Y T.,.and.Wereley,.N M 2002 Performance.analysis.of.ER/ MR.impact.damper.systems.using.Herschel-Bulkley.model Journal of Intelligent

In their 2002 analytical study, Ahmadian, Appleton, and Norris explored the effectiveness of magnetorheological dampers in mitigating fire hazards associated with battery systems, as published in Shock and Vibration Additionally, Lee and Choi (2000) examined the control and response characteristics of magnetorheological fluid dampers specifically designed for passenger vehicles, highlighting their potential applications in enhancing vehicle safety and performance, as detailed in the Journal of Intelligent Transportation Systems.

[16].Lee,.H S.,.Choi,.S B.,.and.Lee,.S K 2001 Vibration.control.of.a.passenger.vehicle.featuring.MR.suspension.units Transactions of the Korean Society for Noise and

[17].Song,.H J.,.Choi,.S B.,.Kim,.J H.,.and.Kim,.K S 2004 Performance.evaluation.of.ER.shock.damper.subjected.to.impulse.excitation Journal of Intelligent

[18].Hong,.S R and.Choi,.S B 2012 Vibration.control.of.a.structural.system.using. magneto-rheological fluid mount Journal of Intelligent Material Systems and

[19].Duncan,.M R.,.Wassgren,.C R.,.and.Krousgrill,.C M 2005 The.damping.performance of a single particle impact damper Journal of Sound and Vibration 286: 123–144.

[20].Lee,.H H 2002 Vibration.control.of.an.engine.mount.featuring.MR.fluid M.S Thesis,.Inha.University,.Incheon,.South.Korea.

[21].Lee,.D Y and.Wereley,.N M 2000 Analysis.of.electro-.and.magneto-rheological flow mode dampers using Herschel-Bulkley model Proceedings of SPIE. 3989: 244–255.

[22].Watten,.J 1989 Fluid Power Systems,.New.York:.Prentice.Hall.

[23].Singh, R., Kim, G., and Ravinda, R V 1992 Linear analysis of automotive. hydro-mechanical.mount.with.emphasis.on.decoupler.characteristics Journal of

[24].Rao,.S S 1995 Mechanical Vibration,.New.York:.Addison-Wesley.

In the pursuit of reducing vibrations in passenger cars, Ooh (1997) conducted a Ph.D dissertation at Inha University, focusing on the optimization of nonlinear engine mounts Complementing this research, Karnopp et al (1974) explored vibration control through semi-active force generators, as published in the ASME Journal of Engineering for Industry Additionally, Leitmann (1994) contributed to the field by examining semi-active control methods for effective vibration attenuation.

[28].Elmarakbi,.A M and.Zu,.J W 2004 Dynamic.modeling.and.analysis.of.vehicle smart structures for frontal collision improvement International Journal of

[29].Kamal,.M 1979 Vibration.analysis.and.simulation.of.vehicle-to-barrier.impact

[30].Weaver,.J R 1968 A.simple.occupant.dynamics.model Journal of Biomechanics.1: 185–191.

Introduction

Magnetorheological (MR) dampers and mounts are extensively researched in automotive engineering, particularly in vehicle brake systems Brakes serve as mechanical devices that inhibit motion, typically applied to rotating wheels or axles to stop vehicles While most conventional brakes utilize friction to convert kinetic energy into heat, there are ongoing efforts to develop variable resistance brakes that employ MR fluids Notably, Webb created an exercise apparatus featuring an MR brake, and Avraam et al proposed an MR brake for wrist rehabilitation The development of MR brakes aims to replace traditional systems, which often involve complex actuating mechanisms, and has garnered significant interest in the field of haptics Additionally, various applications of MR brakes within the automotive industry have been suggested.

Recent studies on magnetorheological (MR) brakes emphasize the importance of optimal design and analysis to enhance performance Researchers, including Karakoc, Park, and Suleman, as well as Li and Du, have proposed various design criteria such as material selection, sealing, mixing schemes, and MR fluid selection to boost efficiency Multi-objective optimization techniques have been explored, particularly in the development of MR brakes for prosthetic knee joints, as demonstrated by Gudmundsson, Jonsdottir, and Thorsteinsson Additionally, Nguyen and Choi focused on geometric optimal design to address heat generation during vehicle operation Park, Luz, and Suleman highlighted that a multidisciplinary optimal design could significantly reduce the mass of MR brakes compared to hydraulic counterparts The finite element method (FEM) is commonly employed to calculate magnetic parameters, providing accurate magnetic field strength at various locations within the device However, the complexity of meshing magnetic models often necessitates the use of commercial software, which can be costly and time-consuming In certain cases, such as MR brakes, it is sufficient to determine the magnetic field strength in MR fluids to derive braking torque, suggesting the need for simpler numerical approaches that streamline the analysis of magnetic fields in desired components.

Recent efforts have focused on analytically modeling the magnetic circuit of MR brakes using an equivalent electric circuit, where it is assumed that the magnetic flux is uniformly distributed across the components' cross-section and the relationship between field strength and flux density is linear A straightforward magnetic circuit method has been developed, applying Kirchhoff’s laws; however, due to several assumptions, this approach may lack accuracy Therefore, it is essential to demonstrate magnetic phenomena by addressing engineering problems, such as optimization, that demand high precision.

Section 7.2 focuses on applying a magnetic circuit model associated with magnetorheological (MR) devices for the optimal design of a novel bi-directional MR (BMR) brake Unlike conventional MR brakes, the proposed BMR brake operates on a distinct principle, comprising a mechanical component modeled using Bingham’s equation and a magnetic circuit developed through a specialized approach This method adopts a unique cross-section of the MR fluid in the brake, where the gap is significantly smaller than its length, allowing for the approximation of constant magnetic parameters along the gap An optimal design is pursued to minimize mass while addressing various constraints and potential local optima, utilizing a particle swarm optimization algorithm alongside a gradient-based repair method The optimal solution is then analyzed and compared with results obtained through finite element method (FEM) software Finally, experiments conducted on the manufactured BMR brake with optimal parameters validate the accuracy of the proposed analysis methodology.

Torsional vibration is a critical issue in mechanical systems with rotating components, particularly in power transmission systems that utilize rotating shafts or couplings If not properly managed, this vibration can lead to system failures In these systems, the torque generated, such as from internal combustion engines in vehicles, may be inconsistent, and the driven components, like reciprocating compressors, may not respond smoothly to this torque Additionally, components transmitting torque can produce non-smooth or fluctuating torques due to factors like worn gears or misaligned shafts Since the components in power transmission systems lack infinite stiffness, irregular torque can induce vibrations around the axes of rotation.

Numerous studies have focused on torsional vibration control, highlighting the effectiveness of the dry friction damper, also known as the Lanchester damper Early research by Den Hartog and Ormondroyd explored both analytical and experimental methods to identify the optimal dry friction needed in a Lanchester damper for effective damping of torsional vibrations in primary systems A key finding from their work is the relationship between the constant optimal friction torque and the excitation torque affecting the primary system.

Ye and Williams developed a precise steady-state solution for a rotational friction damper, utilizing a numerical method to identify the optimal friction torque They explored the application of a magnetorheological (MR) fluid brake for controlling torsional vibrations, proposing a configuration that featured two strategies for managing MR braking torque The first strategy involved applying a constant current to the MR brake coil, allowing it to function as a passive friction damper with variable torque The second strategy employed a modified skyhook damping control algorithm, adjusting MR brake torque based on the absolute and relative velocities between the MR brake and the primary system, thereby enhancing damping effectiveness Experimental results indicated a significant reduction in torsional vibrations, demonstrating the MR brake's potential as an alternative to traditional Lanchester dampers However, the optimal design of the MR brake was not addressed, which may lead to a costly control system due to the need for monitoring both the primary system's absolute velocity and the MR brake disc's velocity Additionally, recent studies by Sun and Thomas investigated the control of torsional rotor vibrations using an electrorheological (ER) fluid dynamic absorber, but challenges such as the requirement for a high electric field and low friction torque have limited the practical application of ER brakes.

Torsional vibration control is crucial in engineering, with the MR brake absorber emerging as a promising solution While extensive research has explored its application for controlling torsional vibrations, optimal design considerations have often been overlooked Previous studies primarily focused on determining the optimal friction torque at the resonance of the shaft system, relying solely on the magnitude of the applied torque However, it is evident that the optimal friction torque is also influenced by the frequency of the applied torque.

Section 7.3 addresses two key issues: determining the optimal friction torque of the friction absorber in relation to the applied torque frequency and designing an MR brake absorber for optimal performance To achieve these objectives, a configuration for controlling torsional vibrations in a rotating shaft using an MR brake absorber is proposed, with the braking torque derived from the Bingham plastic model of MR fluid The investigation includes the optimal design of the MR brake absorber and a proposed procedure for solving the associated optimization problem Additionally, the optimal torque for effective torsional vibration control is examined Following the optimal design procedure, the specific design of a rotating shaft system is conducted, and the vibration control performance of systems using the optimized MR brake absorber is analyzed.

Bi-directional.MR.Brake

Configuration.and.Torque.Modeling

The.configuration.of.the.proposed.BMR.brake.is.demonstrated.in.Figure 7.1

The system comprises two coils, two rotors, and an outer casing filled with MR fluid, which occupies the space between the rotors and the casing To prevent interference between the two magnetic fields, a non-magnetic partition is strategically inserted.

Non-Magnetic Partition Sliding Bearing

The proposed BMR brake features a unique configuration with a middle location for the casing, allowing for independent control of the current magnitudes supplied to two distinct coils Unlike conventional MR brakes that utilize a single rotor and stator, this design incorporates a movable casing fixed to a driving shaft, which can be linked to a one-dimensional handle for haptic applications Additionally, two rotors are attached to their respective shafts, driven by a bi-output source, enabling them to rotate in opposite directions This innovative setup ensures the presence of two relative shear motions between the surfaces of the rotors and the outer casing, even when the casing is stationary.

When current is applied to the coils, magnetic fields are generated in two distinct zones, leading to the rapid solidification of the MR fluid between the rotors and the outer casing This solidification creates shear friction, which produces resultant torque that can be either resistive or repulsive, depending on the excitation of the coils and the rotation direction of the casing Assuming the outer casing remains stationary while the two rotors rotate in opposite directions, exciting only coil 1 or coil 2 causes the casing to be pulled in the direction of the corresponding rotor The braking torque required to maintain the casing's stationary position opposes the rotor's direction Thus, the torque direction can be altered based on which coil is excited If the casing rotates in the same direction as rotor 1 while only coil 1 is energized, the torque generated is repulsive, functioning like a clutch Conversely, if only coil 2 is activated while both the casing and rotor 1 rotate together, the resulting torque is resistive, causing the brake to operate purely as a brake.

The total torque of the BMR brake is derived from three key sources: the friction between the end faces of the rotors and the casing, the friction between the annular faces of the rotors and the casing, and the dry friction resulting from the sealing scheme These frictional forces collectively contribute to the overall torque generated by the brake system.

T T T= 1− 2 (7.1) where.T 1 and.T 2 are.the.induced.torques.contributed.from.the.rotors.1.and.2,. respectively,.whose.expressions.are.given.by

The total torque in the system is represented by the equation T1 = Tai + Tei + Tfi, where Tfi is the torque resulting from dry friction between the rotor shafts and the casing, which can be experimentally determined The induced torques, Tai and Tei, arise from the friction between the magnetorheological (MR) fluid and the surfaces of rotors 1 and 2, as well as the casing faces, with their magnitude influenced by the properties of the MR fluid and the applied magnetic field By utilizing appropriate current sources for the coils, it is possible to completely eliminate the total induced torque The expressions for the field-dependent torques, Tai and Tei, can be derived based on the geometric parameters illustrated in Figure 7.2.

The shear stresses, denoted as τ ai and τ ei, act on the MR fluid at the surfaces of rotors i (where i = 1, 2) and the casing faces These values can be mathematically represented using Bingham's model, which provides a framework for understanding the behavior of the fluid under shear conditions.

τ = τ + γ ei yei K ei (7.6) where.K.is.called.the.consistency;.τ yai and.τ yei are.the.yield.stresses.of.MR. fluid.at.the.surfaces.of.rotors.i.(i =.1,2).and.casing,.respectively The.variation.

The significant geometric dimensions of a BMR brake are influenced by the yield stresses, which depend on the properties of the MR fluid provided in the manufacturer's datasheet and the magnitude of the currents applied to the coils The shear rates of the MR fluid at the gap between the annular faces and the end faces, represented as γ ai and γ ei, can be calculated using Equations (7.5) and (7.6).

= (7.8) where.Ω i and.Ω c are.the.angular.velocities.of.the.rotors.and.casing,.respectively By.substituting.Equation.(7.5).to.Equation.(7.8).in.Equation.(7.3).and. (7.4),.the.field-dependent.torques.can.be.expressed.as:

The yield stress values, τ yai and τ yei, remain relatively constant across the shear surfaces, allowing for simplification in their representation As a result, Equations (7.9) and (7.10) can be reformulated into more straightforward expressions.

The torque generated by viscosity and dry friction is minimal when compared to the torque induced by the magnetic field Consequently, during the optimal design process, the second terms on the right-hand side of the equations, along with the torques from dry friction, can be disregarded.

Magnetic.Circuit

The yield stresses of magnetorheological (MR) fluid in various positions within a BMR brake are influenced by the applied magnetic field Understanding the behavior of magnetic fields relies on two key principles: Ampere’s and Gauss’s laws Ampere's circuital law indicates that the line integral of a magnetic field around any closed loop is equal to the total electric current passing through that loop This relationship can be mathematically represented by a specific equation, highlighting the connection between magnetic fields and electric currents.

∫ ⋅ = nc (7.13) where.I nc is.the.total.net.current.that.penetrates.through.the.surface.that.is. enclosed.by.the.curve.C.

Gauss's law states that the magnetic flux exiting a given volume in space is equal to the magnetic flux entering that same volume This fundamental principle can be expressed in a widely recognized form.

The equation ∫ B ⋅ dS = 0 illustrates the relationship between magnetic density (B) and the closed surface (S) enclosing a volume Solving this equation, along with its counterpart, can be complex due to the intricate surface shapes and varying magnetic parameters in devices However, in the context of an MR brake, only the magnetic parameters of the MR fluid are necessary for calculating braking torque A significant feature is the unique cross-sectional shape of the MR fluid, where the thickness is considerably less than its length By leveraging these characteristics, a new magnetic circuit analysis is proposed, as depicted in Figure 7.3, which illustrates three configurations of the magnetic circuit for a quarter of the BMR brake, each with different coil placements In configuration (a), the coils are partially located at the end of the casing, while in configurations (b) and (c), they are fully positioned in the radial and end portions, respectively Notably, in configuration (c), most of the magnetic flux flows through the casing rather than the MR fluid, resulting in a significantly low induced torque This indicates that configuration (c) does not yield an optimal solution in engineering optimization problems, allowing it to be excluded from consideration by applying specific constraints.

h b≤ R +t f (7.15) where.h.is.the.height.of.the.coil,.and.b R and.t f are.the.thickness.of.the.rotor. and.MR.fluid.elements,.respectively.

In.the.case.of.configuration.(a),.the.volume.is.discretized.into.11.elements. as shown in the figure, whereas elements 8 and 11 are MR fluid and the.

Torsional.MR.Brake

MR.Fan.Clutch

MR.Seat.Damper

Multi-Functional.MR.Control.Knob

MR.Haptic.Cue.Accelerator

Tiêu đề	Magnetorheological Fluid Technology Applications In Vehicle Systems
Tác giả	Seung-Bok Choi, Young-Min Han
Trường học	CRC Press
Thể loại	book
Năm xuất bản	2013
Thành phố	Boca Raton

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Số trang	314
Dung lượng	10,32 MB