An analytical approach to optimally desi

Meccanica DOI 10.1007/s11012-012-9544-3 An analytical approach to optimally design of electrorheological fluid damper for vehicle suspension system Q.H Nguyen · S.B Choi · Y.G Park Received: 18 October 2010 / Accepted: 29 February 2012 © Springer Science+Business Media B.V 2012 Abstract This work develops an analytical approach to optimally design electrorheological (ER) dampers, especially for vehicle suspension system The optimal design considers both stability and ride comfort of vehicle application After describing the schematic configuration and operating principle of the ER damper, a quasi-static model is derived on the basis of Bingham rheological laws of ER fluid Based on the quasi-static model, the optimization problem for the ER damper is built The optimization problem is to find optimal value of significant geometric dimensions of the ER damper, such as the ER duct length, ER duct radius, ER duct gap and the piston shaft radius, that maximize damping force of the ER damper The two constrained conditions for the optimization problem are: the damping ratio of the damper in the absence of the electric field is small enough for ride comfort and the buckling condition of the piston shaft is satisfied From the proposed optimal design, the optimal solution of the ER damper constrained in a specific volume is obtained In order to evaluate performance of the optimized ER damper, simulation result of a quarter-car suspension system installed with the optimized ER damper is presented and compared with that of the non-optimized Q.H Nguyen · S.B Choi ( ) · Y.G Park Smart Structures and Systems Laboratory, Department of Mechanical Engineering, Inha University, Incheon 402-751, Korea e-mail: seungbok@inha.ac.kr url: http://www.ssslab.com ER damper suspension system Finally, the optimal results of the ER damper constrained in different volumes are obtained and presented in order to figure out the effect of constrained volume on the optimal design of ER damper Keywords Electrorheological fluid · Vehicle damper · Optimal design · Vehicle suspension ride comfort · Vehicle stability · Semi-active control Nomenclature area of damper’s piston Ap area of the piston shaft As damping coefficient of the ER damper cvis viscous dampers of the suspension cv system control gain Cs d gap of the annular duct D width of the test bump applied electric field Ea E Young’s modulus of the piston shaft material damping force Fd damping force due to the yield stress of FER the ER fluid I inertia moment of the shaft sectional area k stiffness of the suspension system safety coefficient ks coefficient considering dynamic load kd acting on the shaft Meccanica wheel and suspension stiffness of the quarter car, respectively length of the piston shaft Ls length of the annular duct Ld maximum available ER duct length Ldmax m mass of quarter car mass of vehicle mv unsprung and sprung masses of the m1 , m2 quarter car, respectively pressure in the upper chamber of the P1 damper pressure in the lower chamber of the P2 damper pressure in the gas chamber Pa (accumulator) initial pressure in the gas chamber P0 flow rate of ER flow in the duct Qd radius of the annular duct Rd maximum available radius of the ER duct Rdmax radius of the piston shaft Rs T transmissibility of the suspension system vehicle velocity Vc initial volume of the gas chamber V0 white noise Wn u control input x(t) displacement of the car body displacement of piston xp road surface input x0 (t) x1 (t), x2 (t) unsprung mass and the sprung mass deflection, respectively half of the bump height X0 y(t) input excitation to the suspension system (road profile) α, β intrinsic values of the ER fluid inner electrode thickness pressure drop between the upper Pd chamber and the lower chamber pressure drop between the lower Pa chamber and the gas chamber γ coefficient of thermal expansion μ post-yield viscosity of ER fluid excitation frequency natural frequency of the suspension ωn system ρ road roughness parameter covariance of road irregularity σ2 yield stress of the ER fluid induced by τy the applied electric field k1 , k2 ξ ξmin damping ratio of the suspension system minimum tunable damping ratio Introduction It is well-known that suspension systems take a vital role in automotive technology Traditionally, a passive vehicle suspension system consists of an energy dissipating element which is a damper, and an energystoring element which is a spring Essentially, the suspension supports the weight of the upper part of a vehicle on its axles and wheels, allows the vehicle to travel over irregular surfaces with a minimum of up-and-down body movement (stability), reduces the load from road transferred to occupants (ride comfort), and allows the vehicle to corner with minimum roll or loss of traction between the tires and the road (good handling) These goals are generally at odds Good ride comfort requires a soft suspension, whereas stability of the vehicle requires a stiff suspension Good handling requires a suspension setting somewhere between the two Therefore, the tuning of a passive suspension involves finding the right compromise between the three conflicting criteria This is an inherent limitation of a passive suspension system In order to compensate for the limitations of a passive suspension system, active suspension systems have been developed and applied in the real field [1–5] With an additional active force introduced as a part of a suspension unit, the suspension system is then controlled using appropriate algorithms to make it more responsive to different types of road profile However, this type of suspension requires high power sources, active actuators, sensors and sophisticated control logics A semi-active configuration can address these limitations by effectively integrating a tuning control scheme with tunable passive devices For this, active force generators are replaced by modulated variable compartments such as variable rate damper and stiffness Recently, the possible application of electrorheological (ER) and magnetorheological (MR) fluids to the development of controllable dampers has attracted considerable interest, especially in vehicle suspension application Sturk et al proposed a high voltage supply unit with ER damper and proved their effectiveness via quarter car suspension system [6] Nakano Meccanica constructed a quarter car suspension model using ER damper and proposed a proportional control algorithm in order to isolate vibration [7] Petek et al constructed a semi-active full suspension system installed with four ER dampers and evaluated suspension performance through the implementation of a skyhook control algorithm which considers heave, pitch and roll motions of the car body [8] Choi et al proposed a cylindrical ER damper for passenger car and its controllability of damping force was proved by implementing a skyhook controller [9] More recently, Choi et al have designed an ER suspension system for middle sized passenger vehicle and a field test under bump and random road conditions is undertaken The control responses for the ride quality and steering stability are also evaluated in both time and frequency domains [10, 11] In order to improve ER suspension performance, modern control algorithms have been applied with considerable success [12–15] As is evident from previous works, most of research works have been focused only on design configuration, damping force evaluation and controller design of ER damper However, an optimal design of the ER dampers considering stability, ride comfort and handling of ER suspension system seems to be absent The primary purpose of the current work is to fill this gap Consequently, the main contribution of this work is to develop an analytical approach to optimally design ER dampers for vehicle suspension application considering both stability and ride comfort After describing the schematic configuration and operating principle of the ER damper, a quasi-static model is derived on the basis of Bingham rheological laws of ER fluid Based on the quasi-static model, the optimization problem for the ER damper is built The optimal solution of the ER damper constrained in a specific volume is then obtained based on the proposed optimal design In order to evaluate performance of the optimized ER damper, simulation result of a quarter-car suspension system installed with the optimized ER damper is presented and compared with that of the non-optimized one designed by Choi et al [11] Finally, the optimal solutions of the ER damper constrained in different volumes are obtained and presented in order to figure out the effect of constrained volume on the optimal design of ER damper Quasi-static modeling of ER damper In this study, the cylindrical ER damper for passenger vehicle suspension proposed by Choi et al [11] is considered The schematic diagram of the damper is shown in Fig The ER damper is divided into upper and lower chambers by the damper piston These chambers are fully filled with ER fluid As the piston moves, the ER fluid flows from one chamber to the other through the annular duct between inner and outer cylinders The inner cylinder is connected to the positive voltage produced by a high voltage supply unit, playing as the positive (+) electrode The outer cylinder is connected to the ground playing as the negative (−) electrode On the other hand, a gas chamber located outside of the lower chamber acts as an accumulator of the ER fluid induced by the motion of the piston In the absence of electric fields, the ER damper produces a damping force only caused by the fluid viscous resistance However, if a certain level of the electric field is supplied to the ER damper, the ER damper produces additional damping force owing to the yield stress of the ER fluid This damping force of the ER damper can be continuously tuned by controlling the intensity of the electric field By neglecting the compressibility of the ER fluid, frictional force and assuming quasi-static behavior of the damper, the damping force can be expressed as follows: Fd = P2 Ap − P1 (Ap − As ) (1) where Ap and As are the piston and the piston-shaft cross-sectional areas, respectively P1 and P2 are pressures in the upper and lower chamber of the damper, respectively The relations between P1 , P2 and the pressure in the gas chamber, Pa , can be expressed as follows: P2 = Pa + Pa ; P1 = Pa − Pd (2) where Pa is the pressure drop of ER fluid flow in the connecting pipe between the lower chamber and the accumulator which is small and neglected in this study, Pd is the pressure drop of ER fluid flow through the annular duct The pressure in the gas chamber can be calculated as follows: Pa = P0 V0 V0 − As xp γ (3) Meccanica Fig Schematic configuration of the ER damper where P0 and V0 are initial pressure and volume of the accumulator γ is the coefficient of thermal expansion which is ranging from 1.4 to 1.7 for adiabatic expansion xp is the piston displacement From Eqs (1) and (2), the damping force of the ER damper can be calculated by lows [16]: c = 2.07 + 12Qd μ 12Qd μ + 0.8πRd d τy Plugging Pd from Eq (5) into Eq (4) one obtains Fd = Pa As + cvis x˙p + FER sgn(x˙p ) Fd = Pa As + Pd (Ap − As ) (4) By neglecting minor loss and taking note that radius of the annular duct is much larger than its gap, the pressure drop Pd can be approximately calculated as follows [16]: Pd = 6μLd Ld Qd + c τy d πd Rd (5) where Qd is the flow rate of ER flow in the duct, given by Qd = (Ap − As )x˙p ; τy is the yield stress of the ER fluid induced by the applied electric field; μ is the post-yield viscosity of ER fluid; Ld , Rd and d are length, average radius and gap of the annular duct, respectively c is an coefficient which depends on flow velocity profile and has a value range from a minimum value of 2.07 to a maximum value of 3.07 The coefficient c can be approximately estimated as fol- (6) (7) where, cvis = 6μLd (Ap − As )2 ; πRd d FER = (Ap − As ) cLd τy d The first term in Eq (7) represents the elastic force from the gas compliance This term causes the damping force-piston velocity curve to be shifted vertically and does not affect damping characteristics of the damper The second term represents the damping force due to ER fluid viscosity, thus the damping force when no electric field is applied to the damper The third one is the force due to the yield stress of the ER fluid, which can be continuously controlled by the intensity of the electric field applied to the damper This is the dominant term which is expected to be large enough for suppressing vibration energy Meccanica The commercial ER fluid (Rheobay, TP Al 3565) is used in this study and induced yield stress of the ER fluid can be experimentally estimated by [11] τy = αEaβ (8) Here, Ea is the applied electric field whose unit is kV/mm The α and β are intrinsic values of the ER fluid which are experimentally determined At room temperature, the values of α and β of the above ER fluid are evaluated by 591 and 1.42, respectively The post-yield viscosity of the ER fluid is assumed to be independent on applied voltage and is estimated from experimental results to be 30cSt Optimal design of ER damper In this study, optimal design of the proposed ER damper is considered based on the quasi-static model developed in Sect For vehicle suspension design, the ride comfort and the suspension travel are the two conflicting performance indexes to be considered In order to reduce the suspension travel (i.e., increase stability of the vehicle), high damping force is required On the other hand, for improving ride comfort, low damping force is expected In order to clearly understand the above mentioned, let us consider a simplified one degree of freedom (1-DOF) suspension model shown in Fig 2(a) In this simple idealized model, the vehicle mass mv is supported by four springs k in parallel with four viscous dampers cv of the suspension system The model in Fig 2(a) can be expressed in a more simplified model, quarter car model, shown in Fig 2(b) The motion of the sprung mass in Fig 2(b) can mathematically be expressed as follows: ˙ + k(x − y) = mxă + cv (x y) (9) or mxă + cv x˙ + kx = cv y˙ + ky (10) where m is the mass of quarter car, m = mv /4; x(t) is the displacement of the car body; and y(t) is the displacement of the wheels It is noted that y(t) is considered as an input excitation to the suspension system By assuming a sinusoidal excitation applied to the unsprung mass, the transmissibility of the above 1-DOF suspension system can be obtained as follows: Fig 1-DOF model of a vehicle Fig Transmissibility of 1-DOF quarter car suspension T= + (2ξ ω/ωn )2 (1 − ω2 /ωn2 )2 + (2ξ ω/ωn )2 (11) where ω is the excitation frequency, ωn is √ natural frequency of the suspension system,√ωn = k/m and ξ is the damping ratio, ξ = cs /2 km In practice, the natural frequency of vehicle suspension systems is commonly around 1.5 Hz (ωn = 9.42 s−1 ), and in this case the dependence of the transmissibility on excitation frequency is presented in Fig As shown from the figure, at low damping the resonant transmissibility is relatively large, while the transmissibility at higher frequencies is quite low As the damping is increased, the resonant peaks are attenuated, but vi- Meccanica bration isolation is lost at high frequency The lack of isolation at higher frequencies will result in a harsher vehicle ride This illustrates the inherent tradeoff between resonance control and high frequency isolation associated with the design of passive vehicle suspension systems It is obvious that the damping constant of the damper determines both the stability of the vehicle and the comfort of occupants A high damper (a damper with high damping characteristics) reduces the amplification and provides good stability, keeping the tires in contact with the road and preventing frame oscillations and other problems, but it increases the force transmissibility and transfers much of the road excitation to the passenger, causing an uncomfortable ride On the other hand, a soft damper (a damper with low damping characteristics) increases ride comfort, but it reduces the stability of the vehicle It is noteworthy that the damping force of an ER damper can be controlled continuously by applied electric field Therefore, if the applied electric field is proportional to the sprung mass velocity, the ER damper behaves similarly to a semi-active suspension system with tunable damping ratio and a minimum tunable damping ratio is obtained when no electric field is applied to the damper An inherent challenge in design of ER suspension is the limitation of tuning range of the damping ratio If the ER suspension is designed with large reachable damping ratio to attenuate resonant peak, its ride comfort characteristics is low because the minimum reachable damping ratio can not be tuned to a very small value and via versa Obviously, a wide tunable range of damping ratio can be achieved by using a large sized ER damper However, the large ER damper results in high cost and requires large space In practice, the suspension size is limited depending on practical application Thus, there is an inevitable trade-off between the minimum tunable and the maximum reachable damping ratio in design of ER suspension system From Fig 3, it is observed that the isolation at high excitation frequency approaches to a saturation when the damping ratio is smaller than 0.1 It is also seen from practical application of vehicle suspension that the ride comfort and handling performance are improved very little when the damping ratio decreased to 0.1 or smaller Thus, a smaller value of damping ratio is not necessary and useless Taking the aforementioned into the optimal design of ER damper, the optimization problem can be stated as follows: Find optimum geometric dimen- sions of the ER damper constrained in a specific volume so that the minimum tunable damping ratio can be as small as 0.1 and the damping force is maximized For the ER damper shown in Fig 1, from Eq (7) the damping force can be can be calculated by Fd = Pa πRs2 + cvis x˙p + FER sgn(x˙p ) (12) where, cvis = 6πμLd (Rd − d − )2 − Rs2 ; Rd d FER = π (Rd − d − )2 − Rs2 cLd τy d The minimum tunable damping ratio of the damper is calculated as follows: cvis ξmin = √ km 3πμLd =√ (Rd − d − )2 − Rs2 kmRd d (13) In the above, is the inner electrode thickness From Eqs (12) & (13), it is seen that the damping force Fd and the minimum tunable damping ratio ξmin significantly depends on the duct length Ld , the duct width d, the duct radius Rd and the piston shaft radius Rs of the ER damper The larger value of Rd and Ld is the higher damping force can be obtained However, the large value of Rd and Ld causes an increase of minimum tunable damping ratio which results in a lost of ride comfort Furthermore, the value of Rd and Lp are limited by a constrain in damper size A reduction of duct width d causes an increase of damping force but this significantly increases the minimum tunable damping ratio, especially at small value of d In addition, the duct gap can not be designed too small that results in high cost of fabrication and potential electric short in practical application The piston shaft radius Rs affects not only the damping ratio and the damping force but also the strength of the shaft Under the damping force, the shaft may reach to a buckling state, especially when it is in compression In order to avoid the buckling in the shaft, the following condition must be satisfied [17] Fd ≤ π EI π ERs4 = k s k d Ls ks kd 16L2s (14) Meccanica or ks kd L2s Fd − Rs ≤ π 3E (15) In the above, ks is the safety coefficient which is set by in this study kd is the coefficient considering dynamic load acting on the shaft which is chosen as kd = 1.5 Ls is the length of the shaft, I is the inertia moment of the shaft sectional area and E is the Young’s modulus of the shaft material It is noted again that the first term of the damping force, Eq (12), only causes the damping force-piston velocity curve to be shifted vertically and does not af- 3πμLd (Rd − d − )2 − Rs2 √ kmRd d 24 fect damping characteristics of the damper From the above, the optimization problem of the ER damper is mathematically expressed as follows: – Find the values of Lp , d, Rd and Rs (design variables) that maximize the following objective function: OBJ = 6πμLd (Rd − d − )2 − Rs2 x˙p Rd d + π (Rd − d − )2 − Rs2 cLd τy d (16) – Subject to: − 0.1 ≤ 0; ks kd L2s 6πμLd cLd τy − Rs ≤ 0; Pa πRs2 + (Rd − d − )2 − Rs2 x˙p + π (Rd − d − )2 − Rs2 d π 3E Rd d ≤ Ld ≤ Ldmax ; ≤ Rp ≤ Rpmax ; ≤ d; where Ld is the ER duct length, d is the ER duct width, Rd is the ER duct radius and Rs is the piston shaft radius of the ER damper Ldmax , Rdmax are maximum available values of the ER duct length and the duct radius of the ER damper which are determined from practical application Optimal results and discussion In this study, the above constrained optimization problem is transformed to an unconstrained one via penalty functions The transformed unconstrained optimization problem is then numerically solved using first order method with golden-section algorithm and a local quadratic fitting technique [18] Figure shows optimal solution of the rear ER damper for a middle sized vehicle suspension designed by Choi el al [11] It is noted that, from practical application, Choi et al have determined available space for the ER damper in replacement of the conventional damper of the suspension The maximum available size of the duct length ≤ Rs Ld and duct radius Rd are respectively 280.5 mm and 18 mm In the optimal solution shown in Fig 4, the initial values of the design variables Ld , d, Rd and Rs are arbitrarily selected as follows: Ld = 270 mm; Rd = 15 mm, d = mm and Rs = mm The inner electrode thickness is set equal to that designed by Choi et al., = 3.5 mm The damper piston is assumed to move relatively to the damper housing at a velocity of 0.4 m/s (x˙p = 0.4 m/s) and the applied electric field is KV/mm The higher applied field potentially causes an electric short between the damper electrodes The convergence condition of the objective function is set by 0.2 % In addition, whenever a design variable reaches to its boundary a convergence of that design variable is assumed and the value of the design variable at boundary is set as the optimal value The optimization process is then continued with the other design variables Figure shows that the solution is converged after 23 iterations At the optimum, the damping force reaches up to 1400 N which is around times greater than that at the initial while the damping ratio is constrained to be smaller than 0.1 The optimal values of design variables Ld , d, Rd Meccanica Fig Dependence the optimal solution on the constrained radius of the ER duct Fig Dependence of the optimal solution on the constrained damping ratio Fig Optimization solution of the ER damper constrained in a volume of Rdmax = 18 mm, Ldmax = 280.5 mm and Rs are 280.5 mm; Rd = 18 mm, d = 0.825 mm and Rs = 6.52 mm, respectively It is clearly from the result that the duct length Ld and duct radius Rd are reach to their maximum available values in this case A question arises here that if the duct length Ld and duct radius Rd always reach to their maximum available values in this optimization problem In order to answer this question, optimal solution of the ER damper constrained in many different volumes is considered The results show that the optimal duct radius always reaches to its maximum available value However, this is not always true for the duct length Figure shows the optimal duct length as a function of the constrained duct radius It is noteworthy that no constrain is imposed for the duct length in this case The result shows that there exists an optimal duct length for a constrained duct radius It is also seen that the larger constrained duct radius is the higher values of the optimal duct length, duct gap and shaft radius are obtained However, the optimal duct length is much larger than the maximum available value in vehicle suspension application For instance, the optimal duct length is up to 743 mm if the constrained duct radius is 14 mm This optimal length is obviously much larger than the maximum available length Therefore, in the optimal design of ER damper for vehicle suspension system, the optimal duct radius and duct length can always be selected as large as their maximum available values for simplicity The design variables then can be reduced from four variables (Lp , d, Rd , Rs ) to two variables (d, Rs ) Figure shows the optimal solutions of the ER damper as a functions of the constrained damping ratio (the minimum tunable damping ratio) It is observed Meccanica m2 xă2 Pa As + cvis (x˙2 − x˙1 ) + FMR sgn(x˙2 − x˙1 ) + k2 (x2 − x1 ) = Fig Quarter-car suspension model installed with the ER damper from the figure that the optimal value of the duct length Ld and duct radius Rd are not affected by the constrained damping ratio However, the optimal values of shaft radius and the duct gap are significantly affected The higher value of the constrained damping ratio is the smaller optimal value of the duct gap and the higher value of the shaft radius is It is also observed that the maximum damping force increases almost in proportion with the constrained damping ratio Simulation of the optimized ER damper In order to evaluate the effectiveness of the above optimization solution, performance characteristics of the suspension installed with the optimized ER damper are obtained through simulation and compared with that of the ER suspension designed by Choi et al [11] It is noteworthy that Choi et al have designed the ER suspension based on their experiences and performed a number of simulation results Thus, the design is the best choice from the simulated results By this approach, the ER suspension is expected to be considerably good but not an optimal design Furthermore, it takes time to perform a large number of simulations Figure shows a quarter-car model installed with the ER suspension From the figure, the following governing equations can be derived m1 xă1 + Pa As + cvis (x1 x2 ) + FMR sgn(x˙1 − x˙2 ) + k2 (x1 − x2 ) + k1 (x1 − x0 ) = (17) (18) In the above, m1 , m2 are unsprung and sprung masses of the quarter car; k1 , k2 are wheel and suspension stiffness; x0 (t), x1 (t) and x2 (t) are the road surface input, the unsprung mass deflection and the sprung mass deflection, respectively The parameters of the suspension system are determined based on the parameters of conventional suspension systems For a middle-sized passenger vehicle, the suspension parameters are as follows: m1 = 35 kg, m2 = 310 kg, k1 = 309 kN/m, k2 = 20 kN/m Firstly, bump response of the passenger vehicle equipped with the ER damper is evaluated In this study, the bump profile is mathematically described by x0 (t) = X0 [1 − cos(ωr t)] if t ≤ 2π/ωr if t > 2π/ωr (19) where ωr = 2πVc /D In the above, X0 (= 0.035 m) is the half of the bump height, D (= 0.8 m) is the width of the bump and Vc is the vehicle velocity In the bump test, the vehicle is assumed to travel the bump with constant velocity of 3.08 km/h (Vc = 0.856 m/s) Both the simulation results of the optimized damper and the damper designed by Choi et al (non-optimized damper) are presented It is noted that, based on practical experiences and simulation results, Choi et al have determined geometric dimensions of the ER damper as follows: Ld = 280.5 mm; Rd = 17.88 mm, d = 0.88 mm; Rs = 6.5 mm Figure shows the bump response of the vehicle when no electric field is applied to the electrodes It is noteworthy that there are three main parameters for design and vehicle suspensions evaluation: Sprung mass vibration isolation, which determines ride comfort Suspension stroke, which indicates the limit of the vehicle body motion and tire road contact evaluated through tire deflection, which determines stability and safety It is clearly observed form Fig 8a and 8b that the vibration of sprung mass is better suppressed by using the optimized damper than the non-optimized one It is also observed from Fig 8c and 8d that the suspension deflection and the tire deflection in case of the optimized damper suspension Meccanica Fig Bump responses of the ER suspension system, the applied is E = kV/mm are smaller than those in case of the non-optimized one Thus, optimized suspension can provide a better performance and stability than the non-optimized one This is an important advantage of the optimized damper when the control system in failure condition and the ER damper works similarly to a conventional damper In order to evaluate vibration control characteristics of the optimized suspension, a sky-hook control algorithm is employed to control the ER damper The sky-hook control input is mathematically expressed as follows: u= Cs x˙2 if x˙2 (x˙2 − x˙1 ) > if x˙2 (x˙2 − x˙1 ) ≤ (20) where Cs is the control gain The unit of control input u is kV/mm and the unit of sprung mass velocity x˙2 is m/s Figure shows the bump response of the quartervehicle suspension system featuring the ER damper and the sky-hook controlled algorithm with Cs = 10 It is noted that this value of the control gain is chosen by trial and error It is obvious that different values of Cs results in different performance of the ER suspension However, the relative comparison between the two ER suspensions is not affected by the value of Cs It is observed from Fig that vibration of the sprung mass is significantly reduced by employing the skyhook controller for the ER damper From Fig 9(a), it is seen that the sprung mass acceleration of the optimized suspension and the non-optimized one is almost similar However, the suspension deflection and Meccanica Fig Bump responses of the ER suspension system with sky-hook control, Cs = 10 the tire deflection in case of the optimized suspension are smaller than those in case of the non-optimized one as shown in Fig 9(b) and 9(c) Thus, vibration is better suppressed by using the optimized damper than using the non-optimized one although the control energy is smaller as shown in Fig 9(d) It is worthy to recall that the purposes of suspensions in bump vibration control are to reduce the suspension deflection and the tire deflection The above results show that the optimized ER suspension is favorably better for bump vibration control than the non-optimized one In order to evaluate ride comfort characteristics of the optimized ER damper, a random road response of the ER suspension is obtained and presented The displacement power spectral density of a road profile is described by the following equation [19, 20]: Sz = Aφ −n (21) where Sz is the vertical displacement spectral density measured in (mean vertical displacement)2 /(frequency band), A is the road index, n is the roughness coefficient and φ is the spatial frequency measured in cycles/meter In this study, two different types of road profile are considered: the gravel highway and the Belgian paving road (road with bad terrains) The corresponding parameters of these road profiles are given in Table [21] Figure 10 shows the random road response of the quarter-vehicle suspension system featuring the ER damper and the sky-hook controlled algorithm with Meccanica Table Descriptive parameters of road profiles Description Road Index A Roughness Coefficient n Road RMS value (m) Limiting speed (km/h) Belgian paved road 7.8e–5 4.85 0.003–0.006 >100 Gravel highway 4.4e–6 2.1 0.0003–0.0006 50 Fig 10 Random road responses (gravel highway) of the ER suspension system with sky-hook control (Cs = 10), cruising speed V = 40 m/s Cs = 10 when the vehicle travels on smooth terrain, the gravel highway The limiting cruising speed on this road profile is generally greater than 100 km/h (∼30 m/s) In this simulation, the cruising speed is set by 40 m/s From Fig 10(a), it is seen that the acceleration of the sprung mass in case of the optimized suspension is greater than that in case of the non-optimized one In this case, the root mean square (RMS) of the sprung mass acceleration of the optimized suspension is 0.1315 m/s2 and that of the nonoptimized one is 0.1264 m/s2 It is remarked that if the RMS of sprung mass vertical acceleration is around 0.315 m/s2 or smaller, the vehicle can be considered as good ride comfort (the vehicle causes no uncom- Meccanica fort to passengers) [22] Thus, the smaller RMS of the sprung mass acceleration in the non-optimized case has no significant contribution in ride comfort performance and the purpose of the optimal design is to sacrifice this insignificant ride comfort to improve other performances such as the suspension deflection and the tire deflection From Fig 10(b) and 10(c) it is observed that the suspension deflection and the tire deflection in case of the optimized suspension are smaller than those in case of the non-optimized one From the results it can be calculated that the RMS of the suspension deflection in case of optimized damper is 0.954 mm while that in case of non-optimized one is 1.059 mm and the RMS of the tire deflection in case of optimized damper is 0.725 mm while that in case of non-optimized one is 0.818 mm In addition, The RMS of applied electric field in case of optimized damper is 0.032 kV/mm while that in case of non-optimized one is 0.035 kV/mm The above results clearly depict the purpose of our optimization that is to sacrifice insignificant ride comfort in order to improve suspension performance (suspension stroke), stability and safety It is also noteworthy that the controlled energy of the optimized damper is smaller than that of the non-optimized one In order to clearly show differences between the optimized and non-optimized ER suspension system in response to the random road (the gravel highway), the above simulated results are transformed to frequency domain and presented in Fig 11 From the figure it is observed that, near the two resonances of the vehicle, vibration suppression of the optimized ER suspension is significantly better than that of the nonoptimized one At high frequency of the road profile, the vibration is better suppressed by the non-optimal ER suspension However, the difference is very little Furthermore, at these high frequency of road profile, magnitudes of the sprung mass acceleration, the suspension deflection and the tire deflection are considerately small and not significantly affect human ride comfort Figure 12 shows the random road response of the controlled suspension system when the vehicle travels on bad terrain, the Belgian paving road The limiting cruising speed on this road profile is normally smaller than 50 km/h (∼14 m/s) In this simulation, the cruising speed is set by 14 m/s Again it is observed from the figure that, near the two resonances of the vehicle, vibration suppression of the optimized ER Fig 11 Random road responses (gravel highway) of the ER suspension system in frequency domain suspension is significantly better than that of the nonoptimized one However, at high frequency of the road profile, the vibration is better suppressed by the nonoptimized ER suspension In this case, the root mean Meccanica Fig 12 Random road responses (Belgian paving road) in frequency domain of the ER suspension system with sky-hook control (Cs = 10), cruising speed V = 14 m/s square (RMS) of the sprung mass acceleration of the optimized suspension is 0.54 m/s2 and that of the nonoptimized one is 0.52 m/s2 It is noted that when the vehicle travel on bad terrain, the main purpose of the suspension is to reduce the defection of the suspension and tire for stability and safety It is also worthy to recall that if the RMS of sprung mass vertical acceleration is from 0.315 m/s2 to 0.63 m/s2 , the vehicle causes a little uncomfort to passengers [22] By choosing very small value of the constrained damping ratio (up to 0.01), the RMS of sprung mass vertical acceleration only reduces to 0.51 m/s2 However, the small value of the constrained damping ratio will reduce the maximum damping force which results in large deflection of the suspension and tire The above results again clearly show the purpose of our optimization that is to sacrifice insignificant ride comfort in order to improve suspension performance (suspension stroke), stability and safety On the other hand, the time for ER suspension design can be significantly reduced by using the optimal design because the simulated results are not needed Conclusion In this research work, an analytical approach to optimally design ER dampers of vehicle suspension system has been developed The optimal design considers both stability and ride comfort of vehicle appli- Meccanica cation After describing the schematic configuration and operating principle of the ER damper, the quasistatic modelling of the damper was performed on the basis of Bingham rheological laws of ER fluid The optimization problem for the ER damper was then built based on the quasi-static model The optimization problem was numerically solved using first order method with golden-section algorithm and a local quadratic fitting technique From the proposed optimal design, the optimal solution of the ER damper constrained in a specific volume determined by Choi et al [11] was obtained and presented The optimal results of the ER damper constrained in different volumes and subjected to different constrained damping ratio have shown that, in optimal design of ER damper for vehicle suspension system, the duct length and duct gap can always be set equal to their maximum available values and the design variables can be reduced from four variables to two variables Based on quarter-car simulation results of the ER suspension, a comparison work between the optimized ER damper and the nonoptimized ER damper [11] has been performed The result shows that the optimized damper provides better suspension performance (suspension stroke), stability and safety than the non-optimized does while the ride comfort is nearly similar This directly infers the effectiveness of the proposed optimal design approach It is finally remarked that the proposed optimization technique can be applied to many engineering devices using electrorheological or magnetorheological fluid Acknowledgements This work was supported by Inha University Research Grant This financial support is greatly appreciated References Acker B, Darenburg W, Gall H (1989) Active suspension for passenger cars Dynamics of road vehicles In: Proc 11th IAVSD symp, Kingston, Ontario, Canada Williams RA (1997) Automotive active suspensions Part 1: basic principles Proc Inst Mech Eng Part D, J Automob Eng 211:415–426 Williams RA (1997) Automotive active suspensions Part 2: practical considerations Proc Inst Mech Eng Part D, J Automob Eng 211:427–444 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Van Nostrand, Toronto 18 Nguyen QH, Han YM, Choi SB, Wereley NM (2007) Geometry optimization of MR valves constrained in a specific volume using the finite element method Smart Mater Struct 16:2242–2252 19 Lin YJ, Padovan J, Lu YQ (1992) Toward better ride performance of vehicle suspension system via intelligent control Proc IEEE Int Conf Syst Man Cybern 2:1470–1475 20 International Standards Organisation (1995) Mechanical vibration—road surface profiles reporting of measured data ISO 8608 21 Hall LC (1998) Fundamentals of terramechanics & soil vehicle interaction Terrain profile characteristics In: Proceedings of the wheels and tracks symposium, Cranfield University, UK 22 British Standard Institution (1987) British standard guide to measurement and evaluation of human exposure to whole body mechanical vibration and repeated shock, BS 6841:1987 ... system seems to be absent The primary purpose of the current work is to fill this gap Consequently, the main contribution of this work is to develop an analytical approach to optimally design ER... work, an analytical approach to optimally design ER dampers of vehicle suspension system has been developed The optimal design considers both stability and ride comfort of vehicle appli- Meccanica... and MR fluid in an automotive crash energy absorber Report No MT04.18 17 Timoshenko S (1942) Strength of materials Part II: Advanced theory and problems Van Nostrand, Toronto 18 Nguyen QH, Han