Dynamic investigation and experimental validation of a gear transmission system with damping particles

In addition, the damping particle behavior indicated thathigh rotational speed was more effective in vibration reduction, andincreased filling ratio decreased radial vibration.. Finally,

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Dynamic investigation and experimental

validation of a gear transmission system with damping particles

Yun-Chi Chung, Achmad Arifin, Yu-Ren Wu & Chia-Yuan Wang

To cite this article: Yun-Chi Chung, Achmad Arifin, Yu-Ren Wu & Chia-Yuan Wang (15

Jun 2023): Dynamic investigation and experimental validation of a gear transmissionsystem with damping particles, Mechanics Based Design of Structures and Machines, DOI:10.1080/15397734.2023.2223660

To link to this article: https://doi.org/10.1080/15397734.2023.2223660

Published online: 15 Jun 2023.

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Dynamic investigation and experimental validation of a gear transmission system with damping particles

Yun-Chi Chunga, Achmad Arifinb, Yu-Ren Wua , and Chia-Yuan Wanga

Due to recent technological advances, preciseness in gear transmission has

evolved into an essential goal, in which vibration is one of the critical

issues This study demonstrated an experimental platform for a proposed

construction model on the spur gear pair, in which damping particles were

filled into six through-cavities in gear bodies to verify decreases in

vibra-tion The experimental design involved variable control of input rotational

speed, filling ratio, particle size, and material to investigate the damping

particles ’ behavior Subsequently, operating both ADAMS and EDEM

soft-ware established a reliable two-way coupling analysis model The results

confirmed that the filling damping particles effectively reduced system

vibration, and both simulation and experimental tests demonstrated a

con-sistent impact In addition, the damping particle behavior indicated that

high rotational speed was more effective in vibration reduction, and

increased filling ratio decreased radial vibration Increases in particle

diam-eter only contributed slightly to vibration reduction Moreover, although

density increase significantly reduced vibration, bead elasticity did not

impact vibration Finally, the dynamic behavior of damping particles with

the greatest reduction impact was achieved by applying a soft lead bead

in a 5-mm diameter, 48% filling ratio, and the system working at a

rota-tional speed of 600 rpm.

ARTICLE HISTORY

Received 19 December 2022 Accepted 31 May 2023

KEYWORDS

Damping particles; discrete element method (DEM); gear transmission; multi- body dynamics (MBD); vibration reduction

1 Introduction

Due to recent technological advances, preciseness in gear transmission systems has evolved into

an essential goal The field of technology in industrial manufacturing and automotive applicationprimarily operate precision transmission systems, in which performance standards are becomingincreasingly strict following the demand for enhanced machining accuracy The dynamic modifi-cation of the precision transmission system must be addressed when designing the controller,including the gear components and their specific characteristics (Zhang, Zhong, and Chen 2015;Bao, Mao, and Luo 2016; Yu, Wang, and Zou 2018) Vibration and noise in gear transmissionswill significantly influence the machinery’s precision, performance, and lifetime, and thus thesafety of operators (Xiao, Li, et al.2016) Therefore, research in vibration reduction in gear trans-mission systems to satisfy an actual transference with the smallest potential pulsing has become

an essential task

CONTACT Yu-Ren Wu yurenwu@ncu.edu.tw Department of Mechanical Engineering, National Central University, 300, Zhongda Rd., Zhongli District, Taoyuan City 320317, Taiwan (R.O.C.).

Communicated by Francisco Javier Gonz
alez Varela.

ß 2023 Taylor & Francis Group, LLC

The vibration reduction method applied in gearing systems constitutes two primaryapproaches, including active vibration reduction and passive vibration reduction (Platz and Enss

2015; Baz 2018) Active vibration reduction aims to reduce system vibration by optimizing themechanism itself, such as gear trimming, that can effectively increase installation accuracy.However, the requisite expense will rise significantly with the expanded precision of vibrationrequirements In contrast, passive vibration reduction appends additional energy-consuming devi-ces to absorb the vibration generated during the mechanism’s operation It offers the advantages

of not being restricted by accuracy and having a lower cost, although it does necessitate a lar degree of structural changes Ring damper and squeeze film damper were applied as controlmethods for passive vibration reduction (Tian et al.2022) A typical technique to handle passivevibration is particle damping technology (Papalou and Masri 1998; Friend and Kinra 2000),which was first proposed by Panossian (1992), and is known as non-obstructive particle damping(NOPD) technology Inelastic collisions and friction on particles-to-particles and particles-to-wallsurfaces will restrain energy dissipation by filling the closed cavity with a specific percentage ofdamping particles It possesses numerous strengths, such as effortless installation, minor changes

particu-to the original structure, significant vibration damping effect, and that the vibration dampingimpact is not easily affected by temperature changes Consequently, the fields of civil and mech-anical engineering have widely applied this technology

Particle damping technology was implemented for rotating bodies (Dragomir et al 2012), inwhich the friction and collision generated during the motion of the particles were applied toachieve energy dissipation by mounting the structure with damping particles on the rotatingshaft Another investigation was performed by adding damping particles into the gear cavity andanalyzing the relationship between particles and energy dissipation in distinct operating condi-tions The results revealed that the friction coefficient had a substantial impact on the dampingparticles, and the motion behavior of the damping particles at high and low speeds exhibiteddiverse trends (Xiao, Huang, et al 2016; Xiao et al 2017) Therefore, in this study, all dampingparticles were considered independent units in order to further analyze damping particles’motion, and the discrete element method (DEM) was used to determine the motion behavior ofeach damping particle DEM was first proposed to treat discontinuous material as multiple con-nected discrete spheres (Cundall and Strack 1979) The DEM technique was also applied in agear transmission system to establish MBD-DEM coupling analysis (Xiao, Huang, et al 2016;Xiao et al 2017) The results showed that the particle filling ratio, material, and other particleparameters had a distinct effect on the vibration reduction impact of particles Selecting the mostsuitable particle material can more effectively reduce vibration in gearing systems Consequently,

it provided an essential basis for applying particle selection in centrifugal fields A two-waycoupled dynamic model of multi-body dynamics (MBD) and DEM modeling and experimentalvalidation for the dynamic response of mechanisms containing damping particles, particularly themotion behavior of damping particles in gears were further described (Chung and Wu2019; Wu,Chung, and Wang2021) In that study, the particles had the optimum particle size to ensure thatthe particle grade did not influence the system vibration response Accordingly, an equivalentmass was determined as an additional case to provide an important reference for selecting particleparameters in this study

This study established an experimental platform to measure the vibration of a spur gear pairwith damping particles A tri-axial accelerometer measured the vibration response on the bearingseat of an output shaft The effects of particle size, filling ratio, material, and input rotationalspeed on the vibration effect of the damping particles were investigated according to vibrationenergy values of root mean square (RMS) and gear mesh frequency The effectiveness of dampingparticles on vibration reduction was further confirmed by comparing the model’s RMS values ofvibration increase on the output axis to the experimental measurement results The main andnovel contributions of this study are: (1) incorporate investigation in numerical simulation and

actual experimental methods to achieve more concrete and rigorous results analysis using way coupled MBD–DEM models; (2) elucidate the dynamic behavior of damping particles onvarious characteristic variables to obtain the most suitable parameters; and(3)provides an essen-tial consideration for the future construction design of damping particle applications for spur-gear transmission systems.

two-2 Experimental design and analysis methods

This section describes the equipment construction corresponding to the experimental design andvariable control, including the measurement methods Damping particles were applied, which canreduce the structural vibration in the gear transmission without changing the original structure(Xiao et al 2022; Zhang et al 2022) An experimental platform of a gear transmission systemwith damping particles was established to investigate the dynamic behavior of damping particles

By measuring the vibration signal on the bearing seat of the output shaft with an accelerometer,the changes in system vibration after adding distinct particles were determined, and the optimalparameters of particles were further derived The parameters examined in the experiments of thisstudy were chosen according to previous research

2.1 Experimental construction and equipment

Figure 1 shows the experimental platform used in this research Specifically, the gearbox wasscrewed to the carrier, and the four corners of the carrier were locked with a vibration isolator toavoid vibration transmission to the ground The gear transmission shafts were mounted by deeplygrooved ball bearings with a shaft shoulder and C-type retaining ring as the inner and outer

Figure 1 The construction of the experimental platform.

rings, respectively The gear and the drive shaft were installed tightly to minimize the effect ofmounting on system vibration.

The gear had six through-cavities filled with beads of the same size A transparent acrylic platewas locked on both gear ends to cover the holes, so that the beads’ behavior can be observed dur-ing the experiment A servo motor and magnetic powder brake were connected to the experimen-tal device by coupling with an elastic spider to offset the alignment error and provide a stableinput source through both to guarantee consistent input conditions under different experimentalgroups The vibration signal was measured with a piezoelectric tri-axial accelerometer mountedabove the bearing housing on the output shaft The signal from the accelerometer was sampled

by a DAQ card, transferred to the computer used to control the power input, and analyzed bythe signal analysis software NOVIAN

The computer uniformly controlled the experimental power parameters to ensure that theinput conditions of each experiment were consistent, in which the error span was set up betweenplus-minus 0.5 rpm and 0.05 N-m for motor rotation and load, respectively The input source was

a three-phase AC servo motor with a maximum rotational speed and power of 8000 rpm and7.5 kW, respectively The output side was loaded with a magnetic powder brake, allowing a max-imum rotational speed of 1800 rpm and a maximum permittable torque of 50 N-m It is used due

to its simple structure, rapid response time, and precise torque control

2.2 Signal control and measurement methods

The vibration signal may be interrupted by a noise in measurement during the experiment, soavoiding the influence of extreme values or abnormal signals generated was critical Since it maynot accurately represent the vibration characteristics of the system, the signal needs to be filtered

to guarantee its validity This study designated the driving rotational speed and load to a specifiedvalue prior to recording The total sampling time from the accelerometer’s signal was set to 20 s,and only the middle 10 s was used for analysis after removing the signal’s first and last 5 s Thebandwidth was specified to 2000 Hz with span error of plus-minus 1.5% by considering the char-acteristics of the gear vibration signal and the proposed bandwidth of the magnetic suction accel-erometer A 2 Hz high-pass filter was employed to obstruct noise below 2 Hz, and the samplingfrequency was set to 5120 Hz to avoid the effect of environmental noise on vibration analysis.Data sampling was analyzed by operating root-mean-square value measurements that canindirectly represent the trend of vibration energy and eliminate the effect of positive and negativevalues simultaneously Each set of experiments was repeated five times, and the maximum andminimum extreme values were removed and averaged to represent the results of the set

In addition, since the vibration signals were in the horizontal, axial, and vertical directions, to plify the data and satisfactorily describe the behavior between the gears, the signals in the horizontaland vertical directions were combined, as the square root in the radial direction Since the primaryobject was the spur gear pair, the radial vibration signal can effectively represent the gear mesh vibra-tion, and constituted the primary object of analysis To further determine the effect of particles onthe gear transmission system, this study utilized a fast Fourier transform (FFT) to analyze the change

sim-of vibration energy when damping particles were applied to the system Since FFT treated the domain signal as a complete periodic signal, utilizing a window function, Hann window, was neces-sary to process the time-domain signal during FFT analysis The closer the sampling duration was tothe median value of the measurement time, the higher the Hann window was weighted

finite-2.3 Experimental design and variable control

Figure 2presents the experimental design and the variable control setup There were four bles: input speed; filling ratio; particle size; and material The particle filling ratio was defined as

varia-the particle volume ratio in varia-the cavity to varia-the cavity volume to define varia-the volume of particles inthe cavity, as expressed inEq (1):

e ¼Vp

Vc

(1)where Vp is the volume of particles in the cavity; and Vc is the overall cavity volume

The initial parameters of this experimental analysis, called reference design, were specified todetermine the effect of the particles on vibration reduction Furthermore, an equivalent massexperimental, a set control group without damping particles with the physical parameters consist-ent with those of the reference design experimental, was additionally established since adding par-ticles would change the system mass, which would have a particular effect on the systemvibration behavior It is locked in symmetrical semi-annular equivalent mass pieces from bothgear ends The total mass of the two side masses was identical to the total mass of the particles inthe cavity in the reference design experimental In this way, it avoided the distinction in vibrationbehavior resulting from the difference in system masses The validity of the particle on vibrationreduction could be verified by comparing the reference design with the equivalent mass experi-mental groups and the corresponding simulation model

Furthermore, this study was extended to investigate the damping effect in the experimentalgroup from 200 rpm to 1200 rpm Moreover, the best rotational speed for vibration damping wasapplied as a reference for the subsequent experiments to elucidate the impact of various fillingratio conditions on vibration attenuation The experimental group of filling ratio was determined

as 12%, 24%, 36%, and 48% This experimental group can evaluate the damping effect under tinct filling ratios and be expected to identify the relationship between filling ratio and vibration-reducing impact

dis-Figure 2 The experimental design and the variable control definition.

The obtained optimal filling rate was also applied as a parameter further to narrow the scope

of the rotational speed analysis to investigate the relationship between various particle sizes andthe damping impact Five groups of carbon steel beads with various diameters were examined in

2, 3, 4, 5, and 6 mm By normalizing the experimental data based on the acceleration of theequivalent mass experimental, the damping effect of different particle diameters was compared,and the particle diameter with the most significant impact was identified In previous studies,stainless steel bead has been applied to investigate the damping mechanism and performance onthe nonlinear dynamic characteristics of gears with different speeds and loads (Xiao et al 2021;Guo et al.2022) Therefore, this study utilized five material types to analyze the dynamic behavior

of damping particles concerning their density and elasticity According to the optimum particlesize, five different materials of nitrile butadiene rubber (NBR), polyoxymethylene (POM), glass,carbon steel, and lead beads were manufactured

This section addresses the experimental platform to establish a dynamic analytical model using acoupled MBD-DEM model Since particle motion behavior is a complex system, it requires sim-plifying the model to avoid multiplex conditions for completing the analysis The computationalscheme of the coupled model was launched by operating the multi-body dynamics softwareADAMS and discrete element method EDEM software with appropriate parameters

3.1 Model definition, assumptions, and simplification

Figure 3 illustrates the analytical construction of the simplified dynamic model of a gear mission system with damping particles It consisted of a three-phase servo motor and a steadysource of dynamic force provided by a magnetic powder brake, input and output gears with

trans-Figure 3 Simplified dynamic model of a gear transmission system with damping particles.

damping particles, input and output shafts, and fixed bearings connected to the two transmissionshafts.

During the gear transmission duration, the damping particles in the cavity were affected by thevibrations generated along transmission, resulting in particle-to-particle and particle-to-wall collisionsand friction This drove the system to establish non-linear behavior, however, and made it difficult tocomplete the analysis This study simplified and reduced the degrees of freedom and variables in theanalysis process by restricting the motion behavior of the model to avoid complex conditions.Therefore, this study constructed the following assumptions to simplify the system to build the corre-sponding dynamic analysis model based on the experimental environment inSection 2.1:

a All parts of the system were assumed to be rigid bodies and there was no installation error

b The axial force and the displacement of shafts were sufficiently small to be neglected

c The input rotational speed was steady and did not fluctuate after a period

d The load was stable and invariant with time, making the system power constant

e The outer bearing ring was fully fixed to the ground, and the inner ring was fixed androtated with the shaft

f The radial clearances were identical for all bearings

g The shaft was ideally connected to the coupling and did not constrain the system

h The values for the contact stiffness and damping in similar bearings were identical

i The contact stiffness values and the gear pairs damping were identical

In addition, since the primary excitation source was the spur gear, its vibration in the axialdirection had no significance to the meshing vibration amongst the gear Therefore, as in theexperiment, the effect of axial vibration was ignored in the simulation, and only the radial vibra-tion was considered

3.2 Computational scheme for the coupled MBD–DEM model

The MBD software utilized in this study was established on Lagrange mechanics to solve theMBD issue Newton’s second law of motion described the motion state of the system, and theLagrange equation held when all of the generalized coordinates in the system were assumed to beindependent, which was expressed asEq (2):

ddt

@L

whereq is the generalized coordinate matrix of the system; _q is the generalized velocity matrix ofthe system; and L is the Lagrange used to describe the energy state of the system, which can beexpressed in detail as Lðq, _q, tÞ, or can be expressed asEq (3):

where U is the kinetic energy of the system; and V is the potential energy of the system

Since Lagrange’s equation was too complicated to derive the multi-body dynamic equationsfor all system components in a complex multi-rigid-body system, Cartesian coordinates werederived for establishing the constraint equations according to the system structure In thismethod, q was a non-independent coordinate, where the constraint equation was expressed asuðqÞ ¼ 0: This system problem can be further expressed with the first type of Lagrange equation

asEqs (4)and(5):

ddt

@U

@ _q

uðqÞ ¼ 0 (5)whereu is the constraint equation matrix; k is the Lagrange multiplier; and F is the generalizedforce According to Eqs (4)and (5), the equations were further organized and transformed intoordinary differential equations, and the problem was composed in matrix form, as expressed in

Eq (6)(Flores et al.2008):

M UqT

Uq 0

!

q}k

whereM is the mass matrix of the system; U is the Jacobian matrix of the system constraints; q}

is the generalized acceleration vector; k is the Lagrange multiplier vector corresponding to theconstraints;Q is the generalized force vector; and c is the combined vector used in the equations.The contact force amongst a spur gear pair can be regarded as a non-linear spring dampingproblem and treated as a parallel process in the case of multi-tooth meshing, where the contactforce of a single-tooth contact Fgwas expressed inEq (7):

Fg¼ KgðdgnÞn

where Kg is the single tooth contact stiffness with angular variation; dgn is the normal ation while gear meshing; n is the force exponent; and Dgn is the gear contact damping coeffi-cient Kg can be considered to be composed of three different stiffness, which are furtherexpressed asEq (8) (Liu, Wang, and Wu2020):

stiff-The reasonable value of the force exponent n, when calculated by Hertzian theory of cylinderpair, was presumed to be between 1 and 1.5 (Hunt and Crossley1975) At the same time, somemodels suggested that the force exponent tended to 1.094 (Pereire, Ramalho, and Ambrosio,

2015) or even to linearity when two parallel cylinder pairs were in contact Finally, the forceexponent of the gear mesh was experimentally confirmed to be 1.3, which was in close accord-ance with the MBD model (Dabrowski, Adamczyk, and Plascencia, 2012) The value of the sin-gle-tooth contact stiffness Kg determined the damping factor Dgn: The damping factor must beslighter more than 0.01 times the single-tooth contact stiffness

In the DEM model, all beads were considered independent objects for calculation, and theirphysical properties were assumed to be consistent The Hertz-Mindlin no-slip contact model(Chung and Wu2019) was applied to explore the dynamic behavior of the beads The model div-ided the forces on the particles into normal and tangential directions, as illustrated in Figure 4,where the normal force of the particle was expressed inEq (9):

where Fpn;s is the normal spring force of the particle; Fpn;d is the normal damping force of theparticle Similar to the normal force of a particle Fpn, the tangential force of a particle Fpt wasalso expressed inEq (10):

where Fpt ;s is the tangential spring force of the particle; Fpt ;d is the tangential damping force of

the particle In addition, since the tangential contact force between beads was restricted byCoulomb friction, it must satisfy the coefficient of friction ls of the relationship, expressed in

3.3 Coupled MBD–DEM model parameters and implementation

Due to the characteristics of the MBD model solver, when extreme and discontinuous tions were provided to the model instantaneously, they would lead to discontinuity of the calcu-lated values, which would quickly cause the calculation results to be divergent Furthermore, theSTEP function’s input speed and load were provided to avoid this issue, where it can ensure thatthe MBD model would not crash due to the discontinuity of the value when the model was oper-ated Therefore, the STEP function was provided to intensify speed and load from zero to therated value between 0.2 s and 0.25 s

accelera-Table 1presents the physical properties parameters of the equivalent mass experiment and thepower parameters of the model in the first and second parts, respectively Since both sides ofinput sources considered that it takes time for the DEM model to reach stability after the beadswere filled and dropped, the MBD model needed to be rested for 0.2 s Further, it was navigated

by the STEP function to prevent the DEM model from excessively overlapping through theboundary surface during the particle generation process or starting to operate before the particlesreached stability, which could induce the model to diverge

The last part ofTable 1exhibits the model’s contact parameters, where the gear’s normal tact stiffness was simplified by operating the average K curve of the single tooth mesh stiffness

con-Figure 4 A damped Hertz –Mindlin contact force approach to model the collisions between particles or the collisions between particles and walls (Chung et al 2019 ).

Figure 5 Flowchart for the calculation procedure of a coupled MBD –DEM model (Chung and Wu 2019 ).

Table 1 Parameters for the MBD model of a system comprises a pair of spur gears.

2 Dynamic parameters of MBD model

3 Contact parameters of MBD model

Average normal gear contact stiffness ðK gn;avg Þ 3 :27  10 8 N/m Average normal gear contact damping stiffness ðD gn Þ 3 :27  10 4 N-s/m