In addition, the damping particle behavior indicated thathigh rotational speed was more effective in vibration reduction, andincreased filling ratio decreased radial vibration.. Finally,
Mechanics Based Design of Structures and Machines An International Journal ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/lmbd20 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 transmission system 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 Submit your article to this journal Article views: 132 View related articles View Crossmark data Citing articles: View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=lmbd20 MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES https://doi.org/10.1080/15397734.2023.2223660 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 aDepartment of Mechanical Engineering, National Central University, Taoyuan City, Taiwan (R.O.C.); bDepartment of Mechanical Engineering Education, Universitas Negeri Yogyakarta, Karangmalang, Yogyakarta, Indonesia ABSTRACT ARTICLE HISTORY Received 19 December 2022 Due to recent technological advances, preciseness in gear transmission has Accepted 31 May 2023 evolved into an essential goal, in which vibration is one of the critical issues This study demonstrated an experimental platform for a proposed KEYWORDS construction model on the spur gear pair, in which damping particles were Damping particles; discrete filled into six through-cavities in gear bodies to verify decreases in vibra- element method (DEM); tion The experimental design involved variable control of input rotational gear transmission; multi- speed, filling ratio, particle size, and material to investigate the damping body dynamics (MBD); particles’ behavior Subsequently, operating both ADAMS and EDEM soft- vibration reduction 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 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 application primarily operate precision transmission systems, in which performance standards are becoming increasingly 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 transmissions will significantly influence the machinery’s precision, performance, and lifetime, and thus the safety 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 Gonzalez Varela ß 2023 Taylor & Francis Group, LLC Y.-C CHUNG ET AL The vibration reduction method applied in gearing systems constitutes two primary approaches, including active vibration reduction and passive vibration reduction (Platz and Enss 2015; Baz 2018) Active vibration reduction aims to reduce system vibration by optimizing the mechanism itself, such as gear trimming, that can effectively increase installation accuracy However, the requisite expense will rise significantly with the expanded precision of vibration requirements 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 particu- lar degree of structural changes Ring damper and squeeze film damper were applied as control methods for passive vibration reduction (Tian et al 2022) A typical technique to handle passive vibration 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-wall surfaces will restrain energy dissipation by filling the closed cavity with a specific percentage of damping particles It possesses numerous strengths, such as effortless installation, minor changes to the original structure, significant vibration damping effect, and that the vibration damping impact 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), in which the friction and collision generated during the motion of the particles were applied to achieve energy dissipation by mounting the structure with damping particles on the rotating shaft Another investigation was performed by adding damping particles into the gear cavity and analyzing 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 damping particles, and the motion behavior of the damping particles at high and low speeds exhibited diverse trends (Xiao, Huang, et al 2016; Xiao et al 2017) Therefore, in this study, all damping particles 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 of each 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 a gear 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 particle parameters had a distinct effect on the vibration reduction impact of particles Selecting the most suitable 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-way coupled dynamic model of multi-body dynamics (MBD) and DEM modeling and experimental validation for the dynamic response of mechanisms containing damping particles, particularly the motion behavior of damping particles in gears were further described (Chung and Wu 2019; Wu, Chung, and Wang 2021) In that study, the particles had the optimum particle size to ensure that the particle grade did not influence the system vibration response Accordingly, an equivalent mass was determined as an additional case to provide an important reference for selecting particle parameters in this study This study established an experimental platform to measure the vibration of a spur gear pair with damping particles A tri-axial accelerometer measured the vibration response on the bearing seat of an output shaft The effects of particle size, filling ratio, material, and input rotational speed on the vibration effect of the damping particles were investigated according to vibration energy values of root mean square (RMS) and gear mesh frequency The effectiveness of damping particles on vibration reduction was further confirmed by comparing the model’s RMS values of vibration increase on the output axis to the experimental measurement results The main and novel contributions of this study are: (1) incorporate investigation in numerical simulation and MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES Figure The construction of the experimental platform actual experimental methods to achieve more concrete and rigorous results analysis using two- way coupled MBD–DEM models; (2) elucidate the dynamic behavior of damping particles on various 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 Experimental design and analysis methods This section describes the equipment construction corresponding to the experimental design and variable control, including the measurement methods Damping particles were applied, which can reduce 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 system with 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 optimal parameters of particles were further derived The parameters examined in the experiments of this study were chosen according to previous research 2.1 Experimental construction and equipment Figure shows the experimental platform used in this research Specifically, the gearbox was screwed to the carrier, and the four corners of the carrier were locked with a vibration isolator to avoid vibration transmission to the ground The gear transmission shafts were mounted by deeply grooved ball bearings with a shaft shoulder and C-type retaining ring as the inner and outer Y.-C CHUNG ET AL rings, respectively The gear and the drive shaft were installed tightly to minimize the effect of mounting on system vibration The gear had six through-cavities filled with beads of the same size A transparent acrylic plate was 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 stable input source through both to guarantee consistent input conditions under different experimental groups The vibration signal was measured with a piezoelectric tri-axial accelerometer mounted above 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 by the signal analysis software NOVIAN The computer uniformly controlled the experimental power parameters to ensure that the input conditions of each experiment were consistent, in which the error span was set up between plus-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 and 7.5 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, so avoiding the influence of extreme values or abnormal signals generated was critical Since it may not 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 specified value 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 s The bandwidth 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 Hz high-pass filter was employed to obstruct noise below Hz, and the sampling frequency 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 can indirectly represent the trend of vibration energy and eliminate the effect of positive and negative values simultaneously Each set of experiments was repeated five times, and the maximum and minimum 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 sim- plify the data and satisfactorily describe the behavior between the gears, the signals in the horizontal and vertical directions were combined, as the square root in the radial direction Since the primary object 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 on the gear transmission system, this study utilized a fast Fourier transform (FFT) to analyze the change of vibration energy when damping particles were applied to the system Since FFT treated the finite- 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 to the median value of the measurement time, the higher the Hann window was weighted 2.3 Experimental design and variable control Figure presents the experimental design and the variable control setup There were four varia- bles: input speed; filling ratio; particle size; and material The particle filling ratio was defined as MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES Figure The experimental design and the variable control definition the particle volume ratio in the cavity to the cavity volume to define the volume of particles in the cavity, as expressed in Eq (1): e ¼ Vp (1) Vc 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 to determine the effect of the particles on vibration reduction Furthermore, an equivalent mass experimental, 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 system vibration behavior It is locked in symmetrical semi-annular equivalent mass pieces from both gear ends The total mass of the two side masses was identical to the total mass of the particles in the cavity in the reference design experimental In this way, it avoided the distinction in vibration behavior resulting from the difference in system masses The validity of the particle on vibration reduction 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 experimental group from 200 rpm to 1200 rpm Moreover, the best rotational speed for vibration damping was applied as a reference for the subsequent experiments to elucidate the impact of various filling ratio 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 dis- tinct filling ratios and be expected to identify the relationship between filling ratio and vibration- reducing impact Y.-C CHUNG ET AL Figure Simplified dynamic model of a gear transmission system with damping particles 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 and the damping impact Five groups of carbon steel beads with various diameters were examined in 2, 3, 4, 5, and mm By normalizing the experimental data based on the acceleration of the equivalent 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 on the 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 particle size, five different materials of nitrile butadiene rubber (NBR), polyoxymethylene (POM), glass, carbon steel, and lead beads were manufactured Computational scheme for the coupled MBD–DEM model This section addresses the experimental platform to establish a dynamic analytical model using a coupled 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 computational scheme of the coupled model was launched by operating the multi-body dynamics software ADAMS and discrete element method EDEM software with appropriate parameters 3.1 Model definition, assumptions, and simplification Figure illustrates the analytical construction of the simplified dynamic model of a gear trans- mission system with damping particles It consisted of a three-phase servo motor and a steady source of dynamic force provided by a magnetic powder brake, input and output gears with MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES damping particles, input and output shafts, and fixed bearings connected to the two transmission shafts During the gear transmission duration, the damping particles in the cavity were affected by the vibrations generated along transmission, resulting in particle-to-particle and particle-to-wall collisions and friction This drove the system to establish non-linear behavior, however, and made it difficult to complete the analysis This study simplified and reduced the degrees of freedom and variables in the analysis 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 in Section 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 and rotated 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 axial direction had no significance to the meshing vibration amongst the gear Therefore, as in the experiment, 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 the MBD issue Newton’s second law of motion described the motion state of the system, and the Lagrange equation held when all of the generalized coordinates in the system were assumed to be independent, which was expressed as Eq (2): d @L À @L ¼ (2) dt @q_ @q where q is the generalized coordinate matrix of the system; q_ is the generalized velocity matrix of the system; and L is the Lagrange used to describe the energy state of the system, which can be expressed in detail as Lðq, q_ , tị, or can be expressed as Eq (3): LẳUV (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 equations for all system components in a complex multi-rigid-body system, Cartesian coordinates were derived for establishing the constraint equations according to the system structure In this method, q was a non-independent coordinate, where the constraint equation was expressed as uqị ẳ 0: This system problem can be further expressed with the first type of Lagrange equation as Eqs (4) and (5): d @U @U dt @q_ @q ỵ uqk ẳ F (4) Y.-C CHUNG ET AL uqị ẳ (5) where u is the constraint equation matrix; k is the Lagrange multiplier; and F is the generalized force According to Eqs (4) and (5), the equations were further organized and transformed into ordinary differential equations, and the problem was composed in matrix form, as expressed in Eq (6) (Flores et al 2008): !0 ! M UqT @ q} A ¼ Q (6) Uq c k } where M 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 the constraints; 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 damping problem and treated as a parallel process in the case of multi-tooth meshing, where the contact force of a single-tooth contact Fg was expressed in Eq (7): Fg ẳ Kgdgnịn Dgnd_ gn (7) where Kg is the single tooth contact stiffness with angular variation; dgn is the normal deform- 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 further expressed as Eq (8) (Liu, Wang, and Wu 2020): Kg ¼ À1 (8) À1 À1 Kh ỵ Kde þ Kz Þ where Kh is the contact stiffness between two parallel cylinders;Kde is the body deformation stiff- ness; Kz is the bending stiffness, and the single-tooth contact stiffness Kg will be parabolic with the angle of rotation of the gear The reasonable value of the force exponent n, when calculated by Hertzian theory of cylinder pair, was presumed to be between and 1.5 (Hunt and Crossley 1975) At the same time, some models 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 force exponent 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 be slighter more than 0.01 times the single-tooth contact stiffness In the DEM model, all beads were considered independent objects for calculation, and their physical properties were assumed to be consistent The Hertz-Mindlin no-slip contact model (Chung and Wu 2019) 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 in Eq (9): Fpn ¼ Fpn;s ỵ Fpn;d (9) where Fpn;s is the normal spring force of the particle; Fpn;d is the normal damping force of the particle Similar to the normal force of a particle Fpn, the tangential force of a particle Fpt was also expressed in Eq (10): Fpt ẳ Fpt;s ỵ Fpt;d (10) where Fpt;s is the tangential spring force of the particle; Fpt;d is the tangential damping force of MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES Figure A damped Hertz–Mindlin contact force approach to model the collisions between particles or the collisions between particles and walls (Chung et al 2019) the particle In addition, since the tangential contact force between beads was restricted by Coulomb friction, it must satisfy the coefficient of friction ls of the relationship, expressed in Eq (11): jFptj lsFpn (11) Figure presents a flowchart for the calculation procedure of the coupled MBD–DEM model The coupled MBD-DEM model was mainly established on MBD for analysis of the dynamic behavior of the gear transmission system In the analysis process, the behavior of the particle was calculated by DEM The particle forces were then integrated into the MBD model in the form of generalized forces to assist in the calculation, in which the MBD time step was markedly more significant than the DEM time step If the subsequent MBD time step boundary (tDEM < tMBD þ DtMBD) has not yet been reached, the DEM operation was repeated until the dis- crete model reached the subsequent MBD time step (tMBD ¼ tDEM) 3.3 Coupled MBD–DEM model parameters and implementation Due to the characteristics of the MBD model solver, when extreme and discontinuous accelera- 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, the STEP function’s input speed and load were provided to avoid this issue, where it can ensure that the 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 the rated value between 0.2 s and 0.25 s Table presents the physical properties parameters of the equivalent mass experiment and the power parameters of the model in the first and second parts, respectively Since both sides of input sources considered that it takes time for the DEM model to reach stability after the beads were 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 the boundary surface during the particle generation process or starting to operate before the particles reached stability, which could induce the model to diverge The last part of Table exhibits the model’s contact parameters, where the gear’s normal con- tact stiffness was simplified by operating the average Kg curve of the single tooth mesh stiffness 10 Y.-C CHUNG ET AL Figure Flowchart for the calculation procedure of a coupled MBD–DEM model (Chung and Wu 2019) Table Parameters for the MBD model of a system comprises a pair of spur gears Valve Items (symbol) System System Unit Geometry parameters of MBD model 5.0 5.0 mm 31 37 – Module ðmÞ 20.0 20.0 Gear ðzÞ 35.0 30.0 pressure angle ðagÞ 7.179 7.788 Gear width ðbÞ bearing A bearing B mm System mass ðMsÞ SKF 6207 SKF 6004 kg 72.0 42.0 Bearing type 35.0 20.0 – Bearing outside diameter 17.0 12.0 mm Bearing inner diameter 0.013 0.0125 mm Bearing width mm Radial clearance 800.0 mm 420 Dynamic parameters of MBD model 5.968 rpm W Driving rotational speed ðxÞ 3:27 Â 108 N-m Power 3:27 Â 104 Resistant torque ðTrÞ N/m 0.01 N-s/m Contact parameters of MBD model 1.3 mm Average normal gear contact stiffness ðKgn;avgÞ – Average normal gear contact damping stiffness ðDgnÞ Maximum penetration depth ðdmaxÞ Force exponent ðnÞ MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 11 Table Parameters for the DEM model of a system comprises a pair of spur gears Items (symbol) Value Unit Material parameters of DEM model gear damping particle mm kg/m3 Poisson ration ðÞ 0.29 0.305 GPa Density ðqpÞ 7850.0 7850.0 Young’s modulus ðEÞ 210.0 190.0 – – Contact parameters of particle Bead-bead Bead-wall – Restitution coefficient ðeÞ 0.69 0.85 mm 0.54 mm static coefficient of friction ðlsÞ 0.44 – rotational coefficient of friction g Driven gear Filling parameters of particle Driving gear 2.5 Particle radius ðrpÞ 2.5 6.25 Maximum grid size 6.25 99 Numbers of particles per hole 77 50.86 Filling quality per hole 39.56 in Section 3.2 In the case of multi-tooth contact, the different engagements were considered to be parallel, and the software can determine if contact has occurred At the same time, the max- imum penetration depth dmax was defined to avoid discontinuity resultant from the damping force while gear meshing, and the STEP function was operated to build the hysteresis damping model (Jabłonski and Brezina 2016) Table lists the relevant parameters of the next DEM model for the initial parameters The first part of Table presents the material properties of the DEM model, where a SCM440 steel was utilized for the transmission system In contrast, low-carbon steel was applied for the damp- ing particles, resulting in some differences in Poisson’s ratio and Young’s modulus and density The second part of Table gives the contact parameters applied in the DEM model for the calcu- lation of particle behavior, in which the Hertz-Mindlin (no-slip) contact model did not consider the sliding of the particles during motion Therefore, it was unnecessary to consider the dynamic friction coefficient The maximum grid size referred to the grid size of the model boundary in the DEM model, which was related to the contact point search of the particles It only affected the determination of contact, with no significant impact on the results of calculating the forces on the particles, and was, therefore, set at 2.5 times the minimum particle radius, as recom- mended by default The factor that significantly affected particle behavior in the DEM model was the maximum calculated time step, especially in particle contact force The time step of the DEM model required it to be significantly smaller than the critical time step calculated from the physical prop- erties of the contacting object to describe the particles behavior Furthermore, the Rayleigh time step was typically applied as a reference for calculating the critical time step of particle motion, which was derived from the Rayleigh wave, as expressed in Eq (12): sffiffiffiffiffi TR ¼ prmin qp (12) 0:1631p ỵ 0:8766 Gp where rmin is the minimum particle radius; and qp is the particle density Result and discussion This section presents the experimental and simulation results of the vibration reduction on the proposed construction model Specifically, also discusses the investigation results of the dynamic behavior of damping particles in various input speeds, filling ratios, particle size, and material This experimental study applied a spur gear pair, as shown in Figure 6, with a gear ratio of 31 to 12 Y.-C CHUNG ET AL Figure Experiment gear pair with damping particles 37, a modulus of 5, and a pressure angle of 20, utilizing red cross steel (SCM 440) tempered to HRC 26 The width of the driven gear was 35 mm, and there were six cavities with a radius of 12.5 mm in the pitch circle at a radius of 60 mm from the shaft center The width of the driven gear was 30 mm, and there were six cavities with a radius of 15 mm in the pitch circle at a dis- tance of 50 mm from the shaft center The shaft connected to the gear was constructed of identi- cal SCM 440 steel as the gear, with an average shaft diameter of 25 mm The input and output end connected to the motor and load had thicker shaft diameters and were fitted with SKF-6207 bearings, while the other end was fitted with SKF-6004 bearings 4.1 Validity analysis of damping particle’s impact in vibration reduction on simulation and experimental applications This subsection verified the validity of the damping particle’s impact on vibration reduction in the application of a spur-gear transmission system The experimental designs were assigned to analyze the reference design and the corresponding equivalent mass experiments of carbon steel beads at 800 rpm and 1200 rpm input speed, 420 W equivalent power, and 3-mm particle diam- eter with a 36% filling ratio The calculation procedure of a coupled MBD–DEM model to analyze the validity of the pro- posed construction model was performed corresponding to the flowchart in Figure Firstly, the geometry and physical properties of all components in the system were determined Subsequently, the constraints, load and initial time t0, initial generalized displacement q0, and initial generalized velocity q_ 0, in the MBD model were defined The matrix of parameters was sorted and substi- tuted into the Lagrange equation to solve for the components’ velocity and acceleration vectors, where the components’ displacement vectors were solved by integration Then, the solved param- eters of the displacement and velocity vectors were imported into the DEM model for the solu- tion The displacement and velocity vectors in the DEM model can solve the penetration depth among the boundary surface and the particles at this time By substituting the penetration depth between particles and walls, and the penetration depth between particles and particles, into the Hertz–Mindlin contact force model, the normal and tangential contact forces generated by the particles can be determined for all particles at this discrete time step DtDEM: The calculated force states of all of the particles were analyzed by Newton’s second law of motion to obtain the pos- ition and velocity information of all particles at the next discrete time step and updated MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 13 Figure Avg and RMS values of the translational acceleration of the output shaft for 800 and 1200 rpm Meanwhile, the combined force of the particles on the boundary surface at the discrete time step tDEM was recorded by the force matrix Qp composed of the three-axis translational force and the three-axis rotational moment, where it was expressed in Eq (13): Â Ã Qp ¼ Fx, Fy, Fz, Tx, Ty, Tz (13) The gear model was constructed in Parasolid format to analyze the dynamic behavior of the gearing system, and then the contact depth between the meshes on the actual surface of the con- tact geometry in the MBD model was computed In the contact model, the WSTIFF solver with modified corrector and I3 integral format was applied for the analysis The I3 format had a more acceptable tolerance for discontinuous solution issues, and thus it was less likely to yield abnor- mal returns due to the discontinuous forces in the generalized moment matrix Qp when the DEM model was returned This study’s total simulation time was set to 0.5 s to avoid excessive solving time In addition, the gear model applied in the simulation was a 31 to 37-tooth spur gear at 800 rpm with a mesh frequency fGMF of 413.3 Hz, i.e., 2:4 Â 10À3s for single-tooth inter- meshing, to ensure the accuracy of gear contact behavior The single-tooth mesh duration can be designated to 20000 MBD solution time steps, i.e., DtMBD duration of 2:5 Â 10À5 s, which ensures 100 steps of analysis for each single-tooth mesh Previous investigations found that 67% of the energy waves transmitted during the vibration of an object were transmitted by Rayleigh wave, 26% by shear wave, and only 7% by primary wave (Richart, Hall, and Woods 1970) Another previous study has also reported that DEM simu- lation models with a maximum time step of 20% Rayleigh time step or shorter can yield more accurate results (Chou et al 2012) Consequently, this study designated the 20% Rayleigh time step as the maximum discrete time step utilized of 5:57 Â 10À7 s, which was 45 times the MBD time step DtMBD: In addition, the MBD-DEM two-way coupled vibration analysis model of the spur gear trans- mission system was applied in the experimental and simulation Then both experimental and simulation acceleration results on the bearing seat on the output shaft were compared Figure indicates the results of translational acceleration analysis for the simulation and experimental verification The left side displays the discrepancy between the RMS and the average values of vibration reduction on the simulation and the experimental condition at an input speed of 800 rpm In comparison, the right side shows the result values at 1200 rpm The result indicated that filling damping particles can reduce system vibration significantly compared to applying equivalent mass of a piece of symmetrical semi-annular The experimental and simulated results were consistent regarding numerical values and vibration reduction trends in both 800 rpm and 1200 rpm It was consistent in indicating that the utilization of damping par- ticles is more acceptable than equivalent mass in spur gear transmission systems The discrepancy in the reduction effect of the 800 rpm case among the experimental and simulation application 14 Y.-C CHUNG ET AL Figure The translation acceleration signal on the time domain of an output shaft of simulation and experiment for 800 rpm was 0.2% and 4.9% for RMS and the average values, respectively, in which high similarity was demonstrated Meanwhile, for the 1200 rpm case, the experimental result was more satisfactory than the simulation on both RMS and the average values Nevertheless, both results on 800 rpm and 1200 rpm indicated that the damping particle application can effectively reduce the vibration Therefore, the proposed construction model, shaped of six through-cavities filled with identical beads size in a symmetrical radial position in the gear face, is highly recommended for lessening vibration in spur gear transmission systems The vibration-reducing impact of the damping particle and the trend among the simulated and experimental results were verified The similarity of the equivalent mass model and the particle model was compared in a set of 800 rpm on particular time domain signals for both the experimental and simulated conditions Figure presents the comparison results on the time domain signal of 0.2 s A more detailed illustration is given in 0.02 s for both experimental and simulation methods, and systems of equivalent mass and damping particles According to both detailed charts, the red curve had a smaller deviation range than the black curve, and thus it can be stated that the vibration on the system with particles was lower than the equivalent mass In addition, the signal pattern of both the experimen- tal and simulation conditions exhibited a high degree of similarity, which further confirmed the validity of the vibration-reducing impact of the damping particle 4.2 Analysis of the dynamic behavior of damping particles on various rotational speeds and filling ratios This subsection analyzed the dynamic behavior of damping particles in the gearing transmission system with various rotational speeds and filling ratios according to the vibration reduction MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 15 Figure Effect of the rotational speed on the translational acceleration of the output shaft impact in the previous subsection and the establishment of the MBD-DEM two-way coupled model The experimental designs and the parameters were assigned 420 W equivalent power, car- bon steel beads, and 3-mm particle diameter with a 36% filling ratio Meanwhile, the input speed variable was determined at 200, 400, 600, 800, 1000, and 1200 rpm The calculation procedure of a coupled MBD–DEM model in this subsection refers to the previous subsection Figure illustrates the damping particle impact of vibration reduction in various rotational speeds for equivalent mass and system with particles The RMS value of radial vibration increased as the rotation speed increased, with the lowest value of 1.34 m sÀ2 at 200 rpm and the highest value of 12.48 m sÀ2 at 800 rpm in the equivalent mass experimental group The vibration value decreased gradually due to the input speed decrease and dropped to 10.56 m sÀ2 at 1200 rpm The critical speed that generates the most significant vibration in the system is 800 rpm, indicat- ing an identical result in both investigations of equivalent mass experiments and the design with particles Therefore, the gear transmission system should be operated at a higher rotational speed since the radial vibration at a high input speed is more stable than at a lower input speed Primarily, the result revealed that the particles affected vibration reduction at distinct rotational speeds, which were more suitable at higher rotational speeds Particularly at 600 rpm, the damp- ing particles had the most satisfactory impact on energy dissipation since the span of RMS value among the equivalent mass experiment and system with particles indicated the highest value of 3.10 m sÀ2, which was equal to 30.4% compared to other rotational speeds Moreover, the vibra- tion reduction amplitude was the lowest, decreasing by 0.13 m sÀ2, and the vibration reduction impact was 8.8% at 200 rpm The result indicated that a low rotational speed exclusively presented minor vibration reduc- tion This phenomenon can be explained by the fact that the damping particles cannot closely attach to the wall surface due to the insufficient centrifugal force of rotation at low speed Therefore, it cannot generate adequate friction with the wall surface, and thus the vibration reduction impact at high speed was significantly more satisfactory than at low speed According to the prior result, damping particles’ finest vibration reduction impact in the gear transmission system was at the rotational speed of 600 rpm Accordingly, further investigation involved the input rotational speed between 400 rpm and 800 rpm on various filling ratios of par- ticles with mm in diameter of carbon steel The input rotational speed variable was set in a 100-rpm interval at 400, 500, 600, 700, and 800 rpm At the same time, the filling ratio was deter- mined between 12%$48% with a 12% interval (12%, 24%, 36%, and 48%) Figure 10 indicates the damping particle impact of vibration reduction for various rotational speeds and various filling ratio The result shows that the RMS value of radial vibration for each filling ratio at distinct speeds decreased with the increase of the filling ratio However, the trend 16 Y.-C CHUNG ET AL Figure 10 Effect of the filling ratio on the translational acceleration of the output shaft Figure 11 Translation kinetic energy on frequency domain of gear meshing frequency for 600 rpm of the vibration reduction impact evolved gradually slower with the increasing filling ratio, where the lowest vibration value was shown to be at a 48% filling ratio The result indicated that increasing the contact area between the particles and the wall surface raised vibration attenuation In addition, the experiments with various filling ratios similarly indicated that the most satisfac- tory vibration reduction was achieved at the rotational speed of 600 rpm, where the vibration value changed significantly at higher than 600 rpm The RMS vibration value at the rotational speed of 600 rpm was decreased from 10.19 m sÀ2 to 6.97 m sÀ2 at a 48% filling ratio, which was equivalent to a reduced impact of 31.6% In order to further verify the impact of the filling ratio on vibration reduction, a fast Fourier ana- lysis was conducted for an experimental group of the rotational speed of 600 rpm with a distinct fill- ing ratio for 3-mm diameter carbon steel particles The vibration energy at the gear meshing frequency fGMF ẳ 310Hzị can be determined by applying fast Fourier transform (FFT) analysis The vibration energy in the horizontal and vertical directions was squared and rooted to determine the radial vibration energy The calculation result is presented in Figure 11 The FFT vibration energy analysis and the RMS analysis revealed a similar trend of a higher filling ratio and a lower vibration energy The highest vibration energy value of 1.89 (m sÀ1)2 was reached in the 12% filling ratio, which was 19% higher than the vibration energy in the 48% filling ratio of 1.53 (m sÀ1)2 However, the FFT analysis showed that the vibration energy of carbon steel particles with 48% and 36% filling ratios were identical, which was 1.53 (m sÀ1)2 It was assumed that the enlargement frequency exclu- sively represented the gear vibration, and not the overall system MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 17 Figure 12 Effect of the particle diameter on the vibration reduction rate According to the experimental data and MBD-DEM two-way coupled analysis, the result indi- cates that the most significant impact of the input rotational speed and filling ratio in vibration reduction is 600 rpm and 48%, respectively The result also has been further verified using fast Fourier analysis, demonstrating an identical result Therefore, the dynamic behavior of the damp- ing particles associated with the system setting confirms that the most satisfactory input rotational speed and filling ratio in vibration reduction is 600 rpm and 48%, respectively The subsequent subsection will provide investigation results associated with the particle characteristic of diameter and material types 4.3 Analysis the dynamic behavior of damping particles on diverse particle diameters and material In Subsection 4.2, it was confirmed that the carbon steel particles of 3-mm diameter demon- strated the most satisfactory vibration reduction impact at a 48% filling ratio with various speed conditions and the best vibration reduction effect at 600 rpm Therefore, this section focused on the 48% filling ratio and further reduced the speed analysis range to 600, 700, and 800 rpm, where the particle vibration value was more variable and had a particular vibration reduction impact The equivalent mass experimental group of carbon steel particles with a 36% filling ratio at the identical rotational speed was applied as the standard for normalization, and the vibration reduction impact of carbon steel particles with particle diameters of 2, 3, 4, 5, and mm was compared Figure 12 illustrates the calculation result of vibration reduction rate of damping particle impact for various rotational speeds and various particle diameters The vibration reduction impact gradually increased as the particle diameter increased The most satisfactory vibration reduction was achieved at the rotational speed of 600 rpm, where it is still consistent with the pre- vious result in various filling ratios with a particle diameter of mm, i.e., 33.0% However, the vibration reduction impact declined significantly when the particle diameter increased to mm Furthermore, at the rotational speed of 600 rpm, the discrepancy between the highest and lowest vibration reduction impact of mm and mm particle diameter, respectively, was only 3.2% It can be concluded that, compared to the filling ratio parameter, diverse particle diameters had a minor impact on vibration behavior Since the most satisfactory impact of vibration reduction occurred under a condition of 5-mm particle diameter, five different materials of NBR, POM, glass, carbon steel, and soft lead beads were produced utilizing a 5-mm particle size as the basis for further experiments The vibration reduction trend of various material beads on a 48% filling ratio and 600 rpm speed was 18 Y.-C CHUNG ET AL Table Characteristics of diverse beads materials Material (symbol) NBR POM Glass Steel Lead Unit Young’s Modulus ðEpÞ 0.008 2.90 71.70 190.0 16.0 GPa Density ðqpÞ kg=m3 Poisson’s ratio ðpÞ 1.00 1.42 2.50 7.85 11.34 0.47 0.39 0.25 0.30 0.42 – Bead-Wall coefficient of friction ðlp;wÞ 1.12 0.86 0.62 0.85 1.32 – Bead-wall restitution coefficient ðep;wÞ 0.31 0.70 0.80 0.54 0.20 – Bead-bead coefficient of friction ðlpÞ 0.49 0.39 0.38 0.44 0.42 – Bead-bead restitution coefficient ðepÞ 0.43 0.84 0.77 0.69 0.08 – Figure 13 Impact of the bead’s materials on the translational acceleration investigated in terms of the primary attributes of density and elasticity (see Table 3) Since the static friction coefficient of identical material beads will fluctuate significantly due to the environ- ment, surface roughness, and other factors, an indicator was needed that identifies the physical properties between the particles and other material’s walls These parameters, however, will impact the vibration-damping effect of the particles, and this study additionally carried out par- ticle parameter experiments The static friction and restitution coefficient of various material beads were obtained by slope and drop tests Table presents the final obtained properties of the various materials Moreover, the optimal physical parameters of the particle were achieved by approximating the experimental groups under various material conditions The impact of various material properties, such as density and young’s modulus, on the damping consequence of the particles can also be derived NBR and POM beads with an identical density, but markedly dissimilar Young’s modulus, were selected to determine the impact of elasticity on damping The three remaining materials were assigned to assess the influence of density on damping Among all the materials, the density of lead was the highest, while the density of NBR was the lowest Figure 13 presents the damping particle impact of vibration reduction trend for various bead mate- rials The NBR bead with the lowest density of 1.00 gr cmÀ3 had the smallest vibration reduction impact, where the radial translational acceleration RMS value was 8.09 m sÀ2 The glass bead with a slightly higher density of 2.50 gr cmÀ3 had a more satisfactory vibration reduction impact, where the RMS value was 7.50 m sÀ2, which was reduced by 7.19% compared to the NBR bead The soft lead bead with the most heightened density of 11.34 gr cmÀ3 had the greatest vibration reduction impact, where the RMS value was 6.08 m sÀ2, which was 24.8% lower than that of the NBR bead In other words, bead material with the lowest density had a minor vibration impact In contrast, bead material with the highest density had the most significant vibration reduction impact The elasticity characteristic represented by Young’s modulus of each material bead on the vibration reduction impact was also investigated In the case of NBR and POM beads, which had almost similar densities, although significantly different elasticities (see Table 3), it can be found MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 19 that the discrepancy in translational acceleration was 4.13% In contrast, in the case of glass and lead beads, which had a significant increase in density despite a remarkable decline in elasticity (see Table 3), the vibration reduction impact of the damping particles was significantly increased with the increase in density and did not decrease with a decline in elasticity It was demonstrated that bead elasticity did not have vibration behavior According to the facts presented in the previous paragraphs, it can be confirmed that the vibration reduction impact of the damping particles was primarily affected by their density, which increased significantly with the increase in density However, the elasticity of the bead material is less significant to the vibration behavior reduction Conclusion This study presented an experimental platform on the transmission system of a spur gear pair in which damping particles were filled in the cavities of gear bodies The experimental design was established to verify the validity of damping particles impact on vibration reduction In addition, several experimental groups were determined, corresponding to the variable controls on input rotational speed, filling ratio, particle size, and material to elucidate the dynamic behavior of the damping particles A two-way coupling dynamic analysis model was also proposed Both software of the multi-body dynamics ADAMS and discrete element method EDEM were operated to form a reliable two-way coupling model The results confirmed that the filling damping particles appli- cation effectively reduced the system vibration, and both simulation and experimental test results were consistent in terms of numerical values and vibration reduction The investigation’s results for the dynamic behavior of damping particles applied to the gearing transmission system can be stated as follows: a The vibration reduction impact at high rotational speed was significantly more satisfactory than at low rotational speed b The radial vibration for each filling ratio at distinct rotational speeds decreased with the increase of the filling ratio c The vibration reduction impact gradually increased as the particle diameter increased, although it presented a minor impact compared to the filling ratio parameter d The vibration reduction impact was primarily affected by applied bead density, which rose significantly with the increase in density However, bead elasticity did not impact vibration behavior According to the results, the proposed construction model, shaped of six through-cavities filled with identical beads size in a symmetrical radial position in the gear face, is highly recommended for spur gear transmission systems The dynamic behavior of damping particles with greatest vibration reduction impact was achieved by applying a soft lead bead in a 5-mm diameter, 48% filling ratio, and the system working at a rotational speed of 600 rpm Future works Another research topic concerns deeper analysis in designing particle dampers to dampen gear system vibrations efficiently at higher rotational speeds should be conducted It will consider another cavity shape (not a circle like this paper) that more effectively dampens the system vibration needed to be studied concerning the gear intensity in carry-on force and rotation In addition, a ratio of cavities volume to overall gear body volume should be inves- tigated Then a systematic analysis approaches the particle dynamics within the damper cavities needs to be exam- ined in detail, then the essential influence parameters analyzed