Zhaofeng Shi, Qi Chen, and Tingting Xiong
[1] L.C. Hsieh, T.H. Chen, and H.C. Tang, The engineering designof helical spur gear transmission with single gear pair forelectric scooter, Applied Mechanics & Materials, 764–765, 2015,374–378. [2] J. He, L.C. Shi, C.F. Zhang, J.H. Liu, B. Yang, and X. Zuo,Optimal adhesion braking control of trains based on parameterestimation and sliding mode observer, Mechatronic Systemsand Control, 48(4), 2020, 222–230. [3] T. Ouyang, R. Yang, Y. Shen, J. Chen, and N. Chen,Effect of structural design parameters on nonlinear dynamiccharacteristics of the gear transmission, Proceedings of theInstitution of Mechanical Engineers, Part D: Journal ofAutomobile Engineering, 236(4), 2022, 522–539. [4] V.S. Murty, S. Jain, and A. Ojha, Suitability of linearswitched reluctance motor for advanced electric traction system,Mechatronic Systems and Control, 49(3), 2021, 142–148. [5] X. Wang, B. Wang, Y. Wu, and J. Guo, Adaptive control anddisturbance compensation for gear transmission servo systemswith large-range inertia variation, Transactions of the Instituteof Measurement and Control, 44(3), 2022, 700–715. [6] M. Nakagawa, D. Nishida, D. Sah, and T. Hirogaki, Bond graphsimulation of gear transmission considering tooth meshingstiffness and damping, Proc. of the JSME International Conf.on Motion and Power Transmissions, 2017. [7] Q. Zeng, S. Jiang, L. Wan, and X. Li, Finite elementmodeling and analysis of planetary gear transmission based ontransient meshing properties, International Journal of ModelingSimulation & Scientific Computing, 6(3), 2015, 1–20. [8] Y. Xia, Y. Wan, and Z. Liu, Bifurcation and chaos analysis fora spur gear pair system with friction, Journal of the BrazilianSociety of Mechanical Sciences and Engineering, 40(11), 2018,529. [9] E. Bechhoefer and Y. Ozturk, Signal processing to improvegear fault diagnostics in the presence of gear dynamics,International Journal of COMADEM, 24(4), 2021, 43–48. [10] S. Liu, A. Hu, Y. Zhang, and L. Xiang, Nonlinear dynamicsanalysis of a multistage planetary gear transmission system,International Journal of Bifurcation and Chaos in AppliedSciences and Engineering, 32(7), 2022, 1–16. [12], while the axle isconnected to gear 2, serving as the driven gear.The actual gear transmission system in the multiple-unit train is a continuous elastic system, comprisingan infinite number of degrees of freedom, making itvirtually impossible to calculate as a whole. To simplifythe calculation process and improve accuracy, the finiteelement analysis method is employed. This involvesdiscretising the entire transmission system into variousnodes. Equations of motion are then established for eachnode, and finally, the equations of motion for all nodesare combined to obtain the overall system’s equations ofmotion [13]. To enhance calculation accuracy, the specificnumber of nodes is determined based on time-consumingconsiderations and accuracy requirements. The discretisednode model comprises the beam unit model, gear meshingdynamics model, bearing model, and lumped mass model,which are integrated to form the complete transmissionsystem model. The beam unit model is used to representthe drive shaft within the transmission structure, and theTimoshenko beam unit is utilised in this paper [14].The gear meshing dynamics model is utilised torepresent the gear meshing in the transmission structure,which is the key to the whole transmission structure, andits model equation is:M12X 12 + C12X 12 + K12X12 = F12 + FeC12 = c12 × AT12 × A12K12 = k12 × AT12 × A12Fe = k12 × AT12 × e(t) + c12 × AT12 × e (t), (1)2Figure 2. Schematic diagram of the simplified model of the gearing system of a multiple-unit train.where M12 is the mass matrix of gears 1 and 2, C12 isthe damping matrix of gear meshing, K12 is the stiffnessmatrix of gear meshing, X12, X 12, and X 12 are thedisplacement vector, velocity, and acceleration of the twogears, respectively, F12 is the external load, Fe is theadditional load caused by the transmission error duringthe gear meshing, A12 is the matrix of the relative positionbetween the two gears, e(t) is the transmission error,c12 is the gear mesh damping, and k12 is the gear meshstiffness [15].The bearing is the supporting part of the motor shaftand axle, and its finite element modelling can be simplifiedinto an elastic unit with stiffness and damping. Its modelequation is:Cbu + Kbu = Fb, (2)where Cb represents the damping matrix of the bearing,Kb is the stiffness matrix of the bearing, and u is thedisplacement matrix of the bearing [16].The role of the lumped mass model is to simplify arotating body part located on a rotating shaft in a geartransmission system to a mass node, which is then analysedfor modelling. Its model equation is:Mdu d + ΩGdu d = Fd, (3)where Md stands for the mass matrix of the rotating part,Gd stands for the gyroscopic moment matrix of the rotatingpart, Ω stands for the angular velocity of the rotating part,and ud stands for the displacement vector of the rotatingpart.In summary, before the modelling of the system, nodesare divided first according to the demand, and then thecorresponding dynamics model is constructed according tothe type to which the nodes belong [17] and integrated.The final model equation of the whole traditional systemof the multiple-unit train gear is:Mu + (C + ΩG)u + Ku = F, (4)where M indicates the mass matrix of the whole system,C indicates the damping matrix of the whole system,G indicates the gyroscopic moment matrix of the wholesystem, Ω is the angular velocity matrix of the rotatingparts of the whole system, K is the stiffness matrix of thewhole system, and u stands for the displacement matrix ofthe nodes of the whole system (the displacement containsthe node’s movement in three axial directions and rotationangles) [18].3. Experimental Analysis3.1 Experimental SettingsBased on the basic structure schematic diagram ofthe gear transmission system presented in Fig. 1 and theactual system, a model was constructed. To facilitate thecalculation process, the model was simplified, as depictedin Fig. 2. In this simplified model, some small inverted and3Table 1Parameters of Rotating Parts and BearingsLumped massparameters ofrotating partsName of component Wheel Drive gear Driven gear Coupling Motor rotorMass/kg 320 18 60 25 110Ix = Iy/kg · m219 0.10 1.20 0.20 1.45Iz/kg · m236 0.15 2.35 0.18 1.30BearingstiffnessparametersName of component Axle bearings Driven gear bearings Motor shaft bearings Drive gearbearingskxx (Nm ) 4.7 × 1092.1 × 1091.5 × 1091.5 × 109kyy (Nm ) 4.8 × 1092.0 × 1091.5 × 1091.5 × 109kzz (Nm ) 9.5 × 1090 3.1 × 1090kθxθx(Nmrad ) 8 × 1063 × 1063 × 1063 × 106kθyθy(Nmrad ) 8 × 1063 × 1063 × 1063 × 106rounded corners were disregarded, while some relativelylarge arc parts were divided into multiple segments. Therotating parts in the transmission system were representedby black dots, and the gear meshing was simplifiedto equivalent meshing stiffness and equivalent springdamping. Similarly, the supporting role of the bearingsis also simplified to equivalent spring damping [19]. Thelength of the motor shaft and axle shaft sections within themodel are labelled in Fig. 2 (unit: mm). Table 1 providesthe relevant parameters of the rotating parts and bearings.Furthermore, the relevant parameters of the gear meshin the transmission system are as follows: the drive gearhad 30 teeth with a pitch circle radius of 108 mm, whilethe driven gear had 70 teeth with a pitch circle radius of258 mm.3.2 Experimental Projects3.2.1 Validation of Model ValidityThe simplified model of the transmission system depictedin Fig. 2 served as the basis for constructing the finiteelement model. To solve the kinetic equations of thismodel, the Newmark-β method was employed [20]. Beforeutilising the simulation model to analyse the transmissioncharacteristics during the iterative process of solving, it isessential to verify the validity of the model to guarantee thereliability of the simulation results. The validity verificationexperiments involved setting the stable operating conditionof the system, where the initial speed of the motor andthe external load remain constant. Since the transmissionsystem’s time-related parameters can be neglected, staticanalysis was employed to calculate the system’s parametersin the steady state, including shaft torque, gear rotationspeed, gear mesh force, and mesh frequency. The calculatedvalues were considered as theoretical references to verify theresults obtained from the simulation model. The steady-state condition was set as follows: the motor speed wasfixed at 500 rad/min.3.2.2 Test of Impact of Gear Parameters on TransmissionSystemIn the train gear transmission system, the gear meshingplays a crucial role as the key transmission structure. Anychanges in the gear parameters will impact the inherentcharacteristics of the entire system. Therefore, optimisingthe gear parameters can lead to the optimisation ofthe entire system’s inherent characteristics. The validatedfinite element model can be utilised to simulate thesystem characteristics by adjusting the model parameters,providing a more convenient approach compared toadjusting the parameters of the solid model. In this paper,the structural optimisation focused on the gear meshingstiffness. The gear meshing stiffness was set at 0, 10, 102,103, 104, 105, 106, 107, 108, 109, 1010N/m to test the low-order intrinsic frequency, critical speed, and transmissionerror of the entire system under varying meshing stiffness.3.3 Experimental ResultsBefore assessing the effect of different meshing stiffness onthe transmission system using the finite element model ofthe gear transmission system, the effectiveness of the modelwas verified first. Table 2 presents the validation resultsof the finite element model of the transmission system. Itcan be observed that after the model entered the stableworking condition, the torque error of the motor shaft was0.03%, the torque error of the axle was 0.03%, and therotational speed error of the axle was 0.04%. These resultsindicated that the axle torque and rotational speed of thefinite element model closely aligned with the theoreticalvalues after reaching a stable working condition, confirmingthe validity of the model.After validating the finite element model, the system’sintrinsic frequency, axle critical speed, and transmissionerror were tested under different gear meshing stiffness.The final results are depicted in Figs. 3 and 4, as wellas summarised in Table 3. Due to space limitations, onlythe variations of the first three orders of the intrinsic4Table 2Results of Validity VerificationTransmission shaft Motor shaft AxleSteady state torque simulationresults N · m89.97 209.93Steady state torque theoreticalresults N · m90.00 210.00Torque error/% 0.03 0.03Steady state speed simulationresults rad/min500.00 215.25Steady state speed theoreticalresults rad/min500.00 215.16Rotational speed error/% 0.00 0.04Figure 3. Curve of the first three orders of the transmissionsystem’s intrinsic frequency with the variation of themeshing stiffness.frequency and axle critical speed were presented. FromFigs. 3 and 4, it can be observed that with the increaseof the gear meshing stiffness, only the first-order intrinsicfrequency and the axle critical speed experienced changes,while the second- and third-order intrinsic frequencies andaxle critical speeds remained mostly unaffected. Whenthe meshing stiffness was below 104N/m, the first-order intrinsic frequency was essentially zero, and thecritical speed of the axle also maintained at a low level.This indicated that the system was prone to rigiditydisplacement, mainly demonstrated as the rotationaldisplacement of the axle, and the lower critical speed ofthe axle suggested a higher likelihood of reaching thespeed at which resonance occurs. On the other hand, whenthe meshing stiffness exceeded 104N/m, the first-orderintrinsic frequency and the axle critical speed witnessedsignificant increases; however, they remained relativelyunchanged after 109N/m.Table 3 reveals that as the meshing stiffness increased,the maximum and minimum transmission errors of thetransmission system decreased, and the fluctuation intransmission error also diminished. This behaviour can beattributed to the fact that higher mesh stiffness results inless deformation of the gears during the meshing process,Table 3The Transmission Error of the Transmission System Withthe Changes of the Meshing StiffnessMeshingstiffnessN/mMaximumtransmissionerror/µmMinimumtransmissionerror/µmError fluc-tuation/µm10 61.31 58.64 2.6710253.91 51.58 2.3310346.50 44.51 1.9910439.11 37.44 1.6710531.68 30.37 1.3110624.42 23.30 1.1210717.17 16.23 0.9410810.95 10.16 0.791098.74 8.10 0.6410107.35 6.81 0.54Figure 4. Curve of the first three orders of the axle’s criticalspeed with the variation of the meshing stiffness.and after eliminating errors caused by installation, thetransmission error decreased.In summary, a very low meshing stiffness can lead to ahigh probability of rigid displacement in the transmissionsystem and an increased likelihood of the axle reachingcritical speed and experiencing resonance. Therefore, it isessential for the meshing stiffness to be sufficiently high.Additionally, improving the engagement stiffness can alsocontribute to reducing the transmission error. However,once the meshing stiffness surpasses a certain range,the intrinsic frequency and critical speed of the systemessentially remain unchanged. Although the transmissionerror continues to decrease, considering the cost ofimproving meshing stiffness, it is not necessarily better tohave a higher meshing stiffness. In this study, the besteffect was achieved when the meshing stiffness was 109N/m.54. ConclusionThe fundamental structure of the gear transmission systemin a multiple-unit train is briefly introduced in thispaper, and a corresponding kinetic equation model wasestablished. Subsequently, a corresponding finite elementmodel was constructed using the kinetic equation of thetransmission system. After validating its validity, theintrinsic frequency, axle critical speed, and transmissionerror of the transmission system with different meshingstiffness were tested. The results showed that when themeshing stiffness exceeded 104N/m, both the first-order intrinsic frequency and critical speed of the axleexhibited a significant increase; after reaching 109N/m,these values basically remained constant; as the meshingstiffness increased, the transmission error decreased. Thelimitation of this article lies in the simplification made tothe model for ease of calculation during the simulation ofthe gear transmission system in the multiple-unit train,which results in certain errors in the simulation results.References[1] L.C. Hsieh, T.H. Chen, and H.C. Tang, The engineering designof helical spur gear transmission with single gear pair forelectric scooter, Applied Mechanics & Materials, 764–765, 2015,374–378.[2] J. He, L.C. Shi, C.F. Zhang, J.H. Liu, B. Yang, and X. Zuo,Optimal adhesion braking control of trains based on parameterestimation and sliding mode observer, Mechatronic Systemsand Control, 48(4), 2020, 222–230.[3] T. Ouyang, R. Yang, Y. Shen, J. Chen, and N. Chen,Effect of structural design parameters on nonlinear dynamiccharacteristics of the gear transmission, Proceedings of theInstitution of Mechanical Engineers, Part D: Journal ofAutomobile Engineering, 236(4), 2022, 522–539.[4] V.S. Murty, S. Jain, and A. Ojha, Suitability of linearswitched reluctance motor for advanced electric traction system,Mechatronic Systems and Control, 49(3), 2021, 142–148.[5] X. Wang, B. Wang, Y. Wu, and J. Guo, Adaptive control anddisturbance compensation for gear transmission servo systemswith large-range inertia variation, Transactions of the Instituteof Measurement and Control, 44(3), 2022, 700–715.[6] M. Nakagawa, D. Nishida, D. Sah, and T. Hirogaki, Bond graphsimulation of gear transmission considering tooth meshingstiffness and damping, Proc. of the JSME International Conf.on Motion and Power Transmissions, 2017.[7] Q. Zeng, S. Jiang, L. Wan, and X. Li, Finite elementmodeling and analysis of planetary gear transmission based ontransient meshing properties, International Journal of ModelingSimulation & Scientific Computing, 6(3), 2015, 1–20.[8] Y. Xia, Y. Wan, and Z. 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