Research Article  Open Access
Abdulhameed M. Y. AlTayari, Siyu Chen, Zhou Sun, "A Coupled TorsionalTransition Nonlinear Vibration and Dynamic Model of a TwoStage Helical Gearbox Reducer for Electric Vehicles", Shock and Vibration, vol. 2020, Article ID 8838521, 25 pages, 2020. https://doi.org/10.1155/2020/8838521
A Coupled TorsionalTransition Nonlinear Vibration and Dynamic Model of a TwoStage Helical Gearbox Reducer for Electric Vehicles
Abstract
A coupled torsionaltransition nonlinear dynamic model of a twostage helical gear (TSHG) reduction system for electric vehicles (EVs) is presented in this paper. The model consists of 16 degrees of freedom (DOF), which includes factors such as the nonlinearity of backlash, timevarying mesh stiffness (TVMS), mesh damping, supporting bearings, static transmission error (STE), and the torsional damping and stiffness of the intermediate shaft, in which the fourthorder Runge–Kutta numerical integration method was applied to solve the differential equations. With the help of bifurcation diagrams, timedomain histories diagrams, amplitudefrequency spectrums, phase plane diagrams, Poincaré maps, rootmeansquare (RMS) curves, peakpeak values (PPVs), and Lyapunov exponents, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG system are investigated. The stability properties of steadystate responses are investigated using Lyapunov exponents. The results reveal various types of dynamic evolution mechanisms and nonlinear phenomena such as periodicone responses, quasiperiodic responses, jumps phenomena, and chaotic responses. The research presents useful results and information to vibration control and dynamic design of the TSHG transmission system used in EVs.
1. Introduction
It is known that the power mechanism of EVs is composed of a power coupling system (transmission shafts, gear pairs, bearings, etc.), electrical control systems, motor, and other subsystems. As gear pairs are one of the main parts of EV power coupling system, the dynamic behavior of the gear pairs mechanism has a significant influence on power transmission and even the whole vehicle [1]. Therefore, the analysis of the dynamic characteristics of gear pairs is becoming essential in power transmission systems.
Numbers of dynamical models for both helical and spur gear pairs systems have been presented; a mathematical model and computer simulation of a twostage gearbox including translationaltorsional vibration responses of gear pairs system were presented by Walter Bartelmus [2]. Based on ANSYS software, Yan and Liu [3] developed a new finiteelement modeling method for helical gear transmission with multiple shafts considering the effects of bearing flexibilities and shafts. A dynamic model of a helical gear multipleshafts reduction box was presented by Kubur et al. in [4, 5], which consists of a finiteelement model of shaft structures combined with another threedimensional discrete model of gear pairs. Moreover, Zhang et al. [6] presented an effective dynamic model of a multipleshafts helical gearedrotor system with geometric eccentricity, bearing flexibility, and gear mesh, in which the eigenvalue solution and summation method were applied to estimate the forced responses and the natural frequencies of the geared system.
Since the nonlinear characteristics of gearpair system have been the most significant research fields, many studies on the dynamic model and response of twostage gear systems have been published. For instance, Walha et al. [7] proposed a purely torsional spur gear system with three shafts connected by two spur gear pairs, in which the Newton–Raphson algorithm was used to examine the influence of backlash on twostage spur gear (TSSG) system. Based on the variation of frequency response characteristic, Baek et al. [8] proposed a new technique to predict the contribution ratio of each stage backlash of the TSSG reducer; the validity of the used method was validated in a seeker gimbal with adequate results achieved. Since the dynamic coupling and TVMS between the helical gear pairs have a significant effect on the vibration characteristics of the helical gear system, Wang and Shi [9] proposed a systematic model to analyze the influence of TVMS and helix angle on gear pair system. In [10], Wang and Zhang developed a dynamics model of tensionalbendingswingaxial coupled motion for (TSHG) transmission considering the TVMS, backlash, and deviation, where a fifthorder Runge–Kutta method was used to solve the differential equations. An eighteenDOF model considering the effects of the loading system, driving motor, and supporting bearings was proposed by Brethee et al. [11] to investigate the dynamic response of surface wear on gear including tooth friction and TVMS based on electrohydrodynamic lubrication (EHL) principles.
Considering the effects of TVMS and nonlinear characteristics of bearing, a dynamical model of a TSHG box including the motion of housing was established by Xu et al. [12], and steadystate vibration response of the gearbox has been obtained based on the coupling gearrotorbearing dynamic model. Furthermore, experimental modal analysis was used by Patel and Pathan [13] to derive the natural frequencies of TSHG reducer considering the effect of TVMS on natural frequency, and the modal parameters were obtained by applying the Frequency Response Function (FRF) method. Walha et al. [14] developed a 12DOF dynamic model to study the nonlinear dynamic responses of a TSSG system including mesh stiffness fluctuation, backlash, and bearing flexibility, in which the technique of linearization was used to decompose the system from nonlinear to linear. Ma et al. [15, 16] presented a 14DOF dynamic model with an experimental study of TSSG to investigate the nonlinear dynamic response analysis of the TSSG space driving mechanism under large inertia load taking into consideration TVMS, damping, backlash, and transmission error. And Bin [17] used the model to analyze the effects of the profile modification parameters on the load transmission error, dynamic load coefficient, tooth profile error, and modification. Another 26 DOF of TSSG was established by Jia et al. [18] to study the dynamical modeling of multiple pairs of spur gears in mesh considering the effect of variable tooth stiffness, friction, geometrical errors, localized tooth crack, and pitch and profile errors on one gear.
Systematic modeling and analysis of a TSSG box model with 12 and 26 DOF were used to describe the gear fault features when processed with harmonic wavelet transform (HWT) in [19, 20]. Furthermore, He et al. [21] proposed a TSSG with a twelveDOF dynamic model to study the influences of gear eccentricity on transverse and torsional dynamic responses and the dynamic transmission errors. Dynamic behavior of a threedimensional model TSHG system considering the effect of manufacturing defects was formulated by Walha et al. [22], in which the dynamic response was performed by the Newmark method. Dyk [23] examined the models of TSSG used by the discrete models and considering the effects of the intermediate shaft on dynamic loads in both two stages. Abboudi et al. [24] developed a lampedmass dynamic model of TSHG with twelve DOF used in wind turbines, excited by the variability in wind resources and TVMS fluctuation; the differential equations’ motion of system was solved by the implicit Newmark algorithm. Beyaoui et al. [25] proposed a new methodology considering uncertainties in a gear transmission system of a horizontalaxis wind turbine, in which the dynamic equations of 12 DOF were solved by applying the polynomial chaos method and the ODE45 MATLAB solver. A dynamic model for an automotive train system with 22 DOF was established by Ghorbel et al. [26] to study the kinetic, the vibration mode, and strain modal energies distributions taking into consideration the engine excitation, clutch, gearbox, and disc brake. In order to investigate the nonlinear dynamic response of the TSHG system coupled with an automotive clutch, Walha et al. [27] proposed a dynamic model of 27 DOF considering spline clearance, doublestage stiffness, and dry friction path, in which the equation of motion was solved by Runge–Kutta method.
According to the literature reviews mentioned in Section 1, it is indicated that various dynamical models have been presented to analyze the dynamics of twostage gears pairs. Nevertheless, limited studies have addressed the effect of gear nonlinear dynamic response of TSHG box reduction used in EVs. For this reason, a nonlinear torsionaltranslational dynamic model of a TSHG reduction with a 16 DOF is derived based on Zheng’s method [6], in which the TVMS, mesh damping, support bearings, STE, backlash, and torsional stiffness and damping of the intermediate shaft are considered in this paper.
Consequently, the dynamic model in Section 2 and numerical experiments in Section 3 were carried out to demonstrate the nonlinear dynamic characteristics, where the equation of motion is solved by the Runge–Kutta method. Moreover, the effects of pinion rotational speed, backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG reduction were also studied, which provide an essential understanding of the nonlinear dynamic features of TSHG reduction, as concluded in Section 4.
2. The Dynamic Model of TSHG System
A 16DOF nonlinear dynamic model is presented to simulate the dynamic behavior of a TSHG reduction system, as shown in Figure 1. As revealed by the geometric description in Table 1, the chosen gearbox system is a speed reducer. The system is formed by four helical gears , and ; pinion and gear are denoted with the subscripts and as in ; the two stages of the system are denoted with subscripts 1 and 2 as in , respectively. Each gear is represented as rigid blocks with 4 DOF (one rotation and three translations), in which , and represent the rotating speed, the moment of inertia, and the base circle radius of gears and , respectively. and represent the torque values applied to the pinion and the gear , respectively. The nonlinear backlash function , STE , TVMS, and mesh damping are combined to describe the gear deformation during the meshing process [28]. The resilient elements of bearing supports are represented by the damping and stiffness coefficients, where indicates the four bearings in , and directions (LOA, OLOA, and axial direction), respectively. The intermediate shaft of the TSHG component is described by torsional stiffness and damping components.

The generalized coordinates vectors of the nonlinear dynamic model including 16 DOF can be defined aswhere , and represent the translations of pinion and gear along axes , and ; represents the torsional displacement of pinion and gear around axis .
The STE is generally considered to be a periodic function of displacement, usually spreads out in Fourier series as fundamental frequency part of the harmonics [29], and can be expressed aswhere , and are the constant amplitude of STE, the amplitude of STE, and phase angle, respectively. The excitation meshing frequency of gear pair is determined as shown in [30]where and are the rotational speed of pinion and teeth number.
According to Ishikawa’s method, the TVMS is simplified as periodic waveforms under the meshing frequency and spreads out into Fourier series [31]; the nonlinear TVMS obtained using the Fourier expansion is written as follows:where , and are the mean value of meshing stiffness, the stiffness fluctuation amplitude, which equals , and phase angle, respectively. The mean value of the mesh damping is expressed as
Here, the damping ratio is calculated as Rayleigh damping [32, 33], in which the value generally ranges from 0.03 to 0.17; in this model, the ratio is , where demonstrates the first and second stages.
When presents the relative meshing displacement of the gear under the influence of the gear backlash [34], the nonlinear backlash function is expressed as follows:
Here, represents the total backlash.
The gear mesh is expressed by a nonlinear TVMS and mesh damping . The dynamic mesh force of gear pairs along the LOA can be expressed aswhere and are the dynamic mesh forces of pinions and gears along with the coordinate directions , and , respectively. and indicate the pressure angle and helix angle, respectively.
The governing motion equations of the TSHG system shown in Figure 1 are expressed in the state space form, which is solved by MATLAB ODE45 solver and derived considering the following assumptions:(1)Pinions and gears are modeled as rigid disks(2)Both input and load torque are applied to a system with constant values(3)The gear teeth are considered to be fully involute; assembly and manufacturing errors are ignored(4)Spring and damper are used to represent the torsional stiffness of the intermediate shafts in the midsection of the shaft
It is noted that the modeling method of Zhang in [6] takes into account the effects of geometric eccentricity on gears, though, in this study, the influence of eccentricity is neglected. According to the proposed concept, the rotational and translational motion equations of the TSHG reduction gear pairs are expressed by the following equations, respectively:
With those four DOF for each gear, the twostage gear pair has a total of 16 DOF that define the coupling among the TSHG system, where indicates the mass of pinion and gear in the first and second stages , respectively.
The relative displacement of the firststage and secondstage gear mesh along the LOA is defined as
Equation (8) can be combined and substituted into equation (10); thus, the relative displacement of the first and second stages became as follows:where is the relative displacement of the first gear and second pinion and can be written as follows:with .
For analytical convenience, the dimensionless form of equations (9), (11), and (12) is obtained by assuming the following nondimensional parameters as
Here, represents the natural frequency of gear pairs and is the equivalent mass of gear pairs. The small letter of each variable represents the derivative to time , while the big letter represents the derivative with respect to dimensionless time .
Therefore, the dimensionless form of the nonlinear backlash function in equation (7) becomes
The equations of motion of the entire system in the dimensionless form can be expressed in matrix forms as
The dimensionless mass matrix, damping matrix, stiffness matrix, and the external excitation force vectors of the system are represented by , and , respectively. Consequently, the simplified coordinates vectors of the dimensional nonlinear dynamic model can be defined as :
The mass matrix can be expressed by
Here, represents the mass of the block which can be expressed as follows:
For the sake of simplicity, the stiffness matrix and damping matrix are expressed by as follows:where the damping matrix and stiffness matrix are denoted in with subscripts and , respectively. Here, is the damping and stiffness of the block , which can be expressed by equations (20) and (21) as follows:
However, the vector force can be written as
3. Results and Discussion
The basic parameters of the TSHG reduction gear pairs studied in this paper are shown in Table 2. With those parameters, the set of the secondorder differential equations in equation (16) are solved by the Runge–Kutta method. With the help of the FFT spectrum, phase portrait, Poincaré point section, timedomain history, RMS, PPVs curves, and Lyapunov exponents, the effects of the rotational speed of pinion , backlash, torsional stiffness, and torque fluctuation on the dynamic behavior of TSHG components were studied by applying the numerical integration method considering different conditions. Generally, the dynamic responses are divided into five categories: chaotic, quasiperiodic, subharmonic, periodic harmonic, and periodic nonharmonic response. When the time history is nonperiodic and has unlimited nonrepeating points in the Poincaré section, the response is chaotic. While the response is quasiperiodic, the phase portrait has nonperiodic circles and the points of the Poincaré section formed a closed orbit. When multiple discrete points are formed in the Poincaré section and repeat themselves at the excitation frequencies, the responses become subharmonic response. The system is harmonic period when the phase portrait is circular and repeats itself in Poincaré section. At last, the response is nonharmonic periodic if the phase portrait is noncircular and repeats itself in Poincaré section [35]. The results in this section provide some optimization suggestions for the TSHG reduction gearpair design.

3.1. Effect of the Rotational Speed on the TSHG Dynamic Response
The rotational speed of pinion is considered as one of the main parameters that affect the dynamical behavior of system transmission. In this section, the half backlash of gears is set to ; the other parameters of TSHG transmission system are given in Table 2. Figure 2 presents the forward bifurcation characteristic of the first stage and second stage in dimensionless displacement with regard to the rotating speed of pinion as the control parameter, which increases gradually from 1000 rpm to 20000 rpm, while Figure 3 presents the backward bifurcation characteristic with regard to the decreasing of from 20000 rpm to 1000 rpm. As shown in Figures 2 and 3, the system responses of both and under the variation of contain types of motion forms such as periodicone motion marked with letter P, jumps phenomena, quasiperiodic motion marked with letter Q, and chaotic motion marked with letter C.
(a)
(b)
(a)
(b)
In the case of the foreword bifurcation, the system behaves as a steady periodone motion at low speed and persists until reaches 7110.5528 rpm. The response of timedomain history shows a sine wave, one main peak amplitude is found in FFT spectrum, the phase plane has a single closed circle, a single point is found in Poincaré section, and these characteristics indicate that the response of is harmonic periodicone motion, as shown in Figure 4.
(a)
(b)
(c)
The system keeps transforming between periodic motion and quasiperiodic motion several times until reaches 17040 rpm. However, the system is periodicone motion except when is in the ranges of 7110–7493 rpm, 9974–10452 rpm, 11407–13221 rpm, and 15703–17040 rpm; the system experiences the quasiperiodic motion response. As illustrated in Figure 5, the phase plane diagram does not repeat itself, which forms closed orbit points in Poincaré section and the FFT spectrum is continuous; this means that the system response is in quasiperiodic motion. Meanwhile, a jump phenomenon can be observed when is around the resonant regions at 12744 rpm. When increases to 17040 rpm, the system response changes from quasiperiodic motion to chaotic and remains until reaching 18758 rpm. From Figure 6, it can be seen that system response does not repeat itself in any pattern, the FFT spectrum has continuous broadband, and the phase plane shows disorder circles with many discrete points found in Poincaré section. It is revealed from these properties that the system responses as chaotic motion at the range of 17040–18758 rpm. However, with the increase of pinion speed, the system abandons the chaotic region and returns to quasiperiodic motion response again which preserves from 18758 rpm to 20000 rpm.
(a)
(b)
(c)
(a)
(b)
(c)
As for the backward bifurcation, considering the decreasing of from 20000 rpm to 1000 rpm continuously, a chaotic motion is determined at 20000–18759 rpm and 17613–17040 rpm. As illustrated in Figure 7, the phase plane diagram shows nonrepeated circles with many clustered points of Poincaré map and the diagram of timedomain history also shows a nonperiodic motion. Figure 8 reveals that the system transforms into quasiperiodic response at 18759–17613 rpm, 17040–15703 rpm, and 13221–15512 rpm, where a jump phenomenon occurs at 17613 rpm. The system behaves as periodicone motion response at 15703–13221 rpm and undergoes the chaotic region at 13221–11979 rpm. The chaotic phenomenon causes the vibration and noise issues in the system, which should be avoided during the design. Meanwhile, the system leaves the chaotic region with a jump phenomenon occurring at 11979 rpm. However, the system characteristic responses are the same as the responses in the forwarding bifurcation when is less than 11979 rpm, which is verified by PPVs and RMS curves, as shown in Figures 9 and 10. At this range of speed, the system response of is mostly periodicone motion, as marked in bifurcation diagram with letter P and proved in Figure 11.
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(a)
(b)
(a)
(b)
(c)
It is observed that, under the variation of between 1000 and 20000 rpm, the characteristic of system in forward bifurcation shown in Figure 2(b) is similar to the one in backward bifurcation shown in Figure 3(b), which is verified by RMS and PPVs curves in Figures 9(b) and 10(b). For the sake of simplicity, both bifurcations are discussed in detail as one bifurcation. As illustrated in Figures 2(b) and 3(b), the system behaves as a periodicone motion response at 1000–1763 rpm, as illustrated in Figure 12. The system is in quasiperiodic response at 1763–2241 rpm and returns into periodicone motion again at 2214–5869 rpm. Consequently, the system behaves as quasiperiodic motion and remains in the range of 5869–17040 rpm, where the quasiperiodic motion is indicated by the closed orbit formed in Poincaré section, as shown in Figure 13. And jumps phenomena occur around 3577 rpm and 17040 rpm. In the range from 17135 rpm to 17899 rpm, the system becomes irregular and turns rapidly into chaotic motion, as verified by the disorder circles of phase plane diagram and the discrete points of Poincaré map in Figure 14. In the meantime, with the increasing of the speed, the displacement range of chaotic motion narrows gradually. Eventually, the system behaves as a quasiperiodic motion at 17899–2000 rpm, as illustrated in Figure 15. The results mentioned above reveal that the dynamic behavior of and is steady at lowspeed range. Therefore, it is concluded that the dynamic behavior transforms from linear to nonlinear response with the increases of .
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
From the corresponding PPVs and RMS curves, the conclusions of the relative displacement are obtained that there are two apparent bistable response regions in system at the range of 11979–12839 rpm and 17613–20000 rpm, respectively. The chaotic motion ranges of are indicated at 17040–18758 rpm as for forward case, but at 20000–18758 rpm, 17040–17613 rpm, and 11979–12457 rpm as for backward case, as shown in Figure 9(a), while the chaos response of is indicated at 17135–17899 rpm, as illustrated in Figure 9(b). The response of the system experiences a jump down phenomenon at 12744 rpm in the acceleration process and a jump up phenomenon at 11979 rpm and 17613 rpm in the deceleration process, as illustrated in Figure 10(a), while in system , jump up and down phenomena occur around 3577 rpm and 17040 rpm, as shown in Figure 10(b).
3.2. Effect of the Nonlinear Backlash on the TSHG Dynamic Response
According to the manufacturing accuracy level of gear pair, half of the backlash values of systems and are set to increase from to . For the sake of simplicity, the backward bifurcation diagrams of systems under different backlash values are investigated, as illustrated in Figures 3, 16, and 17, respectively.
(a)
(b)
(a)
(b)
By comparing the bifurcation diagrams of both systems and under different backlash values, it is found that the range of chaotic motion in the rotational speed of pinion enlarges; the critical speed goes frontward along with the increasing of backlash value. In the meantime, the characteristics of chaotic motion are increased in both systems, which increases the difficulty of predicting the vibration of the gear pairs and leads to more noise possibility.
The corresponding RMS and PPVs curves of both systems concerning the variation of under different backlash values are illustrated in Figures 9, 10, and 18–21. It is seen that the displacement amplitudes of the PPV and RMS of both systems and decrease along with the increase of the backlash, and the range of expands as the chaos range extends.
(a)
(b)
(a)
(b)
(a)
(b)
(a)
(b)
Considering the bifurcation diagrams, PPVs, and RMS curves of both systems, we conclude that, under the increases of backlash value, the characteristics of chaotic response could be enhanced which expand the chaotic motion range and enlarge the rotational speed range; as a result, this will lead to more possibility of noise vibration. Nevertheless, the backlash could also lower the amplitudes of RMS and PPVs and deescalate the vibration extension of the system. Consequently, a suitable value of backlash should be selected to meet the requirements of vibration amplitude and reduce the possibility of chaotic behavior extension [32–34].
3.3. Effect of the Torsional Stiffness on the TSHG Dynamic Response
The torsional stiffness of the intermediate shaft is one of the key parameters that affect the dynamic behavior of the transmission system. Therefore, it is essential to investigate the effect of the torsional stiffness of the intermediate shaft on the dynamic characteristics of the TSHG system. In this section, the backlash values of both systems and are set to ; the torsional stiffness is set to increase as , and , respectively.
Figures 22, 23, and 2 illustrate the corresponding forward bifurcation diagrams of the systems and with respect to the rotational speed of pinion under different torsional stiffness values. According to the comparison of the bifurcation diagrams under different torsional stiffness values, the substantial region of periodic, quasiperiodic, and chaotic responses including jump phenomena are observed. At highspeed range of 17000–19000 rpm, the system is stable and behaves as a periodic motion when . The dynamic response transits from periodic to quasiperiodic motion when and eventually turns into chaotic response as increases to . Therefore, the nonlinear system response becomes unstable and the chaotic behavior consequently expands as increases.
(a)
(b)
(a)
(b)
3.4. Effect of the Torque Fluctuation on the TSHG Dynamic Response
According to the BMW i8 electric motor characteristics [36], the ideal torque fluctuation is set to vary from to under the rotational speed range of 1000–20000 rpm. In this section, the backlash values of both systems and are set to , the torsional stiffness is set to , and the torque fluctuation is considered as a control parameter that varies along with the changing of rotational speed. From the corresponding backward bifurcation diagrams of both systems and illustrated in Figure 24, it is observed that the dynamic response and the displacement of both systems at the speed range of 1000–4000 rpm are stable when the torque fluctuation is . As the torque varies among , the corresponding displacement changes clearly at the speed range of 4000–13000 rpm. Then, the displacement is stable again under the torque variation among until the speed reaches 20000 rpm.
(a)
(b)
However, it is observed that chaotic motion for both systems exists at the speed range of 13000–20000 rpm when the torque fluctuates at low values between and . Quasiperiodic motions are observed at the speed range of 7111–17000 rpm when the torque fluctuates between , , , and as for the system and at the speed range of 6000–17000 rpm when the torque fluctuation range is around and