Coupled vibrations of a drive system during automatic transmission

Advances in Vibration Control of Structures and Machinery - Research Article Coupled vibrations of a drive system during automatic transmission Advances in Mechanical Engineering 219, Vol. 11(3) 1 11 Ó The Author(s) 219 DOI: 1.1177/16878141983358 journals.sagepub.com/home/ade Chengyun Su 1,2, Shuhan Wang 1,2, Yanfang Liu 1,3, Peng Dong 1,3 and Xiangyang Xu 1,2,3 Abstract Coupled vibrations are difficult to overcome in automatic transmission with mixed gear trains. In this study, a mathematical model for the bending torsional axial oscillation vibrations of mixed gear trains with parallel-axis gear pairs, a clutch, planetary gear sets and a shaft rotor was developed using the concentrated mass method. The dynamic vibration response and transmission error of the meshing gears in different gears were analysed. The response sensitivity of the vibration parameters was obtained using the design coupling function of the peak peak dynamic transmission error. The degree of coupling of each part with system vibrations was analysed and it was found that the stiffness and damping of gears influenced the coupled vibrations. Finally, the top seven degrees of coupling were obtained, the first two contributions were taken as the target to improve, the thickness of the P1 gear ring was increased by 5 mm and the bushing was added between the P2 planetary and sun gear to increase the stiffness and damping of gears. It could be validated that the vibration performance improved significantly from 3% to 6%. This article presents an analytical method of theoretical and practical value, which can provide a strong basis for reducing the vibrations and noise in automatic transmission. Keywords Automatic transmission, mixed gear trains, bending torsional axial oscillation, coupled vibrations, response sensitivity Date received: 1 June 218; accepted: 1 February 219 Handling Editor: Ali Kazemy Introduction The vibrations and noise occurring during automatic transmission (AT) affect the driving comfort of a vehicle. Rapid developments occurring in the AT technology not only satisfy the demands for a vehicle s dynamic performance but also ensure driving comfort. 1 Vehicle comfort includes not only driving comfort, 2 but also the vehicle s vibrations and noise levels. 3 The connection structure of AT is complex because the transmission system is made up of mixed gear trains with parallel-axis gear pairs, a clutch, planetary gear sets and a shaft rotor. 4 Owing to the relationship between structural coupling and gear meshing frequency, not just the gear meshing but also the entire mixed gear coupling system should be taken into account. Thus far, no comprehensive theoretical methods have been put forth to solve the problem of coupled vibrations and noise of the compound gear train in AT. To improve the vibrations and noise performance of gears, many scholars studied the vibration characteristics of spur gears or planetary gears. J Wei 5 studied the influence of dynamic driving error and vibration 1 School of Transportation Science and Engineering, Beihang University, Beijing, China 2 National Engineering Research Center for Passenger Car Auto- Transmissions, Weifang, China 3 Beijing Key Laboratory for High-efficient Power Transmission and System Control of New Energy Resource Vehicle, Beijing, China Corresponding author: Shuhan Wang, School of Transportation Science and Engineering, Beihang University, Beijing 1191, China. Email: wsh@buaa.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4. License (http://www.creativecommons.org/licenses/by/4./) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).

2 Advances in Mechanical Engineering stability of helical gears. G Liu and RG Parker 6 studied the dynamic behaviour of gear systems under different working conditions. MR Kang and A Kahraman 7 studied the dynamic response of double-helix gear pairs based on experimental and theoretical analyses. Jauregui and Gonzalez 8 developed a single degree-offreedom (DOF) model to study the axial vibrations caused by the manufacturing error of the double-helix gear. A Kahraman 9 analysed the load distribution characteristics of planetary gear transmission. J Lin and RG Parker 1 analysed the inherent characteristics of free vibrations of planetary gear systems. However, these studies focus only on the vibrations or coupled vibrations of gears, shafts and bearings, and do not study the vibrations of compound gear trains in AT. Previously, studies were conducted to evaluate the vibrations and noise performance of transmission systems. LX Chang et al. 11 use the nonlinear lateral torsion coupling mode of a vehicle transmission system to study the response sensitivity and evaluate nonlinear vibrations. HW Lee et al. 12 carried out a study on the vibration characteristics of vehicle transmission. Anderson et al. 13 established a test rig for the twin planetary gear mechanism and measured the vibration amplitude of the system using an accelerometer. However, there is not much information available on the coupling. Sondkar and Kahraman 14 developed a three-dimensional (3D) dynamic model on a dual planetary gear mechanism consisting of a gear mesh and bearing supports; it is concluded that different values of stagger are shown to excite different types of modes, resulting in different dynamic response curves. G Kouroussis et al. 15 conducted dynamic vibration analysis of vehicle and powertrain based on AT. Although these results involve coupled vibrations of AT, there are still three points that should be considered: 1. A large number of studies are available on the vibrations and noise of a single pair gear, while only a few studies are available on vibrations occurring in compound gear sets during transmission; 2. There are more studies on bending torsional vibrations and less on bending torsional axial oscillation vibrations in mixed gear trains; 3. There are more studies on the vibration characteristics of gears and less on the response sensitivity of the vibration parameters. AT involves multiple meshing gears for certain gears 16 and a dynamic response is generated due to transmission speed and torque; 17 furthermore, coupled vibrations are caused by multiple excitation sources and are transmitted by the structure. 18 The major contribution of this work lies in studying the coupled vibration principle of mixed gear trains for AT; a method is proposed to control the vibrations and noise during transmission. First, an eight-speed AT 19 was considered as the research object. Based on the mathematical theoretical analysis, 2 a mathematical model of the bending torsional axial oscillation vibrations of mixed gear trains containing parallel-axis gear pairs, a clutch, planetary gear sets and a shaft rotor was designed. Second, the dynamic vibration response and transmission error of the meshing gears in different gears were analysed along with the degree of coupling of each part to the system vibrations and the response sensitivity of vibration parameters was evaluated. Third, techniques to reduce dynamic transmission error and vibration dynamic response were proposed by optimising gear stiffness and damping; these techniques were validated experimentally. This study is expected to form a theoretical and practical integrated design method for noise/vibration/harshness (NVH) control in AT. Model for coupled vibrations Coupled vibrations are difficult to overcome in AT with mixed gear trains. It was found that there occurred a series noise in the fourth driving gear (D4) during eightspeed AT; the noise sources were gear 4 and gear 7 (G4 and G7, respectively; Figure 1). At first, we only modified the profile modification of gear 4 and gear 7(G47 for short), but did not improve the vibrations. Later, we found that the noise of G47 improved significantly after changing the microscopic parameters of other meshed gears, and at the same time it was found that vibration coupling occurred between meshed gears. Figure 1 shows the transmission layout of the proposed AT. The numbers represent shaft numbers. If there is a gear attached to a shaft, the number of the gear is the same as the number of the shaft (e.g. G2 stands for gear 2 and G5 stands for gear 5). There are five multi-plate shifting elements, three planetary gear sets and three parallel-axis gear pairs connected by two parallel axes. The first planetary gear set (P1) on the first axis splits the input power to the pinion gear of the three parallelaxis gear pairs (G5 and G8, G4 and G7, and G6 and G1), which provide three paths for power flow from the first axis to the second axis. The second (P2) and third planetary gear sets (P3) are on the second axis. The final gears G2 and G11 are arranged between the second axis and the output axis. Gear G2 is arranged on the second axis and gear G11 is connected to the differential housing. Power is transferred to the output shaft, and at the same time the differential balances the different speeds of the two wheels. The five shifting elements link certain gear elements together or to the transmission housing. Brake B1 and clutches C1, C3

Su et al. 3 Table 1. Shift logic of the eight-speed automatic transmission. Gear Engaged clutches B1 C1 C2 C3 C4 1st h h 2nd h h 3rd h h 4th h h 5th h h 6th h h 7th h h 8th h h represents the shift elements closed. h represents the shift elements opened. Figure 1. Layout of the eight-speed automatic transmission. and C4 are on the first axis, while clutch C2 is on the second axis. The shift elements are either closed or opened to achieve a specific certain gear ratio. The shift logic is given in Table 1. A novel structure is suggested for the eight-speed AT; the shift logic is simple shifting (open one clutch and close another clutch) and the gear ratio is perfect for the vehicle speed. Power is input from the left side of the planetary carrier through different clutches and brakes are used as the controls, which can realise different gears. Each gear has five shift elements: three closed and two opened; this results in a specific ratio of transmission. From Table 1, it can be seen that C2, C3 and C4 are closed and B1 and C1 are opened in D4 gear. Figure 1 shows the gear G4 and P1 ring gear is fixed connection as one component, the gear G7 and P2 sun gear is fixed connection as one component, the gear G4 and gear G8 is connected by clutch C3 closed. Different gear and planetary sets act as the fixed connection, thereby forming mixed gears of parallel-axis gear pairs and planetary gear sets. These form coupling points with interconnected elements in the power transfer process. Figure 2 shows the connection of different meshed gears in the D4 gear. Figure 2. Coupling components of D4 gear. w represents the gear pair meshing connection; represents the fixed connection; represents the clutch connection; } represents the spline connection. In this section, we analysed the route of power flow in D4 gears and used the method of substructure decomposition, according to the form of the conjugate gears, to determine the excitation source of the transmission gear system. The vibration model of bent torsional axial oscillation for helical gears Because of the axial dynamic meshing force in helical gears, there are not only bent vibration, torsional vibration and axial vibration, but also oscillation vibration.

4 Advances in Mechanical Engineering Figure 4. Power flow route in a mixed gear system in D4. Figure 3. The vibration model of bent torsional axial oscillation for helical gears. So it is necessary to establish the model for a coupled gear system. In order to simplify the model, the friction force of meshing gear and torsional elastic deformation of the transmission shaft are not considered, and the transverse bending of the transmission shaft is simulated with stiffness and damping. The vibration model of bent torsional axial oscillation for helical gears is shown in Figure 3. From Figure 3, an 8-DOF system is established in the 3D space, and the generalised displacement array fdg can be expressed as fdg= y p, z p, u pz, u px, y g, z g, u gz, u gx ð1þ where y i (i = p, g) is the bending direction of the vibrations of the gear; z i (i = p, g) is the axial vibration of the gear; u ix (i = p, g) is the oscillation of the gear around the central axis and u iz (i = p, g) is the torsional vibration of the gear. In addition, p is driving gear, g is the driven gear and b is the helix angle of gear. Displacement array of the bending torsional axial oscillation vibrations In D4 gears, the clutches C2, C3 and C4 are closed; four pairs of external meshing gears, G58, G47, G61 and G211, are included along with three planetary gears, namely, P1, P2 and P3. Each excitation source has four directional bending torsional axial oscillation vibrations. Furthermore, each excitation source is connected by the clutch and spline, which causes coupled vibrations. The power flow route of D4 is shown in Figure 4. Therefore, for D4 gears, the generalised displacement array of bending torsional axial oscillation vibrations is expressed as T fdg= d 58 d 47 d 61 d 211 d p1 d p2 d p3 d c2 d c3 d c4 ð2þ T d ij = yi z i u iz u ix y j z j u jz u jx ði = 5, 4, 6, 2; j = 8, 7, 1, 11Þ T = zsi u siz z ri u riz y pi u piz u pix u hiz d pi ði = 1, 2, 3Þ fd ci g= fu iiz u ioz g T ði = 2, 3, 4Þ where parallel-axis gear G i (i = 5, 8, 4, 7, 6, 1, 2, 11) has 4 DOFs bending, torsional, axial and oscillation. Sun gear S i (i = 1, 2, 3) has 2 DOFs torsional and axial. Ring gear R i (i = 1, 2, 3) has 2 DOFs torsional and axial. Planet gear P i (i = 1, 2, 3) has 3 DOFs bending, torsional and oscillation. Carrier H i (i = 1, 2, 3) has 1 DOF torsional. Clutch C i (i = 1, 2, 3) has 1 DOF torsional. In addition, y i is the bending direction of the vibration of the gear, z i is the axial vibration of the gear, u iz is the torsional vibration of the gear and u ix is the oscillation of the gear around the central axis. The coupled vibrations of mixed gear trains can be defined as ½mŠ d + ½Š c d _ + ½kŠfdg= fpg ð3þ where [m] is the mass matrix defined as

Su et al. 5 2 ½mŠ= 6 4 m p m p I p 1 2 I 2 p m g m g I g 1 2 I 2 g 3 7 5 ð4þ Bending vibration. The bending vibration equation of G4 is shown below m G4 y G4 + c y _y G4 + cos bc m _y G4 + R G4 _u G4 _y G7 + R G7 _u G7 + k y y G4 + cos bk m (y G4 + R G4 u G4 y G7 + R G7 u G7 ) = cos b(c m _e y + k m e y ) ð8þ [c] is the damping matrix defined as 2 c py + cos bc m sin btgbc m cos bc m R p sin btgbc m R pp ½Š= c cos bc m 6 sin btgbc m 4 cos bc m R g sin btgbc m R gp sin bc m + c pz sin bc m R pp sin bc m sin bc m R gp cos bc m R p sin btgbc m R p cos bc m R 2 p sin btgbc m R pp R p cos bc m R p sin btgbc m R p cos bc m R g R p sin btgbc m R gp R p c upz cos bc m sin btgbc m cos bc m R p sin btgbc m R pp cos bc m + c gy sin btgbc m cos bc m R g sin btgbc m R gp sin bc m sin bc m R pp c gy sin bc m sin bc m R gp cos bc m R g sin btgbc m R g cos bc m R p R g sin btgbc m R pp R g cos bc m R g sin btgbc m R g cos bc m R 2 g sin btgbc m R gp R g c ugy 3 7 5 ð5þ [k] is the stiffness matrix defined as 2 k py + cos bk m sin btgbk m cos bk m R p sin btgbk m R pp ½Š= k cos bk m 6 sin btgbk m 4 cos bk m R g sin btgbk m R gp sin bk m + k pz sin bk m R pp sin bk m sin bk m R gp cos bk m R p sin btgbk m R p cos bk m R 2 p sin btgbk m R pp R p cos bk m R p sin btgbk m R p cos bk m R g R p sin btgbk m R gp R p k upz cos bk m sin btgbk m cos bk m R p sin btgbk m R pp cos bk m + k gy sin btgbk m cos bk m R g sin btgbk m R gp sin bck m sin bk m R pp k gy sin bk m sin bk m R gp cos bk m R g sin btgbk m R g cos bk m R p R g sin btgbk m R pp R g cos bk m R g sin btgbk m R g cos bk m R 2 g sin btgbk m R gp R g k ugy 3 7 5 ð6þ fpg is the loading matrix defined as 8 >< fpg= >: cos b(c m _e y + k m e y ) sin b(c m _e z + k m e z ) T p + R p cos b(c m _e y + k m e y ) sin br pp (c m _e z + k m e z ) cos b(c m _e y + k m e y ) cos b(c m _e y + k m e y ) T g + R g cos b(c m _e y + k m e y ) sin br gp (c m _e z + k m e z ) 9 >= >; ð7þ According to these equations, the vibration of every part is coupled to that of the others. Changing the structural parameters of any part will affect the parts that are coupled. Assuming that damping is not taken into account, the vibration displacement fgcan d be decreased after increasing its stiffness k in different directions. Principle of coupled vibrations Coupled vibration refers to the phenomenon in which the input and output of two or more components interact with each other and transfer energy from one side to the other through interaction. Depending on the deformation direction, vibrations can be divided into bending vibration, torsional vibration, axial vibration and oscillation vibration. The bending vibration equations of other gears are similar. Torsional vibration. The torsional vibration equation of G4 is shown below I G4 u G4z + R G4 cos bc m ðr r1 _u r1 + R p1 _u p1 + R c3 _u 3iz + _y 4 + R G4 _u 4 _y 7 + R G7 _u 7 + _y s2 + R s2 _u s2 Þ + R G4 cos bk m ðr r1 u r1 + R p1 u p1 + R c3 u 3iz + y 4 + R G4 u 4 y 7 + R G7 u 7 + y s2 + R s2 u s2 Þ = T G4 + R G4 cos b(c m _e y + k m e y ) ð9þ The torsional vibration equations of other gears are similar. Axial vibration. The axial vibration equation of G4 is shown below m G4 z G4 + c z _z G4 + sinbc m ½_z G4 tgbð_y G4 + R G4 _u G4z Þ _z G7 + tgbð_y G7 R G7 _u G7z ÞŠ + k z z G4 + sin bk m ½z G4 tgbðy G4 + R G4 u G4z Þ z G7 + tgbðy G7 R G7 u G7z ÞŠ = sinb(c m _e z + k m e z ) ð1þ

6 Advances in Mechanical Engineering The axial vibration equations of other gears are similar. fd G4 g= y 4 + R G4 u 4 y 7 + R G7 u 7 + R r1 u r1 + R p1 u p1 + R c3 u 3iz + y s2 + R s2 u s2 ð14þ Oscillation vibration. The oscillation vibration equation of G4 is shown below 1 2 I G4 2 u G4x + c u x_u G4x + R G4 sin b fc m ½_z G4 tgbð_y G4 + R G4 _u G4z Þ _z G7 + tgb ð_y G7 R G7 _u G7z ÞŠg + k u xu G4x + R G4 sinbfk m ½z G4 tgbðy G4 + R G4 u G4z Þ z G7 + tgbðy G7 R G7 u G7z ÞŠg = sin b(c m _e z + k m e z )R G4 ð11þ The oscillation vibration equations of other gears are similar. Equation (14) shows that the vibration displacement of G4 is positively correlated with the vibration displacement of other gears. If the vibration displacement of other structures is reduced, the vibration of G4 can be reduced by reducing the vibration coupling. Vibration analyses of gear trains As a professional multi-body dynamic simulation tool for transmission systems, Romax can fully consider the bending torsional axial oscillation vibrations. At the same time, the coupled vibrations between different excitation sources can be simulated by the microstructure and dynamic response analysis. Torsional vibration of the clutch. When the clutch is incorporated into the gear drive system for dynamic vibration analysis, only the direction of torsional vibration is considered. Suppose that u i (i = 1, 2) is the torque angle of the plate between the clutch inner and outer hubs; because of the large axial stiffness of the clutch, it is difficult to generate axial vibration and bending vibration. Hence, the torsional vibration equation for clutch C3 is as follows Connection of meshing gear in D4 The connections between different meshing gears of D4 are shown in Figure 2. G58, G47, R1 and R2 are connected by one clutch, while G61 and S1 are connected by a second clutch. The rest of the connections are fixed. I c3 u c3 + R c3 cos bc c R r1 _u r1 + R p1 _u p1 + R c3 _u 3iz + _y 4 + R G4 _u 4 _y 7 + R G7 _u 7 + _y s2 + R s2 _u s2 ð12þ + R c3 cos bk c R r1 u r1 + R p1 u p1 + R c3 u 3iz + y 4 + R G4 u 4 y 7 + R G7 u 7 + y s2 + R s2 u s2 = Tc3 The torsional vibration equations of other clutches are similar. Here u i is the displacement of gear s torsional vibration; _u i is the velocity of gear s torsional vibration; u i is the acceleration of gear s torsional vibration; I is the moment of inertia of the gear; R is the base circle radius of the gear; e is the transmission error of meshing gear; T is the torque of external load; k m is the meshing stiffness of the meshing gear; c m is the meshing damping of the meshing gear; k i (i = x, y, z, u x, u y, u z ) is the support stiffness of a component; c i (i = x, y, z, u x, u y, u z ) is the support damping of a component. According to equations (3) and (9), if the coupling effect between structures is not considered, the vibration displacement in the torsional direction of G4 is only related to conjugate gears fd G4 g= _y 4 + R G4 _u 4 _y 7 + R G7 _u 7 ð13þ If the coupling effect between structures is considered, the vibration displacement in the torsional direction of G4 is Romax model of AT A multi-body dynamic model is built using Romax, in which the gear, bearing and spline are parameterised and the shaft and case are imported by the finite element method. This model takes into account the meshing characteristics of gears and other parts and the supporting characteristics between them. The model is shown in Figure 5. Parameter design In order to ensure the reliability of the obtained results, the micro-modification parameters of five other pairs of meshing gears were also varied; the variation was set at 1 mm (see Table 2 for details). Working condition 1 yields the basic data to calculate the coupling degree. The helix slope deviation of the gear (fhb deviation of condition l) is adjusted by 1 mm in working conditions 2 6. Figure 6 shows a schematic diagram of fhb.

Su et al. 7 Figure 6. Schematic diagram of fhb. Figure 5. Multi-body dynamics model constructed by Romax. For convenience, the transmission error is expressed as the deviation of mesh displacement on the mesh line Table 2. Different working conditions of active gears. TE s = u 1 rb 1 +(u 2 Du 2 ) rb 2 ð17þ Condition Gears Change in fhb (mm) Condition 1 None Condition 2 G58 1 Condition 3 G61 1 Condition 4 G211 1 Condition 5 PG1 1 Condition 6 PG2 1 Condition 7 PG3 1 Vibration analysis of all the gears Transmission error is a parameter used to describe the tolerance of gear transmission. In the process of gear working, the difference position between the actual position and the calculated value is the transmission error. Transmission error is expressed as the deviation of rotation angle u 2 = u 1 rb 1 rb 2 TE s = Du 2 Z 2 Z 1 ð15þ ð16þ The dynamic transmission error of G4 under working condition 1 was calculated as the basic data, after which the parameters of other gears (working conditions 2 6) were changed and the dynamic transmission error of G4 was recalculated (see Table 3 for details). Coupling degree design It can be seen from Table 3 that the dynamic transmission error of G47 is different after the parameters of other gears (working conditions 2 6) were changed. Therefore, the coupling degree of G47 can be defined as the difference in the rate of dynamic transmission error of other meshed gears relative to the transmission error in working condition 1 as shown in the following equation d = TE dyn2 TE dyn1 TE dyn2 3 1% ð18þ where d is the coupling degree of an excitation source with other excitation sources, TE dyn2 is the maximum dynamic transmission error of other excitation sources after the vibration state of a given excitation source is changed, and TE dyn1 is the maximum dynamic Table 3. Dynamic transmission error of G47 under different working conditions. Condition 1 Condition 2 (G58) Condition 3 (G61) Condition 4 (G211) Condition 5 (P1) Condition 6 (P2) Condition 7 (P3) Dynamic 12.88 12.65 12.79 12.86 11.6 11.76 12.77 transmission error Difference.23.9.2 1.82 1.12.11

8 Advances in Mechanical Engineering Table 4. The coupling degrees of other meshing gears with G47. Condition 1 Condition 2 (G58) Condition 3 (G61) Condition 4 (G211) Condition 5 (P1) Condition 6 (P2) Condition 7 (P3) Coupling degree. 5.98 3.73.93 14.13 8.7.85 Figure 7. Increase in the thickness of the P1 ring. Table 5. G47 vibration acceleration with an increase in P1 ring gear thickness. Before Thickness of ring P1 (mm) 5 1 G47 vibration (m/s 2 ) 2.6 2.32 Optimisation Figure 8. Increase in the stiffness of the radial support. increasing its stiffness k. The following two methods are mainly used to reduce the vibration and noise of G47, and the two methods are verified by simulations and experimental testing. transmission error of other excitation sources before the vibration state of an excitation source is changed. The coupling degree of other meshing gears with G47 under different working conditions is obtained using the equation for coupler design. See Table 4 for specific data. It can be seen from Table 4 that the coupling degree of other meshing gears with G47 is in the order of dp1. dp2. dg58. dg61. dg211. dp8 ð19þ As shown in Table 4, G47 exhibits a large coupling degree with other gears because G47 is directly connected to P1 and P2, resulting in a larger coupling degree of the above-mentioned meshing gear with G47. It has been proved that the coupled vibrations of meshing gears are related to structural transmission. Methods of reducing coupled vibrations It has been proved that the coupled vibrations of G47 gear can be reduced by improving the vibration of P1 and P2, and the vibration can be decreased after Simulation verification Method 1: damping and stiffness of the structure. In the transmission system, the thickness of the gear ring of P1 was 5 mm. To improve the vibration and noise level of G47 by Romax simulations, the thickness of the gear ring was increased to 1 mm (Figure 7). As shown in Table 5, as vibrations decreased with an increase in part thickness, the vibration coupling degree of G47 decreased and the vibration acceleration amplitude of G47 decreased from 2.6 to 2.32 m/s 2. Method 2: increasing the support stiffness. The P2 sun gear was supported by planet gears. To improve the vibration and noise level at G47, a bush was added between the P2 sun gear and the P2 planetary carrier. The vibration acceleration amplitude of G47 was simulated and analysed, as shown in Figure 8. As shown in Table 6, after the bush was added between the P2 sun gear and the P2 planetary carrier, the radial support stiffness of the P2 sun gear increased and its vibrations decreased. The vibration coupling degree of P2 sun gear with G47 was reduced, which

Su et al. 9 Table 6. G47 vibration acceleration with a bush was added between the P2 sun gear and the P2 planetary carrier. Before Optimisation P2 sun gear support No bush Bush G47 vibration (m/s 2 ) 2.6 2.7 finally reduced the vibration acceleration of G47 from 2.6 to2.7 m/s 2. Test verification Figure 9 illustrates the vibration test setup of the transmission in a vehicle. The vibration accelerometer was fixed on gearbox housing near the gear G47. The microphone was placed in a seat near the driver s ear. And at the same time, a speed sensor was fixed to monitor the driving speed of wheel. An accelerometer and a microphone were connected to the test equipment, and the test equipment was connected to a computer, as shown in Figure 1. The noise level of the transmission was tested before and after optimisation on the base band modem (BBM) test equipment from Mueller Company, Germany. The key parameters of the equipment are as follows: Number of channels: 16; Maximum sampling frequency: 12.4 khz/channel; Maximum analysis bandwidth: 4 khz; Analogue-to-digital conversion: Sigma-Delta type, 24-bit precision; Phase accuracy: \.2@1 khz. The gear box is driven by an internal combustion engine (ICE), and the wheel will move forward. The speed of ICE is controlled from 75 to 35 r/min in D4 gear, and the vibration data are collected from the accelerometer and microphone, which are time-domain signals. The corresponding order of noise pressure level of each gear is obtained by Fourier transform. The two modes of the vibration coupling reduction described in the previous section were verified experimentally. Method 1: damping and stiffness of the structure. Figure 11 shows the noise data of G47 before and after changing the P1 ring gear thickness (the thickness of P1 gear ring changed from 5 to 1 mm). The red dots represent the noise before improvement and the blue dots represent the noise after improvement. Figure 9. Vibration test setup of transmission in a vehicle. The vibrations of G47 were decreased by 4 8 db, and the noise level was improved by 3% 6% after increasing the P1 ring gear thickness. Method 2: increasing the support stiffness. In the case of vibration coupling between the P2 sun gear and gear G47, the vibration energy of the P2 sun gear is diverted by adding a bearing bush between the P2 sun gear and the P2 carrier. The location of structural changes is shown in Figure 12. As shown in Figure 13, after adding a bearing bush support between the P2 sun gear and the P2 planetary carrier, the noise level of G47, which was caused by a dynamic drive error, was decreased by 2 1 db and the noise level was reduced by 3% 6% after adding the bush. Conclusion This study analysed the coupled vibration principle of AT to obtain an effective method for controlling vibrations and noise. The following conclusions were drawn from the obtained results: 1. A model for the bending torsional axial oscillation vibrations of mixed gear trains containing parallel-axis gear pairs, a clutch, planetary gear sets and a shaft rotor was established and the dynamic response of the gear pairs was studied. 2. The dynamic characteristics and dynamic transmission errors of the gears were analysed and the vibration coupling degree of the system was evaluated, and the top two degrees of coupling were chosen as the target to improve.

1 Advances in Mechanical Engineering Figure 11. Comparison of G47 noise level data. Figure 12. Structure of P2 sun gear. Figure 1. Vibration test setup of transmission in a vehicle. 3. The vibrations reduced and the noise could be improved by increasing the thickness of the P1 gear ring by 5 mm and adding the bushing between the P2 planetary gear and sun gear to increase the stiffness and damping of gears. 4. The optimised results were validated experimentally and it was found that the noise could be improved by 3% 6%. Figure 13. Comparison of G47 noise level data.

Su et al. 11 An integrated method was established for analysing the coupled vibrations and dynamic response of AT; this model has particular significance for dynamic vibration response analysis and can be used to effectively reduce vibrations and vehicle noise. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors acknowledge the financial support provided by the National Science Foundation of China (Grant No. 517512) and The National Project to Strengthen Industrial Foundation Development of China (Grant No. TC15B5C-23). ORCID id Chengyun Su https://orcid.org/-1-683-9272 References 1. Lee DH, Moon KH and Lee WY. Characteristics of transmission error and vibration of broken tooth contact. J Mech Sci Technol 216; 3: 5547 5553. 2. Yakupov RR, Mustafin TN, Nalimov VN, et al. Analysis of transmission error depending on compressor working conditions. Proc IMechE, Part E: J Process Mechanical Engineering 214; 229: 122 129. 3. Litvin FL, Fuentes A, Zanzi C, et al. Design, generation, and stress analysis of two versions of geometry of facegear drives. Mech Mach Theory 22; 37: 1179 1211. 4. Xu XY, Dong P, Liu YF, et al. Progress in automotive transmission technology. Automot Innov 218; 1: 187 21. 5. Wei J, Gao P, Hu XL, et al. Effects of dynamic transmission errors and vibration stability in helical gears. JMech Sci Technol 214; 28: 2253 2262. 6. Liu G and Parker RG. Impact of tooth friction and its bending effect on gear dynamics. J Sound Vib 29; 32: 139 163. 7. Kang MR and Kahraman A. An experimental and theoretical study of the dynamic behavior of double-helical gear sets. J Sound Vib 215; 35: 11 29. 8. Jauregui I and Gonzalez O. Modeling axial vibrations in herringbone gears. In: Proceedings of ASME design engineering technical conference (Paper no. DETC99/VIB- 819), Las Vegas, Nevada, 12 15 September 1999. New York: American Society of Mechanical Engineers. 9. Kahraman A. Load sharing characteristics of planetary transmissions. Mech Mach Theory 1994; 29: 1151 1165. 1. Lin J and Parker RG. Analytical characterization of the unique properties of planetary gear free vibration. J Vib Acoust 1999; 121: 316 321. 11. Chang LX, Huang Y and Liu H. Response sensitivity and the assessment of nonlinear vibration using a nonlinear lateral torsional coupling model of vehicle transmission system. J Vib Acoust 215; 137: 3113. 12. Lee HW, Park SH, Park MW, et al. Vibrational characteristics of automotive transmission. Int J Automot Technol 29; 1: 459 467. 13. Anderson N, Nightingale L and Wagner DA. Design and test of a propfan gear system. J Propul Power 1989; 5: 95 12. 14. Sondkar P and Kahraman A. A dynamic model of a double-helical planetary gear set. Mech Mach Theory 213; 7: 157 174. 15. Kouroussis G, Dehombreux P and Verlinden O. Vehicle and powertrain dynamics analysis with an automatic gearbox. Mech Mach Theory 215; 83: 19 124. 16. Tang XL, Zhang JW, Yu HS, et al. Torsional vibration and acoustic noise analysis of a compound planetary power-split hybrid electric vehicle. Int J Electr Hybrid Veh 213; 5: 18 122. 17. Zhang LN, Wang Y, Wu K, et al. Dynamic modeling and vibration characteristics of a two-stage closed-form planetary gear train. Mech Mach Theory 216; 97: 12 28. 18. Liu YF, Lai JB, Dong P, et al. Dynamic analysis of helical planetary gear sets under combined force and moment loading. Shock Vib 217; 217: 463524. 19. Dong P, Liu YF, Tenberge P, et al. Design and analysis of a novel multi-speed automatic transmission with four degrees-of-freedom. Mech Mach Theory 217; 18: 83 96. 2. Vinayak H, Singh R and Padmanabhan C. Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears. J Sound Vib 1995; 185: 1 32.