Research Article
Effect of clearance configuration on gear system dynamics
Advances in Mechanical Engineering 2018, Vol. 10(4) 1–9 Ó The Author(s) 2018 DOI: 10.1177/1687814018769540 journals.sagepub.com/home/ade
Yanwei Liu1, Yunxue Zhu1, Kegang Zhao2, Zhihao Liang2 and Huibin Lin2
Abstract Depending on the presence and position of a switch control element such as a synchronizer or sleeve in a multi-speed transmission, a gear system with gear backlash can be in one of three clearance configurations: no engaging clearance, engaging clearance at the input shaft, or engaging clearance at the output shaft. In this article, the influence of clearance configuration on gear system dynamics is investigated using the oscillating component of the dynamic transmission error as the dynamic response. The dynamic responses are studied over a wide frequency range by speed sweeping to examine the nonlinear behavior using numerical integration. Gear systems in all three clearance configurations show typical nonlinear behavior, such as multiple resonances, softening nonlinearity, jump phenomena, and chaotic motion. The presence of engaging clearance clearly affects the nonlinear characteristics, including resonance amplitudes, jump frequency, overlap width of primary resonance, and degree of chaos. Compared with the NC configuration, the resonance amplitudes for the CI and CO configurations decrease, while fluctuations increase. The resonance amplitudes and fluctuations for the CI configuration are both greater than those for the CO configuration, indicating that system vibration is greater when the clearance is far from the large inertia. Keywords Gear dynamics, clearance, dynamic response, dynamic transmission error
Date received: 21 February 2017; accepted: 7 March 2018 Handling editor: Luis Baeza
Introduction Gears are widely used as power and motion transmission devices in mechanical systems, and gear system dynamics remains a prominent concern because of issues of noise and durability. A considerable amount of research1–5 has shown that time-varying meshing stiffness and backlash cause vibration of the gear system and that this is the primary source of gear whine noise. The dynamic transmission error (DTE) of the gear pair is widely used as a parameter to represent the dynamic response. Numerical,6 analytical,7 and finite element8 methods, as well as bench tests,9 have been employed to explore the nonlinear dynamical behavior, such as multiple resonances, softening nonlinearity, and jumping and chaotic phenomena. Understanding of gear meshing motion has also improved, with account now being taken not only of
linear no-impact (NI) motions but also of nonlinear single-sided impact (SSI) and double-sided impact (DSI) motions. Parker and colleagues10,11 found that even for high-precision gears with a large work load, the tooth separation is the main source of excitation of nonlinear vibrations. With increasing demand for high speeds, heavy loads, light weight, quiet running, and reliability in vehicle transmissions, reductions in fatigue 1
School of Electromechanical Engineering, Guangdong University of Technology, Guangzhou, People’s Republic of China 2 School of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, People’s Republic of China Corresponding author: Kegang Zhao, School of Mechanical and Automotive Engineering, South China University of Technology, 381 Wushan Road, Tianhe District, Guangzhou 510641, People’s Republic of China. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) 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 damage and noise have become more and more important, necessitating more accurate evaluation of the dynamic tooth load and vibration response related to the nonlinear aspects of geared systems. Gill-Jeong12 examined whether a one-way clutch is effective for reducing the torsional vibration of a gear system, using a paired gear model and considering only rotational motion. Kahraman and colleagues13,14 investigated the dynamic stress factor (DF) and the DTE of unmodified and modified spur gear pairs, in an approach that bridged the gap between studies concerned with dynamics and those focusing on noise and the durability of the gear system. As suggested by Di Nicola et al.,15 the use of a multi-speed transmission instead of a single-speed transmission can lead to improved vehicle performance, together with an enhancement of the overall efficiency of the electric powertrain. Therefore, multi-speed transmissions are being applied not only to traditional engine-driven vehicles, but also to electrical vehicles.16–19 To shift between different gears, switch control engagement elements such as synchronizers or sleeves are essential for multi-speed transmissions. The synchronizer or sleeve can be arranged on either the input or the output shaft of the gear pair, with the engaging clearance accordingly being at the input or the output shaft. The clearance of the synchronizer or sleeve and the backlash of the gear pair together comprise a gear system with a multi-clearance configuration. The clearance configuration of the gear system changes depending on whether a synchronizer or sleeve is present and on its position. It is reasonable to assume that the clearance configuration will influence the gear system dynamics, and also the noise and durability of the gear system. Although there have been many investigations of gear dynamics over the years, there have been no previous reports of studies on the dynamics of a gear pair in which the clearance configuration has been taken into account. In addition, the vehicles are developing rapidly toward electrical drive, such as plug-in hybrid electric vehicles and pure electric vehicles. With the development of high-speed electric powertrains, the improvement on NVH characteristics of the transmission system are becoming research concerns. However, the published investigations of gear dynamics9,10,12–14 were carried out with the meshing frequency limited to about 4000 Hz, which might show some limitations on the study of electrical vehicle transmissions, with drive motor speeds up to 20,000 rev/min. This study focuses on the effect of clearance configuration on the system dynamics for the multi-clearance configuration gear system, in order to provide the theoretical basis for further study on the configuration layout and NVH characteristics of this kind of complex transmission system. The dynamic responses are studied
Advances in Mechanical Engineering over a wide frequency range to confirm and examine the nonlinear behavior, with the oscillating component of the DTE (ODTE) adopted as the main dynamic response for the purposes of comparison.
Dynamics model and response Physical model A multi-speed transmission is constructed using synchronizers or other switch control elements and gear pairs with different speed ratios. A typical two-axle four-speed transmission is shown in Figure 1(a). The sliding part of the synchronizer is driven by a fork (not shown) to move axially, thereby connecting or disconnecting the shaft with the different gear pairs. The synchronizer can be set on the input or the output shaft: as shown in Figure 1(a), the synchronizer of the 3rd and 4th gears is set on the input shaft, while the synchronizer of the 1st and 2nd gears is set on the output shaft. Figure 1(b) shows a schematic representation of the situation where the synchronizer is set on the input shaft, like the 3rd gear pair and its synchronizer in
Figure 1. The gear system (a) and schematic diagrams of different clearance configurations (b, c).
Liu et al.
3 those in the CO configuration are TD r1 k(t)fb (t) + c r1 u_ 1 r2 u_ 2 = (ID + I1 )€u1 r2 k(t)fb (t) + c r1 u_ 1 r2 u_ 2 k2 fb2 (t) c2 r2 u_ 2 rL u_ L = I2 €u2 rL k2 fb2 (t) + c2 r2 u_ 2 rL u_ L TL = IL €uL
ð2Þ and those in the NC configuration are TD r1 k(t)fb (t) + c r1 u_ 1 r2 u_ 2 = (ID + I1 )€u1 r2 k(t)fb (t) + c r1 u_ 1 r2 u_ 2 TL = (I2 + IL )€u2 Figure 2. Physical model of the gear system with different clearance configurations.
Figure 1(a), while Figure 1(c) corresponds to the situation where the synchronizer is arranged on the output shaft, like the 2nd gear pair and its synchronizer in Figure 1(a). Depending on the presence and position of the synchronizer in the transmission, a gear system with backlash can be in one of three clearance configurations: no engaging clearance (NC), engaging clearance at the input shaft (CI), and engaging clearance at the output shaft (CO). To examine the effect of clearance configuration on gear system dynamics, a gear system is constructed with two gears mounted on the input and output shafts, as shown in Figure 2. The gears are perfect involute spur gears with no modifications, and the torsional flexibility of both shafts is neglected. Gears 1 and 2 have base circles of radii r1 and r2 and mass moments of inertia of I1 and I2, respectively. The mass moments of inertia of the driver and load are ID and IL, respectively. The pair of gears is modeled using two disks coupled with nonlinear mesh stiffness and mesh damping; the mesh stiffness and damping coefficient of the gear pair are k(t) and c, respectively. The total backlash is 2b. The vibrations of the driver, load, and gears 1 and 2 about the nominal rigid body rotation are represented by uD, uL, u1, and u2, respectively. The stiffness, damping coefficient, and total clearance of the CI are k1, c1, and 2b1, respectively, and those of the CO are k2, c2, and 2b2, respectively.
System equations The drive torque is assumed to be constant and controllable. The two gears and output shaft are assumed to undergo rotational displacement only. The equations of motion of the system in the CI configuration are TD rD k1 fb1 (t) + c1 rD u_ D r1 u_ 1 = ID €uD r1 k1 fb1 (t) + c1 rD u_ D r1 u_ 1 k(t)fb (t) c r1 u_ 1 r2 u_ 2 = I1 €u1 r2 k(t)fb (t) + c r1 u_ 1 r2 u_ 2 TL = (I2 + IL )€uL
ð1Þ
ð3Þ
where TD is the drive torque and TL is the load torque. In this study, the mesh stiffness and backlash are taken to be nonlinear. The mesh stiffness k(t) is periodic at the gear meshing frequency, so it can be represented by the Fourier series, represented as follows9 k(t) = k0 +
R X
kr cos (2prfm t fr )
r=1
k0 = ICR ktp pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 cos½2pr(ICR 1) kr = pr ktp 1 cos½2pr(ICR 1) fr = sin½2pr(ICR 1)
ð4Þ
where k0 is the average mesh stiffness, ktp is the stiffness of an individual tooth pair in contact, kr and fr are respectively the rth Fourier coefficient and phase angle of k(t), fm is the meshing frequency, and ICR is the involute contact ratio. Here, R = 5. The damping coefficient c of the tooth mesh is calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k0 c = 2z 2 2 r1 =I1 + r2 =I2
ð5Þ
where z is the damping ratio. Gear backlash nonlinearity is modeled as a piecewise linear function, as shown in equation (6). The nonlinearities of the CI and CO are modeled similarly 8 < r1 u1 r2 u2 + b, r1 u1 r2 u2 \ b ð6Þ fb (t) = r1 u1 r2 u2 b, r1 u1 r2 u2 .b : 0, jr1 u1 r2 u2 j b
Response variable and calculation The DTE is defined as the relative displacement at the gear mesh interface DTE = r1 u1 r2 u2
ð7Þ
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Advances in Mechanical Engineering
Table 1. Gear pair parameters. Parameter
Value
Number of teeth Normal module Normal pressure angle Center distance Backlash b Face width Average mesh stiffness k0 Damping ratio z Mass Involute contact ratio
50 3.00 mm 20° 150 mm 200 mm 0.02 m 420 3 106 N/m 0.08 2.8 kg 1.77
The dynamic load carried by the gear teeth at any instant in time is directly related to the DTE, thus making the latter an important indicator for both gear whine and gear durability.2 Since the ODTE is the main source of noise and vibration in a gear system, it is taken as the main dynamic response for comparison.12 The ODTE at a specific constant speed is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X (DTEi + 1 DTEi )2 ODTE = t N i=1
Figure 3. Fitting results of time-varying meshing stiffness k(t).
5 3 104 steps after the start of the calculation, to avoid the unstable speed oscillation phase and ensure the reliability of the response. When data acquisition finishes after 5 3 104 steps, the ODTE at the meshing frequency is calculated according to equation (8). The ODTEs are calculated with 25 Hz frequency interval, and when the final frequency is reached, the numerical integration ends.
Results and discussion ð8Þ
where N is the total number of time steps (1 3 105 in this study). Table 1 shows the gear system parameters. The gear pair parameters are consistent with the gear pair used in previous experiments9 and analytical studies;10,12 thus, the responses presented in this study are compatible with the previously published results. The inertias of the driver and load are initially kept the same as those of the gears, and the load inertia is then changed to realize various load inertia ratios, where the load inertia ratio iL is defined as the ratio of load inertial to driver inertia: IL/ID. The time-varying meshing stiffness k(t) of the gear pair are calculated using equation (4) and the related parameters in Table 1. The calculation results at the meshing frequency of 2500 Hz are shown in Figure 3. The stiffnesses of the engaging clearances k1 and k2 are both 5 3 1010 N/m. In this study, the responses under different drive torques and load inertia ratios are obtained using direct time-domain numerical integration (with a fourth-order Runge–Kutta algorithm). Figure 4 shows the flowchart of the algorithm for numerical integration used in this study. The subroutine is carried out at each meshing frequency for DTE and ODTE calculations. The time step is 1 3 10–6 s for all conditions. In the subroutine, the DTE is calculated with each time step, according to equation (7). Data acquisition at each meshing frequency is carried out
Figure 5 shows the DTE response of the NC configuration at TD = 100 N m. The results exhibit multiple resonances, jump phenomena, and softening nonlinearity (overlap), which are in good agreement with those of earlier published experiments,9 as shown in Figure 6. Figures 7 and 8 show the DTE responses of the CI and CO configurations, respectively. It can be seen that there is little difference between the results in Figures 7 and 8, indicating that the position of the engaging clearance (at the input shaft or the output shaft) has little influence on the dynamic characteristics of the gear system with a load inertia ratio iL = 1. As can be seen from Figure 5, jumps appear around 800, 1300, and 2500 Hz, and a primary resonance is evident at a meshing frequency fm ’ fn. A clear softening nonlinearity (overlap) occurs as the peak bends to the left, and the overlap frequency range is about 900 Hz. After three jumps, chaotic motions can be observed within the frequency range of 3900–5600 Hz. From Figures 7 and 8, it can be seen that the CI and CO configurations also exhibit three jumps, with a narrower overlap frequency range of about 500 Hz, and chaotic motions are observed after 2900 Hz, with a clear increase in the degree of chaos compared with the NC configuration.
Effect of load inertia ratio on different clearance configurations Figure 9 shows the DTE response of the NC configuration for load inertia ratios iL = 1, 10, and 50 at a torque
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Figure 4. Flowchart of the algorithm for numerical integration.
Figure 5. DTE response of the configuration with no engaging clearance (NC).
of 100 N m. With increasing load inertia, the primary resonant frequency decreases to about 1900 Hz for iL = 10 and to about 1700 Hz for iL = 50. The overlap
Figure 6. Measured Arms forced response curves for the three unmodified gear pairs. Natural frequencies fn are 2740, 2900, and 3000 Hz for ICR = 1.37, 1.77, and 2.01, respectively.9
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Advances in Mechanical Engineering
Figure 7. DTE response of the configuration with engaging clearance at the input shaft (CI).
Figure 9. DTE response of the NC configuration for different load inertia ratios iL = 1, 10, and 50.
Figure 8. DTE response of the configuration with engaging clearance at the output shaft (CO).
Figure 10. DTE response of the CI configuration for different load inertia ratios iL = 1, 10, and 50.
frequency range narrows to about 600 Hz for iL = 10 and 50, with chaotic motions being observed after about 2400 Hz. Figure 10 shows the DTE response of the CI configuration for the same values of the load inertia ratio and torque. Again, with increasing load inertia, the primary resonant frequency decreases to about 1900 Hz for iL = 10 and to about 1800 Hz for iL = 50, although the overlap frequency range does not change very much. Chaotic motions are observed after about 2400 Hz for iL = 10 and after about 2000 Hz for iL = 50. Figure 11 shows the DTE response of the CO configuration for the same values of load inertia ratio and torque. The overlap frequency range changes with increasing load inertia. Chaotic motions are observed after about 2900 Hz for iL = 10 and after about 2000 Hz for iL = 50. From the above results, it can be seen that the system dynamics of the NC, CI, and CO configurations are all affected by changes in the load inertia ratio, although the degree of this effect clearly differs among the clearance configurations. Figures 12 and 13 shows comparison of the response results of the NC, CI, and CO configurations for iL = 10 and 50, respectively, and it can be clearly seen that the amplitude and degree of fluctuation are greater for the CI configuration. This
Figure 11. DTE response of the CO configuration for different load inertia ratios iL = 1, 10, and 50.
shows that vibration is greater when the clearance is at the input shaft.
Effect of drive torque on different clearance configurations Figure 14 shows the DTE response of the NC configuration for drive torques TD = 100, 200, and 300 N m at a load inertia ratio iL = 10. With increasing drive torque, the resonance amplitudes clearly increase. The primary resonant frequency does not change much, while the overlap frequency range narrows with
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Figure 12. DTE responses of different configurations with load inertia ratio iL = 10.
Figure 13. DTE responses of different configurations with load inertia ratio iL = 50.
Figure 14. DTE response of the NC configuration for different drive torques TD = 100, 200, and 300 N m at load inertia ratio iL = 10.
increasing drive torque: the width is about 600 Hz for TD = 100 N m, 300 Hz for TD = 200 N m, and 100 Hz for TD = 300 N m. The frequency at which the chaotic state is entered clearly decreases with increasing torque. Figure 15 shows the DTE response of the CI configuration for the same values of drive torque and load inertia ratio. With increasing drive torque, the increase in resonance amplitudes is more pronounced than in the NC configuration. The overlap frequency range narrows with increasing drive torque, but the
7
Figure 15. DTE response of the CI configuration for different drive torques TD = 100, 200, and 300 N m at load inertia ratio iL = 10.
Figure 16. DTE response of the CO configuration for different drive torques TD = 100, 200, and 300 N m at load inertia ratio iL = 10.
narrowness is much smaller compared with the NC configuration. The frequency at which the chaotic state is entered does not change much. Figure 16 shows the DTE response of CO configuration for the same values of drive torque and load inertia ratio. The primary resonant frequency does not change much for TD = 100 and 200 N m, but clearly increases for TD = 300 N m. The chaotic motions are much weaker compared with the CI configuration. The above results show that the resonance amplitudes for the NC, CI, and CO configurations all increase with increasing drive torque. Compared with the NC configuration, the maximum amplitudes for the CI and CO configurations are smaller, but the fluctuations are greater. The amplitudes and fluctuations of the resonances for the CI configuration are greater than those for the CO configuration, indicating that there is greater vibration when the clearance is at the input shaft.
Conclusion With the ODTE adopted as the main dynamic response, this study has examined the effect of the
8 clearance configuration on system dynamics for the multi-clearance configuration gear system, with various load inertia ratios and drive torques. The results show that the presence of engaging clearance, which exists in switch control engagement elements such as synchronizers or sleeves, has obvious impacts on the nonlinear characteristics of the gear system, affecting the resonance amplitudes, jump frequency, overlap width of the primary resonance, and the degree of chaos. In general, compared with the NC configuration, the resonance amplitudes for both the CI and CO configurations decrease, while the fluctuations increase. When the load inertia is larger than the drive inertia, there is a clear dependence of the gear system dynamics on the position of the engaging clearance. Under different torques, the resonance amplitudes and fluctuations for the CI configuration are all greater than those for the CO configuration, indicating that the system vibration is greater when the clearance is located far from the large inertia. Therefore, in engineering practice, the vibration and noise of the system could be reduced if any switch control element such as a synchronizer is positioned near the large inertia. The results provide the theoretical basis for further study on the configuration layout and NVH characteristics of this kind of complex transmission system. 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 work described in this paper was supported by the National Natural Science Foundation of China (Grant No. 51405087) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. 2014KQNCX062).
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Appendix 1 Notation 2b 2b1 2b2 c c1 c2 fb(t) fm fn iL ID, IL, I1, I2 k(t)
total backlash total clearance of the engaging clearance at the input shaft total clearance of the engaging clearance at the output shaft damping coefficient of a gear pair damping coefficient of the engaging clearance at the input shaft damping coefficient of the engaging clearance at the output shaft gear backlash nonlinearity model gear mesh frequency gear pair natural frequency load inertia ratio mass moments of inertia of driver, load, gear 1, and gear 2 mesh stiffness of a gear pair
k2 kr ktp N rD, rL, r1, r2 R TD, TL uD, uL, u1, u2 fr z
average mesh stiffness of a gear pair mesh stiffness of the engaging clearance at the input shaft mesh stiffness of the engaging clearance at the output shaft rth Fourier coefficient of k(t) mesh stiffness of an individual tooth pair in contact total number of time steps base radius of driver, load, gear 1, and gear 2 Fourier series torque of drive and load rotation angles of driver, load, gear 1, and gear 2 phase angle of k(t) damping ratio
