- Original Article
- Open Access
Quasi-Static and Dynamic Behaviors of Helical Gear System with Manufacturing Errors
© The Author(s) 2018
- Received: 23 June 2017
- Accepted: 16 April 2018
- Published: 26 April 2018
Time-varying mesh stiffness (TVMS) and gear errors include short-term and long-term components are the two main internal dynamic excitations for gear transmission. The coupling relationship between the two factors is usually neglected in the traditional quasi-static and dynamic behaviors analysis of gear system. This paper investigates the influence of short-term and long-term components of manufacturing errors on quasi-static and dynamic behaviors of helical gear system considering the coupling relationship between TVMS and gear errors. The TVMS, loaded static transmission error (LSTE) and loaded composite mesh error (LCMS) are determined using an improved loaded tooth contact analysis (LTCA) model. Considering the structure of shaft, as well as the direction of power flow and bearing location, a precise generalized finite element dynamic model of helical gear system is developed, and the dynamic responses of the system are obtained by numerical integration method. The results suggest that lighter loading conditions result in smaller mesh stiffness and stronger vibration, and the corresponding resonance speeds of the system become lower. Long-term components of manufacturing errors lead to the appearance of sideband frequency components in frequency spectrum of dynamic responses. The sideband frequency components are predominant under light loading conditions. With the increase of output torque, the mesh frequency and its harmonics components tend to be enhanced relative to sideband frequency components. This study can provide effective reference for low noise design of gear transmission.
- Manufacturing error
- Mesh stiffness
- Transmission error
- Loaded composite mesh error
- Vibration acceleration
- Sideband frequency
Gear transmission systems are widely used in many industry applications. The prediction and control of gear vibration and noise are always important considerations in recent years. Due to manufacturing and heat treatment process, it is inevitable that gears contain manufacturing errors with different types and magnitudes. As the same as mesh stiffness, manufacturing errors are also one of the two main internal excitations that generate unwanted vibration in gear transmission. To accurately predict vibration of gear system, it is crucial to investigate the coupling relationships between mesh stiffness and manufacturing errors.
As the existence of manufacturing errors, the quasi-static engagement process of mating gear teeth will be not identical with the ideal one. The contact regions will come into contact earlier or later, and overloading or contact loss of mating gear teeth will occur in some engagement positions, which will affect mesh stiffness significantly in different loading conditions, and influence the dynamic behaviors of gear system.
Manufacturing errors include short-term and long-term components. The short-term components mainly refer to profile deviation, helix deviation, as well as pitch deviation and tooth surface modification. Conry and Seireg  proposed an evaluation method of load distribution and optimal modifications for cylindrical gears based on flexibility analysis method and elastic contact theory. Kubo and Kiyono  investigated the effects of different kinds of tooth form errors on vibration of spur and helical gear system, and found that convex tooth form error leads to minimum vibration, but concave and waving tooth form error result in relatively stronger vibration. Later, Kubo et al.  investigated the relationship between tooth contact pattern and transmission error of gears having errors, and developed the fast calculation method through observing the actual tooth contact pattern. Umezawa et al. [4, 5] developed a torsional dynamic model of spur gear system and analyzed the effects of pressure angle error, normal pitch error, and waved form error on vibration of gear system. It is found that the influences of gear errors on system vibration are significant. Vedmar and Andersson  developed a dynamic contact model to investigate the dynamic contact characteristics and vibration of spur and helical gears. Mattar and Velex [7, 8] presented an analytical model for calculating mesh stiffness of cylindrical gear using length of contact line, and developed a dynamic contact model to investigate the effects of shape deviations on quasi-static and dynamic behaviors of narrow-faced helical gears. Matsumura et al.  developed a torsional dynamic model of helical gear system, and studied the gear errors on system vibration under lighter loading condition. The results showed that the partial contact loss due to gear errors in light loading condition has significant effect on system vibration. Munro et al.  presented an approximate formula of transmission error considering corner contact due to manufacturing and assembly errors. Ogawa et al.  performed the theoretical and experimental investigations about the dynamic behaviors of a spur gear pair having helix deviation. The results showed that helix deviation will result in decreased mesh stiffness and lead to lower resonance speed. Wei et al.  employed the finite element method to analyze the effects of five types of flank deviation on load distribution of helical gears, and found the superposition property of the influences of individual flank deviation on load distribution. Fernández-del-Rincón et al. [13, 14] used a global finite element model and a partial finite element model to develop the TVMS calculation model of spur gears based on flexibility analysis method, and investigated the effects of tooth profile deviation and support flexibility on the dynamic behaviors of spur gear system. Wang et al.  employed the thin slice theory and potential energy method to develop the TVMS and contact stress calculation method for helical gears having tooth profile errors. Li  developed a finite element method programs to investigate the influences of manufacturing errors, gear misalignment, as well as assembly errors and gear modifications on TVMS of a spur gear pair. Lin and He  used the finite element method to determine the static transmission error of a spur gear pair with machining errors, assembly errors and modifications, and then established a bending-torsional-axial coupling dynamic model to calculate the dynamic transmission error.
Long-term components of manufacturing errors mainly include eccentricity and accumulative pitch error. Yu et al.  employed a dynamic model of cylindrical geared rotor system to investigate the dynamic coupling behavior of transverse and rotational motions of gears subjected to gear eccentricities. The results indicated that the dynamic coupling behavior will become apparent in the low speed range when the resonances are excited by TVMS or profile errors. Wang et al.  presented theoretical formulas of no loaded static transmission error and time-varying backlash due to gear eccentricity, and developed a calculation method of dynamic transmission error for spur gears in consideration of gear eccentricities based on LS-DYNA3D. Xiang and Gao  studied the coupled torsion-bending vibration of a gear-rotor- bearing system in consideration of TVMS, gear eccentricity and nonlinear bearing force. The results suggested that the eccentricity has more significant effects on system vibration when the rotational speed is relatively lower. Umezawa and Sato  investigated the effect of accumulative pitch error on vibration acceleration of a spur gear pair, and drew the influence chart related with speed and contact ratio when accumulative pitch error is combined with other errors. Fernández-del-Rincón et al.  studied the loaded transmission error, pressure angle, as well as meshing force and bearing force of spur gears having index and run out errors under several transmitted torques. The results indicated that index errors will influence vibration behavior of the system and result in high overloads. Index errors will bring higher amplitude at the rotation frequency of shaft but run out errors will lead to lower values. Handschuh et al. , Talbot et al.  and Inalpolat et al.  performed numerical simulations and experiments to investigate the effect of tooth spacing errors on the root stress, dynamic factors and dynamic transmission error of a spur gear pair, respectively. The results suggested that tooth spacing errors have a direct impact on the root stress and significantly alter the baseline dynamic response. Meanwhile, the frequency spectra of dynamic response are enriched due to amplitude and frequency modulation.
As aforementioned published works, some of them neglected the nonlinear relationship between TVMS and manufacturing errors, which may be not accurate enough for predicting the vibration of gear system, especially in light loading condition. Some models of TVMS and manufacturing errors are too complicated to be widely used, for instance, the contact finite element model. Also, few researches were done on the effect of accumulate pitch error on quasi-static and dynamic behaviors of gear system, and most of them focused on spur gear system.
This study presents an improved LTCA model based on sub-structure technique and elastic contact theory to determine TVMS, LSTE and LCMS of helical gears having manufacturing errors. This model provides sufficient precision and high computation efficiency. Considering the structure of shaft, as well as the direction of power flow and bearing location, a precise generalized finite element dynamic model of helical gear system is developed to obtain the dynamic responses. In order to find out how the manufacturing errors affect the loaded tooth contact characteristics and dynamic behaviors of helical gear system, quasi-static and dynamic behaviors analysis are performed in consideration of short-term and long-term components of gear errors (Additional file 1).
2.1 Improved LTCA Model
The deformation of mating gears consist of global and local contact deformation. The first one is linearly related to applied force, but the second one is nonlinearly related to applied force. A global finite element model and a partial finite element model are used to obtain the global flexibility matrix of potential contact points in the same engagement position based on the sub-structure method . Considering the nonlinear relationship between local contact deformation and applied force, the local contact deformation of interested contact point can be calculated using the analytical formula . The load distribution F and LSTE can be obtained using the iteration algorithm to solve the nonlinear matrix equation .
2.2 Loaded Composite Mesh Error
It can be observed that the LCMS is related to TVMS, distribution of gear errors and applied force.
Parameters of the driving and driven gear
Normal module (mm)
Normal pressure angle (°)
Helix angle (°)
Bottom clearance coefficient
Tooth width (mm)
4.1 Short-term Components of Gear Errors
4.1.1 Description of Gear Errors
The short-term components of gear errors, which include profile deviation, helix deviation and pitch deviation, are considered in this section. It is assumed that profile deviation are distributed along the tooth profile as a parabolic curve. As the inevitable shaft deformation and mounting errors can be regarded as helix deviation, it is defined that helix deviation is distributed along the gear width as a straight line. Neglecting the indexing errors, the pitch deviation alters positive and negative.
4.1.2 Quasi-Static Analysis
4.1.3 Dynamic Analysis
4.2 Long-term Components of Gear Grrors
4.2.1 Description of Gear Errors
4.2.2 Quasi-Static Analysis
4.2.3 Dynamic Analysis
Both short-term and long-term components of manufacturing errors have notable influence on TVMS, LSTE and LCMS of helical gears. Lighter loading conditions lead to smaller mesh stiffness and stronger vibration. The corresponding resonance speed of the system become lower.
Long-term components of manufacturing errors lead to the appearance of sideband frequency components in dynamic responses of the system. The sideband frequency components are predominant under lighter loading condition. The increase of output torque result in the increase of mesh frequency and its harmonics components.
SC was in charge of the whole trial; BY wrote the manuscript; GL and LYW assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Bing Yuan, born in 1987, is currently a PhD candidate at Northwestern Polytechnical University (NWPU), China. He received his bachelor degree from Central South University, China, in 2010. His research interests include gear dynamics, vibration and control. Tel: +86–15389359689; E-mail: email@example.com.
Shan Chang, born in 1965, is currently a professor and supervisor of PhD candidates and director of the Institute for Shaanxi Engineering Laboratory for Transmissions and Controls, Northwestern Polytechnical University (NWPU), China. He received his MS and PhD degrees from Harbin Institute of Technology, China. His research interests include gear modification, gear dynamic and load carrying capacity of gears. E-mail: firstname.lastname@example.org.
Geng Liu, born in 1961, is currently a professor and supervisor of PhD candidates and director of Shaanxi Engineering Laboratory for Transmissions and Controls, Northwestern Polytechnical University (NWPU), China. He received his MS degree from NWPU and PhD degree from Xi’an Jiao Tong University. His research interests include mechanical dynamic design, mechanical systems dynamics, simulation and virtual prototype design, tribology, contact mechanics and numerical methods. Tel: +86–13891999032; E-mail: email@example.com.
Li-Yan Wu, born in 1958, is currently a professor at Shaanxi Engineering Laboratory for Transmissions and Controls at Northwestern Polytechnical University (NWPU), China. He received his BS degree from NWPU. His research interests include mechanical design and mechanical reliability. E-mail: firstname.lastname@example.org.
The authors declare that they have no competing interests.
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Supported by Key Project of National Natural Science Foundation of China (Grant No. 51535009) and 111 Project (Grant No. B13044).
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- T F Conry, A Seireg. A mathematical programming technique for the evaluation of load distribution and optimal modifications for gear systems. Journal of Engineering for Industry, 1973, 95(4): 1115–1122.View ArticleGoogle Scholar
- A Kubo, S Kiyono. Vibrational excitation of cylindrical involute gears due to tooth form error. Bulletin of the JSME, 1980, 23(183), 1536–1543.View ArticleGoogle Scholar
- A Kubo, T Kuboki, T Nonaka. Estimation of transmission error of cylindrical involute gears by tooth contact pattern. JSME International Journal, Series III, 1991, 34(2): 252–259.Google Scholar
- K Umezawa, T Sato, K Kohno. Influence of gear errors on rotational vibration of power transmission spur gears (1st Report, pressure angle error and normal pitch error). Bulletin of the JSME, 1984, 27(225): 569–575.View ArticleGoogle Scholar
- K Umezawa, T Sato. Influence of gear errors on rotational vibration of power transmission spur gear (3rd Report, accumulative pitch error). Bulletin of the JSME, 1985, 28(246): 3018–3024.View ArticleGoogle Scholar
- L Vedmar. A Andersson. A method to determine dynamic loads on spur gear teeth and on bearings. Journal of Sound and Vibration, 2003, 267(5): 1065–1084.View ArticleGoogle Scholar
- M Maatar, P Velex. An analytical expression for the time-varying contact length in perfect cylindrical gears: some possible applications in gear dynamics. Journal of Mechanical Design, 1996, 118(4): 586–589.View ArticleGoogle Scholar
- P Velex, M Maatar. A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour. Journal of Sound and Vibration, 1996, 191(5): 629–660.View ArticleGoogle Scholar
- S Matsumura, K Umezawa, H Houjoh. Rotational vibration of a helical gear pair having tooth surface deviation during transmission of light load. JSME International Journal, Series C, 1996, 39(3): 614–620.Google Scholar
- R G Munro, L Morrish, D Palmer. Gear transmission error outside the normal path of contact due to corner and top contact. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 1999, 213(4): 389–400.Google Scholar
- Y Ogawa, S Matsumura, H Houjoh. Rotational vibration of a spur gear pair considering tooth helix deviation (Development of simulator and verification). JSME International Journal, Series C, 2000, 43(2): 423–431.View ArticleGoogle Scholar
- J Wei, W Sun, L C Wang. Effects of flank deviation on load distributions for helical gear. Journal of Mechanical Science and Technology, 2011, 25(7): 1781–1789.View ArticleGoogle Scholar
- A Fernández, F Viadero, M Iglesias, et al. A model for the study of meshing stiffness in spur gear transmissions. Mechanism and Machine Theory, 2013, 61(61): 30–58.View ArticleGoogle Scholar
- A Fernández, M Iglesias, A de-Juan, et al. Gear transmission dynamic: Effects of tooth profile deviations and support flexibility. Applied Acoustics, 2014, 77(3): 138–149.View ArticleGoogle Scholar
- Q B Wang, Y M Zhang. A model for analyzing stiffness and stress in a helical gear pair with tooth profile errors. Journal of Vibration and Control, 2015, 23(2): 272–289.View ArticleGoogle Scholar
- S L Li. Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears. Mechanism and Machine Theory, 2015, 83(83): 125–136.View ArticleGoogle Scholar
- T J Lin, Z Y He. Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications. Mechanical Systems and Signal Processing, 2017, 91: 167–182.View ArticleGoogle Scholar
- W N Yu, C K Mechefske, M Timusk. The dynamic coupling behaviour of a cylindrical geared rotor system subjected to gear eccentricities. Mechanism and Machine Theory, 2017, 107: 105–122.View ArticleGoogle Scholar
- G J Wang, L Chen, S D Zou. Research on the dynamic transmission error of a spur gear pair with eccentricities by finite element method. Mechanism and Machine Theory, 2017, 109: 1–13.View ArticleGoogle Scholar
- L Xiang, N Gao. Coupled torsion-bending dynamic analysis of gear-rotor-bearing system with eccentricity fluctuation. Applied Mathematical Modelling, 2017, 50: 569–584.MathSciNetView ArticleGoogle Scholar
- K Umezawa, T Sato. Influence of gear error on rotational vibration of power transmission spur gear (3rd report accumulative pitch error). Bulletin of JSME, 1985, 28(246): 3018–3024.View ArticleGoogle Scholar
- A Fernández-del-Rincón, M Iglesias, A De-Juan, et al. Gear transmission dynamics: Effects of index and run out errors. Applied Acoustics, 2016, 108(1–2): 63–83.View ArticleGoogle Scholar
- M J Handschuh, A Kahraman, M R Milliren. Impact of tooth spacing errors on the root stresses of spur gear pairs. Journal of Mechanical Design, 2014, 136(6): 061010.View ArticleGoogle Scholar
- D Talbot, A Sum, A Kahraman. Impact of tooth indexing errors on dynamic factors of spur gears: experiments and model simulations. Journal of Mechanical Design, 2016, 138(9): 093302.View ArticleGoogle Scholar
- M Inalpolat, M Handschuh, A Kahraman. Influence of indexing errors on dynamic response of spur gear pairs. Mechanical Systems and Signal Processing, 2015, 60–61: 391–405.View ArticleGoogle Scholar
- L H Chang, G Liu, L Y Wu. A robust model for determining the mesh stiffness of cylindrical gears. Mechanism and Machine Theory, 2015, 87: 93–114.View ArticleGoogle Scholar
- P Sainsot, P Velex. Contribution of gear body to tooth deflections-a new bidimensional analytical formula. Journal of Mechanical Design, 2004, 126(4): 748–752.View ArticleGoogle Scholar
- L H Chang, Z X He, G Liu. Dynamic modeling of parallel shaft gear transmissions using finite element method. Journal of Vibration and Shock, 2016, 35(20): 47–53. (in Chinese)Google Scholar
- Z H Hu, J Y Tang, J Zhong, et al. Effects of tooth profile modification on dynamic responses of a high speed gear-rotor-bearing system. Mechanical Systems and Signal Processing, 2016, 76–77: 294–318.View ArticleGoogle Scholar