applied sciences Article Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears Yangshou Xiong 1,2,*, Kang Huang 3,*, Fengwei Xu 3,†, Yong Yi 3,†, Meng Sang 3,† and Hua Zhai 4,† 1 School of Materials Science and Engineering, Hefei University of Technology, Hefei 230009, China 2 Anhui Zhongke Photoelectric Color Selection Machinery Co., Ltd., Hefei 231202, China 3 School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China; [email protected] (F.X.); [email protected] (Y.Y.); [email protected] (M.S.) 4 Institute of Industry & Equipment Technology, Hefei University of Technology, Hefei 230009, China; [email protected] * Correspondence: [email protected] (Y.X.); [email protected] (K.H.) † These authors contributed equally to this work. Received: 26 December 2018; Accepted: 7 March 2019; Published: 12 March 2019 Abstract: In light of ignoring the effect of backlash on mesh stiffness in existing gear dynamic theory, a precise profile equation was established based on the generating processing principle. An improved potential energy method was proposed to calculate the mesh stiffness. The calculation result showed that when compared with the case of ignoring backlash, the mesh stiffness with backlash had an obvious decrease in a mesh cycle and the rate of decline had a trend of decreasing first and then increasing, so a stiffness coefficient was introduced to observe the effect of backlash. The Fourier series expansion was employed to fit the mesh stiffness rather than time-varying mesh stiffness, and the stiffness coefficient was fitted with the same method. The time-varying mesh stiffness was presented in terms of the piecewise function. The single degree of freedom model was employed, and the fourth order Runge–Kutta method was utilized to investigate the effect of backlash on the nonlinear dynamic characteristics with reference to the time history chart, phase diagram, Poincare map, and Fast Fourier Transformation (FFT) spectrogram. The numerical results revealed that the gear system primarily performs a non-harmonic-single-periodic motion. The partially enlarged views indicate that the system also exhibits small-amplitude and low-frequency motion. For different cases of backlash, the low-frequency motion sometimes shows excellent periodicity and stability and sometimes shows chaos. It is of practical guiding significance to know the mechanisms of some unusual noises as well as the design and manufacture of gear backlash. Keywords: gear backlash; mesh stiffness; potential energy method; low-frequency; nonlinear dynamics 1. Introduction Gear dynamics is an important method to predict dynamic performance as well as to monitor the status of a gear system. The time-varying mesh stiffness caused by alternately changing the gear pairs in mesh plays a crucial role in dynamic responses. An effective time-varying mesh stiffness model is a basic condition to conduct a dynamic analysis. Thus, it is significant to recognize the mechanism of a gear system’s shock and vibration to investigate the influence of time-varying mesh stiffness on vibration responses, especially with regard to the mechanism of the noise, which is hard to identify. One consistently popular topic in gear dynamics is how to describe the change rule of time-varying mesh stiffness correctly and effectively. The ISO Standard 6336-1 [1] recommends a stiffness formula to calculate the maximum single tooth mesh stiffness for spur and helical gears according to experimental tests. In References [2–4], time-varying mesh stiffness was simplified as a rectangular Appl. Sci. 2019, 9, 1029; doi:10.3390/app9051029 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2404 2 of 13 Appl. Sci. 2019, 9, 1029 2 of 13 experimental tests. In References [2–4], time-varying mesh stiffness was simplified as a rectangular equation which had significant differences from the actual value. The authors in Reference [5] used a equationpolynomial which fitting had method significant to smooth differences the result from of thethe actualtime-varying value. Thecontact authors stiffness in Reference that was obtained [5] used aby polynomial the Hertz contact fitting algorithm. method to The smooth Fourier the series result was of used the to time-varying fit time-varying contact mesh stiffness stiffnessthat function was obtainedapproximatively by the in Hertz Reference contact [6]. algorithm. 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Influence of Backlash on Time-Varying Mesh Stiffness 2. Influence of Backlash on Time-Varying Mesh Stiffness 2.1. Mesh Stiffness Model with Backlash 2.1. Mesh Stiffness Model with Backlash Available mesh stiffness models do not consider the influence of backlash. Precision tooth profile Available mesh stiffness models do not consider the influence of backlash. Precision tooth profile equations play an important role in the analysis method to calculate the mesh stiffness. According to equations play an important role in the analysis method to calculate the mesh stiffness. According