# Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears
## 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; 3; 4], time-varying mesh stiffness was simplified as a rectangularequation which had significant differences from the actual value. The authors in Reference [5] used a polynomial fitting method to smooth the result of the time-varying contact stiffness that was obtained by the Hertz contact algorithm. The Fourier series was used to fit time-varying mesh stiffness function approximatively in Reference [6]. [PERSON] and [PERSON] proposed an equation to calculate the single tooth bending stiffness for the addendum-modified gear and the curve-fitted coefficients took the number of teeth as a variable [7]. In Reference [8], the mesh stiffness between an engaged gear pair was regarded as consisting of local Hertzian stiffness and tooth bending stiffness. [PERSON] and [PERSON] revised [PERSON]'s calculation formula and proposed an analytical method to calculate the mesh stiffness by considering the load condition and the tooth profile modification in References [9; 10]. [PERSON] investigated the differences between the rectangular stiffness and its approximate form on the gear nonlinear dynamic characteristics in Reference [11].  
According to the available literature, the main methods to calculate mesh stiffness are Finite Element Method (FEM) and the potential energy method. These two methods have advantages and disadvantages. The potential energy method was proposed by [PERSON] [12] and has been improved several times [13; 14; 15; 16]. It is widely used in the failure analysis of a gear system [17; 18; 19; 20; 21]. The FEM, which can reflect the real contact status, is often applied to figure out engineering problems [22; 23; 24]. Moreover, the FEM is a common method to verify the validity of the potential energy method while both the FEM and potential energy method always ignore the influence of backlash on mesh stiffness.  
The backlash is very useful to avoid the jam phenomenon in conditions of elastic deformation and thermal expansion, i.e., adaptive backlash is one of the necessary conditions to ensure the normal operation of gear. One way to achieve backlash is to reduce tooth thickness. The reduction in tooth thickness will cause a decrease in the stiffness, further influencing the dynamic performance. In current gear dynamics, the effect of backlash is limited to the definition of the nonlinear backlash function [25; 26; 27; 28; 29; 30].  
Thus, this work will focus on the influence of backlash on time-varying mesh stiffness, carry out nonlinear dynamic analysis, and investigate the dynamic responses for various values of backlash. This is expected to establish the indirect relationship between backlash and dynamic performance as well as direct the design and control of the gear's backlash.