Efficient analytical method to obtain the responses of a gear model with stochastic load and stochastic friction

The friction, which is widely existed in practical, is seldom considered when modelling the gear system with stochastic load. Due to the variation of the temperature and lubrication condition, friction is a stochastic factor to a gear model. In this paper, a gear model with stochastic load, stochastic friction, and some other deterministic factors is considered. Due to the effects of stochastic factors (i.e., load and friction), the gear system faces more vibration and noise than the case with all deterministic factors. Thus, to analyse the variation of responses in the gear dynamical model, the corresponding dynamic equation needs to be solved. However, the statistic characteristics of dynamic responses are hard to obtain by numerical methods. Thus, an efficient analytical method is proposed, and then, an approximate analytical solution of the dynamic equation can be obtained in this paper. By the obtained solution, the vibration and noise of gear systems can be well investigated. Simulation results are provided to demonstrate the superior performance of the proposed method.


Introduction
Gear system which is one of the most widely adopted mechanical parts shows significant important to the development of mechanical theory [1]. Therefore, the research about dynamics and vibrations of gear systems attracts lots of research attention. The gear models under deterministic domain have been investigated in decades. However, stochastic load is one of the main sources of gear vibration and noise in practical [2]. The gear system dynamic modelling with stochastic load attracts more and more attention. In [3], a gear model under stochastic load with transmission error, backlash, and periodic gear mesh stiffness was introduced. Wen et al. [4] investigated a gear dynamic model under stochastic load with backlash, time-varying mesh stiffness (TVMS), and constant damping coefficient. However, besides the load, the internal factors (e.g., friction, backlash, etc.) of a gear system may also have random variation and affect the system's dynamic behaviour greatly [5]. For example, friction, as a main cause of vibration on gear system's transient state, is observed as stochastic in lots of scenarios [6]. Therefore, the stochastic internal factor should be considered when modelling gear systems. Under a gear system model, there are two kinds of forms, analytical expression and numerical solutions, to demonstrate the responses of the gear system. Therefore, some methods have been proposed to obtain the responses of a gear system [7]. For example, Runge Kutta-Monte Carlo (MC) method and statistical Newmark method are proposed to obtain the numerical solutions. Stochastic averaging method and path integration (PI) method are proposed to obtain the analytical expression [7]. Obtaining numerical solution is costly since a large number of samples are required for reasonable accuracy. By contrast, it would save much time if we could obtain the analytical solution [8]. Therefore, we focus on obtaining the analytical solution in this paper.  [9] derived the analytical solution to a gear system considering constant stiffness, constant damping coefficient, and backlash under the excitation of white noise. In [4], the analytical solution to a gear system considering constant damping coefficient, TVMS, and backlash under stochastic load was derived. However, only deterministic internal factors are considered in these works. A gear model with a stochastic internal factor under stochastic load cannot be solved by existing analytical methods.
In this paper, we consider the gear dynamic model with stochastic load, stochastic friction (i.e., the stochastic internal factor), and other deterministic internal factors (e.g., damping, TVMS, backlash, etc.). Then, a method is proposed to derive an approximate analytical solution of the corresponding dynamic equations. By the proposed method, the dynamic characteristics (e.g., vibration, noise, etc.) of gear systems can be well investigated. Compared with the MC method, our proposed method can achieve similar accuracy responses with much less time cost.
The remaining pars of this paper are organized as follows. The gear system with stochastic load and stochastic friction is modelled in Section 2. The method to obtain the responses of the gear model is proposed in Section 3. The simulation results are given in Section 4. Section 5 draws conclusions.

Gear dynamic model
In this work, a gear model considering TVMS, backlash, and friction is introduced and is shown in figure 1. In this model, both the load and the friction (i.e., the single stochastic internal factor) are stochastic. Therefore, the considered model is formulated as: R are the moment of inertia, external torque, arm length of friction, and base circle of gear   1, 2 i  , respectively, F represents the total force between the contact teeth, f F is the sliding friction [2]. Let x , x , and x denote the relative angular displacement, the relative angular velocity, and the relative angular acceleration, respectively. Thus, we have Figure 1. Model of the gear system Simplify equation (1) and equation (2), and then, a normalized equation is obtained and shown as: where c denotes the damping coefficient which is considered as a constant, k denotes the TVMS,  represents the friction coefficient, Lt  is a function which is caused by friction. The expression of Lt  can be obtained as [2]: relates to gear design parameters. Note that can be considered as constant.  is a summation of a deterministic part 0  and a random part 1 () t  . The external load () ft is modeled as a combination of a constant deterministic part 0 f , a periodical deterministic part 1 cos( ) ft  , and a random part () t  [4]. About the random part, it is generally set to a Gaussian white noise. Therefore, the expressions of the external load are given as follows.
where  is a constant frequency,  is also stochastic because of () ft as given in equation (6). To obtain the responses of the gear system, we need to derive the analytical solution of the stochastic differential equation (SDE), i.e., equation (4).

The proposed method to derive the analytical solution
The existing methods can obtain the analytical solution of the differential equation which  (4), a method is proposed in this paper. The basic idea of the proposed method contains the following two steps: 1) Obtain the tentative analytical solution. We first introduce a method to transform the SDE equation (4) into a form that can be solved by PI method. Then, we derive a tentative analytical solution using PI method. 2) Adjust results by adding a modification function. Due to the previous transformation, errors may be brought into the tentative analytical solution, and thus, a modification function is applied to adjust the tentative analytical solution. Supervised learning is used to obtain the modification function.

Obtaining the tentative analytical solution
By considering the deterministic part of the load, equation (9) is obtained.
is the deterministic part of the load. The solution of equation (9), which are x (considered as 1  ) and x (considered as 2  ), can be obtained by solving equation (9). Note  (3), 1  and 2  can be obtained according to the obtained x in equation (9).
After that, we get () L t  as expressed in equation (10) (4), we can obtain Based on this transformation, all parameters in equation (11) become deterministic except for load, and thus, it can be solved by PI method now.
For the case that ( ) 0 gx , the system equation is shown in equation (11). Under the excitation of () t  , an impulse function () x ht satisfies the following equation.
The general solution to the homogeneous equation (12) can be obtained as in equation (17).
The general solution to the homogeneous equation (19) is given as:

Updating the tentative analytical solution
Due to the transformation in Section 3.1, the obtained tentative analytical solution is not the accurate solution of equation (4). Therefore, we try to derive the modification function and then apply it to the tentative analytical solution to obtain the accurate solution of equation (4). Generally, the exact analytical solution of equation (4) is hard to derive. The numerical responses obtained by MC method are regarded as accurate results. Therefore, a modification function is required to demonstrate the rule of errors according to the samples obtained by MC method and the tentative analytical solution. A supervised learning algorithm is applied to obtain the modification function. Finally, we adjust the tentative analytical solution by the modification function and then obtain the final probability distributed function (PDF). The obtained final PDF can be regarded as the analytical solution (i.e., PDF) of the SDE equation (4).
Therefore, a modification to 1 Supervised learning is used to find ( ) t  . Supervised learning is used to find a mapping between a set of input samples and the corresponding output and this mapping is then applied to predict the outputs under other input data [10]. The samples obtained by the tentative analytical solution would be the input data and the samples obtained by MC method would be the corresponding output. We define the input samples and the corresponding output as