The parameters of gear pair are selected as the calculation object, and the contact stiffness of gear pair under Yang sun model and semi empirical formula is calculated. In Fig. 1 (a), the contact stiffness curves of three pairs of single teeth with different tooth width effects (independent and load) are shown. The numerical superposition effect of contact stiffness is carried out according to the series parallel stiffness relationship of single double teeth meshing area, and the corresponding time-varying contact stiffness curves are drawn (Fig. 1 (b)). Combined with the constant mode of the contact stiffness of the formula, it is consistent with the continuous horizontal line of the contact stiffness of each stage shown in the figure.
Taking the gear pair 1 as an example, the load ratio and load distribution curve calculated by the formula are shown in Fig. 2. Fig. 2 (a) shows that the load ratio curve in the double meshing area presents a constant slope straight-line rising and falling mode of y = B ± KX, and in the single meshing area it is expressed as a constant value curve mode of load ratio equal to 1. The formula shows that the load distribution is proportional to the load ratio and normal load form. Fig. 2 (b) shows that the load distribution increases with the increase of torque, and the load distribution and load ratio distribution form are consistent. However, the load increment is not equal at the same meshing point with the same torque increase.
The calculated load distribution in Fig. 2 (b) is substituted into the semi empirical formula model for contact stiffness calculation. The curves of contact stiffness along the tooth profile under different loads and geometric parameters are plotted in Fig. 3. The trend of the curves shows that the contact stiffness in the double meshing area presents a power exponent increasing or decreasing mode, while the contact stiffness in the single meshing area remains the same. The contact stiffness curve increases with the increase of load (Fig. 3 (a)), and also increases with the increase of tooth width (Fig. 3 (b)). However, the contact stiffness increases unevenly with the increase of equal load or tooth width, which further indicates that the contact stiffness calculated by semi empirical formula has nonlinear performance.
To further show the difference between Yang sun model and semi empirical formula, figure 4 draws the curve of time-varying contact stiffness curve under the effect of superposition of parallel stiffness of single and double meshing regions of actual wheel set. The material parameters and geometric parameters of the two models are the same, namely, the elastic modulus, Poisson ratio and tooth width are equal. The torque is lioonm in the semi empirical formula model. The comparison of the contact stiffness amplitude of the two models in Figure 4 shows that the calculated value of Yang sun model is larger than that of the semi empirical formula, among which the stiffness value in the single mesh area is the same level (1000000000), while the difference between the two meshing areas is one order of magnitude. The stiffness value of the torque increases three times even in the semi empirical formula in Fig. 3 (a) is still maintained at the series of 1000000000, which shows that the stiffness value calculated by Yang SIM model is larger than that of the semi empirical formula. At the same time, the results of local amplification of the stiffness curve of the double meshing region of the semi empirical formula in Fig. 4 show that the curve is convex (increase first and then decrease), and the curve variation feature is similar to the curve change characteristic of gear tooth stiffness. The comparison data of the two models show that, from the performance characteristics, the semi empirical formula is better than Yang sun model, while from the point of view of the calculation speed, Yang sun model is better than the semi empirical formula. In the future, the calculation of the time-varying meshing stiffness of gear pair can be calculated according to different calculation requirements.