Mesh stiffness calculation and vibration analysis of the spur gear pair with tooth crack, considering the misalignment between the base and root circles

An improved variable cross‐section cantilever beam model for evaluating the time‐varying mesh stiffness (TVMS) of the perfect gear tooth is developed in which the tooth number of driving gear is less than 42 and that of driven is more than 42. The TVMS obtained by the proposed method is compared with the result without considering the misalignment between the base circle and gear root. Four types of root crack models and changes in TVMS of 13‐crack levels are presented. The fault vibration characteristic of a single‐stage spur gear reducer with root crack is analyzed and the correctness is qualitatively verified by the vibration signals of an experimental gearbox with crack or missing failure. The results presented in this paper are of great significance for a deep understanding of the possible causes of vibration and noise of gears and provide a theoretical foundation for the fault diagnosis of the gearbox.

Tooth crack, as one of the common gear faults, is easy to cause other faults, such as tooth being broken, missing, and so on.
In addition, the existence of tooth crack mainly affects the TVMS and leads to the change of vibration features of the whole gear system. As a result, it needs first to determine the mesh stiffness for prediction or evaluation of the gear dynamic behavior. At present, the potential energy principle proposed by Yang and Lin 6 in 1987 and the improvements based on this method 7,8 are the most frequently used analytical methods to solve the TVMS of healthy or unhealthy gears, where the gear tooth is modeled as a cantilever nonuniform beam. The vibrations of beams have been discussed by many scholars. [9][10][11][12] In earlier studies, [13][14][15] the gear tooth is simplified as a variable cross-section cantilever beam starting from the base circle. In fact, for a real gear tooth, the base circle and the root circle are not exactly coincident. For standard involute spur gears, the base circle and the root circle are equal when the number of teeth is 42; otherwise, the base circle may be bigger or smaller than the root circle. The gear tooth is regarded as a variable cross-section cantilever beam starting from the tooth circle by Liang et al., 16 where the transition curve is expressed as a straight line if the base circle is bigger than the root circle, while the whole tooth profile is assumed to be an involute profile when the gear base circle is smaller than the root circle. It should be noted that the effect of fillet-foundation deflection on the stiffness was not analyzed in his study. Wan et al. 17 put forward a method to improve the TVMS, considering the misalignment between the base and root circles. The multi-DOF gear dynamic model with tooth crack is simulated and verified by an experiment in which the number of teeth of both the pinion and gear is greater than 42, and only the case that the crack depth is above the central line of the tooth is considered. Lei et al. 18 modeled the transition curve between base circle and root circle like an arc when the tooth number is less than 42, which makes the shape of the tooth profile closer to the real shape of the gear; the scheme in which the base circle is smaller than the root circle has not been investigated.
Ma et al. 19 proposed an improved TVMS model where the misalignment of the gear base circle, root circle, and the accurate transition curve is included and a parabolic curve is adopted to simulate the crack propagation, which lacks the influence of crack on the gear system. Additionally, the influence of root crack on the vibration response of the spur gear system has been discussed by many researchers. The dynamic simulation of a 6-DOF gear model with root crack was investigated, in which the crack on gear body deflection was taken into account. 20 The effects of different crack sizes on the change in dynamic response and the natural frequencies related to the gear TVMS were studied by Mohammed and Rantatalo. 15 Wu et al. 21 reported the effect of tooth crack on the vibration response of a one-stage gearbox with spur gears.
However, the gear dynamic models mentioned above are linear timevarying models where the clearance nonlinearity is ignored. Yang et al. 22 worked out the TVMS, taking advantage of the method introduced by Chen and Shao, 23 and analyzed the effect of tooth crack on 3-DOF spur gear with consideration of tooth backlash and bearing clearance nonlinearity. However, there is no detailed analysis of a broken fault on the condition that the crack depth reaches a maximum value.
To overcome the shortcomings of the methods and research mentioned above, this paper takes the single-stage gear reducer as a subject. The gear tooth is assumed to be a cantilever beam with variable cross section starting from the root circle, where the whole tooth profile is regarded as an involute profile when the base circle is less than the root circle, and the transition curve between the root circle and the base circle is considered as an arc when the base circle is more than the root circle, as demonstrated in Figure 1A,B, respectively. The paper is organized as follows. In Section 2,  (1) where U b , U s , U a represent, respectively, the bending, shear, and axial compressive energies stored in a meshing gear tooth, which can be determined as d , shear modulus G, cross-sectional area A x , and area moment of inertia I x can be written as where L is the tooth width.
For convenience, angular displacement is introduced to the calculation. If the angle α is regarded as an independent variable, it can be known from Figure 2 F I G U R E 2 Cantilever beam model of gear tooth when the base circle is equal to root circle The Hertzian contact stiffness k h and fillet-foundation stiffness k f can be obtained by, 24,25 k Significantly, both Young's modulus E and Poisson's ratio v are mental nature properties. The Hertzian contact stiffness is dependent only on tooth width L. Thus, k h is a constant for normal gears during the meshing. Moreover, fillet-foundation stiffness is irrelevant to the position relationship between the base circle and the root circle, and the specific reason is that Equation (19) is always true whatever the position between the two circles.

| The base circle is bigger than the root circle
If the number of teeth is less than 42, that is to say, the base circle is bigger than the root circle, the tooth profile between the root circle and the base circle is described as an arc, as shown in Figure 3. O x z ( , ) 1 0 0 and ρ are the center and radius of the transition curve, respectively, which satisfy the following Under the above assumptions, the bending stiffness, shear stiffness, and axial compressive stiffness when where The calculations for bending, shear, and axial compressive stiffness are still Equations (14)-(16) when x 0 < < d. The total bending, shear, and axial compressive stiffness from root circle to addendum circle can be derived from Equations (14)-(16) to (21)- (23). The algorithm is defined as Method B.

| The base circle is less than the root circle
The cantilever beam model of gear tooth starts from the root circle when the base circle is less than the root circle, which is plotted in Figure 4. The whole gear tooth profile follows the F I G U R E 3 Cantilever beam model of gear tooth when the base circle is bigger than the root circle involute, and the bending, shear, and axial compressive stiffness The integral with respect to x turns into the integral about α in where α 3 and α 4 satisfy The way to calculate the TVMS when the base circle is less than the root circle is defined as Method C.

| Overall TVMS
On the one hand, the total effective mesh stiffness for single-tooth meshing duration can be derived as ( ) ( ) k = 1 where k b1 , k s1 , k a1 , k f1 denote the bending stiffness, shear stiffness, axial compressive stiffness, and fillet-foundation stiffness of driving gear, respectively, and k′ b1 , k′ s1 , k′ a1 , k′ f1 are those of driven gear.
For the double-tooth-pair meshing duration, there are two pairs of gears meshing at the same time. The total effective TVMS can be expressed as where i = 1 for the first pair and i = 2 for the second pair of meshing teeth.

| Comparison of different methods for solving TVMS
To verify the performance of Methods B and C, three standard involute spur gear pairs are chosen and their parameters are listed in  Figure 5, it could be found that the result of Method A is always bigger than that of Method B. Figure 6 shows  between Methods A and C. For gear pair 2, the length of the cantilever beam in Method A is larger than that in Method C; that is, extra deformation energy between the base circle and the root circle is taken into account, which will lead to the lower TVMS of the gear tooth obtained.
The theoretical results are basically consistent with those in Figure 6.
If Methods B and C are used to calculate the TVMS of the pinion and gear, respectively, the results are compared with those procured by Method A, as shown in Figure 7. As can be seen from this figure, the mesh stiffness both in the single-tooth engagement region and in the double-tooth engagement region calculated by Methods B and C are less than those computed by using Method A.

| TVMS of cracked gears
The 13-crack levels corresponding to different crack depths or crack angles are presented in Table 2. In Figure 12, the detailed effects of crack depth and crack angle on TVMS are illustrated in which Figure 12B,D are the local enlargement of Figure 12A,C, respectively. From Figure 12, it could be found that only the TVMS of 1.5 mesh periods may be affected in our model during one revolution of the driving gear, where one mesh period is defined as an angular displacement of driving gear experiencing a double-tooth-pair meshing duration and a single-tooth-pair meshing duration. In Figure 12A The TVMS of the gear tooth is presented in Figure 12C,D, where The vertical motion equations, that is, vibrations in the y direction of driving and driven gears are To eliminate the influence of torsional displacement of gear system, the relative angular displacement y R θ R θ y y e = + + − −

| Vibration response of experimental signals
To qualitatively examine the correctness and reliability of theoretical results, a text rig for drivetrain dynamics simulator is used in this section, where the gear transmission system is composed of a single-stage reduction gearbox. The gears with root cracks or missing faults are shown in Figure 16.  Table 3. The impulses fluctuation caused by the fault is not obvious because of the existence of disturbances during the experiment, as shown in Figure 17A-C. In the frequency domain, Figure 17D-F, one can clearly find the amplitudes of sidebands of gears with cracks or missing faults that are bigger than those of perfect, starting in the resonance region (1440 and 2880 Hz). The vibration trend of simulated signals basically coincides with that of the experiment.

| CONCLUSIONS
In this paper, a modified analytical algorithm is proposed to solve the TVMS of gears by considering the misalignment be- (4) The experiment of single-stage reduction gearbox with defects was designed to analyze the failure features, which agrees qualitatively with the theoretical results.