Dynamic features of planetary gear set with tooth plastic inclination deformation due to tooth root crack

Abstract

Gear tooth root crack, as one of the popular gear tooth failures, is always caused by the dynamic load or excessive load applied to the tooth. It will devastate the working performance of the gear system, by problems such as vibration and noise, or even lead to a broken tooth, which will stop the normal working process of the gear system. It has attracted wide attention from researchers. However, the previous studies focused their concentration only on the mesh stiffness reduction due to tooth root crack, while the tooth plastic inclination due to tooth bending damages like gear tooth root crack is seldom considered. In this paper, a tooth plastic inclination model for spur gear with tooth root crack is developed by regarding the cracked tooth as a cantilever beam. It influences not only the displacement excitation but also the mesh stiffness and load-sharing factor among tooth pairs in mesh. The simulation results obtained by incorporating the tooth plastic inclination deformation model together with the tooth root crack model into a 21-Degree-of-Freedom planetary gear dynamic model indicate that the tooth plastic inclination has a significant effect on the performance of the gear system rather than the mesh stiffness reduction due to tooth root crack.

Appendix

The plastic inclination deformation of a cracked tooth is schematically illustrated in Fig. 1. The solid tooth profile indicates no plastic inclination deformation and the dashed tooth profile exhibits a plastic inclination deformation. Normally, the tooth pair meshes at point E with respect to the angle α ₁₀ when there is no plastic inclination deformation. When a plastic inclination deformation θ _p around the origin o of the local coordinate system xoy happens, point E′ on the undeformed tooth profile corresponding to the angle α ₁ will move to E″ on the LOA where the tooth pair meshes at. Here, the LOA is along line EF which is tangent to the base circle with radius R _b . The angle between line OF and the tooth shape center line OG is represented by α ₁₀ which becomes α ₁ when LOA is through point E′. The symbol α _or is the central angle of the arc between the tangent point of the tooth root fillet circle and the tooth profile and the intersection point of line OOr and the dedendum circle. The point M is the intersection point of involute curve and the base circle. α ₂ indicates the angle between lines OM and OG. The goal of this appendix is to derive the distance of the mesh points E and E″ along LOA, namely δ _EE″.

According to the geometric relationship shown in Fig. 1, the coordinates of the points Or, A, B, D and o in the general coordinate system XOY can be derived, which are illustrated as follows.

The coordinates of point Or, X _Or and Y _Or , are

$$\begin{aligned} &{X_{Or} = (R_{f} + r)\cos \angle O_{r}OM} \end{aligned}$$

(11a)

$$\begin{aligned} &{Y_{Or} = (R_{f} + r)\sin \angle O_{r}OM} \end{aligned}$$

(11b)

where

$$ \begin{cases} \angle O_{r}OM = \pi /2 - \arccos \frac{r}{R_{f} + r}, \\ \quad \mathrm{when}\ R_{f} \le R_{b}\ \mbox{and}\ r \le (R_{b}^{2} - R_{f}^{2} )/(2R_{f}) \\ \angle O_{r}OM = \frac{\pi}{2} - \arcsin \frac{R_{b}}{R_{f} + r} \\ \phantom{\angle O_{r}OM = }- (\tan (\frac{\pi}{2} - \arcsin \frac{R_{b}}{R_{f} + r} ) - r/R_{b} ), \\ \quad \mbox{for others} \end{cases} $$

(12)

where the symbol r is the radius of the tooth root fillet. R _f is the radius of the dedendum circle.

Similarly, the coordinates (X _i ,Y _i ) _i=A,B,D,o of the points A, B, D and o can be obtained based on the geometric relationship shown in Fig. 1, which are not shown here. Points D and A are symmetric about tooth center line OG. And the origin point of the local coordinate system xoy coincides with the point o which is the middle point of line section BD.

And the rotational angle of the local coordinate system relative to the general coordinate system, namely the acute angle between the x axis and the X axis in local and general coordinate system, respectively, can be calculated as,

$$ \theta_{0} = \arctan \frac{X_{B} - X_{D}}{Y_{D} - Y_{B}} $$

(13)

With the general coordinates of point o and θ ₀ in Eq. (13), the coordinates of all the points can be switched between the general coordinate system XOY and the local coordinate system xoy.

The coordinates (X _E′,Y _E′) of the point E′ on the involute tooth profile can be calculated as

$$\begin{aligned} &{ \begin{aligned}[b] X_{E'} &= R_{b}\sqrt{1 + (\alpha_{1} + \alpha_{2})^{2}} \cos \bigl(\alpha_{1} + \alpha_{2} \\ &\quad - \arctan (\alpha_{1} + \alpha_{2})\bigr) \end{aligned}} \end{aligned}$$

(14a)

$$\begin{aligned} &{ \begin{aligned}[b] Y_{E'} & = R_{b}\sqrt{1 + (\alpha_{1} + \alpha_{2})^{2}} \sin \bigl(\alpha_{1} + \alpha_{2} \\ &\quad - \arctan (\alpha_{1} + \alpha_{2})\bigr) \end{aligned}} \end{aligned}$$

(14b)

Transformation of the coordinates (X _E′,Y _E′) of point E′ in XOY to that (x _E′,y _E′) in xoy yields,

$$ \biggl[ \begin{array}{l} x_{E'} \\ y_{E'} \end{array} \biggr] = \biggl[ \begin{array}{l@{\quad}l} \cos \theta_{0} & - \sin \theta_{0} \\ \sin \theta_{0} & \cos \theta_{0} \end{array} \biggr]^{ - 1}\biggl[ \begin{array}{l} X_{E'} - X_{o} \\ Y_{E'} - Y_{o} \end{array} \biggr] $$

(15)

When tooth plastic inclination deformation happens, the tooth rotates around point o by θ _p . At this time, point E′ moves to point E″ whose coordinates (x _E″,y _E″) in xoy are represented as

$$\begin{aligned} &{x_{E''} = \sqrt{x_{E'}^{2} + y_{E'}^{2}} \cos \bigl(\arctan (y_{E'}/x_{E'}) + \theta_{p}\bigr)} \end{aligned}$$

(16a)

$$\begin{aligned} &{y_{E''} = \sqrt{x_{E'}^{2} + y_{E'}^{2}} \sin \bigl(\arctan (y_{E'}/x_{E'}) + \theta_{p}\bigr)} \end{aligned}$$

(16b)

Then, the general coordinate of point E″ (X _E″,Y _E″) can be obtained by transformation from its local coordinates.

The point E″ are just on the line EF, so its general coordinates should meet the equation representing the line EF, which is denoted as

$$ Y_{E''} = - \frac{X_{E''}}{\tan (\alpha_{10} + \alpha_{2})} + \frac{R_{b}}{\sin (\alpha_{10} + \alpha_{2})} $$

(17)

Given a value of α ₁₀ with respect to the ideal mesh point E, a fixed value of the unknown α ₁ could be searched. Thus, the coordinates of the point E″ (X _E″,Y _E″) in the general coordinate system XOY can be obtained.

The coordinates of the ideal mesh position (point E) (X _E ,Y _E ) in the coordinate system XOY can be calculated based on Eq. (14a) and (14b) by substitution of α ₁ with α ₁₀.

Consequently, the distance δ _EE″ between the ideal mesh position (point E) and the actual mesh position (point E″) when the tooth is plastically deformed is obtained as

$$ \delta_{EE''} = \sqrt{(X_{E} - X_{E''})^{2} + (Y_{E} - Y_{E''})^{2}} $$

(18)