# Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

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## Abstract

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.

## Keywords

Gear dynamics Mesh stiffness Profile modification Eccentricity error Force-dependent## List of symbols

- \(\delta _{\mathrm{b}}\), \(\delta _{\mathrm{s}}\), \(\delta _{\mathrm{a}}\)
Bending deformation, shear deformation, axial deformation

- \(\alpha _{1}\)
Angle between the line of action and the line perpendicular to the center line of the tooth

*x*,*l*,*d*Defined in Fig. 1

*h*Distance from the point of contact to the center line of the tooth

*E*Young’s modulus

*v*Poisson’s ratio

*W*Tooth width

- \(I_{x},\,A_{x}\)
Moment of inertia and the area of section at the distance

*x*from the base along the tooth center line- \(k_{\mathrm{b}},\, k_{\mathrm{s}},\, k_{\mathrm{a}}\)
Bending stiffness, shearing stiffness, axial stiffness

- \(\delta _{\mathrm{f}}\)
Additional deformation of the tooth caused gear body

- \(k_{\mathrm{f}}\)
The stiffness related to the additional deformation of gear body

*L**,*M**,*P**,*Q**Coefficients expressing the additional deformation of the tooth

- \(S_{\mathrm{f}}\)
Defined in Fig. 2

- \(\delta _{\mathrm{h}}\)
Local contact compliance

*b*Half width of the contact region on the tooth

- \(r_{\mathrm{p}},r_{\mathrm{g}}\)
Radius of curvature at the point of contact of gear and pinion

- \(v_{\mathrm{p}},\,v_{\mathrm{g}}\)
Poisson’s ratio of pinion and gear

- \(E_{\mathrm{p}},\,E_{\mathrm{g}}\)
Young’s modulus of pinion and gear

- \(h_{x\mathrm{p}},h_{x{\mathrm{g}}}\)
Distance from the point of contact to the center line of the tooth of the gear and pinion along the line of action direction

- \(k_{{\mathrm{h}}}{[F]}\)
Local contact stiffness of tooth pair under load

*F*- \(J_{\mathrm{p}},J_{\mathrm{g}}\)
Mass moments of inertia of pinion and gear

- \(K_{\mathrm{s}}{[F]}\)
Mesh stiffness of single tooth pair under the load

*F**K*[*F*]The overall mesh stiffness of mesh gears under the load

*F*- \(k_{\mathrm{l}}\)
Load-independent stiffness including bending, shear, axial stiffness and additional stiffness caused by gear body

- \(k_{\mathrm{l}i}\)
Load-independent stiffness of tooth pair

*i*- \(k_{{\mathrm{h}}i}{[F]}\)
The local contact stiffness of tooth pair

*i*- \(\delta \)
Total deformation of the tooth at the contact point

- \(e_{i}\)
Initial separation of the potential contact tooth pair

*i*- \(E_{i\mathrm{p}},E_{i\mathrm{p}}\)
Profile error of tooth pair

*i*of pinion and gear- \(\eta _{i\mathrm{p}},\,\eta _{i\mathrm{g}}\)
Profile modification of tooth pair \(i\_\) of pinion and gear

- \(\lambda _{\mathrm{p}},\,\lambda _{\mathrm{g}}\)
Eccentricity error of pinion and gear

- \(\delta _{i}\)
Deformation of the tooth pair

*i*- \(F_{i}\)
Load on the tooth pair

*i*- \(\hbox {Lsf}_i\)
Load sharing of tooth pair

*i*- \(T_{\mathrm{p}}\)
Input torque on the pinion

- \(T_{\mathrm{g}}\)
Output torque on the gear

- \(c_{\mathrm{m}}\)
Linear damping element

*e*No-load transmission error

- \(h_{\mathrm{m}},\,h_{i}\)
Contact coefficient

- \(r_\mathrm{bp},\,r_\mathrm{bg}\)
Radius of the base circle of the pinion and the gear

- \(f_{\mathrm{s}}\)
Rotational frequency of pinion

- \(f_{\mathrm{m}}\)
Mesh frequency

## Notes

### Funding

The authors are grateful for the financial support provided by the National Natural Science Foundation of China (Grant Nos. 51775453 and 51605412), the Sichuan Science and Technology Program (Grant No. 2018JY0159).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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