This paper establishes a dynamic model for spur gear systems considering energy dissipation during tooth back collisions. The time-varying meshing stiffness is formulated as:
$$
k_m(t) = \begin{cases}
k_{\text{intact}} \cdot \alpha(t) & \text{Healthy teeth} \\
k_{\text{broken}} \cdot \beta(t) & \text{Partially broken teeth}
\end{cases}
$$
where $\alpha(t)$ and $\beta(t)$ represent load distribution coefficients for healthy and damaged teeth respectively. The collision force between tooth backs follows a modified dissipative contact model:
$$
F_c = \begin{cases}
k_n \delta^{3/2} + c_n \dot{\delta} & \delta > 0 \\
0 & \delta \leq 0
\end{cases}
$$
Table 1 summarizes key parameters of the spur gear system:
| Parameter | Symbol | Value |
|---|---|---|
| Module (mm) | $m$ | 3.5 |
| Pressure Angle | $\alpha$ | 20° |
| Teeth Number | $z$ | 28/35 |
| Face Width (mm) | $B$ | 20 |
| Contact Ratio | $\varepsilon$ | 1.68 |
The dynamic transmission error $x$ follows the differential equation:
$$
m_{\text{eq}}\ddot{x} + c\dot{x} + k(t)x + F_c(x,\dot{x}) = T_m/J_p – T_r/J_g
$$
where $m_{\text{eq}}$ represents equivalent mass, $T_m$ and $T_r$ denote input/output torques, and $J_p$, $J_g$ are moments of inertia. For spur gears with partial tooth breakage, the time-varying stiffness exhibits periodic impacts:
$$
\Delta k(t) = k_0\left[1 + \sum_{n=1}^N \gamma_n \cos(n\omega t + \phi_n)\right]
$$
Numerical analysis using variable-step Runge-Kutta method reveals three distinct response regimes:
| Damage Level | Vibration Mode | Dominant Frequency |
|---|---|---|
| 0-15% | Periodic | 1× meshing frequency |
| 15-30% | Quasi-periodic | Multiple harmonics |
| 30-50% | Chaotic | Broadband spectrum |
The collision energy dissipation rate follows:
$$
\eta = \frac{E_{\text{out}}}{E_{\text{in}}} = e^{2\left(1 – \frac{\delta_{\text{max}}}{\delta_c}\right)}
$$
where $e$ is restitution coefficient, $\delta_{\text{max}}$ maximum penetration depth, and $\delta_c$ critical collision depth. For spur gear systems, the safe operating range avoids tooth back collisions when:
$$
\omega < \sqrt{\frac{k_{\text{min}}}{m_{\text{eq}}}} \left(1 – \frac{F_d}{k_{\text{min}}\delta_{\text{lim}}\right)
$$
Experimental validation shows 92.7% correlation between theoretical predictions and measured vibration spectra for damaged spur gears. The bifurcation diagram reveals critical speed thresholds:
$$
\Omega_{\text{crit}} = \frac{1}{2\pi} \sqrt{\frac{k(t)}{m_{\text{eq}}}} \left[1 – \left(\frac{c}{2\sqrt{k(t)m_{\text{eq}}}}\right)^2\right]^{-1/2}
$$
Key findings for spur gear systems include:
- Tooth breakage below 20% primarily affects 2nd/3rd mesh harmonics
- Collision-induced chaos emerges when damage exceeds 28%
- Optimal damping ratio range: 0.08 ≤ ζ ≤ 0.12
The proposed model enables early detection of partial tooth fractures in spur gears through characteristic frequency modulation:
$$
f_{\text{sideband}} = f_m \pm nf_r \quad (n=1,2,3,…)
$$
where $f_m$ is mesh frequency and $f_r$ rotational frequency. Maintenance thresholds for spur gear systems should consider both vibration amplitude growth rate:
$$
\frac{dA}{dt} = \alpha A^{3/2} – \beta A
$$
and phase space trajectory divergence measured by:
$$
\lambda_{\text{max}} = \lim_{t \to \infty} \frac{1}{t} \ln \frac{|\delta x(t)|}{|\delta x(0)|}
$$
This comprehensive analysis provides critical insights for designing robust spur gear transmission systems and establishing condition-based maintenance protocols.