This paper investigates the nonlinear dynamic characteristics of spur gear transmission systems considering multi-state meshing behavior and time-varying parameters. A bending-torsional-axial coupled dynamic model is established to capture the complex interactions between gear components under varying operating conditions.
1. Fundamental Formulation
The dynamic model incorporates time-varying mesh stiffness $k(\tau)$, load distribution ratio $L_c(\tau)$, and backlash nonlinearity. The governing equations for spur gear systems are derived as:
$$m_e\ddot{x}_n + c_m\dot{x}_n + \sum_{i=1}^2 L_{ci}(\tau)k(\tau)f(x_n) = F_m + F_e(\tau)$$
Where $f(x_n)$ represents the backlash function:
$$f(x_n) =
\begin{cases}
x_n – D_n & x_n \geq D_n \\
0 & |x_n| < D_n \\
x_n + D_n & x_n \leq -D_n
\end{cases}$$
2. Time-Varying Parameter Calculation
The micro-element method calculates time-varying mesh stiffness considering Hertzian contact, bending, shear, and axial compression:
| Stiffness Component | Formula |
|---|---|
| Hertzian Contact | $$\frac{1}{k_h} = \frac{4(1-\nu^2)}{\pi Eb}$$ |
| Bending | $$\frac{1}{k_b} = \int_{\theta}^{\sigma}\frac{3q^2(\theta-\alpha)\cos\alpha}{2Eb[\sin\alpha+(\theta-\alpha)\cos\alpha]^3}d\alpha$$ |
| Axial Compression | $$\frac{1}{k_a} = \int_{\theta}^{\sigma}\frac{(\theta-\alpha)\cos\alpha\sin^2\sigma}{2Eb[\sin\alpha+(\theta-\alpha)\cos\alpha]}d\alpha$$ |
3. Load Distribution Analysis
The load distribution ratio for spur gears with multi-tooth contact is expressed as:
$$L_c(\tau) = \frac{\sum_{o=1}^{N_j}1/u_o(\xi_{cio})}{\sum_{i=1}^2\sum_{o=1}^{N_j}1/u_o(\xi_{cio})}$$
Where $u_o$ represents the unit potential energy of micro-elements. The time-varying load distribution pattern for spur gears is shown below:
| Meshing State | Load Ratio | Duration |
|---|---|---|
| Single-tooth | 1.0 | $(2-\varepsilon_m)T_0$ |
| Double-tooth | 0.5-0.7 | $(\varepsilon_m-1)T_0$ |
4. Nonlinear Dynamic Behavior
The system exhibits complex nonlinear phenomena governed by dimensionless parameters:
$$ \omega = \frac{\omega_h}{\omega_n},\quad \zeta = \frac{c_m}{2m_e\omega_n},\quad F = \frac{T}{m_e\omega_n^2D} $$
Bifurcation analysis reveals different dynamic regimes:
| Frequency Ratio | Dynamic Behavior | Meshing States |
|---|---|---|
| $\omega < 0.73$ | Period-1 (1-0-0) | Continuous engagement |
| $0.73 < \omega < 1.34$ | Period-2 (2-2-0) | Alternate single/double tooth |
| $\omega > 2.2$ | Chaotic (6-2-2) | Intermittent contact |
5. Parametric Sensitivity Analysis
The system response shows significant dependence on transmission error $\epsilon$:
$$F_e = \epsilon\omega^2\cos(\omega t)$$
Critical transmission error thresholds:
| Error Level | System Response | TL Exponent |
|---|---|---|
| $\epsilon < 0.02$ | Stable periodic | -0.15 |
| $0.02 < \epsilon < 0.26$ | Subharmonic | 0-0.05 |
| $\epsilon > 0.26$ | Chaotic | 0.12-0.18 |
6. Vibration Mitigation Strategies
Key design parameters for spur gear dynamic optimization:
$$Q = \frac{k_{avg}}{\sqrt{m_ek_{avg}}} \propto \frac{1}{\sqrt{\epsilon}}$$
Practical recommendations:
– Maintain $\omega < 0.7$ for stable operation
– Control $\epsilon < 0.02D_n$ through precision manufacturing
– Optimize tooth profile modification to reduce impact forces
The proposed model provides fundamental insights for designing high-performance spur gear transmission systems with improved dynamic characteristics and reduced vibration/noise.