This study investigates the amplitude jump, multi-solution behavior, and tooth surface impact characteristics of spur gear transmission systems considering time-varying stiffness and backlash effects. A dynamic model incorporating harmonic balance methods and continuation algorithms is developed to analyze the parameter sensitivity and solution domain boundary structures.
1. Nonlinear Dynamic Model
The dimensionless governing equations for a 7-DOF spur gear system are expressed as:
$$
\begin{cases}
\ddot{x}_1 + 2\xi_{x1}\dot{x}_1 + k_{x1}x_1 + \mathcal{N}(\lambda) = F_x \\
\ddot{y}_1 + 2\xi_{y1}\dot{y}_1 + k_{y1}y_1 – \mathcal{N}(\lambda) = F_y \\
\vdots \\
\ddot{\lambda} + 2\xi_h\dot{\lambda} + k_h\mathcal{N}(\lambda) = f_e(\tau)
\end{cases}
$$
where $\mathcal{N}(\lambda)$ represents the nonlinear backlash function:
$$
\mathcal{N}(\lambda) =
\begin{cases}
\lambda – b & \lambda > b \\
0 & |\lambda| \leq b \\
\lambda + b & \lambda < -b
\end{cases}
$$
2. Harmonic Balance Implementation
The steady-state response is approximated using first-order harmonic components:
$$
x_i = x_{mi} + x_{ci}\cos(\Omega\tau) + x_{si}\sin(\Omega\tau)
$$
The describing function method converts the piecewise nonlinearity into equivalent harmonic coefficients:
$$
\mathcal{N}_{mi} = \frac{1}{2x_{mi}}[G(\mu_+) – G(\mu_-)]
$$
$$
\mathcal{N}_{ai} = \frac{1}{2}[H(\mu_+) – H(\mu_-)]
$$
where $G(\mu)$ and $H(\mu)$ are defined as:
$$
G(\mu) = \frac{2}{\pi}(\mu\sqrt{1-\mu^2} + \arcsin\mu)
$$
$$
H(\mu) = \frac{2}{\pi}(\sqrt{1-\mu^2} + \mu\arcsin\mu)
$$
3. Parameter Sensitivity Analysis
| Parameter | Amplitude Jump Sensitivity | Impact State Transition |
|---|---|---|
| Backlash (b) | High in [0.1, 0.98] range | 2/2 → 2/1 → 1/0 when b > 0.98 |
| TVMS Factor (α) | Moderate (α > 0.4 reduces jumps) | 1/1 → 1/0 when α > 0.25 |
| Error Excitation (f_e) | Strong (40% increase induces 2nd jump) | 1/0 → 2/0-2 when f_e > 0.21 |
| Load (f_p) | Inverse correlation | 2/1-2 disappears when f_p > 0.57 |
4. Solution Domain Characteristics
The multi-solution/impact (n/I) state matrix reveals critical parameter thresholds:
| Parameter Plane | Stability Threshold | Dominant n/I States |
|---|---|---|
| Ω × b | b > 0.98 stabilizes response | 2/0-1, 1/2, 2/2 |
| Ω × α | α > 0.4 eliminates axial resonance | 1/0, 1/1 |
| Ω × f_e | f_e > 0.34 induces chaos | 2/0-2, 2/1-2 |
| Ω × f_p | f_p > 0.75 linearizes response | 1/0, 1/1 |
5. Dynamic Response Features
The spur gear system exhibits three characteristic regimes:
$$
\text{1. Linear Region } (\Omega < 0.5): \quad \lambda_a \propto \frac{f_e\Omega^2}{\sqrt{(1-\Omega^2)^2 + (2\xi\Omega)^2}}
$$
$$
\text{2. Meshing Resonance } (0.5 < \Omega < 1.2): \quad \lambda_a^{max} = \frac{f_e}{2\xi} + \frac{\alpha b}{4(1-\Omega^2)}
$$
$$
\text{3. Shaft Frequency Resonance } (\Omega > 1.2): \quad \lambda_a \approx \frac{T_p}{k_mr} \sec\left(\frac{\pi\Omega}{2\Omega_n}\right)
$$
This parametric solution domain analysis provides critical guidelines for spur gear design: maintaining backlash b > 1.0mm, error excitation f_e < 0.2, and TVMS variation α < 0.3 ensures stable single-solution operation without tooth surface impacts.