In this paper, I investigate the nonlinear dynamic characteristics of worm gear transmission systems, which are widely utilized in high-tech fields such as aerospace, autonomous vehicles, and smart home systems due to their compact structure, high transmission efficiency, excellent manufacturability, and self-locking properties. As the demand for higher performance in worm gears increases, improving transmission efficiency and load capacity while reducing impact vibrations and noise in gear pairs has become a critical challenge. Impact vibrations during the meshing of worm gear pairs are inevitable and can compromise the safety of the entire mechanical system. Therefore, a nonlinear dynamic analysis of the impact characteristics in worm gear systems is essential. Typically, theoretical derivations and numerical simulations are employed to analyze these systems, with methods like the fourth-order variable-step Runge-Kutta technique used for validation. This study establishes a nonlinear dynamic model incorporating factors such as backlash, comprehensive transmission errors, and time-varying mesh stiffness, and examines the system’s behavior through bifurcation diagrams, phase trajectories, and Poincaré maps.
To model the worm gear system, I adopt a lumped parameter approach that considers bending-torsion-axial coupling, accounting for nonlinearities like tooth flank clearance and support gaps. The system consists of a worm (active gear) and a worm wheel (driven gear), with rolling bearings represented as linear springs and dampers. Applying Newton’s second law, the coupled differential equations of motion are derived. The dimensionless form of these equations is used for analysis to simplify the system dynamics. The equations involve parameters such as mass, damping, stiffness, and间隙 functions, which are defined to capture the nonlinear behavior of worm gears. For instance, the dynamic meshing force and its components along coordinate directions are expressed as follows:
$$ F_n = K_n(\tau) f(X_n) + C_n \dot{X}_n $$
$$ F_x = -F_n \cos \alpha_n \sin \beta $$
$$ F_y = -F_n \sin \alpha_n $$
$$ F_z = F_n \cos \alpha_n \cos \beta $$
Here, the gap function $f(X)$ is defined as:
$$ f(X) = \begin{cases}
X – B, & \text{if } X > B \\
0, & \text{if } -B \leq X \leq B \\
X + B, & \text{if } X < -B
\end{cases} $$
The relative displacement along the meshing line, considering transmission errors and vibrations, is given by:
$$ X_n = (X_1 – X_2 + R_1 \theta_1) \cos \alpha_n \sin \beta + (Y_1 – Y_2) \sin \alpha_n – (Z_1 – Z_2 – R_2 \theta_2) \cos \alpha_n \cos \beta – E_n(\tau) $$
where $E_n(\tau) = E_m \cos(\Omega \tau + \psi_1)$ represents the comprehensive transmission error. By introducing dimensionless parameters, such as $x_i = X_i / b_c$, $t = \omega_n \tau$, and $\omega = \Omega / \omega_n$, the system equations are normalized. The resulting dimensionless differential equations are:
$$ \ddot{x}_1 + 2\xi_{1x} \dot{x}_1 + k_{1x} f(x_1) + a_1 \xi_p \dot{x}_n + a_1 k_p f(x_n) = 0 $$
$$ \ddot{y}_1 + 2\xi_{1y} \dot{y}_1 + k_{1y} f(y_1) + a_2 \xi_p \dot{x}_n + a_2 k_p f(x_n) = 0 $$
$$ \ddot{z}_1 + 2\xi_{1z} \dot{z}_1 + k_{1z} f(z_1) – a_3 \xi_p \dot{x}_n – a_3 k_p f(x_n) = 0 $$
$$ \ddot{x}_2 + 2\xi_{2x} \dot{x}_2 + k_{2x} f(x_2) – a_1 \xi_p \dot{x}_n – a_1 k_p f(x_n) = 0 $$
$$ \ddot{y}_2 + 2\xi_{2y} \dot{y}_2 + k_{2y} f(y_2) – a_2 \xi_p \dot{x}_n – a_2 k_p f(x_n) = 0 $$
$$ \ddot{z}_2 + 2\xi_{2z} \dot{z}_2 + k_{2z} f(z_2) + a_3 \xi_p \dot{x}_n + a_3 k_p f(x_n) = 0 $$
$$ \ddot{\theta}_1 – a_1 (x_1 – x_2) – a_2 (y_1 – y_2) – a_3 (z_1 – z_2) – a_1 k_p f(x_n) – 2\xi_p \dot{x}_n = f_g – f_e \cos(\omega t) $$
The dimensionless parameters are defined as:
$$ \xi_{ij} = \frac{C_{ij}}{2 M_i \omega_n}, \quad k_{ij} = \frac{K_{ij}}{M_i \omega_n^2}, \quad \xi_p = \frac{C_n}{2 M_1 \omega_n}, \quad k_p = \frac{K_n(t)}{M_1 \omega_n^2} $$
$$ f_g = \frac{T_g}{M_1 R_1 b_c \omega_n^2}, \quad f_e = \frac{T_e}{M_2 R_2 b_c \omega_n^2} $$
where $K_n(t) = K_m [1 + k_a \cos(\omega t + \phi_1)]$ represents the time-varying mesh stiffness. The key parameters for the worm gear system are summarized in the table below, which includes design and dynamic properties essential for the analysis of worm gears.
| Parameter | Value |
|---|---|
| Module m (mm) | 4 |
| Number of worm teeth z1 | 3 |
| Worm diameter d1 (mm) | 44 |
| Number of worm wheel teeth z2 | 37 |
| Normal pressure angle αn (°) | 20 |
| Characteristic size bc (m) | 1×10⁻⁴ |
| Dimensionless backlash b | 1 |
| Dimensionless support gap bj | 1 |
| Input torque Tg (N·m) | 300 |
| Comprehensive transmission error Em (m) | 2×10⁻⁵ |
| Average mesh stiffness Km (N/m) | 2×10⁹ |
For the nonlinear dynamic analysis, I use the Poincaré section defined by the meshing period of the worm gear system to examine the system’s state. Numerical simulations are performed with the fourth-order variable-step Runge-Kutta method, and results are visualized through bifurcation diagrams, phase plots, and Poincaré maps. The influence of mesh frequency and damping ratio on the dynamics of worm gears is investigated in detail.
First, I analyze the effect of dimensionless mesh frequency ω on the system’s behavior, with a fixed mesh damping ratio ξ = 0.08. The bifurcation diagram of relative displacement versus ω in the range [0.1, 2.5] reveals distinct dynamic regimes for worm gears. At low frequencies (ω ∈ [0.1, 0.5089]), the system exhibits period-1 motion, indicating full meshing without impact vibrations. A grazing bifurcation occurs at ω = 0.5089, where the relative displacement touches the backlash boundary. As ω increases, impact vibrations begin, leading to period-1 motion with single impacts. A jump bifurcation at ω = 0.5673 alters the trajectory, and at ω = 0.8736, a period-doubling bifurcation transitions the system to period-2 motion. Further increases in ω cause rapid successive period-doubling bifurcations, resulting in high-period orbits and eventually chaotic motion around ω = 1.3, characterized by multiple Poincaré points and increased impacts in worm gears. Beyond this, the system undergoes inverse period-doubling bifurcations, reverting to period-4, period-2, and finally stable period-1 motion for ω ≥ 1.7195, where worm gears operate smoothly in high-frequency regions.
Next, I examine the role of mesh damping ratio ξ in stabilizing worm gear systems. With ξ increased to 1.0 and 1.2, the bifurcation diagrams show that low-frequency behavior remains similar, but high-frequency chaos is suppressed. For instance, in the range ω ∈ [1.35, 1.6], complex chaotic motions diminish as ξ rises, transitioning to stable periodic orbits. This highlights the importance of selecting an appropriate mesh damping ratio to enhance the stability of worm gears, reduce vibrations, and minimize noise. The table below compares the system’s response under different damping ratios, emphasizing the damping effect on worm gear dynamics.
| Damping Ratio ξ | Dynamic Behavior | Impact on Worm Gears |
|---|---|---|
| 0.08 | Chaos and high-period motions | Increased vibrations and noise |
| 1.0 | Reduced chaos, more periodic orbits | Improved stability |
| 1.2 | Stable period-1 motion at high frequencies | Minimized impacts and enhanced reliability |
The equations governing the mesh force and relative displacement are critical for understanding these behaviors. For example, the dimensionless mesh stiffness variation can be expressed as:
$$ k_p(t) = 1 + k_a \cos(\omega t + \phi_1) $$
and the dimensionless damping terms involve parameters like ξ_p, which directly affect the energy dissipation in worm gears. The comprehensive analysis shows that worm gears experience various bifurcations, such as grazing, jump, and period-doubling, which drive the system from periodic to chaotic states. Increasing the mesh damping ratio mitigates these effects, promoting stable operation in worm gear systems.
In conclusion, my study on worm gear systems demonstrates that nonlinear dynamics play a significant role in their performance. The system undergoes grazing bifurcations, jump bifurcations, and consecutive period-doubling bifurcations as mesh frequency increases, leading to chaos, but stabilizes at higher frequencies. Enhancing the mesh damping ratio suppresses chaotic and high-period motions, resulting in stable period-1 behavior. This underscores the importance of optimal damping design in worm gears to ensure reliability, reduce vibrations, and meet advanced industrial requirements. Future work could explore additional parameters or real-world applications to further refine the dynamics of worm gears.