In the field of mechanical engineering, worm gear transmission systems are highly valued for their compact structure, high transmission efficiency, excellent manufacturability, and self-locking characteristics. These systems find extensive applications in high-tech industries such as aerospace, autonomous vehicles, and smart home devices. As the demand for higher performance grows, there is an increasing need to enhance the transmission efficiency and load-bearing capacity of worm gear systems while minimizing gear impact vibrations and noise. The occurrence of impact vibrations during the meshing process of worm gear pairs is inevitable and can compromise the safety of the entire mechanical system. Therefore, conducting nonlinear dynamic research on the impact characteristics of worm gear transmission systems is essential to address these challenges.
Nonlinear dynamic analysis of worm gear transmission systems typically involves theoretical derivation and numerical simulation for validation. Previous studies have explored dynamic responses in parameter spaces, highlighting phenomena such as gear jumping and disengagement under extreme conditions. For instance, research on high-speed heavy-duty herringbone gear systems has examined the correlation between system stability and key parameters, providing insights into factors influencing dynamic responses. In this work, we adopt a lumped parameter method to establish a nonlinear dynamic model for worm gear transmission systems, incorporating factors like backlash, comprehensive transmission errors, and time-varying meshing stiffness. We employ numerical methods, specifically the fourth-order variable-step Runge-Kutta method, to solve the system’s equations and analyze nonlinear behaviors through bifurcation diagrams, phase portraits, and Poincaré maps.
The dynamic model of the worm gear transmission system considers bending-torsional-axial coupling, with nonlinearities arising from backlash and support clearances. The system consists of a worm as the driving gear and a worm wheel as the driven gear. Rolling bearings supporting the system are modeled as linear springs and dampers. Applying Newton’s second law, the coupled motion differential equations are derived. The equations account for vibrations along the X, Y, and Z axes, as well as rotational dynamics. The nonlinear dynamic forces in the worm gear meshing interface are expressed as functions of displacement, stiffness, and damping. The relative displacement along the meshing line is influenced by transmission errors and vibrations, leading to a comprehensive model that captures the system’s complex behavior.
The dimensionless form of the differential equations simplifies analysis and numerical computation. By introducing characteristic frequencies and dimensions, we normalize parameters such as displacement, time, and stiffness. The resulting dimensionless equations facilitate the study of system dynamics under varying conditions. Key parameters include the dimensionless meshing frequency, damping ratios, and backlash values. The time-varying meshing stiffness and transmission errors are modeled as periodic functions, reflecting real-world operational variations. This approach allows for a detailed investigation of how nonlinearities affect the stability and response of the worm gear system.
The nonlinear dynamic characteristics are analyzed using bifurcation diagrams, phase trajectories, and Poincaré maps. The meshing period serves as the Poincaré section, enabling a clear visualization of system states. Numerical simulations are performed with carefully selected parameters to explore the effects of meshing frequency and damping ratio on system behavior. The results reveal transitions between periodic and chaotic motions, influenced by bifurcations such as grazing, jump, and period-doubling bifurcations. These insights are crucial for designing stable worm gear systems that minimize vibrations and noise.
To summarize the key equations, the dimensionless motion differential equations for the worm gear transmission system are given below. The system parameters are defined in subsequent sections, and the equations incorporate nonlinear functions for backlash and time-varying stiffness.
The dimensionless equations are as follows:
$$ \ddot{x}_1 + 2\xi_{x1} \dot{x}_1 + k_{x1} f(x_1) + a_1 \xi_p \dot{x}_n + a_1 k_p f(x_n) = 0 $$
$$ \ddot{y}_1 + 2\xi_{y1} \dot{y}_1 + k_{y1} f(y_1) + a_2 \xi_p \dot{x}_n + a_2 k_p f(x_n) = 0 $$
$$ \ddot{z}_1 + 2\xi_{z1} \dot{z}_1 + k_{z1} f(z_1) – a_3 \xi_p \dot{x}_n – a_3 k_p f(x_n) = 0 $$
$$ \ddot{x}_2 + 2\xi_{x2} \dot{x}_2 + k_{x2} f(x_2) – a_1 \xi_p \dot{x}_n – a_1 k_p f(x_n) = 0 $$
$$ \ddot{y}_2 + 2\xi_{y2} \dot{y}_2 + k_{y2} f(y_2) – a_2 \xi_p \dot{x}_n – a_2 k_p f(x_n) = 0 $$
$$ \ddot{z}_2 + 2\xi_{z2} \dot{z}_2 + k_{z2} f(z_2) + a_3 \xi_p \dot{x}_n + a_3 k_p f(x_n) = 0 $$
$$ \ddot{\theta}_1 + 2\xi_{g} \dot{\theta}_1 + a_1 k_p f(x_n) + a_3 k_p f(x_n) = f_g – f_e \cos(\omega t) $$
$$ \ddot{\theta}_2 + 2\xi_{e} \dot{\theta}_2 – a_1 k_p f(x_n) – a_3 k_p f(x_n) = -f_e \cos(\omega t) $$
Here, the nonlinear backlash 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 \( x_n \) is given by:
$$ x_n = a_1 (x_1 – x_2 + R_1 \theta_1) + a_2 (y_1 – y_2) – a_3 (z_1 – z_2 – R_2 \theta_2) – e_m \cos(\omega t + \psi_1) $$
where \( a_1 = \cos \alpha_n \sin \beta \), \( a_2 = \sin \alpha_n \), and \( a_3 = \cos \alpha_n \cos \beta \). The dimensionless parameters are defined as:
$$ \xi_{ij} = \frac{C_{ij}}{2 M_i \omega_n}, \quad k_{ij} = \frac{K_{ij}}{M_i \omega_n^2}, \quad \xi_p = \frac{C_n}{2 M_1 \omega_n}, \quad k_p = \frac{K_n(\tau)}{M_1 \omega_n^2} $$
$$ f_g = \frac{T_g}{M_1 R_1 b_c \omega_n^2}, \quad f_e = \frac{T_e}{M_2 R_2 b_c \omega_n^2}, \quad \omega = \frac{\Omega}{\omega_n} $$
The time-varying meshing stiffness is modeled as:
$$ K_n(\tau) = K_m \left(1 + k_a \cos(\omega \tau + \phi_1)\right) $$
and the comprehensive transmission error as:
$$ E_n(\tau) = E_m \cos(\Omega \tau + \psi_1) $$
To provide a clear overview of the system parameters, the following table summarizes the key dimensionless values used in the analysis:
| Parameter | Symbol | Value |
|---|---|---|
| Characteristic size | \( b_c \) | 1 × 10⁻⁴ m |
| Dimensionless backlash | \( b \) | 1 |
| Dimensionless support clearance | \( b_j \) | 1 |
| Input torque | \( T_g \) | 300 N·m |
| Comprehensive transmission error amplitude | \( E_m \) | 2 × 10⁻⁵ m |
| Average meshing stiffness | \( K_m \) | 2 × 10⁹ N/m |
| Meshing damping ratio | \( \xi \) | 0.08 (base case) |
In the analysis of nonlinear dynamic characteristics, the meshing frequency plays a critical role in determining the system’s behavior. For a fixed meshing damping ratio of \( \xi = 0.08 \), we vary the dimensionless meshing frequency \( \omega \) from 0.1 to 2.5 and observe the relative displacement responses. The bifurcation diagram illustrates that at low frequencies, such as \( \omega \in [0.1, 0.5089] \), the worm gear system exhibits period-1 motion, indicating full meshing without impact vibrations. At \( \omega = 0.5089 \), a grazing bifurcation occurs, where the relative displacement just touches the backlash boundary, yet no impact vibrations are generated. As \( \omega \) increases further, impact vibrations begin, leading to period-1 motion with single impacts. A jump bifurcation at \( \omega = 0.5673 \) alters the motion trajectory, disrupting system stability without changing the period.
When \( \omega \) reaches 0.8736, a period-doubling bifurcation transitions the system to period-2 motion, characterized by two impacts per cycle. Subsequent increases in \( \omega \) induce another jump bifurcation, maintaining period-2 motion but with altered impact patterns. Near \( \omega = 1.3 \), the system undergoes rapid consecutive period-doubling bifurcations, advancing to period-4, period-8, and eventually chaotic motion. In the chaotic regime, the Poincaré map shows scattered points, reflecting unpredictable behavior and increased gear impacts, which pose risks to system integrity. As \( \omega \) continues to rise, the chaos diminishes through inverse period-doubling bifurcations, reverting to period-4 and period-2 motions, and finally stabilizing into period-1 motion for \( \omega \geq 1.7195 \). This high-frequency stability demonstrates that worm gear systems can operate reliably under certain conditions, with minimal vibrations.
The meshing damping ratio is another vital parameter influencing the dynamic response of worm gear systems. To investigate its effects, we set the damping ratio to \( \xi = 1.0 \) and \( \xi = 1.2 \), while keeping other parameters constant. The bifurcation diagrams for these cases reveal that in low-frequency regions, the system behavior remains similar to the base case, with jump bifurcations occurring unaffected by damping changes. However, in the transition to higher frequencies, impact vibrations intensify twice, potentially compromising stability. Notably, as the meshing damping ratio increases, the complex chaotic behaviors in the range \( \omega \in [1.35, 1.6] \) gradually disappear, replaced by stable periodic motions. This indicates that higher damping ratios effectively suppress chaos and reduce impact vibrations in worm gear systems, enhancing overall stability and performance. Therefore, selecting an appropriate meshing damping ratio is crucial for optimizing worm gear design in high-frequency applications.
For a detailed comparison, the table below summarizes the effects of meshing frequency and damping ratio on system dynamics:
| Parameter Range | System Behavior | Key Observations |
|---|---|---|
| \( \omega \in [0.1, 0.5089] \) | Period-1 motion | Full meshing, no impacts |
| \( \omega = 0.5089 \) | Grazing bifurcation | Displacement touches backlash |
| \( \omega \in [0.5089, 0.8736] \) | Period-1 with impacts | Single impacts, jump bifurcations |
| \( \omega = 0.8736 \) | Period-doubling bifurcation | Transition to period-2 motion |
| \( \omega \in [0.9, 1.3] \) | Period-2 and higher periods | Increased impacts, chaos onset |
| \( \omega \in [1.3, 1.7195] \) | Chaotic motion | Unpredictable, high vibration risks |
| \( \omega \geq 1.7195 \) | Period-1 motion | Stable high-frequency operation |
| Increased \( \xi \) to 1.0-1.2 | Reduced chaos | Enhanced stability in high-frequency range |
In conclusion, our nonlinear dynamic analysis of worm gear transmission systems provides valuable insights into their behavior under varying operational conditions. The worm gear system experiences a series of bifurcations, including grazing, jump, and period-doubling bifurcations, as the meshing frequency increases, leading to transitions from periodic to chaotic motion. However, by increasing the meshing damping ratio, we can suppress these complex behaviors and promote stable periodic operations. This underscores the importance of carefully selecting damping parameters in the design of worm gear systems to minimize vibrations and noise, ensuring reliability in high-performance applications. Future work could explore additional nonlinear factors or optimize control strategies for further improvements in worm gear dynamics.