Abstract
To investigate the nonlinear dynamic characteristics of worm gear transmission systems, this paper establishes a nonlinear dynamic model incorporating nonlinear factors such as backlash, comprehensive transmission errors, and time-varying mesh stiffness using the lumped parameter method. Numerical simulations of the dynamic equations of the worm gear transmission system are conducted using the fourth-order variable step-size Runge-Kutta method. The nonlinear dynamic characteristics of the system are analyzed through bifurcation diagrams, phase portraits, and Poincaré maps. The results indicate that the system undergoes grazing bifurcations, jump bifurcations, and continuous period-doubling bifurcations, transitioning from periodic motion (period-1) to chaotic motion. As the mesh damping ratio gradually increases, high-period and chaotic motions gradually become stable periodic motions. Selecting a reasonable mesh damping ratio during system dynamic design can enhance system stability.
Keywords: worm gear; nonlinearity; bifurcation
1. Dynamic Model
The nonlinear dynamic model of the worm gear transmission system with backlash, support clearance, and coupled bending-torsional-axial dynamics, where the driving gear 1 is the worm, and the driven gear 2 is the worm wheel.
Table 1. System Parameters
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Support stiffness | Kij | – | N/m |
| Support damping | Cij | – | N·s/m |
| Support clearance | Bij | – | m |
| Worm mass | M1 | – | kg |
| Worm wheel mass | M2 | – | kg |
| Backlash | B1 | – | m |
| Time-varying mesh stiffness | Kn(τ) | – | N/m |
| Mesh damping | Cn | – | N·s/m |
| Comprehensive transmission error | En(τ) | – | m |
The differential equations of motion for the coupled bending-torsional-axial dynamics of the worm gear transmission system are given by Equation (1).
2. Nonlinear Dynamic Characteristics Analysis
2.1. Influence of Mesh Frequency
With the design and dynamic parameters fixed, the mesh damping ratio ξ is set to 0.08, and the non-dimensional mesh frequency ω is varied within [0.1, 2.5]. The displacement-frequency bifurcation diagram.
Table 2. Key Observations from Bifurcation Diagram
| Mesh Frequency Range (ω) | System Behavior | Description |
|---|---|---|
| [0.1, 0.5089] | Period-1 Motion | Complete mesh, no impact vibration |
| ω = 0.5089 | Grazing Motion | No impact vibration, virtual line indicates backlash |
| (0.5089, 0.5673) | Single-Impact Period-1 Motion | Backlash leads to impact vibration |
| ω = 0.5673 | Jump Bifurcation | Disrupts smooth operation, changes trajectory |
| (0.5673, 0.8736) | Period-1 Motion with Impacts | – |
| ω = 0.8736 | Period-Doubling Bifurcation | Transition to Period-2 Motion |
| (0.8736, 1.3) | Period-2 Motion with Increasing Impacts | – |
| ω ≈ 1.3 | Chaos | Rapid period-doubling, unpredictable motion |
| (1.3, 1.7195) | Gradual Reduction in Complexity | – |
| ω ≥ 1.7195 | Stable Period-1 Motion | Smooth operation in high-frequency range |
2.2. Influence of Mesh Damping Ratio
The effect of the mesh damping ratio ξ on system dynamics is analyzed by comparing displacement-frequency bifurcation diagrams for ξ = 1.0 and ξ = 1.2 .
Observations:
- In the low-frequency range, system behavior is similar; jump bifurcations are unaffected by changes in ξ.
- In the transition to the high-frequency range, impact vibrations suddenly increase twice, affecting stability.
- As ξ increases, complex chaotic behavior within ω ∈ [1.35, 1.6] diminishes, eventually transitioning to stable periodic motion.
3. Conclusions
This paper establishes a nonlinear dynamic model for worm gear transmission systems incorporating bending, torsion, and axial directions. By studying the system’s dynamics under varying mesh frequencies and mesh damping ratios, the following conclusions are drawn:
- As ω increases, the system undergoes grazing bifurcations, jump bifurcations, and continuous period-doubling bifurcations, rapidly transitioning to high-period and eventually chaotic motion, before reverting to stable period-1 motion through inverse period-doubling bifurcations.
- As ξ increases, complex behaviors such as high-period and chaotic motions gradually disappear, leading to stable operation. Increasing the mesh damping ratio can effectively reduce complex behaviors in the high-frequency range, enhancing system stability and reducing impact vibrations and noise between gears.