Abstract
This paper presents a comprehensive analysis of the dynamic characteristics of a helical gear system considering stochastic disturbance and friction. An 8-degree-of-freedom (DOF) single-stage helical gear transmission model is established. Based on the gear meshing principle, the frictional force and force arm of the system are derived. The differential equations governing the system dynamics are solved using the Runge-Kutta method. Bifurcation diagrams, time history plots, and Poincaré maps are employed to analyze the system’s dynamic behavior under various conditions. The results reveal that as the meshing frequency increases, the system transitions from periodic to chaotic motion. While the friction coefficient does not alter the system’s bifurcation characteristics, it affects the period-doubling phenomena near the chaotic region. On the other hand, stochastic disturbances in the meshing frequency significantly alter the system’s bifurcation characteristics, causing the system to enter chaos prematurely.
1. Introduction
Gear transmissions are widely used in various mechanical systems due to their high efficiency and reliability. However, gear systems are inherently nonlinear, and their dynamic behavior can be complex, especially when subjected to factors such as friction and stochastic disturbances. Friction plays a crucial role in gear dynamics, affecting phenomena such as meshing stiffness and vibration response. Similarly, stochastic disturbances, such as those arising from manufacturing tolerances or operational variations, can significantly alter the system’s performance.
Previous studies have extensively investigated the effects of friction on gear dynamics. For instance, Fang et al. studied the effects of friction and stochastic load on the transient characteristics of a spur gear pair. Mo et al. proposed a new computational model to simplify the calculation of friction torque and time-varying contact lines in helical gears. Lin and Lin analyzed the influence of tooth surface friction on the dynamic characteristics of gear systems. Liu et al. investigated the impact of tooth surface friction on the bifurcation characteristics of bending-torsion coupling gears with multiple clearances. Wang et al. analyzed the dynamic characteristics of helical gear transmissions based on tooth surface friction. Zou et al. conducted a study on the friction dynamics coupling of open helical gear transmissions.
While studies on friction in gear systems are abundant, research on the effects of stochastic disturbances, particularly in helical gears, is relatively scarce. Moreover, few studies have considered the combined effects of friction and stochastic disturbances on gear dynamics. To address this gap, the present study establishes an 8-DOF helical gear transmission model that accounts for both friction and stochastic disturbances in the meshing frequency. The dynamic characteristics of the system under these conditions are then analyzed in detail.
2. Helical Gear Transmission Model
The helical gear transmission system considered in this study exhibits not only torsional vibration but also axial vibration along the z-axis, vertical vibration in the y-direction, and horizontal vibration in the x-direction. An 8-DOF transmission system model considering tooth surface friction is developed. The model includes both the driving (gear 1) and driven (gear 2) gears, with degrees of freedom for translational motion in the x, y, and z directions and rotational motion around the respective axes.
The system equations based on Newton’s second law can be expressed as follows:
- m1x¨1+c1xx˙1+k1xx1=−μηFf
- m1y¨1+c1yy˙1+k1yy1=−Fmcosβ
- m1z¨1+c1zz˙1+k1zz1=Fmsinβ
- I1θ¨1=r1Fmcosβ−μηFzs1−T1
- m2x¨2+c2xx˙2+k2xx2=μηFf
- m2y¨2+c2yy˙2+k2yy2=Fmcosβ
- m2z¨2+c2zz˙2+k2zz2=−Fmsinβ
- I2θ¨2=T2−r2Fy+μηFzs2
Where:
- Ff is the frictional force.
- me is the equivalent mass, given by me=I1r22+I2r12I1I2.
- Fm is the dynamic meshing force, expressed as Fm=khf(x)+cmf˙(x), where f(x) is the backlash function.
- T1 and T2 are the torques applied to gears 1 and 2, respectively.
- xn=(r1θ1−r2θ2+y1−y2)/cosβ−e(t) is the relative displacement of the torsional vibration, with e(t)=em+e1cos(ωnt+ψ) representing the error function.
The system equations are then non-dimensionalized to facilitate numerical solution.
3. Frictional Force Calculation
The frictional force in a helical gear system varies with the relative motion between the gears. the direction of relative motion changes near the pitch line, leading to changes in the direction of the frictional force.
The frictional force Ff can be calculated using the following expression:
Ff=μFm(LLright−Lleft)
Where:
- Lright is the length of the contact line to the right of the pitch line.
- Lleft is the length of the contact line to the left of the pitch line.
- L is the total length of the contact line.
The friction arms s1(t) and s2(t) at the meshing points relative to the driving and driven gears, respectively, are given by:
begin{align*} s_1(t) &= (r_{1b} + r_{2b}) \tan \alpha’ – r_{2a} – r_{2b} + \omega_1 r_{1b} t \\ s_2(t) &= r_{2a} – r_{2b} – \omega_1 r_{1b} t end{align*}
Where η is the friction direction coefficient, taking the value of 1 when the sliding speed is non-negative and -1 otherwise.
4. Numerical Simulation Analysis
4.1 Effect of Meshing Frequency on System Dynamics
To analyze the effect of meshing frequency on system dynamics, the gear parameters listed in Table 1 are used.
Table 1: Helical Gear Parameters
| Parameter | Gear 1 | Gear 2 |
|---|---|---|
| Number of teeth | 28 | 70 |
| Face width (mm) | 70 | 65 |
| Module (mm) | 4 | 4 |
| Helix angle (°) | 18 | 18 |
| Pressure angle (°) | 20 | 20 |
| Meshing stiffness (N/mm) | 3.2×10^5 | 3.2×10^5 |
| Meshing damping (Ns/mm) | 150 | 150 |
The Runge-Kutta method is employed to solve the non-dimensionalized system equations. the bifurcation diagram of the system as the meshing frequency varies within the range of (0.46, 1.6).
It can be observed that the system transitions from periodic to quasi-periodic and then to chaotic motion as the meshing frequency increases. Specific transitions are identified, such as the entry into quasi-periodic motion at a meshing frequency of 0.473 and the onset of chaos at 1.069.
To further illustrate the system’s dynamic behavior, Poincaré maps are generated for selected meshing frequencies.
4.2 Effect of Stochastic Disturbance in Meshing Frequency on System Bifurcation Characteristics
To investigate the impact of stochastic disturbances in the meshing frequency, the meshing frequency is perturbed with normally distributed random variables. presents the bifurcation diagrams for different levels of stochastic disturbance.
As the level of stochastic disturbance increases, the system enters chaotic motion at lower meshing frequencies. This is further illustrated by the Poincaré maps, which show the transition from periodic to chaotic motion with increasing disturbance levels.
The time history plots provide additional insights into the system’s response under stochastic disturbances.
4.3 Effect of Different Friction Coefficients on System Bifurcation Characteristics
The effect of different friction coefficients on the system’s bifurcation characteristics is analyzed. the bifurcation diagrams for friction coefficients of 0 and 0.5.
While the overall bifurcation pattern remains similar, changes are observed in the multi-period states near the chaotic region. This is further analyzed using time history plots and Poincaré maps at a meshing frequency of 1.07.
5. Conclusion
This study presents a comprehensive analysis of the dynamic characteristics of a helical gear system considering both friction and stochastic disturbances in the meshing frequency. An 8-DOF model is established, and the system equations are solved using the Runge-Kutta method. The following conclusions are drawn:
- Effect of Meshing Frequency: As the meshing frequency increases, the system transitions from periodic to chaotic motion, as evident from the bifurcation diagrams and Poincaré maps.
- Effect of Stochastic Disturbance: Stochastic disturbances in the meshing frequency significantly alter the system’s bifurcation characteristics, causing the system to enter chaos at lower frequencies. This highlights the importance of considering meshing frequency perturbations in practical applications.
- Effect of Friction Coefficient: While the friction coefficient does not change the overall bifurcation pattern, it affects the multi-period states near the chaotic region. The friction’s hysteresis effect contributes to the system’s chaotic motion at higher speeds.
Overall, this study provides valuable insights into the dynamic behavior of helical gear systems under complex conditions, which can guide the design and optimization of gear transmissions for improved performance and reliability.