In mechanical transmission systems, the straight bevel gear is widely used due to its ability to transmit power between intersecting shafts. However, nonlinear factors such as backlash, time-varying stiffness, and support clearances can induce chaotic vibrations, which adversely affect system performance and reliability. This study focuses on controlling chaotic motion in a 7-degree-of-freedom straight bevel gear transmission system. We employ a centralized mass method to establish the dynamic model and use the Runge-Kutta method for numerical solutions to analyze the transition from periodic to chaotic motion. For chaotic regions, we design a fuzzy neural network controller optimized by an improved particle swarm optimization algorithm to stabilize the system onto periodic orbits. The controller inputs are the Euclidean distances between points on the Poincaré section, and the output is a perturbation applied to a controllable system parameter. Numerical simulations demonstrate the effectiveness of this approach in suppressing chaos and enhancing system stability.
The straight bevel gear system exhibits complex dynamics due to its inherent nonlinearities. A 7-DOF model is developed, considering vibrations along three translational directions and torsional motion. The dimensionless equations of motion are derived as follows:
$$ \ddot{x}_1 + 2\xi_{x1}\dot{x}_1 + 2a_4\xi_{h1}\dot{\lambda} + k_{x1}x_1 + a_4k_{h1}f(\lambda) = 0 $$
$$ \ddot{y}_1 + 2\xi_{y1}\dot{y}_1 – 2a_5\xi_{h1}\dot{\lambda} + k_{y1}y_1 – a_5k_{h1}f(\lambda) = 0 $$
$$ \ddot{z}_1 + 2\xi_{z1}\dot{z}_1 – 2a_3\xi_{h1}\dot{\lambda} + k_{z1}z_1 – a_3k_{h1}f(\lambda) = 0 $$
$$ \ddot{x}_2 + 2\xi_{x2}\dot{x}_2 – 2a_4\xi_{h2}\dot{\lambda} + k_{x2}x_2 – a_4k_{h2}f(\lambda) = 0 $$
$$ \ddot{y}_2 + 2\xi_{y2}\dot{y}_2 + 2a_5\xi_{h2}\dot{\lambda} + k_{y2}y_2 + a_5k_{h2}f(\lambda) = 0 $$
$$ \ddot{z}_2 + 2\xi_{z2}\dot{z}_2 + 2a_3\xi_{h2}\dot{\lambda} + k_{z2}z_2 + a_3k_{h2}f(\lambda) = 0 $$
$$ -a_1\ddot{x}_1 + a_2\ddot{y}_1 + a_3\ddot{z}_1 + a_1\ddot{x}_2 – a_2\ddot{y}_2 – a_3\ddot{z}_2 + \ddot{\lambda} + 2a_3\xi_h\dot{\lambda} + a_3k_h f(\lambda) = f_{pm} + f_{pv} + f_e \Omega^2 \cos(\Omega \tau) $$
Here, \( x_j, y_j, z_j \) are dimensionless displacements, \( \xi_{ij} \) are damping ratios, \( k_{ij} \) are stiffness coefficients, \( \lambda \) is the relative torsional displacement, and \( f(\lambda) \) is the backlash function defined as:
$$ f(\lambda, b) = \begin{cases}
\lambda – b & \lambda > b \\
0 & |\lambda| \leq b \\
\lambda + b & \lambda < -b
\end{cases} $$
The time-varying mesh stiffness \( k_h(\tau) \) is expressed as a Fourier series:
$$ k_h(\tau) = 1 + \sum_{l=1}^{N} \frac{\Lambda_{kl}}{k_m} \cos(l\Omega \tau + \phi_{kl}) $$
Key parameters for the straight bevel gear system are summarized in Table 1.
| Parameter | Symbol | Pinion | Gear |
|---|---|---|---|
| Number of Teeth | \( z \) | 47 | 53 |
| Pressure Angle (°) | \( \alpha_n \) | 20 | 20 |
| Pitch Cone Angle (°) | \( \delta \) | 41.57 | 48.43 |
| Pitch Diameter (mm) | \( d \) | 94 | 106 |
The dynamic behavior of the straight bevel gear system is analyzed through bifurcation diagrams and phase portraits. For a frequency ratio \( \Omega \) in the range [1.5, 1.7], the system transitions from periodic to chaotic motion via period-doubling and Hopf bifurcations. At \( \Omega = 1.64 \), the system exhibits chaos, as shown by irregular phase trajectories and scattered points on the Poincaré section. This chaotic state necessitates control to ensure stable operation of the straight bevel gear transmission.
To address the limitations of traditional PSO, which is prone to local optima and poor convergence, we propose an improved PSO (IPSO) algorithm. The enhancements include chaotic initialization using the Piecewise map, adaptive inertia weight, dynamic learning factors, and a Lévy flight strategy with dynamic center migration. The position update equation incorporates these elements:
$$ w = (w_{\text{max}} – w_{\text{min}}) \cdot \tan\left(0.875 \cdot \left(1 – \left(\frac{t}{T}\right)^k\right)\right) + w_{\text{min}} $$
$$ c_1 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \left(1 – \frac{t}{T}\right)\right)^2 $$
$$ c_2 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \left(\frac{t}{T}\right)\right)^2 $$
$$ p_{is}(t+1) = \begin{cases}
p_{is}(t) + w v_{is}(t) + c_1 R_1 (p_{\text{pbest}} – p_{is}(t)) + c_2 R_2 (p_{\text{gbest}} – p_{is}(t)) & A \leq 0.5 \\
b_2 \cdot p_{is}(t) + b_1 \cdot p_{\text{gbest}} + 0.01 \cdot S_{\text{Lévy}} (p_{is}(t) – p_{\text{gbest}}) & A > 0.5
\end{cases} $$
where \( S_{\text{Lévy}} = \frac{\mu}{|v|^{1/\beta}} \), with \( \mu \sim N(0, \sigma_\mu^2) \), \( v \sim N(0,1) \), and \( \sigma_\mu \) given by:
$$ \sigma_\mu = \left( \frac{\Gamma(1+\beta) \sin(\pi \beta / 2)}{\Gamma[(1+\beta)/2] \beta 2^{(\beta-1)/2}} \right)^{1/\beta} $$
The fuzzy neural network controller is designed with five layers: input, fuzzification, rule, defuzzification, and output. The inputs are Euclidean distances between consecutive points on the Poincaré section, \( d(k) = \| X(k) – X(k-1) \| \) and \( d(k-1) = \| X(k-1) – X(k-2) \| \), and the output is the perturbation \( \Delta \Omega \) applied to the frequency ratio. The membership functions are Gaussian:
$$ \mu_i^M = \exp\left( -\frac{\| D – C_{iM} \|^2}{b_{iM}^2} \right) $$
The output is computed as:
$$ U(k) = \sum_{i=1}^{M} \bar{a}_i w_i $$
where \( \bar{a}_i \) is the normalized firing strength. The IPSO optimizes the parameters \( w \), \( b \), and \( c \) of the fuzzy neural network to minimize the fitness function:
$$ f(P_i) = \sum_{k=1}^{L} |d^* – \| X(k) – X(k-1) \| | $$
Simulation results demonstrate the controller’s ability to stabilize the straight bevel gear system from chaos to period-1 and period-3 motions. The controlled phase trajectories become closed curves, and the Poincaré sections reduce to distinct points. The optimized parameters for the controller are listed in Table 2 and Table 3.
| Weight \( w \) | Width \( b_{1M}, b_{2M} \) | Center \( C_{1M}, C_{2M} \) |
|---|---|---|
| 3.6333 | 0.3362, -1.0568 | 3.2058, 0.3838 |
| 0.3153 | -1.1805, -0.3960 | 3.0718, 2.4506 |
| 2.5269 | -0.3753, 0.7872 | 3.2917, 0.0920 |
| 0.0590 | 2.5240, 0.3347 | 0.3000, -0.2998 |
| 0.1842 | 2.6771, 3.6370 | -0.6434, 2.5122 |
| Weight \( w \) | Width \( b_{1M}, b_{2M} \) | Center \( C_{1M}, C_{2M} \) |
|---|---|---|
| -2.1479 | 1.9751, 1.9751 | 4.0000, 3.9327 |
| 2.7273 | 0.1322, 0.1322 | 0.5962, 3.8547 |
| 0.4573 | 3.2080, 3.2080 | 4.0000, 3.8477 |
| -0.4460 | 4.0000, 4.0000 | 4.0000, -0.2208 |
| 0.1964 | 2.9706, 2.9706 | 3.3554, -0.2319 |
The improved PSO algorithm is validated using benchmark functions, such as Griewank, Rastrigin, and Ackley, showing superior convergence compared to standard PSO. For instance, the Griewank function is minimized to zero, indicating global optimality. This optimization capability ensures efficient parameter tuning for the fuzzy neural network controller in straight bevel gear applications.
In conclusion, the IPSO-optimized fuzzy neural network controller effectively suppresses chaotic vibrations in straight bevel gear systems. By applying small perturbations to the frequency ratio, the system transitions to stable periodic orbits, enhancing reliability and performance. This method offers a universal solution for nonlinear vibration control in gear transmissions, without requiring Jacobian matrix computations or fixed-point localization. Future work could explore real-time implementation and adaptability to varying operational conditions for straight bevel gear systems.