In mechanical transmission systems, straight bevel gears are critical components due to their ability to transmit power between intersecting shafts. However, nonlinear factors such as gear backlash, time-varying stiffness, and support clearances often induce chaotic vibrations, which can severely impact system performance and reliability. I investigate the chaotic motion control of a 7-degree-of-freedom straight bevel gear transmission system using an enhanced particle swarm optimization (PSO) algorithm to optimize a fuzzy neural network (FNN) controller. The primary goal is to suppress chaotic behavior and stabilize the system onto periodic orbits, thereby improving operational stability. The straight bevel gear system’s dynamics are inherently complex, and controlling its chaotic responses requires a robust and adaptive approach. This study integrates computational intelligence techniques to address these challenges effectively.
The dynamics of the straight bevel gear system are modeled using a lumped mass method, accounting for nonlinearities like backlash and time-varying meshing stiffness. The governing equations are derived and solved numerically using the Runge-Kutta method. Under specific parameter ranges, the system exhibits a transition from periodic to chaotic motion, as observed in bifurcation diagrams. For instance, when the dimensionless meshing frequency ratio $\Omega$ varies between 1.5 and 1.7, the system undergoes period-doubling bifurcations and Hopf bifurcations, leading to chaos. To address this, I design an FNN controller that generates perturbations to the system’s controllable parameter, $\Omega$, based on the Euclidean distance between points on the Poincaré section. The FNN parameters are optimized using an improved PSO algorithm, which incorporates adaptive inertia weights, dynamic learning factors, and a Lévy flight strategy to avoid local optima and enhance convergence. Numerical simulations demonstrate that this approach successfully stabilizes chaotic motion into periodic orbits, offering a universal solution for nonlinear vibration control in straight bevel gear systems.
The straight bevel gear system is modeled with seven degrees of freedom, considering vibrations along the coordinate axes and torsional displacements. The equations of motion are normalized to dimensionless form to simplify analysis. The relative torsional displacement $\lambda$ along the meshing line is defined as a key variable, and the backlash function $f(\lambda, b)$ captures the nonlinearity due to gear clearance. The time-varying meshing stiffness $k_h(\tau)$ and static transmission error $e_n(\tau)$ are represented as Fourier series to account for periodic excitations. The dimensionless equations are given by:
$$ \ddot{x}_1 + 2\xi_{x1}\dot{x}_1 + 2a_4\xi_{h1}\dot{\lambda} + k_{x1}x_1 + a_4k_{h1}f(\lambda) = 0 $$
$$ \ddot{y}_1 + 2\xi_{y1}\dot{y}_1 – 2a_5\xi_{h1}\dot{\lambda} + k_{y1}y_1 – a_5k_{h1}f(\lambda) = 0 $$
$$ \ddot{z}_1 + 2\xi_{z1}\dot{z}_1 – 2a_3\xi_{h1}\dot{\lambda} + k_{z1}z_1 – a_3k_{h1}f(\lambda) = 0 $$
$$ \ddot{x}_2 + 2\xi_{x2}\dot{x}_2 – 2a_4\xi_{h2}\dot{\lambda} + k_{x2}x_2 – a_4k_{h2}f(\lambda) = 0 $$
$$ \ddot{y}_2 + 2\xi_{y2}\dot{y}_2 + 2a_5\xi_{h2}\dot{\lambda} + k_{y2}y_2 + a_5k_{h2}f(\lambda) = 0 $$
$$ \ddot{z}_2 + 2\xi_{z2}\dot{z}_2 + 2a_3\xi_{h2}\dot{\lambda} + k_{z2}z_2 + a_3k_{h2}f(\lambda) = 0 $$
$$ -a_1\ddot{x}_1 + a_2\ddot{y}_1 + a_3\ddot{z}_1 + a_1\ddot{x}_2 – a_2\ddot{y}_2 – a_3\ddot{z}_2 + \ddot{\lambda} + 2a_3\xi_h\dot{\lambda} + a_3k_h f(\lambda) = f_{pm} + f_{pv} + f_e \Omega^2 \cos(\Omega\tau) $$
Here, $x_j, y_j, z_j$ are dimensionless displacements, $\xi_{ij}$ are damping ratios, $k_{ij}$ are stiffness coefficients, and $a_1, a_2, a_3$ are geometric constants related to the straight bevel gear parameters. The backlash function $f(\lambda, b)$ is defined as:
$$ f(\lambda, b) = \begin{cases}
\lambda – b & \lambda > b \\
0 & |\lambda| \leq b \\
\lambda + b & \lambda < -b
\end{cases} $$
The parameters for the straight bevel gear system are summarized in the table below, which includes key values used in simulations.
| Parameter | Symbol | Value |
|---|---|---|
| Damping ratios | $\xi_{i1}, \xi_{i2}$ | 0.01 |
| Meshing damping ratios | $\xi_{h1}, \xi_{h2}$ | 0.0125 |
| Stiffness coefficients | $k_{i1}, k_{i2}$ | 1.0 |
| Time-varying stiffness amplitude | $\alpha$ | 0.2 |
| Dimensionless average load | $f_{pm}$ | 0.5 |
| Backlash | $b$ | 1.0 |
The particle swarm optimization algorithm is enhanced to optimize the FNN controller parameters. Standard PSO tends to converge prematurely and get trapped in local optima, especially for high-dimensional problems. I introduce an improved PSO (IPSO) that combines chaotic initialization, adaptive inertia weights, dynamic learning factors, and a Lévy flight strategy with dynamic center migration. The velocity and position update equations are modified as follows. The adaptive inertia weight $w$ is given by:
$$ w = (w_{\text{max}} – w_{\text{min}}) \cdot \tan\left(0.875 \cdot \left(1 – \left(\frac{t}{T}\right)^k\right)\right) + w_{\text{min}} $$
where $w_{\text{max}} = 0.9$, $w_{\text{min}} = 0.1$, $t$ is the current iteration, $T$ is the maximum iterations, and $k = 0.6$. The learning factors $c_1$ and $c_2$ are dynamically adjusted:
$$ c_1 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \left(1 – \frac{t}{T}\right)\right)^2 $$
$$ c_2 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \frac{t}{T}\right)^2 $$
The Lévy flight strategy is incorporated to enhance global exploration. The step size $S_{\text{Lévy}}$ is calculated as:
$$ S_{\text{Lévy}} = \frac{\mu}{|\nu|^{1/\beta}} $$
where $\mu \sim N(0, \sigma_\mu^2)$, $\nu \sim N(0, \sigma_\nu^2)$, $\sigma_\nu = 1$, $\beta = 1.5$, and $\sigma_\mu$ is defined as:
$$ \sigma_\mu = \left( \frac{\Gamma(1 + \beta) \sin(\pi \beta / 2)}{\Gamma[(1 + \beta)/2] \beta \cdot 2^{(\beta – 1)/2}} \right)^{1/\beta} $$
The position update with Lévy flight becomes:
$$ p_{is}(t+1) = b_2 \cdot p_{is}(t) + b_1 \cdot p_{g\text{best}} + 0.01 \cdot S_{\text{Lévy}} \cdot (p_{is}(t) – p_{g\text{best}}) $$
where $b_1$ and $b_2$ are weights that change dynamically with iterations. A Bernoulli distribution-based selection mechanism is used to choose between this update and the standard PSO update with a probability of 0.5.
The fuzzy neural network controller is designed with five layers: input, fuzzification, rule, defuzzification, and output layers. The inputs are the Euclidean distances between consecutive points on the Poincaré section, $d(k) = \| \mathbf{X}(k) – \mathbf{X}(k-1) \|$ and $d(k-1) = \| \mathbf{X}(k-1) – \mathbf{X}(k-2) \|$, and the output is the perturbation $\Delta \Omega$ to the frequency ratio. The membership functions in the fuzzification layer are Gaussian functions:
$$ \mu_i^M = \exp\left( -\frac{\| D – C_{iM} \|^2}{b_{iM}^2} \right) $$
where $C_{iM}$ is the center and $b_{iM}$ is the width of the $M$-th membership function for the $i$-th input. The rule layer computes the firing strengths $a_M$ as the product of membership degrees, and the defuzzification layer normalizes these strengths. The final output $U(k)$ is:
$$ U(k) = \sum_{M=1}^{5} \bar{a}_M w_M $$
where $w_M$ are the weights optimized by IPSO. The controller parameters, including $w_M$, $b_{iM}$, and $C_{iM}$, are optimized to minimize the fitness function:
$$ f(P_i) = \sum_{k=1}^{L} | d^* – \| \mathbf{X}(k) – \mathbf{X}(k-1) \| | $$
where $d^*$ is the desired distance for the target periodic orbit, and $L$ is the data length.
Numerical simulations are conducted to validate the IPSO algorithm and the control strategy. The IPSO is tested on benchmark functions and compared with standard PSO, showing superior performance in convergence and avoidance of local optima. For the straight bevel gear system, chaotic motion at $\Omega = 1.64$ is controlled to period-1 and period-3 orbits. The controlled system exhibits stable periodic behavior in phase plots and Poincaré sections. The optimization results for the FNN parameters are summarized below.
| Weight $w_M$ | Widths $b_{1M}, b_{2M}$ | Centers $C_{1M}, C_{2M}$ |
|---|---|---|
| 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 |
The control process begins after 200 iterations, applying perturbations $\Delta \Omega$ that remain within bounds. The straight bevel gear system’s response quickly transitions from chaos to periodic motion, demonstrating the effectiveness of the IPSO-optimized FNN controller. This approach eliminates the need for Jacobian matrix computations or fixed-point localization, simplifying implementation. The straight bevel gear system’s dynamics are robustly controlled, highlighting the method’s potential for real-world applications in mechanical transmissions.
In conclusion, I have developed a novel control framework for chaotic motion in straight bevel gear systems using an improved PSO algorithm to optimize a fuzzy neural network controller. The straight bevel gear model’s nonlinearities are accurately captured, and the IPSO enhances global search capabilities, ensuring efficient parameter optimization. Simulations confirm that chaotic vibrations are suppressed, and the system stabilizes onto desired periodic orbits. This method provides a universal and adaptive solution for nonlinear vibration control, contributing to the reliability and performance of straight bevel gear transmissions in industrial settings. Future work could explore real-time implementation and experimental validation on physical straight bevel gear systems.