In mechanical transmission systems, the presence of nonlinear factors such as backlash, time-varying stiffness, and support clearances often leads to chaotic vibrations, which can severely impact performance and reliability. This study focuses on the chaos control in a 7-degree-of-freedom straight bevel gear transmission system, where chaotic motion emerges under specific parameter ranges. We propose a novel control strategy that combines an improved particle swarm optimization (PSO) algorithm with a fuzzy neural network (FNN) controller to suppress chaos and stabilize the system into periodic motion. The approach involves modeling the system dynamics, analyzing chaotic transitions, designing an FNN controller, and optimizing its parameters using an enhanced PSO method. Numerical simulations demonstrate the effectiveness of this method in controlling chaos, providing a universal solution for nonlinear vibration mitigation in straight bevel gear systems.
The straight bevel gear system is widely used in power transmission due to its ability to transmit motion between intersecting shafts. However, the inherent nonlinearities, including gear backlash and time-varying meshing stiffness, can induce complex dynamic behaviors such as chaos. Chaos in straight bevel gear systems manifests as irregular, unpredictable vibrations that exacerbate wear, noise, and fatigue failure. Traditional control methods like OGY require precise knowledge of system fixed points and Jacobian matrices, which are challenging to obtain for high-dimensional, non-smooth systems. To address this, we leverage intelligent control techniques, specifically a fuzzy neural network, which approximates nonlinear mappings without explicit system models. The controller parameters are optimized using an improved PSO algorithm that incorporates adaptive mechanisms and Lévy flight strategies to avoid local optima and enhance convergence. This integrated approach ensures robust chaos control for straight bevel gear systems under various operating conditions.
We begin by establishing a dynamic model for the 7-degree-of-freedom straight bevel gear system using the lumped mass method. The model accounts for translational and torsional vibrations along the coordinate axes, as well as nonlinearities from backlash and time-varying stiffness. The governing equations are derived and normalized to facilitate analysis. Key parameters, such as gear geometry and operating conditions, are summarized in Table 1. For instance, the straight bevel gear pair consists of a pinion and gear with specific tooth numbers, pressure angles, and pitch cone angles. The dynamic model includes differential equations for displacements and a relative torsional displacement along the meshing line. The time-varying meshing stiffness is expressed as a Fourier series, and backlash is modeled using a piecewise linear function. This comprehensive model captures the essential nonlinear dynamics that lead to chaos in straight bevel gear systems.
| Parameter | Symbol | Pinion | Gear |
|---|---|---|---|
| Number of Teeth | z | 47 | 53 |
| Pressure Angle (°) | α_n | 20 | 20 |
| Pitch Cone Angle (°) | δ | 41.57 | 48.43 |
| Pitch Diameter (mm) | d | 94 | 106 |
The dimensionless equations of motion for the straight bevel gear system 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, \( \lambda \) is the relative torsional displacement, \( f(\lambda) \) is the backlash function, and \( \Omega \) is the frequency ratio. The coefficients \( a_1, a_2, a_3, a_4, a_5 \) are derived from geometric relations of the straight bevel gear. The backlash function is defined as:
$$ f(\lambda, b) = \begin{cases}
\lambda – b & \lambda > b \\
0 & |\lambda| \leq b \\
\lambda + b & \lambda < -b
\end{cases} $$
To analyze the chaotic behavior, we numerically solve these equations using the Runge-Kutta method. The bifurcation diagram with respect to the frequency ratio \( \Omega \) reveals transitions from periodic to chaotic motion. For example, in the range \( \Omega \in [1.5, 1.7] \), the system undergoes period-doubling bifurcations, Hopf bifurcations, and chaos. At \( \Omega = 1.64 \), the phase portrait shows non-repeating trajectories, and the Poincaré section exhibits scattered points, confirming chaos. This analysis identifies critical parameter regions where control is necessary for the straight bevel gear system.
Chaos control is achieved by applying small perturbations to a system parameter, such as the frequency ratio \( \Omega \). We design a fuzzy neural network controller that takes the Euclidean distance between consecutive points on the Poincaré section as input and outputs the perturbation \( \Delta \Omega \). The FNN controller consists of five layers: input layer, fuzzification layer, rule layer, defuzzification layer, and output layer. The input variables are \( d(k) = \| X(k) – X(k-1) \| \) and \( d(k-1) = \| X(k-1) – X(k-2) \| \), where \( X(k) \) is the coordinate on the Poincaré section at iteration \( k \). The fuzzification layer uses Gaussian membership functions with centers \( C_{iM} \) and widths \( b_{iM} \). The rule layer computes the firing strengths, and the output layer produces the control signal:
$$ U(k) = \sum_{i=1}^{M} \bar{a}_i w_i $$
where \( \bar{a}_i \) are normalized firing strengths, and \( w_i \) are weight parameters. The controller is designed to minimize the error \( e(k) = d^* – d(k) \), where \( d^* \) is the desired distance for periodic motion.
To optimize the FNN parameters (weights \( w \), centers \( C \), and widths \( b \)), we employ an improved PSO algorithm. Standard PSO tends to converge prematurely and get trapped in local optima, especially for high-dimensional problems. Our enhancements include chaotic initialization using the Piecewise map, adaptive inertia weight, dynamic learning factors, and Lévy flight strategies. The Piecewise map for initialization is given by:
$$ p(t+1) = \begin{cases}
\frac{p(t)}{q} & 0 \leq p(t) \leq q \\
\frac{p(t)-q}{0.5-q} & q \leq p(t) < 0.5 \\
\frac{1-q-p(t)}{0.5-q} & 0.5 \leq p(t) < 1-q \\
\frac{1-p(t)}{q} & 1-q \leq p(t) < 1
\end{cases} $$
with \( q = 1 \). The adaptive inertia weight \( w \) is updated as:
$$ w = (w_{\text{max}} – w_{\text{min}}) \cdot \tan\left(0.875 \cdot \left(1 – \left(\frac{t}{T}\right)^k\right)\right) + w_{\text{min}} $$
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:
$$ c_1 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \left(1 – \frac{t}{T}\right)\right)^2 $$
$$ c_2 = 2 \cdot \sin\left(\frac{\pi}{2} \cdot \frac{t}{T}\right)^2 $$
Lévy flight is incorporated to enhance global exploration. The step size \( S_{\text{Lévy}} \) is calculated as:
$$ S_{\text{Lévy}} = \frac{\mu}{|v|^{1/\beta}} $$
where \( \mu \sim N(0, \sigma_\mu^2) \), \( v \sim N(0,1) \), \( \beta = 1.5 \), and \( \sigma_\mu \) is defined using the Gamma function. The position update with Lévy flight is:
$$ p_{is}(t+1) = b_2 \cdot p_{is}(t) + b_1 \cdot p_{gBest} + 0.01 \cdot S_{\text{Lévy}} \cdot (p_{is}(t) – p_{gBest}) $$
where \( b_1 \) and \( b_2 \) are time-varying weights. A Bernoulli distribution-based selection mechanism chooses between this update and the standard PSO update with probability 0.5. The fitness function for PSO is defined as the sum of absolute errors between the desired and actual Poincaré distances:
$$ f(P_i) = \sum_{k=1}^{L} |d^* – \| X(k) – X(k-1) \| | $$
This ensures that the optimized FNN parameters minimize the deviation from periodic behavior in the straight bevel gear system.
We conduct numerical simulations to validate the control approach. The system parameters are set as follows: damping ratios \( \xi_{i1} = \xi_{i2} = 0.01 \) (for \( i = x, y, z \)), \( \xi_{h1} = \xi_{h2} = 0.0125 \), \( \xi_h = 0.05 \), stiffness coefficients \( k_{i1} = k_{i2} = 1 \), \( k_{h1} = k_{h2} = 0.5 \), time-varying stiffness amplitude \( \alpha = 0.2 \), mean load \( f_{pm} = 0.5 \), dynamic load \( f_{pv} = 0 \), error excitation \( f_e = 0.2 \), and backlash \( b = 1.0 \). The PSO parameters include a population size of 150 and maximum iterations of 100. The FNN parameters are optimized within bounds: \( w \in [-6, 6] \), \( b \in [-6, 6] \), \( c \in [-6, 6] \). Control is initiated after 200 iterations when chaos is evident.
Table 2 summarizes the optimized FNN parameters for controlling chaos to period-1 motion. The controller successfully stabilizes the system, as shown by the phase portrait converging to a closed curve and the Poincaré section reducing to a single point. Similarly, for period-3 motion, the parameters in Table 3 achieve control, with the phase portrait displaying three closed curves and the Poincaré section having three distinct points. The perturbations \( \Delta \Omega \) are small, ensuring minimal impact on system operation. These results demonstrate the efficacy of the improved PSO-optimized FNN in controlling chaos for straight bevel gear systems.
| Weight \( w \) | 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 |
| Weight \( w \) | Widths \( b_{1M}, b_{2M} \) | Centers \( C_{1M}, C_{2M} \) |
|---|---|---|
| -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 benchmarked against standard PSO on test functions like Ackley, Griewank, and Rastrigin. Results show that the enhanced version achieves lower error values and faster convergence, validating its superiority for optimizing FNN parameters in high-dimensional spaces. For instance, on the Griewank function, improved PSO reaches the global optimum of 0, while standard PSO stagnates at a higher value. This performance translates to better control outcomes for the straight bevel gear system.
In conclusion, we have developed a robust chaos control framework for straight bevel gear systems using an improved PSO-optimized fuzzy neural network. The dynamic model captures essential nonlinearities, and the control strategy effectively suppresses chaos by applying minimal perturbations to the frequency ratio. The enhanced PSO algorithm ensures optimal controller parameters through adaptive mechanisms and Lévy flight, avoiding local optima and improving convergence. Simulations confirm that the system transitions from chaos to stable periodic motion, underscoring the method’s practicality. This approach offers a universal solution for managing nonlinear vibrations in straight bevel gear transmissions, with potential applications in automotive, aerospace, and industrial machinery. Future work could explore real-time implementation and adaptation to varying operational conditions for straight bevel gear systems.