In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency ( ω 1 ), detuning parameter ( σ 1 ), feedback signal gain ( β ), control signal gain ( γ ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.

中文翻译：

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通过 1:3 内部共振的复合悬臂梁分岔分析

在本文中，我们研究了用于模拟非线性复合梁模型振动的系统的振动分析和控制的多尺度扰动和数值解。介绍了由二次项和三次项引起的非线性的二阶微分方程组，由参数和外部激励激励。控制器用于控制机械系统可能损坏的初级和参数共振的一个频率。主动控制应用于系统。实施多尺度扰动 (MSP) 方法以获得近似解析解。系统的稳定性分析是通过频率响应（FR）获得的。分岔分析使用各种控制参数进行，例如固有频率 (ω 1 )、失谐参数 (σ 1 )、反馈信号增益 (β)、控制信号增益 (γ) 和其他参数。系统的动态行为在各种分岔参数范围内进行预测。所有稳定稳态（点吸引子）、稳定周期吸引子、不稳定稳态和不稳定周期吸引子都可以使用分岔分析有效地确定。控制器对系统行为的影响是通过数值检验的。为了验证我们的结果，将使用 MSP 方法的近似解析解与使用四阶 Runge-Kutta (RK) 方法的数值解进行比较。使用分叉分析可以有效地确定不稳定的稳态和不稳定的周期性吸引子。控制器对系统行为的影响是通过数值检验的。为了验证我们的结果，将使用 MSP 方法的近似解析解与使用四阶 Runge-Kutta (RK) 方法的数值解进行比较。使用分叉分析可以有效地确定不稳定的稳态和不稳定的周期性吸引子。控制器对系统行为的影响是通过数值检验的。为了验证我们的结果，将使用 MSP 方法的近似解析解与使用四阶 Runge-Kutta (RK) 方法的数值解进行比较。