参数激励下非线性受控系统的动力学和稳定性

研究两自由度参数激励系统的非线性动力学与控制问题．利用Lagrange方程建立含反馈控制的参激捅及其驱动机构组成的系统动力学方程，以多尺度方法获得一阶近似控制方程．然后，对系统受一阶摸态参激主共振与一、二阶模态间3：1内共振联合作用下的幅额响应及其稳定性，以及反馈参数对系统稳态行为的影响作了详细分析．结果表明，响应的稳定域位置和大小取决于位移反馈，位移立方反馈改变了系统的非线性程度，速度反馈类似于阻尼，可使系统呈现自激振动特性．

DYNAMICS AND STABILITY OF A NON-LINEAR CONTROLLED SYSTEM SUBJECT TO PARAMETRIC EXCITATION

The non-linear dynamic system subject to the parametric excitation has received wide attention in connection with interest in application in many fields. When a control strategy is applied to such a system, some new phenomena or special purposes, such as controlling the response, increasing the stable region or decreasing the transient time, can be achieved. Thus, the study on the dynamics of a parametrically excited system is the basis of effective control. As a typical case of a parametrically excited controlled system, a two degree-of-freedom system composed of a parametrically excited pendulum and its driving device, which is controlled by the feedback combination of the displacement, velocity and cubic displacement of the pendulum, is studied in this paper. The Lagrange theory is used to establish the dynamic equations. The method of multiple scales is used to attack directly the governing nonlinear equation and derive a set of first-order nonlinear ordinary differential equations, namely the modulation equations, in the case of principal parametric resonance of the first mode and a three-to-one internal resonance between the two modes. The steady-state response, the corresponding stability of the system and the influences on the system arising from the feedback parameters, are numerically investigated. The results can be summarized as follows.The position and the size of the parametric region for stable response vary along with the parameters of magnitude and frequency of the parametric excitation, the damping coefficient, the detuning parameter of the internal resonance and the feedback coefficients. The evolvement of the dynamic characters is determined by the magnitude of parametric excitation if the system is characterized by the same damping. The trivial solution undergoes the pitchfork bifurcation and the Hopf bifurcation with the variation of the frequency of parametric excitation when no control is applied. The control strategy can change the position and the size of the parametric region for stable response and stability of the solution. For example, the displacement feedback changes the position and size of the parametric region for stable response; the cubic displacement feedback changes the extent of the non-linearity; the velocity feedback plays the same role as the damping, and may change the damping to negative so that some self-excitation may exist in the system.