一般阻尼动力系统非奇异摄动法

借助矩阵摄动理论,将模态叠加法运用于一般阻尼矩阵的动力学方程求解结构的动响应是一种较为理想的方法.但当系统的外荷载激振频率接近于系统的固有频率时,直接将阻尼矩阵作为摄动矩阵,会使解产生奇异,并导致求解失败或误差过大,这是因为模态坐标下的动力学方程是无阻尼方程.为了解决这一问题,本文考虑在模态坐标的动力学方程中保留一定的阻尼.即将阻尼做分解,代入振动方程,得到不同阶次摄动方程,再将摄动方程变换到模态坐标,即采用非奇异摄动方法.最后通过数值算例,得到一阶、二阶摄动,将其与精确解进行比较.精度明显得到改善,基本趋于精确解.从而验证了本方法的精确性和有效性.

Non-singular perturbation method for general damped dynamical system

Using the matrix perturbation theory, it is an ideal method to apply the modal superposition method to the dynamic equation of general damping matrix to solve dynamic response of structure. However, when the frequency of the system is close to the natural frequency of the system, the damping matrix as the perturbation matrix make the solution singular and lead to the solution failure or large error, because the dynamic equation under modal coordinates is the undamped equation. In order to solve this problem, th consider to retain some damping in the dynamic equation of the modal coordinateshe damping is decomposed and substituted into the vibration equationobtain the different order perturbation equation, and the perturbation equation is transformed to modal coordinatesthe nonsingular perturbation method. Finally, the first order and second order perturbation solutions are obtained by numerical examples, which are compared with the exact solutions. The accuracy is improved, basically tend to be the exact solution, which verifies the accuracy and effectiveness of the method.