基于复阻尼模型水平剪切型结构的时域解析算法

A Time-domain Analytical Method for Horizontal Shear Structures Based on the Complex Damping Model

摘要:

近些年来提出的数值方法虽克服了应用复阻尼模型求解动力响应时可能出现的发散现象，但其计算过程繁杂且不能表达出结构动力特性及响应随结构参数的变化规律。论文参照三对角Toeplitz矩阵特征值问题的数学解法，推导出水平剪切型结构各阶自振频率、振型函数的解析形式。通过对运动方程进行Fourier变换，得到复阻尼理论下结构的传递函数解析式，直观地表达出结构动力特性及响应随结构参数的变化规律。最后通过算例对比分析，结果表明：提出的复阻尼模型的解析计算方法克服了时域发散问题，并与时域数值计算方法的位移响应时程对比发现，两种方法得到的时程曲线吻合程度较高，位移峰值也基本一致。同时，对比两种方法在底层刚度变化时的动力响应，解析方法的计算曲线更光滑，避免了数值方法的离散误差问题。

Abstract:

Although the numerical methods proposed in recent years overcome the possible divergence phenomenon when applying the complex damping model to solve the structural dynamic response, the calculation process is complicated and cannot express the structural dynamic characteristics and the change law of response with structural parameters. This paper derived the analytic forms of natural frequencies and mode functions of each order of a horizontal shear structure by referring to the mathematical solution of tridiagonal Toeplitz matrix eigenvalue problem. Through Fourier transform of the motion equation, the analytical equation of transfer function of the structure under complex damping theory was obtained, and the dynamic characteristics of the structure and the variation of the response with structural parameters were expressed intuitively. The results showed that the analytical method based on the complex damping model overcomes the time-domain divergence problem. Then we compared it with the time-domain numerical calculation method and found that the time-history curves of displacement response obtained by the two methods are consistent with each other, and the peak displacements are basically the same. In addition, by comparing the dynamic responses of the two methods when underlying stiffness changes, the curves of the analytical method are smoother and the discrete error of numerical method is avoided.