结构动力学中若干问题辨析

针对结构动力学教学过程中运动方程建立、运动方程求解的若干问题进行讨论,主要包括动力学和静力学刚度系数的区别,频响函数和脉冲响应函数的Fourier变换条件及对动力反应的影响,滞后阻尼体系的频响函数等问题。仅考虑集中质量平动自由度的体系,动力学刚度系数是指平动自由度产生单位位移而转动自由度放松情况下所受的力,静力凝聚方法和单位位移法所得刚度系数是相同的;无论是无阻尼体系还是有阻尼体系,频响函数和脉冲响应函数的Fourier变化关系都精确成立;时域特解包含稳态振动和伴生自由振动,而频域特解仅为体系的稳态解,两者之间的差别主要在振动的初始阶段,自振频率越低,差别越大。对于滞后阻尼体系,负频率的频响函数应为正频率频响函数的共轭函数。

Analysis of several problems in structural dynamics

To discuss several problems in the process of establishing and solving equations of motion in structural dynamics teaching. It mainly includes the difference of stiffness coefficients between dynamic and static mechanics as well as their relationship by the static condensation, the Fourier transformation condition from impulse response function to frequency response function and their effect on dynamic response, the frequency response function of the hysteretic damping system. For the lumped-mass system with only translational degree of freedom, the stiffness in dynamics is the force required along DOF due to unit displacement at translational DOF and relaxation of rotational DOFs, and the stiffness coefficients obtained by the static condensation method and the unit displacement method are identical; Whether a system with damping or not, the Fourier transformation relationship between the frequency response function and the impulse response function is accurate; the particular solution includes steady-state vibration and free adjoint vibration in the time domain, however, only the steady-state vibration in the frequency domain. Their difference is evident for the initial stage of vibration. The difference is more noticeable for small natural frequency. For a system with hysteretic damping, the frequency response function of the negative frequency should be the conjugate function of that of the corresponding positive frequency.