受迫简谐响应分析中的特征值问题（英文）

Eigenvalue problems in forced harmonic responses analysis in structural dynamics

无阻尼结构的受迫振动的共振频率与自由振动的特征值直接相关。在频域响应谱中,共振频率对应于响应峰值位置。指出频谱中的低谷(相对最小值)对应的频率也可用特征值问题求解。当最小值为0时,对应的频率是著名的反共振频率。另一种可能是,处于两个共振频率之间存在非零的最小响应,对应的频率称为最小响应频率。基于特征值问题的列式,反共振频率或最小响应频率的灵敏度分析可以直接通过已有的特征值灵敏度分析方法求解。给出了详细的数学推导并通过数值算例验证。

In structural dynamics, it is well known that resonance frequencies in forced harmonic response of an undamped structure are related to the eigenvalue for free vibration problem. These resonance frequencies are the peaks in the frequency response plots. In this paper,it is shown that the frequencies for the valleys （lowest points in vibration amplitude） in the frequency response plot can also be computed by solving eigenvalue problems associated with the forced harmonic response of the system. When the minimum response is zero, the corresponding frequen- cies are the well-known antiresonsnt frequencies. Otherwise, these are minimum response frequency between two resonances. Using the eigenvalue formulation, the sensitivities of the antiresonant or minimum response frequency can be readily computed using eigenvalue sensitivity results for general non-symmetrical matrices. Detailed formulations are presented in the paper. Several numerical examples are included to illustrate the general formulations.