To investigate the statistical errors inherent in this classical process and its influence on PSDT measurements,based on the perturbation technique,the mean and variance of the quotient of the random variables are approximately expressed with the application of the statistical moment method. Then,the statistical moments of two response spectral estimates are substituted to derive the formulae for the mean and variance of the PSDT’s amplitude regarding the coherence function of the response and the number of averaging segments in spectrum analysis. Based on this,the error law in the amplitude of the PSDT estimate at the resonance frequency is revealed to quantify the estimation error of the mode shape. It is found that the error in the PSDT’s amplitude estimate at the resonances tends to a local minimum value,and the coefficient of variation is smaller than the counterparts of two associated response power spectrum densities. Finally,the accuracy of the derived error formulation in the paper is verified by vibration data from one numerical frame. Moreover,a parameter study regarding the influence of the selection of reference response,the time duration of measurements,and the type of window function on the PSDT and mode shape estimates is performed. The results show that using two basic responses of PSDT as the reference responses is beneficial to reduce the error in PSDT estimates and outcomes of modal analysis,and the standard variance of the estimate also decreases with the increasing time duration of measurements and finally to a specific limit.

摘要

为了研究响应谱估计误差及其传递对振动响应功率谱密度传递比（Power Spectrum Density Transmissibility,PSDT）估计的影响，基于摄动理论和统计矩定义，推导了两个变量比例函数的均值和方差近似表达式；将响应谱估计统计矩代入，可以推导出由响应相干函数、谱估计中信号平均分段数，近似表征的PSDT估计幅值的均值和方差解析公式.基于此，揭示了共振频率处PSDT估计幅值误差规律，并实现了模态振型幅值的精度度量.研究发现，共振频率处PSDT幅值方差存在极小值，且变异系数小于相关响应谱.通过数值框架数据验证了文中误差公式的准确性.此外，还研究了参考响应的选择、响应时长、窗函数类型对PSDT和模态振型估计的影响.结果表明，以PSDT两组响应作为参考响应，能得到较好PSDT和模态分析结果；同时模态振型估计标准差随测试数据时长的增加，也随之降低至一定水平.