In this study, an analytical model is presented to determine the random response of point-driven portal and multi-bay planar frame structures. Coupling effects between bending and longitudinal deformations are taken into account, with the Timoshenko-Ehrenfest beam theory being applied to model the bending deformations. With the excitation taken as band-limited white noise, expressions are derived for the mean square displacements and velocities in terms of the autocorrelation and cross-correlation components. The influence of modal cross-correlations on the overall response is shown to be dependent on the number of bays. For a lightly damped single-bay frame, the natural frequencies are generally well separated and the modal cross-correlations are small. In this situation, the velocity response displays a near symmetric distribution about the center point of the frame. Moreover, narrow zones of intensified response begin emerging as the number of responding modes increases. For frames having two or more bays, the contribution of modal cross-correlations is larger due to the increased occurrence of clusters of natural frequencies. In such cases, modal cross-correlations introduce asymmetry into the overall response distribution of the frame. Additionally, the drive-point velocity of the multi-bay frame can be severely underestimated if modal cross-correlations are ignored. The study also investigates the influence of increased damping on the response characteristics.

中文翻译：

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点驱动门式框架和多开间平面框架的随机振动

在本研究中，提出了一个分析模型来确定点驱动门式和多开间平面框架结构的随机响应。考虑了弯曲和纵向变形之间的耦合效应，并应用 Timoshenko-Ehrenfest 梁理论来模拟弯曲变形。将激励视为带限白噪声，根据自相关和互相关分量导出均方位移和速度的表达式。结果表明，模态互相关对整体响应的影响取决于间隔的数量。对于轻阻尼单隔间框架，固有频率通常分离良好，模态互相关性很小。在这种情况下，速度响应显示出关于框架中心点的近对称分布。此外，随着响应模式数量的增加，强化响应的狭窄区域开始出现。对于具有两个或更多间隔的框架，由于固有频率簇的出现增加，模态互相关的贡献更大。在这种情况下，模态互相关会给框架的整体响应分布带来不对称性。此外，如果忽略模态互相关，多托架框架的驱动点速度可能会被严重低估。该研究还调查了增加阻尼对响应特性的影响。