Complex complete quadratic combination method for damped system with repeated eigenvalues |
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| Authors: | Ruifang Yu Xiyuan Zhou Abduwahit Abduwaris |
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| Institution: | 1.Institute of Geophysics,China Earthquake Administration,Beijing,China;2.Beijing University of Technology,Beijing,China;3.Earthquake Administration of Xinjiang Uyghur Autonomous Region,Wulumuqi,China |
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| Abstract: | A new response-spectrum mode superposition method, entirely in real value form, is developed to analyze the maximum structural response under earthquake ground motion for generally damped linear systems with repeated eigenvalues and defective eigenvectors. This algorithm has clear physical concepts and is similar to the complex complete quadratic combination (CCQC) method previously established. Since it can consider the effect of repeated eigenvalues, it is called the CCQC-R method, in which the correlation coefficients of high-order modal responses are enclosed in addition to the correlation coefficients in the normal CCQC method. As a result, the formulas for calculating the correlation coefficients of high-order modal responses are deduced in this study, including displacement, velocity and velocity-displacement correlation coefficients. Furthermore, the relationship between high-order displacement and velocity covariance is derived to make the CCQC-R algorithm only relevant to the high-order displacement response spectrum. Finally, a practical step-by-step integration procedure for calculating high-order displacement response spectrum is obtained by changing the earthquake ground motion input, which is evaluated by comparing it to the theory solution under the sine-wave input. The method derived here is suitable for generally linear systems with classical or non-classical damping. |
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