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A constrained optimization solution for Caughey damping coefficients in seismic analysis

Danguang Pan (Department of Civil Engineering, University of Science and Technology Beijing, Beijing, China, and State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China)
Chenfeng Li (Zienkiewicz Centre for Computational Engineering, Swansea University, Swansea, UK and Energy Safety Research Institute, Swansea University, Swansea, UK)

Engineering Computations

ISSN: 0264-4401

Article publication date: 2 May 2017




Extended from the classic Rayleigh damping model in structural dynamics, the Caughey damping model allows the damping ratios to be specified in multiple modes while satisfying the orthogonality conditions. Despite these desirable properties, Caughey damping suffers from a few major drawbacks: depending on the frequency distribution of the significant modes, it can be difficult to choose the reference frequencies that ensure reasonable values for all damping ratios corresponding to the significant modes; it cannot ensure all damping ratios are positive. This paper aims to present a constrained quadratic programming approach to address these issues.


The new method minimizes the error of the structural displacement peak based on the response spectrum theory, while all modal damping ratios are constrained to be greater than zero.


Several comprehensive examples are presented to demonstrate the accuracy and effectiveness of the proposed method, and comparisons with existing approaches are provided whenever possible.


The proposed method is highly efficient and allows the damping ratios to be conveniently specified for all significant modes, producing optimal damping coefficients in practical applications.



The research work was supported by the Open Foundation of State Key Laboratory for Disaster Reduction in Civil Engineering (SLDRCE15-01). The authors would also like to thank European Community’s Seventh Framework Programme (Marie Curie International Research Staff Exchange Scheme, Grant No. 612607) and the Sêr Cymru National Research Network in Advanced Engineering and Materials.


Pan, D. and Li, C. (2017), "A constrained optimization solution for Caughey damping coefficients in seismic analysis", Engineering Computations, Vol. 34 No. 3, pp. 682-708.



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