Abstract
The non-proportionally damped system is very common in practical engineering structures. The dynamic equations for these systems, in which the damping matrices are coupled, are very time consuming to solve. In this paper, a modal perturbation method is proposed, which only requires the first few lower real mode shapes of a corresponding undamped system to obtain the complex mode shapes of non-proportionally damped system. In this method, an equivalent proportionally damped system is constructed by taking the real mode shapes of a corresponding undamped system and then transforming the characteristic equation of state space into a set of nonlinear algebraic equations by using the vibration modes of an equivalent proportionally damped system. Two numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method. The numerical results show that: (1) with the increase of vibration modes of the corresponding undamped system, the eigenvalues and eigenvectors monotonically converge to exact solutions; (2) the accuracy of the proposed method is significantly higher than the first-order perturbation method and proportional damping method. The calculation time of the proposed method is shorter than the state space method; (3) the method is particularly suitable for finding a few individual orders of frequency and mode of a system with highly non-proportional damping.
Highlights
Damping is one of critical factors affecting a structure’s responses under dynamic excitations.In practical engineering structures, damped systems are commonly characterized non-proportional damping, such as the soil-structure system [1,2,3], the steel-concrete structures [4,5,6], and structures with supplemental dampers [7,8]
An equivalent proportionally damped system, which is taken as the unperturbed system, is constructed from the real mode shapes of a corresponding undamped system and a non-proportional damping matrix
Non-proportional damping structures widely exist in practical engineering
Summary
Damping is one of critical factors affecting a structure’s responses under dynamic excitations. Even though the dynamic response of a non-proportionally damped system can be solved by the direct integration method, the mode superposition method is frequently used. Wang [31] proposed a perturbation method which can deal with the undamped system with repeated eigenvalues Those perturbation methods are usually suitable to analyze the weakly non-proportionally damped system [27,31,32,33], and need the complete eigensolution set of the unperturbed system [27,30]. A modal perturbation method for non-proportional damping problems applied to the vibrations of a highly non-proportionally damped system was proposed. An equivalent proportionally damped system, which is taken as the unperturbed system, is constructed from the real mode shapes of a corresponding undamped system and a non-proportional damping matrix. Several numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method
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