Chapter 6Reduced Computational Methods for Damped Systems

Dynamics of viscously damped systems and general non-viscously damped systems have been discussed in the previous chapters. It was shown that even when the system is not proportionally damped, the dynamic response can be expressed as superposition of complex modes. Therefore, the determination of the eigensolutions of the system is of crucial importance. Some perturbation-based methods have been proposed for the calculation of the eigenvalues and eigenvectors so far. In this chapter, we explore this problem in further detail. The aim is to develop computational methods for eigensolutions, which are computationally efficient and, at the same time, accurate. These two demands are often conflicting in nature. We develop some iterative methods that can address these issues to some extent. Once the eigensolutions can be obtained using the approach proposed in this chapter, they can be used in the expressions of the dynamic response derived in the previous chapters.

In section 6.1, we consider reduced computational methods for general non-proportionally damped systems with viscous damping. An iterative approach is presented here. In section 6.2, approximate methods for single-degree-of-freedom (SDOF) non-viscously damped systems are discussed. These ideas are extended to multiple-degrees-of-freedom (MDOF) non-viscously damped systems in section 6.3. In section 6.4, a different approach to reduce computational cost is pursued. ...

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