The study on quantification of damping is inspired by a simple question: how much damping is there in a system? The study on damping quantification is important for practical dynamic systems as it is related to dynamic amplification near the resonance frequencies. In the context of a proportionally damped system, the answer to this question is straightforward as the damping is completely quantified by the modal damping factor. However, for non-proportionally damped systems and non-viscously damped systems, the answer to this question is less obvious. One reason for this is due to the complex modes, as discussed in detail in [ADH 14]. In this chapter, we discuss quantification of non-proportional and non-viscous damping.

In section 4.1, methods for the quantification of non-proportional damping are discussed by introducing the idea of optimally normalized complex modes. In section 4.2, the quantification of non-viscous damping is discussed. Four non-viscosity indices are proposed for this purpose. The results obtained in this chapter are summarized in section 4.3.

Both viscous and non-viscously damped systems, in general, have complex modes. When damping is non-proportional in nature, the equation of motion is coupled through the modal damping matrix, **C′** = **X**^{T}**CX**. A common approach in this case is simply to ignore the off-diagonal terms of the modal damping matrix **C′** that couple the equation of motion. ...

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