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Determination of the damping properties of structures is an important problem, often investigated in the search for more satisfactory or accurate solutions. One of the most popular approaches is based on the addition of modal damping to the model. This method is used not only for the sake of analytical simplicity, but also because it is the most favorable way to measure or estimate. This is the way, for example, to estimate the material damping in finite element analysis of large flexible structures. The resulting damping matrix in modal co-ordinates is a diagonal one. The structural damping, for which the modal damping matrix is diagonal, is called classical, or proportional damping. In another approach a damping matrix proportional either to the mass, or to the stiffness matrix, or to both, is introduced. This technique produces proportional damping as well.
Some researchers have sought to replace the non-proportional damping with proportional damping. For example, Cronin [1] used a perturbation technique to approximate a solution for a non-proportionally damped system with harmonic excitation. Chung and Lee [2] applied a perturbation of the proportional damping system by the non-proportional damping matrix to determine the eigensolutions of weakly, non-proportionally damped systems. Bellos and Inman [3] proposed a simple decoupling technique that is not limited either by magnitude of damping or separation of the natural frequencies. The bounds on the response of a non-classically damped system under harmonic and transient excitation were investigated by Yae and Inman [4] and by Nicholson [5], respectively. Falszeghy [6] proposed a new orthogonal co-ordinate transformation that transforms a non-classically damped system into a form in which the error generated by discarding the off-diagonal terms of the damping matrix is minimized. However, the simplest and the most common approach to the problem is to replace the full damping matrix with a diagonal one by neglecting the off-diagonal terms of the non-proportional damping matrix. Several researchers have studied the error bounds generated by this simplified approach (see, for example, Shahruz and Ma [7], Shahruz, [8], Uwadia and Esfandiari [9], Hwang and Ma [10] and Bhaskar [11]). The bounds derived for the case of arbitrary damping are often too conservative for many practical applications (e.g., in example 2 of Shahruz [8], the actual maximal error was 4%, while the bounds spanned the range of 218%). This is the price paid for considering a general case, in which arbitrary values of damping are allowed. However, there is a considerably