-FAMILY OF INTEGRAL/ INTEGRATION OPERATORS FOR DYNAMIC FIELD PROBLEMS WITH NON-PROPORTIONAL DAMPINGIn this section, a formally standarized W-family of integral/integration operators for dynamic field problems with non-proportional damping are described following along the same lines as in the previous discussions for proportionally damped systems.
The response of elastic structures to various types of excitation has often been determined using the mode superposition method. The mode superposition method is advantageous if the response of the structure is dominated by contributions from a relatively small fraction of the lower modes. In order to use this method without approximations, it is necessary that the damping matrix of the structure be orthogonal with respect to the modal vectors of the undamped system. The damping which satisfies this condition is referred to as proportional (or orthogonal or classical). In the case of proportional damping, a change of coordinates from the physical to generalized coordinates leads to a set of uncoupled equations, one for each mode, that can be solved by relatively simple techniques. If the damping is not proportional, the same change of coordinates leads to a set of coupled equations for the determination of the generalized coordinates due to the fact that the damping matrix cannot be diagonalized by the change of coordinates.
Non-proportional damping problems seem to occur in various fields of structural dynamics to include: i) the dynamic analysis of mechanical structures with rubber mounts or vibration absorber, ii) coupled problems of interacting systems such as fluid-structure or soil-structure problems, and iii) in earthquake response analysis and the like.
Some research has been dedicated to the study of approximations that can be used for solving the set of uncoupled equations. A common approach to the problem of non-proportional damping is to diagonalize the modal damping matrix by simply ignoring the off-diagonal elements (see [12]). However, it has been demonstrated that even for lightly damped systems, the damping can be highly non-proportional and such an approximation yields poor results. Conditions for damping distribution and type of excitation under which such approximations do not degrade response accuracy have been found (see [115] and [124]). Efforts towards solving the coupled equations by an iterative process in which the coupling terms are treated as pseudo-forces (see [10]) have resulted in a method with conditional convergence characteristics, therefore restricting the applicability of such methods.
The modal analysis of dynamic systems with non-proportional damping involves the use of the reduced form of the equation of motion and requires the solution of a 
eigenvalue problem, with complex eigenvalues and eigenvectors.
Despite the large computational effort required, many researchers have used this approach for developing methods largely applicable to mechanical systems (see [53], [109]).
In the present exposition, a generalized methodology encompassing both time integral representations and the bridging of the relationships systematically leading to the so-called integration operators in time applicable to non-proportionally damped mechanical systems is overviewed.
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