Response of non‐classically damped structures in the modal subspace

Abstract

SUMMARYThe evaluation of the dynamic response of non-classically damped linear structures requires the solution of an eigenproblem with complex eigenvalues and modal shapes. Since in practice only a small number of complex modes are needed, the complex eigenvalue problem is solved in the modal subspace in which the generalized damping matrix is not uncoupled by classical real modes. It follows that the evaluation of the structural response requires in both cases the determination of complex modes by numerical techniques, which are not as robust as techniques currently used for the solution of the real eigenvalue problem, and the use of complex algebra. In the present paper an unconditionally stable step-by-step procedure is presented for the response of non-classically damped structures in the modal subspace without using complex quantities. The method is based on the evaluation of the fundamental operator in approximated form of the numerical procedure. In addition, the method can be easily modified to incorporate the modal superposition pseudo-static correction terms.

New real‐value modal combination rules for non‐classically damped structures

“…xr xs (10) in which E[·] means the stochastic average of (·). The expression of these last coe cients can be easily obtained once the kind of stochastic input is chosen.…”

“…Recently, modal superpositions not requiring the evaluation of complex eigenproperties have been proposed [7][8][9][10].…”

Response of Beams With Crack of Uncertain-but Bounded Depth Subjected to Deterministic or Stochastic Loads

“…(24) and (25) cannot decoupled by a real coordinate of transformation, then they are referred in literature as non-classically damped systems. It has been recognized that in order to evaluate the response of non-classically damped systems the state variables have to be introduced [11].…”

“…The solution of these equations can be obtained by applying the numerical procedure proposed by [11] writing: …”

Improved dynamic correction method in seismic analysis of both classically and non‐classically damped structures

“…For both classically and non-classically damped structures, it has been recognized that it is more convenient to operate in the modal subspace than in the nodal space [16][17][18]. However, when the degrees of freedom are numerous, only the lowest frequencies and the corresponding lowest modes are usually computed.…”