A nonlinear component mode synthesis method for the computation of steady-state vibrations in non-conservative systems

Abstract

a b s t r a c tThis paper presents an extension to classic component mode synthesis methods to compute the steady-state forced response of nonlinear and dissipative structures. The procedure makes use of the nonlinear complex modes of each substructure, computed by means of a modified harmonic balance method, in order to build a reduced-order model easily solved by standard iterative solvers. The proposed method is applied to a mistuned cyclic structure subjected to dry friction forces, and proves particularly suitable for the study of such systems with high modal density and non-conservative nonlinearities.

Reduced-order modelling using nonlinear modes and triple nonlinear modal synthesis

“…The method relies on a substructuring approach [6] in order to provide both versatility and computational efficiency. It should be reminded that the second step of the procedure and the spectral substitution were already tested and validated in a previous work [25], which has led to the development of the method presented here so as to reveal its full potential on industrial structures.…”

“…In this work, the definition of a nonlinear mode is that proposed by Laxalde and Thouverez in [15], and later used in several papers [24,25,[29][30][31]. This definition comes with a very flexible computation procedure inspired by the HBM, allowing to tackle a broad range of systems without any restriction as to the intensity, smoothness, and dissipation of the nonlinear effects.…”

“…Practically speaking, the real part of λ is neglected in the exponential functions below the integral sign of the inner product. This assumption is not equivalent to neglecting the decay of the modes, and has been validated in numerous papers dealing with nonlinear complex modes [15,24,25]. The procedure yields a set of nonlinear and algebraic eigenproblems,…”

“…In practice, the nonlinear law is usually known as a function of the physical DOFs only, and rebuilding w n (t) to determine the nonlinear forces at each step of the solver could eventually impact the efficiency of the method. The next section shows how this term can be supplemented by an approximation relying on the eigenvalues of the nonlinear modes, referred to as spectral substitution, introduced and validated in a previous work [25].…”

“…The method derived hereafter overcomes this difficulty by retaining nonlinear eigenvectors in the reduction basis, allowing to treat the nonlinear DOFs as slave coordinates of the reduction. Moreover, most of the time frequency-based appoaches require a systematic switch from frequency domain to time domain in order to evaluate the nonlinear forces through a so-called alternating frequency-time (AFT) scheme [24,25], which can be computationally intensive, though providing great flexibility as to the nature of the nonlinearities that can be tackled. To circumvent this numerical cost, it is proposed here to approximate the nonlinearities using the nonlinear eigenvalues of the nonlinear modes in a procedure later referred to as spectral substitution.…”