Nonlinear frequency response analysis of structural vibrations

Abstract

In this paper we present a method for nonlinear frequency response analysis of mechanical vibrations of 3-dimensional solid structures. For computing nonlinear frequency response to periodic excitations, we employ the wellestablished harmonic balance method. A fundamental aspect for allowing a large-scale application of the method is model order reduction of the discretized equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. For an efficient spatial discretization of continuum mechanics nonlinear partial differential equations, including large deformations and hyperelastic material laws, we use the isogeometric finite element method, which has already been shown to possess advantages over classical finite element discretizations in terms of higher accuracy of numerical approximations in the fields of linear vibration and static large deformation analysis. With several computational examples, we demonstrate the applicability and accuracy of the modal derivative reduction method for nonlinear static computations and vibration analysis. Thus, the presented method opens a promising perspective on application of nonlinear frequency analysis to large-scale industrial problems.

On polynomial hyperreduction for nonlinear structural mechanics

“…Thus, various alternative approaches for computation of the matrix Q for the nonlinear case have been suggested. A well‐known method is based on expansion of the eigenmode basis by the so‐called modal derivatives and was successfully applied in several works . However, the approach with modal derivatives has considerable drawbacks.…”

“…As far as the projection step is concerned, it can be shown that modal reduction and Krylov subspaces are not sufficient for reduction of nonlinear equations . In some scenarios, modal derivatives can be used to enrich the common basis and lead to accurate results for real‐life examples . However, we believe that knowledge of the design space has to be exploited to find appropriate projection for a general nonlinear case.…”

“…6 In some scenarios, modal derivatives 11,12 can be used to enrich the common basis and lead to accurate results for real-life examples. [13][14][15] However, we believe that knowledge of the design space has to be exploited to find appropriate projection for a general nonlinear case. One possible approach, which takes this information into account, is application of the proper orthogonal decomposition (POD).…”

Nonlinear model order reduction for flexible multibody dynamics: a modal derivatives approach

“…This choice could be more practical than the nodal fixed frame, as the best choice of the specified fixed nodes in the nodal fixed frame is not straightforward in three dimensional cases. Also, the MDs have been successfully calculated for three dimensional models [29,39] in the inertia frame, to solve nonlinear problems without large rigid body motion. It is worth mentioning that the benefits of the proposed technique will be even more apparent for three dimensional problems that would feature, in general, larger finite element meshes and therefore provide larger gains from the adopted modal approach.…”

A Taylor basis for kinematic nonlinear real‐time simulations. Part II: The Taylor basis

“…To resolve these singularity issues, many modifications have been suggested, see, eg, previous studies. 1,[18][19][20][21][22][23] However, common to all of the suggestions presented is that they represent some kind of approximation. The approximation issue was, however, solved in Part I of the present work where a set of equations governing the complete MDs were derived by use of perturbation methods.…”