Forced harmonic response analysis of nonlinear structures using describing functions

Abstract

The dynamic response of multiple-degree-of-freedom nonlinear structures is usually determined by numerical integration of the equations of motion, an approach which is computationally very expensive for steady-state response analysis of large structures. In this paper, an alternative semianalytical quasilinear method based on the describing function formulation is proposed for the harmonic response analysis of structures with symmetrical nonlinearities. The equations of motion are converted to a set of nonlinear algebraic equations and the solution is obtained iteratively. The linear and nonlinear parts of the structure are dealt with separately, the former being represented by the constant linear receptance matrix [a], and the latter by the generalized quasilinear matrix [A] which is updated at each iteration. A special technique that reduces the computation time significantly when the nonlinearities are localized is used with success to analyze large structures. The proposed method is fully compatible with standard modal analysis procedures. Several examples dealing with cubic stiffness, piecewise linear stiffness, and coulomb friction type of nonlinearities are presented in the case of a ten-degree-of-freedom structure.

An Experimental Characterization of the Impedance and Spectral Content of Multistable Structural Responses During Dynamic Bifurcations

“…On the other hand, it is difficult to generalize the methods of analysis for single-DOF systems to structures of higher dimension, such as for buckled panels, particularly when multiple nonlinearities are present [17] and when responses may be far from equilibria [18]. Seeking to address the challenge, recent advancements have been made in the analytical prediction of multi-DOF nonlinear systems using describing functions [19][20][21]. Yet, these methods rely on assumptions of input/output similarities and are thus inadequate to predict the coexistence of near-to-and farfrom-equilibrium dynamic responses, such as the potential for either small excursions from equilibria or large amplitude snapthrough in multistable structures.…”

Experimental modal analysis of nonlinear systems by using response-controlled stepped-sine testing

“…According to Describing Function Method (DFM) [21], the complex vector of nonlinear internal force amplitude can be expressed as…”

Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT)

“…The identification of nonlinear modal parameters is straightforward via applying standard linear modal analysis techniques to quasi-linear constant-response FRFs measured by RCT. Quasi-linearization of FRFs by keeping the displacement of the test point (equivalently, the modal amplitude) constant is based on the Nonlinearity Matrix concept [17] and the single nonlinear mode theory [9] as explained in Reference [16]. The key formulation of the quasi-linearization concept is as follows:…”

“…keeping the displacement of the test point (equivalently, the modal amplitude) constant is based on the Nonlinearity Matrix concept [17] and the single nonlinear mode theory [9] as explained in Reference [16]. The key formulation of the quasi-linearization concept is as follows:…”

“…The parameters of the system ( The backbone curves obtained from HFS are validated with the ones determined from constant-force simulation. Constant-force and RCT simulations used to extract backbone curves are conducted by using a multiple harmonics version of the Describing Function Method (DFM) details of which can be found in References [16,17,19].…”