$$N-1$$ N - 1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities

Abstract

In this paper the N − 1 nonlinear modal interactions that occur in a nonlinear three-degree-offreedom lumped mass system, where N = 3, are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. ω n1 : ω n2 : ω n3 ≈ 1 : 1 : 1) is considered. However, due to the symmetries of the system under consideration, only N − 1 modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.

ε 2-Order Normal Form Analysis for a Two-Degree-of-Freedom Nonlinear Coupled Oscillator

“…(11) which gives the coefficients for the S 1 and S 2 backbone curves in Eqs. (13). Specifically for the S 1 backbone the coef-ficient producing curvature is A.…”

“…In this paper, we will show that the ǫ 2 terms are required in the direct normal form method of Neild & Wagg [9] to give the correct solutions. Typically the direct normal form method, [9], is applied to systems where the nonlinear, damping and forcing terms are assumed to be of order ε 1 small (or higher orders of ε) when compared to the linear terms [10][11][12][13][14]. The linear terms are the natural frequencies, taken to be of order ε 0 , meaning that the ε 1 nonlinear terms are typically an order smaller than the natural frequencies.…”

Force appropriation of nonlinear structures

“…Investigating the presence of such interactions is therefore key for understanding the dynamic behaviour of many nonlinear structures. The theory of nonlinear normal modes (NNMs) has been successfully used to analyse and predict the presence of modal interactions in nonlinear structures [5][6][7]. NNMs can be defined as families of periodic oscillations of a conservative (i.e.…”

Comparing Analytical Approximation Methods with Numerical Results for Nonlinear Systems

“…An increasingly common application of these techniques is to find the unforced, undamped responses, or backbone curves, of a system [3,6]. Doing so allows the underlying behaviour of the system to be captured and, although they are not related to any particular forced case, they can still be used to indicate the occurrence of internal resonance within the structure [7].…”