Component Mode Synthesis Using Nonlinear Normal Modes

Abstract

This paper describes a methodology for developing reduced-order dynamic models of nonlinear structural systems that are composed of an assembly of component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes. These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface motions). A simple system is used to demonstrate the proof of concept and show the effectiveness of the proposed procedure. Simulations are performed to show that the reduced-order model obtained from the proposed procedure outperforms the reduced-order model obtained from the classical fixed-interface linear component mode synthesis approach. Moreover, the proposed method is readily applicable to large-scale nonlinear structural systems.

Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure

“…The equilibrium solutions of Equation (32) correspond to the normal mode motions or periodic motions of the two degree of freedom system (22). Here R represents the total energy, and R = 0 implies that this is conservative system.…”

“…Comparing the coefficients obtained in equations of motion from the Hamiltonian and the coefficients in Equation (22), we get the following relationships:…”

“…For nonlinear structures considered to be an assembly of component structures, Apiwattanalunggaarn et al [22] developed a nonlinear Component Mode Synthesis methodology by making use of the fixed-interface NNMs in place of the fixedinterface linear normal modes (LNMs) in the traditional Component Mode Synthesis [23]. The NNMs for the various substructures were constructed by the invariant manifolds approach.…”

“…= 4.966, ω 2 = 9.932, H 1 = 1.20710, H 2 = 2.90239, H s = −0.54639, α 1 = 0.61135, χ 1 = 1.26602, α 2 = −2.28580, χ 2 = −5.49601, α s = 0.00296, χ s = −0.09809, β s = 1.2190, δ s = 3.93785, β 1 = −0.36535, δ 1 = −0.76771, β 2 = 0.64377, δ 2 = 0.82340. ξ 1 = 3.93722, κ 1 = 6.07800, ξ 2 = −0.94092, κ 2 = −6.57807, ξ 3 = 3.19007, κ 3 = 11.67807, ξ 4 = −1.46737, κ 4 = −6.64988, ξ 5 = 4.88822, κ 5 = 11.44667, ξ 6 =−1.98945, κ 6 = −9.07079, ξ 7 = 1.14118, κ 7 = 2.76440, ξ 8 = 1.57321, κ 8 = 4.91889, ξ 9 = .28701, κ 9 = −0.23538, η 1 = 32.39759, η 2 = −38.55259, η 3 = 157.32435, η 4 = 5.36998, η 5 = 94.08093.With the above notation, the equations of motion(22) can be rewritten as…”

Analytical and experimental studies on nonlinear characteristics of an L-shape beam structure

“…Crespo da Silva [10] developed a general method, which combined the advantages of both the analytical and numerical methods, to analyze a class of multi-beam structures. Apiwattanalunggarn et al [11] developed a nonlinear component mode synthesis technique based on the traditional component mode synthesis method proposed by Hurty [12] and modified by Craig and Bampton [13], and studied the dynamic properties of complex structures composed of several simple substructures. Wang and Bajaj [14] used theoretical methods and numerical methodology to investigate the bifurcations of the nonlinear normal modes of a two-degree-of-freedom autonomous conservative springmass-pendulum system.…”