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

Abstract

Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.

Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques

“…One of the first approaches was proposed by Slater [14] who combined a shooting method with sequential continuation to solve the nonlinear boundary value problem that defines a family of NNM motions. Similar approaches were considered in Lee et al [15] and Bajaj et al [16]. A more sophisticated continuation method is the so-called asymptotic-numerical method.…”

Nonlinear Normal Modes of Nonconservative Systems

“…Linearizing the energy around each point i of the boundary d, we obtain Equation (12). Solving Equation (13) transforms the energy increment in terms of displacements (∆ui, ∆vi) to apply to the boundary nodes.…”

Reduction of discrete element models by Karhunen–Loève transform: a hybrid model approach

“…In this section we briefly introduce the mathematical background of the KLT procedure and refer to e.g. [28] for a detailed discussion. The KLT is a signal-dependent transform which means that basis depends on the process under investigation: q(t).…”