The objective of the present study is to explore the connection between the
nonlinear normal modes of an undamped and unforced nonlinear system and the
isolated resonance curves that may appear in the damped response of the forced
system. To this end, an energy balancing technique is used to predict the
amplitude of the harmonic forcing that is necessary to excite a specific
nonlinear normal mode. A cantilever beam with a nonlinear spring at its tip
serves to illustrate the developments. The practical implications of isolated
resonance curves are also discussed by computing the beam response to sine
sweep excitations of increasing amplitudes.Comment: Journal pape
The analysis of large, complicated structures can be simplified and made more computationally efficient if smaller, simpler subcomponents can be treated and assembled. Modal substructuring methods allow one to reduce the order of the model at the subcomponent level. Modes are also an intrinsic property of the subcomponent, so they lead to certain physical insights. While modal substructuring is relatively well developed for linear systems, it's counterpart has not yet been developed for nonlinear subcomponent models. This work presents two modal substructuring techniques that can be used to predict the nonlinear dynamic behavior of an assembly. The first method uses the nonlinear normal modes of each subcomponent in a quasi-linear model to estimate the nonlinear modes of the assembly. In the second approach, a small number of linear modes are used to create a nonlinear reduced order model of each substructure, and the reduced models are assembled to build the nonlinear equations of motion of the assembly. Each approach is compatible with the finite element method, allowing for analysis of realistic engineering structures with global nonlinearities. The two methods are validated by using them to predict the nonlinear modes of a simple assembly of geometrically nonlinear beams, and both are found to perform well.
Nonlinearities in structural dynamic systems introduce behavior that cannot be described with linear vibration theory, such as frequency-energy dependence and internal resonances. The concept of nonlinear normal modes accommodates such phenomena, providing a rigorous framework to characterize and design nonlinear structures. A recently developed method has enabled the computation of nonlinear normal modes for structures with hundreds of degrees of freedom, but the formulation is not readily applicable to large scale geometrically nonlinear structures that are modeled within finite element software. This work presents a variation on that approach that can be used to extract the nonlinear normal modes of a structure using commercial finite element software. A model of the structure is created in the finite element package and the algorithm then iterates on the nonlinear transient response in a non-intrusive way to estimate the nonlinear modes. A modal coordinate transformation is used to reduce the order of the Jacobians required by the algorithm. The method is demonstrated on a fixed-fixed beam that is geometrically nonlinear due to coupling between transverse and axial displacements. An alternative procedure is also presented in which static load cases are used to compute a reduced order model of the nonlinear system and then standard continuation is used to find the nonlinear modes of the reduced order model. That approach is explored using both enforced displacements and applied loads and the results obtained are compared with those from the full-order model.
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