Nonlinear normal modes and nonlinear model reduction for discrete and multi-component continuous systems
Abstract
This work addresses two general problems, nonlinear dynamic analysis and model size reduction based on nonlinear normal modes for systems with essential inertial nonlinearities. The two aims of this work are: first building an explicit discretized structural model that retains essential parameters in the model, and performing perturbation analysis and nonlinear dynamic analysis of the discretized model, second then analyzing nonlinear normal modes of essential inertial nonlinear systems, and using these nonlinear normal modes to do further nonlinear model reduction. With a view to understanding the essentials of this approach, nonlinear normal modes and nonlinear model reduction are considered for a simple nonlinear system with essential inertial nonlinearities, this system is the classic spring-mass-pendulum system. Then, the same procedures are applied to a multi-component continuous structural system, the three-beam-tip-mass system. For the three-beam-tip-mass system, nonlinear frequency response of the base excited structure with three mode interactions is investigated. There are four main contributions of the present effort: (1) Calculation of nonlinear normal modes and bifurcation analysis of nonlinear normal modes: Several common methodologies for constructing nonlinear normal modes are reviewed, including the harmonic balance method, the method of invariant manifolds, method of multiple time scales, and the asymptotic method. Besides these methods, the Adomian decomposition method is modified to construct nonlinear normal modes. Finally, a numerical method based on shooting technique is developed, which can obtain nonlinear normal modes even for higher energy case. Bifurcations of nonlinear normal modes for the spring-mass-pendulum system and the three-beam-tip-mass system are both studied by analytical methods and by the numerical method for 1:2 internal resonance case. (2) Generalized procedure to model multi-component continuous structural systems: A general procedure is developed based on the combination of Crespo da Silva's formulation with the substructure synthesis Rayleigh-Ritz method. The quasi-comparison functions bases are utilized as local mode functions and then applied in a global Rayleigh-Ritz method. This greatly reduces the calculations and allows one to obtain discretized system models of continuous structures, models that retain parametric dependence to the maximum possible extent. (3) Model reduction based on nonlinear normal modes: Model reduction for systems under base excitation, which is a combination of external force and parametric force, is investigated. Cases of model reduction for resonantly excited system without and with internal resonance are considered for the spring-mass-pendulum system and the elastic structure. (4) Nonlinear dynamic analysis for three-beam-tip-mass system with three-mode interactions: The discretized model is studied for both external resonance and parametric resonance with three modes interactions. For external excitation case, the 1:2:3 internal resonance cases with force excitation frequency approximately equal to frequency of the second mode is considered. The 1:2:5 internal resonance case for parametric resonance of the 3rd mode is also studied. Stability and bifurcation analysis of the responses carried out by software AUTO 97.
Degree
Ph.D.
Advisors
Bajaj, Purdue University.
Subject Area
Mechanical engineering
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