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A novel modal superposition method with response dependent nonlinear modes for periodic vibration analysis of large MDOF nonlinear systems

Design of complex mechanical structures requires to predict nonlinearities that affect the dynamic behavior considerably. However, finding the forced response of nonlinear structures is computationally expensive, especially for large ordered realistic finite element models. In this paper, a novel approach is proposed to reduce computational time significantly utilizing Response Dependent Nonlinear Mode (RDNM) concept in determining the steady state periodic response of nonlinear structures. The method is applicable to all type of nonlinearities. It is based on the use of RDNM which is defined as a varying modal vector with changing vibration amplitude. At steady-state, due to periodic motion, it is possible to define equivalent stiffness due to nonlinear elements as a function of response level which enables one to create new linear systems at each response level by modifying original stiffness matrix of the underlying linear system. In this method, a new linear system is defined at each response level corresponding to each excitation frequency step, and modal information of these equivalent linear systems is used to construct RDNMs which forms a very efficient basis for the nonlinear response space. The response of the nonlinear system is then written in terms of these RDNMs instead of the modes of the underlying linear system. This reduces the number of modes that should be retained in modal superposition method for accurate representation of solution of the nonlinear system, which decreases the number of nonlinear equations, hence the computational effort, significantly. Dual Modal Space method is employed to decrease the computational effort in the calculation of RDNMs for realistic finite element models, i.e. for large MDOF systems. In the solution, nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations by using Describing Function Method, and the numerical solution is obtained by using Newton’s method with arc-length continuation. The method is demonstrated on two different systems. Accuracy and computational time comparisons are performed by applying different case studies which include several different nonlinear elements such as gap, cubic spring and dry friction. Results show that the proposed method is very effective in determining periodic response of nonlinear structures accurately reducing the computational time considerably compared to classical modal superposition method that uses the modes of the underlying linear system. It is also observed that the variation of natural frequency with energy level in a nonlinear system can be approximately obtained by using RDNM concept.