Chapter 37 A Novel Computational Method to Calculate Nonlinear Normal Modes of Complex Structures Hamed Samandari and Ender Cigeroglu Abstract In this study, a simple and efficient computational approach to obtain nonlinear normal modes (NNMs) of nonlinear structures is presented. Describing function method (DFM) is used to capture the nonlinear internal forces under periodic motion. DFM has the advantage of expressing the nonlinear internal force as a nonlinear stiffness matrix multiplied by a displacement vector, where the off-diagonal terms of the nonlinear stiffness matrix can provide a comprehensive knowledge about the coupling between the modes. Nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations using DFM under harmonic motion assumption. A matrix manipulation based on dynamic stiffness concept was used to localize nonlinearities and reduce the number of nonlinear equations improving the efficiency of the approach, which becomes important in solving large complex structures. The nonlinear algebraic equations are solved numerically by using Newton’s method with Arc-Length continuation. The efficiency of proposed computational approach is demonstrated using a two-degree-of-freedom nonlinear system. The proposed approach has the potential to be applied to large-scale engineering structures with multiple nonlinear elements and strong nonlinearities. Keywords Nonlinear normal modes · Describing function method · Nonlinear vibrations · Nonlinear numerical solver 37.1 Introduction Nonlinear normal modes (NNMs) can be used to explain a wide class of nonlinear phenomena, mathematically and theoretically. However, the majority of structural engineers view it as a concept that is foreign to them. Recent studies show that the concept of NNMs can be used to describe nonlinear behaviors such as jumps, bifurcations, internal resonances, modal interactions, sub- and super-harmonic motions [1, 2]. Such potentials triggered the need to identify NNMs in engineering applications. In much of initial works to identify NNMs, analytical methods were commonly used [3]. However, analytical methods are not useful in analysis of complex high-dimensional structures, i.e. realistic systems, that led to the development of computational methods dedicated to NNMs. Numerical algorithms to compute NNMs of conservative mechanical structures are rare and complex. Many of these algorithms require time integration of the equations of motion, combined with shooting and pseudo-arc-length continuation [4–6], which becomes computationally expensive with increasing number of degrees-of-freedom (DOF) in finite element models. Furthermore, computation of NNMs using shooting method is limited by the capabilities of used time integration algorithms. For example, these algorithms can hardly deal with nonsmooth systems with piecewise linear stiffness. Renson et al. [7] provides a summary of recent frameworks used in calculating NNMs based on time-integration methods. There have been also few recent attempts [8, 9] to experimentally isolate and identify NNMs, however the majority of these attempts are limited to simple structures. Complexity of existed computational tools have prevented engineers from developing a practical nonlinear analog using the concept of nonlinear normal modes. This work focuses on developing a simple and efficient computational approach to identify nonlinear normal modes. The describing function method (DFM) is used to capture nonlinearities of the structure under periodic motion. Nonlinear differential equations of motion are converted into a set of nonlinear algebraic equations using DFM for periodic motion. Since the proposed approach works in frequency domain, it is free from typical challenges H. Samandari ( ) Department of Mechanical and Manufacturing Engineering, Miami University, Oxford, OH, USA e-mail: samandh@miamioh.edu E. Cigeroglu Department of Mechanical Engineering, Middle East Technical University, Ankara, Turkey © Society for Experimental Mechanics, Inc. 2020 G. Kerschen et al. (eds.), Nonlinear Structures and Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-12391-8_37 259
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