Chapter 7 Periodic Response Prediction for Hybrid and Piecewise Linear System s G. Manson Abstrac t This chapter presents a completely novel framework for the prediction and understanding of nonlinear system behavior. The idea is simply that all nonlinear systems can be represented as a combination of linear systems between which information is exchanged. Under harmonic excitation, the periodic responses of such hybrid systems may be easily calculated when the switching between linear systems is specified in terms of time. These time-switching hybrid systems provide a useful stepping stone to more realistic piecewise linear systems where the switching criteria are specified in terms of displacement and/or velocity. This chapter details the framework and illustrates its ability to efficiently predict the periodic responses from piecewise linear systems. The framework is also shown to be capable of predicting both stable and unstable periodic responses for conditions where jump behavior is possible. The extension of the work to continuous nonlinear systems is also briefly discussed. Keyword s Nonlinear · Hybrid systems · Piecewise linear 7.1 Introduction The environmental and economic pressures of the last couple of decades have resulted in an unprecedented effort on the part of structural design engineers to design structures that are more lightweight than, yet are at least as strong as, their structural ancestors. While this drive for strong, lightweight structures is clearly desirable, in many cases, this will also result in the structural response becoming significantly nonlinear. While research has been conducted into nonlinear system behavior for centuries, the current situation appears to be, as Worden and Tomlinson wrote in their monograph “Nonlinearity in Structural Dynamics” [1] that “there is no unique approach to dealing with the problem of nonlinearity either analytically or experimentally and thus we must be prepared to experiment with several approaches.” The primary aim of this monograph was to introduce and explain a number of these approaches that would belong in a “toolbox” for the analysis of nonlinear structural systems and included Harmonic Balance [2], Hilbert Transform [3], NARMAX modeling [4], the Masri-Caughey Restoring Force Surface method [5], Direct Parameter Estimation [6], and the Volterra Series approximation [7]. Approaches not considered in the monograph included perturbation methods [8], multiple scales [9], and nonlinear normal modes [10]. The research conducted using the aforementioned approaches over the last five or so decades has undoubtedly led to much greater understanding of the behavior of nonlinear systems and has resulted in countless applications. That said, it is fair to say that all approaches are not without their limitations. These limitations are often associated with the need for approximation or, where infinite series are being employed, are associated with the related issues of series convergence and truncation. Another limitation with the bulk of these approaches is that they are only capable of representing what is often termed “weak nonlinearity.” One of the reasons as to why there has not yet emerged a General Theory of Nonlinearity may be in the nonlinear dynamicists’ simple system of choice, namely, the Duffing oscillator [11]. While the classical Duffing oscillator model of a linear plus cubic stiffness term lends itself as a good approximation for many real engineering systems, it may not have been the best starting point for the development of a General Theory. The author would instead like to propose that a better starting point may be provided via hybrid systems and, their close counterparts, piecewise linear systems. The reason behind this proposal is that the responses of such systems may be written in terms of linear steady-state and linear transient G. Manson ( ) Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Sheffield, UK e-mail: graeme.manson@sheffield.ac.uk © The Society for Experimental Mechanics, Inc. 2024 M. R. W. Brake et al. (eds.), Nonlinear Structures & Systems, Volume 1 , Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-36999-5_7 41
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