Characterization of Nonlinearities in a Structure Using Nonlinear Modal Testing Methods Mahesh Nagesh, Randall J. Allemang, and Allyn W. Phillips Abstract Using phase-locked loop (PLL) controllers for obtaining frequency response curves (FRC) and backbone curves for nonlinear systems is gaining prominence. Such controllers deliberate a phase lag between the response obtained and the excitation provided. The use of such feedback controllers provides many advantages against traditional sine-sweep methods and helps better characterize nonlinear behavior of the test structure. This paper focuses on obtaining the nonlinear frequency response curves (FRC) and backbone curves of an isolated mode of a structural system exhibiting geometric nonlinearity. The capabilities of testing with such PLL controllers are highlighted and other important characteristics such as stability of the system are discussed. A qualitative comparison is provided between nonlinear modal testing using such feedback controllers as against other traditional methods such as sine-sweep methods. Keywords Nonlinear modal testing · Phase-locked loop · Nonlinear frequency response · Geometric nonlinearity 1 Introduction Study of dynamic mechanical systems with nonlinear force-response relationships is often very challenging. Well-established analytical and experimental techniques exist for linear systems [1–5], but such methods are seldom easily adaptable to nonlinear systems. The study of such nonlinear systems and associated phenomenon is vital in current engineering practices due to large-scale application of lightweight and complex materials that intrinsically exhibit nonlinear behavior [6, 7]. Such nonlinear systems typically exhibit one or more types of nonlinearities; geometric nonlinearity is widely observed and requires adequate understanding when characterizing such nonlinear systems. Distinct features of these systems include the variations of its natural frequencies, deflection characteristics, damping, and other important characteristics with variation in forcing levels. Detailed theoretical analyses and explanations for nonlinear behavior of structures are widely available [8–10]. Experimental techniques that can be employed for detecting, identifying, and characterizing such nonlinearities are seldom straightforward. A comprehensive list of experimental techniques accompanied by a vast theoretical background and their historical evolution is available in [6, 7, 11]. Traditional linear modal analysis techniques leads to several discrepancies in identifying and characterizing nonlinear behavior, and traditional sine-sweep techniques for nonlinear systems are challenging in their own ways, [6, 12] describe some novel excitation techniques that utilize a phase resonance approach to nonlinear testing. Other variations of such techniques using control techniques to ensure stability at or near resonances have since been further developed and have gained wide popularity. Two such techniques are in popular use: phase-locked loop (PLL) and control-based continuation (CBC) [13–17]. Phase-locked loop (PLL), widely used in communication and control applications [18, 19] was successfully adapted for nonlinear modal testing as described in [13–17]. The nonlinear normal mode, a nonlinear counterpart to a linear normal mode, is defined as the condition or frequency when external forcing and response are in phase-lag quadrature, i.e., have a 90◦ phase difference [20, 21]. The behavior of the system is purely conservative in such cases and the measurement of this quadrature condition at several excitation levels constitute a backbone curve (BBC) for the nonlinear system depicting the variation of the underlying conservative system with varying excitation levels. In this paper, a software implementation M. Nagesh ( ) · R. J. Allemang · A. W. Phillips Structural Dynamics Research Laboratory (SDRL), Department of Mechanical and Materials Engineering, College of Engineering and Applied Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: nageshmh@mail.uc.edu; allemarj@ucmail.uc.edu; philliaw@ucmail.uc.edu © The Society for Experimental Mechanics, Inc. 2022 G. Kerschen et al. (eds.), Nonlinear Structures & Systems, Volume 1, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-77135-5_19 167
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