Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped-clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped-clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics' frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped-clamped beam and the effect of the non-ideal input signal.

[1]  L. Azrar,et al.  SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF S–S AND C–C BEAMS AT LARGE VIBRATION AMPLITUDES PART I: GENERAL THEORY AND APPLICATION TO THE SINGLE MODE APPROACH TO FREE AND FORCED VIBRATION ANALYSIS , 1999 .

[2]  Ali H. Nayfeh.  Nonlinear transverse vibrations of beams with properties that vary along the length , 1973 .

[3]  P. D. McFadden,et al.  A review of time-frequency methods for structural vibration analysis , 2003 .

[4]  Vimal Singh,et al.  Perturbation methods , 1991 .

[5]  Alexander F. Vakakis,et al.  Nonlinear normal modes, Part I: A useful framework for the structural dynamicist , 2009 .

[6]  B. P. Patel,et al.  Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams , 2009 .

[7]  K. Worden,et al.  Past, present and future of nonlinear system identification in structural dynamics , 2006 .

[8]  Jean-Jacques Sinou,et al.  Non-linear dynamics and contacts of an unbalanced flexible rotor supported on ball bearings , 2009 .

[9]  F. H. Ling,et al.  An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems , 2007 .

[10]  Gaëtan Kerschen,et al.  Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques , 2009 .

[11]  Annett Wechsler,et al.  Formulas For Natural Frequency And Mode Shape , 2016 .

[12]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[13]  Ludovic Renson,et al.  Nonlinear modal analysis of the SmallSat spacecraft , 2012 .

[14]  Ali H. Nayfeh,et al.  A Parametric Identification Technique for Single-Degree-of-Freedom Weakly Nonlinear Systems with Cubic Nonlinearities , 2003 .

[15]  Pedro Ribeiro,et al.  Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods , 2004 .

[16]  R. Benamar,et al.  a Semi-Analytical Approach to the Non-Linear Dynamic Response Problem of Beams at Large Vibration Amplitudes, Part II: Multimode Approach to the Steady State Forced Periodic Response , 2002 .

[17]  Mahmood Tabaddor,et al.  Influence of nonlinear boundary conditions on the single-mode response of a cantilever beam , 2000 .