Insights on the Bifurcation Behavior of a Freeplay System with Piecewise and Continuous Representations 81 where e is a “tolerance” parameter; in the limit e →∞, the representation approaches the piecewise representation. System parameters are m=5kg, ωn =5Hz, ζ =0.03, α =7 ∗ 10 8 N/m3, p =4N, j 1 =0mm, j2 =0.8mm, Kc =1.4 ∗ 10 4 N/m. 3 Effectiveness of Continuous Representations for System with Freeplay Nonlinearity Figure 1(b) indicates that results can significantly diverge as forcing frequency increases past the primary resonance peak if a large enough value of e is not used, meaning regions of subharmonic resonance may be inaccurately predicted. A graph of the contact force versus displacement for even the coarsest value of e =104 used in Fig. 1(b) appears acceptable (omitted from this extended abstract for brevity), though, and does not indicate that frequency-response results will diverge. Thus, a convergence analysis is necessary. The low-frequency superharmonic resonances and chaotic behavior seem to be relatively unaffected, in addition to the primary resonance peak, for all values of e. However, a good agreement in frequencyresponse results is not always a good indicator that results agree globally and that system physics are not lost [7]. Nonlinear characterization (omitted for brevity) is also performed to determine how well the continuous representations can capture the overall physics of the system response. 4 Conclusions In this work, bifurcation analysis was carried out on a forced Duffing oscillator system with freeplay nonlinearity for different mathematical representations of the freeplay contact force. Results using a hyperbolic tangent representation indicated good frequency-response agreement after a parameter convergence analysis was performed. This convergence was required because the contact-force displacement may look acceptable for an unconverged model, but the frequency response significantly diverges as forcing frequency increases past the primary resonance peak. This is particularly dangerous because subharmonic resonances often lead to high-amplitude responses which can be damaging. Results for other representations (absolute value, polynomial, etc.) are reserved for the final conference presentation. Acknowledgments The authors B. Saunders and A. Abdelkefi gratefully acknowledge the support from Sandia National Laboratories. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the US Department of Energy or the US Government. This paper is also supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc. for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. SAND2020-13766 C. R. Vasconcellos acknowledges the financial support of the Brazilian agency CAPES (grant 88881.302889/2018-01). References 1. Vasconcellos, R., Abdelkefi, A., Marques, F.D., Hajj, M.R.: Representation and analysis of control surface freeplay nonlinearity. J Fluids Struct. 31, 79–91 (2012). https://doi.org/10.1016/j.jfluidstructs.2012.02.003 2. Zhou, K., Dai, L., Abdelkefi, A., Zhou, H.Y., Ni, Q.: Impacts of stopper type and material on the broadband characteristics and performance of energy harvesters. AIP Adv. 9, 035228 (2019). https://doi.org/10.1063/1.5086785 3. Henon, M.: On the numerical computation of Poincaré maps. Physica D. 5(2–3), 412–414 (1982) 4. Paidoussis, M.P., Li, G.X., Rand, R.H.: Chaotic motions of a constrained pipe conveying fluid: comparison between simulation, analysis, and experiment. ASME. J. Appl. Mech. 58(2), 559–565 (1991). https://doi.org/10.1115/1.2897220 5. Alcorta, R., Baguet, S., Prabel, B., Piteau, P., Jacquet-Richardet, G.: Period doubling bifurcation analysis and isolated sub-harmonic resonances in an oscillator with asymmetric clearances. Nonlinear Dynamics. 98, 2939–2960 (2019). https://doi.org/10.1007/s11071-019-05245-6 6. De Langre, E., Lebreton, G.: An experimental and numerical analysis of chaotic motion in vibration with impact. In; ASME 8th International Conference on Pressure Vessel Technology, Montreal, Quebec, Canada (1996) 7. Saunders, B.E., Vasconcellos, R., Kuether, R.J., Abdelkefi, A.: Importance of event detection and nonlinear characterization of dynamical systems with discontinuity boundary. In: AIAA Sci Tech 2021, virtual forum (2021)
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