Summary
Experimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle θ to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed θ less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity map**s due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective dam** value, a striking quantitative agreement is found too.
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Piiroinen, P., Virgin, L. & Champneys, A. Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation. J Nonlinear Sci 14, 383–404 (2004). https://doi.org/10.1007/s00332-004-0616-y
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DOI: https://doi.org/10.1007/s00332-004-0616-y