An Experimental Investigation of Complicated Responses of a Two-Degree-of-Freedom Structure

Abstract

Recent theoretical studies indicate that whereas large excitation amplitudes are needed to produce chaotic motions in single-degree-of-freedom systems, extremely small excitation levels can produce chaotic motions in multi-degree-of-freedom systems if they possess autoparametric resonances. To verify these results, we conducted an experimental study of the response of a two-degree-of-freedom structure with quadratic nonlinearities and a two-to-one internal resonance to a primary resonant excitation of the second mode. The responses were analyzed using hardware and software developed for performing time-dependent modal decomposition. We observed periodic, quasi-periodic, and chaotic responses, as predicted by theory. Conditions were found under which extremely small excitation levels produced chaotic motions.

Chaotic vibrations of non-shallow arches

“…The results given above are similar to those presented by Nayfeh [15] for qmO(e) and .O=co. In fact it is found that the results of reference [15] are exactly obtained as co--*-(2 in equation (34), and the (small) fl2-term of equation (47) is neglected.…”

“…In fact it is found that the results of reference [15] are exactly obtained as co--*-(2 in equation (34), and the (small) fl2-term of equation (47) is neglected. The consequence, however, is that the instability boundary between regions B and D in Figure 5 tilts to the right, away from the results obtained when using both Mathieu-Hill theory and numerical simulations.…”

“…The results of Nayfeh are readily applicable here. However, instead of perturbing around the primary resonance O--co for weak excitation, as was done in reference [15], we shall here perturbate around the secondary resonance O = 2 for hard excitations. To indicate the smallness of damping and non-linearities we introduce a small parameter ~ into equations (20), which in the final equations is set to unity: where foa and Uoa are functions to be determined, To = r is a fast scale characterizing motions at the frequencies I, 2, co and .…”

“…Nayfeh and co-workers [15,19] used the method of multiple scales to study the solutions of general two-degree-of-freedom systems with quadratic non-linearities, of which equation (20) is a special case. It was shown that solutions near and inside the primary region of dynamic instability are dominated by the primary resonance O = oJ, and the internal resonance co = 2.…”

“…Thus, the non-shallow arch possesses a natural example of the so-called autoparametric 239 0022-460x/92/050239+20 $03.00/0 9 1992 Academic Press Limited. case of nearly commensurable linear natural frequencies, treated in particular by Nayfeh and co-workers (see, e.g., reference [15]). Besides becoming unstable through parametric excitation of the fundamental mode, when the excitation frequency 0--.2r the fundamental mode will also become unstable for 0~os, through a non-linear modal interaction between the two modes.…”