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Weakly damped mechanical systems under small periodic forcing tend to exhibitperiodic response in a close vicinity of certain periodic orbits of theirconservative limit. Specifically, amplitude frequency plots for theconservative limit have often been noted, both numerically and experimentally,to serve as backbone curves for the near resonance peaks of the forcedresponse. In other cases, such a relationship between the unforced and forcedresponse was not observed. Here we provide a systematic mathematical analysisthat predicts which members of conservative periodic orbit families will serveas backbone curves for the forced-damped response. We also obtain mathematicalconditions under which approximate numerical and experimental approaches, suchas energy balance and force appropriation, are justifiable. Finally, we deriveanalytic criteria for the birth of isolated response branches (isolas) whoseidentification is otherwise challenging from numerical continuation.

How do Conservative Backbone Curves Perturb into Forced Responses? A Melnikov Function Analysis