Occurrence conditions and properties of multi-frequency resonance
The conditions for the occurrence and the number of kinds of multi frequency resonance along with the analysis procedure were theoretically discussed.In the nonlinear case,the geometric properties of the region where multi frequency resonance occured were analyzed through drawing,and its physical significance was explained from a mechanical perspective.Finally,the theory was verified through examples.The results show that for a linear structure,it can be accurately predicted whether there is no multi-frequency resonance,one multi-frequency resonance or infinite multi frequency resonance between the selected frequencies.A larger nonresonant region can be formed near a larger resonant region.When three frequencies and two ratios were selected as the coordinate axes were considered,there are large areas of multi-frequency resonance occurred near the points whose two coordinates have a small common denominator.In the case of two frequencies,when the truncation number is n,in the process of increasing the nonlinear coupling strength K,the frequency ratio of the latest occurrence of multi frequency resonance is actually the frequency ratio of the earliest occurrence of multi frequency resonance when the truncation number is n+1.Therefore,caution should be exercised in selecting possible scenarios of internal resonance in nonlinear dynamics research.
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