Abstract
An algorithm is proposed to find exact solutions for the free vibration in an asymmetrical Duffing oscillator. The system is composed of a spring function of the Duffing type (with linear and cubic terms with respect to displacement) accompanied by a constant force. It is reduced from the most comprehensive 3rd order polynomial having arbitrary terms from constant to the 3rd power. By employing a bilinear transformation, the system is successfully converted into a regular Duffing equation whose exact solutions already exist and are abstracted here. The whole procedure to obtain the present solution is summarized in a flow chart. Numerical values of conversion constants, skeleton curves and waveforms are computed and illustrated for the typical cases of asymmetrical (hard, soft and snap-through) spring systems. It is demonstrated that the skeleton curves and waveforms are asymmetrical and that, some of them exhibit multivalued responses for a given frequency.
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