An asymmetric Duffing oscillator: hysteretic behaviour and jump phenomena

The primary resonance response of a non-linear oscillatory system that is excited by both a constant and a harmonic force is investigated in this study. The equation of motion is the asymmetric Duffing equation with no linear term and with hardening cubic non-linearity. An approximate solution corresponding to the steady-state response is sought by the harmonic balance method. Its stability is studied by applying Floquet theory. Numerical simulations are also given to confirm the analytical results. It is found that the frequency-response curves of the harmonic response can be single-valued or multi-valued with different shapes depending on the values of the system parameters. First, when they are single-valued, their shape is similar to the response of a linear system. Second, if they are multi-valued, they can have a maximum of three or five steady-state values. In this group, several different shapes can be distinguished: one in which the peak is bent to the right, as is for a hardening Duffing oscillator; another, when it is bent to the left, as is for a softening Duffing oscillator; finally, there exist responses that have a double bend - first towards lower frequencies, and then towards higher frequencies. The multivaluedness of these curves causes the occurrence of the multiple jumps in the system. It is possible for the system to exhibit one or two jumps when increasing frequency and one or two jumps when decreasing frequency. The effects of different parameters on the system behaviour are analysed. The analogy between the asymmetric Duffing equation and the Helmoltz-Duffing equation is also discusse