On rotating vectors, Jacobi elliptic functions and free vibration of the Duffing oscillator

The free vibration displacement of an undamped hardening Duffing oscillator is described in exact form by a Jacobi elliptic function. Unlike an undamped linear oscillator, whose displacement is described by a trigonometric function, a Jacobi elliptic function is difficult to interpret by a simple inspection of the function arguments. The displacement of a linear oscillator is often visualised as a rotating vector, which has two characteristics — a constant amplitude and a phase (or frequency). These parameters are readily related to the physical response of the system. In this paper, a similar approach is applied to the free vibration displacement of a Duffing oscillator. However, the rotating vector description of the motion is much more complicated than for a linear system. It still has two characteristics though — an amplitude and a phase, but in general both these quantities are dependent on the position of the vector, i.e., they are frequency modulated. It is shown that there is not a unique rotating vector representation of the cn Jacobi elliptic function. Indeed, there are an infinite number of elliptical loci bounded between an elliptical and a circular locus of the vector. There are two specific cases. One is where the amplitude of the vector is constant and the phase angle is frequency modulated (the circle), and the other is when the amplitude of the vector is frequency modulated and the angular velocity is constant. In all other cases, both the amplitude and the angular velocity of the rotating vector are frequency modulated. To aid in the visualisation of the rotating vectors that represent the free vibration solution of a highly nonlinear hardening Duffing oscillator, two animations are provided.