On the jump-up and jump-down frequencies of the Duffing oscillator

In this paper, simple approximate non-dimensional expressions, and the corresponding displacement amplitudes for the jump-up and jump-down frequencies of a softening and hardening lightly damped Duffing oscillator with linear viscous damping are presented. Although some of these expressions can be found in the literature, this paper presents a full set of expressions determined using the harmonic balance approach. These analytical expressions are validated for a range of parameters by comparing the predictions with calculations from direct numerical integration of the equation of motion. They are also compared with similar expressions derived using a perturbation method. It is shown that the jump-down frequency is dependent on the degree of nonlinearity and the damping in the system, whereas the jump-up frequency is dependent primarily upon the nonlinearity, and is only weakly dependent upon the damping. An expression is also given for the threshold of the excitation force and the nonlinearity that needs to be exceeded for a jump to occur. It is shown that this is only dependent upon the damping in the system.