Effect of a static force on the dynamic behaviour of a harmonically excited quasi-zero stiffness system

This paper concerns the way in which a static force can dramatically change the dynamic behaviour of a harmonically excited quasi-zero stiffness single-dof system. The primary resonance response of the system is considered using the harmonic balance method and the results are verified numerically. It is found that the system changes its characteristic from being purely hardening to a mixed softening and hardening characteristic and finally to a purely softening characteristic as the static force increases from zero. Consequently for certain values of the static force multiple jumps can occur. Five different cases of a possible response are distinguished and related to the values of the static force. It is also found that just one harmonic is required to qualitatively describe the behaviour of the system. The inclusion of a second harmonic in the solution does result in changes to the response, especially the softening behaviour, but overall it has a small effect for the system parameters chosen in this paper, which is representative of a quasi-zero stiffness isolator. The influence of damping is also considered.