Reid's passive and semi-active hysteretic oscillators with friction force dependence on displacement

In this paper, we study passive and semi-active hysteretic damping oscillators of Reid's type, and find that the model of restoring force–displacement can be viewed in a rotated frame as an isotropic linear hardening element. Under harmonic loadings, new estimates for non-sticking solutions are presented for steady-state responses. The maximum displacement and maximum velocity are dependent linearly on the friction static displacement u0≔p0/k0u0≔p0/k0, but non-linearly on the hysteretic-damping coefficient η≔k0/kη≔k0/k and the frequency ratio rw≔ωd/ω0rw≔ωd/ω0. The border line formula for non-sticking is derived, which is independent of u0u0. Thus, an exact formula is obtained for estimating the maximum hysteretic-damping coefficient needed to prevent sticking. The accuracy is confirmed by comparing with exact time-stepping solutions.