Abstract
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 u 0 ≔ p 0 / k 0 , but non-linearly on the hysteretic-damping coefficient η ≔ k 0 / k and the frequency ratio r w ≔ ω d / ω 0 . The border line formula for non-sticking is derived, which is independent of u 0 . 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.
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