As a typical non-smooth bifurcation, grazing bifurcation can induce instability of elementary near-grazing impact periodic motion in impact oscillators. In this paper, the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities is analyzed, based on which, a control strategy is proposed. The commonly-used leading order zero time discontinuity mapping is extended to a higher order one to aid the perturbation analysis of the characteristic equation. It is shown that the degenerate grazing bifurcation can eliminate the singular term in the characteristic equation, leading to bounded eigenvalues. Based on such a precondition, the bounded eigenvalues are further restricted inside the unit circle, and a continuous transition between non-impact and controlled impact motion is observed. One discrete feedback controller that changes the velocity of the oscillator based on the selected Poincaré sections is adopted to demonstrate the control procedure.
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