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• A new non-smooth model with elastic and rigid impacts and dry friction is discussed. • The motions distribution and transition of the system are studied. • The transition mechanism to sticking-adhesion motions is revealed. • The global motions distribution in parameter-state space offers a detailed insight. Global dynamics of a non-smooth dynamic model is discussed under the combined effects of three non-smooth factors of elastic impact, rigid impact and dry friction,which is a rarely discussed topic before.Six different basic phases, nine events and six segments of the system have been described. The stability of periodic motions in the system is analyzed theoretically. The motions distribution and the transition to the sticking-adhesion motions are studied. Influences of the sliding bifurcation and grazing bifurcation in the way to the transition to sticking-adhesion motions are demonstrated. The transition to sticking-adhesion motions composed of one or more of three basic routes is illustrated. The transition law of three kinds of fundamental periodic motions in the low-frequency and small-gap parameter domain is further analyzed so as to delineate the transition mechanism from fundamental periodic motions to sticking-adhesion motions. In high frequency region, the global motions distribution in the system is obtained through the global analysis method in parameter-state space. The transition and coexistence of the motions including stable and unstable periodic motions are further explored. In addition, the motions distribution and transition of large friction force versus forcing amplitude from the static state are discussed. The results will be beneficial for the motions control of the system. [ABSTRACT FROM AUTHOR]