Shroud devices which are typical cyclic symmetric structures are widely used to reduce the vibration of turbine blades in aero engines. Asymmetric rub-impact of adjacent shrouds or aerodynamic excitation forces can excite the bending-torsion coupling vibration of shrouded blades, which will lead to complex contact motions. The aim of this paper is to study the rub-impact dynamic characteristics of bending-torsion coupling vibration of shrouded blades using a numerical method. The contact-separation transition mechanism under complex motions is studied, the corresponding boundary conditions are set up, and the influence of moments of contact forces and aerodynamic excitation forces on the motion of the blade is considered. A three-degree-of-freedom mass-spring model including two mass blocks with the same size and shape is established to simulate the bending-torsion coupling vibration, and the dynamic equations of shrouded blades under different contact conditions are derived. An algorithm based on the fourth-order Runge–Kutta method is presented for simulations. Variation laws of the forced response characteristics of shrouded blades under different parameters are studied, on the basis of which the method to evaluate the vibration reduction characteristics of the shrouded blade system when the motion of the blade is chaotic is discussed. Then, the vibration reduction law of shrouded blades under bending-torsion coupling vibration is obtained.

The belt velocity is not strictly constant but exhibits periodic oscillations in certain industrial applications. A lumped mass model of a two-dimensional nonlinear friction-induced slider moving on an oscillating belt is established in this paper. Tangential stick-slip motion and normal separation–re-contact behavior are both considered under a symmetrically and uniformly distributed interface force. The analytical expression of the static friction force is deduced in terms of dynamic equations and stick-slip-separate state transition boundaries of the slider. The sliding friction force is modelled by a dynamic model that accounts for relative velocity. The Runge–Kutta algorithm combining the bisection method to capture the transition point between stick and slip motions is adopted to compute the vibration responses of the slider. The numerical results indicate that the belt’s oscillatory angular vibration frequency, the vertical preload, and the nonlinear stiffness have great effects on the dynamic characteristics of the slider, and the slider can experience $\mathrm{p}$-periodic ($\mathrm{p}=1,2,3,4,\mathrm{ect}.$) and chaotic vibration due to the non-smooth behavior at the contact interface.