Vitaly Grishakin

This paper deals with a formulation and a solution of problems of the dynamics of mechanical systems for which solutions that do not take into account the unilateral nature of the constraints imposed on the objects under study have been obtained before. The motive force in all the cases considered is the gravity force applied to the center of mass of each body of the mechanical system. Since unilateral constraints are imposed on all systems of bodies considered in the abovementioned problems, their correct solution requires taking into account the unilateral action of the constraint reaction forces applied to the bodies of the systems under study. A detailed analysis of the motion of the systems after zeroing out the constraint reaction forces is carried out. Results of numerical experiments are presented which are used to construct motion patterns of the systems of bodies illustrating the motions of the above-mentioned systems after they lose contact with the supporting surfaces.

An analysis is made of the motion of a particle on a rotating disk equipped with rectilinear blades for different types of particle collisions with a blade. The dynamics of particle motion is considered using equations describing the dynamics of the relative motion of a particle and is studied for the cases of inelastic and elastic interaction with the blade when the coefficient of friction of the particle against the disk has a value that is the most probable from a practical point of view. The impact interaction of the particle with the blade is studied using the Routh hypothesis, which relates the normal and tangential components of the impact impulse by a dependence similar to the Amonton – Coulomb dependence between the normal and tangential components of the interaction force of two rough bodies. The dynamical coefficient of impact friction is taken to be equal to half the kinematic coefficient of friction of the particle against the disk. After an impact, depending on the values of the above-mentioned coefficients and the angle of inclination of the blade, the particle may or may not possess a velocity component tangential and (or) normal to the surface of the blade. In addition, depending on the overall dimensions of the disk and the inclination angle of the blade the moving particle can repeatedly enter into impact interaction with the blade. In this paper we consider all four patterns of motion of the particle after its first and subsequent impacts against the blade of the rotating disk, present graphs of time dependences of the projections of the velocity of the relative motion of the particle, and plot trajectories of the relative motion of the particle for two practically important cases of installation of a blade on the disk’s surface which ensure the initial collision angles equal to 0${}^\circ$ and 30${}^\circ$. The proposed model is also used to investigate the changes in the antitorque moment acting on the disk. For this purpose, in each of the cases we calculate the resulting moment of impact impulse and the impulse of friction torque throughout the motion of the particle on the disk’s surface. We present the results of numerical experiments and use them to give recommendations on the optimization of mechanical systems of this type.