Dynamics of Two-Mass Vibro-impact Mechanisms with a Fixed Limiter

The first chapter examines the nonlinear dynamics of various two-mass models of a vibration impactor with a fixed limiter. The consideration begins with a general two-mass model, consisting of two masses (impact mass) and (vibrator), fixed by means of springs and interconnected by a connecting spring. An external harmonic force acts on the upper mass. The amplitude of oscillations of the lower mass is limited by a stationary barrier (limiter), upon contact with which an instantaneous, generally speaking, elastic impact occurs, characterized by a coefficient of recovery. In § 2.1, the problem of studying forced oscillations of a vibro-impact system is reduced to the study of a point mapping of a four-dimensional hyperplane (corresponding to the limiter plane) into itself in a five-dimensional phase space. The structure of phase space is considered, and possible types of impact motions are indicated in the case of both elastic and inelastic impacts. It is shown that any periodic movement can be characterized by two numbers m and n, where m is the number of blows during the period of movement, and n is the multiplicity of the period of movement to the period of the external force. In § 2.2, in the parameter space, the coordinates of fixed points corresponding to the periodic single-impact multiple mode of motion are found in parametric form. The boundaries of the existence of these periodic movements are determined. Based on the theory of point mappings and the theory of bifurcations, equations of bifurcation surfaces are given that correspond to the violation of the conditions of existence and stability of fixed points (N-surfaces) and the exit of a closed phase trajectory beyond the domain of definition of the point transformation under study (C-surfaces). Bifurcation C-surfaces, separating single-impact periodic movements from multi-impact ones, correspond to the appearance of additional impacts during the period of movement. It is noted that for parameter values at which the gap is greater than the amplitude of forced oscillations, along with shockless and quasi-periodic movements, movements with a rigid nature of occurrence are possible. In § 2.3, the dynamics of a spring-loaded two-mass vibratory impactor is studied. It is shown that the regions of existence of single-impact multiple periodic modes of motion, previously found from the condition of reality and positivity of the impact velocity, are significantly narrowed due to the loss of their stability. To partition the parameter space into regions of stability, a characteristic polynomial of the fourth degree was obtained, based on which bifurcation N-surfaces for different parameter values. Of the stability regions, the most significant are those located near the resonant frequencies of the corresponding linear system without a limiter. A study of the location of bifurcation C-surfaces showed that they cut off a significant part from the areas of N-stability. A methodology for constructing both N-stability regions and bifurcation C-surfaces is presented. In § 2.4, a model of a two-mass non-spring vibratory hammer was studied. The stability of single-impact multiple periodic movements was studied. The boundaries of the stability regions of these movements are found, and bifurcation C-surfaces are constructed. In § 2.5, stable single-impact multiple periodic movements for a two-mass vibration-impact unspring compactor are studied and found. For an absolutely elastic impact (R = 1), some general properties of bifurcation N-surfaces are given, and a method for constructing bifurcation surfaces Nφ is also indicated. The location of stability regions in three-dimensional parameter space has been studied.