On the basis of the results of numerical analysis, it is shown that the introduction of a force of contact interaction varying according to the Hertz law to the equation of motion makes it possible to model impacts between colliding bodies both under harmonic and stochastic external loads. This enables us to deduce the law of motion of the bodies in a vibroimpact system for the entire time axis, including the period of impact. We perform the numerical optimization of the parameters of the vibroimpact system by the method of gradient projection with correction of the discrepancy in constraints. The comparative analysis of the efficiency of the dynamic nonimpact dampers and shock absorbers of vibrations is presented.Keywords: vibroimpact motion, modeling of an impact, force of contact interaction, Hertz law, optimization, dynamic and shock absorbers. Statement of the Problem.Dynamic processes running in vibroimpact mechanical systems have been studied by many researchers (e.g., [1,2]). The impact interaction of bodies in these systems is described and modeled by using two basic approaches. According to the first approach, the impacts are modeled by the method of boundary conditions by using the hypotheses of the stereomechanical theory of impact and the coefficient of restitution. This approach requires precise recording of the time of impact and the appropriate correction of the initial phase of loading. In the presence of more than one contacting couple in the elastic system, this approach leads to the analysis of multipoint boundary-value problems, which complicates, to a certain extent, the corresponding calculations. The separate description of motion between impacts and the process of impacts may also turn into a serious problem if it is necessary to use more general concepts of the theory of collisions of solid bodies. This approach can be used for the solution of simple problems. However, it does not allow one to realize a single form of representation of the equations of motion on the entire time axis and obtain general results describing the qualitative properties of motion. This is why it is also reasonable to use the other approach in which it possible to deduce and analyze a single form of the equations of motion for elements of vibroimpact systems on the entire time axis that describe the entire collection of realized motions by the application of forces (modeling the process of force interaction of colliding bodies) to the bodies. This approach seems to be more general and significantly simplifies the processes of construction connected with the investigation of vibroimpact modes of oscillation of elastic systems. Thus, in particular, the problems of external impacts against an immobile obstacle and of internal impacts between the bodies in the system are identical in this statement.It is shown that the description of the force of contact interaction by the Hertz formulas enables one to obtain the law of oscillatory motion of complex nonlinear objects on the entire time axis both during the impacts a...

We consider different methods of modeling impact in vibroimpact systems by contact interaction force which can be regarded as elastic force as well as force corresponding to the Hertz law and with the help of the boundary condition method by using the coefficient of restitution. Comparison of the results of the modeling by means of these methods is performed, and recommendations for their application are proposed.Keywords: vibroimpact motion, shock-vibration platform, modeling of an impact, contact interaction force, Hertz law, coefficient of restitution. Introduction.Of special interest are dynamic processes running in vibroimpact mechanical systems. This predetermines studying the motion and interaction forces between the bodies in vibroimpact systems [1,2]. The basic problem of such investigations is modeling of the impact [3]. In this paper, the following methods of modeling impact are analyzed: by contact interaction force considered as elastic force and force described by Hertz law [3][4][5] and with the help of the boundary conditions by using the coefficient of restitution R.The object of investigations is a two-mass model with two degrees of freedom corresponding to the shock-vibration platform which is widely used in construction industry [6-8]. Design Model and Equations of Motion.To study the dynamics of the platform and impact phenomenon between its main body and table, we consider the simplest variant of the mathematical model presented in the form of the two-mass vibration system without a fixed main body (Fig. 1).Table of the platform with mass m 1 is mounted on linear vibroisolation springs with total stiffness k 1 and is bolted to the foundation. An excitation force F t ( ) is generated by electric motors mounted under the table. Elastic gasket (an oscillation limiter) with thickness h and linear stiffness k 0 is attached to the table. Main body with mass m 0 is placed on this elastic gasket, but not attached to the elastic gasket and thus can be lifted off.At the equilibrium state the main body and table start moving simultaneously until the moment when the main body pulls away. When two bodies are far apart, they move separately until the main body falls on the elastic gasket. Impact phenomenon is observed during which bodies move simultaneously until the main body moves away from the elastic gasket.Let us consider three states of the platform: combined initial motion of the main body and table until one body tears away from another, separate motion of the bodies during the uplifting of the main body, and simultaneous motion of the bodies at the moment of impact due to the main body falling on the table.While moving simultaneously (first time when the main body uplifts), the following forces are exerted on the bodies of the system: on the main body -its weight with concrete P 0 and resistance forces in the concrete F damp2 , elastic force in the gasket F k 0 and resistance force in it F damp0 ; on the table of the platform -table weight P 1 , elastic force and resistance forces in vibroisolati...

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Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency in particular are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in non-smooth dynamical systems is of the special scientists’ interest. In this paper we study the type-III intermittency route to chaos in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We apply relatively new mathematical tool – continuous wavelet transform CWT – for investigation this phenomenon. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate intermittency route to chaos, to distinguish and analyze the laminar and turbulent phases.

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