Theory of Vibro-Impact Systems and Applications

We shall examine some aspects of impact theory for solids [60, 80, 103, 115], limiting the analysis to those points necessary to understand the processes occurring in vibro-impact systems, and generate mathematical models for them. We shall mainly be interested in the force changes and the kinematic characteristics of the motion of solids and their elements: particles, plane sections, etc.

When we constructed exact solutions we took one mathematical model to describe motion between impacts and another for the instant of collision. The first we described by linear differential equations; and the second using finite expressions from stereomechanical impact theory, used as the boundary conditions of the process. Such a description hampers the general analysis of vibro-impact systems, largely because it is impossible to predict the successive instants when the bodies will collide. Using the procedure of finding a periodic solution, which makes it possible to postulate regularity of collisions, does, in some measure, make it possible to overcome these difficulties, but, even in these cases, the solution constructed does not appear completely adequate due to the additional possibility of the body trajectory crossing the boundary levels. Separate descriptions of inter-impact motion and the impact process are even more problematic when using a more general concept of collisions between solids. For this reason, we will turn to the construction and analysis of single forms to record the equation of motion for the colliding elements of a vibro-impact system on all time axes, describing the fully combined motion occurring. This is possible by introducing nonlinear expressions reflecting the process of force interaction of colliding bodies or their elements.

Using the methods of harmonic linearisation described in §7, we will continue by analysing forced vibration of a linear oscillator with rigid stops, which will occasionally be referred to as impact oscillators later on. We will analyse the basis of using the method of harmonic force and motion linearisation, and compare the results with the exact solutions obtained in §2 to §3. We will begin by studying the non-symmetrical system (Fig. 2.1, a), assuming as before, that the stop is arranged with gap (interference) A relative to the equilibrium condition of the system without stop.

We shall now look at vibro-impact interactions in systems with multiple degrees of freedom, implying either lumped or continuous models. As we showed in §6, using dynamic compliance operators makes it possible to develop uniform equations of motion in both cases which differ only in the actual form of the dynamic compliance. The description of analysis methods for problems of existence and stability of periodic motion in the previous chapter can be extended to the class of system described here. A solution for such problems can often be successfully obtained in the general form with the final formula [39].