Equivalent linearisation of vibro-impact systems

Abstract

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.

For this relief much thanks.