Dynamics of non-linear vibro-impact systems
Dynamical systems with unilateral constraints, turning on and off in their motion, have peculiarities associated with successive variation of their kinematical structure and change of their number of degrees of freedom. Description of their periodical vibrations should be performed via joint consideration of constitutive differential equations and a set of inequalities transformed into equalities at the times of impact interactions unknown in advance. So it is proposed for the system solution to introduce additional unknown variables characterizing the time values when the impacts occur and magnitudes of the velocities discontinuities. For non-linear vibro-impact systems this approach permits to transform the starting two-boundary in time problem for a system of differential equations and inequalities into the multi-boundary in time problem for differential equations without inequalities. In this case, the intermediate boundaries are displacing with the vibration intensity change and additional unknown variables variation.
To construct periodical solutions to the gained systems of non-linear differential equations and to study their stability method of continuation by parameter is used jointly with Newton’s method and Liapunov’s and Floquef’s theory of stability. The approach includes sequential linearization of the equations and construction of a transfer matrix at each step of the leading parameter variation. Applied problems are solved with the use of the elaborated technique.
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