BRIEF SKETCH THE DYNAMICS OF ELASTIC SYSTEMS WITH MOVABLE INERTIAL LOAD NONCLASSICAL BY SEPARATION OF VARIABLES To the 170 anniversary of the problem - Demianenko, A. G.; Guridova, V. A. (Computer Science and Applied Mathematics, 2017)
In May 2017, the 170th anniversary were carried out from the day of the first formulation of the problem of the dynamic load influence on elastic structures and buildings that is related to destruction of Chester bridge in May 1847 in United Kingdom. After that, the mechanical engineers asked the question what are the differences between effects from acting of movable and fixed loads applied to the same elastic structure. Gradually the main interest was being moved from the applied aspects of this problem to the area of its physical and mathematical bases and was causing the necessity developing of the respective theory to replace a wide variety of the incomplete problem formulations and solving methods that caused to contradictory, paradoxical or even incorrect results so frequently. This problem is closely related to the dynamic problems of objects with time-dependent dimensions and some nonholonomic constraints. Masses and velocities of the systems that have been significantly increased in XX-XXI centuries put new tasks before engineers, require their solutions, originates new approaches in mechanics and mathematical modelling combined with on-going development of existing ones to discover quantitative and qualitative properties of kinematical and dynamical characteristics of a system motion. The main features of mathematical models describing the dynamic behavior of the elastic objects under movable inertia loading is the presence of an inertia operator in one of its forms that determines the force influence of the movable inertia loading on the elastic objects. The operator usually depends on intensity and velocity v(t) of loading stream, elastic strain w(x,y,t), and it can be clearly tracked the dependency of the acceleration of the deformation wtt(x,y,t), of force influence from both bending velocities wtx(x,y,t) and from curvature wxx(x,y,t) of object’s surface, so the movable inertia loading is a follower load. It is the second feature of the dynamics of elastic systems in the field of movable inertia loads The third feature is that the odd – order mixed derivatives by the time should be contained in the system due to the Coriolis’ acceleration of the movable load, that makes some difficulties for getting the solution applying of Fourier schema to the variable separation on the field of real numbers. A brief review of research on the dynamics of elastic systems with movable inertia loads solved by method of two-wave superposition of the eigen-oscillations and accompanying oscillations that allows to get analytical solutions for initial hypotheses. Accompanying oscillations are caused by the presence of moving inertia loads. The vibration and stability of the Timoshenko beam on a prism basis compressed by axial forces under moving load is considered. It is shown that the eigenoscillation forms and accompanying oscillations depend on the velocity of motion and the ratio between moving and stationary masses of the system. The dependency of the beam oscillations on the movable loads velocity and the ratio of moving and stationary masses of the system has been researched. Two critical velocities have been discovered and two matched eigenfrequencies for the second critical velocity. We have to note that the above features of the beam behavior cannot be obtained by numeric methods of mathematical physics from simplified models without considering of the Coriolis’ inertia forces.