Короткий нарис досліджень динаміки пружних систем з рухомим інерційним навантаженням некласичним методом відокремлення змінних

Abstract

Наведено короткий огляд досліджень динаміки пружних сиcтем з рухомим інерційним навантаженням методом двохвильового подання коливань у вигляді суперпозиції власних та супровідних коливань, який у деяких випадках дозволяє побудувати точні розв’язки задач. Супровідні коливання зумовлені наявністю рухомого інерційного навантаження. Як приклад, розглянуто коливання та стійкість балки Тимошенко за дії рухомого інерційного навантаження. 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. 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. 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.