# Dynamics of a Plate with Elastically Attached Mass

### Аuthors

Nigar E. S.

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russia

e-mail: nig-dig@unesp.co.uk

### Abstract

The article regards the problem of a beam dynamic loading by an impact body in the presence of an intermediate damper, namelly, a spring of a given stiffness. The purpose of the study consisted in determining the joint motion of a mechanical system of a beam-spring-body type, neglecting herewith the spring mass. The beam movement is being modeled by the equations of the plate cylindrical vibrations. The obtained equations for the joint movement of the beam–spring–body system consist of equations for the beam deflection and the equation of the body motion, with account for the spring stiffness. The system of equations, modelling the motion, consists of a fourth-order partial differential equation in coordinate, and a second-order equation in time, one of the boundary conditions of which is an ordinary second-order differential equation in time. The problem is solved by the integral Laplace transform method in time. The Durbin numerical method is used for the obtained solution inversion. Graphs of solutions, allowing observing the body behavior and calculating the beam deflection at a time instant, were plotted using this method. The graphs of analytical and numerical solutions coincide for small initial times. The dependence of the sought functions on the main parameters of the problem, such as spring stiffness and the beam bending stiffness, is demonstrated as well. It can be seen from the illustrating graphs that the beam deflection and body motion functions are directly proportional to the spring stiffness, and inversely proportional to the bending stiffness of the beam.

### Keywords:

beam deflection, beam vibration, spring, tension, deformation, system balance, Durbin’s method

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