Free vibrations of a Continuous-discrete multi-span Beam taking into account inertial Rotation Forces

Abstract

Objective. The aim of the study is to estimate free vibrations of a continuousdiscrete multi-span beam taking into account the inertial forces of rotation. The goal is to determine the spectra of natural frequencies, damping coefficients and natural modes. Method. The study is based on the methods of linear mechanics of structures; numerical and numericalanalytical calculation methods. The solution to the problem is found using the method of separation of variables. Rotational movements of particles of continuous sections are taken into account according to one of the models of the Timoshenko beam. The D'Alembert principle and hypotheses on the smallness of displacements and angles of rotation of sections are used. Result. A system of equations in matrix-vector form is obtained. The mathematical model of transverse vibrations consists of three systems of differential equations. The equation includes transverse forces, external concentrated forces, d'Alembert inertial forcesi, and linear-viscous resistance forces. The inertial forces of rotation of concentrated masses are taken into account. The boundary and other additional conditions to the equations correspond to the calculation scheme. The left end of the beam is hinged. The conjugation conditions are met at the junction of the sections. Conclusions. This random process of disturbances is very close to the processes used in deterministic problems. The amplitudes and standard deviations of displacements in the deterministic and stochastic problems almost coincide, which confirms the reliability of the proposed calculation theory. Analysis of the curves shows that the standard deviations significantly depend on the degree of correlation of the components of the vector random process of disturbances. The use of modern computing computer systems such as Matlab allows us to successfully combine the advantages of both numerical and graphical methods.

Keywords

• transverse and free vibrations of a beam
• continuous-discrete multi-span beam
• translational motion
• rotational motion
• d'alembert's principle
• linear-viscous resistance forces