Exact solution of the differential equation of transverse oscillations of the rod taking into account own weight

Yurii Krutii,Mykola Suriyaninov,Victor Vandynskyi,G Vatulia,A Plugin,O Darenskyi

doi:10.1051/matecconf/201711602022

Abstract

The free transverse oscillations of the rod of uniform cross section taking into account its own weight are considered in this work. The appropriate partial differential equation of transverse oscillations of a rod was reduced to two ordinary differential equations for the time function and the amplitude function of deflections. Concurrent with the differential equations for the amplitude state, the equivalent system of differential equation is considered. In total, the exact solution of the initial partial differential equation of transverse oscillations of a rod, expressed in nondimensional fundamental functions and initial parameters is attained. The method of power series was used for the construction of fundamental functions. Due to the exact solution, the formulas in an explicit form for dynamic variables of the state of a rode – deflections, angular displacement, bending moment and transverse force – were defined. The analytical form for equation of free oscillation frequency is defined. That has limited the finding of frequency to definition the unknown non-dimensional parameter through the frequency equation. As a result, the presence of derivative exact solutions provides the possibility to investigate the free oscillations of rod with various types of boundary conditions.

Highlights

That oscillation is the most widespread type of motion. There is no any branch of technology without phenomena where the vibrations take place
It is generally known, that oscillation is the most widespread type of motion
List of symbols: EI − the flexural rigidity of a rod; m − the intensity of the distributed mass of the rod; N(x) = qx − the variable axial force, where q − weight per unit length of beam; y(x, t) − the cross motion of the axis point of the rod with coordinate x at time t; φ(x,t) − the dynamic angular displacement; M (x,t) − the dynamic bending moment; Q(x, t) − the dynamic transverse force; f (x,t) − the intensity of inertial forces that appear during oscillation (D’Alembert force)

Summary

Introduction

That oscillation is the most widespread type of motion. There is no any branch of technology without phenomena where the vibrations take place. It is known [3, 6], that the equation of free transverse oscillations taking into account own weight is written as: EI The solution of the Eq (6), expressed in terms of parameters of initial motion conditions T (0), T (0) , is given by 2.2 The exact solution of amplitude equation of oscillation.

Results

Conclusion