THE EXACT SOLUTION OF A PROBLEM OF VIBRATION

OF TIMOSHENKO BEAM

J. Cabański,

K. Cabańska - Płaczkiewicz,

In this paper an axact method of a solutin of a problem of free vibration of Timoshenko beam is presented.

The phenomenon of transverse - rotational vibration of Timoshenko beam is descriebed the following coupled system of partial differential equations:

(1)

with the following phisical and geometrical dependences:

(2)

where:

w=w(x,t) - deflection of a beam, - rotational angle of a transversal cross - section of a beam, - an angle of non - dilatational strain, - continuous moment,

- continuous force, - bending moment, - transversal force, - Kirchhoff's modulus of material of a beam, I - moment of inertia of a transversal cross - section of a beam, A - area of a cross section of a beam, r - density of material of a beam, k - coefficient of shearing, x - coordinate of a beam, t - time.

Substituting :

(3)

to the differential equations (1) is accomplish an analysis of a solution of the system of differential equations of vibration are accomplished and the ortogonality of a mode of free vibration is indicated. The problems of the free vibration and the forced vibration are solved by arbitrary initial conditions and the arbitrary input.

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