FREE VIBRATION ANALYSIS OF EULER-BERNOULLI BEAM WITH DISCONTINUITIES BY MEANS OF DISTRIBUTIONS

Abstract eng:
The general equation for the transverse vibration of Euler-Bernoulli beam has been used since it was derived by means of classical derivatives of the shear force, the bending moment, the slope and the deflection of the beam. However these derivatives are not defined at such points of center-line between ends of the beam in which there is a concentrated support or a concentrated mass or a concentrated moment of inertia or an internal hinge connecting beam segments. We have applied the distributional derivative for a discontinuous shear force, a discontinuous bending moment, and a discontinuous slope of the beam in order to derive generalized mathematical model for free transverse vibration as a system of partial differential equations. We have computed a general solution to the generalized mathematical model for prismatic beam by means of a symbolic programming approach via Maple. As a result of this approach, computing natural frequencies and modal shapes of a slender beam, it is not necessary for continuity conditions to be put together at discontinuity points mentioned. As an example we have used this approach to obtain a frequency equation of a beam on three pin rigid supports, which would be more complicated if we tried to apply the transfer matrix method.

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