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Use of Lagrange multiplier formalism to solve transverse vibrations problem of stepped beams according to Timoshenko theory

Cekus D.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2011

Vol. 10, nr 2

49-56

In this paper Lagrange multiplier formalism has been used to find a solution to a free transverse vibrations problem of stepped beams. The beams have been circumscribed according to the Timoshenko theory. The sample numerical calculations for a cantilever two-stepped beam have been carried out to illustrate the validity and accuracy of the present method.

Free vibrations of a stepped beam with two uniform and/or tapered parts

El-Din S. S., Mahmoud A. A., Nassar M. A.

Mechanics and Mechanical Engineering

2000

Vol. 4, nr 2

165--181

Closed form solutions for free vibrations of a stepped beam of two parts are given. Each part is of rectangular cross-sectional area. The parts may be of uniform cross-section and/or tapered with both equal and different tapered ratio in the horizontal and vertical planes. General constraints at the ends are possible at the three ends of the beam. The equations of motion of the beam are given in terms of trigonometric functions, hyperbolic functions, and the well known Bessel functions. Various special cases are deduced from the present solution and showed complete agreement with previous closed form solutions for these special cases.