VERTICAL AND LATERAL VIBRATIONS OF EULER BEAMS IN A FLUID

An Integral Equation formulation of the coupled Euler Equation and three dimensional Laplace Equation boundary value problem for a uniform Euler beam vibrating in a fluid is given. The solution for a circular uniform beam is obtained by solving the three dimensional Laplace Equation for the velocity potential as a function of position along the length of the beam, time and the unknown deflection shape of the beam. The effective force exerted on the beam is then determined from the linearized Bernoulli Equation. The Euler Equation and its boundary conditions are converted to a Fredholm Integral Equation of the 2nd kind for the unknown beam deflection and solved by Fourier Transforms and matrix methods to obtain the eigenvalues and eigenvectors which then yield the natural frequencies and mode shapes of vibration. The solution has been computerized and computations carried out for beams with free and pinned ends.; The results of the computations are compared to classical methods utilized in predicting the natural frequencies of ship hull girders modelled as beams. These methods treat the effects of the fluid as an "added mass" and avoid dealing directly with the coupled boundary value problem. Conclusions are offered with regard to the applicability of the classical methods. Comparisons with other limited data on natural frequencies in a fluid, available in the literature, are also given.; The effects of the fluid on the mode shape deflections are investigated. The mode shape deflections of circular uniform beams vibrating in fluids of various densities are presented and the implications considered.