The study of vibrations of elastic shells and rods began with the pioneering works of Daniel Bernoulli and Euler. They derived the one-dimensional differential equations of the flexural vibrations of beams by what we now call the variational principle of stationary action. They determined the eigenfunctions and the eigenfrequencies of a beam in the six cases of boundary conditions corresponding to the free, clamped or fixed edges. The BernoulliEuler theory preceded the exact three-dimensional linear elasticity discovered by Navier, Cauchy and Lame. Immediately after this great discovery Poisson applied three-dimensional elasticity to the derivation of one-dimensional equations of vibrations of thin rods. Regarding the rod as a circular cylinder of small cross section, he expanded all the quantities in powers of the distance from the central line of the cylinder. When terms above a certain order (the fourth power of the radius) are neglected, the equations for flexural vibrations turn out to be identical with those of Bernoulli-Euler. The equation for the longitudinal vibrations was derived by Navier; that for the torsional vibrations was first obtained by Poisson. Saint-Venant proposed the semi-inverse method for solving the problems of torsion and flexure of beams within 3-D elasticity. Although his method is not directly related to the dynamics, its influence on the development of shell and rod theories cannot be overlooked. Concerning the bending of beams, Saint-Venant adopted two assumptions: i) extensions and contractions of the longitudinal fibres are proportional to their distances from the plane drawn through the central line at right angles to the plane of bending, and ii) there is no normal traction across any plane parallel to the central line. The application of the theories rests upon a principle introduced by Saint-Venant and bearing his name, according to which statically equivalent tractions applied to the end of the bent beam or twisted bar produce the same stresses far from their end.1 Kirchhoff generalized Saint-Venant’s ideas for rods undergoing large displacements. He deduced an approximate expression of the strain in an element of the rod, and then found the one-dimensional energy functional. He obtained the equations of equilibrium or motion by varying the energy functional. From the three-dimensional elasticity, the problem of wave propagation in an infinite cylinder of the circular cross section was treated by Pochhammer and Chree, who obtained the dispersion relation for the axisymmetric longitudinal waves.
KeywordsFlexural Vibration Edge Resonance Piezoelectric Shell Thickness Vibration Bent Beam
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