ON TWO VARIATIONAL PRINCIPLES FOR THE NATURAL VIBRATIONS OF ELASTIC BODIES

Consider the natural vibrations of elastic bodies. For the natural frequencies, we have obtained two variational principles. In the first variational principle, an allowable state is defined as follows. The stress components and their variations have the formsand satisfy the stress-free boundary conditions. The corresponding displacement components are derived from equations of motion:Then it is proved that the natural frequencies ω of the body correspond to the extreme values of the following functional: where V is the strain energy per unit volume expressed in terms of stress components.In the second Variational principle, an allowable state is defined by any continuous stress and displacement distributions, not necessarily satisfying any equation and boundary condition. In this case, it is proved that the natural frequencies ω of the body correspond to the extreme values of the following functional:where H has the expressionPx,Py,Pz are the surface tractions, and Su is that part of