Equations for large deflections of homogeneous and layered beams with application to chaos and acoustic radiation

Abstract

This work is a continuation of our early efforts on the dynamics and vibration of homogeneous and layered plates and shells. It consists of a comprehensive report on results obtained during the past year on large deflections of beams, although our recent efforts have also covered plates and shallow shells. The variational equation of motion for a new simplified case for large rotations in nonlinear elasticity theory is presented. Based on this, the equations of motion and simple dynamical models are derived systematically for beams of various types, by considering the effects of transverse shear and buckling as well as large deflection. These include homogeneous beams of both classical and Timoshenko types, laminated composite beams, and sandwich beams with isotropic or orthotropic core and isotropic or laminated composite facings. As applications we look into the area of chaos, by paying special attention to the effect of damping, and the area of acoustic radiation generated by nonlinear and chaotic vibrations in beams, by proposing a new method of analysis. Further use of the package of tools and models developed here for the purpose of avoiding chaos is also indicated.