Concept of Coupled Displacement Field for Large Amplitude Free Vibrations of Shear Flexible Beams

Continuum solutions for solving the large amplitude free vibration problem of shear flexible beams using the energy method involves assuming suitable admissible functions for the lateral displacement and the total rotation. Use of even, single-term admissible functions leads to two coupled nonlinear temporal differential equations in terms of the lateral displacement and the total rotation, the solution of which is rather involved. This situation can be effectively tackled if one uses the concept of a coupled displacement field wherein the fields for lateral displacement and the total rotation are coupled through the static equilibrium equation of the shear flexible beam. This approach leads to only one undetermined coefficient, in the case of single-term admissible functions, which can easily be used in the principle of conservation of total energy, neglecting damping, to solve the problem. Finally, one gets a nonlinear ordinary differential equation of the Duffing type which can be solved using any available standard method. The effectiveness of the concept discussed above is brought out through the solution of the large amplitude free vibrations, in terms of the fundamental frequency, of uniform shear flexible beams, with axially immovable ends, using single-term admissible functions.