Large Amplitude Free and Forced Vibrations of Functionally Graded Timoshenko Beams Using Coupled Displacement Field Method

The large amplitude vibrations of functionally graded (FG) beams consisting of metal rich layers at the bottom, ceramic rich layers at the top, and a concentrated mass at the mid-span have been studied using coupled displacement field method. Unlike traditional methods, the coupled displacement field method reduces the 2n undetermined coefficients problem, one each for total rotation and transverse displacement distribution of the beam at n modes, to n undetermined coefficients using a coupling equation obtained from the minimization of potential energy principle. A suitable admissible function having single undetermined coefficient has been assumed for total rotation distribution and the corresponding transverse displacement distribution of the beam has been obtained at each mode for hinged-hinged and clamped-clamped FG beams. The equations of motion for large amplitude vibrations of FG beams at each mode in terms of the undetermined coefficients are derived from the conservation of total energy principle. The free vibration problem is solved using harmonic balance method whereas the forced vibration problem is solved using the Newmark-β method to obtain the time response of the undetermined coefficients and the dynamic response of the beam has been computed from the modal superposition method. The proposed coupled displacement field approach has been successfully applied for the first time to study the large amplitude vibrations of FG beams with suitable validations, and the influence of power law index, slenderness ratio, harmonic load, and concentrated mass has been investigated.