Vibrational energy flow model for functionally graded beams

In this paper, the vibrational energy flow model for functionally graded (FG) beams is developed, which can be used for high frequency response prediction. The motion governing equation for FG beams is derived using the Hamilton's principle under the assumption of the Euler-Bernoulli beam theory. The beam properties, including Young's modulus, mass density and structural damping factor, are assumed to vary continuously in the thickness direction according to the power-law and exponential forms. The dispersion relation for the FG beam is obtained and the wavenumber is derived. Considering the variation of the damping, the effective damping factor is introduced and obtained by considering the total dissipated power due to the damping over the beam cross-section. The energy density governing formula is obtained by considering the energy balance of the infinitesimal element in the beam. To validate the proposed energy flow model for FG beams, the results from the energy flow model are compared with modal solutions for different physical parameters, and both good correlations and accuracy are observed. The results show that the dynamics characteristics of FG beams can be affected by the material variation profile and alter the energy level along the beam in turn.