High frequency vibration analysis of beams on elastic foundation by a developed energy finite element method

This paper develops an Energy Finite Element (EFE) model for analysing the high-frequency vibration of beams on elastic foundations. By introducing the potential energy density and energy intensity associated with the foundation reaction, the energy transmission and dissipation equations are derived. In addition, the point impedance at the excitation point of the beam on an elastic foundation is derived to calculate the input power. Both the Winkler and the Pasternak foundation models are considered. Combined with the energy transmission and dissipation equations and the input power, the governing equation for energy density is established. Based on this, an EFE model is constructed to predict the response of the beam on elastic foundation in the high-frequency range. The proposed model is validated by comparison with exact analytical solutions, demonstrating its effectiveness in predicting high-frequency responses. Finally, the influence of the elastic foundation on wave propagation characteristics, energy distribution, and total energy of the beam under high-frequency excitation is discussed. The results demonstrate that the elastic foundation has a significant impact on wave propagation, and the vibration energy of the beam. Both the foundation's normal stiffness and the shear stiffness have significant influence on the high-frequency vibration of the beam. It is shown that at 2000 Hz, changes of the foundation stiffness and shear stiffness in reasonable ranges can lead to up to 60 % and 40 % variations on the total energy of the beam. It is demonstrated that vibration energy in the structure can be effectively controlled by adjusting the elastic foundation properties.