A Semi-analytical Approach for Dynamic Characteristics of Beams with the Effect of Static Load

This work introduces a semi-analytical approach to illustrate the linearized vibration of clamped–clamped beams in the nonlinear regime due to the effect of static load. The von Karman strain and Hamilton’s principle are utilized to derive the genernal nonlinear equations of beams under static and acoustic pressure load. The nonlinear dynamic problem is analyzed in two parts: the nonlinear static problem and the linearized vibration around the nonlinear static equilibrium state. The modal equation under initial large deflection is a variable coefficient partial differential equation and is difficult to obtain an analytical solution. An approximate solution is performed by the transfer-matrix method and local homogenization. The analysis shows that the variation of pressure load affects the static deflection and the dynamic characteristics of the beam. With the gradual increase of the pressure load, the deflection of the beam has a great influence on the higher-order modal shapes of the beam. And the peak value of the modal shapes near the center of the beam is lower than the sides.