Softening-Spring Phenomenon in Large Amplitude Vibration of Two-Layer Bi-Material Beams

Abstract

The purpose of this study is to analyze the nonlinear vibration behavior of two-layer bi-material beams by using the Euler beam theory and Timoshenko beam theory. It is assumed that material properties satisfy a power law distribution along the thickness direction. The differential quadrature (DQ) method in conjunction with an iterative algorithm is employed to solve the nonlinear governing equations for two-layer bi-material beams. Numerical results are presented to show the influence of Young’s modulus ratio and thickness ratio on nonlinear frequency ratio–amplitude curves of two-layer bi-material beams. It is observed that the nonlinear frequency ratio–amplitude curves are unsymmetrical with the hinged end because of bending-extension coupling effect in two-layer beams. It is also observed that the hinged–hinged two-layer beams may occur the softening-spring nonlinearity when the amplitude of oscillation is small. This softening-spring nonlinearity is dependent on the thickness ratio, Young’s modulus ratio, vibration amplitude, layer number and boundary condition. The greater the difference of material properties between the two layers is, the more obvious the softening-spring nonlinearity becomes.