Vibration analysis of complex fractional viscoelastic beam structures by the wave method

Abstract

The vibration analysis of fractional viscoelastic beam structures with complex shapes and boundaries is a difficult task. In this paper, the wave method is applied to analyse the dynamic characteristics and steady-state response of fractional viscoelastic beam structures, and the effects of fractional order on wave propagation, transmission and reflection are discussed. Based on the fractional constitutive equation for viscoelastic materials, a fractional partial differential equation for a uniform viscoelastic beam is established. In the vibration analysis process, the beam is discretized according to the discontinuous feature positions, and then the vibration amplitude of the beam is obtained by a matrix calculation of the wave propagation, transmission and reflection associated with the beam. To verify the effectiveness of this method in studying fractional-order structures, a fractional-order viscoelastic beam is used as an example, and the results obtained by this method are compared with those obtained by the modal superposition method. In addition, the natural frequencies, mode shapes, frequency response function and amplitude response of viscoelastic beams with uniform symmetrical supports and beams with support positions that are optimized by a particle swarm optimization(PSO) are obtained with different fractional constitutive models. Moreover, the relationship between the dynamic characteristics and fractional order is also analysed. Actually, the model of viscoelastic material is changed under external excitation, which can be described by the variation of fractional order. The results show that this method has high accuracy in calculating the vibration properties of fractional viscoelastic beam structures.