Novel energy-based analysis approach for determining elastic wave complex band of damped periodic structures using virtual springs

It is of importance to determine the complex band property of damped periodic structures for the evaluation of their wave attenuation performance. In view of this, the current paper proposes a new analysis approach based on the energy method and the virtual spring model for the calculation of the complex band. Its essence is to use a virtual spring to simulate periodic boundary conditions such that the wave numbers will only appear in the stiffness matrix of the virtual spring. Subsequently, the previously existed nonlinear eigenvalue solution problem is transformed into a linear eigenvalue solution problem by decoupling the wave numbers of the stiffness matrix and by reducing the order. The calculation procedure of the proposed approach is demonstrated by a case study of a periodically discrete-supported Euler beam, and then extended to deal with two-dimensional periodic structures. The accuracy of the proposed approach is verified by comparison the results with those in existing studies. The effects of the material frequency variation and damping on the propagation and attenuation of vibration waves are investigated. The results reveal that the material frequency variation and damping have a significant effect on the range and rate of wave attenuation. The proposed method has excellent applicability and promising application potential in calculating the complex band structures analysis of coupled periodic structures.