Dynamic analysis of frame structures with free viscoelastic layers: New closed-form solutions of eigenvalues and a viscous approach

Materials of viscoelastic nature are of great importance for damping and vibration control in the civil engineering, automotive and aircraft fields. The frequency dependence of their mechanical properties results in integral–differential equations of motion in the time domain along with a nonlinear eigenvalue problem in the frequency domain. The main challenge of this paper is to develop closed-form expressions for the eigenvalues of framed general structures with bonded unconstrained viscoelastic layers, whose constitutive relations are based on the fractional derivative. The developed expressions allow us to obtain an equivalent viscous model which dynamic matrices explicitly depend on the viscoelastic damping parameters. Then, the time response problem solution is reduced to a simpler system of second order linear differential equations decoupled in the modal space of the undamped problem. The proposed methodology is validated by two numerical examples, a cantilever bean and a frame. Very good agreement is found between the proposed and exact (from other iterative methods) eigenvalues and transfer functions for a wide range of viscoelastic materials.