Non-local finite element analysis of damped beams

Non-local viscoelastic beam models are used to analyse the dynamics of beams with different boundary conditions using the finite element method. Unlike local damping models the internal force of the non-local model is obtained as weighted average of state variables over a spatial domain via convolution integrals with spatial kernel functions that depend on a distance measure. In the finite element analysis, the interpolating shape functions of the element displacement field are identical to those of standard two-node beam elements. However, for non-local damping, nodes remote from the element do have an effect on the energy expressions, and hence on the damping matrix. The expressions of these direct and cross damping matrices may be obtained explicitly for some common spatial kernel functions and Euler-Bernoulli beam theory. Alternatively numerical integration may be applied to obtain solutions. Examples are given where the eigenvalues are compared to the exact solution for a pinned-pinned beam to demonstrate the convergence of the finite element method. The results for beams with other boundary conditions are used to demonstrate the versatility of the finite element technique.