A discrete convolutional Hilbert transform with the consistent imaginary initial conditions for the time-domain analysis of five-layered viscoelastic sandwich beam

A discrete convolutional Hilbert transform (DCHT) with the consistent imaginary initial conditions, together with the development of 2-node 8-DOF damped beam element, are presented for the reliable DOF-efficient time-domain analysis of five-layered viscoelastic sandwich beam. Motivated by the fact that the longitudinal displacements of three metallic layers can be replaced with the transverse shear strains of two viscoelastic core layers, a DOF-efficient damped beam element with the nodal DOFs composed of the deflection and rotation of beam and shear strains of two viscoelastic core layers is derived according to the virtual work principle and the compatibility relation. The standard Hilbert transform using Fourier and inverse Fourier transform of impulse signals produces the totally different results from the analytically derived ones near the end of time period, and the non-conjugate complex eigen values in a state-space formulation cause the unbounded growth in the time response of the damped structural dynamic system when a standard time integration scheme is used. To resolve these numerical problems, the imaginary external force is obtained by dividing the real external force into a finite number of rectangular impulses and by superposing Hilbert transforms of each rectangular impulse. And the time response of the damped sandwich beam subject to arbitrary external force is obtained by the convolution of time response to unit impulse. Meanwhile, the consistent imaginary initial conditions which can provide the bounded damped time response are numerically derived by splitting each decoupled complex second-order differential equation in the mode superposition approach into real and imaginary ones and by solving general solutions of each two split equations in the space-state formulation. The proposed method is validated through the numerical experiments composed of analytic and five-layered damped sandwich beam examples.