The balanced proper orthogonal decomposition applied to a class of frequency-dependent damped structures

Abstract

A model reduction technique aimed at computing efficiently Frequency Response Functions of damped structures is presented. The frequency-dependent complex moduli are approximated by a mini-oscillators model, known as the Golla–Hughes–MacTavish (GHM) model, which permits to recast the original problem as a more familiar second-order, constant-coefficient system of equations. The matrix system, although much larger, is then treated by application of the Balanced Proper Orthogonal Decomposition (BPOD) which aims at approximating the transfer function matrix, or equivalently the admittance matrix, connecting forces and displacements at a specified set of points of the vibrating structure. All the necessary ingredients of the reduction strategy as well as its efficiency measured in terms of data reduction, accuracy and computational cost are shown. Two illustrative examples of increasing complexity involving a clamped cantilever beam and a realistic windshield are presented. It is shown that the admittance matrix can be approximated by matrices of very small size which computation can be speeded up via diagonalization. It is concluded that the application of BPOD combined with the GHM decomposition of the frequency-dependent algebraic system proves extremely efficient for the modeling of vibrating structures made of different materials, either viscoelastic or purely elastic.