DEVELOPING POD OVER THE COMPLEX PLANE TO FORM A DATA PROCESSING TOOL FOR FINITE ELEMENT SIMULATIONS OF STEADY STATE STRUCTURAL DYNAMICS

摘要

To analyze the steady state response of structural dynamical systems with multi-field response (example, Timoshenko shearable rod) given complex-valued databases (finite element simulations of complexified equations of motion), we have developed a Complex Proper Orthogonal Decomposition (C-POD) transform. Like the regular multi-field POD, the development of the C-POD is based on the primitive space and frequency auto-correlation operations. These data fusion operations give rise to complex Hermitian operators whose solution determines the C-POD transform. The eigen-values of the complex Hermitian operators are strictly positive and it is shown that they represent the energy fractions of the autocorrelation energy contained in the POD modes. The POD modes have both amplitudes and shapes that are complex-valued scalar functions. The C-POD transform is verified by applying it to characterize the finite element simulations of the steady state dynamics of planar beams and arches. It turns out that the real part of the shape of a POD mode coincides with the shape of the linear POD; whereas its amplitude is a localized function of frequency at a critical frequency which is identical to a natural frequency.