On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

Proper orthogonal decomposition (POD) is a useful experimental tool in dynamics and vibration. A common application of POD to structures involves sensed displacements, $x1t,$$x2t,…,xMt,$ at M locations on the structure. When the displacements are sampled N times at a fixed sampling rate, we can form displacement-history arrays, such that $xi=(xit1,$$xit2,…,xitN)T,$ for $i=1,…,M.$ The mean values are often subtracted from the displacement histories. These displacement histories are used to form an $N×M$ ensemble matrix,