The “round trip” theory for reconstruction of Green's functions at passive locations

An expression for the Green's function at an arbitrary set of passive locations (no applied force) is derived and validated by experiment. Three sets of points are involved, the passive reconstruction points, c, which lie on a virtual boundary and two sets of auxiliary points, denoted a and b, located either side. The reconstruction is achieved using Green's functions forming a "round trip" from and to the reconstruction points via a and b. A two stage measurement procedure is described involving excitation at b and a but with no excitation required at the reconstruction points. A known "round trip" relationship is first introduced which is theoretically exact for points on a multi-point interface between two linear, time invariant subsystems. Experimental results for frequency response functions of a beam-plate structure show that this relationship gives good results in practice. It is then shown that the theory provides an Nth order approximation for the Green's function at arbitrary points, where N is the number of points at b. The expression is validated by reconstructing point and transfer frequency response functions at two passive points on an aluminum plate.