Displacement and strain of vibrating structures using time-average holography

A technique for computing the bending strain resulting from the resonant modal deformation of vibrating plate-like structures is described. Interferometric fringes obtained by time-average holography are used as the basis for generating a mathematically continuous series approximation of a plate-like structure's normal displacement. The terms of the series consist of the clamped-free or free-free eigenfunctions of a simple beam. The bending strain is then obtained by computing the second derivative of the displacement series. The coefficients of the terms of the displacement series are computed for a given segment of a cantilevered plate-like structure based upon the holographic-fringe values lying along the same plate segment. A linear least-squares-solution routine is used to solve for the series coefficients, called modal weighting coefficients, in terms of the normal displacement values obtained from the holographic-fringe value. A ‘best fit’ solution is thus obtained for the plate displacement. This least-squares approach in conjunction with the fact the beam-series functions exactly satisfy the plate's geometric boundary conditions and approximately satisfy the plate's natural boundary conditions, results in a displacement series that yields quite accurate displacement and bending strain values.The technique described above has been programmed for use on a Tektronix 4010 interactive computer terminal. The accuracy and effectiveness of the computer algorithm, called HOLOCURVE, is checked by determining the displacement and bending strain at selected segment locations for three different modes of vibration of a cantilevered plate. The results are compared to those of an eigenvalue analysis carried out on a finite-element model of the plate using the finite-element computer program, NASTRAN. The HOLOCURVE results are in excellent agreement with the finite-element analysis except for cases where the bending strain is essentially zero as in the case of chordwise segment for a torsional-mode shape.