4 Full-Field Strain Shape Estimation from 3D SLDV 35 Fig. 4.6 Element nodes and coordinate system 4.3.2 Strain Post-processor In this work, the displacements are arbitrary shapes (mode shapes or ODS) rather than directly measured displacements. This posed an issue for using the Polytec strain post-processor, which operates on Band Data within their proprietary scan file format. Rather than creating a tool to write arbitrary shapes to a Band Data scan file, a standalone 2D strain post-processing script was written in MATLAB (note, 3D implementations will work equally well). Measured data from the SLDV system were exported in Universal File Format, which included the test geometry. The geometry node coordinates were in 3D (x,y,z) but were reduced to 2D surface coordinates (x,y) for each surface of interest. Node coordinates (x,y) and connectivity information were extracted from these files to create elements. Elemental deformations were either the experimental mode shapes (E) estimated from the SLDV data using Synthesize Modes and Correlate (SMAC) [6] or ODS ( O) that were extracted directly from Polytec “Fastscan” data. With geometry and displacement data, strains were then calculated using a bilinear quadrilateral element formulation as detailed in [7] and summarized below. Strains were computed at the center of the element, (ξ, η) =(0, 0), and interpolated to a common set of points with the FEM so the measured results could be directly compared to analytical results. The elemental coordinate system is shown in Fig. 4.6 and the element shape functions are given in Eq. 4.3: N1 = 1 4 (1−ξ)(1−η) N2 = 1 4 (1+ξ)(1−η) N3 = 1 4 (1+ξ)(1+η) N4 = 1 4 (1−ξ)(1+η) (4.3) The Jacobian matrix of (x,y) with respect to (ξ, η), denoted asJ, is used to establish the relationship between the derivatives of physical and elemental displacement: ∂Ni ∂x ∂Ni ∂y =J−1 ∂Ni ∂ξ ∂Ni ∂η ;J = ∂x ∂ξ ∂y ∂ξ ∂x ∂η ∂y ∂η (4.4) where the differential entries of Jare obtained through Eq. (4.5) below. ∂x ∂ξ = i xi ∂Ni ∂ξ , ∂y ∂ξ = i yi ∂Ni ∂ξ , ∂x ∂η = i xi ∂Ni ∂η , ∂y ∂η = i yi ∂Ni ∂η (4.5) Finally, strains are calculated using the strain-displacement matrix, B, composed of the differentials calculated above and measured displacements, denoted here as (u,v) representing either ODS or mode shape coefficients. τ = ⎡ ⎣ τxx τyy 2τxy ⎤ ⎦= ⎡ ⎢ ⎢ ⎣ ∂N1 ∂x 0 ∂N2 ∂x 0 . . . ∂Nm ∂x 0 0 ∂N1 ∂y 0 ∂N2 ∂y . . . 0 ∂Nm ∂y ∂N1 ∂y ∂N1 ∂x ∂N2 ∂y ∂N2 ∂x . . . ∂Nm ∂y ∂Nm ∂x ⎤ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ u1 v1 u2 v2 u3 v3 u4 v4 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭ =Bu (4.6)
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