Calculating the Dynamic Strain Tensor Field Using Modal Analysis and Numerical Differentiation

Dynamic Traction Vector Field Estimation in a Structure Using Hybrid Strain Analysis

“…The method is an extension and application of the results reported by Sehlstedt [2], where a technique for obtaining dynamic strain tensor "elds was presented. This technique, called hybrid strain analysis (HSA), is based on input from hybrid modal analysis (HMA).…”

“…Approximation of the residual and the consequence of neglecting the residual were discussed in Dovstam [3] and Sehlstedt [2].…”

“…The term hybrid should be interpreted here as a mixture of measurements and continuous mode series expansion. For details of HMA and HSA see Dovstam [3] and Sehlstedt [2] respectively.…”

“…An indication of this will be noticed when validating the coe$cient spectra, as in item 5; (4) Fourier coe$cient spectra estimation, using equation (10); (5) Validation of coe$cient spectra using the truncated version of equations (7), and measured validation spectra not used in the estimation of the Fourier coe$cient spectra, as in item 4; (6) Assume new values for the properties for the "ctitious part, DGAR , and carry out 2}5 until the convergence criterion is ful"lled, i.e., the error between the predicted and separately measured validation spectra is minimized; (7) Displacement response prediction at 3n#1 points, i.e., the point of interest and n adjacent points in each (Cartesian) co-ordinate direction, using the truncated version of equation (7); (8) Strain tensor calculation using "nite di!erence schemes of order n and equation (A.6). It is assumed that the numerical di!erentiation is performed by n#1 point "nite di!erence schemes (see reference [2]); (9) Traction vector estimation using the calculated strain tensor and equation (6).…”

“…The true traction vector is obtained using de"nition (6) and direct FE computed displacements. The spatial derivatives in equation (6) are calculated using numerical di!erentiation, in three point non-equidistant "nite di!erence schemes (see reference [2]). Also in HTA three point non-equidistant "nite di!erence schemes are used.…”