Abstract
Cubic-spline and discrete-quadratic polynomial techniques are presented for reliably computing up to third-order derivatives of experimental information. The concept is demonstrated by stress analyzing from measured displacements a transversely loaded plate and a beam under four-point bending. The respective displacement fields were recorded using holography and moiré. The accuracy of the employed numerical-differentiation techniques is indicated.
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Abbreviations
- x :
-
independent variable
- y=y(x) :
-
theoretical relationship
- y′, y″, y‴ :
-
analytical derivatives from theoretical coordinates (x, y)
- R((Y; x) :
-
cubic-spline polynomial
- I (Y; x) :
-
cubic-spline interpolation polynomial
- L (Y; x) :
-
discrete-quadratic polynomial
- R′(x, y), L′(x, y) :
-
numerical derivatives from theoretical coordinates (x, y)
- R′, L′ :
-
numerical derivatives from smoothed input data represented byR(Y; x) orL(Y; x), respectively
- R″, L″ :
-
numerical second derivatives from smoothedR′ orL′ input data, respectively
- \(\frac{{\partial ^2 R}}{{\partial x^2 }}, \frac{{\partial ^2 L}}{{\partial x^2 }}\) :
-
numerical second derivatives computed directly from smoothed input data represented byR (Y; x) orL (Y; x), respectively
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Rowlands, R.E., Liber, T., Daniel, I.M. et al. Higher-order numerical differentiation of experimental information. Experimental Mechanics 13, 105–112 (1973). https://doi.org/10.1007/BF02323967
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DOI: https://doi.org/10.1007/BF02323967