Experimental Mechanics

, Volume 13, Issue 3, pp 105–112 | Cite as

Higher-order numerical differentiation of experimental information

Cubic-spline and discrete-quadratic polynomials are described for numerically computing up through third-order derivatives. Concept is demonstrated by stress analyzing, from moiré and holographically recorded displacements, loaded plates and beams
  • R. E. Rowlands
  • T. Liber
  • I. M. Daniel
  • P. G. Rose


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.


Mechanical Engineer Fluid Dynamics Displacement Field Stress Analyze Experimental Information 
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List of Differentiation Symbols


independent variable


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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Copyright information

© Society for Experimental Mechanics, Inc. 1973

Authors and Affiliations

  • R. E. Rowlands
    • 1
  • T. Liber
    • 1
  • I. M. Daniel
    • 1
  • P. G. Rose
    • 2
  1. 1.Stress Analysis, IIT Research InstituteChicago
  2. 2.Northwestern UniversityEvanston

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