Dr. Paul Bosomworth (CEO/President, BuzzMac International) re-analyzed some of the original frequency equations for rectangular and circular bars by Spinner, Reichard, and Tefft1,2, Goens3, and Pickett4 - the data that ASTM E1876 and C1259 are based upon - to see if these frequency equations could be improved. Much improved frequency equations were derived. An example is shown here.

The results will be published in more detail in Journal Of Materials Science

The equation relating young's modulus to the fundamental flexural resonant frequency can be written1,2:

eq1........................................................1

Where:

f = flexural resonant frequency
beta1 = eigenvalue for the first (fundamental) mode of vibration
k = radius of gyration of the cross-section
L = sample length
E = Young’s modulus
rho= bulk density

For a rectangular or square cross-section, k = t/√12 where t = thickness.

 

and T1 is given by:

eq2........................................................2

Where:

μ = Poisson’s ratio

 

T1 is the correction factor given in ASTM E1876 and C1259. However, in the original work by Spinner, Reichard, and Tefft2, T1 given by equation 2 begins to deviate from an experimentally determined value for k/L > 0.08. By a new analysis at BuzzMac International of the original equations of Goens3 and Pickett4 and the experimental data of Spinner et al2, it was found that an improved expression for T1 could be used (figure 1).

E vs T

Fig. 1. comparison of correction factors (T1) for the fundamental flexure mode of vibration


Note: the Poisson's ratio of the samples was determined by Tefft et al to be 0.292


Equation 2, as used in ASTM E1876/C1259, is represented by the green line. The new equations are represented by the red and blue lines. All these equations were plotted with a Poisson's ratio of 0.29. It can be seen that the new equation covers a wider range of aspect ratios or reciprocal slenderness ratios (k/l).

1A. Spinner, and W. E. Tefft, "A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies," Proceedings ASTM, 61 1221-1239 (1961)

2S. Spinner, T. W. Reichard, and W. E. Tefft, "A Comparison of Experimental and Theoretical Relations Between Young's Modulus and the Flexural and Longitudinal Resonance Frequencies of Uniform Bars," J. Res. of the National Bureau of Standards-A. Physics and Chemistry, 64 [2] (1960)

3E. Goens, "Euber die Bestimmung des Elastizitätsmoduls von Stäben mit Hilfe von Biengungsschwingungen," Annalen der Physik, B. Folge, Band 11 649-678 (1931)

4G. Pickett, "Equations for Computing Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prisms and Cylinders," Proceedings ASTM, 45 846-865 (1945)