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Complex-modulus measurement by longitudinal vibration testing

Paper presents a method to measure the complex modulus of suitably rigid viscoelastic materials by measurement of the end-displacement ratio when one end of a rod of the material is driven harmonically

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Abstract

A simple method is presented to measure the complex modulus of suitably rigid, linear viscoelastic materials over the audio-frequency spectrum. The case is considered where one end of a rod of the material is driven harmonically and the complex displacement ratio is measured. The effect of a rigid end mass on the free end is accounted for. It is shown that, at specific frequencies near resonance, it is easy to obtain modulus data with standard equipment usually found in the vibration laboratory.

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Abbreviations

A :

cross-sectional area of the bar

c :

phase velocity\(\sqrt {E*/\rho } \sec \delta /2\)

E(iw) :

complex modulus

E * :

magnitude of complex modulus

i :

\(\sqrt { - 1} \)

Im :

imaginary part of the displacement ratio

l :

length of the test bar

M :

end mass

n :

mode of vibration

p :

ω/c(1−i δ/2)

Q :

displacement ratio of free end to driven end

R :

mass ratiom/ρAl

Re :

real part of the displacement ratio

t :

time

u :

displacement relative to the moving base\(\bar u(x, i\omega )\) exp (iωt)

\(\bar u\) :

amplitude of relative displamementu

u o :

displacement at the fixed end of the barU o exp (iωt)

U o :

amplitude of displacementu o

x :

axial coordinate along bar

δ:

angle by which strain lags stress δ(ω)

ɛ:

strain\(\bar \in (x, i\omega )\) exp [i(ωt−δ)]

\(\bar \in \) :

amplitude of strain ɛ

ζ:

frequncy ratio ωl/c

ρ:

mass density

σ:

stress\(\bar \sigma (x,i\omega )\) exp (iωt)

\(\bar \sigma \) :

amplitude of stress σ

ϕ:

phase angle between bar-end displacements

ω:

exciting angular frequency

References

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Authors and Affiliations

  1. Department of Mechaical Engineering, University of New Hampshire, Durham, N. H.

    D. M. Norris Jr. (Associate Professor of Mechanics)

  2. Applied Mechanics Group, Combustion Engineering, Inc., Windsor, Conn.

    Wun-Chung Young (Project Engineer)

Authors
  1. D. M. Norris Jr.
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  2. Wun-Chung Young
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Norris, D.M., Young, WC. Complex-modulus measurement by longitudinal vibration testing. Experimental Mechanics 10, 93–96 (1970). https://doi.org/10.1007/BF02320139

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  • DOI: https://doi.org/10.1007/BF02320139

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