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Analysis of elastic waves from two-point strain measurement

Strain, particle velocity, power transmission and related quantities can be determined at an arbitrary section of a cylindrical rod from measurement of strains at two different rod sections

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Abstract

Provided that the one-dimensional wave equation applies, strain measurement at two sections of a linearly elastic cylindrical rod makes it possible to determine a number of important quantities at an arbitrary section of the rod; for example, strain, particle velocity and power transmission. The equations needed are derived, and the design of an analogue real-time analyzer is presented. The influence of some principal sources of error is analyzed and it is shown that it should be possible to perform accurate evaluation (errors less than a few percent) during a time interval which is not very long compared to the travelling time for a wave between the two gage positions. Comparisons are made between direct measurement and digital evaluation of strain, and between digital and analogue evaluation of particle velocity and power transmission. The discrepancies are typically less than ten percent during a time interval of 20 travelling times between gages. Although these results do not represent what is achievable, the accuracy is sufficient in several applications and demonstrates the feasibility of the method used.

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Abbreviations

A :

cross-sectional area (m2)

c :

bar velocity (m/s)

E :

Young's modulus (N/m2)

F 0(t):

compressive force atx=0 (N)

f(t) :

arbitrary function (m)

g :

gage factor

h(t) :

arbitrary function (m)

P 0 (t),P 1 (t) :

power transmissions atx=0 andx=x 1, respectively (W)

R 0 :

resistance (ohm)

T,T 1,T 2 :

time constants (s)

T R :

rise time (s)

t :

time (s)

U A :

calibration voltage (V)

U P0 ,U P1 :

voltages representingP 0 andP1. respectively

U r :

calibration constant (V)

U v0 ,U v1 :

voltages representingv 0 andv 1, respectively (V)

U ε0,U ε1,U ε2 :

voltages representing ε0, ε1, ε2, ε A and ε B

U εA,U εB :

respectively (V)

U 0 :

factor of proportionality (V)

u(x, t) :

displacement (m)

v 0 (t), v 1 (t) :

particle velocities atx=0 andx=x 1, respectively (m/s)

x, x 1 ,x 2 :

position coordinates (m)

Z 0 :

characteristic impedance (ohm)

ΔC :

calibration error for strain measurement

ΔG :

error in open-loop gain

ΔT :

error in time distance between gages (s)

ε0 (t), ε1 (t), ε2 (t), ε A (t), ε B (t) :

strains atx=0,x=x 1, ‘A’, and ‘B’, respectively

σ0 (t) :

stress atx=0 (N/m2)

References

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Authors and Affiliations

  1. Mechanical Engineering Dept., University of Lulea, S-951 87, Lulea, Sweden

    B. Lundberg (Professor of Solid Mechanics)

  2. Instramentation Section, Institut CERAC S.A., C11-1024, Ecublens, Switzerland

    A. Henchoz (Head)

Authors
  1. B. Lundberg
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  2. A. Henchoz
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Lundberg, B., Henchoz, A. Analysis of elastic waves from two-point strain measurement. Experimental Mechanics 17, 213–218 (1977). https://doi.org/10.1007/BF02324491

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  • DOI: https://doi.org/10.1007/BF02324491

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