Automating Manufacturing Systems - Process Control and ...

continuous sensors - 23.16
After the wire in Figure 23.16 has been deformed it will take on the new dimensions
and resistance shown in Figure 23.17. If a force is applied as shown, the wire will
become longer, as predicted by Young’s modulus. But, the cross sectional area will
decrease, as predicted by Poison’s ratio. The new length and cross sectional area can then
be used to find a new resistance.
w’
L’
t’
σ
F
--
F
= = ----- = Eε
A wt
∴ε
=
---------
F
Ewt
F
R' ρ------- L' ρ ---------------------------------------------
L( ⎛ 1 ⎞ + ε)
= =
w't' w( 1 – νε ⎝
⎠ )t( 1 – νε)
∴∆R
= R' – R = R
( 1 + ε)
( --------------------------------------- –
1 – νε) ( 1 – νε)
1
where,
ν = poissons ratio for the material
F = applied force
E = Youngs modulus for the material
σε , = stress and strain of material
Aside: Gauge factor, as defined below, is a commonly used measure of stain gauge
sensitivity.
GF
=
------ ⎛ ⎞ ∆R
R ⎝ ⎠
------------
ε
Figure 23.17
The Electrical and Mechanical Properties of the Deformed Wire