Page 796 - Mechatronics with Experiments

782 MECHATRONICS
lengthening and shortening of fibers of material. The amount of extension is called “strain,”
, and is defined as

Δl
= (11.105)
l
As long as the deformation of the material stays within the linear elastic region, the relation
between stress and strain is provided by the constitutive law of the material:

= E ⋅ (11.106)

where E is the Young’s modulus of the material.
There is a linear relationship between the applied force and the induced strain,
as long as the deformation is limited to the elastic region of the deformation of the
material. For a cantilever beam with rectangular cross-section we have the following
relations,
M ⋅ (h∕2)
max = (11.107)
I
F ⋅ l ⋅ (h∕2)
s
= (11.108)
(1∕12) ⋅ b ⋅ h 3
6 ⋅ F ⋅ l ⋅ h
s
= (11.109)
b ⋅ h 3
6 ⋅ F ⋅ l s
= (11.110)
b ⋅ h 2
1
max = max (11.111)
E
6l s
= ⋅ F (11.112)
Eb h 2
where b, h are the width and thickness of the cantilever beam used in this experiment, l s
is the distance from the force application point to the strain gauge (measurement) point. I
is the area moment of inertia of the cross-section of the beam around the neutral axis of
1 3
bending, I = ⋅ b ⋅ h .
12
Equation 11.112 shows that measuring the strain results in an indirect measure of the
applied force. If we can measure strain, then we can estimate/calculate force if we have the
geometric and material property information of the beam (l , b, h, E).
s
( 2 )
Eb h
F = max (11.113)
6 l
s
Strain Gauge The strain measurement may be performed by strain gauge sensors.
These devices are applied on the test part in such a way that they are subjected to the same
strain (deformation) as the test part. In other words, the strain-gauge is bonded (glued)
on the test surface with a high strength adhesive. The resistance of the small wires that
constitute the strain gauge increases when they lengthen, and decreases when they shorten.
It follows that strain variations results in small resistance variations.
The resistance of a conductor of length L and cross-section A is given by:
L
R = (11.114)
A