Department of Mechanical Engineering
University of Bahrain
MEG 381 Instrumentation:
Lab: Cantilever Stiffness using LVDT
Objective
To determine a cantilever stiffness using Linear Variable Differential Transformer
(LVDT)
Theory:
The LVDT is a series of inductors in a hollow cylindrical shaft and a
solid cylindrical core, See the figure below. The LVDT produces an
electrical output proportional to the position of the core. The lack of
friction between the hollow shaft and the core increases the life of
the LVDT and enables very good resolution. In addition, the small
mass of the core allows for good sensitivity in dynamic tests.
In this experiment, the deflection at the end of cantilever beam will
be measured using LVDT- sensor. When the beam is loaded with
different weights, the end of the beam will deflect.
The displacement of the end of the beam can be measured For each
applied load. Since the LVDT has been previously calibrated, this
measurement will provide a measure of displacement. The
relationship between the applied load and displacement of the beam
at the end is given by Hook's law which states that
F ky
Where:
F:
applied load
Dr. Mostafa S. Habib
MEG 381
Cantilever Deflection using LVDT
1
K:
y:
stiffness of the beam
vertical displacement of the beam at the applied force
Therefore, by plotting the data (force vs. displacement), the stiffness of
the beam (k) can be determined experimentally as the slope of a
best-fit linear approximation. Recall that for a cantilevered beam of
a uniform cross-section, the stiffness is given by the equation
k
Where:
3E I
L3
E: Young's modulus
I:
Moment of inertia
L: is the length from the support to the point of force application
The stiffness obtained from this equation will be compared to the
stiffness determined experimentally. If there are significant
differences, it will be necessary to determine the most likely
dominant source(s) of error.
Theoretical Beam stiffness:
Beam geometry:
b  0.036 m ;
I
h  0.002 m ;
L  0.245 m
b h 3 (0.036 ) (0.002 ) 3

 2.4 *10 11 m 4
12
12
3 E I 3 (200 *10 9 ) (2.4 *10 11 )
k 3 
L
(0.245) 3
k  960 N / m
Experimental Procedure:
1- Setup the LVDT to measure the end deflection of the cantilever
beam.
Dr. Mostafa S. Habib
MEG 381
Cantilever Deflection using LVDT
2
2- Setup and obtain all other necessary equipment (Clamps,
weights etc).
3- Apply the known weights to the end of the beam, and measure
the corresponding LVDT output voltage.
Note that the LVDT has been previously calibrated and the null
point was obtained.
4- Construct a table between the applied load and LVDT output
voltage.
5- Obtain a linear relationship between the applied force and
displacement of the cantilever free end.
6- Determine the stiffness of the beam using linear regression.
(show the best fit line together with the obtained data).
7- Compare the stiffness obtained analytically to the stiffness
measured experimentally.
8- Comment on your results.
Experimental Results:
Note that from previous lab we found that:
V  0.084 y
(Sensitivity = 0.084 volt / mm )
Force, F (N)
Dr. Mostafa S. Habib
V (volt)
MEG 381
Deflection, y
(mm)
Cantilever Deflection using LVDT
3
Force, F (N)
V (volt)
Deflection, y (mm)
0.00
0.002
0.024
0.981
0.067
0.798
1.962
0.134
1.595
2.943
0.200
2.381
3.924
0.265
3.155
4.905
0.330
3.930
5.886
0.395
4.700
6.867
0.454
5.405
7.848
0.521
6.202
8.829
0.583
6.941
9.810
0.646
7.691
10.791
0.696
8.286
11.772
0.730
8.700
%Finding the deflection by LVDT theory
%AMMAR AHMED 20003417
%MEG381
%-------------------------------------% Data
F=[0.00 0.981 1.962 2.943 3.924 4.905 5.886 6.867 7.848...
8.829 9.810 10.791 11.772];
y=[0.0000 0.0008 0.0016 0.0024 0.0032 0.0039 0.0047...
0.0054 0.0062 0.0069 0.0077 0.0083 0.0087];
plot(y,F,'x');grid on
xlabel('Deflection (m)');ylabel('Force (N)');
title('Deflection as function of applied load');
Dr. Mostafa S. Habib
MEG 381
Cantilever Deflection using LVDT
4
% error=
1300  960
* 100 %  35 %
960
Conclusion:
In this experiment we measured the deflection of a cantilever beam using LVDT fixed
at the end tip of the beam. This was done by applying different loads and measuring
the output voltage of the LVDT. Since the LVDT has been calibrated in the previous
lab, we were able to convert the LVDT output voltage to corresponding displacement
of the LVDT by dividing over the sensitivity. By graphing the relationship between
the force and the deflection, we found that the beam stiffness was equal to
k  1300 N / m .And from analytical method the beam stiffness was k  960 N / m .
The percentage error calculated indicates that there is a significant difference between
the two values; this difference is due to sources of error present in the experiment.
Dr. Mostafa S. Habib
MEG 381
Cantilever Deflection using LVDT
5
Dr. Mostafa S. Habib
MEG 381
Cantilever Deflection using LVDT
6