Using the Frequency of Vibration of a Loaded Beam to
Determine the Young's Modulus E, or to Determine g, the
Rate of Acceleration
By Patrick Bruskiewich
Abstract
The frequency of a vibrating beam can be used to determine the Young's modulus of the
beam, or g, the rate of acceleration . This paper was written in 1981. The manuscript
remained lost in the author's papers for years, until recently rediscovered with other
manuscripts.
1.0 A Loaded Vibrating Beam
Consider a beam of length 21 and negligible mass supported at either end, with a weight
W at the centre. The beam is of negligible mass and has a moment of inertia I, and a
Young's Modulus E (refer to Fig. 1: Loaded, vibrating beam) [1]
Fig. 1 : A Loaded Vibrating Beam
With a weight W placed at the centre of the beam, the centre is depressed by an amount y
from its unloaded equilibrium level given by
f 73 A
y =
1
v
K EIj
W
1.1
By Newton's Third Law, the beam exerts an equal but opposite force on the weight, such
that the restoring Force F res is given by
r
K,
6
V
/ 3
y
1.2
Set the beam vibrating with a slight push to the weight W. Given that the weight W is
subject to the acceleration g, the equation of motion for the vibrating beam is
W d 2 y
g dt 2
( \EI\
V
y + W
J
1.3
which is a second order differential equation with constant coefficients,
d 2 y , g(*Ei}
+
dr wy v )
6
y-g =
1.4
To solve this equation use a trial function of the form
y = A + B cos(cot + a)
1.5
where a> is an angular frequency and a is the initial phase. By inspection we find that
f 73 N \
y =
K EIj
W + Bcos
f
VL
g( 6 m
w
V
V
J
\
t + a
1.6
J
2.0 Using the Period to determine the Young's Modulus
The frequency f of the vibrating beam is given by
/ = - ^
2x\W
6
V
V
2.1
J
Provided there is no deformation of the beam, for a known weight W, and a known
moment of inertia I, if we can measure the frequency of oscillation experimentally,
ceteris paribus, we can determine the Young's Modulus E at that loading, namely
f
f ;3 A
4tt ? '
6gl
W
f
An
2 m
3\
V
6gl
= E
2.2
3.0 Determination of the Value for g using a Vibrating Beam
If we have a known beam, with an accurately determined W, E and I we can use the
frequency of the vibration of the beam to determine g, the rate of acceleration, namely
/
(
An
I
3 N \
V
6/
J
(W^
\z J
= g
3.1
The inquiring and intrepid mind can try their hand at these types of experiments. This
technique to determine g is used extensively in mineral and petroleum exploration.
References:
,rd
[1] H.B. Phillips, Differential Equations, 3 ra ed. John Wiley and Sons, 1951, p. 132-133
1981 Patrick Bruskiewich