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```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.

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

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