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Full text of "Using the Frequency of Vibration of a Loaded Beam to Determine the Young's Modulus E, or to Determine g, the Rate of Acceleration"

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