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Apparatus Construction

In order to obtain a beat period of 20 s, the following dimensions were determined experimentally:

 

 

Bolt mass,  

Nut mass,  

Beat Nodes

In vibrational analysis, a node is a point that remains stationary. Due to beating, when one of the nut

masses is stationary, the other nut mass will obtain its highest amplitude. The following graph

indicates the maximum recorded amplitudes for various mass combinations.

48 50 52 54 56 58 60 62 64 66 6810

12

14

16

18

20

22

24

26

28

30

Mass (g)

   B  e  a   t   A  m  p   l   i   t  u   d  e   (  m  m   )

Measured Beat Amplitudes for Various Mass Combinations

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Modified System Configuration

The recorded dimensions of the modified system are as follows:

 

 Nut mass,  

Dowel Stick mass,  

Figure 1: Representative Diagram of Modified System

Analytical Derivation of Modified System

Lagrange’s Equations shall be used to determine an analytical derivation of the behaviour of the

modified system.

Since the wooden dowel stick is a rigid body, the summation of the kinetic energy of each mass

segment, , along the length of the stick is equal to the total linear kinetic energy of the stick.

Figure 2: Dowel Stick in Modified System

 

 

    

 

 

  

  

 

 

 

 

 

,

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The velocity can be determined along the length of the stick as a function of :

 The kinetic energy summation is represented as follows:

 ∫    

   ∫     

Where     for each element and    is the linear density.

Substituting for

 and simplifying:

     ∫    

     ∫          

         

Substituting       and simplifying:

 [     ] 

  [    ] 

Therefore the total kinetic energy is:

  [    ] The potential energy of the nut masses:

   

From Figure 2,   can be defined in terms of  and  for each nut mass.

 

 

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Therefore, the total potential energy,

  √    √   

  (  )  (  ) 

( )  (  ) 

√   Referring to Figure 1,  √   

Using small angle approximation,

 

Therefore

  Similarly,

  Thus, to determine the coupled equations of motion:

 

 

 (  )  (  )    

 ( )  (   )    

The matrix equation is therefore represented below:

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       {}    

   

The following mode shapes have been obtained on MATLAB for each degree of freedom:

     

      

This correctly indicates that the nut masses move either in the same direction, or in the opposition

direction. The mode shapes can thus be represented as:

     

     

The fundamental frequencies obtained are,

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0 10 20 30 40 50 60 70 80 90 100-0.01

-0.005

0

0.005

0.01

t(s)

  y   1

   (   t   )

Motion of Nut Mass 1

0 10 20 30 40 50 60 70 80 90 100-0.01

-0.005

0

0.005

0.01

t(s)

  y   2

   (   t   )

Motion of Nut Mass 2

The motion of the nut masses is represented graphically below:

Classic beating is demonstrated, with

 peaking when

 troughs and vice versa. Additionally,

the period is equal to that obtained experimentally.