Engineering Acoustics/Simple Oscillation

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Part 1: Lumped Acoustical Systems1.11.21.31.41.51.61.71.81.91.101.11

Part 2: One-Dimensional Wave Motion2.12.22.3

Part 3: Applications3.13.23.33.43.53.63.73.83.93.103.113.123.133.143.153.163.173.183.193.203.213.223.233.24

The Position EquationEdit

This section shows how to form the equation describing the position of a mass on a spring.

For a simple oscillator consisting of a mass m attached to one end of a spring with a spring constant s, the restoring force, f, can be expressed by the equation

 

where x is the displacement of the mass from its rest position. Substituting the expression for f into the linear momentum equation,

 

where a is the acceleration of the mass, we can get

 

or,

 

Note that the frequency of oscillation   is given by

 

To solve the equation, we can assume

 

The force equation then becomes

 

Giving the equation

 

Solving for  

 

This gives the equation of x to be

 

Note that

 

and that C1 and C2 are constants given by the initial conditions of the system

If the position of the mass at t = 0 is denoted as x0, then

 

and if the velocity of the mass at t = 0 is denoted as u0, then

 

Solving the two boundary condition equations gives

 


 


The position is then given by

 


This equation can also be found by assuming that x is of the form

 

And by applying the same initial conditions,

 


 


This gives rise to the same position equation

 



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Alternate Position Equation FormsEdit

If A1 and A2 are of the form

 
 


Then the position equation can be written

 


By applying the initial conditions (x(0)=x0, u(0)=u0) it is found that

 


 


If these two equations are squared and summed, then it is found that

 


And if the difference of the same two equations is found, the result is that

 


The position equation can also be written as the Real part of the imaginary position equation

 


Due to euler's rule (e = cosφ + jsinφ), x(t) is of the form

 


Example 1.1

GIVEN: Two springs of stiffness,  , and two bodies of mass,  

FIND: The natural frequencies of the systems sketched below

 

 

 



 

 






 

 

 

 

 

 



 

 










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