Engineering Acoustics/Simple Oscillation

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Part 1: Lumped Acoustical Systems1.

Part 2: One-Dimensional Wave Motion2.12.22.3

Part 3: Applications3.

The Position Equation


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




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 Forms


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

    Simple Oscillator-1.2.1.a




    Simple Oscillator-1.2.1.b



    Simple Oscillator-1.2.1.c







    Simple Oscillator-1.2.1.d



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