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 Equation[edit]

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 Forms[edit]

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

  1. Simple Oscillator-1.2.1.a



  1. Simple Oscillator-1.2.1.b






  1. Simple Oscillator-1.2.1.c

Simple Oscillator-1.2.1.c-solution



  1. Simple Oscillator-1.2.1.d










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