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Composition of 2 simple harmonic motions with different angular veloci

  1. Apr 19, 2014 #1
    1. The problem statement, all variables and given/known data

    I have 2 simple harmonic motions and I want to compose them on the same axis. So:

    x1 = A1*sin(ω1*t+θ1)
    x2 = A2*sin(ω2*t+θ2)

    The goal is to find the resultant motion of these 2 in the form:
    X = A*sin(ω*t+Θ), so to find A,ω and Θ as functions of A1,A2,ω1,ω2,θ1 and θ2.

    2. Relevant equations

    I believe there is an analytical way and a graphical way(Fresnel graph) to solve this .



    Thanks,
    Radu
     
  2. jcsd
  3. Apr 19, 2014 #2
    Use the trigonometric formulas for the addition and/or subtraction of two sines.
     
  4. Apr 19, 2014 #3

    Simon Bridge

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  5. Apr 19, 2014 #4
  6. Apr 19, 2014 #5
    The idea is to find the resultant motion of these two harmonic motions.
     
  7. Apr 19, 2014 #6
    Have you tried my suggestion on post #2?
     
  8. Apr 19, 2014 #7
    The thing is that x1+x2 is not a solution for the differential equation, because the two motions do not have the same pulsation(angular velocity).
     
  9. Apr 19, 2014 #8

    Simon Bridge

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    There are many ways that two harmonic motions can be combined.
    You used the work "composition" to describe how they are combined.

    How are they to be combined?

    In English ##\omega## is "angular velocity" or "angular frequency".
    This is the first time you have mentioned a differential equation.
    Please provide all information that is important to the problem.

    Note: if x1 and x2 are solutions to the same linear DE, then x1+x2 is also a solution.

    Please understand:
    (1) If you leave out important information, we cannot help you.
    There is no way to tell of a suggestion is any good if we don't know what conditions the solutions have to satisfy.

    (2) If you do not answer questions, we cannot help you.
    The questions we ask you are not arbitrary or rhetorical - the answers to the questions help us to know how best to help you.
     
  10. Apr 19, 2014 #9

    gneill

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    If the two harmonic motions are to be combined additively and they do not have the same angular frequency then the resultant cannot be represented by a single sinusoid. Consider the Fourier transform of the combined (added) functions; both frequencies will be represented in the power spectrum as separate components. As a consequence the inverse transform must contain terms for each, too.

    Things might be more interesting if the motions were directed along different axes. Then the combined motion might be represented by a 2D Lissajous figure.
     
  11. Apr 19, 2014 #10
    You will have to allow the amplitude of the combined motion to be a function of time. Have you tried my suggestion from post #2?
     
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