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Math Help - Integration Unsolvable Problem

  1. #1
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    Integration Unsolvable Problem

    A rotating, solid body that is simulated on a computer is done by calculating each frame after a miniscule amount of time.

    I believe that it is possible to predict the actual frame of orientation of a rotating body with a high degree of accuracy by using theorectical equations. Well, as far as I know, there is no solution for a theorectical equation. I am trying to find one where the property of a roating body can be calculated after any given amount of time.

    I am trying to solve this equation at the moment. It describes the propety of a rotating body against time.

    \frac{d^2\theta}{dt^2} = K sin (2\theta)

    where I believe that

     \theta = k F(kt + k) + kt + k

    where F is a function of time for a periodic waveform and all instances of k are different constants. Eg F could be sin(kt) but a simply sine function does not satisfy the differential.

    Anyone knows of any solutions!
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  2. #2
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    Quote Originally Posted by eulinator
    Anyone knows of any solutions!
    Maybe you can express the sine function as an approximate taylor polynomial.
    \theta ''=2\theta -\frac{4}{3}\theta ^3
    Problem is how to work with it, since it is a non-linear diffrencial equation.
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  3. #3
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    Quote Originally Posted by eulinator
    A rotating, solid body that is simulated on a computer is done by calculating each frame after a miniscule amount of time.

    I believe that it is possible to predict the actual frame of orientation of a rotating body with a high degree of accuracy by using theorectical equations. Well, as far as I know, there is no solution for a theorectical equation. I am trying to find one where the property of a roating body can be calculated after any given amount of time.

    I am trying to solve this equation at the moment. It describes the propety of a rotating body against time.

    \frac{d^2\theta}{dt^2} = K sin (2\theta)

    where I believe that

     \theta = k F(kt + k) + kt + k

    where F is a function of time for a periodic waveform and all instances of k are different constants. Eg F could be sin(kt) but a simply sine function does not satisfy the differential.

    Anyone knows of any solutions!
    If K is negative this is an equation that is similar to the exact simple pendulum equation (set x = 2 theta and simplify) so I can say immediately that the solution can be expressed as an elliptic integral. (If K is positive I don't know what the solution would be, but I imagine it is similar enough to the negative case that a similar comment would apply.) In other words, there is no closed form solution for the exact equation. As ThePerfectHacker suggests the standard method is to approximate the equation with a Taylor series. If you take only the first term approximation you get a simple harmonic oscillator (again with negative K), which can indeed be solved using a sine function. I don't know if the second order approximation has a closed form or not.

    -Dan
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  4. #4
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    Thanks

    Thanks for your helpful inputs.

    Quote Originally Posted by ThePerfectHacker
    Maybe you can express the sine function as an approximate taylor polynomial.
    \theta ''=2\theta -\frac{4}{3}\theta ^3
    Problem is how to work with it, since it is a non-linear diffrencial equation.
    I have very limited knowledge on simple linear differencial, but I see where you are going with this by using Taylor's approximation. For my benefit (link1)(link2)

    I think that I can get more clues on what the equation form is more precisely. Without clues, it is very difficult to solve

    I analysed another property of the rotating body, and it seemed to be composed of three superimposed sinusoids. This was done using a numerical Fourier transform on the proerty. I still do not know a general equation for this property either, but I may find actual numerical values for the constants of the composite sinusoids via further software analysis. Then use an actual numerical instance of the equation, not general form, to calculate an actual numerical instance of theta. (theta has a known relation with the composite sinusoid property. If the composite sinusoid instance is 3sin(t)+4sin(2t)+3sin(t+3) then an instance for theta should be calculable, eg theta = sin(5t) - 10e^t + 7t + ...) . Then try to get a general form for theta from its instance form. The analysis will take me some time, because of the time I am away from the developing computer.

    Here are some graphs of theta, simulated numerical values.





    I will soon post, hopefully, a particular instance for what theta is.
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  5. #5
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    Do you have software which draws slope-fields?
    I would really like to look at the curve.
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  6. #6
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    Software Tool Needed

    I do not have any software on (Slope Fields) but I have the curve data in an MS Excel document, about 1 MB. I can email it to you.

    This would also be useful. Does anyone know of a software tool for doing fourier analysis. I have a text data of a composite sinusoid. Eg
    Code:
    time      f(time)
    0.001      0.030
    0.002      0.031
    0.003      0.032
    ...
    If 3 sinusoid waves are present in the signal, f(time), then which software tool can calculate the frequency, amplitude, and phase shift of each individual sinusoid. It would be helpful to display the graphs of the individual sinusoids for checking purposes.

    I already have an adhoc C++ program which can display the frquency, graphically, but I would like the amplitude for each sine wave, as well.

    I can generate various files with the f(time) data for extremely small changes in time, but I am having software difficulty in analysis the files. I am a software developer, but I would like a software that can do this already and better than adhoc.
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  7. #7
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    Attatched file

    [QUOTE=ThePerfectHacker]Do you have software which draws slope-fields?
    I would really like to look at the curve.QUOTE] Here is an xls file that has time in the first column and theta in the second column.

    Not sure if that is what you wanted, but I know that figuring it out without any clues is extremely difficult.
    Attached Files Attached Files
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  8. #8
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    Code:
    File:  small.zip  
    Status:  OK  
    MD5  12c9995490089914623d7ab8ca716419  
    Packers detected:  - 
    Scanner results  
    AntiVir  Found nothing 
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    Can never be too careful about these .rar/.zip files.
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