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Math Help - Mode Shape Function Integration

  1. #1
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    Mode Shape Function Integration

    Hi

    I am really finding it difficult to integrate the square of the mode shape function(please see the attachment) with respect to r and theta. Basically I am trying to fidn the modal mass of a clamped edge circular plate whose mode shape function is known (consists Bessel Functions).

    modal mass= rho * thickness * integration of (mode shape function)^2 with respect to r and theta

    I havespent almost three weeks trying to solve this problem. Still no luck. If someone could really help me with it I would be very grateful. How would I approach this problem??

    Thanks
    Attached Thumbnails Attached Thumbnails Mode Shape Function Integration-mode-shape.jpg  
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  2. #2
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    Re: Mode Shape Function Integration

    Hey ruzfactor.

    Do these Bessel functions have simplifications due to n being 0? Taking a look at Wolfram Alpha suggests there is:

    Bessel Function of the Second Kind -- from Wolfram MathWorld

    Another theorem that may be interested in is Fubini's Theorem:

    Fubini's theorem - Wikipedia, the free encyclopedia

    So pretty much if you look at the two separate terms (after you multiply everything out), the first will give you an integral (and I mention Fubini's theorem since if you can use it, then the problem becomes solving a particular integral for that term) and the other term is just a similar kind of integral since you can write the second order Bessels in terms of a first order:

    Bessel Function of the Second Kind -- from Wolfram MathWorld

    I know I haven't really given a full response, but there are lots of different relationships between the Bessel functions, their derivatives, and the different orders. If you can use Fubini's theorem to get a double integral for each term, then you'll have the integrals at least which will give you an expression for the integral of U(r,lambda) which is going to be another integration on top (so it will be two triple integrals maybe more).

    The good thing about having this is that at least you can use a computer to get a good enough estimate if you can't get an analytic solution to evaluate.

    Another thing to use when comparing solutions is to treat U(r,theta) as a derivate and then use a numerical scheme that takes into account the nature of periodic functions (like the Bessel) to get a good approximation of the integral using that technique.

    I emphasize the right scheme because without it, you will get stability and error issues if you don't take into account this periodic nature of the Bessel functions.

    A final point is that if you can relate U(r,lambda) as dU/dr in terms of a Bessel function relationship like the ones in the above Wolfram Alpha pages, then the task becomes solving a particular kind of DE-equation if the relationship is "DE-like" in its form.
    Last edited by chiro; October 9th 2012 at 07:13 PM.
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  3. #3
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    Re: Mode Shape Function Integration

    Thanks for the reply. I found this webpage where it gives the integration of bessel functions. I am not sure how accurate would be the results. but it gives an expression for integrands involving bessel functions.

    Wolfram Mathematica Online Integrator
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