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JBlackhole16-Pearl
16-Pearl
January 23, 2018
Solved

Duhamel integral for a damped SDOF with mathcad

  • January 23, 2018
  • 4 replies
  • 6587 views

to all,

 

I am trying to set up the standard convulsion /Duhamel integral for a damped SDOF subject to an arbitrary input p(t) but can’t figure out the syntax required in MathCAD to make it work. Could someone help me on this one? (see attached mcad file)

 

Thanks

Regards

Best answer by Fred_Kohlhepp

I think part of your problem stems from the definition of your impulse function p.  You impulse frequency is much higher than the natural frequency  of your system, and the pulse is over very fast.  But the top illustration in your image shows an excitation imposed on a system with a higher frequency than the excitation.

 

Attached is an example with the same damped system, with a half-sine pulse with a frequency much lower than the natural frequency of your system.  And the integral works (sort of.)

 

If you had an impulse defined by data points you would have to fit a function to the data before you integrated.

4 replies

F
Fred_Kohlhepp23-Emerald I
23-Emerald I
January 23, 2018

Mathcad REALLY likes units.

 

See if this helps

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    -MFra-21-Topaz II
    21-Topaz II
    January 23, 2018

    CONVOLUTION! please! You have to define x.0 and v.0

    convolution JXB.jpg

    W
    Werner_E25-Diamond I
    25-Diamond I
    January 23, 2018

    While sure Fred is right about units you see in his sheet that you have to strip out the units for the odesolve-block (one of the very few advatages of Prime is that you can use units here.).

     

    But maybe we should explain why your attempt failed. The reason is that in your initial conditions you used x0 and v0, but forgot to define them.

     

    EDIT: Sorry, Francesco already explained that. His reply did not show up until I posted mine.

    J
    JBlackhole16-PearlAuthor
    16-Pearl
    January 24, 2018

    Thanks Fred, Francesco and Werner for the replies

     

    Convolution indeed! Will call it Duhamel from now on Smiley Happy

     

    I didn't use unit because all the inputs are consistent (unit wise) and I did notice that the odesolve didn't like units. I also realized that I used m to define the mass

     

    Thanks for pointing the error on the way I defined the Duhamel integral

     

    Questions:

    1.  Why is the Duhamel integral which should give the response x(t) different to the x(t) obtained with the odesolve?

     

    2. I am testing with a well defined p(t) function. How does one handle a function define with data points? given time step I suspect I need to define t[i

     

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    -MFra-21-Topaz II
    21-Topaz II
    January 24, 2018

    ...Indeed, In all branches of engineering and physics, is used the term - convolution - ......

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    -MFra-21-Topaz II
    21-Topaz II
    January 26, 2018

    Hi JXB, 

    I would say that the convolution was not applied correctly. In fact, by applying the Laplace transform to the equation and then anti-transforming, I obtain a very large solution that can not be displayed, but coincides with that calculated using odesolve. See below:

    SDOF.jpgSDOF0.jpgSDOF1.jpgSDOF2.jpg

     

      J
      JBlackhole16-PearlAuthor
      16-Pearl
      January 26, 2018

      Thanks for that insight. Struggling to follow the math but it is interesting to note that what is a simple academic example is throwing some "issues" once defined/set-up in Mathcad. I was simply doing some checks

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