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KenFarley21-Topaz II
21-Topaz II
March 28, 2014
Solved

Model with simple frustrum - volume calculation wrong

  • March 28, 2014
  • 1 reply
  • 7746 views

I make a lot of models that involve pyramidal frustrums (transition of a rectangular section to another rectangular section, linearly). Rather than use the measured value calculated by Creo, I thought I'd have a go at calculating the volume directly from the dimensions.

When I check my calculations, the volume calculated directly from the dimensions differs from the Creo analysis result.

I minimized the accuracy value, but still get the exact same result.

Here are my simple values

Base rectangle: 5 X 3 -> Base area: 15

Top rectangle: 4 X 2 -> Top area: 8

Height: 1

Frustrum of a pyramid volume is

Volume = ( Area Base + Area Top + SQRT ( Area Base * Area Top ) ) * Height / 3 = 11.3182

Creo, on the other hand, gives a volume of 11.3333.

Anyone have any insights into this?

This is simple shape, and kind of scares me about what might be happening with more complicated geometry.

    Best answer by dschenken

    I think Creo is right.

    I'm not sure where the formula came from but ...

    L = l_base - ((l_base-l_top)/delta_z) * Z

    W = w_base - ((w_base-w_top)/delta_z) * Z

    A = L * W =

    l_base*w_base

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z

    Integrating:

    V =

    l_base*w_base*Z

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z*Z/2

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z*Z/3

    Plugging it all in I get 11.33...

    1 reply

    D
    dschenken1-VisitorAnswer
    1-Visitor
    March 28, 2014

    I think Creo is right.

    I'm not sure where the formula came from but ...

    L = l_base - ((l_base-l_top)/delta_z) * Z

    W = w_base - ((w_base-w_top)/delta_z) * Z

    A = L * W =

    l_base*w_base

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z

    Integrating:

    V =

    l_base*w_base*Z

    - (l_base*((w_base-w_top)/delta_z) + w_base*((l_base-l_top)/delta_z)) * Z*Z/2

    + ((l_base-l_top)/delta_z)*((w_base-w_top)/delta_z)*Z*Z*Z/3

    Plugging it all in I get 11.33...

    D
    dschenken1-Visitor
    1-Visitor
    March 29, 2014

    I should be able to simplify the integral to be the same as the original formula, but so far no luck. I looked at the Wikipedia page that showed the derivation of the original formula, but it doesn't yield terms I can yet convert.

    I get 11.3182 from the original formula, but don't see the reason for the discrepancy.

    K
    KenFarley21-Topaz IIAuthor
    21-Topaz II
    March 29, 2014

    I think I figured it out. The frustrum equation, though not described as such, must only apply to a square pyramid, i.e. where L and W are the same for each height, z. When that assumption is applied to the integrated solution, the resultant equation collapses to the one from the CRC tables.

    I wish I'd noticed that earlier.

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