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phazelwood1-Visitor
1-Visitor
October 9, 2014
Question

F(z) with dimensionless (z) : causing issues on integration

  • October 9, 2014
  • 2 replies
  • 1365 views

Hi,

Pretty new user to mathcad, was introduced to it as a way of formatting equations for validation and checking purposes.

Please see the attached mathcad page for a page of calcs that I'm struggling with.

Its calculations for a variable section beam, done as part of an ANSYS validation exercise

Calculations are on deflection of a beam per unit of force applied.

Hopefully my equations are understandable, I've included references for relevant dimensions and theory.

My question:

When writing functions f(z), how do you define the units of the z variable?

Is there a simple way of doing this that I have missed?

I've set up calculations for Moment of inertia: I(z), using the height of the section: (z) as a variable.

The issue that I seem to be having:

z is dimensionless, but I want units of this variable to be mm

Not having a unit assigned to the z variable causes several issues:

  • I later use the unit load method to calculate deflection (per unit force: mm/N)
    • This involves a definite integral in terms of z, which would introduce the extra mm unit from integration of f(z) dz
    • As you can see on the example, results of this integration come out with a 1/N unit, where the units SHOULD be mm/N
  • Plotting MOI as a function of z:
    • z is dimensionless
    • Plot z axis cannot have units of mm assigned without screwing up display

2 replies

W
Werner_E25-Diamond I
25-Diamond I
October 10, 2014

My question:

When writing functions f(z), how do you define the units of the z variable?

You don't.

If you define M(z):=(h+z) mm, then you will get an error when z does not have the same dimension as h.

If you define h:=7 mm and M(z):=z+h, then you get an error when z does not have dimension length.

Otherwise there is no need to set the dimension of a function argument. You decide which unit when you call the function.

F
Fred_Kohlhepp23-Emerald I
23-Emerald I
October 10, 2014

First of all, the variables M and m are moments (Newton-meters) not lengths.

The equation you boxed in yellow is derived from the basic beam equation:

E I d^2y/dx^2 = M

(the change in slope of the beam is the moment divided by the stiffness.)

It's far easier to use Mathcad to solve this differential equation directly, Define the applied moment due to the load as a function of position along the beam. You may also vary I as a function along the beam. For a second order DE you need two conditions; for your cantilever deflection and slope at the fixed end are zero. Research "Odesolve."

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