How can I find dimensions that would give me the same geometrical properties?

[yiviw97057]

To the left I have two similar T-profiles that have given geometrical properties (moment of inertia, radii of gyration, centroid etc). I need to combine the two into one bigger profile that will have the same geometrical properties. How could I go about finding dimensions of the new bigger profile?

I have been told I can use AutoCad, but im not sure how to go about doing it; I know I can use the command MASSPROP to find geometrical properties of any given geometry. Should I perhaps just try changing the dimensions of the new profile until the geometrical properties are about the same? - because thats the only way I can think of.

Left: Two original profiles. Right: New profile with the same geometrical properties.

[fatal.error]

Please follow these steps:

1. Type PLINE (polyline command) to draw each individual T-Profile.

2. Type REG (region command), select the 3 previous polys and hit Enter to convert them into regions. You should read "3 loops extracted. 3 Regions created."

3. Type UNI (union command), select the first 2 individual profiles and hit Enter, to merge them.

4. Type MASSPROP (massproperties command), select any of the 2 regions and hit Enter. A full report will be shown.

Please see the attached step-by-step file.

Furthermore, you can automate the process with some custom-made LISP routine.

Please also check the following link: www.cadforum.cz/en/qaID.asp?tip=4861

[leeminardi]

There are two tools I would consider using to solve the problem you stated.  One is the parametric features of AutoCAD and the other Excel and its numerical solvers.

The image below shows a version of the combined shapes with  dimensional and geometric constraints.   Note that the massprop command gives the centroid location and other properties relative to the active UCS. I have assumed that the UCS is at the minimum x,y location of the shape (lower left).

I created an Excel spreadsheet with the driving dimensions and calculations for the area and the x coordinate of the centroid (assuming the lower left corner).

The area is simply the base x height for the 3 rectangles or:

=dim1_*dim7_+2*((dim6_-dim7_)*dim3_  )

The centroid formula for a composite shape relative to the lower left corner is the area of each shape times the distance to its centroid from the left edge or for the x location:

=(dim1_*dim7_*(dim1_/2)+dim3_*(dim6_-dim7_)*(dim2_+dim3_/2)+dim3_*(dim6_-dim7_)*(dim1_-dim2_-(dim3_/2)  ))/( dim1_*dim7_+2*(dim3_*(dim6_-dim7_)   )     )

Let's say that the centroid location is correct for x but the area is wrong and should be 157.5 and not 152.5. In such a case we would want to modify dim3.  This would change the area but not the centroid's x location.

Using Excel's What-If Analysis we can ask to modify dim3 to yield an area of 157.5.

The result for dim3 is 2.7 as seen below.

You can now use this value to update the geometry in AutoCAD.

You could also add an expression for area (and the other properties) in AutoCAD's Parameter Manager and experiment with the dimensions.  For example,

In the case where your goal is to manipulate serval dimensions to get desired results for several parameters (e.g., centroid, radii of gyration, etc.) I would use Excel's Solver Add-In.

[yiviw97057]

Thank you @leeminardi , would it be possible to get a copy of the excel and Autocad file you made for this question? I tried to do it myself but struggled setting up the Autocad model, I think it would be nice to have your file and then work from there with other profiles. 

Edit: To use command "massprop" I need to make the geometry into a region by using the command "region", but doing to seems to remove all of my parametric dimensions?

[leeminardi · Accepted solution]

@yiviw97057 parametric constraints cannot be assigned to regions. To counter this I have a polyline outline of the shape on layer 0 and a region version of the shape on layer "regions" (red).  After changing a parametric dimension make a region of the shape so that you can use massprop to check the geometric properties. 

In the attached file I added user parameters in the Parametric Manager to calculate the area and the location of the centroid relative to the lower left corner. This enables you to change a dimensional constraint and immediately see the new location of the centroid.  Value for moments and radii of gyration could be included in a similar manner. 

Note that the values for the area and centroid agree for the three methods, massprop, parametric manager, and Excel. Also note that I used named cells in Excel to make interpreting the formulas for area and centroid more readable.  E.g., "dim1_" refers to cell B3.

To determine appropriate dimensions for the merged shape make an initial educated guess for the various dimensions.   Then either tweak a parametric dimension or use Goal Seek in Excel. 

Let me know if you have any questions.

Lee