Diametral Area Moment of Inertia from rotor spider spokes 1

[electricpete]

This is the rotor for a 2500 hp 1800 rpm horizontal sleeve bearing motor.

In the core area of the rotor, the 7.25" OD shaft has 6 spider spokes equally spaced around the periphery. The spider spokes are 2.5" wide x 3.625" in the radial direction (figure attached).

I would like to calculate the diametrical area moment of inertia of the combination shaft + spider spokes as an input to a critical speed calculation.

I'm pretty sure the sum of the area moment of inertia contributions from the spokes doesn't change as the spokes rotate (that's why the number 3 or 6 is selected). So I can declare that one of the spokes is at the 12:00 position and find it's contribution about the 3:00-9:00 diameter using the parallel axis theorem... but then the spokes at 10:00 and 2:00 are more problematic since they are at an angle to the 3:00<->9:00 diameter.

Any suggestions?


=====================================
Eng-tips forums: The best place on the web for engineering discussions.

 

[zekeman]

".........I would like to calculate the diametrical area moment of inertia of the combination shaft + spider spokes as an input to a critical speed calculation.........."

Each spoke contributes the same to the total diametric momment of inertia, i.e. the numerical n*Js
n=number of spokes
Js inertia of one spoke.

 

[electricpete]

Thanks.

If I were calculating polar area moment of inertia (such as for calculationg polar mass moment of inertia for acceleration calc), then I would allocate each spoke the same amount.

However I am calculating diametral area moment of inertia (as a measure of stiffness). That requires that we freeze the rotor at a moment in time and select a diameter line about which to calculate Area MOI ... for example 3:00-9:00 diameter as discussed above. Each spoke does not look the same with respect to that diameter.

=====================================
Eng-tips forums: The best place on the web for engineering discussions.

 

[IDS]

Pete - I'd calculate the coordinates of the corners, with the origin at the axis of the rotor, then use a section properties Excel function from here:

http://newtonexcelbach.wordpress.com/2008/03/14/section-properties-udf-and-in-cell-charting/

or here:

http://newtonexcelbach.wordpress.com/2008/03/25/section-properties-of-defined-shapes-spreadsheet/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[electricpete]

Thanks Doug. That is exactly what I was looking for and it worked like a charm.

Attached is the worksheet which computes Id = 790 inch^4

Also I did a ballpark calc using parallel axis theorem (used an approximation regarding centroid of the 2:00/4:00/8:00/10:00 spokes) in the "work" tab and it gives the same neighborhood (826 inch^4), so it passes the sanity check.


=====================================
Eng-tips forums: The best place on the web for engineering discussions.

 

[electricpete]

Whoops, last spreadsheet used 2" instead of the actual 2.5" for the width of the spokes. Attached is corrected version. Coordinate spreadsheet now calculates 953 and approximate check now calculates 999. (attached).

=====================================
Eng-tips forums: The best place on the web for engineering discussions.

 

[electricpete]

In the unlikely event that anyone follows behind trying to re-create what I did (or more me... two months from now), I have posted a revised version which corrects the sketch to make it easier to follow what I did when I created the coordinates.

Also poking around Doug's spreadsheet, I see there is a tool to "rotate" shapes and see effect upon moment of inertia. I used that tool to try to improve my "manual" calculation based on parallel axis theorem, adding up the shaft and spokes. Results of that calc are now 983 against program output of 953, That is as good an agreement as can be expected considering I used some approximations in defining the coordinates (I assumed the outer corners of the spokes were located at an angle alpha/4 from the inside corners of the spokes which is not exact but pretty darned close). Good enough for my purposes.

The result is negative because I chose to define my coordinates going CCW, would be positive if I chose CW.

Another comment about the result: Ix = Iy, even though visually the shape looks different if you flip it 90 degrees. That is again the property of rotors with 3*integer number of spokes... diametral moment of rotation stays constant no matter what angle you rotate it.

Seems like a great tool for general shapes. The process that Doug's program used was a little mysterious until I found the page on his site that talks about summing contributions from trapezoidal shapes defined by each pair points. Makes a little more sense now.

Thanks again Doug!


=====================================
Eng-tips forums: The best place on the web for engineering discussions.

 

[electricpete]

Correction in bold:
That is again the property of rotors with 3*integer number of spokes... diametral moment of rotation stays constant no matter what angle you rotate it.
Should have been:
That is again the property of rotors with 3*integer number of spokes... diametral moment of inertia stays constant no matter what angle you rotate it.

I'm done now...


=====================================
Eng-tips forums: The best place on the web for engineering discussions.