# Moment of Inertia Formula

### Help Support HomeBuiltAirplanes.com:

#### plncraze

##### Well-Known Member
HBA Supporter
http://zoomaviation.com/design/propeller/reduction_70hp.pdf

I would like to apologize for the broad nature of my question but can't wrap my head around the answer.
On page 22 of this paper they do the moment of inertia calcs for a prop. They use the falling weight method. The author's give the formula but when I try to follow along the answers never come close. There are tutorials for this on the web but those formulas don't get these answers either. Any of you guys have their formula? Thnks

#### DeepStall

##### Well-Known Member
The authors left out a few common equations:
1. The angle the system rotates thru is the length of the string divided by the radius it's wrapped around. This is from s=r*theta so theta=s/r where s is the arc length and r the radius
2. The torque produced by the falling weight is constant, mass*g*radius. g=9.81 m/s/s
3. Assuming constant angular acceleration, the total angle of travel is given by theta=0.5*alpha*t^2, where theta is the total angle, alpha the angular acceleration, t is time. Rearranging, 2*theta/t^2 = alpha. Theta is computed by my item 1) above and t is measured. ~
4. Rearranging T=I*alpha, I=T/alpha, where T is computed in 2) above and alpha in 3).
Using this, I can reproduce their numbers for MoI.

#### plncraze

##### Well-Known Member
HBA Supporter
Thank You! boy this thing was making me crazy. I have theta and can plug and chug for I but how do you write g for Excel? Is it 9.81 times the distance I am using or am I completely missing something?

#### DeepStall

##### Well-Known Member
Yes, if you are using metric, then the torque is mass in [kg] * 9.81 [m/s/s] * radius in [meters]. The first bit is just F=ma and gives you Newtons. In US units, we'd use T=weight in lbf * radius in inches or feet. The lbf is a force and has the gravity term baked in already. Careful with US units for dynamics...!

#### wsimpso1

##### Super Moderator
Staff member
Log Member
Doing constant torque accels is one way of doing this, but it tends to underestimate MMOI because there can be significant windage losses. The other common method is the torsional pendulum, where you hang the prop from a single wire, hang it from the ceiling, and time it through a few oscillations while knowing the wire diameter and length. Torsional pendulum can reduce windage losses (run at modest deflections) and was still in use at major automakers when I retired in 2015.

The authors completely missed out on an opportunity to check resonant frequencies of each system, and importantly, to design their now stiff system to carry the dynamic loads from firing that the whole system will now experience...

Billski

#### plncraze

##### Well-Known Member
HBA Supporter
DeepStall thank you. I now have a spreadsheet for this. Billski, how does the torsional pendulum compare to the bifilar (spelling) pendulum used in the Rotax service Info letter? While hunting info down it seemed all the teachers who put their lesson plans for MMOI online had the two wire pendulum.

#### rv6ejguy

##### Well-Known Member
DeepStall thank you. I now have a spreadsheet for this. Billski, how does the torsional pendulum compare to the bifilar (spelling) pendulum used in the Rotax service Info letter? While hunting info down it seemed all the teachers who put their lesson plans for MMOI online had the two wire pendulum.
I believe they're one in the same. I used the bifilar method to measure the MMOI on all my engine, flywheel and redrive parts when I did the math model on my system. Very repeatable and accurate

I had a video here: http://s1105.photobucket.com/user/rv6ejguy/media/propvid0002_zpsb22bb67b.mp4.html?sort=3&o=1 if you can open Photobucket.

Last edited:

#### wsimpso1

##### Super Moderator
Staff member
Log Member
DeepStall thank you. I now have a spreadsheet for this. Billski, how does the torsional pendulum compare to the bifilar (spelling) pendulum used in the Rotax service Info letter? While hunting info down it seemed all the teachers who put their lesson plans for MMOI online had the two wire pendulum.
Forum etiquette lesson - Since the folks you chat with invariably have different life histories than you, they also are unlikely to know about all the things you know. If you want them to comment on a specific document, you increase the likelihood of getting answers about it by including a link to it.

Single wire method was used for things like projectiles for rifle through artillery sizes with appropriate sized wires. Oh, and you need to know the Inertia of your fixture too. The fixture should have small Inertia, say 1% or less than the item measured, have feature for centering and for preventing torsional slip at tested torques.

We used a single steel wire on the order of 0.080" hung from the ceiling for flywheel/ring gears, manual clutch assemblies, and torque converter assemblies. Ford and Chrysler/FIAT both did it this way. Very low friction in the mechanism, you can run with very low surface speeds for very low air drag, and if you want more significant digits in your estimate, increase the number of cycles you run each time by an order of magnitude.

The constant torque method has bearing drag, aero drag increases rapidly with increasing run time, timing of movement accuracy is limited. Gotta make a lot more runs and find a away to correct for bearings and aero drag. Running a known reference part of known aero drag, weight, and Inertia is helpful, but still makes this method fussy.

As to two and three wire schemes, well, let's see the document.

Billski

Last edited:

#### plncraze

##### Well-Known Member
HBA Supporter
Dan is that a Metro crank? We have never met but you have helped me in more ways than I can count. There is a Chevy Metro engine partially assembled in my garage with a couple of Lovejoy couplings just waiting for some money and time. I especially enjoyed your descriptions of failing reduction drives and discussions with Ross and others on the VAF site. I appreciate your warning of torsionals in these little engines. Welcome to the forum! If all you do is repeat what you have written on the VAF site it will still be helpful.

Billski, thank you for joining in this discussion and your advice on supporting documentation. I am embarrassed to admit I did not read the Rotax letter itself but instead had watched this video: https://video.eaa.org/detail/video/2010764613001/measuring-propeller-mass-momemt-of-inertia

#### rv6ejguy

##### Well-Known Member
Hey Dan, nice to see you here on HBA. I'm sure others here will appreciate your knowledge in many fields.

BTW, HBA members, Dan taught me all I know on this subject and helped me complete a mathematical TV analysis on my Subaru redrive system which led to a huge reduction in the lower rpm ranges.

Last edited:

#### DanH

##### Member
Dan is that a Metro crank?
Sprint, previous generation, but I'm pretty sure it's the same crank.

Hey Dan, nice to see you here on HBA. I'm sure others here will appreciate your knowledge in many fields.
Is the Son of RAH? Good place way back when.

Can't hang here too much. Let me know if something interesting comes up.

Shout out: Billski is Bill Peyton? You've been writing some good stuff. Nice work. Wish I had that professional experience.

BTW, HBA members, Dan taught me all I know on this subject and helped me complete a mathematical TV analysis on my Subaru redrive system which led to a huge reduction in the lower rpm ranges.
Now if I could just convince you to ditch that bushing-and-pin coupler <g>

#### wsimpso1

##### Super Moderator
Staff member
Log Member
Dan is that a Metro crank? We have never met but you have helped me in more ways than I can count. There is a Chevy Metro engine partially assembled in my garage with a couple of Lovejoy couplings just waiting for some money and time. I especially enjoyed your descriptions of failing reduction drives and discussions with Ross and others on the VAF site. I appreciate your warning of torsionals in these little engines. Welcome to the forum! If all you do is repeat what you have written on the VAF site it will still be helpful.

Billski, thank you for joining in this discussion and your advice on supporting documentation. I am embarrassed to admit I did not read the Rotax letter itself but instead had watched this video: https://video.eaa.org/detail/video/2010764613001/measuring-propeller-mass-momemt-of-inertia
The two wire pendulum shown in the EAA video works too. Process is similar to a single wire, probably less critical to set up. Understand that you are measuring the inertia of the prop in the video as well as the fixture, but given as the fixture can be some standard shape, you can probably calculate its inertia pretty closely or rationalize that it is "small enough" to neglect. Rotax appears to have done the math for you and put up a nomograph to finish the process. In all a WAY better process than the constant torque process the students used in the above cited paper.

Billski

#### wsimpso1

##### Super Moderator
Staff member
Log Member
Sprint, previous generation, but I'm pretty sure it's the same crank.

Is the Son of RAH? Good place way back when.

Can't hang here too much. Let me know if something interesting comes up.

Shout out: Billski is Bill Peyton? You've been writing some good stuff. Nice work. Wish I had that professional experience.

Now if I could just convince you to ditch that bushing-and-pin coupler <g>
Not Bill Peyton, sorry. Just the roll of the dice that I got the experience I did. Who is Bill Peyton?

Billski

#### DanH

##### Member
Not Bill Peyton, sorry. Just the roll of the dice that I got the experience I did. Who is Bill Peyton?

Billski
Brain fade, sorry, Mr Peyton is an RV-10 builder. I'm told you're Bill Simpson. Keep swinging the bat!

#### plncraze

##### Well-Known Member
HBA Supporter
https://www.messiah.edu/departments/engineering/projects/ipc/pdf/L/Light Sport Aircraft Belt Reduction Drive (2009 FR).pdf

I am still playing around with making a spreadsheet that will match their answers which are in the paper in the link above. Now I am trying to figure out what they call "processional speed" on page 23 of the document linked above. They mean precessional I believe and the formula for this is (I think) 2 times gravity over radius times rotational speed in rad per sec. I am having a hard time duplicating their answer. They left out the propeller diameter in their paper. Is this part of my problem? Thanks in advance!!