# Finding Moment of Inertia of axle

## Homework Statement

Not really a HW problem (no diagram), I just need to figure out this number for a design.

I have an axle with two crank arms attached to either end, pointing in opposite directions. The entire thing will spin. I need to find the equivalent moment of inertia of the axle with these cranks attached, but I'm confused over what method i should use. Classes never really go deeper with MOI than an I-beam problem or just looking up bh^3/12, etc.

I have all of the specifications of the lengths of axle, cranks, diameters, etc. So should I try:

A) Using the geometry of the part(s) to find I

B) Analyze the part more to find the work and angular velocity so I can find MOI

With method A) i thought I could just use an s-beam geometry to find it:
http://www.efunda.com/DesignStandards/beams/RectangularSBeam.cfm
the numbers seem to make sense when I did it, but for some reason this seems really wrong since the axle is a "horizontal" s-beam so to speak. (The axle is parallel with the ground)

Would the Iyc value in the link above be justified here? Or do I need to start looking at method B?

Any help is greatly appreciated.

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SteamKing
Staff Emeritus
Homework Helper
The MOI used to find angular velocity is NOT the same as the MOI used to find bending stress.

bh^3/12 is the area MOI of a rectangle. I think what you are looking for are mass moments of inertia about a particular spin axis. These are quite different to evaluate.

The MOI used to find angular velocity is NOT the same as the MOI used to find bending stress.
I will be doing fatigue analysis, so I figured that wasn't the same, thanks.

bh^3/12 is the area MOI of a rectangle. I think what you are looking for are mass moments of inertia about a particular spin axis. These are quite different to evaluate.

From what I looked up for mass MOI (3rd Figure):
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html
I have a few questions:

For the horizontal axle and two cranks that basically looks like this (ignore the a's):

|_____________
aaaaaaaaaaaaaa|

Axis:

y-y: perp to axle through center of axle
y'-y': perp to axle through one crank
x-x: parallel to axle through both axle and cranks

Using solid cylinders MOI's from link (since I have the diameters for them):
where M = mass of the axle, L = length of the axle, R = radius of axle, m = mass of one crank, l = length of one crank

Iy-y = (1/4)MR^2 + (1/12)ML^2 + (2m/2)(L/2)^2
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa^Since either crank is only on one side of the x-x axis?

Iy'-y' = (1/4)MR^2 + (1/3)ML^2 + (m/2)(L^2)
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa^Same question?

Ix-x = (1/2)MR^2 + 2m(l^2)

I guess i just don't know for sure how the two cranks only being "one half" length (on one side of the x-x axis) would affect the calculations.

Also, if I use Mass MOI for these calculations, but the mass of the rod/cranks are being assumed negligible for the problem, is it still applicable?

Any help is very appreciated.

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SteamKing
Staff Emeritus
Homework Helper
If you are doing a fatigue analysis on this 'axle', then I'm confused about what the OP was asking. Are loads applied to the crank arms at the end of the axle, which then introduce shear stresses into the connecting shaft? How is the axle supported? Are the applied loads cyclic in nature?

Sorry, should have specified more:

The axle in question is the spindle of a bicycle, with the two crank arms attached to it. The two crank arms each have bike pedals of negligible mass that receive loads from the rider. So yes on the loads on the crank shafts. There is also a gear on the axle that will work the drive the chain, but that is assumed negligible here as well. The axle is supported by two bearings on either side of the center of mass, each between the center and end crank respectively. Also, the loading is cyclic in nature as each side will be of the axle will be experiencing a torque ranging from 0 to Tmax.

SteamKing
Staff Emeritus
Homework Helper
In this case, since you are analyzing for fatigue, the AREA moment of inertia of the pedal axle is what you are concerned with. The cross section thru the axle at the bearings is going to be circular, since the axle rotates. The formula is I = pi * r^4.

The cyclic stresses which should be central to you analysis are produced by the bending moments arising from the force applied to the pedals. The bending moment magnitude will be equal to the force applied to the pedal times the distance from the point of application to the shaft bearing.

the crank will be under "torsion." since the axle is a cylinder (the crank like a cylinder too right?) i think you should be using:

tmax= Tr/Ip or maximum stress = (torque * radius of bar)/ (polar moment of inertia)

you have to consider both the torque applied by the person and torque in the other direction caused by the bike "not wanting to move forward" and pulling back.

the polar moment of inertia for a circular bar is:

(pi) (diameter^4)/32

then given material properties you should compare the two. i dont know about cyclical stresses/ fatigue and if what i just wrote applies tho.

if the crank isn't a cylindrical shape you'll have to treat it using nonuniform torsion which requires integration over the length of the crank. if this is the case it'll be easier to model it in a program, assign a torque, and have it tell you the max stress.

hope that helps

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