Area moment of inertia

[PhilipFry]

Hey,

I'm trying to calculate the stiffness of a spur gear tooth, but am having a little trouble finding the second moment of inertia (I). Do I need to pull out my calculus book or is there already an equation for it laying around? I've looked in the machinery's handbook without much success.

 

[electricpete]

One formula I have used for calculating critical speed of shafts:

I=(pi*d^4)/64

This represents a moment of inertia about a line passing through the center (diameter). i.e. the moment of inertia that would be important if you were to grab shaft at both bearings and try to bend it. A picture would be easier.

Not sure if this is what you're after.



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[PhilipFry]

Well, let me try to pin down what I'm looking for...

Looks like there is some confusion around the terminology I used. Treating the gear tooth as a cantilevered beam, you would use the equation: sigma=Mc/I, or something like that to find the bending stress. The stiffness of the tooth would be: k=3*E*I/L^3. This moment of inertia deals (I think) entirely with the cross-sectional area of the gear tooth profile and not where the center of mass is. I don't know if we are talking about the moment of inertia or not.

Is it sufficient to treat the gear tooth as a trapezoidal beam?

 

[desertfox]

Hi mae1778

If your considering bending on the profile of the tooth then your formula are correct as the teeth are treated as a cantilevered beam however you need to look for the Lewis Equation which introduces a form factor for the tooth most
design books will have something on this.
Just to clear up some terminology "moment of inertia" is resistance of a body to move against a force so for instance
take a circular disc mounted on a central shaft one would after overcome its "moment of inertia" to get it to rotate about its central shaft. Very often "moment of inertia" is used incorrectly even in text books to describe the resistance of beams to bending loads. The beams resistance to bending loads should correctly be called "second moment of area".


hope this helps

regards

desetfox

 

[PhilipFry]

Yes, I understood that to be some area of confusion in that matter. From what I've seen, many textbooks actually print both versions, which leads to further confusion as to which one is correct.

Regarding the Lewis Equation...

I was under the impression that the form factor dealt with correcting the bending stress value on the gear tooth, and was separate from the moment of area.

I'm just trying to get the moment of area so I can calculate the stiffness of a pair of gear teeth. I found this equation one online that removes the moment of area completely:

keq=gear_thickness/9*(E1*E2/(E1+E2)) which is the equivalent stiffness between to gear teeth, but I haven't been able to verify that from any reputable source.

Anyone ever seen that formula before? If so, where did it come from?

 

[desertfox]

Hi mae1778

According to Schaum's Outline series "Machine Design" the second moment of area =


I= b*t^3/(12)


where b = face width

and t is equal to the thickness at a section where a parabola drawn inside the tooth outline is at a tangent with the tooth outline, if you look in the book you will see
what it means.

best I could do in the time I have to spare.

regards desertfox

 

[PhilipFry]

Thanks,

I'll try to get my hands on that book.

 

[desertfox]

HI mae1778

Have a look at this website it may help you.

http://www.mech.uwa.edu.au/DANotes/gears/bending/bending.html

regards desertfox

 

[PhilipFry]

That's what I'm talkin about! I'd actually visited that site some time ago, but my mind wasn't on the moment of area at the time. I will certainly use this in the future.

I'm desperate to keep my model as simple as possible without compromising too much on accuracy. I'd originally used the parallel axis theorem to combine a trapezoid and a rectangular cross-section, which seemed to approximate the tooth decently well. It seems to me (unless I'm reading this wrong) that the geometry factor is actually independent of the geometry of the gear tooth? Or is it valid for one diametral pitch and pressure angle only?

Thanks for the help!