How Do I Solve for Planetary Differentials?

[one_raven]

OK.
Here's the situation:

I am trying to build a spreadsheet to solve for planetary gear configurations.

I have done most of the work already, but I am getting stuck when solving for planetary differentials.

If you don't know, a planetary gear differential is basically a gear set-up that has three shafts that can be used as inputs or outputs.

Let:
S = Speed (in RPM)
N = Number of teeth on the gear
r = ring gear
c = planet carrier
s = sun gear

The relationship can be described by the following equation:

(Sr-Sc)/(Ss-Sc)=Ns/Nr

There are six different configurations I am trying to solve for.

Unfortunately, Algebra and I have been apart for far too long.
Think maybe you can help?

Configuration 1
Ns: known
Nr: known
Sr: known
Sc: known
Ss: not known

Configuration 2
Ns: known
Nr: known
Sr: known
Sc: not known
Ss: not known

Configuration 3
Ns: known
Nr: known
Sr: known
Sc: not known
Ss: known

Configuration 4
Ns: known
Nr: known
Sr: not known
Sc: known
Ss: not known

Configuration 5
Ns: known
Nr: known
Sr: not known
Sc: known
Ss: known

Configuration 6
Ns: known
Nr: known
Sr: not known
Sc: not known
Ss: known

Any help would be greatly appreciated.
Especially if you show me how you did it.

 

[NateTG]

Usually, you can only solve for as many unknown quantities as you have equations. Since all you've got is
(Sr-Sc)/(Ss-Sc)=Ns/Nr

That means that you'll need to bring in some additional information for the examples that you give where there are multiple unknowns (2,4, and 6). Perhaps there is some additional relationship that you missed?

In general, you'll probably want to assume that (Ss-Sc) and Nr are both not zero.

Then you can easily convert the equation above into:

Ns=Nr(Sr-Sc)/(Ss-Sc) (Multiply both sides by Nr)
Nr=Ns(Ss-Sc)/(Sr-Sc) (Take reciprocal of both sides, and multiply by Ns)

And these are fairily straightforward after taking a cross product:

(Sr-Sc)/(Ss-Sc)=Ns/Nr
=>
Ns(Ss-Sc)=Nr(Sr-Sc)

Sr=Ns(Ss-Sc)/Nr +Sc
Ss=Nr(Sr-Sc)/Nr +Sc
Sc=(SrNr-SsNs)/(Nr-Ns)

Hope this helps

 

[Kenneth Mann]

Equation check (Sorry, should have attached to this one)

I compared your equation to a generic equation for planetaries, which is:

(P-1)(Sr/Ss) = P(Sc/Ss) - 1

and which, for a ring-gear planetary:

P = (Nr = Ns)/Ns

If the author of these equations was correct, then the following was derived:

(Sr - Sc)/(Sc - Ss) = Ns/Nr

If so, then your directions will be reversed. I'd like to know which is correct.
Kenneth Mann is online now Reply With Quote