# How to model a planetary gear system with transformation matrices?

I need to model the planetary gear-train with transformation matrices of 1 sun and 2 planets system, with no ring and an "arm" or carrier holding the gears ,like in the second picture. I already have a dynamical model of the simple system derived analytically, but I need to model it dimensionally to be able to specify the spatial relationships—the relative positions and orientations of the epicyclic geartrain to simulate it.

I only have experience deriving homogeneous transformation matrices of robot joints but I don't know where to star for a gear train system or what method to use.

I'm not sure about which frames of reference to use and how to set up the coordinate system between the gears correctly, for example; what would be the difference between considering a fixed frame of reference for the sun vs a rotating frame of reference? Also, I don't know which and how to include variables like mass, stiffness and damping, like shown in the first picture (for example).

• Not sure that I understand how you can have a planetary gear system with no ring gear. Which is your input and which is your output? Are any of the gears fixed? A typical combination would be for the sun to be input and the arm to be output. But without a ring, then the planets would just spin freely and the output shaft won't rotate. What am I missing? May 9, 2017 at 1:50
• @DanielKiracofe basically I am interested in studying the vibrations caused by the meshing of the gears, I have a physical set up of a gear train with only one sun and two planets, where the input is either the sun or the carrier. I plan to add a ring later but now the one described is what I have. So technically you are correct, I can either spin the carrier with the planets around the sun, or spin the sun and let the planets spin freely. May 9, 2017 at 2:19
• There is something weird about this question. Why would you use transformation matrices for anything else than visualisation? I mean thats not actually modeling anything else than the position of the gears, which is hardly relevant since the underlying calculations already account for that. Anyway you dont need to have a frame of reference as such. May 9, 2017 at 6:12
• @joojaa I need to follow this approach because this representation will be the input to my software model for which I'll be able to generate the equations of motion automatically. May 9, 2017 at 13:50
• Are you sure, i mean the matrix has no dynamic information in it. May 9, 2017 at 14:00

in studying the vibrations caused by the meshing of the gears

So I take that to mean you are interested in a forced response vibration problem. Well, actually because it's gear system, the real driver of the response is the time varying mesh stiffness which leads to "parametric excitation", and you will also have to contend with the non-linear behavior of the gear tooth contact.

Start by building a linear model for free vibration, i.e. natural frequencies. This paper Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration by Jian Lin and R. G. Parker is one of the classic references on planetary gear free vibration. If you follow Parker's modeling approach, the answer won't necessary be physically meaningful, since you have no ring gear, but it will get you in the right direction.

Once you have that basic free vibration model, then you will want to move on to the forced response / parametric instability modeling. Here are a few references

There are plenty of other references out there. These are just some of the ones I'm most familiar with. Google scholar and the bibliography of these papers should help.

If you are not familiar with parametric excitation and non-linear dynamics, you'll probably want pick up a basic non-linear dynamics textbook. I recommend the one by Jordan and Smith.

• Thank you, that helps a lot. Do you know of any software that might aid in the automatic generation of the equations of motion? May 10, 2017 at 17:14
• Well, there is this software ansol.us/Products/Planetary2D That's akin to a finite element program (although strictly speaking, it's more than just FEM). But in terms of automatically generating the equations of motion, no I do not know of any. But getting the EOM written down is probably only 5% of the job. Properly done, simulating non-linear gear dynamics is a hard problem. So even if you could automate generating the EOM, you still have lot of work to do. May 11, 2017 at 0:28