# What does natural frequencies with multiplicities m=1,2 or n mean?

If I have a planetary gear set consisting of one sun and N planets, what does it mean if it says that typical vibration modes of 6 natural frequencies always have multiplicity m = 1 for different N, except for the zero natural frequency? How would this multiplicities look in a frequency domain plot?

## 1 Answer

I cannot find a freely available PDF of it, but if you have access to ASME journals, then this paper lays out the basics of planetary gear natural frequencies (at least in the most basic case, there are many later papers with different variations in assumptions).

Lin J, Parker RG. Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration. ASME. J. Vib. Acoust. 1999;121(3):316-321. doi:10.1115/1.2893982.

As outlined in this paper, planetary gears have three types of modes: rotational modes, translational modes, and planet modes. Rotational modes always have multiplicity 1, translational modes always have multiplicity 2, and planet modes have multiplicity N-3. What the multiplicity means in that there is more than one vibration mode (eigenvector) with the same natural frequency (eigenvalues).

For translational modes, here is how to think about the multiplicity: there could be one mode where everything is translating vertical (e.g. just up down... in reality it might not be exactly up/down, but let's just assume it is for now). There is a completely separate mode where everything is translating laterally (just left right). Now, let's call the stiffness in the vertical direction $k_v$ and the stiffness in the horizontal direction $k_h$. The mass is the same for both cases, let's call it $m$. So the vertical natural frequency is $\sqrt{k_v/m}$ and the horizontal natural frequency is $\sqrt{k_h/m}$. But because planetary gears are axisymmetric, $k_v = k_h$. So the two natural frequencies are identical. They are two completely separate modes, that just happen to have the same frequency. That is what is meant by multiplicity 2. If you were to make the bearings or some other part of the structure slightly stiffer in one direction, such that $k_v>k_h$, then it's no longer symmetric and you would have two different frequencies each of multiplicity 1.

• This is exactly what I was looking for, thank you. What does the zero natural frequency refer to? – spe4ker Mar 23 '17 at 0:07
• Zero natural frequency is what is called the "rigid body mode". It's not actual vibration at all. It's just the gears turning. I.e. the sun gear turns, which causes the planets to turn, which causes the carrier to turn, which causes the output shaft to turn. In general, a "rigid body mode" means a way in which the system can move without introducing any strain energy into the components. if you want to eliminate the zero frequency mode mathematically, just introduce a small torsional stiffness to ground at your sun or carrier. – Daniel K Mar 23 '17 at 0:34