# 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?

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.