# What does multiplicity of a multiple natural frequency mean?

I'm doing some vibration tests on planetary gear sets and a lot of the information I´ve read mentions a lot about mode shapes and multiplicities of the natural frequencies. I understand how one can derive the natural frequencies from the equations of motion of the gears and even the eigenvalues, but I don't understand what the multiplicity of the natural frequencies mean. Does it refer to the harmonics of each natural frecuency? If for example there's a fault it will probably be a multiple of the 1st harmonic? Also, does the eigenvalues have to do with the modal modes?

## 1 Answer

"Multiplicity" usually refers to several different modes with the same natural frequency. For example, ignoring any fault conditions and tolerances, if there are several identical planet gears, each one will have the same mode shapes and frequencies.

If you are measuring the vibration response, you may not be able to distinguish these modes - you will measure one "mode" which is a combination of all of them.

If there is a fault involving just one of the nominally identical parts, then you may be able to identify different modes for that part.

This situation can occur in any structure which is symmetrical - it's not restricted to planetary gearboxes.

• 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? – spe4ker Mar 21 '17 at 18:38
• If you give a reference to "it", somebody might be able to help. Otherwise, we are just guessing what you are asking about. – alephzero Mar 22 '17 at 4:20