critical speed/natural frequency confusion

[crowbar14]

Hi,

I'm looking to understand how to analyze a low speed (100s of RPM) rotating shaft system. It has two radial bearings, one thrust bearing, and two rotors. I'm doing background reading to understand what a critical speed is, and I'm getting confused about how to go from analyzing a shaft in the stationary, non-rotating reference frame to the rotating frame. If I analyze the system without any rotation, I can get the torsional and lateral natural frequencies for the shaft just by doing a hand calculation or running a static eigenvalue FEA (SOL103). For example, this formula appears to provide the torsional frequencies:



However, in reading about critical speeds of shaft, the first critical speed for a simple system looks the same as the first lateral natural frequency mode shape. Am I right in thinking that gyroscopic and centrifugal forces seem to influence the lateral natural frequencies of the shaft in the rotating reference, as they essentially provide a "stiffening" effect in the rotating frame, and therefore I can't use a simple eigenvalue FEA to evaluate frequencies of concern? Are these first and second critical speeds the same as the 1st and 2nd natural bending frequencies in the static frame, or are they separate modes that require a mass unbalance to calculate and occur?

Any resources would be greatly appreciated, thank you. I haven't had much exposure to dynamic vibration analysis before. I see a lot of resources presenting the case of an off-center mass on a stiff shaft, but I'm more looking to see whether the modes calculated in the static reference frame have any bearing on the modes of the same shaft system when it is rotating at a given speed, even for a hypothetically perfectly balanced system about the rotational axis.

 

[GregLocock]

Yes the first critical speed of the shaft is the first bending mode of the non rotating shaft.

Am I right in thinking that gyroscopic and centrifugal forces seem to influence the lateral natural frequencies of the shaft in the rotating reference, as they essentially provide a "stiffening" effect in the rotating frame, and therefore I can't use a simple eigenvalue FEA to evaluate frequencies of concern?

No.

Are these first and second critical speeds the same as the 1st and 2nd natural bending frequencies in the static frame, or are they separate modes that require a mass unbalance to calculate and occur?


Yes they are the same and due to the positive feedback effect of the shaft bending, and clearances etc in the bearings, can occur even in balanced shafts. On the other hand if you built and balanced 10 supposedly identical propshafts for a RWD car, one of them will probably be OK to drive through that first critical speed.


As to resources

Rotor Dynamics - J. S. Rao​

is probably good.
https://books.google.com.au/books/about/Rotor_Dynamics.html?id=kX9FiJnqRoQC
https://books.google.com.au/books/about/Rotor_Dynamics.html?id=kX9FiJnqRoQC

 

[dgeesaman]

Hi,

I'm looking to understand how to analyze a low speed (100s of RPM) rotating shaft system. It has two radial bearings, one thrust bearing, and two rotors. I'm doing background reading to understand what a critical speed is, and I'm getting confused about how to go from analyzing a shaft in the stationary, non-rotating reference frame to the rotating frame. If I analyze the system without any rotation, I can get the torsional and lateral natural frequencies for the shaft just by doing a hand calculation or running a static eigenvalue FEA (SOL103). For example, this formula appears to provide the torsional frequencies:

View attachment 7572

However, in reading about critical speeds of shaft, the first critical speed for a simple system looks the same as the first lateral natural frequency mode shape. Am I right in thinking that gyroscopic and centrifugal forces seem to influence the lateral natural frequencies of the shaft in the rotating reference, as they essentially provide a "stiffening" effect in the rotating frame, and therefore I can't use a simple eigenvalue FEA to evaluate frequencies of concern? Are these first and second critical speeds the same as the 1st and 2nd natural bending frequencies in the static frame, or are they separate modes that require a mass unbalance to calculate and occur?

Any resources would be greatly appreciated, thank you. I haven't had much exposure to dynamic vibration analysis before. I see a lot of resources presenting the case of an off-center mass on a stiff shaft, but I'm more looking to see whether the modes calculated in the static reference frame have any bearing on the modes of the same shaft system when it is rotating at a given speed, even for a hypothetically perfectly balanced system about the rotational axis.

When the shaft system critical speeds can be accurately calculated using a static solution it's called a 'rigid' rotor. This doesn't mean there is zero deflection in the system, rather, it means for the purpose of calculating critical speeds the deflections stabilize against centrifugal imbalance forces and don't shift the critical speeds. For shafts under 1000rpm that carry significant load, rigid rotor analysis is generally quite good.

Sometimes the elastic deflection caused by normal imbalance forces, low bearing stiffness, etc change the shape of the rotating shaft enough to affect its critical speeds. Then it's called a 'flexible' rotor and more detailed math is required.

In my segment of industry, the rotors are analytically rigid and occasionally a customer insists on a full flexible rotor analysis and they waste thousands of dollars and several weeks for our rotordynamics subcontractor to perform the analysis. So if you're on the buying side, consider asking the vendor if flexible rotor analysis is relevant.

Another consideration is whether you intend to operate this rotating assembly above first critical. If so, pinning down exact frequencies is more important and flexible rotor analysis is more relevant.

 

[GregLocock]

If you are messing about with running close to or above the natural frequency (here there be dragons, but it is common with big turbines) then the other thing to watch for is half speed whirl.