#### Introduction to Performing Lateral Rotor Dynamics Analysis

Hello and welcome to the Lateral Analysis section of our Rotor Dynamics Blog Series! If you haven’t had a look at the other entries in this series, you can find them here: Series Preface

We’ve finally made it to the analysis part of the rotor dynamics and bearing analysis intro series! Let’s get into it, this blog will have a lot to cover!

For starters, it’s important to note that all rotating machinery is subjected to a variety of lateral excitations and forces. These can include:

• Residual unbalances
• Gearing
• Rotor Rubs (rubbing against blades, seals, etc.)
• Electromagnetic or fluid-flow thrust There can also be other sources of vibration and self-excitation, such as:
• Sleeve bearings
• Labyrinth seals
• Rotor rubs

So, having listed some of the different kinds of excitations and vibrations that a rotor can be subject to, a question comes up. Just what goes into a lateral analysis? Per the API requirements that are found in API 684 as well as in the respective sections that cover different kinds of rotating machines, the standard lateral analysis should consist of 3 analyses:

1. Undamped critical speed analysis
2. Damped undamped response analysis
3. Stability (damped eigenvalue) analysis (depending on the type of rotor)

The steps for performing these analyses can be found in this helpful list below:

Preparation Steps:

Step-by-step of the lateral analysis:

• Calculate critical speed map to analyze possibility of resonances
• Calculate undamped natural frequencies to identify mode shapes (select unbalances to be applied based on mode shape at each critical speed of interest)
• Run damped unbalance response analysis for each mode shapes identified above
• Determine if a critical speed may be an issue based on amplification factor and if relevant that it features a sufficient separation margin
• Obtain amplitude limits at probe locations
• Calculate correction factor
• Calculate predicted vibration amplitudes during start–up / shut-down and compare against clearances

• Calculate destabilization factor (cross-coupled coefficient)
• Obtain damped modes as function of rotor speed and its logarithmic decrement
• Compare against the specified minimum

The first step once we have performed all the preparation steps, will be to generate the critical speed map, and subsequently perform an undamped critical speed analysis.

The undamped critical speed analysis looks at the rotor’s behavior if there were no external forces or damping. It can be thought of as the preliminary estimation of the machine’s critical speeds and mode shape characteristics. This analysis is useful because it’s a quick, rapid analysis of the general dynamic behavior of the machine.  Keep in mind though that since we’re not taking into account the damping forces and other external forces acting on the rotor, that this isn’t the complete picture. Remember, the critical speed of the machine and their associated mode shapes are mostly influenced by the support (bearing and pedestal structure) stiffness magnitudes, the support locations, and the rotor’s mass and stiffness properties.

Undamped mode shapes help by providing an approximate indication of the shaft’s relative displacements at associated critical speeds. This can be helpful since it indicates which unbalance distribution should be applied to excite a given rotor’s critical speed.

And through the undamped critical speed analysis, we are able to create a critical speed map, like the one below where different support stiffnesses are accounted for and the critical speed values are calculated accordingly! As seen on the figure it allows easily seeing what type of system we have and therefore if a change in bearing will influence the critical speed or not, in which case the rotor would have to be modified to shift an unwanted critical speed away from operating conditions

The Unbalance Response Analysis will help us determine whether or not the machine is going to meet the required separation margins and vibration limits while taking into account damping present in the overall system. Per API 684 .

In an unbalance response analysis, the following criteria need to be met:

1. Adequate separation margin between critical speeds and operating speeds if the AF (Amplification Factor) is greater than 2.5
2. The probe vibration limit is not exceeded within the specified operating speed range even with twice the maximum allowable residual unbalance present
3. No rubbing will occur even if the rotor’s balance state degrades to the probe vibration limit

An important factor in performing unbalance response is to ensure that the unbalance itself is in the proper place in the rotortrain. According to API 684 SP6.8.2.7: Unbalance must analytically be placed at the locations that have been determined by the undamped analysis to affect the particular mode most adversely. This is pretty straightforward, and you can see in the diagrams below where the unbalance should be placed for overhung machines and between-bearing machines for different types of modes (mostly rigid bearing vs. mostly rigid rotor) as these determine for instance for the first mode of between bearing machines if the vibrations will show a translatory or bending shape.

The end result of the unbalance response will be a Bode plot, which will show the frequency, phase, and amplitudes generated at the vibration probe locations through the range of each critical speed resulting from the unbalance.

From here, one should determine if the amplification factor is greater than 2.5 If it is below 2.5 , the design is considered critically damped, and can be run at the critical speed. If the amplification factor is 2.5 or more, and depending upon whether the machine is operating above or below the critical speed, the separation margins need to be considered. You can see in the diagram below the correct functions to determining the required separation margins, based on the machine’s operating speed and critical speeds:

Additional results from the unbalance response analysis should include:

• Rotor deflected shapes for each critical speed
• The minimum design diametral running clearance of the seals
• Additional Bode plots that compare absolute shaft motion with shaft motion relative to the bearing housing for machines (support stiffness is less than 3.5 times the oil-film stiffness)

Another important factor that needs to be considered following the unbalance response analysis, is the critical clearance. To do so, according to API 684 SP6.8.2.11, the following Correction Factor (CF) calculation should be used:

• The calculated unbalanced peak to peak amplitudes shall be multiplied using the CF calculation.
• CF shall have a value greater than 0.5.

Al = amplitude limit (μm) peak to peak

A4X = amplitude at the probe location (μm) peak-to-peak

N = maximum continuous speed

According to API 684 SP6.8.2.12, the calculated major-axis, peak-to-peak, corrected rotor response amplitudes shall not exceed 75% of the minimum design diametral running clearances throughout the machine.

One last supplemental analysis that can/should be performed depending on the machine is the Level 1 Stability analysis. For certain machines, a stability analysis isn’t required, but it is always recommended that they be performed. A stability analysis is:

• Not required for API 611/612
• Recommended in API 684
• Requirement for API 617

If you want some clarification on which API code covers which machine, head on over to this blog from a few months ago, and you can see for yourself!

So what are level 1 and 2 stability analyses, and what do they do? A stability analysis is meant to be a screening process in which a quick and simple analysis can be conducted to filter out machines that are well away from the instability threshold. Some factors that can cause destabilization include aerodynamic forces from impellers, force being exerted by seals, and oil whip. This flowchart below is helpful in seeing where a stability analysis fits in, and the proper procedure.

A stability analysis shall be performed on all axial and/or radial flow rotors for:

• Centrifugal compressors
• Steam turbines
• Except those rotors whose maximum

If the results of a level 1 stability analysis are not satisfactory, then a level 2 stability analysis should be performed, or the rotor-bearing system should be redesigned. Although it is more in-depth, it will reflect the true behavior of the rotor due to a much more detailed and time-consuming calculation. It reflects the true behavior of the rotor because it will include the following sources that influence stability:

• Labyrinth seals
• Balance piston
• Shrink fits
• Shaft material hysteresis

The final logarithmic decrement should be no more than δf> 0.1, otherwise the rotor will need to undergo redesigns to make the logarithmic decrement compliant or measures will need to be taken to prove that the machine will still operate safely.

Although this blog entry is quite lengthy, we have only scratched the surface on what goes into a proper lateral rotor dynamics analysis for a rotating machine. If you want to learn more, I’d encourage you to head over to our webinar we hosted where we apply API Standards to lateral rotor dynamics analysis. You can find it on our educational website, SoftInWay Turbomachinery University.

Coming Up in Next Month’s Rotation…

Next month, we’ll cover torsional rotor dynamics and ensuring the rotor’s characteristics meet the proper API standards.

If you want to learn more about the importance of rotor dynamics, or about the tools our engineers and thousands of others around the world rely on for their turbomachinery designs, reach out to us at info@softinway.com