nice paper on rotor dynamics

Introduction to Rotor Dynamics

Short Course at GMRC 2005 of 23

Introduction to Rotor Dynamics:A Physical Interpretation of the Principles and Application of Rotor Dynamics

by

Krish Ramesh, Ph.D.,Senior Product Technology Engineer,Dresser-Rand, Houston, TX – 77043.

ABSTRACT

The intent of this short course is to:• Explain the basic concepts of rotor dynamics• Describe the application of rotor dynamics to rotating equipment• Present an overview of the newer technologies (hardware) available to solve rotor

dynamic problems

The objective of the short course is to provide a physical understanding to rotordynamics, specifically the vibration characteristics of rotating machinery. This coursecovers the principles of lateral vibration of turbomachinery. Because most of thesemachines operate in critical services in the oil and gas industries, they are designed tooperate reliably when used properly. The dynamic characteristics of the turbomachineryneed to be completely understood before a machine is placed in service. A basicknowledge of the underlying principles of rotor dynamics will help promote a betterunderstanding of the behavior of rotating machinery.

Contents:• Review of basic vibration principles• Terminology used in rotor dynamics• Discussion of journal bearings• Introduction to rotor dynamics• Analytical methods: Critical speeds, unbalance response and stability• Interpretation of results• Overview of hardware used in solving stability problems in turbomachinery



Table of Contents

Abstract 1

1.0 Introduction 3

2.0 Basic Vibration Principles and Definitions 3

3.0 Discussion of Journal bearings 6

3.1 Motion of the shaft in the bearing 7

3.2 Bearing stiffness and damping coefficients 7

3.3 A closer look at bearing instability (Oil Whirl) 8

4.0 Entering the World of Rotor Dynamics 10

4.1 Rotor supported on rigid supports 10

4.2 Rotor supported on flexible supports 11

5.0 Rotor Dynamic Analyses 12

5.1 Undamped critical speed analysis 12

5.2 Unbalance response analysis 15

5.3 Damped eigenvalue analysis 17

5.4 Stability analysis 17

6.0 Technologies to Improve the Stability of Rotor-bearing Systems 20

7.0 References 22



1.0 Introduction

In very simple terms, the turbomachinery consists of a rotor (with impellers/bladed disks,etc) supported on bearings and rotating in the bearing clearance space. Basically there arethree forms of vibrations associated with the motion of the rotor: torsional, axial andlateral. Torsional vibration is the dynamics of the shaft in the angular/rotational direction.Normally, this is little influenced by the bearings that support the rotor. Axial vibration isthe dynamics of the rotor in the axial direction and is generally not a major problem.Lateral vibration, the primary concern, is the vibration of the rotor in lateral directions.The bearings play a huge part in determining the lateral vibrations of the rotor. In thisshort course, we will study the basic concepts of the lateral rotor dynamics ofturbomachinery. [1,2,3]

2.0 Basic Vibration Principles and DefinitionsThe rotor dynamic terminology that is commonly used is as follows:• Rotor: rotating element consisting of the shaft, impellers/bladed disks, shrunk-on

components like sleeves, balance piston, etc• Bearings: journal bearings that support the rotor in the lateral direction. Thrust

bearings support the axial forces generated during the operation of theturbomachinery.

• Damper: a device usually in series with a journal bearing used to provide additionaldamping to the rotor-bearing system

• Stiffness: a property of a spring defined as force per unit displacement (units: lb/in.).The effect of stiffness is to cause a sinusoidal motion as shown in Figure 1A.

• Damping: a property (typically of dampers and bearings) defined as force per unitvelocity (units: lb-s/in.). The effect of damping is to cause an exponential decrease inmotion as shown in Figure 1B.

Fig. 1A. Effect of stiffness (spring) Fig. 1B. Effect of damping (dashpot)

• Natural frequency: the frequency of vibration of a system (e.g.. rotor-bearing system)under free conditions (i.e., without external forces). This is a function of the system.Each system has its own natural frequencies. Consider a very simple system – a mass

Effect of Damping

Time

Am

plitu

de

Effect of Stiffness

Time

Am

plitu

de



supported by a spring, as shown in Figure 2. The natural frequency of this system isgiven by:

mk n =ω (Equation 1)

Fig.2. A simple mass-elastic system

• Damped Natural frequency: In real systems, the damping is almost always present,though in many cases it may be relatively small. The frequency of vibration of asystem (e.g., a rotor-bearing system) with damping is called the “damped naturalfrequency”. Consider a simple system – a mass supported by a spring and damper, asshown in Figure 3A. The variation of amplitude with time, for the mass-spring-damper system is shown in Figure 3B. The sinusoidal part of the curve is a result ofstiffness and the exponentially decreasing part of the curve is the result of damping.The combined effect is an exponentially decreasing sine wave. The assumption madehere is that the damping ratio ζ, is less than 1. The damped natural frequency (again,in simple terms) of this system is given by:

)ς - (1 2nωω =d (Equation 2)

where, m * k2 c and , cc cc

==ς

Fig.3A. A simple mass-spring-damper system Fig. 3B. Damped vibration

• Critical speed: When the operating(running) speed of a machine coincides with thedamped natural frequency, it is termed “critical speed” by definition as given by API617. [4]

m = mass (lb-s2/in)

k = stiffness (lb/in.)

m=mass (lb-s2/in)

k=stiffness (lb/in.) c=damping (lb-s/in.)

Damped Vibration of Mass-Spring-Damper System

Time

Am

plitu

de



In most cases, this simple equation (Equation 1) provides significant insight into thephysics of the problem. Let us examine this equation more closely. Consider the caseof small machinery. These machines have relatively small mass, but have large valuesof stiffness resulting from the combination of shaft stiffness and bearing stiffness.Hence the damped frequency will be relatively high. This design permits the smallermachines (assuming the speed of operating is low) to operate in a range below theirdamped natural frequencies. This kind of operation is known as subcritical operationand is highly desirable if it can be attained.

On the other hand consider the case of large rotating machinery like centrifugalcompressors, gas turbines and steam turbines. The mass of the rotor is usually largeand by design there is a limit of the shaft diameter that can be used. Using Equation 2,it can be seen that as the mass increases, the damped natural frequencies of the largermachines are much lower. Hence during the operation, these machines typically haveto pass through one critical speed before they reach their actual operating speeds. Thisis known as supercritical operation. The main problem is that the machine has to gothrough the critical speed during start-up and shutdown. The challenge in designingsuch machines would be to properly locate the bearings and ensure that the systemhas enough damping to pass through the critical speed.

• Response: The classical dynamic behavior of the rotor-bearing system is plotted inthe form of response plots. The amplitude of the rotor at a particular axial location(usually the probes, located near the journal bearings) is plotted as a function of therunning speed of the rotor. The term “forced response” or “unbalance response” isalso used to refer to these plots. The external force acting on the rotor is in the formof unbalance. Unbalance is caused by the mass distribution of the rotor, i.e., a resultof the manufacturing process, when geometric center and mass center do notcoincide. This leaves behind what is known as “residual unbalance”. In the realworld, there is always a finite amount of residual unbalance in the rotor system.

• Stability: The stability of a system is defined as the “reaction” to any externalperturbation. In other words, consider a rotor that is rotating in journal bearings.Imagine “tapping” the rotor with a “hammer” (perturbation). If the rotor comes backto its “original” position in a finite amount of time, the rotor is said to be stable. If therotor amplitude increases with time (eventually distressing the machine), the rotor issaid to be unstable. Later in this paper we shall discuss in detail the causes for rotorinstability and the various methods and technologies that are available to solve thisproblem.



3.0 Discussion of Journal bearingsThe fluid (oil) film bearings are the most common means of supporting turbomachinery.Of course, rolling element bearings are also used in many gas turbine applications. Butwe will discuss the effect of journal bearings on the rotor dynamic behavior later in thispaper. The term journal applies to the portion of the shaft that is supported by a bearing.An understanding of the role played by the bearings on the dynamic behavior of the rotoris an essential requirement for engineers who deal with the design and operation ofbearings for turbomachinery.

At zero speed (non-rotating) the shaft is at rest at the bottom of the clearance space. Asthe rotor picks up speed, it tends to “climb” on to the inner surface of the bearing. Theconvergent wedge formed between the rotating shaft and the inner surface of the bearinghousing acts as a “pump”, pumping the oil beneath the shaft. This lifts the shaft and atspeed, the shaft occupies an equilibrium position. These phases are shown in Figure 4.

Figure 4. Motion of the shaft in the bearing from rest to speed

The rotating shaft is supported by a thin film of oil. The thin oilfilm that is “squeezed” between the shaft and the housing,generates a pressure that supports the rotor weight. Figure 5shows the pressure profile of a simple sleeve bearing. Thedistance between the geometric center of the bearing and thecenter of the rotating shaft is known as eccentricity.

Figure 5. Pressure profile in a simple sleeve bearing



3.1 Motion of the Shaft in the Bearing:Figure 6 shows the schematic of a simple type of journal bearing (also known as sleevebearing). The shaft rotates in the clearance space (exaggerated) of the bearing as shown.The clearance is usually about 1.5 mils/inch of the journal diameter. Typically for a 4-in.journal diameter, the bearing would have a clearance of 6 mils. When the shaft rotates inthe bearing clearance space, it has two kinds of motions. • The first motion is the rotation or the spin of the shaft. This is the same as the running

speed of the shaft. • The second type of motion is the precession which is the rotation of the center of the

shaft with respect to the geometric center of the bearing. (An analogy would be themotion of the earth which rotates about its axis and also revolves round the sun) Thisprecession (in rotor dynamic terminology) is more commonly known as whirl. Thiswhirl motion is further classified as forward whirl (shown in Figure 6) and backwardwhirl. Forward whirl is the motion in which the center of the shaft moves in the samedirection as the rotation of the shaft. The backward whirl is the motion in which thecenter of the shaft moves in the opposite direction as the rotation of the shaft. Ingeneral, the whirl orbit is elliptical.

Figure 6. Example of forward whirl in the bearing clearance space

3.2 Bearing stiffness and damping coefficients:When the shaft is not rotating it is resting at the bottom of the clearance space in thebearing. With the clearance space filled with oil, as the shaft starts to rotate, it acts as a“pump”, “pushing” the oil underneath itself! This generates the lift of the shaft. At anyconstant rotating speed, the center of the shaft is located away from the geometric centerof the bearing as shown in Figure 5. This is known as the eccentricity of the journal. Theoil “wedge” supports the shaft. The properties associated with the oil film are stiffnessand damping. These are inherent properties of the oil and are a function of oil type,viscosity, temperature, etc. For analytical purposes the stiffness and damping are orientedtowards the horizontal and vertical axes – hence, the bearing is said to have a horizontalstiffness and vertical stiffness (same for the damping). The horizontal stiffness isindicated by Kxx and the vertical stiffness by Kyy. Similarly the damping is indicated asCxx and Cyy. To complicate (real life!) things, because the oil film is continuous aroundthe shaft, there exist components of the stiffness and damping in the x-y direction also!

+++++

Whirl = VibrationAmplitude

Bearing inner surface

Spin or Rotation

Whirl direction(motion of the marker)

Marker on the shaft



The role played by this cross-coupled stiffness (Kxy) is very important in theunderstanding of the stability of rotor-bearing systems.

Figure 7. A simple journal bearing geometry

3.3 A closer look at bearing instability (Oil Whirl):The definition of Kxx is the “force of the rotor in x direction caused by a displacement inthe x direction”. Similarly, Kxy is the “force of the rotor in x direction caused by adisplacement in the y direction”. Now, Kxy and Kyx are complementary. Because of the“link” in the x and y forces and displacements, these are called the cross-couplingcoefficients.

Let’s take the example of the plain sleeve bearing in which the Kxy and Kyx values arerelatively high. Recalling the definitions of Kxy and Kyx, a displacement in x-directioncauses a force in y-direction which causes the rotor to move in the y-direction. Thismotion in y-direction causes a force in the x-direction, which results in the movement ofthe rotor in x-direction! This “feed-forward” mechanism eventually grows to a significantamount and finally results in instability. In a plain sleeve-type journal bearing, thishappens when the whirl speed coincides with the natural frequency of the rotor. The oilfilm looses its capacity to support the load. This could result in a catastrophic failure ofthe bearing. The design of the journal bearing has since evolved to in order to improvethe stability of the bearing. Table 1 gives a list of most common types of journal bearingsin increasing order of stability.

Table 1. Table showing Journal bearing hierarchy with respect to stability

+

Bearing clearancespace with oil

Kxx, Cxx

Kyy, Cyy

Relative stability of different types of Journal bearings:

Stability Design

1. (Least stable) Plain Cylindrical2. (More stable) Lemon, Multi-lobe3. (More stable) Offset halves, Pocket, Pressure Dam4. (Most stable) Tiltpad-design



The different types of fixed pad (sleeve-type) journal bearings that are used forsupporting rotors are as shown in Figure 8.

Figure 8. Different journal bearing geometries

The latest type of journal bearings are the tiltpad journal bearings where the rotor issupported by four or five small radial pads that are pivoted inside the bearing housing asshown in Figure 9.

Figure 9. Tiltpad Journal bearing

The tiltpad bearing design allows each of the pads to rotate about its pivot and attain anequilibrium position with respect to the rotating shaft. Also, the oil film exists only alongthe pad in the circumferential direction. This has practically eliminated the cross-couplingstiffness in the journal bearing. Therefore the tiltpad journal bearings are the most stablebearings.



4.0 Entering the World of Rotor DynamicsAs defined earlier, turbomachinery consists of a rotor (with impellers, bladed disks, etc)supported on bearings and rotating in the bearing clearance space. To understand thebasic principles of the dynamic behavior of the rotor, let us look at a simple rotor-bearingsystem and then extend these principles to the more complicated real-worldturbomachinery.

4.1 Rotor Supported on Rigid SupportsFigure 10 shows a classical Jeffcott rotor – a rotor with an external concentrated mass atthe center, a massless shaft, and supported by bearings at each end.

Figure 10. Jeffcott Rotor – concentrated mass at the midspan and supported at the ends by rigid bearings.

Let us assume that the mass is concentrated at the midspan. The bearings are assumed tobe rigid supports. Thus the rotor can be assumed to be simply supported. Using the theoryof beams, the stiffness of the simply supported beam can be written as,

where, l = “length” refers to the bearing span (axial distance between the bearingcenterlines) and d = diameter of the shaft.

Using the equation 1 for natural frequency, we obtain,

m 64

)d ( E 48 mk 3

4

n lπω ==

and thus proportional to d2/ l1.5. Then for rotors with small (slender) shafts with largeexternal masses the critical frequency is directly proportional to the square of thediameter of the shaft and inversely proportional to the 1.5 power of the bearing span. Thisimportant relationship can be used to physically understand the effects of the designchanges to these machines.

For distributed system, the derivation of stiffness becomes a little more complicated. Ifwe assume no (or negligible) external mass on the shaft of diameter d and length l, andthe shaft mass m, the equation for natural frequency can be written as,

=

=== 24

24

23

44eqn

d d 16

E d 4 *

64 )d ( E

mk

l

flll ρ

π

ρπ

ππω

( )

64d E 48 I E 48 k 3

4

3 ll

π==

m



In other words, for rotors with small external masses compared to the shaft, the naturalfrequency is directly proportional to the diameter and inversely proportional to the secondpower of the length. This important relationship also can be used to physically understandthe effects of the design changes to the machine.

In both cases as the diameter of the shaft increases the natural frequency increasesand as the bearing span increases the natural frequency decreases.

4.2 Rotor Supported on Flexible SupportsIn the real world, the support is not infinitely rigid. There is a finite amount of stiffnessthat a journal bearing provides at the supports. Figure 11 shows the rotor supported onactual bearings. The bearing stiffness and damping are given by Kb and Cb respectively.

Figure 11. Jeffcott Rotor – concentrated mass at the midspan and supported at the ends by actual bearings.

Adding the bearing stiffness in series with the shaft stiffness, reduces the effectivestiffness as in the equation,

K1

2K1

K1

sbeff

+=

Hence, using the equation for natural frequency (ωn = sqrt(Keff/m)), we can see thatadding the bearing stiffness, reduces the natural frequency of the system!

m

Kb Cb Kb Cb



5.0 Rotor Dynamic Analyses

A rotor dynamic analyses consists of:• Undamped critical speed analysis• Unbalance response analysis• Damped eigenvalue analysis• Stability Analysis

5.1 Undamped Critical speed analysisIn simple terms “undamped critical speed analysis” means “calculating the critical speedsof the machines without the effects of damping”! The inclusion of damping adds a hugedegree of complexity in formulating and solving the problem. Historically, the undampedanalysis was the first and in many cases the only analysis done. Most of the calculationsin rotor dynamics are based on matrix manipulations. Because of the lack of high-speedcomputers, the matrix calculations had to be done, to a large extent, manually in a time-consuming manner.

Let’s look at the physics of the analysis and what the output looks like. As defined above,the output of the analysis is the undamped critical speeds of the rotor-bearing system. Themethodology is to vary the support stiffness from a very low value (flexible supports) to avery high value (rigid supports) in discrete steps. At each step (or, value of the supportstiffness), the undamped natural frequency of the rotor is determined. The result is plottedon a chart that is classically known as the “undamped critical speed map” in theturbomachinery industry. A typical undamped critical speed map is shown in Figure 12.

Figure 12. A typlical undamped critical speed map.

Rigid supports/Flexible rotorFlexible supports/Rigid rotor

1 2 3 4

f1-1

f1-2

f1-3

f1-4

Actual bearing stiffness



Now, let’s look into how this plot can be “physically” generated. Let’s start the“experiment” with the rotor supported on very soft supports. Also the rotor is assumed tobe appropriately instrumented. An accelerometer is placed on the rotor, say at midspanand connected to a FFT analyzer. This instrument can measure the frequency at which therotor vibrates. Let’s assume the support stiffness is 1.0 x 103 lb/in. as shown by “1” inFigure 12. This represents a very soft spring (“flexible support”). Imagine “striking” therotor with a hammer and allowing the rotor to vibrate. The frequency at which it willvibrate is its natural frequency.

Now, the rotor being a continuous system (as compared to a point or concentrated mass),it will have many natural frequencies. Usually we would be interested in the first fournatural frequencies. With the help of the instrumentation we measure the first fourfrequencies of the rotor. Also, if we measure the movement of the shaft at thesefrequencies, we could get the mode shape that corresponds to each of these frequencies.These four frequencies are shown in figure 12 as f1-1, f1-2, f1-3 and f1-4. The correspondingmode shapes are shown in figure 13.

Mode shape at f1-1 Mode shape at f1-2 Mode shape at f1-3 Mode shape at f1-4

Figure 13. The Mode shapes of the rotor for the first four frequencies at low support stiffness

Repeat the same experiment of striking with a hammer and measuring the first fourfrequencies, now at a increased value of the support stiffness (say, 1.0 x 105 lb/in. asshown by “2” in Figure 12. This gives four more points on the plot. Repeating the aboveprocess for increasing value of support stiffness gives the four frequencies at each of thesupport stiffness. Hence the four curves can be generated. Now, we can see that as thesupport stiffness increases, the natural frequencies increase. Let’s look at how the modeshape for the first critical speed varies with increasing support stiffness, as shown inFigure 14.A. Figure 14.B shows the mode shape for the second critical speed.

Figure 14.A. The variation of mode shape of the rotor for the first frequency



Figure 14.B. The variation of mode shape of the rotor for the second frequency

With low support stiffness, the rotor does not bend much. The rotor is referred to as RigidRotor. As the support stiffness increases, the rotor does get to bend. With very highvalues of support stiffness (1.0 x 108 lb/in.), the rotor is bent to an extent that the bearingsessentially “lock” the rotor with negligible displacement at the bearing locations. Thiscondition is referred to a Flexible Rotor, or Rigid Supports. As you can see from Figure12, the natural frequency of the rotor-bearing system is a function of the support stiffness.The common terms used in rotor dynamic world are “first rigid-bearing critical speed” or“first undamped critical speed on rigid supports” and “second rigid-bearing criticalspeed” or “second undamped critical speed on rigid supports”. These refer to the first andsecond natural frequencies at the condition when the support stiffness is extremely high –in other words, rigid supports.

Most turbomachinery operates above the first critical speed. Hence, the criteria widelyused in the industry are based on the first rigid critical speed. The ratio, which is verypopularly used in the turbomachinery industry is:

speed critical bearing RigidFirst machine the of Speed ContinuousMax

NcMCOS

1r

=

This ratio gives a rough idea of how low the first critical speed of the machine is, withrespect to the maximum continuous speed. API 617 [4] has defined the rules for rotordynamic acceptability for centrifugal compressors based on a plot of this ratio vs. theaverage gas density of the application. See [5] for an extensive database of centrifugalcompressor applications on this plot.

The bearing has a finite amount of stiffness and damping values. The bearing coefficientsare typically calculated by a “bearing program” that takes in the bearing geometry andchurns out the coefficients. As we have seen in the bearing discussion, the position of theshaft in the bearing clearance space is a function of speed. Thus the oil film thickness is afunction of speed. Thus the bearing coefficients (K and C) are a function of speed!



If we plot the bearing stiffness on the undamped critical speed plot, we can gain a wealthof information about the possible damped critical speeds of the rotor! As shown in Figure12, the bearing coefficients are plotted on the critical speed map. The intersection of thebearing stiffness line with the rotor frequency line indicates a potential damped criticalspeed of the rotor. The details of this “potential damped critical speed” will be discussedin the section on unbalance response calculations.

5.2 Unbalance response analysisDue to nature of manufacturing of the rotor and its components, unbalance always exists,although in small quantities. The amount of unbalance can be reduced to a tolerable levelby different unbalance techniques. Nevertheless, the residual unbalance that is alwayspresent in the rotor system acts as a forcing function on the rotor and tends to “pull” therotor away from the undeformed rotor centerline during operation. Figure 15 shows thecentrifugal force generated by an unbalance. Typically, the unbalance is indicated in “oz-in”. Assume an unbalance of mass “m” located at a radius “r” and the rotor spinning at“ω” rad/s. The unbalance force acting on the rotor, F = m*ω2*r. This displacement of therotor is the “response of the rotor to the unbalance”.

Figure 15. Unbalance force on a rotor

The response plot is a plot of the displacement of the rotor (at a particular location) as afunction of running speed. The main purpose of an unbalance response calculation is todetermine the actual critical speed and the corresponding amplitude as the rotor increasesfrom zero speed to its running speed. In the analytical world, to simulate a forcedresponse, a known amount of unbalance is located on the rotor at specific locations. Weknow from previous discussion (Figure 14A) that the first critical speed will have a “half-sine wave” as its mode shape. Therefore to “force” the rotor to bend at the midspan, theappropriate location of the “theoretical” unbalance would be at the midspan! Theresponse of the rotor, typically at the probe locations, is noted down as a function ofspeed. Figure 16 shows a typical response plot of a rotor for a midspan unbalance. As therotor increases in speed, let us look at one of the probe locations of the machine. This isthe probe located next to the bearing housing (in most cases) at the intake end of themachine (typical in a compressor). As the speed increases, the amplitude of vibrationincreases. The amplitude is maximum at the critical speed of the machine and thendecreases. Typically, machines are designed such that the critical speed is well belowtheir operating speed. API 617 [4] has rules on how far the peak of the critical speed canbe from the operating speed range.

r

mω

F = m*ω2*r mF=m ω2r



Figure 16. Forced response of the rotor for a midspan unbalance

There are a few parameters that are checked using the unbalance response plot.

• Amplification factor: When the rotor response goes through the critical speed, theresponse follows a peak. Amplification factor, in simple terms, is the “sharpness ofthe peak”. The popular half-power method is used to calculate the amplificationfactor. This is described in Figure 17.

Figure 17. Calculation of Amplification Factor at the Critical Speed

0

6

0 5 10

x1

0.707 * x1

N1 N2Np

The half-power method is used toobtain the “sharpness” of the peak.Draw a horizontal line at (x1/sqrt(2))(or in other words, at 0.707*x1) toget the intercepts N1 and N2.

A.F. = Np / (N2-N1)



• Separation margin: This is the distance of the peak of the critical speed with respectto the nearest operating speed. API 617 [4] has defined the separation margin as givenbelow. Let us assume that the critical speed peak is at Nc1 rpm. If NMIN (Minimumoperating speed) rpm is the minimum operating speed of the machine and assumingNMIN > Nc1, the separation margin is given by:

NMIN

)N - (NMIN * 100 (%) margin Separation c1=

API 617 [4] has rules on the minimum separation margin based on the AmplificationFactor.

• Amplitude at critical speed: The amplitude at critical speed is indicative of theseverity of the vibration as the rotor coasts through the critical speed. The limit of theamplitude is determined by the turbomachinery manufacturer, the end user and theprocess conditions under which the machines are operated. API 617 [4] has rules onthe limit of amplitude.

5.3 Damped Eigenvalue analysisIn this type of analysis, the natural frequencies of the rotor, supported on its bearings, aredetermined. For the analysis, no external forces are assumed. It is similar to allowing therotor to rotate, then “hitting” it with a hammer and recording the frequencies of vibration.It is quite similar to the undamped critical speed analysis, but now the rotor is supportedon actual bearings. Usually the first eight frequencies are noted down as a function ofrunning speed. The outputs of this analysis are the damped natural frequencies and thecorresponding logarithmic decrement.

5.4 Stability analysisThis is a measure of how the rotor system responds to external excitation. It is measuredby how fast the vibration “decays” with respect to time. There are three definitions ofstability in systems as shown in Figure 18.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.5

-1

-0.5

0

0.5

1

1.5

Stable system: Exponentially decreasing Sine wave,Logdec > 0

Neutrally Stable system: Sine wave, Logdec = 0



Figure 18. Stability of systems

The quantity that is used to measure the stability is called logarithmic decrement, or inshort, log dec. Log dec is the natural logarithm of the ratio of the amplitude of one peakover the amplitude of the next peak as shown in Figure 19.

Figure 19. Definition of logarithmic decrement

In rotor-bearing systems, an unstable behavior is characterized by an increase in vibrationat a particular frequency, over time. Eventually the amplitude of vibration could be largeenough to damage the bearings, seals, etc.

The gases, while passing through narrow passages in the aero-path in the compressor,generate forces on the rotor. Imagine small molecules of the gases passing through thenarrow paths, twisting and turning. These molecules have velocity in all the threedirections. The tangential velocity is the one that generates the “exciting” forces on theshaft. This is also known as the Swirl velocity. The forces generated by the swirlcomponent can be modeled as cross-coupling stiffness (“Kxy terms”). (Recall that thecross-coupling forces are “bad” elements and that such forces reduce the stability of thesystem.)

These aero-induced forces are almost always present in the system. To quantify theseforces, an empirical relation is used to estimate them, as given by API [4] and Memmott[6]. It was derived from the Wachel [7] and Alford [8] numbers. It gives the cross-coupling forces created by the aero forces (function of stage horse-power, gas density,speed, etc.). Let’s call this the Wnumber. These aero-dynamic influences can beconsidered as “stresses” that are already present in the system. The purpose of the

-10

-8

-6

-4

-2

0

2

4

6

8

Unstable system: Exponentially increasing Sine wave,Logdec < 0

Log dec = ln(x1/x2)

Stable : if log dec > 0Unstable: if log dec <0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time

Am

plitu

de

x1x2



stability analysis is to determine how much “cross-coupling” forces can the rotor-bearingsystem withstand before the system becomes unstable. For analytical purposes, the rotoris simulated with an “artificial cross-coupling” at midspan. As this value of cross-coupling is increased, the frequency (the first forward mode) and the corresponding logdec are noted down. As the cross-coupling increases, the stability will drop (indicated bya decrease in log dec). This is done until the log dec becomes zero. This is the thresholdof instability. The amount of cross-coupling gives the maximum aero-forces the systemcan withstand before it goes unstable. Let’s call this value “Kxy-threshold”. From the abovediscussion we know that the system already has a cross-coupled force of Wnumber.Now the limit that the system can withstand is Kxy-threshold.

Thus the “Stability Margin” = Kxy-threshold / Wnumber.

API [4] and each manufacturer of rotating machines have acceptable values for the logdec and Stability Margin based on their experience. The results of the stability analysisare as shown in Figure 20.

Figure 20. Results of Stability analysis

“Inherent” cross-couplingin the system

Threshold cross-coupling that thesystem can withstand = x2

Margin = x2 / “Inherent” Kxy



6.0 Technologies to improve the stability of rotor-bearing systemsLack of stability is one of the cause for poor operation of turbomachinery. It is criticalthat the components selected (bearings, seals, dampers, etc) are able to satisfactorilyoperate under the given conditions without failure. [9-17]

The two major sources of potential instability in a turbomachinery are the aero-flow paththrough the stages (always present) and the passage of the gas through the clearancespace between the laby and the shaft. The cross-coupling generated when the gas passesthrough the clearance space under the laby can be reduced by new technologicaladvances like the swirl brakes and damper seals.

• Damper bearings: In the turbomachinery world, a common way to increase thestability of rotor-bearing systems would be the addition of damping to the system byadding a damper. Functionally, the damper that goes behind the journal bearings, is aring (also known as the cage) that houses the bearings and is in turn housed and freeto move in the bearing housing. The schematic representation of the damper bearingconfiguration is shown in Figure 21. There is an oil film that is trapped between theoutside diameter of the cage and housing. The axial leakage of the oil is prevented byO-rings. This oil film is historically known as “squeeze film” and hence theterminology “squeeze film damper”. The oil is fed through the housing to the spacebetween the cage and housing and then onto the journal. This damper acts as a“cushion” for the vibrations and hence “dampens” the vibration amplitude. Thestability of the rotor is also increased by this additional damping. The newer design ofthis damper assembly has alternate load-carrying components instead of the O-rings.

Figure 21. Damper bearing configuration

• Damper seals: The new technology of damper seals, used to replace toothedlabyrinths at the balance piston or division wall of a centrifugal compressor, has



proven to be a good means to add additional damping to the rotor system to improvestability. In principle, the damper seal is like a silencer for a gun. It has radial holeswhich “trap” the gases and produce small “pockets of damping”. These technologieshave been put to good use in the recent years and have reduced the number ofstability problems in the turbomachinery industry.

• Swirl brakes: The cause for aero-induced exciting forces, as noted above, is the swirlor the tangential velocity component of the gas. If we are able to redirect the gas toreduce the swirl velocity, we could potentially reduce the excitation forces. The swirlbrake is a stationary ring with blade-like axial “teeth” along its circumference. Theteeth redirect the gas flow and reduce the swirl velocity. This device improves thestability of the rotor-bearing system. It is used at the impeller eye seals and at thebalance piston or division wall seal of a centrifugal compressor.

• Shunt holes: Similar to swirl brakes, taking gas off the diffuser of the last stageimpeller and injecting it before the balance piston or division wall will also reduce theswirl velocity. They are typically used at the division wall of a back-to-backcompressor.

AcknowledgementThe information contained in this presentation consists of factual data and technicalinterpretations and opinions which, while believed to be accurate, is offered solely forinformation purposes. No representation or warranty is made concerning the accuracy ofsuch data, interpretations and opinions.



7.0 References:

1. Kirk, R.G., 1980, “Stability and Damped Critical Speeds: How to Calculate and Interpret theResults,” CAGI Technical DIGEST, Vol. 12, No. 2, pp. 375-383

2. Ruddy, A.V., and Summers-Smith, D., 1980, “An Introduction to the influence of the bearingson the dynamics of rotating machinery”, Tribology International, October.

3. White, D.C., 1972, “The Dynamics of Rigid Rotor Supported on Squeeze Film Bearings,”I.Mech.E., Conference on Vibration in Rotating Systems, pp. 213-229.

4. API Standard 617 7th Edition, July 2002, “Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services”

5. Memmott, E. A., 2002, “Lateral Rotordynamic Stability Criteria for Centrifugal Compressors,”CMVA, Proceedings of the 20th Machinery Dynamics Seminar, Quebec City, pp. 6.23-6.32,October 21-23

6. Memmott, E. A., 2000, “Empirical Estimation of a Load Related Cross-Coupled Stiffness andThe Lateral Stability Of Centrifugal Compressors,” CMVA, Proceedings of the 18th MachineryDynamics Seminar, Halifax, pp. 9-20, April 26-28.

7. Wachel, J. C. and von Nimitz, W. W., 1981, "Ensuring the Reliability of Offshore GasCompressor Systems," Journal of Petroleum Technology, pp. 2252-2260, Nov.

8. Alford, J. S., 1965, "Protecting Turbomachinery from Self-Excited Whirl," ASME Journal ofEngineering for Power, Vol. 38, pp. 333-344

9. Memmott, E. A., 1992, "Stability of Centrifugal Compressors by Applications of Tilt PadSeals, Damper Bearings, and Shunt Holes," IMechE, 5th International Conference on Vibrationsin Rotating Machinery, Bath, pp. 99-106, September 7-10.

10. Marshall, D. F., Hustak, J. F., and Memmott, E. A., 1993, "Elimination of SubsynchronousVibration Problems in a Centrifugal Compressor by the Application of Damper Bearings, TiltingPad Seals, and Shunt Holes," NJIT-ASME-HI-STLE, Rotating Machinery Conference andExposition, Somerset, New Jersey, November 10-12.

11. Memmott, E. A., 1994, "Stability of a High Pressure Centrifugal Compressor ThroughApplication of Shunt Holes and a Honeycomb Labyrinth," CMVA, 13th Machinery DynamicsSeminar, Toronto, Canada, pp. 211-233, September 12-13.

12. Kuzdzal, M. J. and Hustak, J. F., 1996, "Squeeze Film Damper Bearing Experimental vs.Analytical Results for Various Damper Configurations," Proceedings of the Twenty FifthTurbomachinery Symposium, Turbomachinery Laboratory, Department of MechanicalEngineering, Texas A&M University, College Station, Texas, September 17-19.

13. Ramesh, K and Kirk, R. G., 1999, “Nonlinear Response of Rotors Supported on Squeeze FilmDampers,” Proceedings of DETC99 – 17th Biennial ASME Vibrations Conference, ASME PaperDETC99/VIB-8294, Las Vegas, Nevada, September 12-15



14. Moore, J. J. and Hill, D. L., 2000, “Design of Swirl Brakes for High Pressure CentrifugalCompressors Using CFD Techniques,” 8th International Symposium on Transport Phenomenaand Rotating Machinery, ISROMAC-8, Vol. II, Honolulu, Hawaii, pp. 1124-1132, March 26-30

15. Ramesh, K, 2002, "A State-of-the-art Rotor Dynamic Analysis Program," Proceedings of the9th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery,Honolulu, Hawaii, February 10-14

16. Moore, J.J, Walker, S.T., Kuzdzal, M.J., 2002, “Rotordynamic Stability Measurement DuringFull-Load, Full-Pressure Testing of a 6000 PSE Re-Injection Centrifugal ”, 32nd TurboMachinery Symposium, Houston, TX.

17. Moore, J. J. and Soulas, T. A., 2003, “Damper Seal Comparison in a High Pressure Re-Injection Centrifugal Compressor During Full-Load, Full-Pressure Testing Using DirectRotordynamic Stability Measurement,” ASME, Proceedings of the 19th Biennial Conference onMechanical Vibration and Noise, International 2003 DETC, Chicago, Illinois, DETC2003/VIB-48458, September 2-6

nice paper on rotor dynamics

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