Gas Turbine Engineering Handbook 2 Episode 5 pps

Critical Speed Calculations for Rotor Bearing SystemsMethods for calculating undamped and damped critical speeds thatclosely follow the works of Prohl and Lund, respectively, are listed

This very special case is known as critical damping The value of c for thiscase is given by:

c2 cr

4m2ˆmk

c2

mˆ 4mkThus,

ccrˆp4mkˆ 2m



km

r

ˆ 2m!n

Underdamped system If c2=4m2< k=m, then the roots r1 and r2 areimaginary, and the solution is an oscillating motion as shown in Figure 5-9.All the previous cases of motion are characteristic of different oscillatingsystems, although a specific case will depend upon the application Theunderdamped system exhibits its own natural frequency of vibration.When c2=4m2< k=m, the roots r1 and r2 are imaginary and are given by

r1;2ˆ i



km

c2

4m2

r

…5-15†Then the response becomes

Figure 5-8 Critical damping decay

which can be written as follows:

Forced Vibrations

So far, the study of vibrating systems has been limited to free vibrationswhere there is no external input into the system A free vibration systemvibrates at its natural resonant frequency until the vibration dies down due

to energy dissipation in the damping

Now the influence of external excitation will be considered In practice,dynamic systems are excited by external forces, which are themselves periodic

in nature Consider the system shown in Figure 5-10

The externally applied periodic force has a frequency !, which can varyindependently of the system parameters The motion equation for thissystem may be obtained by any of the previously stated methods TheNewtonian approach will be used here because of its conceptual simplicity.The freebody diagram of the mass m is shown in Figure 5-11

Figure 5-9 Underdamped decay

Figure 5-10 Forced vibration system

The motion equation for the mass m is given by:

and can be rewritten asmx ‡ c _x ‡ kx ˆ F sin !tAssuming that the steady-state oscillation of this system is represented bythe following relationship:

where:

D ˆ amplitude of the steady-state oscillation

 ˆ phase angle by which the motion lags the impressed forceThe velocity and acceleration for the system are given by the followingrelationships:

mD!2 sin …!t † cD! sin !t  ‡ 2

Inertia force ‡ Damping force ‡ Spring force ‡ Impressed force ˆ 0From the previous equation, the displacement lags the impressed force

by the phase angle , and the spring force acts opposite in direction to

Figure 5-11 Free body diagram of mass (M)

displacement The damping force lags the displacement by 90 and is fore in the opposite direction to the velocity The inertia force is in phasewith the displacement and acts in the opposite direction to the acceleration.This information is in agreement with the physical interpretation of harmo-nic motion The vector diagram as seen in Figure 5-12 shows the variousforces acting on the body, which is undergoing a forced vibration withviscous damping Thus, from the vector diagram, it is possible to obtainthe value of the phase angle and the amplitude of steady oscillation

ccˆ 2 m!nˆ critical damping coefficient:

From these equations, the effect on the magnification factor (D=F=k) andthe phase angle () is mainly a function of the frequency ratio !=!n and thedamping factor  Figures 5-13a and 5-13b show these relationships Thedamping factor has great influence on the amplitude and phase angle in theregion of resonance For small values of !=!n 1:0, the inertia and dampingforce terms are small and result in a small phase angle For a value of

!=!nˆ 1:0, the phase angle is 90 The amplitude at resonance approachesinfinity as the damping factor approaches zero The phase angle undergoesnearly a 180shift for light damping as it passes through the critical frequencyratio For large values of !=!n 1:0, the phase angle approaches 180, andthe impressed force is expended mostly in overcoming the large inertia force.Design Considerations

Design of rotating equipment for high-speed operation requires carefulanalysis The discussion in the preceding section presents elementary analy-sis of such problems Once a design is identified as having a problem, it is analtogether different matter to change this design to cure the problem Thefollowing paragraphs discuss some observations and guidelines based on theanalysis presented in the previous sections

Natural frequency This parameter for a single degree of freedom isgiven by !nˆpk=m Increasing the mass reduces !n, and increasing thespring constant k increases it From a study of the damped system, thedamped natural frequency !d ˆ !n



1 2

p

is lower than !n.Unbalances All rotating machinery is assumed to have an unbalance.Unbalance produces excitation at the rotational speed The natural fre-quency of the system !n is also known as the critical shaft speed From thestudy of the forced-damped system, the following conclusions can be drawn:

(1) the amplitude ratio reaches its maximum values at !mˆ !n 

1 22

p

,and (2) the damped natural frequency !d does not enter the analysis of theforced-damped system The more important parameter is !n, the naturalfrequency of the undamped system

In the absence of damping the amplitude ratio becomes infinite at ! ˆ !n.For this reason, the critical speed of a rotating machine should be kept awayfrom its operating speed

Small machinery involves small values of mass m and has large values ofthe spring constant k (bearing stiffness) This design permits a class ofmachines, which are small in size and of low speed in operation, to operate

in a range below their critical speeds This range is known as subcriticaloperation, and it is highly desirable if it can be attained economically.Figure 5-13a Amplitude factor as a function of the frequency ratio r for variousamounts of viscous damping

The design of large rotating machineryÐcentrifugal compressors, gas andsteam turbines, and large electrical generatorsÐposes a different problem.The mass of the rotor is usually large, and there is a practical upper limit

to the shaft size that can be used Also, these machines operate at highspeeds

This situation is resolved by designing a system with a very low criticalspeed in which the machine is operated above the critical speed This isknown as supercritical operation The main problem is that during start-upand shut-down, the machine must pass through its critical speed To avoiddangerously large amplitudes during these passes, adequate damping must

be located in the bearings and foundations

The natural structural frequencies of most large systems are also in the frequency range, and care must be exercised to avoid resonant couplings betweenthe structure and the foundation The excitation in rotating machinery comesfrom rotating unbalanced masses These unbalances result from four factors:

low-1 An uneven distribution of mass about the geometric axis of thesystem This distribution causes the center of mass to be differentfrom the center of rotation

2 A deflection of the shaft due to the weight of the rotor, causing furtherdistance between the center of mass and the center of rotation Add-itional discrepancies can occur if the shaft has a bend or a bow in it.Figure 5-13b Phase angle as a function of the frequency ratio for various amounts

of viscous damping

3 Static eccentricities are amplified due to rotation of the shaft about itsgeometric center.

4 If supported by journal bearings, the shaft may describe an orbit sothat the axis of rotation itself rotates about the geometric center of thebearings

These unbalance forces increase as a function of !2, making the designand operation of high-speed machinery a complex and exacting task Balan-cing is the only method available to tame these excitation forces

Application to Rotating MachinesRigid Supports

The simplest model of a rotating machine consists of a large disc mounted

on a flexible shaft with the ends mounted in rigid supports The rigidsupports constrain a rotating machine from any lateral movement, but allowfree angular movement A flexible shaft operates above its first critical.Figures 5-14a and 5-14b show such a shaft The mass center of the disc ``e''

is displaced from the shaft centerline or geometric center of the disc due tomanufacturing and material imperfections When this disc is rotated at arotational velocity !, the mass causes it to be displaced so that the center of

Figure 5-14a Rigid supports Figure 5-14b Flexible supports

the disc describes an orbit of radius r, from the center of the bearingcenterline If the shaft flexibility is represented by the radial stiffness (Kr),

it will create a restoring force on the disc of Krr that will balance thecentrifugal force equal to m!2(r‡ e) Equating the two forces obtains

Flexible Supports

The previous section discussed the flexible shaft with rigid bearings In thereal world, the bearings are not rigid but possess some flexibility If theflexibility of the system is given by Kb, then each support has a stiffness of

Kb=2 In such a system, the flexibility of the entire lateral system can becalculated by the following relationship:

ˆ !n



Kb

Kb‡ Krr

…5-28†

It can be observed from the previous expression that when Kb Kr(veryrigid support), then !ntˆ !n or the natural frequency of the rigid system.For a system with a finite stiffness at the supports, or Kb><Kr, then !nis lessthan !nt Hence, flexibility causes the natural frequency of the system to belowered Plotting the natural frequency as a function of bearing stiffness on

a log scale provides a graph as shown in Figure 5-15

When Kb Kr, then !ntˆ !nKb=Kr Therefore, !ntis proportional to thesquare root of Kb, or log !nt is proportional to one-half log Kb Thus, thisrelationship is shown by a straight line with a slope of 0.5 in Figure 5-15.When Kb  Kr, the total effective natural frequency is equal to the naturalrigid-body frequency The actual curve lies below these two straight lines asshown in Figure 5-15

The critical speed map shown in Figure 5-15 can be extended to includethe second, third, and higher critical speeds Such an extended critical speedmap can be very useful in determining the dynamic region in which a givensystem is operating One can obtain the locations of a system's critical speeds

by superimposing the actual support versus the speed curve on the criticalspeed map The intersection points of the two sets of curves define thelocations of the system's critical speeds

Figure 5-15 Critical speed map

When the previously described intersections lie along the straight line onthe critical speed map with a slope of 0.5, the critical speed is bearingcontrolled This condition is often referred to as a ``rigid-body critical.''When the intersection points lie below the 0.5 slope line, the system is said

to have a ``bending critical speed.'' It is important to identify these points,since they indicate the increasing importance of bending stiffness over sup-port stiffness

Figures 5-16a and 5-16b show vibration modes of a uniform shaft ported at its ends by flexible supports Figure 5-16a shows rigid supports and

sup-a flexible rotor Figure 5-16b shows flexible supports sup-and rigid rotors

To summarize the importance of the critical speed concept, one shouldbear in mind that it allows an identification of the operation region of therotor-bearing system, probable mode shapes, and approximate locations ofpeak amplitudes

Critical Speed Calculations for Rotor Bearing SystemsMethods for calculating undamped and damped critical speeds thatclosely follow the works of Prohl and Lund, respectively, are listed here-

in Computer programs can be developed that use the equations shown inthis section to provide estimations of the critical speeds of a given rotor for arange of bearing stiffness and damping parameters

The method of calculating critical speeds as suggested by Prohl and Lundhas several advantages By this method, any number of orders of criticalfrequencies may be calculated, and the rotor configuration is not limited inthe number of diameter changes or in the number of attached discs Inaddition, shaft supports may be assumed rigid or may have any values ofdamping or stiffness The gyroscopic effect associated with the moment of

Figure 5-16a Rigid supports and a flexible rotor

Figure 5-16b Flexible supports and rigid rotors

attached disc inertia may also be taken into account Perhaps the greatestadvantage of the technique, however, is the relative simplicity with which allthe capabilities are performed.

The rotor is first divided into a number of station points, including theends of the shafting, points at which diameter changes occur, points at whichdiscs are attached, and bearing locations The shafting connecting the stationpoints are modeled as massless sections which retain the flexural stiffnessassociated with the section's length, diameter, and modulus of elasticity Themass of each section is divided in half and lumped at each end of the sectionwhere it is added to any mass provided by attached discs or couplings.The critical-speed calculation of a rotating shaft proceeds with equations

to relate loads and deflections from station n 1 to station n The shaftshear V can be computed using the following relationship:

where ˆ flexibility constant

The vertical linear displacement is

Ynˆ n Mn 1

Mn6

The initial boundary conditions are V1ˆ M1ˆ 0 for a free end and, toassign initial values for Y1and 1, the calculation proceeds in two parts withthe assumptions given as

Pass 1 Y1ˆ 1:0 1ˆ 0:0Pass 2 Y1ˆ 0:0 1ˆ 1:0For each part, the calculation starts at the free end and, using Equations(5-29) through (5-35), proceeds from station to station until the other end isreached The values for the shear and moment at the far end are dependent

on the initial values by the relationship:

relation-V0 x

V0 y

M0 x

M0 y

264

375

375

375

n

‡ …K ‡ sB†n

XY



264

375

E ˆ Young's modulus of elasticity

I ˆ sectional moment of inertia

G ˆ shear modulus

 ˆ logarithmic decrement of internal shaft damping divided

by shaft vertical position ˆ cross-sectional shape factor ( ˆ :75 for circular cross section)

Electromechanical Systems and AnalogiesWhere physical systems are so complex that mathematical solutions arenot possible, experimental techniques based on various analogies may beone type of solution Electrical systems that are analogous to mechanicalsystems are usually the easiest, cheapest, and fastest solution to the prob-lem The analogy between systems is a mathematical one based on thesimilarity of the differential equations Thomson has given an excellenttreatise on this subject in his book on vibration Some of the highlightsare given here

A forced-damped system is shown in Figure 5-17 This system has a mass

M, which is suspended on a spring K with a spring constant and a dash pot

to produce damping The viscous damping coefficient is c

repre-Figure 5-17 Forced vibration with viscous damping.

Figure 5-18 A force-voltage system

represent the mass, spring constant, and the viscous damping, respectively,can be written as follows:

Ldi

dt‡ Ri ‡

1C

Z t

A force-current analogy can also be obtained where the mass is sented by capacitance, the spring constant by the inductance, and the resist-ance by the conductance as shown in Figure 5-19 The system can berepresented by the following relationship:

repre-Cde

dt‡ Ge ‡

1L

Z t

Comparing all these equations shows that the mathematical relationshipsare all similar These equations convey the analogous values For conveni-ence, Table 5-1 also shows these relationships

Forces Acting on a Rotor Bearing System

There are many types of forces that act on a rotor-bearing system Theforces can be classified into three categories: (1) casing and foundationforces, (2) forces generated by rotor motion, and (3) forces applied to arotor Table 5-2 by Reiger is an excellent compilation of these forces.Forces transmitted to casing and foundations These forces can bedue to foundation instability, other nearby unbalanced machinery, pipingstrains, rotation in gravitational or magnetic fields, or excitation of casing or

Figure 5-19 Force-current analogy

foundation natural frequencies These forces can be constant or variablewith impulse loadings The effect of these forces on the rotor-bearing systemcan be great Piping strains can cause major misalignment problems andunwanted forces on the bearings Operation of reciprocating machinery inthe same area can cause foundation forces and unduly excite the rotor of aturbomachine.

Forces generated by rotor motion These forces can be classified intotwo categories: (1) forces due to mechanical and material properties, and (2)forces caused by various loadings of the system The forces from mechanicaland material properties are unbalanced and are caused by a lack of homo-geneity in materials, rotor bow, and elastic hysterisis of the rotor The forcescaused by loadings of the system are viscous and hydrodynamic forces in therotor-bearing system, and various blade loading forces, which vary in theoperational range of the unit

Forces applied to a rotor Rotor-applied forces can be due to drivetorques, couplings, gears, misalignment, and axial forces from piston andthrust unbalance They can be destructive, and they often result in the totaldestruction of a machine

Rotor Bearing System Instabilities

Instabilities in rotor-bearing systems may be the result of different forcingmechanisms Ehrich, Gunter, Alford, and others have done considerablework to identify these instabilities One can divide these instabilities intotwo general yet distinctly different categories: (1) the forced or resonantinstability dependent on outside mechanisms in frequency of oscillations;

Table 5-1 Electromechanical System Analogies Mechanical

Parameters Force-Voltage AnalogyElectrical ParametersForce-Current Analogy

Velocity _x or v

Displacement x ˆR0tvdt Charge q ˆR0tidt

Coefficient

Table 5-2 Forces Acting on Rotor Bearing Systems

1 Forces transmitted

to foundations, casing,

or bearing pedestals.

Constant, unidirectional force Constant linear acceleration.Constant force,

rotational magnetic field.Rotation in gravitational orVariable,

unidirectional or foundation-motion.Impressed cyclic groundImpulsive forces Air blast, explosion,

or earthquake Nearby unbalanced machinery Blows, impact Random forces

2 Forces generated

by rotor motion. Rotating unbalance:residual, or bent shaft. Present in all rotating machinery.

Coriolis forces Motion around curve of varying

radius Space applications.

Rotary-coordinated analyses Elastic hysteresis

of rotor appears when rotor is cyclicallyProperty of rotor material, which

deformed in bending, torsionally

or axially.

Coulomb friction Construction damping arising from

relative motion between shrunkfitted assemblies.

Dry-friction bearing whirl.

Fluid friction Viscous shear of bearings.

Fluid entrainment in turbomachinery Windage.

Hydrodynamic forces, static. Bearing load capacity.Volute pressure forces.

Hydrodynamic forces, dynamic. Bearing stiffness and dampingproperties Dissimilar elastic beam

Stiffness reaction forces stiffnesses Slotted rotors,Rotors with differing rotor lateral

electrical machinery, Keyway Abrupt speed change conditions Gyroscopic moments Significant in high-speed flexible

rotors with discs.

3 Applied to rotor Drive torque Accelerating or constant-speed

operation Cyclic forces Internal combustion engine torque

and force components.

Oscillating torques Misaligned couplings Propellers.

Fans.

Internal combustion engine drive table continued on next page

and (2) the self-excited instabilities that are independent of outside stimuliand independent of the frequency Table 5-3 is the characterization of thetwo categories of vibration stimuli.

Forced (resonant) vibration In forced vibration the usual driving quency in rotating machinery is the shaft speed or multiples of this speed.This speed becomes critical when the frequency of excitation is equal toone of the natural frequencies of the system In forced vibration, the system

fre-is a function of the frequencies These frequencies can also be multiples ofrotor speed excited by frequencies other than the speed frequency such asblade passing frequencies, gear mesh frequencies, and other componentfrequencies Figure 5-20 shows that for forced vibration, the critical fre-quency remains constant at any shaft speed The critical speeds occur at one-half, one, and two times the rotor speed The effect of damping in forcedvibration reduces the amplitude, but it does not affect the frequency at whichthis phenomenon occurs

Typical forced vibration stimuli are as follows:

1 Unbalance This stimulus is caused by material imperfections, ances, etc The mass center of gravity is different from the geometriccase, leading to a centrifugal force acting on the system

toler-2 Asymmetric flexibility The sag in a rotor shaft will cause a periodicexcitation force twice every revolution

3 Shaft misalignment This stimulus occurs when the rotor center lineand the bearing support line are not true Misalignment may also be

Table 5-2 continued

Transient torques Gears with indexing or

positioning errors.

Heavy applied rotor force Drive gear forces.Misaligned 3-or-more rotor-

bearing assembly.

Nonspatial applications Magnetic field:

stationary or rotating Rotating electrical machinery.Axial forces Turbomachine balance piston

Cyclic forces from propeller, or fan Self-excited bearing forces Pneumatic hammer

caused by an external piece such as the driver to a centrifugal pressor Flexible couplings and better alignment techniques are used

com-to reduce the large reaction forces

Periodic loading This type of loading is caused by external forces thatare applied to the rotor by gears, couplings, and fluid pressure, which istransmitted through the blade loading

Self-Excited Instabilities

The self-excited instabilities are characterized by mechanisms, which whirl

at their own critical frequency independent of external stimuli These types

of self-excited vibrations can be destructive, since they induce alternatingstress that leads to fatigue failures in rotating equipment The whirlingmotion, which characterizes this type of instability, generates a tangentialforce normal to the radial deflection of the shaft, and a magnitude propor-tional to that deflection The type of instabilities, which fall under thiscategory, are usually called whirling or whipping At the rotational speedwhere such a force is started, it will overcome the external stabilizing damp-

Table 5-3 Characteristics of Forced and Self-Excited Vibration

Forced or Resonant Vibration Instability VibrationSelf-Excited orFrequency/rpm

relationship NFrational fractionˆ Nrpmor N or Constant and relativelyindependent of rotating speed Amplitude/rpm

relationship Peak in narrow bandsof rpm Blossoming at onset and continueto increase with increasing rpm Influence of damping Additional damping Additional damping may defer to

a higher rpm Will not materially affect amplitude.

Reduce amplitude

No change in rpm at which it occurs System geometry Lack of axial sym Independent of symmetry.

External forces Small deflection to an

axisymmetric system.

Amplitude will self-propogate Vibration frequency At or near shaft

critical or natural frequency

Same.

Avoidance 1 Critical freq Above

running speed 1 Operating rpm below onset.

2 Axisymmetric 2 Eliminates instability.

3 Damping Introduce damping.

ing force and induce a whirling motion of ever-increasing amplitude Figure5-21 shows the onset speed The onset speed does not coincide with anyparticular rotation frequency Also, damping results from a shift of thisfrequency, not in the lowering of the amplitude as in forced vibration.Important examples of such instabilities include hysteretic whirl, dry-frictionwhip, oil whip, aerodynamic whirl, and whirl due to fluid trapped in therotor In a self-excited system, friction or fluid energy dissipations generatethe destabilizing force.

Figure 5-20 Characteristic of forced vibration or resonance in rotating machinery.(Ehrich, F.F., ``Identification and Avoidance of Instabilities and Self-Excited Vibra-tions in Rotating Machinery,'' Adopted from ASME Paper 72-DE-21, GeneralElectric Co., Aircraft Engine Group, Group Engineering Division, May 11, 1972.)

Hysteretic whirl This type of whirl occurs in flexible rotors and resultsfrom shrink fits When a radial deflection is imposed on a shaft, a neutralstrainaxis is induced normal to the direction of flexure From first-order consid-erations, the neutral-stress axis is coincident with the neutral-strain axis, and

a restoring force is developed perpendicular to the neutral-stress axis Therestoring force is then parallel to and opposing the induced force In actu-ality, internal friction exists in the shaft, which causes a phase shift in thestress The result is that the neutral-strain axis and neutral-stress axis aredisplaced so that the resultant force is not parallel to the deflection The

Figure 5-21 Characteristics of instabilities or self-excited vibration in rotating ery (Ehrich, F.F., ``Identification and Avoidance of Instabilities and Self-Excited Vibra-tions in Rotating Machinery,'' Adopted from ASME Paper 72-DE-21, General ElectricCo., Aircraft Engine Group, Group Engineering Division, May 11, 1972.)

machin-tangential force normal to the deflection causes whirl instability As whirlbegins, the centrifugal force increases, causing greater deflectionsÐwhichresult in greater stresses and still greater whirl forces This type of increasingwhirl motion may eventually be destructive as seen in Figure 5-22a.Some initial impulse unbalance is often required to start the whirl motion.Newkirk has suggested that the effect is caused by interfaces of joints in arotor (shrink fits) rather than defects in rotor material This type of whirlphenomenon occurs only at rotational speeds above the first critical Thephenomenon may disappear and then reappear at a higher speed Somesuccess has been achieved in reducing this type of whirl by reducing thenumber of separate parts, restricting the shrink fits, and providing somelockup of assembled elements.

Dry-friction whirl This type of whip is experienced when the surface of

a rotating shaft comes into contact with an unlubricated stationary guide.The effect takes place because of an unlubricated journal, contact in radialclearance of labyrinth seals, and loss of clearance in hydrodynamic bearings.Figure 5-22b shows this phenomenon When contact is made between thesurface and the rotating shaft, the coulomb friction will induce a tangentialforce on the rotor This friction force is approximately proportional to the

Figure 5-22a Hysteretic whirl (Ehrich, F.F., ``Identification and Avoidance ofInstabilities and Self-Excited Vibrations in Rotating Machinery,'' Adopted fromASME Paper 72-DE-21, General Electric Co., Aircraft Engine Group, Group Engin-eering Division, May 11, 1972.)

radial component of the contact force, creating a condition for instability.The whirl direction is counter to the shaft direction.

Oil whirl This instability begins when fluid entrained in the spacebetween the shaft and bearing surfaces begins to circulate with an averagevelocity of one-half of the shaft surface speed Figure 5-23a shows themechanism of oil whirl The pressures developed in the oil are not symmetricabout the rotor Because of viscous losses of the fluid circulating through thesmall clearance, higher pressure exists on the upstream side of the flow than

on the downstream side Again, a tangential force results A whirl motionexists when the tangential force exceeds any inherent damping It has beenshown that the shafting must rotate at approximately twice the critical speedfor whirl motion to occur Thus, the ratio of frequency to rpm is close to 0.5for oil whirl This phenomenon is not restricted to the bearing, but it also canoccur in the seals

The most obvious way to prevent oil whirl is to restrict the maximumrotor speed to less than twice its critical Sometimes oil whip can be reduced

or eliminated by changing the viscosity of the oil or by controlling the oil

Figure 5-22b Dry friction whirl (Ehrich, F.F., ``Identification and Avoidance ofInstabilities and Self-Excited Vibrations in Rotating Machinery,'' Adopted fromASME Paper 72-DE-21, General Electric Co., Aircraft Engine Group, Group Engin-eering Division, May 11, 1972.)

temperature Bearing designs that incorporate grooves or tilting pads arealso effective in inhibiting oil-whirl instability.

Aerodynamic whirl Although the mechanism is not clearly understood,

it has been shown that aerodynamic components, such as compressor wheelsand turbine wheels, can create cross-coupled forces due to the wheel motion.Figure 5-23b is one representation of how such forces may be induced.The acceleration or deceleration of the process fluid imparts a net tangen-tial force on the blading If the clearance between the wheel and housingvaries circumferentially, a variation of the tangential forces on the bladingmay also be expected, resulting in a net destabilizing force as shown inFigure 5-23b The resultant force from the cross-coupling of angular motionand radial forces may destabilize the rotor and cause a whirl motion.The aerodynamic cross-coupling effect has been quantified into equivalentstiffness For instance, in axial-flow machines

Kxyˆ KyxˆD T

Figure 5-23a Oil whirl (Ehrich, F.F., ``Identification and Avoidance of Instabilitiesand Self-Excited Vibrations in Rotating Machinery,'' Adopted from ASME Paper72-DE-21, General Electric Co., Aircraft Engine Group, Group EngineeringDivision, May 11, 1972.)

M0 y

26 4

3 75

3 75

3 75

n

‡ …K ‡ sB†n

XY



26 4

3 75