A Practical Review Of Rotating Machinery Critical Speeds .
A Practical Review of RotatingMachinery Critical Speeds and ModesErik Swanson, Xdot-Consulting, Chapel Hill, North CarolinaChris D. Powell, Structural Technology Corporation, Zoar, OhioSorin Weissman, Alfa Wasserman, Inc., West Caldwell, New JerseyThe goal of this article is to present a practical understanding of terminology and behavior based in visualizing how ashaft vibrates, and examining issues that affect vibration. Itis hoped that this presentation will help the nonspecialist better understand what is going on in the machinery, and that thespecialist may gain a different view and/or some new examples.For the engineer unfamiliar with some of the unique characteristics of rotating machinery vibration, the terminology andbehavior of a machine can appear to be overwhelming. Likemost specialty areas, there are a number of excellent texts, butit can be difficult to quickly pull out the practical insightneeded. At the other end of the spectrum, there is also a largenumber of troubleshooting resources that focus on identification of problems and characteristics, but only offer limitedinsight. Discussion of a recent combined experimental andanalytical effort raised the possibility of an article that wouldattempt to provide a deeper insight into some of the basic characteristics of rotating machinery vibration from a less mathematical perspective.Thus, we have set out to discuss several issues that are basic to an understanding of rotating machinery vibration:y What are “critical speeds”?y How do critical speeds relate to resonances and natural frequencies?y How do natural frequencies change as the shaft rotationalspeed changes?y How are shaft rotational natural frequencies different frommore familiar natural frequencies and modes in structures?y What effects do bearing characteristics have?Vibration IntuitionAs a part of exploring the world as children, everyone is familiar with the idea that banging on a structure will make itbounce back and vibrate. Some items vibrate more easily thanothers (a metal rod versus a wooden stick, for example). We alsohave intuition that it is easier to get things to vibrate or moveback and forth at certain frequenies. For example, we tend tolearn that a swing with long ropes moves back and forth moreslowly than a swing with short ropes. ‘Pumping’ the swing ata rate that matches the rate at which it wants to naturally moveback and forth will get you swinging much higher than ratesthat are faster or slower than the swing’s natural frequency.Many of us have also had some experience with stringed instruments. From this experience, we develop some idea thatheavy objects (thick strings) tend to vibrate at a lower frequencythan light objects (thin strings). We learn that increasing stiffness (tightening the string) raises the frequency of its vibration.Finally, we also learn that decreasing a major dimension(shorter string) results in higher frequency vibration.A Brief Review of Structural VibrationAs engineers, we learn that vibration characteristics are determined by a structure’s mass and stiffness values, with damping (ability to dissipate vibrational energy) playing an integralrole by controlling amplitudes. This education generally startswith the simplest possible system – a rigid mass attached to aspring as shown in Figure 1.With this simple system, we quantify our intuition about10vibrational frequency (heavier objects result in lower frequency, stiffer springs yield higher frequency). After somework, we reach the conclusion that the free vibration frequencyis controlled by the square root of the ratio of stiffness to mass.StiffnessMassNatural Frequency (1)Experimentally, we could (in principle) build a single degreeof freedom system consisting of a rigid block sitting on a spring.Were we to push the block down and release it, we would findthat the displacement versus time is a sinusoidal function at asingle frequency, which is equal to the natural frequency as predicted by Equation 1 and shown in Figure 2.We could then add a viscous damper parallel with the spring,and provide a sinusoidal force as shown in Figure 3. By carefully applying a constant amplitude sinusoidal force thatslowly increases in frequency and recording the amplitude ofthe motion, we could then generate the classic normalized frequency responses of a spring-mass-damper system. By repeating the test with a variety of dampers, the classic frequencyresponse shown in Figure 4 can be developed. Assuming weknew the mass, stiffness and damping of our system, this response is also predicted quite well by the standard frequencydomain solution to the differential equation of motion for thissystem shown in Equation 2.Amplitude F0k2(2)2Êmw 2 ˆÊ cw ˆÁ 1 - k ÁË k ËThere are several noteworthy points about these frequencyresponses. The first is that at low excitation frequencies, theresponse amplitude is roughly constant and greater than zero.The amplitude is governed by the ratio of the applied force tothe spring stiffness. The second is that the response increasesto a peak, then rapidly decreases in the low and medium damping cases.This peak frequency is approximately the damped naturalfrequency, (more technically correct, it is the peak responsefrequency, which moves down in frequency from the dampednatural frequency as damping increases). The system is said tobe “in resonance” when the excitation frequency matches thedamped natural frequency. Very large amplitudes are possiblewhen the excitation frequency is close to this frequency. Theamplitude is controlled by the magnitude of the damping (moredamping reduces the amplitudes). The high damping case hasno real peak, and is said to be ‘overdamped.’ Finally, the amplitude continues to decrease for all higher frequencies. Thesecharacteristics will be contrasted with the response of a rotating system to unbalance excitation in a later section.Moving from the simple single mass system to multimasssystems, the basics do not change. Natural frequencies are stillprimarily related to mass and stiffness, with some changes dueto damping. Excitation frequency equal to a damped naturalfrequency is a resonance. Excitation near a resonance can result in large amplitude responses. Response amplitudes arecontrolled by damping. With enough damping, the responsepeak can be completely eliminated. The biggest change is thatthere are now multiple natural frequencies and that each natu-SOUND AND VIBRATION/MAY 2005
Displacement (Y)Amplitude ( Y )Mass (M)Spring Stiffness (K)Low DampingMedium DampingFigure 1. Simple spring-mass system.High DampingAmplitude ( Y )Exitation Frequency (w)Displacement (Y)Figure 4. Frequency response of spring-mass-damper system to constantamplitude force.Time (T)First Mode(i 1)Second Mode(i 2)Third Mode(i 3)Figure 5. First three mode shapes of pinned-pinned beam.Figure 2. Free response of simple spring-mass system.660.4 mmIdeal Force F sin (wt)304.8 mm Diameter50.8 mm ThickCentered, Rigid DiskY50.8 mmMIsotropicBearingIdeal Viscous Damper (C)KFigure 3. Simple spring-mass-damper system with forcing.ral frequency has a corresponding unique “mode-shape” withdifferent parts of the structure vibrating at different amplitudesand differing phases relative to one another.Real structures can be viewed as a series of finer and finerlumped mass approximations that approach a continuous massdistribution. The continuous structure has an infinite numberof natural frequencies, each with its own characteristic vibration shape (mode).As an example, consider a simple beam structure supportedby pin joints at each end. This structure is simple enough thata closed-form solution to the natural frequencies and modeshapes is possible. Equation 3 presents the resulting equationfor natural frequencies, and the first three mode shapes areshown in Figure 5.fi 2(modulus of elasticity)(area moment of inertia)(ip )(3)mass per unit length2p (Length)2In essence, this equation is still just the square root of the ratio of stiffness to mass. The mode shapes shown in Figure 5and throughout the article are the shape of the beam at the position of maximum displacement for a given (damped) naturalfrequency. The dashed lines show the positions of the beamduring the vibration cycle. These intermediate positions are notSOUND AND VIBRATION/MAY 2005IsotropicBearingFigure 6. Basic machine model cross section.shown in the remainder of the figures.All of this background and intuition carries over into therotating machinery world – with a few important differences,especially once the rotor starts to spin.A Simple Rotating MachineThe rotating machinery equivalent to the single spring-massdamper system is a lumped mass on a massless, elastic shaft.This model, historically referred to as a ‘Jeffcott’ or ‘Laval’model, is a single degree of freedom system that is generallyused to introduce rotor dynamic characteristics. For the purposes of this article, a slightly more complex multi-degree offreedom model corresponding to a physical rotor will be used.This model, shown in cross-section in Figure 6, consists of arigid central disk, a shaft (with stiffness and mass) and tworigidly mounted bearings. To make the examples more concrete,dimensions shown were selected. Physically, this is somewhatsimilar to a center-hung fan, pump or turbine.Nonrotating DynamicsSuppose that our simple machine is not spinning, that thebearings have essentially no damping, and that the bearingshave equal radial stiffness in the vertical and horizontal directions (all typical characteristics of ball bearings). Let us alsosuppose that there are three versions of this machine, one eachwith soft, intermediate and stiff bearings.Through either analysis or a modal test, we would find a setof natural frequencies/modes. At each frequency, the motionis planar (just like the pinned-pinned beam). This behavior iswhat we would expect from a static structure. Figure 7 shows11
WhirlWhirlMode 11345 rpm4168 rpmShaft SpinT 06458 rpmShaft SpinT 0T 1T 6Mode 23755 rpm13,441 rpm26,781 rpmT 2T 5T 4Soft Bearings33,999 rpmIntermediate Bearings89,416 rpmT 3T 3Forward Whirl(Shaft spin directionsame as whirl direction)Stiff BearingsT 5T 2Mode 327,485 rpmT 6T 1T 4Backward Whirl(Shaft spin directionopposite of whirl direction)Figure 7. Mode shapes versus bearing stiffness, shaft not rotating.Figure 9. Whirl sense.Shaft Shape7Stiff: 1st Mode Forward WhirlPath/Orbit ShapeStiff: 1st Mode Backward Whirl1345 rpm4168 rpm6458 rpmSoft BearingsIntermediate Stiffness BearingsStiff BearingsFigure 8. Shaft rotating at 10 rpm, 1st mode shapes and frequencies inrpm.the first three mode shapes and frequencies for the three bearing stiffnesses. As with the beam, the thick line shows the shaftcenterline shape at the maximum displacement. As it vibrates,it moves from this position to the same location on the opposite side of the undisplaced centerline, and back.Note that the ratio of bearing stiffness to shaft stiffness has asignificant impact on the mode-shapes. For the soft and intermediate bearings, the shaft does not bend very much in thelower two modes. Thus, these are generally referred to as “rigidrotor” modes. As the bearing stiffness increases (or as shaftstiffness decreases), the amount of shaft bending increases.One interesting feature of the mode shapes is how the central disk moves. In the first mode, the disk translates withoutrocking. In the second mode, it rocks without translation. Thisgeneral characteristic repeats as the frequency increases. If wemoved the disk off-center, we would find that the motion is amix of translation and rocking. This characteristic will give riseto some interesting behavior once the shaft starts rotating.If we repeated the constant amplitude excitation frequencysweep experiment, we would get very similar behavior as withthe spring-mass-damper system plot shown previously. Therewould be a spring-controlled deflection at low frequencies, apeak in amplitude, and a decay in amplitude with further increases in frequency.Rotating Dynamics – Cylindrical ModesSince rotating machinery has to rotate to do useful work, let’sconsider what happens to the first mode of our rotor once it isspinning. Again, we will have three different versions with increasing bearing stiffness, and we will assume our support bearings have equal stiffness in all radial directions. Let’s repeatour analysis/modal test with the shaft spinning at 10 rpm, andlook at the frequency and mode shape of the lowest natural frequency. Figure 8 below shows the frequencies and mode shapesfor the lowest mode of the three machines.Note that the shape of the motion has changed. The frequencies, though, are quite close to the nonrotating first mode. Asin the nonrotating case, the bearing stiffness to shaft stiffnessratio has a strong impact on the mode-shape. Again, the casewith almost no shaft bending is referred to as a rigid mode.These modes look very much like the nonrotation modes, butthey now involve circular motion rather than planar motion.To visualize how the rotor is moving, first imagine swinging ajumprope around. The rope traces the outline of a bulging cylinder. Thus, this mode is sometimes referred to as a ‘cylindri-12Natural Frequency (kcpm)65Int.: 1st Mode Forward Whirl4Int. 1st Mode Backward Whirl(hidden by forward)32Soft: 1st Mode Forward WhirlSoft: 1st Mode Backward Whirl(hidden by forward)1005101520253035Operating Speed (krpm)404550Figure 10. Effect of operating speed on 1st modes.cal’ mode. Viewed from the front, the rope appears to be bouncing up and down. Thus, this mode is also sometimes called a‘bounce’ or ‘translatory’ mode.Unlike most jump-ropes, however, the rotor is also rotating.The whirling motion of the rotor (the ‘jumprope’ motion) canbe in the same direction as the shaft’s rotation or in the opposite direction. This gives rise to the labels “forward whirl” and“backward whirl.” Figure 9 shows rotor cross sections over thecourse of time for both synchronous forward and synchronousbackward whirl. Note that for forward whirl, a point on thesurface of the rotor moves in the same direction as the whirl.Thus, for synchronous forward whirl (unbalance excitation, forexample), a point at the outside of the rotor remains to the outside of the whirl orbit. With backward whirl, on the other hand,a point at the surface of the rotor moves in the opposite direction as the whirl to the inside of the whirl orbit during thewhirl.To see how a wider range of shaft speeds changes the situation, we could perform the analysis/modal test with a range ofshaft speeds from nonspinning to high speed. We could thenfollow the forward and backward frequencies associated withthe first mode. Figure 10 plots the forward (red line) and backward (black dashed line) natural frequencies over a wide shaftspeed range. This plot is often referred to as a “Campbell Diagram.” From this figure, we can see that the frequencies of thiscylindrical mode do not change very much over the speedrange. The backward whirl mode drops slightly, and the forward whirl mode increases slightly (most noticeably in the highstiffness case). The reason for this change will be explored inthe next section.Rotating Dynamics – Conical ModeNow that we have explored the cylindrical mode, let’s lookat the second set of modes. Figure 11 shows the next frequencies and mode-shapes for the three machines. The frequenciesare close to the nonrotating modes where the disk was rockingwithout translating. The modes look a lot like the nonrotatingmodes, but again involve circular motion rather than planarmotion.To visualize how the rotor is moving, imagine holding a rodSOUND AND VIBRATION/MAY 2005
Shaft ShapeAdded MassStiff BearingsFigure 11. Shaft rotating at 10 rpm, 2nd mode shapes and frequencies in rpm.25201510500204060Operating Speed, krpm3025201510500204060Operating Speed, krpmHeavier DiskMass 2x NominalNominal50Natural Frequency, kcpm26,773 rpm (F)26,789 rpm (B)13,437 rpm (F)13,445 rpm (B)Intermediate Bearings30Natural Frequency, kcpm3752 rpm (F)3758 rpm (B)Soft BearingsNatural Frequency, kcpmPath/Orbit Shape201510500204060Operating Speed, krpmFigure 13. Comparison of different disk properties, center disk configuration.Stiff: 2F30Stiff: 2BAdded Mass20Int: 2F15Int.: 2B50Soft: 2B051015202530Operating Speed, krpm35404550Figure 12. Effect of operating speed on 2nd natural frequencies.stationary in the center, and moving it so that the ends traceout two circles. The rod traces the outline of two bulging conespointed at the center of the rod. Thus, this mode is sometimesreferred to as a ‘conical’ mode. Viewed from the side, the rodappears to be rocking up and down around the center, with theleft side being out-of-phase from that on the right. Thus, thismode is also sometimes called a ‘rock’ mode or a ‘pitch’ mode.As with the first mode and the nonrotating modes, the lowbearing stiffness mode is generally referred to as a rigid mode,and a high bearing stiffness pulls in the rotor ends. As with thecylindrical mode, the whirl can be in the same direction as therotor’s spin (“forward whirl”), or the opposite direction (“backward whirl”).To see the effects of changing shaft speeds, we could againperform the analysis/modal test from nonspinning to a highspin speed and follow the two frequencies associated with theconical mode. Figure 12 plots the forward (red line) and backward (black dashed line) natural frequencies over a wide speedrange. From this figure, we can see that the frequencies of theconical modes do change over the speed range. The backwardmode drops in frequency, while the forward mode increases.The explanation for this surprising behavior is a gyroscopiceffect that occurs whenever the mode shape has an angular(conical/rocking) component. First consider forward whirl. Asshaft speed increases, the gyroscopic effects essentially act likean increasingly stiff spring on the central disk for the rockingmotion. Increasing stiffness acts to increase the natural frequency. For backward whirl, the effect is reversed. Increasingrotor spin speed acts to reduce the effective stiffness, thus reducing the natural frequency (as a side note, the gyroscopicterms are generally written as a skew-symmetric matrix addedto the damping matrix – the net result, though, is a stiffening/softening effect).In the case of the cylindrical modes, very little effect of thegyroscopic terms was noted, since the center disk was whirling without any conical motion. Without the conical motion,the gyroscopic effects do not appear. Thus, for the soft bearingcase, which has a very cylindrical motion, no effect was observed, while for the stiff bearing case, which has a bulgingcylinder (and thus conical type motion near the bearings), aslight effect was noted.Exploring Gyroscopic and Mass EffectsNow that we have seen how gyroscopic effects act to changeSOUND AND VIBRATION/MAY 200516141210864200204060Operating Speed, krpmNominal161412108642Natural Frequency, kcpmSoft: 2FNatural Frequency, kcpm10Natural Frequency, kcpmNatural Frequency, kcpm402525Diametral Inertia 0.65 IdPolar Inertia 0.53 IpIp/Id Ratio 0.79M Nominal45353000204060Operating Speed, krpmHeavier DiskMass 2xNominal16141210864200204060Operating Speed, krpmDiametral Inertia 0.65 IdPolar Inertia 0.53 IpIp/Id Ratio 0.79M NominalFigure 14. Comparison of different disk properties, overhung configuration.the rotating natural frequency whenever there is motion withsome conical component, let’s look at two sets of single diskrotors. In each case, there will be a nominal rotor, a heavy diskrotor, and a smaller diameter, longer disk rotor. The heavy diskdiffers from the nominal in that a fictitious mass equal to thedisk mass is attached (i.e., mass increases, but mass momentof inertia is unchanged) The smaller, longer disk is the sameweight, but smaller in diameter and greater in length. Thissmaller disk has reduced the mass moment of inertia about thespin axis (‘polar’ moment Ip
rotating machinery world – with a few important differences, especially once the rotor starts to spin. A Simple Rotating Machine . our analysis/modal test with the shaft spinning at 10 rpm, and look at the frequency and mode shape of the lowest natural fre-SOUND AND VIBRATION/MAY 2005 8.
Balancing involves redistributing the mass which may be carried out by addition or removal of mass from various machine members Balancing of rotating masses can be of 1. Balancing of a single rotating mass by a single mass rotating in the same plane. 2. Balancing of a single rotating mass by two masses rotating in different planes. 3.
rotating damping matrix is able to predict the decay rate of the first forward mode. Keywords Rotating machinery Rotating damping Experimental modal analysis Damping matrix estimation Decay rate 16.1 Introduction High speed applications are trending in industry. These applications result in smaller component dimensions and lead to
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The rotating bending fatigue testing machine was the first machine used to generate fatigue data in greater effect. The most widely used fatigue-testing device is the R.R Moore high-speed rotating beam machine. This machine subjects the specimen to pure bending (no transverse shear). The R.R. Moore, recognized as the standard for rotating beam .
