International Review of Applied Engineering Research. ISSN 2248-9967 Volume 4, Number 3 (2014), pp. 241-250 © Research India Publications http://www.ripublication.com/iraer.htm
A Review of Rotating Machinery Critical Speeds and Modes of Vibrations Mili J. Hota and D.P. Vakharia Sardar Vallabhbhai National Institute of Technology, Surat.
Abstract
The aim of this paper is to present a practical understanding of terminology and behavior based in visualizing how a shaft vibrates, and examining issues that affect vibration. The paper also presents the synchronous critical speeds of rotor-bearing systems. The majority of attention is on the unbalance excitation of natural frequencies with whirling and spinning spinning in same direction. Most rotor dynamic dynamic analyses typically ignore the potential for critical speeds to be created by traversing a backward processional whirl mode. While not commonly recognized, a backward mode can be excited using unbalance as the driving force. One of the major areas of interest in the modern-day condition monitoring of rotating machinery is that of vibration. If a fault develops and goes undetected, then, at best, the problem will not be too serious and can be remedied quickly and cheaply; at worst, it may result in expensive damage and down-time, injury, or even loss of life. By measurement and analysis of the vibration of rotating machinery, it is possible to detect and locate important faults such as mass unbalance, shaft alignment, rub and cracked shafts vibration. This paper helps to learn the basic modes of vibrations and predicting critical speed by practical approach and also concludes some important practical recommendations on rotor-bearing design so that unique critical speed situation may be avoided. Index Terms: Critical speed, Gyroscopic, Gyroscopic, high-speed, natural Frequency, Frequency, resonances, resonances, stiffness, vibration mode.
1. Introduction The unique characteristics of rotating machinery vibration, the terminology and behavior of a machine are been discussed. Like most specialty areas, there are a
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number of excellent texts, but it can be difficult to quickly pull out the practical insight needed. At the other end of the spectrum, there is also a large number of troubleshooting resources that focus on identification of problems and characteristics, but only offer limited insight. Discussion of recent combined experimental and analytical effort raised the possibility of an article that would attempt to provide a deeper insight into some of the basic characteristics of rotating machinery vibration from a less mathematical perspective. Thus in this article several issues that are basic to an understanding of rotating machinery vibration are been discussed: • What are “critical speeds”? • How do critical speeds relate to resonances and natural frequencies? • How do natural frequencies change as the shaft rotational speed changes? • How are shaft rotational natural frequencies different from more familiar natural frequencies and modes in structures? • What effects do bearing characteristics have?
2. Vibration Intuition 2.1 A Brief Review of Structural Vibration As 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 integral role by controlling amplitudes. This education generally starts with the simplest possible system – a rigid mass attached to a spring as shown in Fig. 1[1]. With this simple system, we quantify our intuition about vibrational frequency (heavier objects result in lower frequency, stiffer springs yield higher frequency). After some work, we reach the conclusion that the free vibration frequency is controlled by the square root of the ratio of stiffness to mass as shown in Fig. 1. We could then add a viscous damper parallel with the spring, and provide a sinusoidal force as shown in Fig. 2[3]. By carefully applying a constant amplitude sinusoidal force that slowly increases in frequency and recording the amplitude of the 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 frequency response shown in Fig. 3 can be developed. Assuming we knew the mass, stiffness and damping of our system, this response is also predicted quite well by the standard frequency domain solution to the differential equation of motion for this system shown in Equation 1[2].
F 0 Amplitude =
k 2
⎛ mω 2 ⎞ ⎛ cω ⎞ ⎜⎜1 − ⎟⎟ + ⎜ ⎟ k k ⎝ ⎠ ⎝ ⎠
2
(1)
There are several noteworthy points about these frequency responses. The first is that at low excitation frequencies, the response amplitude is roughly constant and greater than zero. The amplitude is governed by the ratio of the applied force to the
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spring stiffnes [6]. The econd is t at the resp nse increases to a pe k, then ra idly dec eases in the low and medi m dampi g cases. This pea frequenc is approximately the dampe natural fr quency, ( ore techni ally corre t, it is the eak res onse freq ency, whi h moves down in requency rom the amped natural fre uency as d amping in reases). T e system is said to b “in reson nce” whe the exc tation freq ency matc es the da ped natura frequency. Very larg amplitudes are possible when the excitation freque cy is clos to this fr quency. T e amplitu e is controlled by the magnitude of the d amping (more dampi g reduces he amplitu es). The high dam ing case h s no real peak, and is said to be ‘overdamp d.’ Finally, the am litude continues to de rease for a l higher frequencies. These chara teristics will be contrasted wit the response of a r tating system to unb lance exci ation in a later sec ion. Moving from the simple sin le mass system to multimass sys ems, the b sics do ot change. Natural frequencies ar e still prim rily related to mass and stiffness, with so e changes due to d mping. E citation fr equency e ual to a amped natural fre uency is a resonance. Excitation near a re onance ca result in arge amplitude res onses. Res onse amplitudes are controlled b damping. With enou h damping, the res onse peak can be co pletely eli inated. The biggest c ange is th t there are now multiple natur l frequen ies and that each natural frequ ncy has a corresponding uni ue “mode-shape” w th differe t parts of the structure vibrating at diff rent am litudes and differing phases relative to one a other.
ig. 1: Sim le spring- ass- syste with a d without amper.
Fig. : Frequenc response f spring mass damp r system to constant a plitude force.
Fi . 2: Free r sponse of imple spri g- Fig. 4: Basic ma hine mode cross section. ass- syste
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2.2 A Simple otating achine The rotating achinery equivalent to the sin le spring- ass damper system is a lu ped mass n a massless, elastic shaft. Thi model, h storically eferred to as a ‘Je fcott’ or ‘Laval’ model, is a singl degree of freedom s stem that i generally used to i troduce rotor dynami characteristics. This model, shown in cros -section in Fig. 4[5 , consists f a rigid c ntral disk, a shaft (wi h stiffness and mass) and two ri idly mo nted beari gs. No rotating ynamics Sup pose that o ur simple achine is ot spinnin , that the bearings ha e essentially no da ping, and that the bea ings have qual radial stiffness in the vertical and horiz ntal directions (all typical characteristics of ball bearings). Let us also su pose that here are three versi ns of this achine, o e each wit soft, inter mediate and stiff bear ngs. Thr ough eith r analysis or a m dal test, we woul find a set of natural fre uencies/m des. At each frequency, the m tion is planar (just l ke the pinned pin ed beam). This behav or is what e would expect from a static str cture. Fig. 5[3] sho s the first three mod shapes an frequencies for the t ree bearin stiffnesse . As wit the bea , the thic line sho s the sha t centerline shape at the maxi um dis lacement. s it vibra es, it mov s from thi position t the same location o the opposite side of the displaced cent rline, and back. Not that the atio of be ring stif ness to sha t stiffness as a signif cant impact on the mode-shapes. For the sof and intermediate bearings, th shaft doe not bend very muc in the lo er two m des. Th s, these ar generally referred to as “rigid r otor” modes. As the earing stif ness inc eases (or a shaft stiff ness decreases), the a ount of shaft bendin increases. One interesting feature of the ode shape is how th central disk moves. I the first ode, the disk transl tes without rocking. n the seco d mode, it rocks wit out translation. This general c aracteristi repeats a the frequ ncy increases. If we moved the disk off-center, we would fin that the otion is a mix of tra slation and rocking. This cha acteristic ill give rise to some interesting ehavior o ce the shaf t starts rotating. If e repeate the constant amplit de excitation frequency sweep experiment we wo ld get ver similar b havior as ith the spring-mass- amper sys em plot s own pre iously. Th re would be a spring- ontrolled eflection at low frequ ncies, a pe k in am litude, and adecay in mplitude ith further ncreases i frequency.
F g. 5: Mode shapes ver sus bearing stiffness, shaft not r otating.
Fig. 6: Shaft rotat ng at 10 rp m,1st mod s ape and fr quencies i rpm.
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Fig. : Whirl se se.
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Fig. 8: Effect of perating s eed on 1st odes.
3. Rotating Dynami s – Cylindrical M des Sin e rotating machinery has to rotate to do useful work, l t’s consid r what hap pens to t e first mo e of our r tor once it is spinnin . Again, e will hav three diff rent ver ions with ncreasing earing sti fness, and we will as ume our support bea ings have equal sti fness in all radial directions. Let’s repeat o r analysis/modal test with the shaft spinn ng at 10 r m, and look at the frequency and mode sha e of the lowest nat ral freque cy. Fig. 6 elow sho s the frequencies and mode shapes or the lowest mo e of the t ree machines. Note hat the sh pe of the motion ha s changed. The fre uencies, th ugh, are q ite close t the nonrotating first ode. As in the nonrot ting cas , the bearing stiffnes to shaft stiffness rat o has a st ong impact on the modesha e. Again, the case with almost o shaft bending is referred to as a rigid ode. These modes look very much like the non r tation mod es, but th y now in olve circular motio rather tha planar mo ion. To visualize how the otor is mo ing, first i agine swinging a ju p rope around. The rope trac s the outl ne of a b lging cylinder. Thus, this mod e is somet mes referred to as a ‘cylindri al’ mode. Viewed fr m the fro t, the rop appears to be bouncing up nd down. Thus, this mode is lso sometimes called a ‘bounc ’ or ‘tra slatory’ mode. The whirling mot on of the r tor (the ‘j mprope’ motion) can e in the same direc ion as the haft’s rotation or in the opposite irection. This gives rise to the labels “for ard whirl and “back ward whirl..” Fig. 7[8] shows rot r cross sec ions over the cour e of time for both s nchronous forward a d synchronous back ard whirl. Note t at for for ard whirl, a point on the surface of the rot r moves i the sa e direction as the whir l. Thus, for s nchronous forward w irl (unbalance excitation, for exa ple), a point at the outside of the rotor r mains to the outside of the whi l orbit[9]. With back ard whirl, on the ther hand, a point at the surface of the rotor moves in the opposite direction as the whirl to the inside of he whirl or bit during t e whirl. To see ho a wider range of sha t speeds c anges the situation, w could per orm the analysis/m dal test with a range of shaft sp eds from non spinnin to high s eed. We could then follow the forward a d backwar frequencies associat d with the first mo e. Fig. 8 lots the f rward (red line) and backward ( black dashed line) natural
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frequencies over a wide shaft speed range. This plot is often referred to as a “Campbell Diagram.” From this figure, we can see that the frequencies of this cylindrical mode do not change very much over the speed range. The backward whirl mode drops slightly, and the forward whirl mode increases slightly (most noticeably in the high stiffness case). The reason for this change will be explored in the next section.
4. Rotating Dynamics – Conical Mode Now that we have explored the cylindrical mode, let’s look at the second set of modes. Fig. 9 shows the next frequencies and mode-shapes for the three machines. The frequencies are close to the nonrotating modes where the disk was rocking without translating. The modes look a lot like the nonrotating modes, but again involve circular motion rather than planar motion. To visualize how the rotor is moving, imagine holding a rod stationary in the center, and moving it so that the ends trace out two circles. The rod traces the outline of two bulging cones pointed at the center of the rod. Thus, this mode is sometimes referred to as a ‘conical’ mode. Viewed from the side, the rod appears to be rocking up and down around the center, with the left side being out-of-phase from that on the right. Thus, this mode is also sometimes called a ‘rock’ mode or a ‘pitch’ mode. As with the first mode and the nonrotating modes, the low bearing stiffness mode is generally referred to as a rigid mode, and a high bearing stiffness pulls in the rotor ends. As with the cylindrical mode, the whirl can be in the same direction as the rotor’s spin (“forward whirl”), or the opposite direction (“backward whirl”). To see the effects of changing shaft speeds, we could again perform the analysis/modal test from non spinning to a high spin speed and follow the two frequencies associated with the conical mode. Fig. 10 plots the forward (red line) and backward (black dashed line) natural frequencies over a wide speed range. From this figure, we can see that the frequencies of the conical modes do change over the speed range. The backward mode drops in frequency, while the forward mode increases. The explanation for this surprising behavior is a gyroscopic effect that occurs whenever the mode shape has an angular (conical/rocking) component. First consider forward whirl. As shaft speed increases, the gyroscopic effects essentially act like an increasingly stiff spring on the central disk for the rocking motion. Increasing stiffness acts to increase the natural frequency. For backward whirl, the effect is reversed. Increasing rotor spin speed acts to reduce the effective stiffness, thus reducing the natural frequency (as a side note, the gyroscopic terms are generally written as a skewsymmetric matrix added to the damping matrix – the net result, though, is a stiffening/ softening effect). In the case of the cylindrical modes, very little effect of the gyroscopic 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 bearing case, which has a very cylindrical motion, no effect was observed, while for the stiff bearing case, which has a bulging cylinder (and thus conical type motion near the bearings), a slight effect was noted.
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Fig. 9: Shaft otating at 0 rpm,2nd mode shapes nd freque cies in rpm
Fi . 10: Effec of oppera ing speed n 2nd natural frequ ncies
Fig. 11: Comparison of different isk properties, center dis configuration
Fi . 11: Com arison of ifferent disk properties, overhung configuration
2.3 Exploring Gyroscopic and Mas Effects No that we have seen how gyros copic effects act to change the rotating natural fre uency whenever there is motion with some conical co ponent, l t’s look at two sets of single d isk rotors. In each case, there will be a nominal rotor, a eavy disk r otor, and a smaller iameter, l nger disk r otor. The eavy disk iffers fro the nominal in that a fictitiou mass equal to the dis mass is attached (i.e., mass inc eases, but ass mo ent of in rtia is unchanged). T e smaller, longer disk is the same weight but sm ller in dia eter and greater in ength. This smaller isk has re uced the ass mo ent of in rtia about the spin a is (‘polar’ moment Ip) by a fact or of 0.53, and red ces the ass mome t of inertia about the disk dia eter (Id) by a factor of 0.6 [11]. For the first case, let’ use a sym etric, cen er disk rot r again. Fig. 12 show the three models, nd the thr ee sets of atural freq uencies ve sus speed. Comparin the no inal model to the two modified v rsions, note that: • The in reased ma s lowers t e first mode frequen ies (mass is at a poi t of large whirling motion). • The increased mass leaves th second m de unchanged (increased mass is at a point o little whirling motion).
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•
The reduced mass moment of inertia version does not change the first mode (disk center of gravity has very little conical motion). • The reduced mass moment of inertia increases the frequency of the second mode, and decreases the strength of the gyroscopic effect (disk center of gravity has substantial conical motion). For the second case, let’s move the disk to the end, and move the bearing inboard to result in an overhung rotor with the same mass and overall length. Fig. 14 shows the three models and the three sets of natural frequencies versus speed. Comparing the nominal model to the two modified versions, the important things to note are: The increased mass lowers the first mode frequencies and very slightly lowers the second mode frequencies. The reduced mass moment of inertia version increases the frequency of both the first and second modes, and decreases the strength of the gyroscopic effect. If we looked at the mode shapes and these plots, we would again see that the reasons are the same as for the center disk rotor. Changes in affect the natural frequency of that mode but have little effect if it is at a node. Changes to mass moment of inertia at a location of large whirl orbit, on the other hand, have little effect. Changes to mass moment of inertia at a node with large conical motions have a strong effect on the corresponding mode. Although not entirely obvious from the plots presented, changes in the ratio of polar mass moment of inertia to diametric mass moment of inertia change the strength of the gyroscopic effect. Indeed, for a very thin disk (a large ratio), the forward conical mode increases in speed so rapidly that the frequency will always be greater than the running speed. Indeed, there will be no conical critical speed as defined below. 2.4 Critical Speeds With some insight into rotating machinery modes, we can move on to “critical speeds.” The American Petroleum Institute(API), in API publication 684 (First Edition, 1996), defines critical speeds and resonances as follows: • Critical Speed – A shaft rotational speed that corresponds to the peak of a noncritically damped (amplification factor > 2.5) rotor system resonance frequency. The frequency location of the critical speed is defined as the frequency of the peak vibration response as defined by a Bodé plot (for unbalance excitation). • Resonance – The manner in which a rotor vibrates when the frequency of a harmonic (periodic) forcing function coincides with a natural frequency of the rotor system. Thus, whenever the rotor speed passes through a speed where a rotor with the appropriate unbalance distribution excites a corresponding damped natural frequency, and the output of a properly placed sensor displays a distinct peak in response versus speed, the machine has passed through a critical speed. Critical speeds could also be referred to as “peak response” speeds.
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Numerically, these are distinct from critical speeds as defined by the API spe ification. As a critic l speed ex mple, we ill use the medium s iffness, ce ter disk m del, and add an un alance dist ibution that excites th first three modes. W will also add a sm ll amount of dampi g at the earings. ig. 13 sh ws the resulting ve tical dis lacement r esponse due to the un alance for es at the l ft bearing s a functi n of spe d. The da ped natur l frequenc versus sp ed plot (C mpbell Di gram) is d awn bel w for refer ence.
Fig. 13: Co parison be ween natu al Fig. 4: Phase r lationship f center of frequenci s and criti al speeds. orbit v rsus center of mass th ough critical peed. 3.1 Character stics of U balance E citation A c areful com arison of the previou set of unbalance response plot i Figures 1 for the center disk machine, nd the fre uency response plot f or the spri g-mass-da per structure in Fig. 3 reveals wo signifi ant differe ces. At lo frequenci s, the structural plo shows a response equal to the static respons , whereas the unbalan e response plot star ts out with no response. Likewi e, at higher frequencies, the str ctural response dec ys, while the unbalan e response tends to a onstant value at highe speeds. These two differences are the re ult of the requency ependency of the constant am litude sinusoidal force versus unbalance excitation. The str ctural excitation was assumed to be a const nt force at all frequencies, • while unbalance e citation has a speed-squared. At zero rpm, there is no orce from u balance excitation, w ich explains the first d fference n ted. • The se ond diffe ence – th t the unb lance resp nse amplitude goes to a consta t value above the criti al speed – as a more nteresting xplanation. We can se what occurs by referr ing to Fig. 14. This Fi . plots the relative an ular relationship between the nbalance location an rotor response as rotor speed p sses thr ugh a criti al speed. elow the ritical spe d, the unb lance acts to pull the disk out into an or it that gro s increasingly large with speed . At the critical speed , the rot r response lags the unbalance by approximately 90°. However after pa sing
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through the critical speed, the phase between the unbalance force and the response direction has changed by 180°. As a result, the disk now rotates around the mass center of the disk/unbalance. Once the disk achieves this state, further increases in speed do not change the amplitude until the effects of the next mode are observed.
5. Conclusions It was shown that cylindrical rotor modes are not influenced by gyroscopic effects and remain at a fairly constant frequency versus rotor speed. Conversely, conical rotor modes are indeed influenced and caused to split into forward and backward whirl components that respectively increase and decrease in frequency with increased rotor speed.
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Michael I. Friswell et al. Dynamics of rotating machines, 1 edition 2010. {79} AKPOBI, J.A.; OVUWORIE, “G.C. Computer–Aided Design of the Critical Speed of Shafts” J. Appl. Sci. Environ. Manage. December, 2008 Vol. 12(4) 79 – 86 Den Hartog, J. P. Mechanical Vibrations, 4th ed. New York:McGraw-Hill,1996. Eshleman, R. Torsional vibration of machine systems. Proceedings of the 6th Turbo machinery Symposium, Texas A&M University, 2010, p.13. Farouk Hamdoon O, “Application of finite element package for modeling rotating machinery vibration”, Engg and Tech Journal, vol. 27, No 12, 2009. Henrich Sprysl, Gunter Ebi, “Bearing stiffness determination through vibration analysis of shaft line of hydro power plant”, International Journal of hydro power & dams, 1998, pp. 437-447 John M Vance, Brain T Murphy, “Critical speeds of turbo machinery computer prediction vs. Experimental measurements”, ASME Journal of Engg for power, 1981, pp.141-145. J.S. Rao.” Rotor Dynamics”. Wiley Eastern Ltd., 1983. Lund, J. W. “Destabilization of Rotors from Friction in Internal Joints with Microslip”. International Conference in Rotordynamics, JSME, 1986, pp. 487–491. Mathuria P.H, Sainagar A, “Lateral natural frequency of a shaft rotor system by the transfer matrix method” Citeseerxa scientific literature digital library, Pennsylvania state University. 2010 Michael I. Friswell et al. Dynamics of rotating machines, 1 edition 2010. {79} P. Srinivasan. “Mechanica Vibration and Analysis”. Tata McGraw-Hill Co., 1982. Steidel, R. F., Jr. An Introduction to Mechanical Vibrations. New York: Wiley, 1989. Thomson, W. T. Theory of Vibration with Applications, 4th ed. Englewood Cliffs, NJ: Prentice Hall, 1993. Vance, J. M. and French, R. S. “Measurement of torsional vibration in rotating machinery. Journal of Mechanisms”, Transmissions, and Automation in Design 108:565–577 (1986).