Crack Detection In A Rotor Dynamic System By Vibration .

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Philip Varney1Graduate Research Assistante-mail: pvarney3@gatech.eduItzhak GreenProfessore-mail: itzhak.green@me.gatech.eduWoodruff School of Mechanical Engineering,Georgia Institute of Technology,Atlanta, GA, 303321Crack Detection in a RotorDynamic System by VibrationMonitoring—Part II: ExtendedAnalysis and ExperimentalResultsAn increase in the power-to-weight ratio demand on rotordynamic systems causesincreased susceptibility to transverse fatigue cracking of the shaft. The ability to detectcracks at an early stage of progression is imperative for minimizing off-line repair timeand cost. The vibration monitoring system initially proposed in Part I is employed herein,using the 2X harmonic response component of the rotor tilt as a signature indicating atransverse shaft crack. In addition, the analytic work presented in Part I is expanded toinclude a new notch crack model to better approximate experimental results. To effectively capture the 2X response, the crack model must include the local nature of thecrack, the depth of the crack, and the stiffness asymmetry inducing the gravity-forced 2Xharmonic response. The transfer matrix technique is well suited to incorporate thesecrack attributes due to its modular nature. Two transfer matrix models are proposed topredict the 2X harmonic response. The first model applies local crack flexibility coefficients determined using the strain energy release rate, while the second incorporates thecrack as a rectangular notch to emulate a manufactured crack used in the experiments.Analytic results are compared to experimental measurement of the rotor tilt gleaned froman overhung rotor test rig originally designed to monitor seal face dynamics. The test rigis discussed, and experimental angular response orbits and 2X harmonic amplitudes ofthe rotor tilt are provided for shafts containing manufactured cracks of depths between0% and 40%. Feasibility of simultaneous multiple-fault detection of transverse shaftcracks and seal face contact is discussed. [DOI: 10.1115/1.4007275]IntroductionThe demand placed on modern rotating machinery results inhigh operating stresses, which increases susceptibility to transverse cross-section fatigue cracking of the shaft. The capability todetect these cracks at an early stage of progression is imperativenot only for safety but also for economy. Early shaft crack detection allows the operator to plan accordingly for repair, without theneed to prematurely take the system off-line for an extendedperiod of time.The first step in development of an online crack detectionsystem is the identification of unique system response characteristics induced by a shaft crack. Numerous crack models havebeen developed that attribute the appearance of a 2X shaftspeed harmonic to the presence of a transverse shaft crack[1–10]. The 2X harmonic, appearing at a frequency equal totwice the shaft speed and reaching resonance at a shaft speedequal to half of a natural frequency, arises due to stiffnessasymmetry in the system in the presence of a fixed or stationaryforcing such as gravity [1,2]. For this reason, shafts with dissimilar area moments of inertia also display a prominent 2Xharmonic [1–3,6,8,9].The extent to which a globally asymmetric shaft model can beused to approximate a system with a highly localized shaft crackis limited. It is well known [5–7] that the position of the crack1Corresponding author.Manuscript received May 16, 2012; final manuscript received July 28, 2012;published online September 20, 2012. Assoc. Editor: Jaroslaw Szwedowicz.along the shaft greatly influences the magnitude of the 2X harmonic response. Any reasonable model of a cracked shaft mustaccount for this localization.Cracks can be categorized into gaping (open) cracks[1,3,6,10,11] and breathing cracks [3–10]. Gaping cracks remainopen regardless of the angle of rotation of the shaft. This assumption is generally valid in systems with small static displacementsand vibrational amplitudes [6,9]. As such, it is reasonable toassume that the dominant characteristic of a gaping crack is thelocalized reduction in stiffness.In Part I, Casey and Green [1] employ two models to approximate the characteristics of a gaping crack. The first model usesthe depth of the crack to create a system displaying global shaftasymmetry; the global nature of the asymmetry allows for an analytical steady-state solution of the system equations of motion.Four degrees of freedom are used: two displacements and twotilts. A gravity-forced response analysis indicates that the 2Xharmonic resonant magnitude increases as crack depth (or levelof asymmetry) increases, while the frequency at which the 2Xharmonic resonance occurs decreases.The next model approximates the localization of the crack viacrack flexibility coefficients, as per the strain energy release rate(SERR) [1,3,4,9–13]. The additional flexibilities caused by thecrack are incorporated into a transfer matrix allowing for thelocalization of the crack along the shaft. The extended transfermatrix, as provided by Lee and Green [14] and originallyproposed by Pestel and Leckie [15], allows for expansion of thetransfer matrix to include forcing. As crack depth increases,the gravity-forced response of the system predicts a decrease inthe 2X resonant frequency with an increase in magnitude, thoughJournal of Engineering for Gas Turbines and PowerC 2012 by ASMECopyright VNOVEMBER 2012, Vol. 134 / 112501-1Downloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

the decrease in frequency is less pronounced when compared tothe globally asymmetric shaft model.Contrary to gaping cracks, breathing cracks open and closeperiodically as a function of the shaft’s angle of rotation, causingthe stiffness to vary as a function of rotation angle. The impactand rubbing induced by the opening and closing motion of thebreathing crack gives rise to increased damping caused by frictional dissipation [4]. The aforementioned characteristics of abreathing crack are responsible for the appearance of additionalharmonics occurring at integer multiples of the shaft speed [4–8].A myriad of models employing time-dependent stiffness coefficients are available to describe breathing cracks: various nonlinearequations of motion [8], an adapted SERR approach [4], andstepped stiffness functions [12,16,17].The ability of a model to emulate the actual nature of the crackis imperative for quantitatively predicting the response. Carefulconsideration must be given to deduce which crack model bestsuits the system. In this work, an experimental procedure for crackdetection using an existing vibration monitoring system is discussed. The vibration monitoring system was originally constructed to monitor flexibly mounted rotor (FMR) mechanicalface seal dynamics [14,18–20]. The ability of the test rig to detecthigher harmonics due to seal face contact is discussed inRef. [20], and an active control system is proposed by Dayan,Zou, and Green [21] to eliminate the contact. Modifications to thetest rig allowing shaft crack detection, as first proposed in Part I,are provided herein with the goal of discussing the feasibility ofsimultaneous multiple-fault detection.The experimental and analytic results of two gaping crack models are compared. The first model incorporates local crack flexibility coefficients derived from the SERR, while the secondapproximates the manufactured crack as a small rectangularnotch. Plots of the 2X harmonic tilt response versus shaft speedare provided for several crack depths, as well as experimentallymeasured angular response orbits.2Forced Response: Local Crack Flexibility ModelAn appropriate crack model must account for the additionalcompliance caused by the crack. One widely used methodemploys the SERR along with linear elastic fracture mechanicstheory to estimate the additional compliance caused by thecrack [3,13]. The method was first proposed by Irwin [22] andsubsequently extended to rotordynamic systems of six degrees offreedom by Dimarogonas and Paipetis [13]. The additional displacement ui , along the direction of force Pi , caused by a transverse crack of depth a isð@ aJðyÞdy(1)ui ¼@Pi 0where JðyÞ is the SERR [4] and y denotes the direction parallelto the crack depth, as shown in Fig. 1. The xy frame shown in thefigure is a shaft-fixed reference frame that rotates with the shaftand always maintains its orientation relative to the crack edge.The uncracked section of the circular shaft of radius R is represented by hatching, and the half-width of the crack is b.The SERR JðyÞ depends on several factors: the elastic modulusand the Poisson ratio of the shaft material, the stress intensity factors (SIF) corresponding to the geometry of the cracked section,and the applied loads. The SIFs are provided by Papadopoulosand Dimarogonas [3] for a circular cross section containing atransverse crack. The additional compliance cij caused by thecrack is a flexibility in the ith direction caused by application of aforce in the jth direction, and is found by integrating the definitionof displacement, Eq. (1), along the length of the crack edge:cij ¼@ui@2¼@Pj @Pi @PjFig. 1Cross section of shaft containing transverse crackThe dimension of the crack compliance matrix is reduced fromsix to four, as axial and torsional deflection are neglected. Thecrack compliances of interest relate the shear force V and bendingmoment M to the linear and angular displacements u and h (indirections x and y) according to8 9 293800 Vx ux c22 0 u 6 0 c 00 7y337 Vy6¼6(3)7 4 00 c44 c45 5 hy My : ;:;hx00 c54 c55Mxwhere c45 ¼ c54 . The compliance matrix shown above can easilybe rearranged into a transfer matrix, as demonstrated by Caseyand Green in Part I [1]. The crack transfer matrix updates the statevector fSg from the left side of the crack to the right side according tofSgRight ¼ ½Fcrack fSgLeft(4)where ½Fcrack is the crack transfer matrix provided in Part I [1],and summarized in Appendix A. The state vector fSg isfSg ¼ fux hy My Vx uy hx Mx Vy gT(5)where the direction of the state vector quantity is indicated by thesubscript. A more comprehensive treatise on the state vector termscan be found in Ref. [14].An overhung rotordynamic system of shaft speed n and lengthL is shown in Fig. 2, with a crack located a distance L1 from theclamped support. The remaining distance to the rotor is designatedL2 , and the width of the crack is assumed to be negligible. Theshaft-fixed xyz frame shown in the figure is equivalent to thatshown in Fig. 1.It is well known that forcing due to gravity gives rise to the 2Xharmonic response. The transfer matrix was adapted to includeðb ðaJðyÞdydx b0112501-2 / Vol. 134, NOVEMBER 2012(2)Fig. 2Overhung rotor system with transverse shaft crackTransactions of the ASMEDownloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

forcing by Lee and Green [14] and described in detail by Caseyand Green in Part I [1] for a cracked system. In order to incorporate forcing due to gravity, the transfer matrix must be expandedfrom 8 8 to 9 9, and an additional entry of unity is concatenated onto the end of the state vector:fSg ¼ fux hy My Vx uy hx Mx Vy 1gT(6)The field matrices and point matrices for a rotating referenceframe transfer matrix analysis, incorporating forcing due to gravity, are provided and discussed in Part I [1]. The field matrices tothe left and right of the crack are designated ½F1 and ½F2 , respectively, while the point matrix corresponding to the lumped inertiaof the rotor and damping effects is designated ½P . Note that theinertia of the shaft is neglected in comparison to that of the rotor.For convenience, ½P and ½F are summarized in Appendix A. Thetotal transfer matrix ½U is found via successive multiplication ofelement transfer matrices:½U ¼ ½P ½F2 ½Fcrack ½F1 (7)where ½U is size 9 9. The overall transfer matrix in Eq. (7)updates the state vector from the left support to the right support.As such, application of clamped-free boundary conditions resultsin expressions used to obtain the forced response of the system:899 8 9382U39 0 U33 U34 U37 U38 My U 0 7 6U496 43 U44 U47 U48 7 Vxþ¼76 4 U73 U74 U77 U78 5 Mx U79 0 ::;; : ;Vy SUU83 U84 U87 U88U890 FE(8)899382U13 U14 U17 U18 My ux h 7 6Uy6 23 U24 U27 U28 7 Vx¼(9)76 4 U53 U54 U57 U58 5 Mx uy ::;;Vy SUU63 U64 U67 U68hx FEFig. 3 Cross section of shaft containing transverse notchcrackare easier to measure and quantify than the width and depth of acrack generated through prolonged fatigue testing.An overhung rotordynamic system displaying an exaggeratednotch crack is shown in Fig. 3. As before, the xyz frame shown inthe figure is a shaft fixed reference frame that rotates with theshaft. The width of the notch is Lc , the length of the beam to theleft of the notch is L1 , and the length of the beam to the right ofthe notch is L2 . The cross section of the shaft follows the designation provided in Fig. 1.A notch crack transfer matrix model is derived by following theprocedure dictated by Lee and Green [14] for generating a generalfield matrix. A shaft-fixed centroidal set of axes x y is defined parallel to the xy frame shown in Fig. 1, though attached to the centroid of the cracked cross-section. The asymmetric field matrix fora system of centroidal area moments of inertia Ix and Iy is26½Fnotch ¼ 4 ½0 4 4(10)where the subscripts r and i denote real and imaginary responsecomponents, and t represents time. The appearance of a 2X harmonic response component of the tilt is immediately evident; themagnitude of this component is 12jhxr þ ihxi þ ihyr hyi j.The procedure outlined above provides the ability to includeboth the localization of the crack and the crack depth. The crackmodel presented above is accurate for cracks obeying several criteria. First, the crack must terminate in a sharp tip along the lengthb of the crack edge [23]. Also, the crack must remain open andhave a negligible width [4,24]. Finally, the crack flexibility coefficients calculated in Eq. (2) are valid only for cracks up to 80%depth [3].3Forced Response: Notched Crack ModelAs stated above, the SERR approach is valid only for narrowcracks terminating in a sharp tip. However, for experimental purposes, it is easier to manufacture a finite-width notch than to subject the shaft to lengthy fatigue testing to generate a crack. Inaddition, the width and depth of a crack manufactured as a notchJournal of Engineering for Gas Turbines and Power½0 4 403705½FY 4 4001(11)9 9where2where SU designates the state vector at the support and FE designates the state vector at the free end. The procedure for obtainingthe forced response is discussed in detail in Part I [1]. Transformation into an inertial reference frame results in the magnitude of thetotal tilt c of the rotor, which when neglecting constant offset isfound to be1c ¼ ½ðhxr þ ihxi þ ihyr hyi Þei2nt 2½FX 4 4½FX 4 46 1 Lc6666¼ 60 16660 040 03L2cL3c2EIy 6EIy 777LcL2c 777EIy 2EIy 771Lc 7501 4 4(12)and2½FY 4 46 1 Lc666¼660 16640 0003L2cL3c2EIx 6EIx 777LcL2c 77EIx 2EIx 7771Lc 501(13)4 4The width of the notch beam element is Lc and the elastic modulusis E. The centroidal area moments of inertia Ix and Iy for thecracked circular cross section shown in Fig. 1 are provided inAppendix B.As per the discussion on the local flexibility crack model, theoverall transfer matrix ½U is found through successive multiplication of elemental transfer matrices:½U ¼ ½P ½F2 ½Fnotch ½F1 (14)The magnitude of the forced response of the rotor tilt is obtainedin a manner identical to that shown in Eqs. (8)–(10).NOVEMBER 2012, Vol. 134 / 112501-3Downloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 4 Dimensionless crack flexibilityFig. 6 Local crack flexibility model: magnitude of 2X tiltresponse versus shaft speed and crack depthFig. 5 Local crack flexibility model: magnitude of 2X tiltresponse4 Analytic Results: A Comparison Between the LocalCrack Flexibility Model and the Notch ModelThe degree to which the crack model emulates the real crackedsystem at hand is crucial towards predicting quantitative characteristics of the response. Perhaps most importantly, a suitablejudgment must be made as to what degree the particular crackedsystem under consideration displays breathing behavior. It isemphasized that the aforementioned models are only valid for agaping crack.As stated previously, it is shown in Part I [1] that as crack depthincreases the magnitude of the 2X harmonic response at resonanceincreases, while the frequency at which resonance occursdecreases. The degree to which these phenomena are observed ina cracked system will be investigated using the aforementionedmodels for cracks of varying depth.4.1 Local Crack Flexibility Model. The first step in theanalysis is to determine the crack flexibility coefficients given byEq. (3). Dimensionless flexibilities for cracks up to 75% of thediameter are shown in Fig. 4. As expected, the flexibility inducedby the crack increases as the crack depth increases.The parameters for the analysis are provided later in thedescription of the test rig and the manufactured shaft. The lengthof the shaft is decreased by approximately 3% such that the 0%crack depth response for both models matches that observedexperimentally. The elastic modulus and Poisson ratio are takento be 207ð10Þ9 Pa and 0.33, respectively.As per Eqs. (8) and (9), the magnitude of the 2X harmonicresponse of the rotor tilt is plotted in Fig. 5 versus shaft speedand crack depth. A two-dimensional view of Fig. 5 is provided inFig. 6 for qualitative discussion; darker shades correspond to alarger response magnitude. The frequency at which the 2X harmonic tilt response reaches resonance is plotted along with the 2Xtilt resonance magnitude in Fig. 7. As the crack becomes deeper,the shaft speed at which the 2X response reaches resonance diminishes, while the magnitude of the maximum tilt increases (seeFigs. 5–7).112501-4 / Vol. 134, NOVEMBER 2012Fig. 7 Local crack flexibility model: magnitude and frequencyof 2X tilt resonanceTable 12X resonant shaft speedsShaft speed (Hz):c% Crack depth01020304050607075Local crackNotch 3.5673.5072.9271.6770.09————The shaft speed at which the 2X harmonic tilt response reachesresonance is one half of a system natural frequency. As the frequency of shaft rotation deviates from this value, the 2X harmonictilt response magnitude becomes markedly diminished.It is clear from the second column of Table 1 that from 0%crack depth to 40% crack depth, the frequency of the 2X harmonictilt resonance diminishes by approximately 11%. However, from0% crack depth to 60% crack depth, the frequency of the 2X harmonic tilt resonance diminishes by approximately 31%. It is clearthat as the crack grows beyond 40% of the shaft diameter, the frequency of the 2X harmonic resonance decreases substantially,accompanied by a substantial increase in magnitude, as seen inFig. 7.4.2 Notch Crack Model. The response due to a notch crackis intrinsically tied to the area moments of inertia about differentaxes. As the ratio of the area moments of inertia increases, themagnitude of the 2X harmonic response is predicted to increase.Transactions of the ASMEDownloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 8Notch area moments of inertiaFig. 11 Notch crack model: magnitude and frequency of 2Xharmonic tilt resonanceFig. 9 Notch crack model: magnitude of 2X tilt responseFig. 12Test rig cross sectionfor safety reasons), the 2X resonance frequency decreases by only2.4%. However, from 0 to 60% depth the 2X resonance frequencydecreases by approximately 9.8%.5Fig. 10 Notch crack model: 2X tilt response versus shaftspeed and crack depthTo assist in interpretation of the results, nondimensional areamoments of inertia for a cracked circular cross section are shownin Fig. 8, generated using the relations provided in Appendix B. Itis clear from the figure that as notch crack depth increases, theratio of the centroidal area moments of inertia likewise increases:It is, therefore, expected that the response magnitude increases asnotch depth increases.The magnitude of the 2X harmonic response of the rotor tilt isplotted in Fig. 9 versus shaft speed and notch crack depth. A twodimensional view of Fig. 9 is provided in Fig. 10 where, onceagain, darker shades correspond to a higher response magnitude.The frequency at which the 2X harmonic tilt reaches resonance isplotted along with the magnitude of the resonant response inFig. 11. As the notch crack becomes deeper, the shaft speedat which the 2X harmonic response reaches resonancediminishes, while the magnitude of the resonant peak increases(see Figs. 9–11]. Compared to the local crack flexibility model,however, the rate at which the 2X resonance frequency drops asdepth increases is much less pronounced.The 2X resonance frequencies for notch cracks between 0%and 75% depth are provided in Table 1. It is clear from the tablethat from 0 to 40% depth (maximum depth used in experiments,Journal of Engineering for Gas Turbines and PowerExperimental Test RigThe objective of the experimental work is to investigate the feasibility of using the 2X harmonic tilt response component todetect a crack in an existing FMR mechanical face seal contactmonitoring system.Higher harmonics are present in the frequency spectrum ofmost real rotor dynamic systems. These harmonics can beattributed to characteristics such as asymmetric bearings, rubbingcontact within the system, and general nonlinear behavior[7,20,25]. For this reason, the mere presence of a higher harmonicdoes not indicate a particular fault. As such, further signaturesmust be employed to distinguish faults.5.1 Test Rig Overview. Though only a concise descriptionis given here, a comprehensive description of the test rig used todetect seal face contact is found in [14,23]. The cross section ofthe test rig is shown in Fig. 12. The test rig consists of a precisionspindle into which a shaft is screwed. The three part housingassembly is labeled “Part I,” “Part II,” and “Part III.” The spindleis driven by a dc motor with a maximum speed of 1750 rpm. A1:4 gear ratio of the motor to the spindle provides a maximumspindle rotation speed of 7000 rpm.Gaping cracks varying from 0% to 40% of the shaft diameterare manufactured in the shaft 6.35 mm from the base using electrical discharge machining (EDM). It is important to note that theEDM process can create corners within the notch that are notexactly rectangular but instead have a finite corner radius. Theeffect of this radius is assumed to be small, however, and theNOVEMBER 2012, Vol. 134 / 112501-5Downloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 14Adapted probe configurationThe probes are capable of measuring tilts cg and cn abouteach axis. The rotating reference frame xy used in the analyticdevelopment is shown in the figure to provide reference. A plot ofcg versus cn provides the angular response orbit of the rotor.5.3 Shaft Damping. A frequency-independent structuraldamping model is used to incorporate energy dissipation. Viscousdamping constants equivalent to the structurally damped systemare incorporated into the point matrix provided by Casey andGreen [1] (the point matrix is summarized in Appendix A). Theenergy dissipated per cycle via viscous damping isEdisv ¼ pxcjXj2Fig. 13 Monitoring system block diagram(15)while the energy dissipated via structural damping per cycle iscorners assumed to be sharp. The width of the crack is approximately 1.0 mm. The manufactured rectangular shape of the crackindicates that the crack is perhaps best modeled using a notchcrack model, such as that discussed above.The shaft is composed of AISI 4140 steel of diameter 10.16 mmand length 88.9 mm. The rotor has a mass of 0.5733 kg, polar massmoment of inertia of 3:847ð10Þ 4 kg m2 and a transverse massmoment of inertia of 2:371ð10Þ 4 kg m2 . The center of gravity ofthe rotor is axially offset from the end of the shaft by a distance of10:4 mm. The surface of the rotor is polished to a surface roughnessof 0:1lm (rms) to provide accurate probe data readings.5.2 Monitoring System. A block diagram schematic of themonitoring system is shown in Fig. 13. The dynamic response ofthe rotor is measured via three eddy-current proximity probes(one of which is shown in Fig. 12). The probes produce a voltageproportional to the distance from the end of the probe to the surface of the rotor. The bandwidth of the probes is approximately10 kHz, and the signal from the probes is first passed through a1 kHz low-pass filter to prevent high frequency cross-talk andaliasing. Following the low-pass filter, the signal passes through avoltage divider that drops the voltage from 21.2 V to 10 V,which is required for input to the control board. The control boardis a dSPACE DS1102 floating-point controller board with bothanalog-to-digital and digital-to-analog conversion, as well as fullyprogrammable processing capabilities [23].The probe layout used to measured seal face contact is providedin Ref. [23]. The orientation of the probes is adjusted to bestobserve the gravity induced tilt about the g axis: an optimum layout places probes a maximum distance from the g axis. The probelayout is shown in Fig. 14 about an inertial g n frame that does notrotate with the shaft. Gravity acts in the n direction, as shown inthe figure. Probe C is inclined 60 deg above the g axis, and probesA and B are rotated 90 deg and 180 deg counterclockwise fromthis position, respectively.112501-6 / Vol. 134, NOVEMBER 2012Ediss ¼ pbkjXj2(16)where x is the response frequency, c is the equivalent viscousdamping coefficient, k is the stiffness, b is the structural dampingconstant, and jXj is the magnitude of the response. A value for anequivalent viscous damping coefficient ceq is obtained by relatingEqs. (15) and (16) and solving for c (which is now the equivalentviscous damping coefficient ceq ):ceq ¼bkx(17)An accelerometer is placed on the end of the nonrotating shaft andthe system is set into oscillatory motion; the output from theaccelerometer is used to measure the response of the system. Alog-decrement approach is used, in conjunction with Eq. (17), toprovide an estimate for b of 0.00981. Appendix A discusses therelationship between b and the damping coefficients dij appearingin the equations of motion.Several assumptions are made in the development of the dampingmodel. Though structural damping has been shown to occur independent of frequency over a wide frequency range, there is still significant difference between the frequency of response at which thedamping experiments are conducted (700 Hz) and the frequency atwhich crack detection experiments are conducted (60–150 Hz). Also,the stiffness was approximated using the globally asymmetric shaftmodel as a worst-case scenario. It is assumed that the crack does notintroduce additional damping into the system.6Experimental ResultsExperimental results for manufactured shaft notch cracks up to40% depth are provided for comparison against analytic results(though analytic results are given for crack depths up to 75%, theexperiments were only carried out to 40% crack depth for safetyTransactions of the ASMEDownloaded 01 Nov 2012 to 130.207.153.60. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms Use.cfm

Fig. 15 Low speed range of experimental 2X responseFig. 16 High speed range of experimental 2X responsereasons). It is shown analytically that the 2X harmonic response ismaximum at shaft speeds equal to one-half of a natural frequency.An increase in the depth of the crack causes the natural frequencies to decrease (as seen by a decreased 2X resonance frequency)while the amplitude of the resonance increases.6.1 2X Tilt Response Results. The following procedure isemployed to generate experimental plots of the magnitude of the2X tilt response component as a function of shaft speed. The shaftspeed is incrementally adjusted, and a time sample of the probedata is taken by the DS1102 board. The power spectral density(PSD) of the time data is computed, and the 2X harmonicresponse magnitudes of the tilt are obtained via filtering of thetotal signal. The process is repeated for cracks varying between0% and 40% depth. A single shaft specimen is utilized in theexperiments, and the crack depth is incrementally adjusted foreach subsequent set of experiments. A single shaft is used in orderto mitigate potential variations in a set of shaft specimens and toisolate the effect of crack depth on the system response.Figures 15 and 16 show the scaled 2X PSD amplitude of theresponse provided by one of the probes as a function of shaftspeed for the low and high shaft speed ranges, respectively. Theexperimentally observed 2X resonance frequencies are extractedand provided in the last column of Table 1. As the crack increasesfrom 0% depth to 40% depth, the frequency of the 2X resonantpeak decreases by approximately 4.7%.Recall that from 0 to 40% crack depth, the local crack flexibility model predicted an 11% decrease in the 2X resonant peakfrequency while the notch model predicted a 2.4% decrease. It isclear that the experimentally measured decrease of 4.7% liesbetween the analytically predicted values. A plausible conjecturefor this result is that the manufactured notched crack was ne

vector fSg from the left side of the crack to the right side accord-ing to f SgRight ¼½F crack Left (4) where ½F crack is the crack transfer matrix provided in Part I [1], and summarized in Appendix A. The state vector fSg is fSg¼fu x h y M y V x u y h x M x V yg T (5) where the direction of the state vector quantity is indicated by the .

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