Journal of Applied Mechanical Engineering

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1 Journal of Applied Mechanical Engineering ISSN: Journal of Applied Mechanical Engineering Wu et al., J Appl Mech Eng 016, 5:6 OI: / Research Article Article OMICS Open International Access A Full Spectru Analysis Methodology Applied to an Anisotropic Overhung Rotor Xi Wu*, Caeron Naugle and Ji Meagher Mechanical Engineering epartent, California Polytechnic State University, San Luis Obispo, CA93407, USA Abstract This paper deonstrates the ethodology and usefulness of full spectru analysis on theoretical odels and experiental easureents. This ethod establishes another bridge between odel and data that is not currently being used to its greatest advantage. Soe experiental devices display full spectru vibration signals but ost often half spectru is reported. Correlating analytical odels with experiental data using full spectru analysis is siilarly underutilized. In this paper, 3 full spectru plots fro x and y signals (fro either experiental or theoretical results) are shown to atch very well with the plots generated by experiental ARE software. Tracking filters are used to isolate synchronous and non-synchronous vibration and allow for accurate phase angle easureent. The strategy developed by this paper can accurately convert the ath angle fro theoretical odel to experiental instruentation phase angle. Then we draw direct coparison of experiental results of a flexible overhung rotor with theoretical results to estiate unknown syste paraeters such as bearing stiffness and skew angle due to the gyroscopic effect. Finally, our paper provides other significant results such as 3 orbit plots. The tools developed here directly connect the experient to the theoretical odel and can be used to verify theoretical odels with greater confidence. Keywords: Overhung rotor; Full spectru cascade plots; Backward whirl; Anisotropic rotor; Gyroscopic effects Noenclature Mass of the isk kg x irection perpendicular to axis of rotation y irection perpendicular to axis of rotation z Axis of rotation k Spring Matrix coponent in x -direction fro force in the x xx -direction k Spring Matrix coponent in y -direction fro force in the y yy -direction kx θ y ky θ x Spring Matrix coponent in x -direction fro oent about the y -axis Spring Matrix coponent in y -direction fro oent about the x -axis N / N / N / N / θ x Rotation angle of the disk about the x -direction rad θ y Rotation angle of the disk about the y -direction rad ε Eccentricity of disk center of ass fro the center of rotation E Modulus of elasticity N / F x Forcing function due to eccentricity in the x -direction N F y Forcing function due to eccentricity in the y -direction N M x Moent function due to skew about the x -axis N* M y Moent function due to skew about the y -axis N* K A Bearing A stiffness N / K B Bearing B stiffness N / L istance fro Bearing A to Bearing B C istance fro Bearing B to disk A Location of Bearing A N / A B Location of Bearing B N / A shaft Flexibility atrix due to the bending of the shaft / N bearing Flexibility atrix due to the suspension of the bearings / N χ Skew angle fro disk axis of rotation to shaft axis of rotation rad Ω Rotational speed of the disk RPM δ Phase angle of the skew vector easured fro the positive x χ -axis rad Phase angle of the eccentricity vector easured fro the positive δ ε x -axis rad t Tie s J iaetric ass oent of inertia of the disk kg * d J p Polar ass oent of inertia of the disk kg * 4 I Area oent of inertia of the shaft *Corresponding author: Xi Wu, Mechanical Engineering epartent, California Polytechnic State University, San Luis Obispo, CA93407, USA,Tel: ; E-ail: xwu@calpoly.edu Received August 05, 016; Accepted Septeber 3, 016; Published Septeber 9, 016 Citation: Wu X, Naugle C, Meagher J (016) A Full Spectru Analysis Methodology Applied to an Anisotropic Overhung Rotor. J Appl Mech Eng 5: 3. doi: / Copyright: 016 Wu X, et al. This is an open-access article distributed under the ters of the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original author and source are credited.

2 Page of 10 Flexibility atrix due to the suspension of the bearings in the x - z Bxz plane Flexibility atrix due to the suspension of the bearings in the y Byz - z plane / N / N K bearing Stiffness atrix due to the suspension of the bearing N / Introdution Overhung rotors are widely used in industrial turbo-achines. For certain gas turbines, gyroscopic effects of the disks ay alost double the critical speeds of the rotor systes, copared to noral echanical vibration systes. In addition, the asyetry of the bearing stiffness will bring ore coplication to a rotor syste. Even though there are any established publications about how to theoretically odel an overhung rotor with anisotropic bearings [1-6], very few papers copared the theoretical odels with experiental results. Ishida al etc. [6] derived a sophisticated atheatical odel to theoretically investigate the nonstationary vibration of a flexible rotor with nonlinear spring paraeters during acceleration. They eployed an asyptotic ethod to copute the first approxiate solution to vibration response. Then, they calculated the aplitude variation curves of each oscillation coponent using coplex-fft. Furtherore, they inspected how each nonlinear coponent in polar forat affected dynaic vibration. They proposed a unique signal processing ethod called coplex-fft where the rotor whirling plane is coincident with the coplex plane. The whirling direction of the rotor can be judged by filtering the vibration signals at different frequencies using coplex-fft ethod. However, the coplex-fft doesnot provide instruentation phase angle inforation and full spectru cascaded plots. Gunter et al. [7,8] perfored investigations about the forward and backward odes of overhung rotors through theoretical odels and Finite Eleent Analysis. Like the ajority of the publications, they use FFT to analyze their siulation results. In 1993, Southwick, Goldan, and Muszynska [9-1] introduced the new powerful â œfull spectruâ plot in rotating achinery vibration and rotor dynaics. Since then, Bently Nevada Corporation has installed full spectru cascade plots in all of their ajor achinery data acquisition software package entitled as ARE syste. Copared to â œtraditional (half) spectruâ -FFT plot, full spectru can extract ore significant diagnostic inforation fro the original signals generated by X, Y transducers, this allows engineers to deterine whether the vibration response is forward or backward with respect to rotational direction of the shaft. As a powerful tool for interpreting the vibration signals of rotating achinery, the full spectru plots display the correlation between the vibration signals fro the X and Y transducers. Based on reference [9,13], Bently Nevada of GE Energy builds a ultichannel signal processing and data acquisition syste, the ARE Sxp Software and the 408 SPi (ynaic Signal Processing Instruent) respectively. Unlike any other data acquisition systes, ARE 408 SPi is an extreely versatile syste designed for real-tie highly parallel signal processing and presentation. Cascade full spectru plots are ebedded into all ARE software which are installed on the ajority of turbo-achines in industry. Muszynska [14] ethodically investigated the dynaic behaviors of a vertically configured overhung ibalanced rotor supported by flexible anisotropic bearings by theory and experients. She concluded that the interaction of ibalance and shaft bow causes the synchronous forced precession of the rotor to be forward or backward. Furtherore, she investigated the situation in which the id-span rotor sections precess one way, while the outboard disk precesses the other. The phenoenon is partly attributed to the relationship between the unbalances in ters of their relative direction. Ishida et al. [15] theoretically investigated internal resonances near both the priary and the gravity critical speeds for a rotor syste of an asyetric shaft supported at both ends and a disk installed in the iddle; nonlinearities of the syste induced by bearing clearances were thoroughly explored. Nagasaka et al. [16] did further research by studying internal resonance between the forward and backward whirl odes near both the priary and the secondary critical speeds for a siply-supported rotor syste with asyetric shaft. Based on above research, Nandakuar et al. [17] extended their research to an overhung rotor with substantial gyroscopic effects and relatively large lateral vibration using the ethod of ultiple scales (MMS). The authors derived the theoretical nonlinear equations of otion for an asyetrical overhung rotor running near its gravity critical speed. The authors exained how gyroscopic effects contribute on axiu resonant aplitudes, which disclose soe interesting phenoena induced solely by nonlinearities of the syste. Yi et al. [18] studied the dynaic response of a flexible shaft with a disk subjected to axial forces for two overhung rotor systes using transfer atrix ethod. They concluded that, under the force load, the gyroscopic effect not only increases the critical axial force, but also changes the instability type fro divergence to flutter. Ma et al. [19] experientally investigated the vibration response of an overhung rotor under sudden unbalance excitation due to the blade loss and quantitatively assessed the ipact effect. The effect of the sudden unbalance is tested in both the subcritical state and the supercritical state. The results deonstrate that the response of the syste to sudden ibalance contains frequencies fro the ipact response as well as frequencies due to rotational speed. A flexible rotor was shown to excite ore fro sudden ibalance than a ore rigid rotor. Ma et al. [19] presents a finite eleent odel of the oil fil instability of an overhung rotor with both parallel and angular isalignents including the gyroscopic effect. Oil fil bearings are siulated using a non-linear oil-fil force odel, assuing short-length bearings. The validity of the odel is verified by coparison to experiental results in published literature. Results show that isalignent of the coupling can delay the first ode of vibration and even reduce its aplitude [0]. Unfortunately, to the best of the authorsâ knowledge, there are no publications about how to actually generate the cascade full spectrus directly fro either X, Y transducers or siulation results. If we successfully solve this proble, we can predict what will happen in experients fro theoretical odels. In this research, we will solve the following 4 probles. 1) We construct 3 full spectru cascade plots using coplex FFT through MATLAB prograing. ) We use tracking windows to filter the transducer data to nx coponents of rotor speed when the rotor starts up or runs down. 3) We experientally copare our results with ARE data. The results are directly coparable and the MATLAB plots, using our ethod, provide opportunity of further post processing full spectru data. 4) Theoretical results of overhung rotor are converted into 3 full spectru plots which will be copared with experiental data. The results presented in this paper atch with experients with confidence. More iportantly, effects of different coponents of the theory are able to be deterined. This allows for better diagnoses of real rotor systes. Stiffness of the bearings is deterined by coparing theoretical natural frequencies to natural frequencies deterined by experientation. Using a Bode plot for each xz and yz plane, (Figures 1 and ), the natural frequency of the experiental apparatus is deterined for each plane. The equations

3 Page 3 of 10 of otion 1, with disregard for forcing and oent equations on the right hand side, are coupled with the total stiffness atrix 10 to provide a syste of equations that is dependent only on tie, speed and the unknown bearing stiffness paraeters. This syste of equations is used to solve for the natural frequencies in ters of the speed and the bearing stiffness using the eigenvalue proble. ifferent values of bearing stiffness are chosen in each plane and a Capbell diagra is used to solve for the natural frequency as a function of speed. Values of stiffness are iterated until the theoretical and experiental natural frequencies atch. Skew angle is deterined by coparing theoretical and experiental angular displaceent vibrations. educing the angular displaceent fro experient is not easy to do, as transducers are observing the Figure 3a: Scheatic of rotor geoetry. Figure 3b: Overview of apparatus. Figure 1a: isplaceents, rotations, forces and oents. Figure 4: Exaple Capbell diagra. Figure 1b: epiction of skew angle χχ. Figure : iagra showing iportant paraeters. vibration of the shaft and not the angle of the disk. A ethod for deterining the angle was eployed in which displaceents fro two sets of transducers is copared. To the best of the authors knowledge, there are no papers which provide direct coparison between theoretical odels and experiental results using full spectru analysis. Experiental work soeties uses full spectru but ost often it uses half spectru. Analytical work that reports full spectru does not correlate phase angel to instruent easureent. Theoretical Model The theoretical odel used to siulate the experiental apparatus is depicted in Figures 3 and 4. Equations of otion 1 are deterined using conservation of oentu and a dynaic analysis of the disk ass and inertial changes. The forcing functions on the right hand side

4 Page 4 of 10 of 1 include forces due to acceleration of the shaft so that the siulation can include change in speed of the shaft and gyroscopic effects, such as a start-up or run-down. Moent equations are presented as pertaining to the change in angular oentu of the disk [1,3,8] (Figure 3). x + kxxx kx θ θ ( cos( ) sin( )) y y = ε Ω Ω t + δε α Ω t + δε y + k yyx k yθ θ ( sin( ) cos( )) x x = ε Ω Ω t + δε + α Ω t + δε J dθx + JpΩ θy + ky θ y+ k ( ) sin( )) x yθ θ x x = Ω Jd Jp χ Ω t + δχ J dθy JpΩ θx + kx θ x+ k ( ) cos( ) y θθθ y y y =Ω Jd Jp χ Ω t + δχ The total stiffness constants of the syste are deterined fro geoetry and properties of the rotor bearings, and divided into two ain contributions fro the bending of the shaft and the suspension of the bearings. These two contributions are considered independent to solve for each contribution and then are cobined in series to deterine the total stiffness (Figure 4). Case (A) Flexible shaft with rigid bearings. Flexible Influence coefficients ethod in the general for of is used to derive flexibility atrix which can be applied in both xz and yz plane, respectively. (1) Since the shaft is considered to be isotropic, the atrix is identical for the xz and the yz plane [6,13]. Linear isplaceent Force = [] () Angular isplaceent Moent C ( L+ C) C(L+ 3 C) 3EI 6EI shaft = (3) C(L+ 3 C) ( L+ C) 6EI 3EI Case (B) Rigid shaft with flexible bearing. The stiffness atrix is derived in the general for of 4 using stiffness influence coefficients ethod. Force Linear isplaceent = [ K] (4) Moent Angular isplaceent ka + kb ka( L+ C) kc B KBearing = (5) ka( L+ C) kc B kck B A( L+ C) In order to cobine the bearing stiffness with the shaft stiffness, both are added as flexibility atrices. 1 kck( L+ C) k( L+ C) + kc 1 B A A B Bearing = KBearing = LkAkB ka( L+ C) + kc B ka + kb (6) Apply to xz and yz plane, respectively to account for anisotropy of bearing stiffness for different planes. Figure 5: 3 cascading plot deonstrating the ipact of stiffness anisotropy on positive/negative frequencies. Bxz Bxz 1 kbyc kay ( L+ C) kay ( L+ C) + kbyc = Lk kay ( L C) k Ayk By ByC kay k By 1 kbxc kax( L+ C) kax( L+ C) + kbxc = LkAxkBx kax( L+ C) + kbxc kax + kbx Total flexibility atrices in xz and yz plane are: (7) (8) xx x θ y yy y θ x = + = + θ shaft Bxz shaft Bxz xθy θθ y y θθ x x y x (9) Figure 6: 3 cascading plot deonstrating the ipact of stiffness anisotropy on horizontal/vertical frequencies. Figure 7: Full spectru plot produced with the ARE sxp software. Total stiffness atrices are shown in 10, which are applied in equations of otion xx xθy xx xθ k y yy ky θ x yy yθ x k k = = k x k θ y θθ y y xθ k y θθ y y yθ k x θθ x x yθ θθ x x x Experiental Apparatus (10) Our experiental apparatus consists of the GE (Forally Bently Nevada) RK4 rotor kit, the ARE 408 spi ata asset condition onitoring equipent, and a laptop with the ARE sxp software (Figures 5-7). Four eddy current displaceent transducers are used to easure the vibration of the shaft. The ARE 408, when coupled with ARE sxp, is capable of providing real-tie signal processing fro the rotor syste in eaningful figures. Identify the Bearing Stiffness Paraeters fro Experiental ata With the experiental data we were able to identify the natural frequency in the horizontal and vertical directions fro the Bode plot of each horizontal and vertical transducer. Fro the theoretical equations of otion, the eigenvalue proble can be used to solve for the natural frequencies of the syste. Since the syste includes

5 Page 5 of 10 gyroscopic oents that depend on the rotational speed of the rotor, the natural frequency is a function of the speed of the rotor. The Capbell iagra is a plot of the natural frequency as it changes with the increasing speed of the rotor. This diagra can be used to tune the bearing stiffness. The paraeters that affect the natural frequency that are not evident fro the description of the syste is the stiffness of the bearings in each xz and yz plane. To solve for the stiffness values, a Capbell iagra is created for each value of stiffness until the point at which the natural frequency line intersects the rotor speed line is the sae value as the experiental natural frequency. An exaple of a Capbell diagra can be seen in Figure 8. The Capbell diagra gives a good depiction of how the natural frequency changes with speed Figure 8: Full spectru plot produced with using our ethod. Figure 9a: Rotational speed change over tie. Figure 9b: Rotational speed change over a window in tie. and how negative natural frequencies decrease in frequency, while the positive natural frequency increases. The stiffness of the bearings was deterined to be about N/ in the xz plane and about N/ in the yz plane (Figure 8). To further understand the effect that stiffness anisotropy has on the syste, Figures 9 and 10 display a cascade of aplitude plots with changing stiffness anisotropy between the horizontal and vertical planes. Viewing the theoretical data in this way allows for coparison of the two natural frequencies to experiental values just as the Capbell iagra does. In addition, the cascaded plots depict how the shape of the aplitude plots change with varying anisotropy. Figure 9 uses the aplitude of positive and negative frequencies to display the effect of anisotropy. The interaction between the first and second natural frequencies can be seen as the two get closer to one another. Negative whirling is doinant only between the two natural frequencies. When the two natural frequencies coincide, there is a highly circular positive whirl orbit. Experiental aplitude plots are used in conjunction with Figures 9 and 10 to choose the correct anisotropy at which the characteristic shape atches. The skew angle, χ, and the eccentricity, ε, are deterined next by independent ethods. Eccentricity can be deterined by siulating the theoretical odel under varying eccentricities until the agnitude of the vibration is equivalent to that of the experiental results. The 4 value was deterined to be The skew angle can be quantified using a siilar technique. But the skew angle is closely related to the aplitude of vibration in the two orthogonal directions, x and y, as well as the disk tilting angles, θ x and θ y. In order to easure the tilting angles in the experient, there were two sets of orthogonal transducers. One before the disk on the shaft and the other after the disk. This displaceent between the two sets of transducers allows for an approxiation of the angles of tilt. Assuing no bending of the shaft, the angle is deduced fro a pivot about Bearing B, with the angle being equal to the inverse tangent of the ratio of difference in displaceent between the transducers, and the distance between the transducers. Then the siulation is run with varying values of χ to atch the aplitude of angles and displaceents to the experient. The value was deterined to be 0.09 radians. Produce Full Spectru Plots fro Orthogonal Transducers The purpose of this work was to produce figures that represent vibration aplitude data fro theoretical odels in a way that is coparable to figures produced by the ARE Sxp software. This allows the analysis of theoretical odels with confidence in the representation of the data with figures such as the Bode plot, the Cascade plot, and full spectru plots. The ajor achieveent in producing the figures was obtaining phase lag angles between two signals. Another achieveent was producing spectru plots with both positive and negative frequencies (Full Spectru plots). Full spectru plots are ore coplete in displaying the inforation contained in a Fourier Transfor. In the frequency doain, negative frequencies pertain to vibration of the rotor that is precessing in the opposite rotational direction than that of the rotor rotational direction. The full spectru plot is produced by first coupling the orthogonal transducer signals into a coplex signal and then perforing a Fourier transfor. The transfor on the coplex data results in frequency inforation both negative and positive. Negative frequencies tell a lot about the syste, especially when they are greater than their positive counterpart. When the negative

6 Page 6 of 10 Figure 10: Bode plot for verification of the phase angle technique. Figure 11: Experiental cascade plot directly produced by the ARE sxp software. frequency vibration is greater in aplitude, the precession is said to be in the reverse direction. This can be vital inforation in diagnosing a proble in a real rotor syste. In the case explored here it can be a sign of both, strong gyroscopic oents, as well as anisotropy of the stiffness fro the shaft or the bearings. Another exaple is the rub in a fluid fil bearing which anifests itself as a negative precession at twice the running speed of the rotor. This proble would be ore difficult to diagnose without the knowledge of negative frequencies. In order to verify that the MATLAB code of our ethod would produce accurate frequency data, the Fast Fourier Transfor, or FFT, of a steady state signal was copared to the FFT produced by the ARE 408 with ARE Sxp software. Both the aplitude and frequency of positive and negative frequency coponents was verified by coparison as deonstrated by the coparison of Figures 11 and 1. Produce Bode Plots fro Orthogonal Transducers The process for producing phase lag angles begins with segenting the data into windows of a fixed nuber of saples. This is because ost rotordynaic data coes in the for of a startupor a run-down. In both of these scenarios, the rotational speed of the rotor is continuously changing, as deonstrated by Figure 13. This change deters capturing both tie and frequency inforation. Segenting the data provides sall windows on which analysis can be perfored as if the signal was a constant frequency within the window, as deonstrated in Figure 14. For a segent of data in which the frequency is not changing, a Fourier Transfor can be applied to obtain the frequency spectru of the signal. In this frequency doain provided by the Fourier Transfor, the phase angle of each frequency can be obtained as well as the aplitude of vibration of each frequency. In order to calculate the phase lag fro one signal to another, this phase angle is copared for the sae frequency of each signal in the sae window in tie. This process is done for each window through the entire length of data producing a phase angle for each window. The Bode plot is ubiquitous in rotating achinery diagnostics and can be used to characterize a syste fairly thoroughly. Phase angle is a vital part of the bode plot and can shed light on any phenoenon One of the ain goals in this paper is to deonstrate the ability to produce accurate phase lag inforation fro two signals (Figure 14). As seen in the Bode plot of Figure 16, the MATLAB code based on coplex FFT strategy produces phase lag inforation nearly identical to the phase lag inforation produced with the ARE syste using the sae transducer data./par (Figure 15). This ability to produce phase lag inforation fro any transducer data allows accurate coparison of theoretical and experiental data. The MATLAB code that produces these plots also allows the ability of tuning the resolution of the phase and aplitude inforation, gaining

7 Page 7 of 10 Figure 1: Re-constructed 3 experiental full-spectru cascade plot using our strategy. Figure 13: 3 theoretical full-spectru cascade plot using our strategy. Figure 14: Bode plot, experiental vs. theoretical.

8 Page 8 of 10 Figure 15: Positive/negative frequency coparison, experiental vs. theoretical. Figure 16a: Experiental 1333 RPM Orbit. Figure 16b: Experiental 1433 RPM Orbit. Theoretical and Experiental Coparison The coparison of experient and theoretical odel developed in this research were perfored. The geoetric and other paraeters of the rotor kit are shown in Table 1 (Figures 17 and 18). The full-spectru cascade plot is a vital tool for analysis in rotating achinery. The cascade plot depicts how the vibrations change with tie. With the addition of the full spectru in these cascade plots, the negative frequencies provide insight for diagnostics of certain rotor issues, such as isalignent of disk axis with the rotor axis. Just as with the FFT and Bode plots, the cascade plot produced with our MATLAB code using coplex FFT strategy is copared with the cascade produced with the ARE software to validate the ethod. Fro Figures 17 and 18 we can see the results are siilar and good enough to validate our full-spectru strategy through experient. Furtherore, MATLAB has the ability to display data in three diensions as shown in the Figures 11, 1 and 13. With proper tuning of window size and sapling frequency, the cascade plot can display a host of rotordynaic phenoenon. The Bode plot of Figure 14 copares the startup aplitude and phase angle of the theoretical odel and the experient. Because both the experiental and the theoretical results were processed using our strategy with the sae MATLAB code to produce these plots, coparisons can be confidently drawn. The phase angles are of a phase lag fro the X transducer and fro the Y transducer. The two transducers are 70 positive degrees apart around the shaft. Observing the coparison of positive and negative frequencies as done in Figure 11 allows the detection of backward or forward whirl. When the negative frequency aplitude is greater than that of the positive frequency, the whirl is backward. Backward whirl describes the phenoenon of precession of the shaft vibration rotating opposite to the rotation of the shaft. Coparison to the typical horizontal vs. vertical aplitude plot shows that the crossover fro positive to Figure 16c: Theoretical 1358 RPM Orbit. accuracy in either tie or frequency. The phase lag calculation accuracy is deterined by the nuber of cycles analyzed in each window, so the larger the window, the better the phase inforation. But, as the window gets larger, the accuracy of speed is diinished (Figure 16). Figure 16d: Theoretical 148 RPM Orbit.

9 Page 9 of 10 escription Value Unit istance Fro Bearing 1 to Bearing (LL) 3 c istance Fro Bearing to Mass(CC) 13 c Mass of isk() 800 g iaeter of the Shaft 1 c Shaft Material Steel N/A Table 1: Rotor paraeters. Figure 17: Experiental 3 Orbit plot produced by our strategy. Figure 18: Theoretical 3 Orbit plot. negative whirl occurs when either the horizontal or vertical vibration aplitudes drop. Orbit plots of Figure 16 are created contain the key-phasor dot to ark the beginning of a rotation fro a fixed location on the shaft. This allows the deterination of the orbit rotation direction. For all of the orbit plots, the shaft was rotating in the counter-clockwise direction. Therefore, an orbit in the clockwise direction is in the reverse direction. Experiental data was filtered using a tracking filter for the orbit plots. Since a once per turn reference is unavailable for theoretical data, the orbits are plotted with a reference dot at the axiu x value for that orbit. Orbit plots are cascaded in increasing rotational speed in the 3 orbit plots of Figures 17 and 18. This cascade of orbits gives a sense of how the natural frequencies are physically anifested in these startups. This 3 orbit plot lends a ore intuitive representation of the data and at the sae tie, iportant characteristics such as phase lag, aplitudes and reverse precession can be discerned fro the plot. This plot was also useful in coparing the theoretical odel to the experiental data, because it deonstrates agnitude as well as general shape of the response of the entire orbit in one plot. Also Figures 17 and 18 akes clear the effect of gyroscopic oents, as the shape of the elliptical orbits during peak aplitude vibrations are bent. The ARE Sxp software does not produce this plot. iscussion and Conclusion 3 cascade plots (Figures 1 and 13) produced using our

10 Page 10 of 10 coplex FFT strategy clearly display the full vibration spectru and copleent those produced with the ARE Sxp software (Figure 17). Both the theoretical odel and the experiental results display a sudden dip in aplitude of positive frequencies between the two natural frequencies (Figures 1 and 13). This dip is coupled with an increase in negative frequencies, resulting in reverse orbits for a period in tie. These negative orbits can be seen in Figure 16. The theoretical odel suggested a strong influence fro the relative phase angle of the skew to the eccentricity. Soe values of phase difference offered copletely positive orbits through the natural frequency, while others offered highly negative orbits. This is the greatest contribution fro the skew angle, as without the consideration of it, the aount of negative to positive vibration could not be easily altered in the theoretical odel. In this paper, the dynaic behavior of a flexible shaft with an overhung disk has been studied by eploying full spectru analysis. Anisotropic bearing stiffness and the effect of gyroscopic oents are considered in our classic theoretical odel. Soe of the paraeters which are difficult to directly easure are estiated by cobination of theoretical calculation and experiental results. Full spectru analysis is applied to the syste to reveal soe interesting backward whirl vibration phenoena due to the interaction of anisotropy of the stiffness of the bearings, flexibility of the shaft and gyroscopic effect. Phenoena such as these cannot be disclosed by traditional half spectru FFT. 3 cascade full spectru plots fro theoretical odel are copared to those fro experients. Our full spectru cascade plots using coplex FFT strategy capture ore detail in the vibration response of the syste than the typical cascade plots. Filtered orbit plots with the key-phasor dots at different speed deonstrate forward and backward whirls. Methods for identifying the eccentricity and skew angle of the disk are presented. Also, the effect of the skew angle is deterined by coparison of experient and theory. The skew angle is deterined by coparing theoretical angular aplitudes to experiental angular aplitudes. Bearing stiffness is deterined through a ethod of atching theoretical and experiental natural frequencies. Consideration of the gyroscopic oents introduces speed as another dependent variable in the equation for natural frequency, aking the process of atching an iterative one. Acknowledgent The authors acknowledge the onald E. Bently Center for Engineering Innovation at California Polytechnic State University San Luis Obispo for support of this work. References 1. Genta G (005) ynaics of rotating systes.. Childs (013) Turboachinery rotor dynaics with case studies. 3. Muszynska A (005) Rotor dynaics. 4. Kraer E (1993) ynaics of rotors and foundations. 5. Yaaoto T, Ishida Y (001) Linear and nonlinear rotor dynaics. 6. Ishida Y, Yasuda K, Murakai S (1997) Nonstationary oscillation of a rotating shaft with nonlinear spring characteristics during acceleration through a ajor critical speed (A discussion by the asyptotic ethod and the coplex-fft Method). ASME J Vibration and Acoustics 119: Gunter E, olans J (000) Calculation of the forward and backward odes of an overhung rotor using MSC/NASTRAN for Windows. 8. Gunter E (1993) Forward and backward critical speeds and forced response of an overhung rotor with asyetrical bearing supports. 9. Bently (00) Fundaentals of rotating achinery diagnostics. 10. Goldan P, Muszynska A (1999) Application of full spectru to rotating achinery diagnostics. Orbit First Quarter 0: Southwick (1993) Using full spectru plots. Orbit 14: Southwick (1994) Using full spectru plots. Orbit 15: iarogonas A (1996) Vibration for engineers. 14. Muszynska A (1996) Forward and backward precession of a vertical anisotropically supported rotor. J Sound and Vibration 19: Ishida Y, Liu J, Inoue T, Suzuki, A (008) Vibrations of an asyetrical shaft with gravity and nonlinear spring characteristics (isolated resonances and internal resonances). ASME J Vibration and Acoustics. 16. Nagasaka I, Ishida Y, Liu J (008) Forced oscillations of a continuous asyetrical rotor with geoetrical nonlinearity (ajor critical speed and secondary critical speed). ASME J Vibration and Acoustics. 17. Nandakuar K, Chatterjee A (010) Non-linear secondary whirl of an overhung rotor. Proceedings of the royal society of london A 466: Yi KB, Yi J (01) ynaic behavior of overhung rotors subjected to axial forces. International J precision engineering and anufacturing 13: Ma Y, Liang Z, Zhang, Yan W, Hong J (015) Experiental investigation on dynaical response of an overhung rotor due to sudden unbalance. Proceedings of ASME turbo expo 015: Turbine technical conference and exposition. 0. Ma H, Wang X, Niu H, Wen B (015) Oil-fil instability siulation in an overhung rotor syste with flexible coupling isalignent. Archive of Applied Mechanics 85: Citation: Wu X, Naugle C, Meagher J (016) A Full Spectru Analysis Methodology Applied to an Anisotropic Overhung Rotor. J Appl Mech Eng 5: