[mechanisms and machine science] vibration engineering and technology of machinery volume 23 ||...

Forward and Backward Whirlingof a Rotor with Gyroscopic Effect

Ali Fellah Jahromi, Rama B. Bhat and Wen-Fang Xie

Abstract The determination of whirling frequencies of high speed turbines isalways challenging in rotor dynamics. The natural frequencies of a Jeffcott rotor aresplit in the presence of gyroscopic effect. It is quite well known that the lowerbranch corresponds to the backward whirl and the upper branch corresponds toforward whirl. The forward whirl mode of the rotor has been observed experi-mentally, however, the backward whirl has not been observed. In this study it isshown that the backward whirl can be observed when the rotor is coasting down torest from above the critical speed corresponding to the backward whirl. In order toillustrate the forward and the backward critical speeds of a simple Jeffcott rotor, thenatural frequencies are obtained analytically for the second natural frequency of thesystem because of the large gyroscopic effect present in that mode. An experimentalset up was used to verify the presence of backward whirl while the rotor is coastingdown to rest. The rotor is also simulated using finite element method by ANSYS,and Campbell diagram is plotted. The analytical, experimental and ANSYS sim-ulations confirm the existence of the backward whirl when the rotor is coastingdown.

Keywords Backward whirling � Gyroscopic effect � Finite element method �Campbell diagram

A.F. Jahromi (&) � R.B. Bhat � W.-F. XieDepartment of Mechanical and Industrial Engineering, Concordia University,1455 de Maisonneuve Blvd. W., Montreal H3G 1M8, Canadae-mail: [email protected]

R.B. Bhate-mail: [email protected]

W.-F. Xiee-mail: [email protected]

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_78

879

1 Introduction

The analysis of the critical speeds of high speed rotors is necessary for definingdesign acceptance criteria of the industrial rotors. The forward whirl mode has beenobserved experimentally for rotors supported on isotropic and symmetric bearings[1]. However, a survey of the literature reveals that the backward whirling mode hasnot been observed experimentally. Difference in bearing stiffness in horizontal andvertical directions, considering negligible damping effect, exhibits backwardwhirling mode of rotor [2]. Subbiah et al. [3] observed backward whirling of a rotorsupported by stiffness asymmetry of the bearings through experimental procedure.The backward whirl mode for the dynamic imbalance of rotor supported by fluidfilm damping and shaft structural damping is studied in the literature. Rao [4]studied the conditions for backward whirling for a flexible rotor supported by thehydrodynamic bearings. Moreover, the occurrence of the backward whirl is alsostudied for the cylindrical journal bearings [5, 6].

The start-up or coast-down results in a time variant frequency signal [7]. In thepresent paper, the occurrence of the backward whirling mode is studied consideringthe coast-down phenomenon for a rotor supported on isotropic ball bearings. Ananalytical model is developed to study the backward and forward critical speeds inpresence of the gyroscopic effect. Moreover, the critical speeds of the Jeffcott rotorare calculated using Finite Element Method (FEM). Furthermore, an experiment isconducted to observe the backward whirling mode in a coast down phenomenon.

The paper is organized as follows: First, the analytical and FEM model areillustrated in Sect. 2. Then, the explanation of the experimental set-up is presentedin Sect. 3. The results of the analytical model, the FEM and experimental study arepresented in Sect. 4. Finally, the conclusions and remarks are given is Sect. 5.

2 Analysis and Simulation

In order to study the forward and backward critical speeds, a simple Jeffcott rotor ismodeled considering the gyroscopic effect using analytical and FEM methods. Thegeometrical description and material properties of the Jeffcott rotor are presented inFig. 1 and Table 1.

Fig. 1 Schematic of simple Jeffcott rotor

880 A.F. Jahromi et al.

2.1 Analytical Model

The disk is located in the middle of the shaft shown in Fig. 1. Therefore, thegyroscopic effect would be more prominent in the second mode compared to thefirst mode for the shaft with simply-supported boundary conditions. Consequently,based on the literature [8] the analytical model is derived to study the secondbending mode of the system in Eq. (1). The straightening tendency of the shaftbending is modeled using torsional spring Kt which is shown in Fig. 2.

€h� jXa _hþ x2t h ¼ MejXt ð1Þ

where θ = ψ + jφ with ψ and φ being the rotations about the z and y axes,respectively, Ω is the angular velocity of shaft, ωt and α are the natural frequencyand ratio of mass moment of inertia respectively. M is the magnitude of the har-monic moment of excitation, and is given by M = meΩ2 where me is the unbalance.The e is considered 0.0001 (m) due to the static deflation of the shaft. Further:

Kt ¼ JTx2t ð2Þ

a ¼ JpJT

ð3Þ

Jp ¼ 18mD2 ð4Þ

Table 1 Rotor parameters

Symbol Description Value

l Shaft length 0.4 (m)

d Shaft diameter 0.006 (m)

T Disk thickness 0.025 (m)

D Disk diameter 0.08 (m)

E Modulus of elasticity 210 (GPa)

m Disk mass 1 (kg)

Fig. 2 Schematic of the rotor with torsional spring

Forward and Backward Whirling of a Rotor with Gyroscopic Effect 881

JT ¼ m12

34D2 þ l2

� �ð5Þ

Equation (1) can be presented as:

€wþ j€u� jXað _wþ j _uÞ þ x2t ð _wþ j _uÞ ¼ MejXt ð6Þ

By separating the real and imaginary terms in Eq. (6), it is written in the form ofEqs. (7) and (8) as:

€wþ Xa _uþ x2t w ¼ M cosðXtÞ ð7Þ

j €u� Xawþ x2t u

� � ¼ jM sinðXtÞ ð8Þ

In order to study the forward and backward whirling, let Ω = βt be the speedwhen the rotor starts from rest and increases towards the operating speed, andconsidering Ω = Ω0 − βt when the rotor is coasting down to rest. Based on theproperties of the rotor studied, the β and Ω0 are chosen as 300 (rad/s) and 4,500(rad/s), respectively. Equations (9) and (10) represent the coupled differentialequations for analyzing the forward critical speed and Eqs. (11) and (12) representthe mathematical model for the backward critical speed.

€wþ ba _ut þ x2t w ¼ M cosðbt2Þ 0\ t\15 (s) ð9Þ

€u� ba _wt þ x2t u ¼ M sinðbt2Þ 0\ t\15 (s) ð10Þ

€wþ ðX0 � btÞa _uþ x2t w ¼ M cosðX0t � bt2Þ 0\ t\15 (s) ð11Þ

€u� ðX0 � btÞa _wþ x2t u ¼ M sinðX0t � bt2Þ 0\ t\15 (s) ð12Þ

Equations (9)–(12) are solved in the time domain in Matlab using Runge-Kuttamethod for ordinary differential equations.

2.2 ANSYS Modeling

The model is defined using volume elements in ANSYS as shown in Fig. 3. In orderto determine the gyroscopic effect, the model is meshed using the element SOLID186 which is a 10-node tetrahedral element for homogenous structures which can beused for the calculation of Coriolis effect. The material model is assumed to be linearisotropic steel which is identified in Table 1. Simply-supported boundary conditionsare applied at both ends of the shaft by fixing the center point at the ends of the shaftin y and z directions. In order to determine the forward and backward critical speeds


the angular velocity of the shaft is defined in range of 0 to 3,500 (rad/s) with steps of,0 (rad/s), 400 (rad/s), 600 (rad/s), 1,000 (rad/s), 1,500 (rad/s), 2,000 (rad/s), 2,500(rad/s), 3,000 (rad/s) and 3,500 (rad/s).

3 Description of the Experimental Set-Up

The experimental set-up consists of a rotor supported by the two identical isotropicball bearings and is driven by a DC motor with variable speed range of 0 to 6,000(RPM) as shown in Fig. 5. The rotor is connected to the motor using a flexiblecoupling. The deflection of the beams is measured by two displacement sensors,TM 620 and GUNT HAMBURG with 90° phase difference for monitoring thewhirling orbit (Fig. 4). The DC motor is connected to the driver for controlling theangular velocity of the shaft. The proximity displacement sensors are connected tothe signal conditioner (TM 151 GUNT HAMBURG) which sends signal to theoscilloscope. The data sampling time of the oscilloscope for recording the data is1.6e-5 (s). The assembly of the experimental set-up is shown in Fig. 5.

4 Results and Discussions

The analytical model of a Jeffcott rotor which is presented in Eqs. (9)–(12) is solvedusing Matlab for 15 s with a time interval of 1e-4 s. In order to determine theforward and backward critical speeds the real part of the system response ψ in time

Fig. 3 Finite element model of the rotor in ANSYS

Fig. 4 Rotor and theproximity displacementsensors


domain is transformed to the frequency domain using Fast Fourier Transform(FFT). Figures 6 and 7 show the frequency responses for decreasing and increasingangular velocities of the shaft, respectively. Based on the calculated frequencyresponse for the second natural frequency of the rotor, the backward critical speed is105 (Hz) and forward critical speed is 445.6 (Hz). Figures 6 and 7 validate thesplitting of the second natural frequency of the rotor by the proposed analyticalmodel for the backward and forward critical speeds.

In order to calculate the variation of the first two natural frequencies of the rotorwith changing rotational velocity of the shaft, the Campbell diagram of the rotor isplotted using ANSYS for the range of 0–3,500 (rad/s). Figure 8 shows that thegyroscopic effect is not significant in the first natural frequency of the rotor becausethe disk is located in the middle of the shaft and therefore, the disk does not rotateabout the lateral axis in the first bending mode. Consequently, the gyroscopic effectis not significant in the first bending mode and the backward and forward criticalspeeds have same value which is 17.16 (Hz). The second natural frequency of therotor is separated into forward and backward branches. Based on the FEM analysis,the backward critical speed is 97.2 (Hz) and forward critical speed is 458.4 (Hz)which are approximately equal to the analytical results.

The experimental set-up can reach up to rotational speed of 100 (Hz) which is inthe range of the fundamental frequency of the system. Considering this limitation,

Fig. 5 Assembly of experimental set-up for whirl monitoring

Fig. 6 Frequency response ofthe ψ for coasting down of therotor from 4,500 (rad/s) to rest


the experiment is designed for observing the whirling direction of disk in the firstnatural frequency of the system. The critical speeds for the forward and backwardwhirls have the same value due to the position of the disk. In order to observe thebackward whirl the rotor is coasting down from 1,600 (RPM) to rest and the signalsof the displacements sensors are recorded in 1.6e-5 (s) sampling time. Figure 9presents the deflection of the beam in y and z directions, and the region showninside the ellipse indicates the change in the whirl direction. Within the ellipse inFig. 9 the backward whirl happens right after the critical speed. The phase change inthe sensor signals confirms the existence of the disk backward whirling.

In order to illustrate the whirling direction of the disk, the deflection of the shaftin y and z directions is plotted against each other as a phase portrait for the portionof the data which are marked by ellipse in Fig. 9. The black arrows in Fig. 10 depictthe forward whirling and red arrows present the backward whirling. Figure 10confirms the changing whirl direction.

In order to quantitatively compare the determined frequency in the analytical study,FEM and experimental study, the difference between the results are presented as anerror in Table 2. Table 2 presents the convergence of the results which are achievedfrom different methods for studying the forward and backward critical speeds.

Fig. 7 Frequency response ofthe ψ for increasing angularvelocity from 0 to 4,500 (rad/s)

Fig. 8 Campbell diagram forfirst two bending modes ofcase study rotor


Fig. 9 Deflection of the beam in y and z directions

Fig. 10 Phase portrait of the shaft deflection in y and z direction

Table 2 Quantitative comparison among analytical, FEM and experimental results

Forward Backward

First mode Analytical – Analytical –

ANSYS 17.16 (Hz) ANSYS 17.16 (Hz)

Experiment 19.18 (Hz) Experiment 19.18 (Hz)

Error 10.53 % Error 10.53 %

Second mode Analytical 444 (Hz) Analytical 105 (Hz)

ANSYS 458.4 (Hz) ANSYS 97.4 (Hz)

Experiment – Experiment –

Error 3.15 % Error 4.69 %


The analytical results differ slightly from those obtained using FEM analysiswhich may be because the analysis did not include shear deformation inertia effect.

5 Conclusions

An analytical model is derived to simulate the second mode of a Jeffcott rotor instart-up and run-down phases in the presence of the gyroscopic effect. The solutionof differential equations for the case study rotor confirms the existence of backwardwhirl which occurs while coasting-down. The critical speed of the case study rotoris determined using ANSYS for the first two bending modes of the rotor. Thecomparison between analytical and FEM models shows approximately a differenceof 4 % in critical speeds of the rotor. The proposed idea for the observation of thebackward whirl which occurs during coast-down of the rotor to rest is experi-mentally demonstrated for a laboratory rotor supported on isotropic bearings. Thebackward whirl is experimentally observed after coasting down through the criticalspeed which is shown in Fig. 9. The comparison of the analytical, FEM andexperimental results shown in Figs. 6, 7, 8, 9, and 10 and Table 2 confirm theexistence of the backward whirling mode which manifests during coast-down.

Acknowledgement Authors wish to thank Mr. Gilles Huard and Mr. Dan Juras for their technicalsupport and guidance in carrying out the experiments.

References

1. Yamamoto Y (1954) On the critical speeds of a shaft, vol 6. Memoirs of the Faculty ofEngineering, Nagoya University, Japan, p 160–174

2. Downham E (1957) Theory of shaft whirling: a fundamental approach to shaft whirling.Engineer 204:518–522, 552–555, 660–665

3. Subbiah R, Bhat RB, Sankar TS, Rao JS (1985) Backward whirl in a simple rotor supported onhydrodynamic bearings. In: Proceedings of a symposium sponsored by Bently Rotor DynamicsResearch Corporation and held in Carson City, June 1985, in NASA conference publication

4. Rao JS (1982) Conditions for backward synchronous whirl of a flexible rotor in hydrodynamicbearings. Mech Mach Theory 17(2):143–152

5. Rajalingham C, Ganesan N, Prabhu BS (1986) Conditions of backward whirling motion of aflexible rotor supported on hydrodynamic journal bearings. J Sound Vib 111:29–36

6. Rao C, Bhat RB, Xistris GD (1996) Experimental verification of simultaneous forward andbackward whirling at different points of Jeffcott rotor supported on identical journal bearings.J Sound Vib 198(3):379–388

7. Sekhar AS (2004) Detection and monitoring of crack in a coast-down rotor supported on fluidfilm bearings. Tribol Int 37:279–287

8. Lee CW (1993) Vibration analysis of rotors. Springer, Dordrecht


[mechanisms and machine science] vibration engineering and technology of machinery volume 23 ||...

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