Experimental Analysis of Flexibly Coupled Two Rotors

Yohta KUNITOH†, Hiroshi YABUNO††, Hossain Md. Zahid‡, Tsuyoshi INOUE‡, and Yukio ISHIDA‡

†Graduate School of Systems and Information Engineering, University of Tsukuba1-1-1 Tennodai, Tsukuba-shi, Ibaraki 305-8573, Japan,

††Institute of Engineering Mechanics and Systems, University of Tsukuba1-1-1 Tennodai, Tsukuba-shi, Ibaraki 305-8573, Japan,

Email: kunitoh@aosuna.esys.tsukuba.ac.jp, yabuno@esys.tsukuba.ac.jp‡Department of Electronic-Mechanical Engineering, School of Engineering, Nagoya University

Furo-cho, Chikusa-ku, Nagoya 464-8603 , Japan,Email: ishida@nuem.nagoya-u.ac.jp

Abstract—Rotating machineries are the most widelyused elements in mechanical systems. In many systems,such as power plant and jet engine, the rotating machineryis multi-span rotor, which is supported by multiple bear-ings. In this research, we consider a 2-DOF nonlinear ro-tary system in which the two nonlinear rotors are coupledby means of a weakly linear stiffness. We show nonlinearnormal modes in this system using the method of multiplescales. Furthermore, we indicate the occurrence of modelocalization in this system.

1. Introduction

Rotating machineries, such as steam turbines, gas tur-bines and motors, are the most widely used elements inmechanical systems. However, the rotating parts of suchmachineries often become the main source of vibration.Hence, analyzing the source of vibration is the critical is-sue, in order to enhance the stability and the reliabilityof mechanical systems. In many systems, such as powerplant and jet engine, the rotating machinery is multi-spanrotor, which is supported by multiple bearings. There-fore, the multi-span rotor is characterized as the systemin which each rotor with nonlinear spring characteristic iscoupled by means of a weakly linear stiffness. In thesesystems, there is a possibility of the occurrence of modelocalization[1]∼[3] as indicated in the multiple degree offreedom spring-mass systems. Recently, also in the non-linear systems such as multi-span rotor, nonlinear normalmodes has attracted attention [4]∼[6].

In this research, we consider a 2-DOF nonlinear rotarysystem in which the two nonlinear rotors are coupled bymeans of a weakly linear stiffness. We show nonlinearnormal modes in this system using the method of multi-ple scales. Then, we indicate the occurrence of mode lo-calization in this system. Furthermore, we experimentallyobserve the orbits of rotors.

m mCoupling

Rotor1 Rotor2

l l

Figure 1: Analytical model

y1

x1

G(xG1, yG2)

M(x1, y1)

edωt

(a) Rotor1

O

y2

x2

M(x2, y2)

(b) Rotor2

O

Figure 2: Coordinate system

2. Analytical model and dimensionless equation of mo-tion

As shown in Fig. 1, we consider the behavior of the sys-tem in which the two Jeffcott rotors are coupled by meansof a weakly linear stiffness. Here, Jeffcott rotor is the rotorin which a rigid disk is mounted at the center of a masslesselastic shaft supported at both ends by ball bearings. Weintroduce the coordinate system as shown in Fig. 2. Theorigin O of the coordinate system O-xy coincides with thebearing centerline. In Fig. 2, the point G of rotor 1 devi-ates slightly (ed) from the geometrical center M, where Gand M are the center of gravity and the geometrical cen-ter, respectively. On the other hand, the setup of rotor 2 iswell-assembled. Moreover, considering the cubic nonlin-earity from the characteristic of spring of bearing and theshaft elongation, the equation of motion of the system can

2004 International Symposium on NonlinearTheory and its Applications (NOLTA2004)

Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

461

be written as follows:

mx1+cd x1+kx1+γd x2+βd(x21+y2

1)x1 =medω2 cosωt(1)

my1+cdy1+ky1+γdy2+βd(x21 + y2

1)y1 =medω2 sinωt (2)

mx2+cd x2+kx2+γd x1+βd(x22 + y2

2)x2 = 0 (3)my2+cdy2+ky2+γdy1+βd(x2

2 + y22)y2 = 0, (4)

where ω, cd, k, βd and γd are the angular velocity, theviscous damping coefficient, the linear spring constant ofthe elastic shaft, the nonlinear spring constant and the linearspring constant of the coupling, respectively.

Next, we rewrite Eqs. (1)-(4) in the dimensionless form.Using the inverse value of the frequency Ω =

√k/m as the

representative time Tr and the span l as the representativelength, we set the dimensionless parameters as follows:

t = (1/Ω)t∗, x1 = lx∗1, y1 = ly∗1, x2 = lx∗2, y2 = ly∗2.

Hereafter the asterisk is omitted.Hence, we obtain the following dimensionless equa-

tions:

x1 + cx1 + x1 + γx2 + β(x21 + y2

1)x1 = eν2 cos νt (5)y1 + cy1 + y1 + γy2 + β(x2

1 + y21)y1 = eν2 sin νt (6)

x2 + cx2 + x2 + γx1 + β(x22 + y2

2)x2 = 0 (7)y2 + cy2 + y2 + γy1 + β(x2

2 + y22)y2 = 0, (8)

where the dimensionless parameters, e∗, c∗, γ∗, β∗ andν∗, are expressed as follows:

e∗ =ed

l, c∗ =

cd√mk, γ∗ =

γd

k, β∗ =

βdl2

k, ν∗ =

ω

Ω.

3. Derivation of the approximate solution using themethod of multiple scales

We analyze Eqs. (5)-(8) as 4-DOF equations of oscilla-tions. By using a small parameter ε (| ε | 1) as a bookingdevice, we quantitatively set the magnitudes of the param-eters as follows:

e = ε3e, c = ε2c, γ = ε2γ,

where ( ˆ ) denotes ”of the order O(1)”. We seek theapproximate solutions of Eqs. (5)-(8) in the form

x1 = εx11 + ε3 x13 + · · · (9)

y1 = εy11 + ε3y13 + · · · (10)

x2 = εx21 + ε3 x23 + · · · (11)

y2 = εy21 + ε3y23 + · · · . (12)

We introduce the multiple time scales as follows:

t0 = t, t2 = ε2t,

where t0 is the fast time scale, and t1 is slow time scale.By applying the method of multiple scales, the followingapproximate solutions are obtained:

x1 = ax1 cos(νt + ϕx1) + O(ε3) (13)y1 = ay1 cos(νt + ϕy1) + O(ε3) (14)x2 = ax2 cos(νt + ϕx2) + O(ε3) (15)y2 = ay2 cos(νt + ϕy2) + O(ε3). (16)

Then, ax1, ϕx1, ay1, ϕy1, ax2, ϕx2, ay2 and ϕy2 are com-puted by solving the following set of eight modulationequations:

ddt

ax1 =−12

cax1 +12γax2 sin (ϕx1 − ϕx2)

+18βax1a2

y1 sin 2(ϕx1 − ϕy1) − 12

e sin ϕx1 (17)

ax1ddtϕx1 =−σax1 +

12γax2 cos(ϕx1 − ϕx2)

+38βa3

x1 +14βax1a2

y1

+18βax1a2

y1 cos 2(ϕx1 − ϕy1) − 12

e cosϕx1 (18)

ddt

ay1=−12

cay1 +12γay2 sin(ϕy1 − ϕy2)

−18βa2

x1ay1 sin 2(ϕx1 − ϕy1) − 12

e cosϕy1 (19)

ay1ddtϕy1 =−σay1 +

12γay2 cos(ϕy1 − ϕy2)

+38βa3

y1 +14βa2

x1ay1

+18βa2

x1ay1 cos 2(ϕx1 − ϕy1) +12

e sin ϕy1 (20)

ddt

ax2 =−12

cax2 − 12γax1 sin(ϕx1 − ϕx2)

+18βax2a2

y2 sin 2(ϕx2 − ϕy2) (21)

ax2ddtϕx2 =−σax2 +

12γax1 cos(ϕx1 − ϕx2) +

38βa3

x2

+14βax2a2

y2 +18βax2a2

y2 cos 2(ϕx2 − ϕy2) (22)

ddt

ay2=−12

cay2 − 12γay1 sin(ϕy1 − ϕy2)

−18βa2

x2ay2 sin 2(ϕx2 − ϕy2) (23)

ay2ddtϕy2 =−σay2 +

12γay1 cos(ϕy1 − ϕy2) +

38βa3

y2

+14βa2

x2ay2 +18βa2

x2ay2 cos 2(ϕx2 − ϕy2), (24)

where ν = 1 + σ.

4. Mode localization

In Fig. 3, the frequency response curves are depictedwhen c = 2.14 × 10−3, γ = −1.09 × 10−2, β = 2.52 ×

462

6

4

2

0

100x10-3500-50

6

4

2

0

100x10-3500-50

8x10-3 8x10-3

σ σ

ax

1

ax

2

(a) x direction of rotor1 (b) x direction of rotor2

Figure 3: Frequency response curve (c = 2.14 × 10−3, γ = −1.09 × 10−2, β = 2.52 × 103, e = 3.85 × 10−5, —— : stable,- - - - : unstable)

103 and e = 3.85 × 10−5, these values correspond to thoseof the subsequent experiment. The solid and dashed linesdenote stable and unstable equilibrium points, respectively.From Eqs. (5)-(8), the motions in the x and y-directions aresymmetric, the vibrations in the x-direction has the sameamplitude as in the y-direction, and the phase difference ofπ/2 from in the y-direction. It can be found that bifurcatednonlinear normal modes are localized. Mode localizationoccurs as shown with the symbols© and in Fig. 3.

Figures 4-6 express the orbits of rotors at the rotor speedsshown with the symbols , © and in Fig. 3. At the con-dition of the symbol , where the nonlinear normal modesis not bifurcated, the amplitude of rotor 1 is almost equal tothat of rotor 2 as shown in Fig. 4. At the symbol ©, eventhough the rotor 1 has unbalance, the amplitude of rotor 2is much larger than that of rotor 1 as shown in Fig. 5. Dueto the occurrence of such a response, there is a possibilityof wrong diagnosis that rotor 2 has unbalance. On the otherhand, this phenomenon indicates that the rotor 2 can be uti-lized as a dynamic vibration absorber. At the symbol , theamplitude of rotor 1 is much larger than that of rotor 2 asshown in Fig. 6. The usage of this phenomenon preventsthe influence of the oscillation in the rotor 1 caused by theunbalance on the rotor 2.

5. Experiments

5.1. Experimental setup

Figure 7 shows the experimental setup. An elastic shaftwith circular cross section with a length l = 0.708 m and adiameter 1.2 × 10−2 m is supported at both ends by a self-aligning double-ball bearing (1200) and a single-row deepgroove ball bearing (6804). A disk is mounted at the cen-ter of the shaft. The disk is 0.3 m in diameter and 8.21kg. The two rotors like this are coupled by spring in imita-tion of a flange type shaft coupling. The shaft is driven bythe three-phase induction motor (Meidensha Corp., TIS85–NR) through V-belt and V-pulley. The motions of disks andthe angular velocities are measured from the laser sensorsand the rotary encoders, respectively. Other parameters areas follows:

k = 3.28 × 104 N/m, βd = 1.65 × 108 N/m3, γd = −358 N.

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6x10-3

-6 -4 -2 0 2 4 6x10-3

x10-3

-6

-4

-2

0

2

4

6x10-3

x2

y 2

x1

y 1

(a) Rotor1 (b) Rotor2

Figure 4: Theoretical orbit (σ = 0.0015, at the symbol inFig. 3)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6x10-3

x10-3

x2

y 2

-6 -4 -2 0 2 4 6x10-3

-6

-4

-2

0

2

4

6x10-3

x1

y 1

(a) Rotor1 (b) Rotor2

Figure 5: Theoretical orbit (σ = 0.0207, at the symbol ©in Fig. 3)

-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6x10-3

x10-3

x2

y 2

-6 -4 -2 0 2 4 6x10-3

-6

-4

-2

0

2

4

6x10-3

x1

y 1

(a) Rotor1 (b) Rotor2

Figure 6: Theoretical orbit (σ = 0.0597, at the symbol inFig. 3)

463

Rotor1 Rotor2

Coupling

Three-phase

Induction Motor

ωmControl

Signal

Rotary

Encoder

ω1

Rotary

Encoder

ω2

Motor Inverter

x1

x2

y2

y1

Laser Sensor (1) Laser Sensor (2)

PC

FFT Analyzer

Figure 7: Experimental setup

5.2. Experimental results and discussion

In Figs. 8-10, we show the experimantal orbits of therotors. In previous section, we analyzed the motion of rotoras circular motion. Indeed, the experimental motion can beexpressed the elliptical motion as a summation of forwardwhirling motion and backward whirling motion. Therefore,we compare the amplitudes of orbits afterward. In the caseωr(angular velocity)×60/2π = 600 rpm, the amplitude ofrotor 1 is almost equal to that of rotor 2 as shown in Fig. 8.Moreover, in the case ωr × 60/2π = 641 rpm, even thoughrotor 1 has unbalance, the amplitude of rotor 2 is largerthan that of rotor 1 as shown in Fig. 9. On the other hand,in the case ωr × 60/2π = 624 rpm, the amplitude of rotor1 is much larger than that of rotor 2 as shown in Fig. 10.We experimentally confirm the occurrence of theoreticallypredicted two types of mode localizations.

6. Conclusions

In this research, we investigate the nonlinear normalmodes in 2-DOF nonlinear rotary system in which the twononlinear rotors are coupled by a weakly linear stiffness.Analyzing the equations of motion as 4-DOF equations ofoscillations, we show the occurrence of mode localizationsin this system. Moreover, based on mode localization, weindicate the possibility of vibration suppression and wrongdiagnosis. Furthermore, we experimentally observe theo-retically predicted mode localizations.

Acknowledgments

This work was supported by TEPCO Research Founda-tion.

References

[1] Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuk,V. N., and Zevin, A. A., Normal Modes and Localization inNonlinear Systems, (1996), John Wiley & Sons, Inc., NewYork.

-1.0

0.0

1.0

-1.0 0.0 1.0

x1 [mm]

y 1 [

mm

]

-1.0

0.0

1.0

-1.0 0.0 1.0

x2 [mm]

y 2 [

mm

]

(a) Rotor1 (b) Rotor2

Figure 8: Experimental orbit (ωr × 60/2π = 600 rpm)

-1.0

0.0

1.0

-1.0 0.0 1.0

x1 [mm]

y 1 [

mm

]

-1.0

0.0

1.0

-1.0 0.0 1.0

x2 [mm]

y 2 [

mm

]

(a) Rotor1 (b) Rotor2

Figure 9: Experimental orbit (ωr × 60/2π = 641 rpm)

-1.0

0.0

1.0

-1.0 0.0 1.0

x1 [mm]

y 1 [

mm

]

-1.0

0.0

1.0

-1.0 0.0 1.0

x2 [mm]

y 2 [

mm

](a) Rotor1 (b) Rotor2

Figure 10: Experimental orbit (ωr × 60/2π = 624 rpm)

[2] Aubrecht, J. and Vakakis, A. F., Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span BeamWith Geometric Nonlinearities, Journal of Vibration andAcoustics, 118 (1996), 553–542.

[3] Pierre, C., Tang, D. M. and Dowell, E. H., Localized Vibra-tions of Disordered Multispan Beams: Theory and Experi-ment, AIAA Journal, 25-9 (1987), 1249–1257.

[4] Pesheck, E., Pierre, C. and Shaw, S. W., Accurate Reduced-Order Models for a Simple Rotor Blade Model Using Nonlin-ear Normal Modes, Mathematical and Computer Modeling,33 (2001), 1085–1097.

[5] Yabuno, H. and Nayfeh A. H., Nonlinear Normal Modes of aParametrically Excited Cantilever Beam, Nonlinear Dynam-ics, 25 (2001), 65–77.

[6] Kanda, R., Yabuno, H., Zahid, H. Md., Inoue, T., and Ishida,Y., Nonlinear Normal Modes in 2-DOF Nonlinear Systemunder Forced Excitation, D&D Conference 2003 CD-ROM,No. 419 (2003) (in Japanese).

[7] Yamamoto, T., and Ishida, Y., Linear and Nonlinear Rotor-dynamics, (2001), John Wiley & Sons, Inc., New York.

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