Nonlinear vibration analysis of a cracked rotor-ball bearing system during flight maneuvers

Mechanism and Machine Theory 105 (2016) 515–528

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Nonlinear vibration analysis of a cracked rotor-ball bearing system during flight maneuvers Lei Hou a,b,⁎, Yushu Chen a, Qingjie Cao a, Zhenyong Lu a a b

School of Astronautics, Harbin Institute of Technology, Harbin 150001, PR China School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e

i n f o

Article history: Received 4 September 2014 Received in revised form 29 July 2016 Accepted 30 July 2016 Available online xxxx Keywords: Nonlinear vibration Cracked rotor-ball bearing system Maneuver load Climbing-diving flight

a b s t r a c t This paper focuses on the nonlinear responses of a cracked rotor-ball bearing system caused by aircraft flight maneuvers. The equations of motion of the system are formulated with the consideration of the breathing mechanism of a transverse crack and the maneuver load of a climbing-diving flight. The fourth order Runge-Kutta method is employed to detect the nonlinear responses of the system, which are reflected by bifurcation diagrams, power spectrums, maximum Lyapunov exponent, phase portraits and Poincaré sections. It is shown that the super-harmonic responses of the system are affected significantly by the maneuver load under sub-critical speeds. Plenty of quasi-periodic motions are obtained, and a variety of complex nonlinear behaviors including bifurcations and jumping phenomenon are observed near 1/4, 1/3, 2/5 and 1/2 critical speeds when the maneuver load increases from 0 to 10 g. The nonlinear responses of the system influenced by crack stiffness, bearing clearance and rotor eccentricity are also investigated. Chaotic motions are demonstrated when the crack stiffness or the bearing clearance increases across a critical value. However, the responses maintain quasi-periodic when the rotor eccentricity changes. The results will contribute to a better understanding of the nonlinear dynamic behaviors of cracked rotors in flight maneuvers. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Considerable attention has been paid to crack fault, which is one of the most serious damage in aircraft engines and other rotating machines, in the last three decades [1]. Wauer [2], Gasch [3] and Dimarogonas [4] reviewed the dynamical behavior of rotor systems with transverse cracks, in which, highly nonlinear vibrations were shown. Switching crack model (also known as hinge model) [3] and response-dependent breathing crack model [5,6] were proposed in early times, based on which, the dynamical comparison of the two models [7], the critical speed influenced by the nonlinear breathing of the crack and the imbalance orientation angle of the rotor [8], and the stability of periodic movements in cracked rotor systems [9–11] were investigated to detect the nonlinear dynamics of cracked rotor systems. Finite element models [12–14] were presented afterwards to study the dynamic behaviors of rotor systems affected by crack depth, position and type (transverse or slant). Harmonic balance method [15], alternate frequency/time domain approach [16] and experimental methods [17,18] were also developed to gain an insight into the dynamical characteristics of cracked rotors. From the above studies, the 2 × and 3 × super-harmonic frequency components are shown as distinct signals, based on which, the diagnostic tools that the changes in natural frequencies and evolution of the non-linear behavior of the system at the super-harmonic frequency components are proposed to gain crack detection strategies [19–24]. ⁎ Corresponding author at: School of Astronautics, Harbin Institute of Technology, Harbin 150001, PR China. E-mail address: [email protected] (L. Hou).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.024 0094-114X/© 2016 Elsevier Ltd. All rights reserved.

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L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

In recent years, a significant amount of research has been conducted in the area of coupling problems [25–30] and multicracks [31,32] in cracked rotor systems. It is shown that the excitation in one mode may lead to an interaction between all the modes due to the coupling between longitudinal, lateral and torsional vibrations [25], and coupled modes induce a quasiperiodic motion or even non-periodic behavior in the region of internal resonance [26,27]. Unique features of nonlinear vibrations are shown in cracked rotor systems respectively coupled with rub excitation [28], bow [29] and misalignment [30], which are useful for the identification of crack faults. It is also shown that the dynamic response in the Jeffcott rotor system with two transverse surface cracks is affected significantly by the angular orientation of one crack relative to the other [31]. Sekhar [32] summarized the different studies on double/multi-cracks to bring out the state of the research on multiple cracks and their identification methods in vibration structures. Generally, gravity that plays an important role named weight dominance in the crack breathing in normal rotor systems is a constant force [33,34]. In an aircraft rotor system, however, the maneuver load that plays the same role as gravity may increase from 0 to highly 10 times of gravity in flight maneuvers [35,36], which makes a great influence on the dynamic behaviors of nonlinear rotor system. Lin et al. [37] has investigated the nonlinear dynamics of a cracked rotor by considering the flight maneuver model with a constant flight speed or a constant acceleration, in which, it was shown that the climbing angle, acceleration, and other flight parameters make significant influences on the parameter range for bifurcation, quasi-periodic response and chaotic response as well as system stability. Yang et al. [38] found three different ways for the vibration response going to chaos: quasi-periodic, intermittence and period-3 bifurcation, in a cracked rotor system under the maneuver load of hovering flight. Hou et al. [39,40] found that sub-harmonic resonance is a nonlinear mode inducing rub-impact in maneuvering rotor systems. The motivation of this paper is to detect the nonlinear dynamic behaviors of an aircraft cracked rotor-ball bearing system under maneuver load. The equations of motion proposed herein enable us to investigate the nonlinear responses of the system considering the effects of the maneuver load induced by the climbing-diving flight, which is a maneuver in the vertical plane. Numerical technique is employed to detect the nonlinear dynamic behaviors of the system when the crack is not very deep. Nonlinear behaviors such as super-harmonic resonances, jumping phenomenon, and plenty of quasi-periodic and chaotic motions are obtained due to the effect of the maneuver load. The paper is organized as follows. Firstly, the model of a ball bearing supported rotor system with a breathing transverse crack is presented, in which, the crack breathing model combining both switching and cosine functions, and the maneuver load of a divingclimbing flight model are considered. Secondly, the super-harmonic responses of the system affected by the maneuver load are analyzed in detail from different aspects by using bifurcation diagrams, power spectrums, maximum Lyapunov exponent, phase portraits and Poincaré sections. Finally, the nonlinear responses of the system influenced by crack stiffness, bearing clearance and rotor eccentricity are respectively investigated to give an insight into the evolution of the system behavior as the increase of the parameters. 2. Mathematical model 2.1. Cracked rotor-ball bearing system Consider the ball bearing supported rotor system with a transverse crack on the shaft at the bottom of the disk as shown in Fig. 1, where o is the left endpoint of the shaft, and it is assumed that o is also the gravity center of the aircraft, o' and om are, respectively, the geometric center and the mass center of the disk, and e is the rotor eccentricity between om and o'. The motion of the system is modeled by six degrees of freedom: the vertical displacement y and the horizontal displacement z of the disk, and the vertical displacement yb1 and yb2, the horizontal displacement zb1 and zb2 of the two bearings. k and c are the stiffness and the damping of each side of the shaft. m, mb1 and mb2 are the masses of the disk and the two bearings respectively. 2.2. Maneuver load Fig. 2 shows the flight maneuver model discussed in this study, where it is assumed that the angular velocity ωmaneuver and the speed v of the aircraft are constants in the climbing and diving flight, and o is the gravity center of the aircraft corresponded with Fig. 1. The maneuver loads of the system at the six degrees of freedom can be written as T

F maneuver ¼ ½mωmaneuver v 0 mb1 ωmaneuver v 0 mb2 ωmaneuver v 0 :

Fig. 1. Schematic diagram of a cracked rotor-ball bearing system.

ð1Þ

L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

517

Fig. 2. Climbing-diving flight model.

2.3. Ball bearing force ri ω is Fig. 3 is a schematic diagram of the ball bearing model with a clearance δ0, in which, Nb is the number of balls, ωb ¼ ri þr o the rotation speed of the cage, where ri and ro are the inner and the outer race curvature radiuses, ω is the rotation speed of the rotor. It is supposed that the load–deformation relationship between the ball and raceways satisfy the Hertzian contact of elastic points, then the restoring forces generated by the ball bearings can be obtained as follows [41–43].




fy fz

 ¼ −K b

Nb X

δj

1:5

j¼1

 
cos θ  j ; H δj sin θ j

ð2Þ 

1; N 0 is Heaviside function which shows the 0;  ≤ 0 contact situation between the balls and outer ring of the bearing, Kb is the Hertzian contact stiffness. where δj(x, y, t) = y cos θj + z sin θj − δ0, θ j ¼ N2πb ðj−1Þ þ ωb t, j = 1 , … , Nb, HðÞ ¼

2.4. Crack stiffness A transverse crack model is presented in Fig. 4, where o′ζη is the rotating coordinate system with respect to the crack orientation, αis 1/2 of the crack angle, and φ ¼ tan−1 yz. The growth of the crack is not considered in this paper, in other words, it is supposed that the crack keeps the same size for a certain simulation condition. The stiffness of the cracked shaft is denoted as follows. 2

ky ¼ k−f ðθÞΔk cos ðθ þ φÞ;

ð3aÞ

Fig. 3. Ball bearing model.

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Fig. 4. Schematic diagram of the cracked cross-section of the shaft.

2

kz ¼ k−f ðθÞΔk sin ðθ þ φÞ;

ð3bÞ

kyz ¼ −f ðθÞΔk sinðθ þ φÞ cosðθ þ φÞ;

ð3cÞ

in which, f(θ) is the breathing of the crack, Δk is the crack stiffness implying the depth of the crack, θ =ωt+ β − φ, where β is the angle between the crack and the unbalance of the rotor.

Fig. 5. Bifurcation diagrams of horizontal displacement with different values of maneuver load. (a) G = 0, (b) G = 1, (c) G = 3, (d) G = 5.

L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

519

Formula (4) denotes the crack breathing model used in this paper, which combines both switching and cosine functions. This model describes not only the open and closed lasting states, but also the transition procedure of the two states, and it has been proved to be accurate by experimental investigations [6].

f ðθÞ ¼

8 > > 1; > > > 0 1 > > π > > θ− þ α > 1 > B C 2 > > πA; @1 þ cos > > 2α > < 2 > > 0; > > > 0 1 > > 3π > > θ− −α > > 1 B C > 2 > π A; @1 þ cos > > 2α >2 :

π π − þ α ≤ θ b −α; 2 2 π π −α ≤ θ b þ α; 2 2 ð4Þ

π 3π þα ≤ θb −α; 2 2 3π 3π −α ≤ θ b þ α: 2 2

2.5. Equations of motion The equations of motion are formulated as follows by using the Lagrange equation [39] 2

€ þ cð2y− _ y_ b1 −y_ b2 Þ þ ky ð2y−yb1 −yb2 Þ þ kyz ð2z−zb1 −zb2 Þ ¼ mω e cos ωt þ mωmaneuver v; my

ð5aÞ

2 m€z þ cð2z_ −z_ b1 −z_ b2 Þ þ kz ð2z−zb1 −zb2 Þ þ kyz ð2y−yb1 −yb2 Þ ¼ mω e sin ωt;

ð5bÞ

€b1 þ cðy_ b1 −y_ Þ þ ky ðyb1 −yÞ þ kyz ðzb1 −zÞ ¼ f yb1 þ mb1 ωmaneuver v; mb1 y

ð5cÞ

20 Amplitude of Z

Amplitude of Z

20

10

0 1000

10

ω

2ω

1.5ω

2.5ω

0 1000 2000 500

ω (rad/s)

(a)

2000 500

1000 0

Frequency (rad/s)

ω (rad/s)

(b)

0

Frequency (rad/s)

20 2ω

10

ω

0 1000

2.5ω 3ω

Amplitude of Z

Amplitude of Z

20

1000

4ω 5ω

2ω 10

ω

0 1000

2.5ω 3ω

4ω 5ω

2000 500

(c)

ω (rad/s)

2000 500

1000 0

Frequency (rad/s)

(d)

ω (rad/s)

1000 0

Frequency (rad/s)

Fig. 6. Power spectrums of horizontal displacement for ω varying from 0 to 1000 rad/s with different values of maneuver load. (a) G = 0, (b) G = 1, (c) G = 3, (d) G = 5.

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L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528 4.64

Y

Y

Z

4.2 -1 2.662

Amplitude of Z

4.62

4.4 0

4.6

4 2.664 τ

(a)

4

4.6

1

2.666

-1 4

x 10

(b)

0 Z

4.58 0.7

1

2 ω

0 0.8

0.9

1

Z

(c)

4ω (ω c)

0

2ω

5ω

500

1000

1500

Frequency (rad/s)

(d)

Fig. 7. Dynamical response for ω=263 rad/s. (a) Horizontal displacement, (b) phase portrait, (c) Poincaré section, (d) power spectrum of horizontal response.

mb1 €zb1 þ cðz_ b1 −z_ Þ þ kz ðzb1 −zÞ þ kyz ðyb1 −yÞ ¼ f zb1 ;

ð5dÞ

€b2 þ cðy_ b2 −y_ Þ þ ky ðyb2 −yÞ þ kyz ðzb2 −zÞ ¼ f yb2 þ mb2 ωmaneuver v; mb2 y

ð5eÞ

mb2 €zb2 þ cðz_ b2 −z_ Þ þ kz ðzb2 −zÞ þ kyz ðyb2 −yÞ ¼ f zb2 ;

ð5fÞ

where c = 2ξmω. The dimensionless equations of Eqs. (5a)–(5f) can beq obtained as follows by letting Y ¼ yδ, Z ¼ δz, Y 1 ¼ yδb1, Z 1 ¼ zδb1, Y 2 ¼ yδb2, Z 2 ¼ zδb2, ffiffiffiffi mg m e 2k , n ¼ , E ¼ , where δ ¼ , ω ¼ τ = ωt, s ¼ ωωc0 , K ¼ Δk , and it is supposed that the two bearings are the same with the mass c0 mb δ m 2k k mb1 = mb2 = mb. i  0 1 Kh G ″ 0 0 2 Y þ 2ξ 2Y −Y 1 −Y 2 þ 2 ð2Y−Y 1 −Y 2 Þ−f 2 cos ðθ þ φÞð2Y−Y 1 −Y 2 Þ þ sinðθ þ φÞ cosðθ þ φÞð2Z−Z 1 −Z 2 Þ ¼ E cos τ þ 2 ; s s s

ð6aÞ

i  0 1 Kh ″ 0 0 2 Z þ 2ξ 2Z −Z 1 −Z 2 þ 2 ð2Z−Z 1 −Z 2 Þ− f 2 sinðθ þ φÞ cosðθ þ φÞð2Y−Y 1 −Y 2 Þ þ sin ðθ þ φÞð2Z−Z 1 −Z 2 Þ ¼ E sin τ; s s

ð6bÞ

pffiffiffi i  0 n Kh G δK b ″ 0 2 F y1 þ 2 ; Y 1 þ 2nξ Y 1 −Y þ 2 ðY 1 −Y Þ−f 2 cos ðθ þ φÞðY 1 −Y Þ þ sinðθ þ φÞ cosðθ þ φÞðZ 1 −Z Þ ¼ − s s s mb ω2

ð6cÞ

i pffiffiffiffiffiffiffiffi  0 δK b n Kh ″ 0 2 Z 1 þ 2nξ Z 1 −Z þ 2 ðZ 1 −Z Þ−f 2 sinðθ þ φÞ cosðθ þ φÞðY 1 −Y Þ þ sin ðθ þ φÞðZ 1 −Z Þ ¼ F z1 ; s s mb ω2

ð6dÞ

5.5

5

5

15 Amplitude of Z

6

Y

0

Y

Z

5

4

4.5

3

-5 2.662

2.664 τ

(a)

2.666

-5 4

x 10

(b)

0 Z

4 -1.6

5

(c)

3ω ( ω c) 10 5 ω 0

-1.4 Z

-1.2

0

(d)

2ω

500 1000 1500 Frequency (rad/s)

Fig. 8. Dynamical response for ω=352 rad/s. (a) Horizontal displacement, (b) phase portrait, (c) Poincaré section, (d) power spectrum of horizontal response.

5

8 6

-5 2.662

(a)

2.664 τ

2.666 4

x 10

(b)

Y

Y

Z

6

0

Amplitude of Z

8

4

4

2

2

0 -5

0 -5

0 Z

5

(c)

0 Z

5 ω 0

5

(d)

2.5ω ( ω c)

10

0

2ω

500 1000 1500 Frequency (rad/s)

Fig. 9. Dynamical response for ω=431 rad/s. (a) Horizontal displacement, (b) phase portrait, (c) Poincaré section, (d) power spectrum of horizontal response.

L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

Y

Z

0 -5

5

0.6

0

-10 2.662

0.8 Y

5

10

0.4

-5 2.664

τ

2.666

-10 4

x 10

(b)

Amplitude of Z

1

10

(a)

521

0 Z

0.2 -9

10

(c)

-8.8

-8.6 Z

20 10

ω 0

-8.4

(d)

2ω (ω c)

30

0

500 1000 1500 Frequency (rad/s)

Fig. 10. Dynamical response for ω=542 rad/s. (a) Horizontal displacement, (b) phase portrait, (c) Poincaré section, (d) power spectrum of horizontal response.

pffiffiffi i  0 n Kh G δK b ″ 0 2 Y 2 þ 2nξ Y 2 −Y þ 2 ðY 2 −Y Þ−f 2 cos ðθ þ φÞðY 2 −Y Þ þ sinðθ þ φÞ cosðθ þ φÞðZ 2 −Z Þ ¼ − F y2 þ 2 ; s s s mb ω2

ð6eÞ

pffiffiffi i  0 n Kh δK b ″ 0 2 F z2 ; Z 2 þ 2nξ Z 2 −Z þ 2 ðZ 2 −Z Þ−f 2 sinðθ þ φÞ cosðθ þ φÞðY 2 −Y Þ þ sin ðθ þ φÞðZ 2 −Z Þ ¼ − s s mb ω2

ð6fÞ

v in which, G ¼ ωmaneuver is the dimensionless maneuver load, the value of which refers to the multiple of g, Fyi and Fzi refer to the g dimensionless bearing forces that are as follows.




F yi F zi

 ¼

Nb  1:5  
cos θ  X 2π ri j ; θj ¼ Y i cos θ j þ Z i sin θ j −1 H Y i cos θ j þ Z i sin θ j −1 ð j−1Þ þ τ; j ¼ 1; …; N b : sin θ j r þ ro N b i j¼1

ð7Þ

Fig. 11. Bifurcation diagrams for horizontal displacement versus G with different values of rotating speed. (a) ω=263 rad/s, (b) ω=352 rad/s, (c) ω=431 rad/s, (d) ω=542 rad/s.

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The approximate solutions of Eqs. (6a)–(6f) can be obtained by using numerical methods. The calculation parameters of the system are as follows [44,45] −1

7

m ¼ 32:1 kg; e ¼ 0:01 mm; k ¼ 1:25  10 N m ; ξ ¼ 0:01; 10 −1:5 ; mb1 ¼ 4 kg; mb2 ¼ 4 kg; Nb ¼ 8; K b ¼ 1:334  10 N m ∘ ∘ δ0 ¼ 0; r i ¼ 40:1 mm; r o ¼ 63:9 mm; K ¼ 0:1; β ¼ 60 ; α ¼ 30 :

ð8Þ

In this paper, the 4th order Runge-Kutta method is employed to carry out the numerical calculations for Eq. 6, and the vibration responses of the system are obtained accordingly and discussed in detail.

3. Numerical results and discussions 3.1. Vibration response of the cracked rotor system under maneuver load

2ω

0.2 0.1

3ω

ω

1

5ω 4ω ( ω c)

Amplitude of Z

Amplitude of Z

The bifurcation diagrams of the system with different values of maneuver load are shown in Fig. 5, from which, the significant effects of maneuver load on the nonlinear vibrations of the system under sub-critical speeds are observed, furthermore, the nonlinear responses are getting stronger as the maneuver load increases from G = 1 to G = 5. The power spectrums of the system corresponded to Fig. 5 for ω varying from 0 to 1000 rad/s are shown in Fig. 6. Compared with Fig. 6(a), the super-harmonic responses are apparent in Fig. 6(b), (c) and (d), and a peak value of super-harmonic response is reached at some certain speeds, e.g., four peak values are observed in Fig.6(c) for G = 3 at ω = 263 rad/s, ω = 352 rad/s, ω = 431 rad/s and ω = 542 rad/s that are near 1/4, 1/3, 2/5 and 1/2 critical speed ωc respectively. Figs. 7 to 10 show the dynamical responses of the system by horizontal displacements, phase portraits, Poincaré sections and power spectrums at ω = 263 rad/s, ω = 352 rad/s, ω = 431 rad/s and ω =542 rad/s for G = 3, where quasi-periodic motions are obtained.

0 10

0.5

2ω ω

0 10

1500

5

3ω ( ω c)

1500

1000 G

500 Frequency (rad/s)

0 0

(a)

5 G

2ω

3ω

ω

0 10

0

500 Frequency (rad/s)

2ω ( ω c)

3 Amplitude of Z

Amplitude of Z

1.5

0.5

0

(b)

2.5ω (ω c)

1

1000

2 1

ω

0 10 1500 5 G

(c)

0

0

1500

1000

5

500 Frequency (rad/s)

G

(d)

1000 0

0

500 Frequency (rad/s)

Fig. 12. Power spectrums of horizontal displacement for G varying from 0 to 10 with different values of rotating speed. (a) ω= 263 rad/s, (b) ω = 352 rad/s, (c) ω=431 rad/s, (d) ω=542 rad/s.

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3.2. Effect of maneuver load Taking maneuver load as the control parameter, the bifurcation diagrams of the horizontal response of the system for G varying from 0 to 10 at ω= 263 rad/s, ω = 352 rad/s, ω =431 rad/s and ω = 542 rad/s are shown in Fig. 11. A variety of complex nonlinear behaviors containing bifurcations and jumping phenomenon are observed, which demonstrates that the maneuver load affects the nonlinear response of the system significantly. Fig. 12 gives the power spectrums for the same parameters as in Fig. 11, in which, the super-harmonic response components affected by maneuver load are shown. Among all the super-harmonic responses, the component near the critical speed shows the most dramatic change as the varying of the maneuver load, and a peak value is reached under some certain maneuver load, which implies the occurrence of the super-harmonic resonance. The maximum Lyapunov exponent corresponded to Fig. 11 are shown in Fig. 13, where almost no positive value can be seen in the whole regions of G, which indicates that the complex nonlinear behaviors the system are quasi-periodic in the main. As mentioned in [37] and [38], due to the effect of the flight maneuver, the response of the rotor system with crack faults will be changed, and may vary among periodic, quasi-periodic and chaotic motions. Besides, the stability of the rotor system may also be changed. These properties can also be found from Figs. 11, 12 and 13. While apart from these, the bearing nonlinearity makes the rotor response more complicated and more unstable. It can be seen that there exists jumping phenomenon in Fig. 11(b), (c) and (d), and that the rotor system behaves quasi-periodic motions during large regions of maneuver load for all of the four different situations in Fig. 11. Moreover, several peaks and complex variations of the maximum Lyapunov exponent can be found in each of the four subfigures in Fig. 13. 3.3. Effect of crack stiffness Taking crack stiffness as the control parameter, the responses of the system for ω = 352 rad/s and G= 3 are shown in Fig. 14 by bifurcation diagram (Fig. 14(a)), maximum Lyapunov exponent (Fig. 14(b)) and power spectrums (Fig. 14(c) and (d)). Positive value of the maximum Lyapunov exponent is obtained in Fig. 14(b) and continuous spectra of the power spectrums is observed in Fig. 14(c) and (d) when K N 0.135, which implies the response of system gets into chaotic. The Poincaré sections for (Z, Y) with different values of K are shown in Fig. 15 to give an insight into the evolution of the system behavior as the increase of crack

-3

5 Maximum Lyapunov exponent

Maximum Lyapunov exponent

0.01

0.005

0

-0.005

-0.01

(a)

0

5 G

10

0

-5

-10

(b)

x 10

0

Maximum Lyapunov exponent

Maximum Lyapunov exponent

(c)

4

-2 -4 -6

0

6

8

6

8

10

-3

x 10

0

-8

4 G

-3

2

2

2

4

6 G

8

2 0 -2 -4 -6 -8

10

(d)

x 10

0

2

4

10

G

Fig. 13. Maximum Lyapunov exponent for G varying from 0 to 10 with different values of rotating speed. (a) ω=263 rad/s, (b) ω=352 rad/s, (c) ω=431 rad/s, (d) ω=542 rad/s.

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L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

Maximum Lyapunov exponent

0.03

0.02

0.01

0

-0.01

(a)

3ω ( ω c) Amplitude of Z

Amplitude of Y

0.1

0.2

0.3

K

3ω (ω c)

3 2 1

2ω

ω 0

4

2

2ω

ω 0

1500

0.2 K

0

0

500 Frequency (rad/s)

1500

0.2

1000

0.1

(c)

0

(b)

1000

0.1

(d)

K

0

0

500 Frequency (rad/s)

Fig. 14. Responses of the system versus K for ω =352 rad/s and G =3. (a) Bifurcation diagram of horizontal displacement, (b) maximum Lyapunov exponent, (c) power spectrums of vertical displacement, (d) power spectrums of horizontal displacement.

stiffness, where quasi-periodic motions for K = 0.1 and K = 0.125 (Fig. 15(a) and (b)) and chaotic motions for K = 0.15, K = 0.2, K = 0.225, K = 0.25, K = 0.275 and K= 0.3 (Fig. 15(c) to (h)) are demonstrated. For the purpose of comparison, the basic properties of system without considering the maneuver load, namely G = 0, are also obtained. The evolutions of the vibration behavior and the stability of the rotor system for ω= 352 rad/s with respect to the crack stiffness are shown in Fig. 16, where Fig. 16(a) and (b) respectively perform the bifurcation diagram of horizontal displacement and the maximum Lyapunov exponent. It is obvious shown that the deeper the crack is, the higher nonlinear the rotor system would perform, and the easier the rotor response may lose stability. This property has also been confirmed by many researches (see [9] and [10]). Considering the maneuver load, however, the responses of the system (see Fig. 14) are more irregular and more unstable, and their bifurcation behaviors are more complicated in comparison with that shown in Fig. 16. Thus it is indicated

Fig. 15. Poincaré sections for (Z, Y) with different values of K. (a) K=0.1, (b) K=0.125, (c) K=0.15, (d) K=0.2, (e) K=0.225, (f) K=0.25, (g) K=0.275, (h) K=0.3.

L. Hou et al. / Mechanism and Machine Theory 105 (2016) 515–528

525

-3

x 10

Maximum Lyapunov exponent

10

5

0

-5

(a)

0

0.1

0.2

0.3

K

(b)

Fig. 16. Responses of the system versus K for ω=352 rad/s and G=0. (a) Bifurcation diagram of horizontal displacement, (b) maximum Lyapunov exponent.

that the maneuver load and its coupling with crack stiffness make the system response more unstable, and lead to complex dynamic behaviors. 3.4. Effect of the clearance of ball bearing Taking bearing clearance as the control parameter, the responses of the system for ω = 352 rad/s, G= 3 and K = 0.1are shown in Fig. 17 by bifurcation diagram (Fig. 17(a)), maximum Lyapunov exponent (Fig. 17(b)) and power spectrums (Fig. 17(c) and (d)), where a variety of strong nonlinear responses are observed. The 3 × super-harmonic exponent of vertical response in

Maximum Lyapunov exponent

0.15

0.1

0.05

0

-0.05

(a)

2 δ0 (m)

1 ω

2ω

ω

0.5 0 3

1500

2

1000

x 10

δ0 (m)

x 10

3ω ( ω c)

1

0 -5

3 -5

2ω

1.5 3ω ( ω c)

Amplitude of Z

Amplitude of Y

1

(b)

2

(c)

0

0

0

500 Frequency (rad/s)

-5

2

x 10

(d)

1 δ0 (m)

1000 0

0

1500

500 Frequency (rad/s)

Fig. 17. Responses of the system versus δ0 for ω=352 rad/s, G=3 and K=0.1. (a) Bifurcation diagram of horizontal displacement, (b) maximum Lyapunov exponent, (c) power spectrums of vertical displacement, (d) power spectrums of horizontal displacement.

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Fig. 18. Poincaré sections for (Z, Y) with different values of δ0. (a) δ0 =2.5 μm, (b) δ0 =5 μm, (c) δ0 =10 μm, (d) δ0 =12.5 μm, (e) δ0 =20 μm, (f) δ0 =25 μm, (g) δ0 =27.5 μm, (h) δ0 =30 μm.

Fig. 17(c) and the 2× super-harmonic exponent of horizontal response in Fig. 17(d) are taking the most important role when δ0 N 13 μm. Positive value of the maximum Lyapunov exponent is obtained in Fig. 17(b) and continuous spectra of the power spectrums is observed in Fig. 17(c) and (d) when δ0 N 0.7 μm, which implies chaotic motion appears in the response of the system. The Poincaré sections for (Z, Y) with different values of δ0 are shown in Fig. 18 to give an insight into the evolution of the system behavior with the increase of bearing clearance, where quasi-periodic motions for δ0 = 2.5 μm and δ0 = 12.5 μm (Fig. 18(a) and (d)) and chaotic motions for δ0 = 5 μm, δ0 = 10 μm, δ0 = 20 μm, δ0 = 25 μm, δ0 = 27.5 μm and δ0 = 30 μm (Fig. 18(b), (c) and (e) to (h)) are demonstrated.

-3

Maximum Lyapunov exponent

2 0 -2 -4 -6 -8 -10

(a)

x 10

0

1

2 e (m)

(b)

3 -5

x 10

3ω ( ω c)

ω

0.2

2ω

0 3 -5

1 0.5

ω

2ω

0 2

x 10

1500 -5

1000

1 e (m)

(c)

1.5

3ω ( ω c) Amplitude of Z

Amplitude of Y

0.4

0

0

500 Frequency (rad/s)

1500

2

1000

x 10

e (m)

(d)

0

0

500 Frequency (rad/s)

Fig. 19. Responses of the system versus e for ω=352 rad/s, G=3, K=0.1 and δ0 =0. (a) Bifurcation diagram of horizontal displacement, (b) maximum Lyapunov exponent, (c) power spectrums of vertical displacement, (d) power spectrums of horizontal displacement.

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It can be obviously seen that the bearing clearance plays an important role in inducing the instability of the system response in flight maneuvers. Consequently, the bearing clearance must be controlled in a sufficient small magnitude for the safe and stable operation of the rotor system. 3.5. Effect of rotor eccentricity Taking rotor eccentricity as the control parameter, the responses of the system for ω= 352 rad/s, G =3, K = 0.1 and δ0 = 0 are shown in Fig. 19 by bifurcation diagram (Fig. 19(a)), maximum Lyapunov exponent (Fig. 19(b)) and power spectrums (Fig. 19(c) and (d)), where a variety of complex dynamical behaviors are observed. Fig. 19(c) and (d) show that the harmonic exponents increase, but the super-harmonic exponents almost remain unchanged with the increase of rotor eccentricity. In Fig. 19(b), no positive value but several bifurcation points can be seen, which implies that the motion of the system remains quasi-periodic. In other words, the eccentricity makes minor effect on the vibration behavior of the maneuver rotor system. 4. Conclusions The nonlinear responses of a cracked rotor-ball bearing system caused by aircraft flight maneuvers has been investigated in this paper. The equations of motion of the system have been formulated with the consideration of a crack breathing model combining both switching and cosine functions and the maneuver load of a climbing-diving flight. The 4th order Runge-Kutta method has been employed to detect the vibration behaviors of the system influenced by maneuver load, crack stiffness, bearing clearance and rotor eccentricity. It has been shown that the super-harmonic responses of the system are affected significantly by the maneuver load under sub-critical speeds even when the crack stiffness is small, i.e., the crack is not very deep. Quasi-periodic motions have been obtained, and a variety of complex nonlinear behaviors including bifurcations and jumping phenomenon have been observed near 1/4, 1/3, 2/5 and 1/2 critical speed when the maneuver load increases from 0 to 10 g. Moreover, chaotic motions have been demonstrated when the crack stiffness or the bearing clearance increases across a critical value. However, the motion of the system remains quasi-periodic with the increase of the rotor eccentricity. In comparison with the recent publications by the current authors [46,47], the results obtained in this paper give a more deep insight into the evolution of the system behavior as the change of the parameters, and will provide a better understanding of the nonlinear dynamic behaviors of aircraft cracked rotor systems during flight maneuvers. Acknowledgement The authors would like to acknowledge the financial supports from the National Basic Research Program (973 Program) of China (Grant No. 2015CB057400), the China Postdoctoral Science Foundation (Grant No. 2016M590277) and the National Natural Science Foundation of China (Grant No. 11372082). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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