Response evaluation of imbalance-rub-pedestal looseness coupling fault on a geometrically nonlinear rotor system

Mechanical Systems and Signal Processing 118 (2019) 423–442

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Response evaluation of imbalance-rub-pedestal looseness coupling fault on a geometrically nonlinear rotor system Yang Yang a,⇑, Yiren Yang a, Dengqing Cao b, Guo Chen a, Yulin Jin c a

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China School of Astronautics, Harbin Institute of Technology, PO Box 137, Harbin 150001, China c School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, China b

a r t i c l e

i n f o

Article history: Received 24 February 2018 Received in revised form 30 June 2018 Accepted 30 August 2018 Available online 10 September 2018 Keywords: Dynamic performance Rotor system Rub Pedestal looseness Geometrical nonlinearity

a b s t r a c t In order to achieve the purpose of condition monitoring and the appropriate design of rotating structure, analyzing the dynamic performance associated with rotor-stator rub coupling fault is of high significance. Due to large imbalance excitation and pedestal looseness, the whirling motion appears with larger amplitude and then the geometrical nonlinearity of shaft becomes impossible to ignore. In order to reveal the inner interaction between coupling fault commonly appearing in the rotating machine and geometrical nonlinearity of shaft, a geometrical nonlinear rotor system with imbalance-rub-pedestal looseness coupling fault is proposed in this paper. The mechanical mechanism of rotor-stator normal impact is represented in terms of a novel force model and its different modifications. Meanwhile, the friction between them is assumed to be a tangential dry Coulomb force, which is proportional to the impact force. After that the vibration features of the rotor system are analyzed with respect to the effects of geometrical nonlinearity, rotorstator rub and pedestal looseness. The change rules of resonant characteristic and rub region are revealed under different loose stiffness. What is more, the dynamic variation routes of the rotor system are analyzed by the bifurcation diagram, time waveform, whirl orbit, and Poincaré section, respectively. At last, the vibration experiment is performed on a rotor test rig and the typical signals of coupling fault are obtained at different rotational speeds. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Thrust-weight ratio and efficiency of rotating machine can be enhanced via precisely manufactured bearings with reduced clearances. However, under this engineering circumstance, the probability of rotor-stator rub is increased sharply, which may result in decreased machine life and adverse thermal effects. The rub is termed as the contact between rotor and stator, which may be a dominant factor of rotordynamic behavior [1]. As one of secondary faults occurring in the rotating machine, rotor-stator rub is usually represented in the form of coupling failure. The sources for primary causes could be rotor imbalance, misalignment, fluid forces, shaft crack and pedestal looseness [2–6]. As far as the stability and safety of the rotating machine are concerned, the coupling failures are more harmful and uncertain than single faults.

⇑ Corresponding author. E-mail addresses: [email protected] (Y. Yang), [email protected] (D. Cao). https://doi.org/10.1016/j.ymssp.2018.08.063 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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That is to say it is essential to master the unique vibration signature of the rotor system with rub coupling fault from state monitoring point of view. During the past decades, the rotor-stator rub coupling fault has been under comprehensive investigations. Considering this issue that the Jeffcott rotor was subjected to imbalance and rub, Chu et al. [7] revealed the distribution rule among periodic region, quasi periodic region and chaotic region at different rotational speed. As for an asymmetric double-disc rotorbearing system, Xiang et al. [8] adopted the numerical method to study the nonlinear dynamic behavior of the system varying with the model parameters. The electromagnetic vibration of electrical machines with an eccentric rotor was addressed in the reference [9], in which the electromagnetic excitation, mass imbalance and rub were taken into account. Hou et al. [10] focused on the influence of aircraft hovering flight on the rub rotor system and they investigated the nonlinear dynamic phenomenon. AlZibdeh et al. [11] proposed a three degree-of-freedom extended Jeffcott rotor for describing the drill string, and then they captured the vibration response of the rotor system with rub fault in an approximate solution. Under the periodic excitation caused by the mass eccentricity of disc, Vlajic et al. [12] analytically and numerically investigated the torsional vibration of a Jeffcott rotor system subjected to continuous contact of stationary components. For a rotating continuous flexible shaft-disc system with rotor-stator rub, Khanlo et al. [13] emphasized the torsional coupling effect on the chaotic characteristic and gave the conclusion that this effect could primarily change the speed ratios at which rub occurred. Popprath and Ecker [14] presented the nonlinear dynamic response of a Jeffcott rotor system having intermittent contact with a stator and discussed the effect of the visco-elastically suspended stator on the rotor motion. Wang et al. [15] theoretically studied the sudden unbalance and rub-impact caused by blade loss, in particular investigated the response of the rotor on a rotor test rig. Cong et al. [16] proposed an Impact Energy Model (IEM) to evaluate the probability or severity of rub-impact fault. Meanwhile, they conducted the experiment in two steps i.e. hammer test and rub-impact fault validation. Based on variational mode decomposition, Wang et al. [17] gave a novel method of rubbing fault diagnosis and proved the effectiveness of the method. Ma et al. [18] investigated the fault characteristics of a single span rotor system with two disc when the rubimpact between a disc and an elastic limiter occurred. By using conventional scalograms and reassigned scalograms, Peng et al. [19] explained the cause of rubbings, its occurrence and phenomenon if the severity of rubbing became serious. Thus, there is no denying that the complicated nonlinear phenomenon is generally associated with a rub rotor and then this is supposed to be worthy of intensive study [20–25]. Because of the poor quality of installation or long period of vibration, the pedestal looseness becomes one of the common faults that happen in rotating machine [26]. The looseness fault will reduce the elastic constraint stiffness of pedestal and cause the violent vibration of the rotor system. It is suggested that the work on the topic of pedestal looseness is indeed significant to aviation industry in terms of safe operation. Ma et al. [27,28] established a single-span rotor model with two discs, where the looseness fault was described by a piecewise linear spring-damper model, and analyzed the nonlinear vibration characteristic. For a rotating machine with only one pedestal looseness, Goldman and Muszynska [29,30] observed the synchronous and sub-synchronous frictional components referring to the numerical results and experimental data. Jiang et al. [31] developed a nonlinear measure to quantify the degree of nonlinear behaviors in a bearing-rotor system and predicted the dynamic behavior under different looseness clearances. In the reference [32], a method of multiple scales was adopted to analyze the free vibration and forced vibration of the nonlinear rotor-bearing system. Besides, the influences involved in this bearing pedestal model were also revealed in detail. According to the vibration sensitive time-frequency feature, Chen et al. [33] proposed a novel method to recognize the pedestal looseness extent of rotating machine and then successfully examined the validity of the method. It should be noted that the looseness fault has a higher potential risk to induce the rotor-stator rub and causes the complicated nonlinear vibration. In other word, there is a close relation between pedestal looseness and rub in the most of actual cases. Meanwhile, the coupling fault of looseness-rub can easily aggravate the whirling motion, so that the geometrical nonlinearity of shaft should not be ignored. Meanwhile, it becomes an extremely crucial component in dynamic design of rotating machine. In the previous authors’ work [34], the geometrically nonlinear relation between strain and displacement of flexible shaft was characterized by the equivalent spring and equivalent damper. However, the pioneering contributions to the dynamic response of the rotor system considering geometrical nonlinearity of shaft, rotor-stator rub and pedestal looseness have not been observed in existing literature. In view of this case, the main contribution of this paper is to investigate the close interaction between geometrical nonlinearity and coupling fault acting on the rotor system. According to the Hamilton principle, a general dynamic model for geometrically nonlinear rotor system subjected to imbalance-rub-pedestal looseness coupling fault is established in this paper. To reveal the normal impact mechanism in the condition of thermal barrier coatings, a novel force model and its modified forms [35,36] are employed at the different penetration stages. In the tangential direction, the Coulomb model [37] is used to describe the friction characteristic. Then the numerical simulation is applied to obtain the nonlinear vibration response of the rotor system at different rotational speed. Briefly speaking, there are five parts in the present work, including (1) sweep frequency analysis of linear/nonlinear rotor system without any fault, (2) imbalance-rub coupling fault under different initial clearance, (3) imbalance-pedestal looseness coupling fault under different looseness stiffness, (4) nonlinear dynamic characteristic of the rotor system with imbalance-rub-looseness coupling fault, (5) hammering test and vibration test on the rotor test rig. To some extent, this work can enrich our understanding to the vibration mechanism of rotating machine and may promote the development of fault diagnosis.

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2. Formulation of vibrating equations The mathematical formulation of the geometrically nonlinear rotor system with single pedestal looseness is briefed in this section. Due to the severe imbalance excitation and pedestal looseness, the whirling motion with larger amplitude usually happens. Thus, for the flexible shaft, the relation between strain and displacement becomes nonlinear rather than linear. This may change the resonant frequency of the system and lead to the complicated vibration behavior. At the same time, the rub-impact fault is more likely to occur in the condition of smaller initial clearance between rotor and stator. Considering the actual aviation background that the multifarious coatings are painted on the rotating components and stationary components, the incorporation of constraint imposed by stator on the rotor while formulating the vibrating equations is considered as well. 2.1. Stator-rotor-looseness pedestal Taking into account the coupling faults, i.e., imbalance-rub-pedestal looseness, a schematic diagram of the rotor-stator system capable of bending vibration is shown in Fig. 1. The curvature radius and mass of the disc are R1 and m1 , respectively. The disc coincides with the casing center with a clearance d0 in the static configuration. As shown in Fig. 1(a), the green components represent the thermal barrier coatings, which are painted on the surfaces of disc and casing, respectively. Besides, Fig. 1(b) gives that there is a looseness fault in the left pedestal, where the maximum looseness gap is set to dl and the looseness mass is set to m2 . Since that the mass eccentricity exists in the rigid disc, the rotor system is subjected to the centrifugal action and the whirling motion happens synchronously. If the rigid disc is installed in the middle of the flexible shaft, the normal of disc is parallel to the axis oz. Correspondingly, there are only two translational degrees of freedom (x1 and y1 ) in the process of bending deformation. This means that the gyroscopic effect of disc does not exist in the rotor system, as shown in Fig. 2(a). However, if the rigid disc is installed in the other position of shaft, there will be four degrees of freedom in the process of bending deformation, including two translations (x1 and y1 ) and two rotations (hx1 and hy1 ), respectively. In other word, the gyroscopic effect of disc should be considered in this kind of condition, as shown in Fig. 2(b). In conclusion, for the rotor system with multi coupling faults shown in Fig. 1, there is no gyroscopic effect of disc. In addition, the geometrical relationship of strain-displacement of shaft is supposed to be nonlinear, while the material relationship of stress-strain of shaft is assumed to be linear. Referring to the authors’ previous work [34], the transverse vibration equation of the massless shaft without geometrical nonlinearity has been obtained, so that

0
00  1 2 2 3 EIw00  2EIðw0 Þ w00 þ 2EIðw0 Þ w0  EAðw0 Þ ¼ Q nc 2

ð1Þ

where w denotes the transverse deformation of shaft, E denotes the elastic modulus of shaft, I denotes the inertia moment of cross section, A denotes the area of cross section. In addition, Q nc denotes the generalized force, which contains damping force of shaft and acting force generated by disc, respectively. Combing with the actual case that the operating speed of aero-engine is higher than the first order critical speed and lower than the second order critical speed, the first order modal function of the simply-supported beam is adopted to disperse the above partial differential equation. Therefore, in the process of whirling motion, the elastic restoring force can be expressed as

F r ¼ ke dr þ ad3r þ ce d_ r

Fig. 1. Schematic diagram of a rotor system with multi coupling faults: (a) rotor model and (b) left looseness pedestal model.

ð2Þ

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a)

b)

y (x) o

y (x)

z y1 (x1)

o

z y1 (x1)

θ x1 (θ y1) Fig. 2. Schematic diagram of whirling motion of the rotor system: (a) without gyroscopic effect and (b) with gyroscopic effect.

where ke denotes the equivalent linear stiffness, a denotes the equivalent nonlinear stiffness, ce denotes the equivalent damping, including

8 4 k ¼ EI 2lp 3 > > < e 3p4 p6 a ¼ EA 16l 3  EI 5 2l > > : ce ¼ cl2

ð3Þ

In the above equation, c is the structure damping of shaft, l is the length of shaft, A is the area of cross section, I is the inertia moment, namely

(

A ¼ pr 2

ð4Þ

4

I ¼ p4r

where r represents the radius of cross section of shaft. According to the schematic diagram of whirling motion shown in Fig. 2(a), the radial displacement of the middle part of the shaft can be written as

dr ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ y21

ð5Þ

Generally speaking, the pedestal looseness fault has an immediate impact on the support condition and leads to the support nonlinearity. According to the existing literature [27], the nonlinear support stiffness and nonlinear support damping of the pedestal can be written in the piecewise form. As shown in Fig. 3(a), there is a loose gap between bolt and pedestal. When the vertical vibration displacement of pedestal is in the interval, the support stiffness is set to kb1 . Otherwise, the support stiffness is changed to kb2 . Therefore, the support stiffness of the left pedestal with looseness fault can be expressed as

8 > < kb2 kb ¼ kb1 > : kb2

y2 > dl ð6Þ

0 6 y2 6 dl y2 < 0

where y2 denotes the vertical vibration displacement of the loose pedestal, and dl denotes the loose gap, respectively. Similarly, the expression of support damping considering the effect of pedestal looseness obeys the following form.

8 > < cb2 cb ¼ cb1 > : cb2

y2 > dl 0 6 y2 6 dl y2 < 0

a)

ð7Þ

b)

kb

kb 2

cb

cb 2

kb 2

cb1

kb1 0

cb 2

δl

y2

0

δl

Fig. 3. Loose characteristic of left pedestal: (a) support stiffness and (b) support damping.

y2

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where cb1 denotes the constraint damping of the loose bolt and cb2 denotes the constraint damping of the locked bolt, respectively. On this basis, an equivalent dynamic model for the rotor system shown in Fig. 1 is further proposed to investigate the coupling effect of geometrical nonlinearity and pedestal looseness. As mentioned in the references [26,38], the lateral looseness of pedestal is much weaker than the vertical looseness of pedestal, and then it can be ignored to some extent. Therefore, the dynamic model contains two translational displacements of disc and one vertical loose displacement of pedestal, as shown in Fig. 4(a). If the effect of pedestal looseness is not taken into consideration, the novel dynamic model can be reduced to an existing one, which has been introduced and discussed in the reference [34]. As shown in Fig. 4(b), the simplified model includes two translational displacements of disc. After considering the influence of self-gravity, the static equilibrium position is selected for dynamic modeling. Then, using the Newton’s second law, the vibration equations of the 3-DOF model are expressed as

  8 m1 €x1 þ ce x_ 1 þ ke x1 þ ax1 x21 þ y21 ¼ F x þ mex2 cosxt > > >
 < €1 þ ce ðy_ 1  y_ 2 Þ þ ke ðy1  y2 Þ þ aðy1  y2 Þ x21 þ ðy1  y2 Þ2 ¼ F y þ m1 ex2 sinxt m1 y >
 > > :m y _ 2 þ ðke þ kb Þy2  ce y_ 1  ke y1  aðy1  y2 Þ x21 þ ðy1  y2 Þ2 ¼ 0 2 € 2 þ ðc e þ c b Þy

ð8Þ

where m1 denotes the disc mass, m2 denotes the pedestal mass, e denotes the disc eccentricity, x denotes the rotational speed, F x and F y denote the components of rub-impact force in the two directions of o  x and o  y, respectively. 2.2. Rub-impact models at the different penetration stages Due to the disc eccentricity and pedestal looseness, to a remarkable extent delivered, the rub-impact fault between disc and casing may be induced. If the radial displacement of the disc is larger than the initial clearance, the rub-impact fault happens. Otherwise, there is no rub-impact fault in the dynamic model. Further more, the thermal barrier coatings are widely used in the surfaces of aero-engine components [39,40]. In this condition, a novel force model and its modified forms recently proposed by Yang and Cao et al. [35,36] are used to describe the

Fig. 4. Schematic diagram of a rotor-stator equivalent dynamic model considering thermal barrier coatings: (a) with pedestal looseness and (b) without pedestal looseness [34].

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impact mechanism between disc and casing. As depicted in Fig. 5, there are three stages in the rub-impact process, including low stage, general stage and high stage. Specifically, a minor variable v is defined to illustrate the minor penetration. Thus, at

vk2c the low stage d0 ; d0 þ 27k 2 , the impact deformation is mainly exhibited as small contact deformation of coatings. At the h
 2

vk2c vkc þ 1 27k general stage d0 þ 27k2 ; d0 þ p2ffiffiffiffi 2 , the small structure deformation of casing and contact deformation of coatings 27 h h


 2 v kc 2 þ 1 coexist. At the high stage d0 þ pffiffiffiffi 2 ; þ 1 , the impact deformation consists of large structure deformation of cas27 27k h

ing and large contact deformation of coatings. Thus, the different impact forces F N at different penetration stages can be respectively written in the following forms.

8 1 > kc kh d2h > > 1 ðdr  d0 Þ > > > kc þkh d2h > > > < 2ð43=2 v3=2 Þ k ðd  d0 Þ F N ¼ 3pffiffiffiffi 27ð4vÞþ2ð43=2 v3=2 Þ c r > >


pffiffiffiffiffiffi > pffiffiffiffi pffiffiffiffiffiffi pffiffiffi 3 > > 2 27 54 54 > > pffiffiv  v ln 27 þ v þ v ln 27 kh ðdr  d0 Þ2 > > : 0

dr P d0 þ




p2ffiffiffiffi 27

2

þ1

v kc d0 þ 27k 2 6 dr < d0 þ h



4k2c

27k2h




p2ffiffiffiffi 27

þ1



4k2c

27k2h

ð9Þ

vk2c

d0 6 dr < d0 þ 27k2 h

dr < d0

where kc denotes the structure stiffness of casing, and kh denotes the local contact stiffness of coatings, which can be calculated by the Hertz theory.

 1 4 R1 R2 2 kh ¼
1m2 1m2  R1 þ R2 3 E1 1 þ E2 2

ð10Þ

Two variables E1 , E2 express the elastic modulus of coatings, and two variables m1 , m2 express the Possion ratios of coatings, respectively. Meanwhile, according to the deformation relationship, the local contact deformation of coatings dh can be expressed by the total penetration, namely

0 B B dh ¼ B @

12

  13 pffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:59 kc 27ðdr d0 Þk2h 2k2c þ3 3 ðdr d0 Þð27ðdr d0 Þk2h 4k2c Þkh 6kh

 

6kh kc

2:52k2c

13 pffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 27ðdr d0 Þk2h 2k2c þ3 3 ðdr d0 Þð27ðdr d0 Þk2h 4k2c Þkh

þ C C C  kc A

ð11Þ

3kh

In the tangential direction, the friction characteristic between disc and casing is supposed to the Coulomb friction law.

F T ¼ lF N

ð12Þ

where l is the Coulomb friction coefficient that is mainly determined by the surface smoothness and material characteristic of coatings. According to the geometrical relationship shown in Fig. 6, the components of the impact force and friction force in the two directions of o  x and o  y obey that



F x ¼ F x1 þ F x2

ð13Þ

F y ¼ F y1 þ F y2

Fig. 5. Schematic diagram of three penetration stages in the rub-impact process.

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y FN

Casing

Fx 2

o1

o x1

Fx1 Fy 2

y1

Rotation

Fy1

FT

x Coatings Rotor

Fig. 6. Schematic diagram of rub-impact force components in the two directions of o  x and o  y.

When the above equation is used to describe the mechanical mechanism of rub-impact between disc and casing, the thermal phenomenon will be ignored. Meanwhile, the four variables F x1 , F x2 , F y1 and F y2 can be written as

8 T y1 F ¼ pFffiffiffiffiffiffiffiffiffi > > x21 þy21 > x1 > > > F T x1 > > < F y1 ¼  pffiffiffiffiffiffiffiffiffi 2 2 x1 þy1

N x1 > F x2 ¼  pFffiffiffiffiffiffiffiffiffi > > x21 þy21 > > > > > N y1 : F y2 ¼  pFffiffiffiffiffiffiffiffiffi 2 2

ð14Þ

x1 þy1

3. Results and discussion Because of these features, such as geometrical nonlinearity of shaft, mass imbalance, rotor-stator rub and pedestal looseness, performing the theoretically qualitative analysis becomes relative difficult, so that this case becomes impossible to obtain the solutions in a closed form. So the numerical methods have to be resorted in this paper. Referring to the existing research on the nonlinear vibration, the Runge-Kutta method is chosen, in which the time step of direct numerical integration is set to p=1000. In order to guarantee the data being used is in a steady state, the time series data of the first 800 revolutions is not used. The main parameters of the rotor system with coupling faults (see Fig. 1) are presented in Table 1. 3.1. Sweep frequency analysis of a rotor system subjected to imbalance excitation In order to master the dynamic characteristic of the rotor system more comprehensively, conducting the sweep frequency analysis is necessary in advance. To be specific, the first order resonant characteristic of the rotor system is discussed in this section. As shown in Fig. 7, the horizontal axis is the rotational speed and the vertical axis is the vertical displacement of disc, respectively. When the nonlinear factors involved in the rotor system (see Fig. 3(b)) are put on the back burner, the amplitude-frequency curve is depicted in Fig. 7(a). By the sweep frequency analysis, it can be observed that the first order resonant frequency is about xo ¼ 184 rad=s and the corresponding vibration amplitude is y1 ¼ 0:089 m. This phenomenon gives the resonant behavior, which means that the excitation frequency is equal to the natural frequency of the rotor system. When the geometrical nonlinearity of shaft is considered, the sweep frequency analysis is also performed in the same condition. Compared with the first order resonant frequency shown in Fig. 7(a), that shown in Fig. 7(b) obviously increases from 184rad=s to 303 rad=s. Meanwhile, the resonant amplitude decreases from 0:089m to 0:015 m. In addition, a special jump phenomenon happens at x ¼ 303 rad=s. This suggests that the geometrical nonlinearity of shaft has the ability to change the resonant characteristic of the rotor system. The contrastive analysis between Fig. 7(a) and (b) shows that when the rotational speed is in the range ½100; 215 rad=s, the vibration amplitude of the linear rotor system is larger than that of the nonlinear rotor system. However, when the rotational speed belongs to ½215; 400 rad=s, the vibration amplitude of the linear rotor system is less than that of the nonlinear rotor system. Two cases of rotational speed are used to further study the response difference, which is caused by the geometrical nonlinearity of shaft. At x ¼ 150 rad=s, the vertical vibration displacement of the linear rotor system is about y1 ¼ 4:4 mm,

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Y. Yang et al. / Mechanical Systems and Signal Processing 118 (2019) 423–442 Table 1 Main parameters of a rotor system with multi coupling faults. Physical parameter

Value

Mass of disc m1 (kg) Eccentricity of disc e (mm) Length of shaft l (mm) Radius of shaft r (mm) Elastic modulus of shaft E (GPa) Damping of shaft ce (N.s/m) Elastic modulus of coatings E1 ; E2 (GPa) Poisson ratio of coatings m1 ; m2 Mass of loose pedestal m2 (kg) Constraint stiffness of loose bolt kb1 (MN/m) Constraint stiffness of locked bolt kb2 (MN/m) Constraint damping of loose bolt cb1 (N.s/m) Constraint damping of locked bolt cb2 (N.s/m) Initial clearance between disc and casing d0 (mm) Initial loose clearance of pedestal dl (mm)

58.3613 2.2 448.8 12.2 210 261.8 200 0.3 116.723 42 220 350 500 5 15

Fig. 7. Sweep frequency analysis: (a) linear rotor system and (b) nonlinear rotor system.

while that of the nonlinear rotor system is about y1 ¼ 3:3 mm, as shown in Fig. 8. At x ¼ 250 rad=s, the vertical vibration displacement of the linear rotor system changes to y1 ¼ 4:8 mm, while that of the geometrically nonlinear rotor becomes y1 ¼ 11 mm, as shown in Fig. 9. What is more, the effects of the different eccentricities of disc on the amplitude-frequency curve of the linear rotor system and nonlinear rotor system are respectively discussed. When the disc eccentricity is relative smaller (i.e., e ¼ 0:44 m), the sweep frequency analysis from x ¼ 100rad=s to x ¼ 400rad=s is given in Fig. 10. According to Figs. 7(a) and 10(a), it can

Fig. 8. Vertical vibration displacement of disc at x ¼ 150 rad=s: (a) linear rotor system and (b) nonlinear rotor system.

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Fig. 9. Vertical vibration displacement of disc at x ¼ 250 rad=s: (a) linear rotor system and (b) nonlinear rotor system.

Fig. 10. Effect of smaller disc eccentricity on the sweep frequency investigation: (a) linear rotor system with e ¼ 0:44mm and (b) nonlinear rotor system with e ¼ 0:44 mm.

be seen that there is no obvious change about the first order resonant frequency of the linear rotor system. However, as shown in Figs. 7(b) and 10(b), that of the first order resonant frequency decreases from 303 rad=s to 212 rad=s. In the condition of larger eccentricity of disc (i.e., e ¼ 4:4 m), the amplitude-frequency curves of the rotor system with/ without geometrical nonlinearity are given in Fig. 11. By contrastive analysis between Figs. 10 and 11, it can be further concluded that despite the apparent increase of disc eccentricity, the first order resonant frequency of the rotor system without

Fig. 11. Effect of larger disc eccentricity on the sweep frequency investigation: (a) linear rotor system with e ¼ 4:4mm and (b) nonlinear rotor system with e ¼ 4:4 mm.

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geometrical nonlinearity remains unchanged. However, considering the effect of geometrical nonlinearity of shaft, the first order resonant frequency of the rotor system increases from 212 rad=s to 445 rad=s. The above dynamic variations reveal that there is a close relation between geometrical nonlinearity and resonant characteristic in the nonlinear rotor system. For the flexible rotating machine, both the geometrical nonlinearity and material nonlinearity should be considered in the structure design.

3.2. Dynamic characteristic of a rotor system with imbalance-rub coupling fault The bifurcation diagram, which is a useful way of observing nonlinear dynamic behavior, provides a summary of the essential dynamics. Then the bifurcation diagrams are used to evaluate the influence of rotor-stator rub on the nonlinear behavior of the rotor system. In this section, the pedestal looseness is not accounted provisionally, and then the initial clearance between disc and casing is set to d0 ¼ 7:5 mm and d0 ¼ 5 mm, respectively. It should be emphasized that due to the reason that the disc is installed in the middle of the flexible shaft, the gyroscopic effect does not exist in the rotor system (see Figs. 1(a) and 2(a)). At the same time, by referring to the existing literature [41,42], the dynamic model (see Eq. (8)) is established based on the static equilibrium position. Therefore, the whirl orbit of the rotor system is usually exhibited as circle rather than ellipse. This indicates that at the lower rotational speed, the full annular rub will happen rather than partial rub. Fig. 12(a) shows the bifurcation diagram of the rotor system in the condition of d0 ¼ 7:5 mm. It can be seen that in the range of rotational speed ½100; 203rad=s, the vibration displacement of disc is always less than the initial clearance, which suggests no rub fault. With the increase of rotational speed, the full annular rub happens and lasts until x ¼ 309 rad=s. At this interval, the whirling motion tends to be larger and the rub-impact fault gradually becomes more serious. In the range ½309; 400rad=s, the rub-impact fault disappears and the vibration amplitude decreases gradually. Overall, the rotor system is mainly exhibited as 1 T-periodic motion in the condition of larger initial clearance. When the initial clearance is reset to d0 ¼ 5 mm, the new bifurcation diagram of the rotor system with imbalance-rub coupling fault is given in Fig. 12(b). Due to the smaller initial clearance, the rub-impact fault happens at the lower rotational speed (i.e.,x ¼ 172 rad=s). When the rotational speed reaches x ¼ 285 rad=s, the previous full annular rub is replaced by the partial rub and then this new fault form can remain until x ¼ 330 rad=s. Since then, the partial rub disappears and the vibration response enters into the window of the regular periodic motion. To our knowledge, compared with full annular rub, the partial rub is more common and more likely to happen, which significantly affects the motion state of the rotor system. During the evolution process shown in Fig. 12(b), two typical rotational speeds corresponding to full annular rub and partial rub are respectively chosen, namely x ¼ 250 rad=s and x ¼ 293 rad=s. Fig. 13(a) shows that, at the rotational speed x ¼ 250 rad=s, the disc is always in contact with the casing and the vibration response of the rotor system with full annular rub is period 1. At this moment, the vibration response is similar to that of the rotor system with only imbalance fault. When the rotational speed mounts up to x ¼ 293 rad=s, the rub-impact between disc and casing occasionally happens, as shown in Fig. 13(b). Due to no gyroscopic effect in the rotor system, the whirl orbit of the rotor system keeps a regular circle before rubimpact. With the increase of rotational speed, the radius of the circle gets larger. In this case, the full annular rub happens at the lower rotational speed and the amplitude of whirl orbit is usually smaller. As the rotational speed further increases, the penetration between disc and casing becomes serious and the vibration behavior of the rotor system changes obviously. Then the partial rub will happen at the higher rotational speed. Because that the imbalance force is proportional to quadratic of rotational speed, the amplitude of whirl orbit under partial rub is usually larger.

Fig. 12. Considering no pedestal looseness, bifurcation diagram of the rotor system with imbalance-rub coupling fault: (a) d0 ¼ 7:5 mm and (b) d0 ¼ 5 mm.

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Fig. 13. Considering no pedestal looseness, whirl orbit of the rotor system in the condition of dl ¼ 5 mm: (a) x ¼ 250 rad=s and (b) x ¼ 293 rad=s.

3.3. Dynamic characteristic of a rotor system with imbalance-looseness coupling fault In this section, considering no rub-impact, the dynamic characteristic of a rotor system with imbalance-looseness coupling fault is analyzed. More specifically, the loose stiffness kb1 is taken as a key parameter to investigate the influence of pedestal looseness. For convenient analysis, a new dimensionless variable is defined as b and then the new loose stiffness can be expressed as kb1 b. When the whirling amplitude of disc is equal to the initial clearance, the rotational speed at this moment is defined as xc . After that, the relation between dimensionless variable b and rotational speed xc can be obtained by the numerical calculation, as shown in Fig. 14. When the variable b is in the range ½0:5; 3, it means that the pedestal looseness is relatively large and the rotational speed xc increases sharply with the development of variable b. When the variable b is in the range ½3; 9, the pedestal looseness is relatively small and the rotational speed xc will tend to be stable in the end. In Section 3.1, the first order resonant frequency and resonant amplitude of the rotor system have been conducted. Based on this point, the effects of the pedestal looseness on the resonant characteristic are further discussed in detail. As shown in Fig. 15(a), the pedestal looseness can make the resonant frequency decrease from xo ¼ 303 rad=s to xo ¼ 288 rad=s. However, it can intensify the whirling motion and the resonant amplitude increases from y1 ¼ 0:015 mm to y1 ¼ 0:033 mm, as shown in Fig. 15(b). The above phenomena suggests that the resonant characteristic of the rotor system is closely related to pedestal looseness. On behalf of small looseness and large looseness, two kinds of loose stiffness are chosen to reveal the change law of the bifurcation characteristic. Fig. 16(a) shows the bifurcation diagram of the rotor system with small pedestal looseness. The quasi period and chaos appear at the interval of ½240; 260 ðrad=sÞ and ½343; 400 ðrad=sÞ, which is obviously different from Fig. 7(b). Keeping the other parameters constant, the loose stiffness is reset to kb1 ¼ 14 MN=m, which indicates that the larger pedestal looseness happens. In this case, the bifurcation diagram of the rotor system is shown in Fig. 16(b). It is clear that the vibration responses are exhibited as alternative form of period 1, quasi period and chaos. The speed range of quasi period and chaos depicted in Fig. 16(b) is wider than that depicted in Fig. 16(a). Under the action of imbalance-looseness coupling fault, the equilibrium position offset of the vertical vibration response is also studied at the different rotational speed. In the condition of x ¼ 150 rad=s and kb1 ¼ 42 MN=m, the equilibrium position offset of the vertical vibration response does exist in Fig. 17(a). At the same rotational speed, the large looseness can

Fig. 14. Relation between dimensionless variable b and rotational speed of critical rub xc .

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Fig. 15. Effects of pedestal looseness on the resonant characteristic of the rotor system: (a) first order resonant frequency xo and (b) vertical resonant amplitude y1 .

Fig. 16. Considering no rub, bifurcation diagram of the rotor system with imbalance-looseness coupling fault: (a) small looseness kb1 ¼ 42 MN=m and (b) large looseness kb1 ¼ 14 MN=m.

Fig. 17. Considering no rub, vertical vibration displacement of disc with imbalance-looseness coupling fault at x ¼ 150 rad=s: (a) smaller looseness kb1 ¼ 42 MN=m and (b) larger looseness kb1 ¼ 14 MN=m.

cause more apparent response offset, as shown in Fig. 17(b). With the increase of rotational speed, i.e., x ¼ 280 rad=s, the vertical vibration displacement of disc is further studied in the two conditions of smaller looseness and larger looseness, respectively. Fig. 18 suggests that the offset phenomenon can be seen as a typical feature of imbalance-pedestal coupling fault to some extent.

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Fig. 18. Considering no rub, vertical vibration displacement of disc with imbalance-looseness coupling fault at x ¼ 280 rad=s: (a) smaller looseness kb1 ¼ 42 MN=m and (b) larger looseness kb1 ¼ 14 MN=m.

3.4. Dynamic characteristic of a rotor system with imbalance-rub-looseness coupling fault As introduced in the previous part, the severe imbalance and pedestal looseness can easily lead to the whirling motion with large amplitude. Under this circumstance, the rotor-stator rub is susceptible to imbalance fault and pedestal looseness. Therefore, it is of significance to analyzing the geometrically nonlinear rotor system with imbalance-rub-looseness coupling fault. For the different case of b, the detailed distribution of no rub and rub is shown in Fig. 19. From this, it can be concluded that the rotor-stator rub condition is easily affected by the pedestal looseness. In other word, the existence of pedestal looseness, especially for larger looseness, has a detrimental effect on the motion stability of the rotor system. Fig. 20 shows the bifurcation diagram of the rotor system with imbalance-rub-looseness coupling fault, where the lateral axis is rotational speed and vertical axis is vibration amplitude of disc, respectively. Figs. 20(a) and 12(a) illustrate that even though the pedestal looseness is not very serious, it may also cause the complicated dynamic behaviors. For example, the Poincaré section of the rotor system at x ¼ 230 rad=s includes closed loop curve, which is made up of numerous discrete points, as shown in Fig. 21(a). However, that of the rotor system without pedestal looseness consists of only one point, as shown in Fig. 21(b). When the rotational speed mounts up to x ¼ 300 rad=s, considering the coupling fault of imbalance-rub-looseness, there are lots of irregular dense points in the Poincaré section (see Fig. 22(a)). Nevertheless, without the effect of pedestal looseness, only one point appears in the Poincaré section, as shown in Fig. 22(b). Next, the nonlinear dynamic characteristic of the rotor system with imbalance-rub-looseness coupling fault is further analyzed in the condition of smaller initial clearance (i.e., d0 ¼ 5 mm). As described in Fig. 20(b), it is certain that both vibration complexity and vibration amplitude are significantly changed by the initial clearance. Additionally, the smaller initial clearance enhances the probability of rub-impact fault.

4. Experimental operation on a rotor test rig In order to show the effectiveness of modeling method and relavant dynamic phenomena observed in this paper, the hammering test and vibration experiment are conducted on the rotor test rig, which is set up in the ADVC (Aircraft Dynamics Vibration and Control) Laboratory, HIT. There are two main research contents in the vibration experiment: (1) measuring the

Fig. 19. Distribution of no rub and rub in the different conditions of b.

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Fig. 20. For the case of smaller looseness kb1 ¼ 42 MN=m, bifurcation diagram of the rotor system with imbalance-rub-looseness coupling fault: (a) initial clearance d0 ¼ 7:5 mm and (b) initial clearance d0 ¼ 5 mm.

Fig. 21. In the condition of x ¼ 230 rad=s and d0 ¼ 7:5 mm, Poincaré section of whirling motion of disc: (a) with imbalance-rub-looseness and (b) with imbalance-rub.

Fig. 22. In the condition of x ¼ 300 rad=s and d0 ¼ 7:5 mm, Poincaré section of whirling motion of disc: (a) with imbalance-rub-looseness and (b) with imbalance-rub.

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vertical vibration under imbalance fault and (2) measuring the vertical vibration under imbalance-pedestal looseness coupling fault. There is an urgent need for special protective equipment in the field of rubbing experiment, especially in the rubbing-looseness coupling experiment. Unfortunately, this kind of protective equipment is not provided in the ADVC Laboratory. Thus the vertical vibration experiment under imbalance-rub-pedestal coupling fault is not performed in this paper. Fig. 23 shows a typical single rotor test rig, which is composed of several key components, such as rigid disc, flexible shaft, pedestal, bolt and electric machinery. According to the structure design and experiment purpose, the disc contains the imbalance mass and the looseness fault exists in the left pedestal. Meanwhile, the highest rotational speed of the electric machinery is set to 6000 rpm, which is higher than the first order resonant frequency of the test rig. 4.1. Modal hammering experiment In order to obtain the natural frequency of the rotor test rig, the modal hammering test is carried out in this section. The main parameters of the rotor test rig are listed in Table 2. The basic principle of hammering experiment on the rotor test rig is illustrated in Fig. 24. It can be seen that the eddy current sensor is used to measure the change of voltage signal, which is caused by impact force. Then the voltage signals are transformed to the displacement signals and analyzed by the Dewesoft software. In the FFT response of disc, the first order natural frequency of the rotor test rig is about 33:57 Hz, as shown in Fig. 25(a). According to Eq.(3) and Table 2, the first order natural frequency of the rotor test rig can be obtained by the theoretical calculation, namely 33:69 Hz. By the contrast analysis, the relative error between experimental result and theoretical result is about 0:36% . What is more, the vertical vibration response of the disc is depicted in Fig. 25(b). Because of the structure damping, the vibration response of the rotor test rig reduces gradually. 4.2. Whirling response experiment To master the vibration feature of the rotor system with coupling fault more accurately, it is also necessary to perform the whirling motion experiment on the rotor test rig. As shown in Fig. 26, four eddy current sensors are applied to measure the lateral and vertical displacements of disc and left pedestal, respectively. When dealing with the left pedestal, two work conditions are designed, including no bolt looseness and bolt looseness. For the case of no bolt looseness, the vertical vibration signals of the disc subjected to the imbalance excitation can be gathered at the rotational speed x ¼ 234 rad=s. In the same work condition, the numerical results of the rotor system are also given in Fig. 27. It is clear that the numerical results are similar to the experimental results. When the rotor test rig

Fig. 23. Rotor test rig and eddy current sensor.

Table 2 Main structure parameters of the rotor test rig. Physical parameter

Value

Length of shaft l (mm) Radius of shaft r (mm) Elastic modulus of shaft E (GPa) Mass of rigid disc m1 (kg) Eccentricity of disc e (mm)

423 5 210 1.48 0.084

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Fig. 24. Principle diagram of hammering experiment on the rotor test rig.

Fig. 25. Measurement signals of hammering experiment: (a) FFT response of disc and (b) vertical vibration displacement of disc.

Fig. 26. Principle diagram of whirling motion experiment on the rotor test rig.

is subjected to only imbalance force, the equilibrium position is y1 ¼ 0 mm. Thus the vibration experiment proves that the offset phenomenon of equilibrium position does not happen. On this basis, the coupling effect of imbalance and pedestal looseness is further analyzed. The looseness fault exists in the left pedestal, and then the corresponding vibration signals are obtained by the numerical simulation and experiment, respectively. As shown in Fig. 28, the offset of theoretical result is about 13:65lm and that of experimental result is about

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Fig. 27. Vertical vibration displacement of disc with only imbalance effect at x ¼ 234 rad=s: (a) numerical results and (b) experimental results.

Fig. 28. Vertical vibration displacement of disc with coupling effect of imbalance and pedestal looseness at x ¼ 215 rad=s: (a) numerical results and (b) experimental results.

Fig. 29. Spectrum diagram of vertical response of disc with coupling effect of imbalance and pedestal looseness at x ¼ 196:3 rad=s: (a) numerical results and (b) experimental results.

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Fig. 30. Spectrum diagram of vertical response of disc with coupling effect of imbalance and pedestal looseness at x ¼ 249 rad=s: (a) numerical results and (b) experimental results.

Fig. 31. Spectrum diagram of vertical response of disc with coupling effect of imbalance and pedestal looseness at x ¼ 325:8 rad=s: (a) numerical results and (b) experimental results.

10:92 lm. This means that the offset phenomenon can be seen as one of the typical features of pedestal looseness to some extent. In the aspect of fault diagnosis and fault recognition, the spectrum feature analysis is an essential work, which can identify the specific frequency components. Therefore, by simulation and experiment, the spectrum digrams corresponding to the above coupling fault are obtained at different rotational speed. When the rotational speed is set to 196:3 rad=s, the spectrum diagram is shown in Fig. 29. Except for the component synchronous to rotational speed (1X component), the higher order harmonics, such as 2X and 3X, are observed in the spectrum diagram. Meanwhile, there are similar frequency components between numerical results and experimental results. Keeping the other conditions constant, the rotational speed of the test rig is reset to 249 rad=s. It can also be found that 1X component and 2X component coexist in the spectrum diagram, as shown in Fig. 30. At last, the spectrum characteristic of the rotor test rig at x ¼ 325:8 rad=s is further discussed as well. As described in Fig. 31, the main frequency component is equal to the rotational speed. In addition, there is a 2X component, whose amplitude is far less than that of 1X component. What is more, the theory and experiment can match well, which further verifies the correctness of the research results. 5. Conclusion In this paper, taking into account the coupling fault of imbalance-rub-pedestal looseness, the dynamic model of a threedegree-of-freedom rotor system has been proposed, in which the geometrically nonlinear property of the shaft has been presented. Under the influence of severe imbalance and pedestal looseness, the rotor-stator rub has a higher potential risk and then the mechanical mechanism has been described by the novel impact force model and Coulomb friction model. By using the Runge-Kutta method, the vibration equations of the rotor system have been numerically solved and analyzed in the form of bifurcation diagram, time waveform, whirl orbit and Poincaré section. Finally, the fault features of imbalance-pedestal

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looseness are further identified by the vibration experiment, which is performed on the rotor test rig. Some conclusions are summarized as follows: (1) For the three-degree-of-freedom rotor dynamic model, the first order resonant frequency is obviously affected by the geometrical nonlinearity of shaft. With the increase of disc eccentricity, the geometrical nonlinearity becomes obvious and the resonant frequency turns to be larger. (2) In the different range of rotational speed, the existence of geometrical nonlinearity may intensify whirling motion or restrain whirling motion. (3) Pedestal looseness may slightly change the resonant characteristic of the geometrically nonlinear rotor system while it can intensify the whirling motion. (4) Both offset phenomenon of vibration equilibrium position and higher order harmonics can be seen as the typical characteristics of imbalance-pedestal looseness coupling fault. (5) Compared with single fault and two coupling fault, the imbalance-rub-pedestal looseness coupling fault can lead to more complicated nonlinear dynamic phenomena in the response of the geometrically nonlinear rotor system.

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