Casing vibration response prediction of dual-rotor-blade-casing system with blade-casing rubbing

Mechanical Systems and Signal Processing 118 (2019) 61–77

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Casing vibration response prediction of dual-rotor-blade-casing system with blade-casing rubbing Nanfei Wang a, Chao Liu a,b,⇑, Dongxiang Jiang a,c, Kamran Behdinan d,e a

Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China c State Key Laboratory of Control and Simulation of Power System and Generation Equipment, Tsinghua University, Beijing 100084, China d Advanced Research Laboratory for Multifunctional Light Weight Structures, Department of Mechanical & Industrial Engineering, University of Toronto, Toronto M5S3E3, Canada e Department of Mechanical & Industrial Engineering, University of Toronto, Toronto M5S3E3, Canada b

a r t i c l e

i n f o

Article history: Received 9 May 2018 Received in revised form 17 July 2018 Accepted 8 August 2018

Keywords: Aero-engine Dual-rotor-blade-casing system Blade-casing rubbing Casing acceleration signal Finite element method

a b s t r a c t This paper focuses on the dynamic responses of whole aero-engine with blade-casing rubbing. The vibration response of casing is simulated and the dynamic behaviors of casing acceleration under blade-casing rubbing are investigated to diagnose the blade-casing rubbing fault effectively. Firstly, the finite element model of a dual-rotor-blade-casing (DRBC) system with inter-shaft bearing is proposed. The shear deformations and inertias of the rotor and casing are taken into account. The gyroscopic moments of the rotors are evaluated. Secondly, the effects of the blade-casing clearance and the number of blades on the vibration responses of DRBC system with blade-casing rubbing are considered. Finally, the casing acceleration responses of the DRBC system with blade-casing faults are solved numerically, and the influences of some variables, such as the rotating speed ratio, eccentricity of disk and rubbing stiffness, on the dynamic behaviors have been investigated by time waveform of acceleration, frequency spectrum and waterfall. The results indicate that (1) blade-casing rubbing can cause impulsive load to the casing and rotor, and lead to abrupt increase of the vibration amplitude; (2) the obvious periodic impact characteristics are contained in the casing vibration acceleration signals, and the impact frequency equals the product of the rotational frequency and the numbers of blades; (3) the fraction frequency component of rotating speed difference of dual rotors is excited on both sides of impact frequency and its multiple frequency components. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The dual-rotor structure has been widely used in aero-engine, which consists of many rotary and stationary accessories. To improve the performance of aero-engine, reducing the blade-casing clearance is one of most widely used methods [1]. However, minimizing clearances increases the possibility of blade-casing rubbing, which has long been identified as the main cause of engine malfunction. The blade-casing rubbing fault may result in excessive vibration of the whole machine, high maintenance cost and even catastrophic accidents [2,3]. Hence, in order to detect fault effectively, it’s necessary to develop the whole aero-engine model and analyse the vibration characteristics of blade-casing rubbing. ⇑ Corresponding author. E-mail address: [email protected] (C. Liu). https://doi.org/10.1016/j.ymssp.2018.08.029 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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Due to the complexity of the rubbing fault, the blade-casing interaction have attracted increasing attention of scholars. Considering the influences of different parameters, Padovan and Choy [4] analyzed the dynamic behaviours under singleand multiple-blade rubbings. Sinha [5–7] investigated the vibration behaviours of a rotating Timoshenko beam exposed to a periodic pulse because of local rubbing between rigid casing and radial blade. Ma et al. [8,9] introduced an improved blade-casing rubbing model and evaluated the correctness of the new model by using experiments, and adopted the pulse loading model to study the dynamic behaviours of the single rotor system under blade-casing rubbing. Batailly et al. [10] investigated the blade-casing rubbing characteristics by utilizing the reduced models to detect the modal interaction. Legrand et al. [11–13] analyzed the modal interaction phenomenon and the effects of blade-casing interaction on the dynamic behaviours. Ma et al. [14] analyzed the effects of different variables on the dynamic responses of the shaft-diskblade system with blade-tip rubbing. Petrov [15] developed a method for frequency domain analysis in gas turbine engines with rubbing between bladed disk and casing. Based on a revised rotor-blade dynamic model, Ma et al. [16] simplified the casing as a lumped mass point and revealed the vibration characteristics of the single- and four-blade rubbing faults. Thiery et al. [17] proposed the finite element model of a Kaplan turbine and evaluated the dynamics of the initially misaligned rotor with blade-casing rubbing. Cao et al. [18] introduced a novel rubbing force model to describe the blade-casing rubbing fault in aero-engine. Thiery et al. [19] adopted numerical and experimental methods to investigate the nonlinear vibration of a bladed Jeffcott rotor. Wang et al. [20,21] adopted finite element method, envelop demodulation and empirical mode decomposition to investigate the dynamic behaviours of a single rotor system with rub-impact and extract fault features. Chen et al. [22,23] investigated the dynamic responses of the casing under blade-casing rubbing, and the numerical results are verified by experiments. Qin et al. [24,25] verified the correctness of finite element models to predict dynamic characteristics of rotating bolted disk-drum type rotors. Ma et al. [26] investigated the vibration response resulting from blade-casing rubbing during the run-up process and steady-state process. From above literatures, emphasis is put on the dynamic behaviours of single rotor with blade-casing rubbing and the actual dual-rotor structure of aero-engine is not fully considered. Wang et al. [27] revealed the dynamic characteristics of dual-rotor system by using finite element models and experiments. Yang et al. [28,29] adopted numerical and experimental methods to analyse the vibration responses of a dual-rotor system under fixed point rubbing condition. Wang et al. [30] proposed the dynamic model of dual-rotor and investigate the dynamic characteristics of the system under pedestal looseness fault. Lu et al. [31] investigated the complicated vibration response of a dual-rotor system with cracked highpressure rotor. Xu et al. [32] developed rub-impact fault model and the finite element model of a dual-rotor system, and obtained the vibration responses under faulty condition by using theoretical and experimental methods. Hou et al. [33] carried out the resonance analysis of a dual-rotor system with two rotor unbalance excitation by using HB-AFT method. Wang et al. [34,35] performed the investigation on the dynamic behaviours of a dual-rotor system with rubimpact and misalignment faults. Sun et al. [36] adopted theoretical model to predict the steady-state responses and stability of a dual-rotor system under rub-impact condition. Yu et al. [37] investigated the vibration response of the dualrotor system at instantaneous and windmilling statuses when fan blade out event occurs. However, the elastic casing and blades are not considered in these dual-rotor models, and the dynamic characteristics are investigated by means of the displacement responses of rotors. Considering the complex structure of actual aero-engine, some characteristics are rarely taken into account in the existing experiment and theoretical studies at the same time, such as dual-rotor structure and thin-walled casing, and thus the current results don’t truly reflect the vibration fault features of whole aero-engine. In addition, various rubbing types are not fully taken into account in the models. Besides, for aero-engine, only the casing acceleration signal can be obtained to diagnose rubbing fault. Therefore, it is of great importance to reveal the dynamic behaviours and laws of casing acceleration response for the precise blade-casing rubbing fault diagnosis and stability of aero-engine. The paper is organized as follows. Based on the actual structure of aero-engine, a simplified dual-rotor-blade-casing (DRBC) system is introduced in Section 2.1. The blade-casing rubbing model, the rotor-casing clearance simulation and the finite element model of the DRBC system are proposed in Section 2.2, Section 2.3 and Section 2.4, respectively. In Section 3, the dynamic characteristics and the effects of rotating speed ratio, eccentricity of disk and rubbing stiffness on the dynamic behaviours of DRBC system are investigated in detail. Finally, the conclusions are drawn in Section 4.

2. Mathematical formulation 2.1. DRBC system model with blade-casing rubbing CFM56 is one of most widely used aero-engines, and its rotor-support structure is shown in Fig. 1(a). Fig. 1(b) illustrates the structural coupling relationship of low-pressure rotor, high-pressure rotor and casing in CFM56. Based on the basic structure of CFM56, a dual-rotor-blade-casing (DRBC) system consisting of a high-pressure rotor (outer rotor), a low-pressure rotor (inner rotor), blades, casing and elastic support is developed, as shown in Fig. 1(c), where Bearing 1 and Bearing 2 (Fig. 1(a)) are simplified as one bearing. Each of the rotors consists of elastic shaft and two disks representing compressor and turbine, respectively. The inter-shaft bearing is employed to connect low-pressure rotor and high-pressure rotor. Two engine mounts are used to support the whole system. The attention is paid to the rubbing between disk 1 and casing. x1 and x2 are the rotating speeds of inner and outer rotors, respectively. Perfect balance can’t be achieved, and assuming that

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(c)

Engine mount 1

Bearing 1

Bearing 2

x

y

Engine mount 2

Bearing 4

2

Outer shaft

z

Bearing 3

1

Inter shaft

(Inter-shaft bearing) Disk 4 Disk 1

Disk 2

Disk 3

Unbalance position Fig. 1. (a) The rotor-support structure of CFM56 [38]; (b) The coupling diagram of rotors and casing in CFM56; (c) The sketch map of a DRBC system proposed in this paper.

the two turbine disks have two eccentricities, the initial phase angle is zero and the phase angle difference between two unbalance forces is zero. 2.2. Mathematical model for blade-casing rubbing The effects of blade number and variation of the rotor-casing clearance are not fully taken into account in the ordinary elastic rub-impact model, as shown in Fig. 2(a). The proposed blade-casing rubbing investigates the influences of the number of blade and changes of rotor-casing clearance in detail, as shown in Fig. 2(b) and (c) [22]. Assuming that N blades are uniformly installed on the disk, the angle between the ith blade and the x axis is hbi ¼ 2pi=N þ xt at time t, and x is the rotating speed (rad/s). The vibration displacements of rotor at time t are represented by xr and yr , and the vibration displacements of casing are represented by xc and yc . The radial displacement of theith blade and casing can be obtained as follows:




r bi ¼ xr cosðhbi Þ þ yr sinðhbi Þ rci ¼ xc cosðhbi Þ þ yc sinðhbi Þ

Fig. 2. Rubbing model: (a) Common elastic rubbing model, (b) rubbing clearance and (c) blade-casing rubbing model.

ð1Þ

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The normal impact force resulting from the rubing between blade and casing is written as:


F iN ¼

fn

r bi  r ci > cðhbi Þ

0

rbi  r ci  cðhbi Þ

ð2Þ

The casing in aero-engine has large diameter and small thickness, which indicates that its stiffness is usually lower than the stiffness of blade. It is acceptable to assume that the rubbing stiffness kc is equal to the casing stiffness. The deformation of casing caused by rubbing at the contact point is much more than the deformation of blade. Hence, the specific expression of f n is derived as follows [22,39]:

fn ¼

    3 3 1  c2e r_ 1  kc  f½r bi  r ci  cðhbi Þ þ jr bi  r ci  cðhbi Þjg2 1 þ 2 4r_ 

ð3Þ

where r_ denotes the velocity of disk and r_  is the initial impact velocity, respectively; ce is the restitution coefficient of velocity. The normal force components in the x and y directions can be expressed as follows:




F iNx ¼ F iN cosðhbi Þ

ð4Þ

F iNy ¼ F iN sinðhbi Þ It is assumed that the tangential rub obeys the Coulomb friction model, and the tangential force can be given as:

F iT ¼ lF iN

ð5Þ

The tangential force components in the x and y directions are obtained as follows:




F iTx ¼ F iT sinðhbi Þ

ð6Þ

F iTy ¼ F iT cosðhbi Þ At time t, the force on the ith blade of rotor resulting from blade-casing rubbing is listed as follows:




F ix ¼ F iNx þ F iTx

ð7Þ

F iy ¼ F iNy þ F iTy

At time t, the sum of the rubbing forces imposing on the N blades composes the rubbing force on the rotor, which can be derived by the sum of the force components:

(

Fx ¼ Fy ¼

PN

1 F ix

ð8Þ

PN

1 F iy

2.3. Rotor-casing clearance simulation The simulation of single-point rubbing between blade and casing can be achieved by using a local deformation of casing. Let A denote the deformation amplitude corresponding to angle h, D denote the initial gap, anda is the central angle. The rub^

impact in the angle range of b is simulated by a cosine function. If b < 5A , the simulation function is described in Formula (9). Otherwise, the simulation function is given in Formula (10) [22]. In this paper, the clearance function described by Formula (9) is used. Meanwhile, multi-point rubbing can also be simulated by using clearance function.

( cðaÞ ¼

D;

h i D  A 0:5 þ 0:5cos pðabhÞ ;

ja  hj > b ja  hj 6 b

0 < a 6 2p

8 DA ja  hj < b=2 > > < h i pðahÞ b=2 < ja  hj 6 b 0 < a 6 2p cðaÞ ¼ D  A 0:5  0:5cos b=2 > > : D ja  hj> b

ð9Þ

ð10Þ

Formula (10) has larger deformation range than Formula (9). The relationship between clearance and cycle angle is shown in Fig. 3, where the parameters are defined as A = 10 mm, h = 90° and D = 7.5 mm. The deformation ranges in Figs. 3(a) and 3 (b) areb = 5° and 15°, respectively. Physical meanings of different parameters shown in clearance function is exhibited in Fig. 3(c). 2.4. Finite element discretization of the DRBC system Based on literature [40], beam element that does not rotate has enough accuracy and high calculation efficiency to describe the dynamic characteristics of casing. As illustrated in Fig. 4, the DRBC system is divided into 59 Timoshenko beam

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65

Fig. 3. Clearance between rotor and casing for blade-casing rubbing: (a) clearance curve underb = 5°, (b) clearance curve underb = 15° and (c) the diagram of clearance parameter.

Fig. 4. FE model of the DRBC system (beam-based model).

elements, four disk elements and six support elements. The Timoshenko beam element has four degrees of freedom at each node, consisting of two translational and two rotations, as shown in Fig. 5. The bearings and mounts are simplified as springdamping elements. The numbers of beam elements of inner rotor, outer rotor and casing are 27, 12 and 20, respectively. The blades are considered as concentrated masses attached to the first disk. The two node numbers of support element used to simulate inter-shaft bearing are 22 and 41. The similar spring elements are utilized to simulate the elastic supports between rotors and casing. The node positions of disks, bearings and mounts are presented in Fig. 4. Accordingly, the generalized displacement vectors of inner rotor can be described as follows:

 T uL ¼ x1 ; y1 ; hy1 ; hx1    ; x28 ; y28 ; hy28 ; hx28

ð11Þ

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Fig. 5. Sketch map of discrete elements: (a) Timoshenko beam element, (b) disc element.

where xi and yi (i = 1,  , 28) represent the translations of nodes 1–28, hxi and hyi (i = 1,  , 28) represent the rotations of nodes 1–28, respectively. Then the governing equation of the inner rotor in matrix form can be expressed as

€ L þ ðC L  x1 GL Þu_ L þ K L uL ¼ Q L ML u

ð12Þ

where M L , C L , GL and K L are the mass, damping, gyroscopic and stiffness matrices of the inner rotor, respectively. The generalized force vectors of the inner rotor conform to

 T Q L ¼ 0; 0; 0; 0    ; F x ; F y ; 0; 0    ; md1 e1 x21 cosx1 t; md1 e1 x21 sinx1 t; 0; 0    0; 0; 0; 0

ð13Þ

where F x and F y are exerted on node 4 of inner rotor, md1 and e1 are the mass and eccentricity of disk 4 (node 24), respectively. Similarly, the generalized displacement vectors of outer rotor are defined as

 T uH ¼ x29 ; y29 ; hy29 ; hx29    ; x41 ; y41 ; hy41 ; hx41

ð14Þ

Hence, the governing equation of the outer rotor can be obtained

€ H þ ðC H  x2 GH Þu_ H þ K H uH ¼ Q H MH u

ð15Þ

where M H ,C H , GH and K H are the mass, damping, gyroscopic and stiffness matrices of the outer rotor, respectively. The generalized force vectors of the outer rotor obey

 T Q H ¼ 0; 0; 0; 0    ; md2 e2 x22 cosx2 t; md2 e2 x22 sinx2 t; 0; 0    0; 0; 0; 0

ð16Þ

where md2 and e2 are the mass and eccentricity of disk 3 (node 35), respectively. Adopting the same method, the generalized displacement vectors of the casing can be written as follows

 T uC ¼ x42 ; y42 ; hy42 ; hx42    ; x62 ; y62 ; hy62 ; hx62

ð17Þ

The casing is considered as a non-rotating beam. The vibration equation of the casing can be expressed as

€ C þ C C u_ C þ K C uC ¼ Q C MC u

ð18Þ

where M C ,C C and K C are the mass, damping and stiffness matrices of the casing, respectively. The generalized force vectors of the casing obey

 T Q C ¼ 0; 0; 0; 0    ; F x ; F y ; 0; 0    0; 0; 0; 0

ð19Þ

where rubbing reaction forces are applied on the node 44 of casing. Based on the previous analysis, the differential equations of the DRBC system can be further written as

2

ML 6 4 0 0

0 MH 0

3 2 €L C L  x1 G L u 76 € 7 6 0 0 54 uH 5 þ 4 €C MC 0 u 0

32

0 C H  x2 G H 0

3 2 KL u_ L 76 _ 7 6 0 5 4 uH 5 þ 4 0 CC 0 u_ C 0

32

0 KH 0

3 2 3 uL QL 76 7 6 7 0 5 4 uH 5 ¼ 4 Q H 5 KC uC QC 0

32

ð20Þ

Hence, the governing equation of the DRBC system can be derived by assembling three subsystems.

€ þ C u_ þ Ku ¼ Q Mu

ð21Þ

where M; C; K; Q and u are defined as follows

M ¼ diagfM L ; M H ; M C g

ð22Þ

C ¼ diagfC L  x1 GL ; C H  x2 GH ; C C g

ð23Þ

N. Wang et al. / Mechanical Systems and Signal Processing 118 (2019) 61–77

K ¼ diagfK L ; K H ; K C g

67

ð24Þ

2

3 2 3 QL uL 6 7 6 7 Q ¼ 4 Q H 5; u ¼ 4 uH 5 QC uC

ð25Þ

In modeling the dual-rotor-blade-casing system, the boundary conditions (i.e., bearing support in this case) are not taken into account in the first step (till Eq. (25). The stiffness and damping of the nodes with bearings installed are modified in the next part. Since the spring elements used to simulate the engine mounts are elastic supports, the stiffness values are added to K (173,173), K (174,174), K (253,253) and K (254,254), respectively. Similarly, the damping of the mounts is added to C (173,173), C (174,174), C (253,253) and C (254,254), respectively. Due to the coupling effect of rotors and casing, the stiffness values of Bearing 1, Bearing 2, Bearing 3 and Bearing 4 are added to K (1,1), K (2,2), K (165,165), K (166,166), K (113,113), K (114,114), K (197,197), K (198,198), K (85,85), K (86,86), K (161,161), K (162,162), K (109,109), K (110,110), K (245,245), K (246,246), respectively. The same operation is applied to the total damping matrix. In addition, the stiffness values of Bearing 1, Bearing 2, Bearing 3 and Bearing 4 needs to be subtracted from K (1,165), K (165,1), K (2,166), K (166,2), K (113,197), K (197,113), K (114,198), K (198,114), K (85,161), K (161,85), K (162,86), K (86,162), K (109,245), K (245,109), K (110,246), K (246,110), respectively. The same operation is applied to the total damping matrix. 3. Response analysis and discussion The differential equations of the DRBC system are calculated by means of the Newmark method and time step is Dt = 0.0001 s. According to the design and basic structure of actual dual-rotor test bench [32], the specific physical parameters used to describe the system are listed in Table 1. For the convenience of discussion, the rotating speed ratio is defined as

j¼

X2 X1

ð26Þ

where X1 and X2 are the rotating speeds of inner rotor and outer rotor (unit: rev/min), respectively.

Table 1 Model parameters of the DRBC system. Physical parameter

Value

Length of inner shaft LI (m) Length of outer shaft LO (m) Length of casing LC (m) Outer and inner radii of casing RC , r C (m) Elastic modulus of rotational shafts, disks and casing E(GPa) Density of rotational shafts, disks and casing E(kg/m3) Poisson ratio of rotational shafts , disks and casingl Outer and inner radii of inner rotor compressor disk RIc , r Ic (m) Outer and inner radii of inner rotor turbine disk RIt , r It (m) Outer and inner radii of outer rotor compressor disk ROc , r Oc (m) Outer and inner radii of outer rotor turbine disk ROt , r Ot (m) Thickness h1 of disk 1 (m) Thickness h2 of disk 2 (m) Thickness h3 of disk 3 (m) Thickness h4 of disk 4 (m) Support stiffness ks1 (N/m) and support damping cs1 (N.s/m) of bearing 1 Support stiffness ks2 (N/m) and support damping cs2 (N.s/m) of bearing 2 Support stiffness ks3 (N/m) and support damping cs3 (N.s/m) of bearing 3 Support stiffness ks4 (N/m) and support damping cs4 (N.s/m) of bearing 4 Support stiffness kIb1 (N/m) and damping cIb1 (N.s/m) of engine mount 1 Support stiffness kIb2 (N/m) and damping cIb2 (N.s/m) of engine mount 2 The eccentricity e1 of disk 4 (m) The eccentricity e2 of disk 3 (m) The friction coefficientl The restitution coefficient of velocityce The initial clearance D (m) between blade-tip and casing The deformation A (m) at angle 90° ^) The rubbing range b (A

0.64 0.266 0.64 0.12,0.115 193 7754 0.3 0.07,0.009 0.07,0.009 0.07,0.015 0.07,0.015 0.02 0.01 0.02 0.02 6.11  106, 200 5.02  106, 200 3.12  106, 200 4.38  106, 200 3  108, 200 3  108, 200 0.0001 0.0003 0.3 0.9 0.0075 0.01 5

The number of blades

6

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3.1. Modal validation based on natural characteristics In order to verify the effectiveness of the beam-based DRBC system, the solid finite element model is established by using SOLID 185 element in ANSYS, where the casing is the cylinder. The first four natural frequencies and mode shapes are shown in Figs. 6 and 7, respectively. Only part casing is shown for a better view of dual-rotor structure in Fig. 7. The natural frequencies calculated from two models are listed in Table 2. It can be found that the corresponding modal shapes of both models are same. The satisfactory agreement exists by comparing the natural frequencies, as shown in Table 2. However, the calculation

Fig. 6. The first four natural modals and frequencies of DRBC system.

Fig. 7. The first four natural modals and frequencies of DRBC solid model.

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Table 2 Comparison of natural frequency components. Order

Beam-based model (Hz)

Solid model (Hz)

Error (%)

1 2 3 4

48.6424 123.6543 221.7833 306.6804

50.1624 126.342 234.02 320.926

3.03 2.13 5.23 4.44

Fig. 8. Campbell diagram of DRBC system (beam-based model).

efficiency of beam-based model is much higher than that of solid model. The main reason is the neglect of coupling effect between the disks and shafts. The Campbell diagram of the DRBC system is shown in Fig. 8. It can be found from Fig. 8 that with the increase of rotating speed of inner rotor, the forward whirling frequencies increase and the backward whirling frequencies decrease due to gyroscopic effect, but the fundamental modes of the DRBC systems remain unchanged. The potentially dangerous speeds are marked in Fig. 8, and the three critical speeds (A1, A2 and A3) mainly excited by inner rotor are 3112 rpm, 7894 rpm and 13702 rpm, respectively. Due to the effects of gyroscopic effect, the whirl eigenvalues for the forward whirl increase and for the backward whirl decrease. However, whether it is forward whirl or backward whirl, the mode shapes remain same in the interest speed range. 3.2. Effects of rotating speed ratio In the case of X1 ¼2400 rpm and X2 ¼3840 rpm ðj ¼ 1:6Þ, the vibration responses of node 4 and node 44 without fault and with blade-casing rubbing are shown in Fig. 9 and Fig. 10 (kc = 4  107N/m, e1 = 0.2 mm), respectively. Since the inner rotor is one of the direct excitation source and the vibration of casing results from these excitation sources under normal condition, the amplitude of node 4 is greater than that of node 44, as shown in Fig. 9(a) and (b). However, when bladecasing rubbing occurs, the rubbing force acts on inner rotor and casing, therefore, the difference between vibration amplitudes of node 4 and node 44 is reduced, as displayed in Fig. 10 (a) and (b). By comparison, it can be observed that the time waveforms of node 4 and node 44 have periodic impulses and the amplitudes obviously increase under faulty condition. Figs. 9(c) and 10(c) indicate that the orbit of node 4 is smoother under normal condition. The reason of this phenomenon is that the periodic rubbing between blade and casing has periodic interference to the motion of the inner rotor. The spectrums calculated from the vibration acceleration signals of node 4 and node 44 are presented in Fig. 9(d) and 10(d), where the same frequency components in both normal and fault conditions are illustrated, respectively. Due to blade-casing rubbing, the spectrums shown in Fig. 10(d) are more complicated and contain more frequency components. It also can be seen from Fig. 10(d) that the impulse frequency (240 Hz) and its multiple frequency components are prominent. The impulse frequency is equal to the product of the number of blades and the rotating frequency of inner rotor (240 Hz = 6  40 Hz).

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Fig. 9. Vibration responses and corresponding spectrums of node 4 and node 44 under normal condition (X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz), j=1.6, kc =4  107N/m, e1 = 0.2 mm; —— Node 4; —— Node 44), (a) time waveform of node 4, (b) time waveform of node 44; (c) orbit; (d) spectrums.

Fig. 10. Vibration responses and corresponding spectrums of node 4 and node 44 under blade-casing rubbing (X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz), j=1.6, kc =4  107N/m, e1 = 0.2 mm; —— Node 4; —— Node 44), (a) time waveform of node 4, (b) time waveform of node 44; (c) orbit; (d) spectrums.

The local enlargements of Fig. 10(d) are shown in Fig. 11. Under the effects of unbalance forces, the rotating frequencies f 1 and f 2 of dual rotors (f i ¼ Xi =60; i ¼ 1; 2) are excited, as illustrated in Figs. 9(d) and 11(a). It can be observed from Fig. 11(a) that some combinations of rotating frequencies of inner rotor and outer rotor, such as 2f 1 , 1.5f 2 , 2f 1 þ 1=2f 2 and 5f 1  f 2 , are

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71

Fig. 11. Spectrum (the enlargements of A and B in Fig. 10(d)).

exhibited. Due to the excitation of unbalance forces and nonlinear blade-casing rubbing, the whirling frequencies (48 Hz and 113 Hz) with small amplitudes are exhibited in Fig. 11(a). Fig. 11(b) shows that there is a sideband family whose frequency spacing is the 1=6 of speed difference (4 Hz = 1=6  ð64  40ÞHz) on both sides of the blade passing frequency and its multiple frequency components. In order to clearly observe the vibration responses of different positions on the casing, the dynamic behaviours of five nodes are compared, as shown in Fig. 12. It can be observed that the time waveforms are different, and the impact amplitude of node 42 is greatest, which results from the direct connection with the inner rotor via bearing support. However, since the blade-casing rubbing information is passed throughout the whole system, the frequency components contained in the spectrums are similar. In particular, due to the strong rubbing force, the amplitudes of impact frequency in all of the nodes are most prominent. Therefore, it’s reasonable to select the node 44 as the represent of casing. When the rotating speed of low-pressure rotor is X1 ¼ 2400rpm and rotating speed ratio is j=1.1, namely X2 ¼ 2640rpm, the vibration responses of the DRBC system without and with blade-casing rubbing are displayed in Figs. 13 and 14 (kc = 4  107N/m, e1 = 0.2 mm), respectively. It can be observed from Fig. 13 that due to small difference between rotating speeds, the time waveforms of inner rotor and casing present beat vibration phenomena. However, when blade-casing rubbing occurs, the amplitudes of beat vibration is much smaller than the amplitudes caused by periodic strong impact, and the beat vibration is weaken. Therefore, it can be observed from Fig. 13 that the vibration amplitudes obviously increase and periodic impulse are presented in the time waveforms of inner rotor and casing. Meanwhile, under the influence of periodic rubbing force, the orbit of normal condition is smoother than that of faulty condition, as shown in Figs. 13(c) and 14(c). Besides, the spectrums of node 4 and node 44 contain same frequency components in the same condition. Under normal condition, only rotating frequencies (40 Hz and 44 Hz) and whirling frequencies (47.33 Hz and 113 Hz) are presented, as

Fig. 12. Vibration responses and corresponding spectrums of different node on the casing (X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz), j=1.6, kc =4  107N/m, e1 = 0.2 mm).

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(b)

0

y acceleration (m/s )

-100 -200 -300 -400 1.9

1.92

1.94 1.96 Time/s

1.98

0 -100

(d)

0.2

2

0.1 0 -0.1 -0.2 -0.2

100

-200 1.9

2

Amplitude (m/s )

y displacement (mm)

(c)

200

2

100

2

(a) y acceleration (m/s )

Fig. 13. Vibration responses and corresponding spectrums of node 4 and node 44 under normal condition (X1 ¼ 2400rpm (40 Hz), X2 ¼ 2640rpm (44 Hz), j=1.1, kc =4  107N/m, e1 = 0.2 mm; —— Node 4; —— Node 44), (a) time waveform of node 4, (b) time waveform of node 44; (c) orbit; (d) spectrums.

-0.1 0 0.1 x displacement (mm)

0.2

1.92

1.94 1.96 Time/s 480

240

720

1.98

2

960

0

10

-5

10

A 0

Node 44 Node 4

B 200

400 600 Frequency (Hz)

800

1000

Fig. 14. Vibration responses and corresponding spectrums of node 4 and node 44 under blade-casing rubbing (X1 ¼ 2400rpm (40 Hz), X2 ¼ 2640rpm (44 Hz), j=1.1, kc =4  107N/m, e1 = 0.2 mm; —— Node 4; —— Node 44), (a) time waveform of node 4, (b) time waveform of node 44; (c) orbit; (d) spectrums.

shown in Fig. 13(d). Nevertheless, more frequencies are excited under blade-casing rubbing, as shown in Fig. 14(d), where the impulse frequency (240 Hz) and its multiple frequencies (480 Hz, 720 Hz and 960 Hz) are apparently presented other than rotating frequencies and whirling frequencies. The periodic impulse frequency is equal to the product of blade number and the rotating frequency of inner rotor (240 Hz = 6  40 Hz).

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The local enlargements of Fig. 14(d) are shown in Fig. 15. Under the excitation of unbalance forces, the rotating frequencies f 1 and f 2 of dual rotors (f i ¼ Xi =60; i ¼ 1; 2) are excited, as shown in Fig. 13 (d) and Fig. 15 (a). Because of the excitation of unbalance forces and nonlinear blade-casing rubbing, the whirling frequency 47.33 Hz with small amplitudes are displayed in Fig. 15(a). It can be found from Fig. 15(a) that some combinations of rotating frequencies of inner rotor and outer rotor are illustrated. Some obvious amplitudes and frequencies in Fig. 15(a) are given in Table 3. Fig. 15(b) shows that there is a sideband family whose frequency interval is the rotating speed difference of inner rotor and outer rotor (4 Hz= (44  40ÞHz). 3.3. Effects of eccentricity of disk The effects of the mass eccentricity e1 of Disk 4 for five cases, which are e1 = 0.2, 0.25, 0.3, 0.35, 0.4 mm, are investigated here, where other parameters remain constant, namely X1 ¼ 2400rpm (40 Hz), j=1.6, kc =4  107N/m. The relationship between the amplitudes variation of some main frequency components and the eccentricity is given in Fig. 16, which shows the following dynamic phenomena. (1) Due to the increase of unbalance force, the amplitude of rotating frequency of inner rotor (40 Hz) is enhanced. (2) With the increasing eccentricity, the peaks of some combined frequency components increase, such as 120, 128, 136, 144, 216, 224 and 232 Hz, which explains the severity of the blade-casing rubbing.

Fig. 15. Spectrum (the enlargements of A and B in Fig. 14(d)).

Table 3 Vibration frequency component analysis (j=1.1). Frequency (Hz)

Components

Frequency (Hz)

Components

Frequency (Hz)

Components

40 44 84 88 92

f1 f2 f 1 +f 2 2f 2 3f 2 -f 1

104 108 112 116 120

3=2f 1 +f 2 1=2f 1 +2f 2 3f 2 -1=2f 1 4f 2 -3=2f 1 3f 1

124 128 132 136 148

2f 1 +f 2 f 1 +2f 2 3f 2 4f 2 -f 1 3=2f 1 +2f 2

Fig. 16. The vibration responses of node 44 with different eccentricities (X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz), j=1.6, kc =4  107N/m).

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N. Wang et al. / Mechanical Systems and Signal Processing 118 (2019) 61–77

Fig. 17. The vibration responses of node 4 and node 44 with different rubbing stiffness, (a) 5  106N/m, (b) 2  107N/m, (c) 2  108N/m (X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz), j=1.6, e1 = 0.1 mm).

(3) The amplitudes of some combined frequency components are also influenced but not obvious, such as 160, 176 and 240 Hz. 3.4. Effects of rubbing stiffness Effects of casing stiffness on the vibration responses of the DRBC system with blade-casing X1 ¼ 2400rpm (40 Hz), X2 ¼ 3840rpm (64 Hz) and e1 = 0.1 mm are discussed in this section. The vibration responses of inner rotor and casing under

different rubbing stiffness, kc =5  106, 2  107, 2  108N/m, are shown in Fig. 17, which shows the following dynamic phenomena.

(1) With the increase of rubbing stiffness, the time waveform of node 4 have more obvious changes, and periodic impulse characteristics are more and more prominent. (2) When the rubbing stiffness is smaller and smaller, the time waveforms gradually degenerates into the smooth curve under normal condition. By comparison, the vibration characteristics of node 44 changes a little, and periodic impulses are illustrated. Since the large rubbing stiffness produces a large rubbing force, the vibration amplitudes of inner rotor and casing gradually increase when the rubbing stiffness is enhanced. (3) Under the periodic interaction between blade and casing, the impact frequency (240 Hz) is clearly presented. With the increasing rubbing stiffness, more frequency components are excited, especially in the green patches of Fig. 17, which demonstrates the deterioration of blade-casing rubbing to some extent. 4. Conclusions In this paper, a dynamic model of a dual-rotor-blade-casing (DRBC) system with blade-casing rubbing is proposed. The acceleration vibrations of inner rotor and casing are obtained. The main conclusions are summarized as follows: (1) Considering the effects of blade number and rotor-casing clearance, a blade-casing rubbing model is introduced to describe the rub-impact mechanism between blade and casing. By calculating the modal characteristics of the DRBC system, the proposed beam-based model has satisfactory accuracy and calculation efficiency. (2) The periodic impact features are presented in the casing vibration acceleration signals under single-point rubbing on the casing. The impact frequency is equal to the product of the rotating frequency and the number of blades. The impulse is modulated by the fraction of rotating speed difference of dual rotors. (3) Blade-casing rubbing levels are related with the eccentricity of disk and rubbing stiffness. The increase of disk eccentricity and rubbing stiffness deteriorate the rub-impact under blade-casing rubbing. In this paper, the numerical simulation of single-point blade-casing rubbing is carried out. The focus is put on the dynamic response of casing under blade-casing rubbing. In future work, attention will be paid to more types of rubbing. Besides, more experiments will be conducted to confirm the validity of simulated results.

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5. Conflict of interest None. Acknowledgment We would like to thank the support from the National Natural Science Foundations of China (Nos. 11572167 and 11802152). Appendix The relative matrices of Timoshenko beam element are given as follows. The stiffness matrix:

2

3

12

6 0 12 6 6 2 6 0 6l ð4 þ UÞl 6 6 6 6l 0 0 EI 6 K eB ¼ 3 6 12 0 0 l ð1 þ Us Þ 6 6 6 0 12 6l 6 2 6 0 6l 2l 4 6l

0

7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

symm ð4 þ UÞl 6l

12

0

0

12

0

0

6l

ð4 þ UÞl

6l

0

0

ð2  UÞl

0

2

2

2

ð4 þ UÞl

2

Us ¼ 12EI=GAs l2 The translational inertial matrix:

2

3

M T1

6 0 6 6 6 0 6 6 M l q A l 6 T2 s e MT ¼ 26 820ð1 þ Us Þ 6 6 M T3 6 0 6 6 4 0

7 7 7 7 7 7 7 7 7 7 7 7 7 5

M T1 2

M T2 l

M T5 l

0

0

M T5 l

0

0

M T4 l

MT1

symm 2

M T3

M T4 l

0

0

M T1

M T4 l

M T6 l

2

0

0

M T2 l M T5 l

MT4 l

0

0

MT6 l

2

M T2 l

0

0

2

M T5 l

2

MT1 ¼ 312 þ 588Us þ 280U2s ; M T4 ¼ 26 þ 63Us þ 35U2s MT2 ¼ 44 þ 77Us þ 35U2s ; MT5 ¼ 8 þ 14Us þ 7U2s

:

MT3 ¼ 108 þ 252Us þ 140U ; M T6 ¼ 6 þ 14Us þ 7U 2 s

As ¼

10 9

2 s

A

1 þ D1:6Dd 2 2 þd

The rotational inertial matrix:

2

3

M R1

6 0 6 6 6 0 6 6 M l qI 6 R2 M eR ¼ 26 30ð1 þ Us Þ 6 6 M R1 6 0 6 6 4 0 MR2 l

7 7 7 7 7 7 7 7 7 7 7 7 7 5

MR1 2

M R2 l

M R3 l

0

0

MR3 l

0

0

MR2 l

M R1

0 0

0 0

M R1 MR2 l 2 M R2 l MR4 l 0

0

symm 2

M R4 l

2

M R2 l

MR1 2 MR2 l MR3 l 0

0

M R3 l

2

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N. Wang et al. / Mechanical Systems and Signal Processing 118 (2019) 61–77

MR1 ¼ 36 ; MR2 ¼ 3  15Us ; MR3 ¼ 4 þ 5Us þ 10U2s

p D4  d4 MR4 ¼ 1 þ 5Us  5U2s ; I ¼ 64 The gyroscopic matrix:

2

3

0

6 M R1 0 6 6 6 M R2 l 0 6 6 0 M q I R2 l 6 Ge ¼ 6 0 M 15lð1 þ Us Þ2 6 R1 6 6 M 0 6 R1 6 4 M R2 l 0 0

0

7 7 7 7 7 7 7 7 7 7 7 7 7 5

antisymm 2

MR3 l

0

MR2 l 0

0 M R2 l

0

M R4 l

M R2 l M R4 l

2

0

2

0 M R1

0

M R2 l

0

0

0 2

M R2 l MR3 l

0

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