Cracked Continuous Rotors Vibrating on Nonlinear Bearings

7/28/2019 Cracked Continuous Rotors Vibrating on Nonlinear Bearings

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Cracked Continuous Rotors Vibrating onNonlinear Bearings

Papadopoulos C.A., Chasalevris A.C., Nikolakopoulos P.G.

Department of Mechanical Engineering & Aeronautics,

University of Patras

Patras 26504

GREECE

Abstract The dynamic behavior of cracked rotors continues to attract the interestof both designers and maintenance engineers. In this work, a continuous mechanicsapproach is used to simulate rotor vibration. The case of the cracked continuous ro-tor is examined by introducing suitable complex boundary conditions. The shaft ro-tates on two journal bearings that are simulated as forces acting on it and the boun-dary conditions are expressed accordingly. When the angular velocity passesthrough critical speeds, these forces become highly nonlinear. Identifications ofcracking and wear of the bearing are separately investigated.

The current challenge for designer engineers is to provide lighter, quieter, more ef-ficient, compact, and stable, as well as less expensive and ecologically friendly, ro-

tating machines, operating even in severe conditions. In other words, new targetsmust be seen from the following three points of view: (a) analysis and design, (b)

new material technology, and (c) new production techniques.

Keywords: rotor, shaft, crack, bearing, wear, nonlinear

1 IntroductionRotor vibrations are expressed by the Timoshenko differential equation which in-cludes the effects of the transmitting torque, the rotary inertia, the transverse shear

and the gyroscopic moments, as described by Eshleman-Eubanks [1]. The rotating

crack is modeled using the Strain Energy Release Rate (SERR) method as a func-tion of both the crack depth and the angle of rotation. A state-of-the-art review ispresented by Papadopoulos [2]. The complex boundary conditions for the rotatingcrack and for the bearings are also introduced by Chasalevris and Papadopoulos in[3]. The rotor is supported by two journal bearings that operate in nonlinear condi-tions. The journal bearings are modeled by two forces calculated under the validity


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2 Papadopoulos C.A., Chasalevris A.C., Nikolakopoulos P.G.

Fig. 1: Two step cracked rotor-bearing system carrying a disk

of Reynolds equation for laminar, isothermal and isoviscous flow using the finitedifference method. Highly nonlinear bearing forces are present when the rotor-bearing system operates near or at resonance. These forces affect the dynamic be-havior of the rotor-bearings system, and conversely the bearing hydrodynamicfunctionality takes into account the dynamic properties of the entire shaft instead ofthe journal mobility, thus resulting in more precise journal mobility.

The main aim of the present paper is to construct an accurate continuous rotormodel that is mount-bounded from the finite bearing boundary conditions, which

enable importing of the entire model and provide accurate properties of nonlinearforces regardless of where or how the journal trajectories are developed. The re-sults include time frequency analysis of the resulting time series (rotor response),rotor orbits and frequency response computation. Methods are presented for crackidentification and wear assessment (using the model of Dufrane et al. [4]) by ex-

ploitation of the vibration at the bearings.

2 Continuous model of a cracked rotorIn this approach, the equations of a continuous rotating shaft are used, and the

boundary conditions of the rotating crack are introduced, thus enabling the conti-nuous modeling of a cracked rotor [5]. Let us assume a uniform, homogenous andcracked rotating Timoshenko shaft (Fig. 1), with Youngs modulus E, shear mod-ulus G, density , moment of inertia of the cross-section aboutXaxisI, shear factork = 10/9, length L, radius R, surface of cross section A, radius of gyra-

tion0

/r I A= , and Poisson ratio . The shaft is rotating with an angular velocity

, whirling with , and transmitting an axial torque .

Consider also a transversely located disk in the mid-span (x = L / 2) of the shaft ofthe same material, with radiusRd, mass md, and thicknessLd. A breathing crack, of

depth /a a R= , is located at the mid-span, adjacent to the disk. IfY(x,t) andZ(x,t)are the vertical and horizontal responses at an axial coordinatex and time t, respec-


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Cracked Continuous Rotors Vibrating on Nonlinear Bearings 3

tively, then, by supposing the complex notation ( , ) ( , ) ( , )U x t Y x t i Z x t = + , the

coupled governing equation of motion is given by Eq. 1 [1, 6]:

4 3 4 3

2 2

0 04 3 2 2 2

3 4 3 22 2 2 2

0 0

2 4 3 2

2

2 0

j j j j

j j j j

U U U U E IE I iT A r i A r

k Gx x x t x t

U U U U Ar ArTi i A

k G k G k Gx t t t t

+ + +

+ + + =

(1)

wherej = 1 for the first part of the shaft from the left end up to the crack and j = 2for the part from the crack up to the right end (Fig. 1). Eq. 1 is a complete fourthorder complex partial differential equation of motion for the Uj. The solution pro-cedure of the above equation and the usual boundary condition are presented in [5].For the boundary condition due to the crack the Strain Energy Release Rate(SERR) method, introduced by Dimarogonas and Paipetis [7], was applied to the

calculation of compliance due to a rotating crack by Chasalevris and Papadopoulos[8]. Crack breathing could be linear with periodically varying coefficients when theweight deformation dominates the response amplitude, or nonlinear when the in-verse occurs. Numerical analysis must follow the resulting bending moment in thetwo main directions relative to the crack in the rotating coordinate system at eachtime step of the integration. Afterwards, the decision of whether the crack is open,

closed or partially open could be made, and the respective compliances, in thefixed coordinate system, should be used.

At the crack position if1(L1,t) and 2 (L1,t) are the complex slopes before and af-

ter the crack and { } { }2 2 44 45 54 55, , ,x c c c c= C is the well-known compliance ma-trix, then the boundary condition due to the rotating crack is described by Eq. 2:

( ) ( )2 1 1 1 1 1

2 1 1 1

( , ) ( , ) ( , ) ( , ) open crack

( , ) ( , ) 0 closed crack

L t L t i M L t i M L t

L t L t

= + + +

=

(2)

where ( )55 44 / 2c c = + , ( )45 54 / 2c c = , ( )55 44 / 2c c = , ( )45 54 / 2c c = + , and

( ),x t

is the conjugate ofM(x,t).

3 J ournal bearing support worn bearingThe nonlinear fluid film forces generated by the journal bearing are derived fromthe solution of the Reynolds equation, which, for laminar, isothermal, and isovisc-ous flow, is written as Eq. 3 [9, 10]:


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Fig. 2: Worn journal bearing. Loads and wear zone for a specific equilibrium position.

3 3

2

( , ) ( , )( ) 1 ( ) ( ) ( )2

6 6

k kt tP l P lh h h h

l x tR

+ = +

(3)

In Eq. 3, the term ( , )kt

P A is the developed oil pressure at time tk, is the lubricant

viscosity,Ris the journal radius, and is the angular coordinate relative to the atti-

tude angle axis. The fluid film thickness ( )h is given by Dufrane [4] as in Eq. 4:

( )

( )

1

0

0

1

0

1 cos( ( )), for 0 ,( )

1 cos( ( )) ( ), for

3cos 1

2( ) 1 cos and

32cos 1

2

a b

h a b

h

b

h

+ =

+ + <


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( ) ( ), ,0 0 0 0

sin( ) and cos( )k kt t r t

F P R l F P R l

= = = =

= = (5)

Besides the previous solutions, FLUENT package was also used in order to obtainthe journal bearing characteristics and to test the results of this code. The continui-ty and momentum conservation equations have been solved and the method, aswell as the results, are presented by Gertzos et al. in [11, 12].

4 Rotor bearing systemIn this chapter the fluid film impedance forces are applied in the rotor at the points

of the two journals in order to construct a system of equations using boundary con-ditions. In Eq. 3, there are four variables as inputs for the calculation of the bearingimpedance force, which must be expressed as functions of the rotor (journal) re-

sponse. These variables, i.e., the eccentricity , ki te and the attitude angle , ki t of

each journal (i = 1, 2) together with their respective velocities, ,

,k ki t i t

e , are ex-

pressed as functions ofYi(x,tk) and Zi(x,tk) for the time tk, in Eq. 6:

( ) ( ) ( )

( ) ( )

2 2 1

, ,

2

, , , , ,

(0, ) (0, ) , tan (0, ) / (0, )1,2

/ , (0, ) (0, ) (0, ) (0, ) /

k k

k k k k k

i t i k i k i t i k i k

i t i t i t t i t i k i k i k i k i t

e Y t Z t Y t Z t i

e e e t Z t Y t Y t Z t e

= + = =

= =

(6)

From the above equations, it is clear that the pressure deviation in each bearing is a

function ofYandZthat depends on the solution constants qi(t). Thus, a system of32 equations (16 of real and 16 of imaginary parts) is obtained using the 16 com-plex boundary conditions for displacements, slopes, bending moments, and shear-ing forces [5].

Fig. 3: Nonlinear system due to the interdependency of the bearing forces and the rotor response


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The boundary conditions at both ends of the system (bearings) are expressed as theequality of fluid film impedance forces to the journal shearing force. The imped-ance moment developed of fluid film as a reaction to journal misalignment is nottaken into account yielding boundary conditions of bending moment equal to zero.Under this consideration the shearing force boundary conditions become a functionof rotor response and consequently the unique variable incorporated in the systemis the time (Fig. 3). The resulting dynamic system yields nonlinear oscillations pre-

senting sometimes quasi-periodic or even chaotic motions, especially near reson-ance operation. The solution of the system is achieved numerically in discrete timewith the time interval to be the significant parameter. In brief, the 32x32 system ofequations (boundary conditions) is solved using a modified Newton-Raphson me-thod, providing the ability of random initial guess. The evaluation result in the de-finition of parametersp

iand q

i(Fig. 3) at every time step and thus the response is

calculated. The main benefit of this consideration is that no bearing coefficientsare used since no journal equilibrium position has to be defined. Additionally, the

bearing properties are incorporated at any operational condition no matter what thetrajectory of the journal is inside the bearing.

5 Experimental crack identification us ing external exciterAn external electromagnetic excitation device was designed, constructed, and used(Fig.4), as suggested by Lees et al. [13], to externally excite the rotor in the hori-zontal direction during its operation for identification purposes. The applicability

of this method depends on the possibility to install on the system an external exci-ter. Instead of this exciter, the method can be applied in cases where magnetic

bearings are used, as it is easy in such cases to impose an excitation to the rotatingsystem using the controller of the magnets.The shaft is rotating at n = 500 RPM, the excitation is of a steady frequency, in thehorizontal direction, and the steady-state vertical response should be measured inorder to develop a method of crack detection using the dynamic coupling betweenhorizontal and vertical response due to crack breathing. When the horizontal exter-nal excitation is introduced in the system at nEX= 4000 cycles / min, the vertical re-

sponse is altered (it is suggested to use nEX = 8n). The magnitude of the electro-magnetic force used is estimated to be approximately 10% of total system weight(here is about 40N). The horizontal excitation intrudes on the vertical response sig-nal through the mechanism of coupling of the system due to both the bearing

asymmetry and the crack. Subsequently, the responses of the intact and cracked ro-tor (/R = 20%) are subtracted, the resulting difference is transformed using theContinuous Wavelet Transform (Morlet Wavelet), and the corresponding compo-nent (Scale 61, resulting from Eq. 7) of the frequency of external excitation is ex-

tracted.


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(a) (b)

Fig. 4: a) The electromagnets arrangement in the horizontal direction provides the external excita-

tion sinusoidal force, with variable excitation frequency, b) The electromagnet operation scheme.

0.8125 60.93 61(4000 / 60) (1/ 5000)

c

a

F HzaF rpm Hz

= = =

(7)

where a is the monitored scale of the wavelet, Fc is the centre frequency of Morletwavelet,Fa = nEX / 60 Hz is the excitation frequency and is the sampling period(here the sampling frequency is 5000 samples / sec).Fig. 5 shows the plot of the extracted component for both the experiment and thesimulation. The wavelet coefficient of scale corresponding to the external excita-

tion frequency properly demonstrates the coupling due to the crack, during the ro-tation of the shaft. It contains only one frequency (4000 RPM), the amplitude ofwhich is well localized in the time necessary to determine whether or not the cou-pling exists. The coupling presence during rotation is a function of the crack rota-tional angle, and Fig. 5(b) clearly shows that the coupling intensifies at the timesteps when the crack is totally open, near samples at 400, 1000, and 1600 s (Fig.

5). This fact enables the detection of a crack, since only the defect of a crack canyield this dynamic coupling.

In the experimental case, the variation of the amplitude of the wavelet coefficient isalso noticed during the rotation but not as clearly as in simulation. The differencesbetween Fig 5 (a) and (b) are due to two reasons: (a) the experimental crack (a cutwas used) remains open during the rotation and does not breathe as the crack doesin the simulation and (b) the force in the simulation is of constant magnitude

(a) (b)

Fig.5: Extraction of wavelet coefficient of Scale 61 (Pseudo frequency 4000RPM). (a) Experi-

ment and (b) simulation.


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(-40 N to 40 N) while in the experiment this force is also depending on the fluctua-tions of the gap between the rotor and the magnets. Thus it was expected for theexperiment to give higher values to the coefficient.Thus, the coupling due to a cut exists for most of the time needed for an entire rota-tion. However, the current wavelet coefficient is judged to be very sensitive tocrack depth variation and can be used for detection of cracks as small as 20% ofthe radius as shown in Fig. 5. As it was proven it is highly beneficial that bearing

measurements can also yield crack detection as this facilitates the applicability ofthe method in real machines, since bearing measurements are widely used in largemachine monitoring.

6 Wear assessmentThe wear assessment can be done by weighting the bearing before and after its use.The difference indicates the material lost due to wear. This method cannot be doneduring operation. Saridakis et al. [14] used artificial neural networks in order todetect the wear percentage and the misalignment angles for a journal bearing dur-

ing its operation. Gertzos et al. [12] investigated the operational and easily measur-able characteristics, such as eccentricity ratio, bearing attitude angle, lubricant sideflow, and friction coefficient that could be used for bearing wear assessment with-out stopping the machine. They used Computational Fluid Dynamics (CFD) analy-sis in order to solve the Navier-Stokes equations. A graphical detection methodwas used to identify the wear depth associated with the measured characteristics.

The Archards model was also used in order to predict the wear progress when thejournal is in full contact with the bearing pad or wears out the bearing under the ab-

rasive mechanism, and finally to predict the volume loss of the bearing material.Nikolakopoulos et al. [15, 16] also proposed a mathematical model and an experi-mental setup in order to investigate the wear influence on the dynamic response ofthe system and on other dynamic characteristics of the frequency and time domain.A numerical application with the physical and geometric properties listed in Table1 is used here in order to investigate the effects of a worn bearing on the dynamicproperties of the system.

Table 1: Geometric and physical properties of the current rotor bearing system

Item Symbolandvalue Item SymbolandvalueShaft Radius R = 0.025 m Material Loss Factor = 0.001

1st

Step Length L 1 = 1 m Bearing Length L b = 0.05 m

2nd

Step Length L 2 = 1 m Bearing Radial Clearance c r= 100 m

Disk Radius R d = 0.19 m External Load EF = Wd N

Shaft/Disk Density = 7832 kg/m3

Oil Viscosity = 0.005 Pa.s

Disk Width L d = 0.022 m Youngs modulus E = 206 GPa


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(a) (b)

Fig.6: Log modulus of STFT of time history through first critical in Journal 1 for relative wear

depths of a) 0% and b) 40 %

The system start-up is performed from the initial rotational speed of = 30 rad/s

to the maximum of = 100 rad/s, with an acceleration of24rad/s = , while the

sampling frequency is 1/ 800Samples/st = . Note that the sampling frequency is a

significant parameter and is a result of various tests performed to render the algo-

rithm computable. A time step oft = 0.00125is used in all evaluations. The ma-

terial loss factor is set arbitrarily to this low value to cut the infinite response just

enough to make the start-up computable.In this work, the variable loss factor is not included because the internal damping istreated as a tool in order to avoid the infinite response that cannot be damped bythe bearing damping coefficients. The left journal (Journal 1) vertical response iscalculated for wear depths of 0% and 40%. A time frequency analysis using Short

Time Fourier Transform (STFT) is applied to these signals, and the result is shownin Fig. 6. The development of 1/2X, 3/2X, 5/2X etc harmonics can be easily ob-served. These harmonics are due to the wear defect.

7 ConclusionsA continuous approach is used here to simulate the dynamic behavior of a rotor-bearing system. The finite difference method is used to solve the Reynolds equa-tion. A crack of the rotor and the wear of the bearing are considered as defects anddynamic methods are presented for their identification, both analytically and expe-rimentally. In the future, rotordynamics is expected to be influenced by the use ofnew and better materials, whether composites or conventional. In the era of nano-

technology, micro- and nano-rotors are expected to open new horizons in this field.New smart materials and fluids are also expected to be used in journal bearings toconfront the problem of friction and wear minimization.


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Cracked Continuous Rotors Vibrating on Nonlinear Bearings