Cracked Continuous Rotors Vibrating on Nonlinear Bearings

Cracked Continuous Rotors Vibrating on Nonlinear 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 interest of both designers and maintenance engineers. In this work, a continuous mechanics approach 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 passes through critical speeds, these forces become highly nonlinear. Identifications of cracking 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 targets must 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 Introduction

Rotor 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 is presented by Papadopoulos [2]. The complex boundary conditions for the rotating crack 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

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 finite difference 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 hydrodynamic functionality takes into account the dynamic properties of the entire shaft instead of the journal mobility, thus resulting in more precise journal mobility. The main aim of the present paper is to construct an accurate continuous rotor model that is mount-bounded from the finite bearing boundary conditions, which enable importing of the entire model and provide accurate properties of nonlinear forces 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 crack identification 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 rotor

In 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 and cracked rotating Timoshenko shaft (Fig. 1), with Young’s modulus E, shear mod-ulus G, density ρ, moment of inertia of the cross-section about X axis I, shear factor k = 10/9, length L, radius R, surface of cross section A, radius of gyra-tion 0 /r I A= , and Poisson ratio ν. The shaft is rotating with an angular velocity Ω, whirling with ω, and transmitting an axial torque Τ. Consider also a transversely located disk in the mid-span (x = L / 2) of the shaft of the same material, with radius Rd, mass md, and thickness Ld. A breathing crack, of depth /a a R= , is located at the mid-span, adjacent to the disk. If Y(x,t) and Z(x,t) are the vertical and horizontal responses at an axial coordinate x and time t, respec-

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 32 2

0 04 3 2 2 2

3 4 3 22 2 2 20 0

2 4 3 2

2

2 0

j j j j

j j j j

U U U UE IE I iT Ar i Ark Gx x x t x t

U U U UAr ArTi i Ak G k G k Gx t t t t

ρ ρ ρ Ω

ρ ρ Ωρ ρ

∂ ∂ ∂ ∂⎛ ⎞− − + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂ ∂+ + − + =

∂ ∂ ∂ ∂ ∂

(1)

where j = 1 for the first part of the shaft from the left end up to the crack and j = 2 for the part from the crack up to the right end (Fig. 1). Eq. 1 is a complete fourth order 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 the weight deformation dominates the response amplitude, or nonlinear when the in-verse occurs. Numerical analysis must follow the resulting bending moment in the two main directions relative to the crack in the rotating coordinate system at each time 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 the fixed coordinate system, should be used. At the crack position if Θ1(L1,t) and Θ2 (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 crackL t L t i M L t i M L tL t L t

Θ Θ β γ δ εΘ Θ

− = + + +

− = (2)

where ( )55 44 / 2c cβ = + , ( )45 54 / 2c cγ = − , ( )55 44 / 2c cδ = − , ( )45 54 / 2c cε = + , and

( ),M x t is the conjugate of M (x,t).

3 Journal bearing support – worn bearing

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


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

3 3

2

( , ) ( , )( ) 1 ( ) ( ) ( )26 6

k kt tP l P lh h h hl x tR

θ θθ θ θ θΩμ θ μ θ θ

∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂+ = +⎜ ⎟ ⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ (3)

In Eq. 3, the term ( , )kt

P θ is the developed oil pressure at time tk, μ is the lubricant viscosity, R is the journal radius, and θ is the angular coordinate relative to the atti-tude angle axis. The fluid film thickness ( )h θ is given by Dufrane [4] as in Eq. 4:

( )

( )

10

01

0

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

1 cos( ( )) ( ), for

3 cos 12( ) 1 cos and

32 cos 12

a b

h a b

h

b

h

α

ε θ ϕ π θ θ θ θ πθ

ε θ ϕ π δ θ θ θ θ

πθ δπδ θ δ θ

πθ δ

−

−

+ − − ≤ ≤ ≤ ≤⎧= ⎨ + − − + < <⎩

⎧ = − −⎪⎛ ⎞ ⎪⎛ ⎞= − + − ⎨⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎪ = + −

⎪⎩

(4)

In previous equations, ε is the eccentricity ratio / re cε = , φ is the attitude angle of

the journal, ε and ϕ are their respective time derivatives, rc is the radial clear-ance, and 0d and 0 0 / rd cδ = are the absolute and relative wear depths (see Fig. 2). Note that all angles are defined with respect to the coordinate system in Fig. 2. Three loads exist in each journal: a percentage of the weight Wg of the rotor, a per-centage of external force EF (here, the gravity force of the disk, F dE W= ), and a percentage of the unbalance force Fu that acts in the plane of the disk. The fluid film hydrodynamic reaction in this dynamic load results from the radial, Fr and tangential, Ft, forces which consequently are functions of time. The fluid film hy-drodynamic reaction in this dynamic load consists of a tangential force Ft and a radial force Fr, which consequently are functions of time. The finite difference me-thod is used here to solve the Reynolds equation.


( ) ( ), ,0 0 0 0

sin( ) and cos( )k kt t r tF P R l F P R l

ν ν ν νη λ η λ

η λ λ η λ λη λ η λ

θ θ θ θ= = = =

= Δ Δ = Δ Δ∑∑ ∑∑ (5)

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

4 Rotor – bearing system

In 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 bearing impedance 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 tϕ of

each journal (i = 1, 2) together with their respective velocities , ,,k ki t i te ϕ , are ex-

pressed as functions of Yi(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 ti

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

ϕ

ϕ

−

−Δ

⎫= + = ⎪ =⎬= − Δ = − ⎪⎭

(6)

From the above equations, it is clear that the pressure deviation in each bearing is a function of Y and Z that depends on the solution constants qi(t). Thus, a system of 32 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


The boundary conditions at both ends of the system (bearings) are expressed as the equality 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 not taken into account yielding boundary conditions of bending moment equal to zero. Under this consideration the shearing force boundary conditions become a function of rotor response and consequently the unique variable incorporated in the system is 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 time with the time interval to be the significant parameter. In brief, the 32x32 system of equations (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 parameters pi and qi (Fig. 3) at every time step and thus the response is calculated. The main benefit of this consideration is that no bearing coefficients are used since no journal equilibrium position has to be defined. Additionally, the bearing properties are incorporated at any operational condition no matter what the trajectory of the journal is inside the bearing.

5 Experimental crack identification us ing external exciter

An 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 rotating system using the controller of the magnets. The shaft is rotating at n = 500 RPM, the excitation is of a steady frequency, in the horizontal direction, and the steady-state vertical response should be measured in order to develop a method of crack detection using the dynamic coupling between horizontal 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 (α/R = 20%) are subtracted, the resulting difference is transformed using the Continuous Wavelet Transform (Morlet Wavelet), and the corresponding compo-nent (Scale 61, resulting from Eq. 7) of the frequency of external excitation is ex-tracted.


(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 HzΔ

= = = ≈×

(7)

where a is the monitored scale of the wavelet, Fc is the centre frequency of Morlet wavelet, Fa = nEX / 60 Hz is the excitation frequency and Δ 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 the simulation. 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 of which 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 time steps 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 can yield this dynamic coupling. In the experimental case, the variation of the amplitude of the wavelet coefficient is also noticed during the rotation but not as clearly as in simulation. The differences between Fig 5 (a) and (b) are due to two reasons: (a) the experimental crack (a cut was used) remains open during the rotation and does not breathe as the crack does in 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.


(-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 the experiment 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 to crack depth variation and can be used for detection of cracks as small as 20% of the 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 of the method in real machines, since bearing measurements are widely used in large machine monitoring.

6 Wear assessment

The 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 done during operation. Saridakis et al. [14] used artificial neural networks in order to detect 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 side flow, 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 method was used to identify the wear depth associated with the measured characteristics. The Archard’s model was also used in order to predict the wear progress when the journal 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 of the system and on other dynamic characteristics of the frequency and time domain. A numerical application with the physical and geometric properties listed in Table 1 is used here in order to investigate the effects of a worn bearing on the dynamic properties of the system.

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

Item Symbol and value Item Symbol and value

Shaft Radius R = 0.025 m Material Loss Factor η = 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 μm

Disk Radius R d = 0.19 m External Load E F = W d N

Shaft/Disk Density ρ = 7832 kg/m3 Oil Viscosity μ = 0.005 Pa.sDisk Width L d = 0.022 m Young’s modulus E = 206 GPa


(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 Ω = 30 rad/s to the maximum of Ω = 100 rad/s, with an acceleration of 24rad/sΩ = , while the sampling frequency is 1/ 800Samples/stΔ = . 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 of Δt = 0.00125 is 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 is treated as a tool in order to avoid the infinite response that cannot be damped by the bearing damping coefficients. The left journal (Journal 1) vertical response is calculated 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 shown in 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 Conclusions

A 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 and dynamic methods are presented for their identification, both analytically and expe-rimentally. In the future, rotordynamics is expected to be influenced by the use of new 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 to confront the problem of friction and wear minimization.


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Cracked Continuous Rotors Vibrating on Nonlinear Bearings