The influence of drillstring-borehole interaction on backward whirl K. Vijayan 1 , N. Vlajic 2 , M.I. Friswell 1 1 College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK Singleton Park e-mail: [email protected] 2

Department of Engineering Mathematics, University of Bristol, Queen’s Building, Bristol BS8 1TR, UK

Abstract A major concern within the oil drilling industry remains the high risk associated with the drilling bit and tool failure from the build-up of damaging vibrations. Effective understanding of the drillstring dynamics is essential for efficient drilling operation. Complex dynamic behaviour is often observed in the drillstring due to friction, impact, unbalance eccentricity and energy exchange between different modes of vibrations. The interaction between the drillstring and borehole wall involves nonlinearities in the form of friction and contact. A two disk model is used to study the behaviour of the system. The effects of impact, friction and mass unbalance are included in the model. The drillstring borehole interaction induces whirling behaviour of the drillstring causing forward whirl, backward whirl or intermittent bouncing behaviour depending on the system parameters. A critical steady state behaviour within the system is the backward whirling of the drillstring, which reduces the fatigue life of the drillstring. The theoretical model analyses the influence of different parameters, such as the eccentricity and rotational speed, on the induced backward whirl within the system. The influence of rotor speed on the system dynamics is explored using a run up and run down and is analysed using a two dimensional waterfall plot. The waterfall plot indicated the frequency of maximum response for each rotor speed. Depending on the whirling behaviour the dominant frequency was observed at the natural frequency, the rotational speed or the backward whirl frequency. The behaviour of the system by varying the eccentricity and coupling stiffness was analysed. The results from the study indicated that depending on the eccentric mass and the coupling stiffness the backward whirl could be localized to a single rotor.

1

Introduction

A major concern in the oil drilling industry is the high cost and lead time caused due to the drillstring and bit failure from the the build up of damaging vibration. The system behaviour is non-linear due to impact and friction at borehole/drill string and bit formation interfaces. The excitation is provided from the unbalance eccentricity or initial curvature in drillstring sections. There are also geometric non-linearities in the inertia terms due to coupled between the torsional and lateral motions. The length of the drill assembly increases the possibility of energy exchange between various modes of vibrations. Lateral oscillations which could results in whirling behaviour within the drillstring can be the most destructive generating shocks as high as 250g [1]. If the shaft has one or more rotors attached, more complicated whirl phenomena can occur. Various authors [2, 3, 4, 5, 6, 7, 8, 9] have described the non-linear phenomena that characterize the dynamics of a rotor. Whirling of a shaft or rotor can result in interaction with the enclosure or bearing. This interaction in turn can change the nature of the whirl. Rubbing is a commonly

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encountered problem in rotor dynamics. Problems of rubbing involve investigation of two main effects: determination of local casing rotor interaction and global vibration of the rotor casing/bearing. The frequencies present in the measured vibration signal constitute some of the most useful information for diagnosing rotordynamics problems. But not all shaft whirling is synchronous; the more destructive rotordynamic problems involve non-synchronous whirl. Basically there are two common steady state vibration regimes of rotor motion which are created by rub. These steady states are usually reached through some transient motions involving partially rubbing surfaces. The first steady state regime is due to unbalance and the second is a self excited vibration. The former is less dangerous but the latter can often cause catastrophic failure. Steady state whirl due to unbalance usually occurs during transient conditions of start-up and/or shutdowns when the rotor passes through the resonance speed. This regime is often referred as full annular rub. The second of the quasi stable steady state regimes, which could be more serious in its destructive effect, is the self excited backward full annular rub known as “backward whirl”. One of the simplest models that can be used to study the flexural behaviour of rotors consists of a point mass attached to a massless flexible shaft. This model is often referred to as the Jeffcott rotor [8, 10]. Several authors have tried to formulate the equations of motion in polar [7, 11] and Cartesian coordinates[3, 12]. In the present study the equations of motion of a two DOF system is formulated in Cartesian coordinates.

2

Theoretical model

The model consists of a massless shaft with two rotor disks having eccentric mass as shown in Fig. 1. In the torsional stiffness model one end of the rotor is attached to the drive and the other represents the free end of the drillstring. The mass located eccentrically rotates at rotor speed ω as shown in Fig. 2. The rotor is displaced from the geometric center of the “borehole” by a distance (δ) and a linear spring of stiffness ki is used to model the restoring force which is due to the bending of the shaft. The angular position of the shaft centre is denoted by θi . The position vector of the shaft centre and eccentric mass for RiM and Rim are given by RiM

= xi ˆj + yi kˆ

Rim = [xi + ei cos(ψ + θi + φi )]ˆj + [yi + ei sin(ψ + θi + φi )]kˆ

(1) (2)

where (xi , yi ) is the position of the shaft centre for rotor i, ei is the position of the eccentric mass with respect to the shaft centre, θi is the torsion angle of the rotor i, ψ is the rotational angle of the drive which for a constant rotational speed (ω) will be ωt and φi is the phase angle of the eccentric mass. The Kinetic energy (T ) of the system is given as T

=

2 1X 2 ˙ + θ˙i )2 + mi (R˙ 2 ) Mi (R˙ iM ) + Ji (Ψ im 2 i=1

(3)

Mi is the mass of rotor i and mi is the eccentric mass of rotor i. Ji is the moment of inertia of rotor i. The potential energy (V ) of the system is V

=

k1 2 k2 kc kc kt ktc (x1 + y12 ) + (x22 + y22 ) + (x2 − x1 )2 + (y2 − y1 )2 + (θ1 )2 + (θ2 − θ1 )2 (4) 2 2 2 2 2 2

where k1 , k2 are the first and the second rotor stiffness in the X and Y directions respectively. kc is the coupling stiffness between the two rotor disks in the X and Y directions. The stiffness is assumed to be identical in the X and Y directions for both k1 + k2 and kc . kt and ktc are the torsional stiffness and torsional coupling stiffness respectively of the shaft. Using the Lagrange formulation the equations of motion are

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formulated in the X, Y and θ directions as dT dV d ∂T )− + ( dt ∂ x˙i dxi dxi d ∂T dT dV ( )− + dt ∂ y˙ dy dy dT d ∂T dV )− ( + dt ∂ θ˙ dθ dθ

= Fx(i)

(5)

= Fy(i)

(6)

= Mθ(i)

(7) (8)

˙ + θ˙1 )2 cos(Ψ + θ1 + φ1 ) (M1 + m1 )x¨1 + c1 x˙1 + k1 x1 + kc (x1 − x2 ) = m1 e1 [(Ψ ¨ + θ¨1 ) sin(Ψ + θ1 + φ1 )] +(Ψ ˙ + θ˙1 )2 sin(Ψ + θ1 + φ1 ) (M1 + m1 )y¨1 + c1 y˙1 + k1 y1 + kc (y2 − y1 ) = m1 e1 [(Ψ ¨ + θ¨1 ) cos(Ψ + θ1 + φ1 )] −(Ψ (J1 + m1 e2 )θ¨1 + ct1 θ˙1 + kt1 θ1 + ktc (θ1 − θ2 ) = m1 e1 [(x¨1 sin(Ψ + θ1 + φ1 ) − y¨1 cos(Ψ + θ1 + φ1 )] ˙ + θ˙2 )2 cos(Ψ + θ2 + φ2 ) (M2 + m2 )x¨2 + c2 x˙2 + k2 x2 + kc (x2 − x2 ) = m2 e2 [(Ψ ¨ + θ¨2 ) sin(Ψ + θ2 + φ2 )] +(Ψ ˙ + θ˙2 )2 sin(Ψ + θ2 + φ2 ) (M2 + m2 )y¨2 + c2 y˙2 + k2 y2 + kc (y1 − y2 ) = m2 e2 [(Ψ ¨ + θ¨2 ) cos(Ψ + θ2 + φ2 )] −(Ψ (J2 + m2 e2 )θ¨2 + ct2 θ˙2 + kt2 θ1 + ktc (θ2 − θ1 ) = m2 e2 [x¨2 sin(Ψ + θ2 + φ2 ) −y¨2 cos(Ψ + θ2 + φ2 )]

(9)

where c1 and c2 are the damping coefficients of first and second rotor disks.

2.1

Contact modelling

The next phenomenon that needs to be modelled is the effect of the contact between the borehole wall and the whirling drillstring especially at the stabilizer location. Similar to the impact of a rotor against a stator within the rotor-dynamic context the whirling drillstring can impact the borehole wall. In addition to the plastic deformation the contact between the system also causes dissipation due to the frictional force. There are a plethora of models available for friction. For the present model the frictional force is assumed to be Coulomb [13, 14] with no Stribeck or viscous effect even though there is an uncertainty in the type of model to be used [7]. For numerical stability in the simulation, the variation in the friction coefficient with relative velocity (Vrel ) was smoothed using a continuous function, such that Vrel µ = µd tanh V0

(10)

where µd is the dynamic friction coefficient, Vrel is the relative velocity at the contact interface given by ˙ + θ˙i )Ri − x˙i yi − y˙i xi Vrel(i) = (Ψ ρi ρi ˙ ˙ = (Ψ + θi )Ri − x˙i sin αi − y˙i cos αi

(11)

q

where ρi = x2i + yi2 and αi = arctan( xyii ). V0 is a constant and its value can be varied to obtain different velocity profiles at the contact region. The model, is a reasonable first approximation representation of the forces acting in the real system except that the effect of drilling mud influences the response in the form of fluid-structure interaction, which is neglected. Contact with the borehole generates a normal force in the radial direction (Fn ) and a frictional force in the tangential direction (Ft ). Contact is modelled using a spring of relatively high stiffness. Damping is included

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in the model to incorporate the effect of a coefficient of restitution. The contact force generated in the radial direction is given by Fn(i) = ks (ρi − δi ) + cs ρ˙i = 0

for ρi > δi for

ρi ≤ δi

(12)

where δi is the clearance. The frictional force is modelled using Coulomb friction which generates the tangential force (Ft(i) ) Ft(i) = µFn(i)

(13)

Transforming the forces to the Cartesian coordinate we obtain the forces in the X and Y directions as Fx(i) = −Ft(i) sin(α) − Fn(i) cos(α) Fy(i) =

Ft(i) cos α − Fn(i) sin α

(14)

The frictional moment generated in the torsional direction is: Mext = Ri Ft(i)

3

(15)

Results and Discussion

Using the theoretical model the system was simulated for the parameters given in Table 1. Torsional damping is assumed to be small for the present study similar to rotor dynamic systems [13]. However within an actual drilling environment the torsional damping could be higher due to the drilling mud and formations. The natural frequencies of the system and the mode shapes are given in Table 2. It can be observed that the two lateral modes are well separated in frequency. A case study was performed by altering the relative phase of the eccentric mass location. A frequency sweep was carried out by varying the rotor speed from 0.9 to 2.4 Hz. The range was chosen to cover the lateral natural frequencies of the system. Initially the eccentric mass was located at the same angular position for both disks. This implies that the external excitation forces due to unbalance acts in phase for both of the disks, therefore exciting the first mode of the system. The temporal variations in the responses of both rotors are shown in Fig. 3(a). For clarity only the run up is shown in Fig. 3(a), which clearly shows that the system establishes contact at nearly the same instant on both rotors. Figure 3(b) shows the variation of the normalised clearance between the drillstring and borehole, with a value of one implying contact between the drillstring and borehole. Both the run up and run down is shown in Fig. 3(b) and it can be observed that the drillstring is continuously in contact with the borehole during run down. A value less than one indicates a bouncing behaviour since intermittent contact is observed. A waterfall spectrum may be used to study the key response frequencies. A waterfall spectrum is a three dimensional frequency response spectrum corresponding to various rotational speeds shown in Fig. 4(a). The frequency response was averaged by sampling across sections of the time series. The frequency corresponding to the maximum response for each rotor speed was extracted and a two dimensional version of the waterfall plot was generated, as shown in Fig. 4(b). It can be observed from Fig. 4(b) that the maximum response frequency jumps when the rotor speed is near the lateral natural frequency of the system. The subsequent peaks are observed at Rδ ω where ω is the rotor speed which corresponds to the backward whirl frequency with the drillstring rolling along the borehole surface. A hysteresis in the frequency content near the jump frequency was also observed. During run down the backward whirl frequency was persistent beyond the lateral frequency. Next the initial position of the eccentric mass on the second rotor was altered to be 180 degree out of phase to that on the first rotor. This excites the second lateral mode and thus delays the establishment of continuous

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contact between the drillstring and borehole for the same rotor speed sweep as shown in Fig. 5(a). The variation in the normalised clearance shown in Fig. 5(b) and is similar to Fig. 3(b) for the reverse sweep. However during run up there are instants where the system is undergoing forward whirl with no contact within the same rotor speed sweeping range since the first mode is not excited. Figure 6 confirms that the system establishes continuous contact and the jump in frequency content occurs when the rotor speed approaches the second lateral natural frequency of the system. For the study carried out until now the modes were well separated. In order to understand the influence of natural frequencies of the system the lateral coupling stiffness was reduced to 0.1 N/m. This reduces the spacing between the two lateral natural frequencies of the system [15, 16, 17]. The second out of phase lateral mode is now reduced to 0.17 Hz. The temporal variation in the response of the two rotors shown in Fig. 7 and indicates that the continuous contact is established at different instants on the two rotors. The waterfall plot shows similar behaviour with the natural frequency dominating the frequency content before contact and once continuous contact is established the backward whirl frequency Rδ ω dominates the response. The influence of natural frequency was analysed further by reducing the mass and the position of the eccentric mass on the second rotor to 0.1 kg and 0.01 m respectively. This reduces the magnitude of forced excitation on the second disk. It can be observed from Fig. 8 that the response of the two disks are more asymmetric with disk 1 undergoing continuous contact and disk 2 is rotating within the constraint with no contact during both the forward and reverse sweep. Therefore the response is localized to a single disk.

4

Conclusion

The backwhirl behaviour of a two disk rotor was analysed and different steady state regimes were identified. It was observed that the mode excited by the system can be altered by varying the initial phase difference between the eccentric masses of the two disks. This controlled the initiation of backward whirl or continuous contact between the drillstring and the borehole. The frequency content of the rotor response during run up and run down was analysed using the waterfall spectrum by extracting the frequency which produced the peak in the averaged lateral response. It was observed that the natural frequency dominates the response when the drillstring undergoes forward whirl with no contact with the borehole. However with continuous contact a jump in the frequency contents was observed. The jump in frequency content is observed when the rotor speed is near the lateral frequency. The frequency contents is then dominated by the backward whirl frequency. A hysteresis in the jump behaviour of the system was observed with the backward whirl frequency dominating the response even at rotor speeds below the lateral resonance during run down. The influence of natural frequency and excitation within the non-linear system was controlled by altering the lateral coupling stiffness and the position and mass of the eccentric mass. This altered the frequency spacing between the two lateral frequencies of the system as well as the unbalance excitation force acting on the rotor. It was observed that depending on the parameters the backward whirl could be localized to a single disk. Future work will analyse the behaviour on a MDOF system and a bifurcation analysis will be carried out to identify various steady state regimes by varying system parameters within the system.

Acknowledgments The authors acknowledge the support of the Engineering and Physical Sciences Research Council through grant number EP/K003836.

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References [1] S. Jardine, L. Malone, and M. Sheppard. Putting a damper on drilling’s bad vibrations. Oil Field Review, pages 15–20, 1994. [2] A. Muszynska, D. E. Bently, W. D. Franklin, J. W. Grant, and P. Goldman. Applications of sweep frequency rotating force perturbation methodology in rotating machinery for dynamic stiffness identification. Journal of Engineering for Gas Turbines and Power-Transactions of the ASME, 115(2):266–271, 1993. [3] P. Goldman and A. Muszynska. Chaotic behavior of rotor-stator systems with rubs. Journal of Engineering for Gas Turbines and Power-Transactions of the ASME, 116(3):692–701, 1994. [4] A. Muszynska and P. Goldman. Chaotic response of unbalanced rotor-bearing stator system with looseness or rubs. Chaos Solitons & Fractals, 5(9):1683–1704, 1995. [5] Agnieszka Muszyska. Rotordynamics. Taylor and Francis, 2005. [6] Edwards, S., Lees, A.W., and Friswell, M.I. The influence of torsion on rotor-stator contact in rotating machinery. Journal of Sound and Vibration, Vol. 225, No. 4:pp. 767–778, August 1999. [7] F. K. Choy and J. Padovan. Nonlinear transient analysis of rotor-casing rub events. Journal of Sound and Vibration, 113(3):529–545, 1987. [8] Giancarlo Genta. Dynamics of rotating systems. Springer, 2004. [9] Friswell, M.I., Penny, J.E.T., Garvey, S.D., and Lees, A.W. Dynamics of Rotating Machines. Cambridge University Press, 2010. [10] H. H. Jeffcott. The lateral vibrations of loaded shafts in the neighbourhood of a whirling speed - the effect of want of balance. Philosophical Magazine, 37:304–314, 1919. [11] J. D. Jansen. Nonlinear rotor dynamics as applied to oilwell drillstring vibrations. Journal of Sound and Vibration, 147(1):115–135, 1991. [12] F. Chu and Z. Zhang. Bifurcation and chaos in a rub-impact jeffcott rotor system. Journal of Sound and Vibration, 210(1):1–18, 1998. [13] N. Vlajic, X. Liu, H. Karkic, and B. Balachandrana. Torsion oscillations of a rotor with continuous stator contact. International Journal of Mechanical Sciences, Vol. 83:pp.65–75, 2014. [14] Chien Min Liao, Nicholas Vlajic, Hamad Karki, and Balakumar Balachandran. Parametric studies on drill-string motions. International Journal of Mechanical Sciences, 54(1):260 – 268, 2012. [15] K. Vijayan and J. Woodhouse. Shock transmission in a coupled beam system. Journal of Sound and Vibration, 332(16):3681 – 3695, 2013. [16] K. Vijayan and J. Woodhouse. Shock amplification, curve veering and the role of damping. Journal of Sound and Vibration, 333(5):1379 – 1389, 2014. [17] K. Vijayan. Vibration and shock amplification of drilling tools. PhD thesis, University of Cambridge, 2012.

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M1 m1 e1 J1 k1 kt1 c1 ct1 R1 δ1 kc ktc

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1 0.05 0.05 0.0025 1 0.5 0.02 1.83E-04 0.05 0.01 1 0.5

M2 m2 e2 J2 k2 kt2 c2 ct2 R2 δ2

1 0.05 0.05 0.0025 1 0 0.02 1.83E-04 0.05 0.01

Units kg kg m kg m2 N/m N/m Ns/m Ns/m m m N/m N/m

Table 1: System parameters corresponding to the two disk model.

Natural frequency (Hz) x1 x2 y1 y2 θ1 θ2

ω1 0.16 -0.69 -0.69 0.00 0.00 0.00 0.00

ω2 0.16 0.00 0.00 -0.69 -0.69 0.00 0.00

ω3 0.27 0.00 0.00 -0.69 0.69 0.00 0.00

ω4 0.27 -0.69 0.69 0.00 0.00 0.00 0.00

ω5 1.36 0.00 0.00 0.00 0.00 -0.38 -0.92

ω6 3.42 0.00 0.00 0.00 0.00 -0.92 0.38

Table 2: Natural frequencies and mode shapes corresponding to the system parameters, without contact.

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Figure 1: Schematic of the two disk rotor model

Figure 2: Model of the two disk system including the contact model.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

0

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0

500

1000

1500

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3500

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x2(m)

0.01 0 −0.01 −0.02

(a) Run up behaviour of the system time response.

(b) Variation in the normalised clearance during run up (red) and run down (green).

Figure 3: Temporal variation in the lateral response of the two rotors when the eccentric masses are in phase.

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Avg. x1 (dB)

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0 −100 −200 0.4

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ωe (Hz)

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(a) Waterfall spectrum during run up (blue) and run down (green). Subplot 1: Rotor 1 Frequency (Hz)

2 1.5 1 0.5 0 0.05

0.1

0.15

0.2 0.25 Rotor speed (Hz) Subplot 2: Rotor 2

0.3

0.35

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0.2 0.25 Rotor speed (Hz)

0.3

0.35

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Frequency (Hz)

2 1.5 1 0.5 0 0.05

(b) The variation of the frequency of maximum response with rotor speed during run up (red) and run down (green). A jump in frequency is observed near the first lateral mode.

Figure 4: The variation in frequency content of the response with change in rotor speed.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

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x2(m)

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(a) Run up behaviour of the system time response.

(b) Variation in the normalised clearance during run up (red) and run down (green).

Figure 5: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase.

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Avg. x1 (dB)

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(a) Waterfall spectrum during run up (blue) and run down (green). Subplot 1: Rotor 1 Frequency (Hz)

2 1.5 1 0.5 0 0.05

0.1

0.15

0.2 0.25 Rotor speed (Hz) Subplot 2: Rotor 2

0.3

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0.2 0.25 Rotor speed (Hz)

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Frequency (Hz)

2 1.5 1 0.5 0 0.05

(b) The variation of the frequency of maximum response with rotor speed during run up (red) and run down (green). A jump in frequency is observed near the second lateral mode.

Figure 6: The variation in frequency content of the response with change in rotor speed.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

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x2(m)

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(a) Run up behaviour of the system time response.

(b) Variation in normalised clearance of the system in the Run up(red) and Run down (green).

Figure 7: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase and the lateral coupling stiffness is reduced.

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x1(m) 0.02

x1(m)

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x2(m)

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(a) Forward sweep behaviour of the system time response.

(b) Variation in normalised clearance of the system in the forward sweep(red) and reverse sweep (green).

Figure 8: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase and by reducing the eccentric mass and its position for the second rotor and also reducing the lateral coupling stiffness.

Department of Engineering Mathematics, University of Bristol, Queen’s Building, Bristol BS8 1TR, UK

Abstract A major concern within the oil drilling industry remains the high risk associated with the drilling bit and tool failure from the build-up of damaging vibrations. Effective understanding of the drillstring dynamics is essential for efficient drilling operation. Complex dynamic behaviour is often observed in the drillstring due to friction, impact, unbalance eccentricity and energy exchange between different modes of vibrations. The interaction between the drillstring and borehole wall involves nonlinearities in the form of friction and contact. A two disk model is used to study the behaviour of the system. The effects of impact, friction and mass unbalance are included in the model. The drillstring borehole interaction induces whirling behaviour of the drillstring causing forward whirl, backward whirl or intermittent bouncing behaviour depending on the system parameters. A critical steady state behaviour within the system is the backward whirling of the drillstring, which reduces the fatigue life of the drillstring. The theoretical model analyses the influence of different parameters, such as the eccentricity and rotational speed, on the induced backward whirl within the system. The influence of rotor speed on the system dynamics is explored using a run up and run down and is analysed using a two dimensional waterfall plot. The waterfall plot indicated the frequency of maximum response for each rotor speed. Depending on the whirling behaviour the dominant frequency was observed at the natural frequency, the rotational speed or the backward whirl frequency. The behaviour of the system by varying the eccentricity and coupling stiffness was analysed. The results from the study indicated that depending on the eccentric mass and the coupling stiffness the backward whirl could be localized to a single rotor.

1

Introduction

A major concern in the oil drilling industry is the high cost and lead time caused due to the drillstring and bit failure from the the build up of damaging vibration. The system behaviour is non-linear due to impact and friction at borehole/drill string and bit formation interfaces. The excitation is provided from the unbalance eccentricity or initial curvature in drillstring sections. There are also geometric non-linearities in the inertia terms due to coupled between the torsional and lateral motions. The length of the drill assembly increases the possibility of energy exchange between various modes of vibrations. Lateral oscillations which could results in whirling behaviour within the drillstring can be the most destructive generating shocks as high as 250g [1]. If the shaft has one or more rotors attached, more complicated whirl phenomena can occur. Various authors [2, 3, 4, 5, 6, 7, 8, 9] have described the non-linear phenomena that characterize the dynamics of a rotor. Whirling of a shaft or rotor can result in interaction with the enclosure or bearing. This interaction in turn can change the nature of the whirl. Rubbing is a commonly

1275

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encountered problem in rotor dynamics. Problems of rubbing involve investigation of two main effects: determination of local casing rotor interaction and global vibration of the rotor casing/bearing. The frequencies present in the measured vibration signal constitute some of the most useful information for diagnosing rotordynamics problems. But not all shaft whirling is synchronous; the more destructive rotordynamic problems involve non-synchronous whirl. Basically there are two common steady state vibration regimes of rotor motion which are created by rub. These steady states are usually reached through some transient motions involving partially rubbing surfaces. The first steady state regime is due to unbalance and the second is a self excited vibration. The former is less dangerous but the latter can often cause catastrophic failure. Steady state whirl due to unbalance usually occurs during transient conditions of start-up and/or shutdowns when the rotor passes through the resonance speed. This regime is often referred as full annular rub. The second of the quasi stable steady state regimes, which could be more serious in its destructive effect, is the self excited backward full annular rub known as “backward whirl”. One of the simplest models that can be used to study the flexural behaviour of rotors consists of a point mass attached to a massless flexible shaft. This model is often referred to as the Jeffcott rotor [8, 10]. Several authors have tried to formulate the equations of motion in polar [7, 11] and Cartesian coordinates[3, 12]. In the present study the equations of motion of a two DOF system is formulated in Cartesian coordinates.

2

Theoretical model

The model consists of a massless shaft with two rotor disks having eccentric mass as shown in Fig. 1. In the torsional stiffness model one end of the rotor is attached to the drive and the other represents the free end of the drillstring. The mass located eccentrically rotates at rotor speed ω as shown in Fig. 2. The rotor is displaced from the geometric center of the “borehole” by a distance (δ) and a linear spring of stiffness ki is used to model the restoring force which is due to the bending of the shaft. The angular position of the shaft centre is denoted by θi . The position vector of the shaft centre and eccentric mass for RiM and Rim are given by RiM

= xi ˆj + yi kˆ

Rim = [xi + ei cos(ψ + θi + φi )]ˆj + [yi + ei sin(ψ + θi + φi )]kˆ

(1) (2)

where (xi , yi ) is the position of the shaft centre for rotor i, ei is the position of the eccentric mass with respect to the shaft centre, θi is the torsion angle of the rotor i, ψ is the rotational angle of the drive which for a constant rotational speed (ω) will be ωt and φi is the phase angle of the eccentric mass. The Kinetic energy (T ) of the system is given as T

=

2 1X 2 ˙ + θ˙i )2 + mi (R˙ 2 ) Mi (R˙ iM ) + Ji (Ψ im 2 i=1

(3)

Mi is the mass of rotor i and mi is the eccentric mass of rotor i. Ji is the moment of inertia of rotor i. The potential energy (V ) of the system is V

=

k1 2 k2 kc kc kt ktc (x1 + y12 ) + (x22 + y22 ) + (x2 − x1 )2 + (y2 − y1 )2 + (θ1 )2 + (θ2 − θ1 )2 (4) 2 2 2 2 2 2

where k1 , k2 are the first and the second rotor stiffness in the X and Y directions respectively. kc is the coupling stiffness between the two rotor disks in the X and Y directions. The stiffness is assumed to be identical in the X and Y directions for both k1 + k2 and kc . kt and ktc are the torsional stiffness and torsional coupling stiffness respectively of the shaft. Using the Lagrange formulation the equations of motion are

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formulated in the X, Y and θ directions as dT dV d ∂T )− + ( dt ∂ x˙i dxi dxi d ∂T dT dV ( )− + dt ∂ y˙ dy dy dT d ∂T dV )− ( + dt ∂ θ˙ dθ dθ

= Fx(i)

(5)

= Fy(i)

(6)

= Mθ(i)

(7) (8)

˙ + θ˙1 )2 cos(Ψ + θ1 + φ1 ) (M1 + m1 )x¨1 + c1 x˙1 + k1 x1 + kc (x1 − x2 ) = m1 e1 [(Ψ ¨ + θ¨1 ) sin(Ψ + θ1 + φ1 )] +(Ψ ˙ + θ˙1 )2 sin(Ψ + θ1 + φ1 ) (M1 + m1 )y¨1 + c1 y˙1 + k1 y1 + kc (y2 − y1 ) = m1 e1 [(Ψ ¨ + θ¨1 ) cos(Ψ + θ1 + φ1 )] −(Ψ (J1 + m1 e2 )θ¨1 + ct1 θ˙1 + kt1 θ1 + ktc (θ1 − θ2 ) = m1 e1 [(x¨1 sin(Ψ + θ1 + φ1 ) − y¨1 cos(Ψ + θ1 + φ1 )] ˙ + θ˙2 )2 cos(Ψ + θ2 + φ2 ) (M2 + m2 )x¨2 + c2 x˙2 + k2 x2 + kc (x2 − x2 ) = m2 e2 [(Ψ ¨ + θ¨2 ) sin(Ψ + θ2 + φ2 )] +(Ψ ˙ + θ˙2 )2 sin(Ψ + θ2 + φ2 ) (M2 + m2 )y¨2 + c2 y˙2 + k2 y2 + kc (y1 − y2 ) = m2 e2 [(Ψ ¨ + θ¨2 ) cos(Ψ + θ2 + φ2 )] −(Ψ (J2 + m2 e2 )θ¨2 + ct2 θ˙2 + kt2 θ1 + ktc (θ2 − θ1 ) = m2 e2 [x¨2 sin(Ψ + θ2 + φ2 ) −y¨2 cos(Ψ + θ2 + φ2 )]

(9)

where c1 and c2 are the damping coefficients of first and second rotor disks.

2.1

Contact modelling

The next phenomenon that needs to be modelled is the effect of the contact between the borehole wall and the whirling drillstring especially at the stabilizer location. Similar to the impact of a rotor against a stator within the rotor-dynamic context the whirling drillstring can impact the borehole wall. In addition to the plastic deformation the contact between the system also causes dissipation due to the frictional force. There are a plethora of models available for friction. For the present model the frictional force is assumed to be Coulomb [13, 14] with no Stribeck or viscous effect even though there is an uncertainty in the type of model to be used [7]. For numerical stability in the simulation, the variation in the friction coefficient with relative velocity (Vrel ) was smoothed using a continuous function, such that Vrel µ = µd tanh V0

(10)

where µd is the dynamic friction coefficient, Vrel is the relative velocity at the contact interface given by ˙ + θ˙i )Ri − x˙i yi − y˙i xi Vrel(i) = (Ψ ρi ρi ˙ ˙ = (Ψ + θi )Ri − x˙i sin αi − y˙i cos αi

(11)

q

where ρi = x2i + yi2 and αi = arctan( xyii ). V0 is a constant and its value can be varied to obtain different velocity profiles at the contact region. The model, is a reasonable first approximation representation of the forces acting in the real system except that the effect of drilling mud influences the response in the form of fluid-structure interaction, which is neglected. Contact with the borehole generates a normal force in the radial direction (Fn ) and a frictional force in the tangential direction (Ft ). Contact is modelled using a spring of relatively high stiffness. Damping is included

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in the model to incorporate the effect of a coefficient of restitution. The contact force generated in the radial direction is given by Fn(i) = ks (ρi − δi ) + cs ρ˙i = 0

for ρi > δi for

ρi ≤ δi

(12)

where δi is the clearance. The frictional force is modelled using Coulomb friction which generates the tangential force (Ft(i) ) Ft(i) = µFn(i)

(13)

Transforming the forces to the Cartesian coordinate we obtain the forces in the X and Y directions as Fx(i) = −Ft(i) sin(α) − Fn(i) cos(α) Fy(i) =

Ft(i) cos α − Fn(i) sin α

(14)

The frictional moment generated in the torsional direction is: Mext = Ri Ft(i)

3

(15)

Results and Discussion

Using the theoretical model the system was simulated for the parameters given in Table 1. Torsional damping is assumed to be small for the present study similar to rotor dynamic systems [13]. However within an actual drilling environment the torsional damping could be higher due to the drilling mud and formations. The natural frequencies of the system and the mode shapes are given in Table 2. It can be observed that the two lateral modes are well separated in frequency. A case study was performed by altering the relative phase of the eccentric mass location. A frequency sweep was carried out by varying the rotor speed from 0.9 to 2.4 Hz. The range was chosen to cover the lateral natural frequencies of the system. Initially the eccentric mass was located at the same angular position for both disks. This implies that the external excitation forces due to unbalance acts in phase for both of the disks, therefore exciting the first mode of the system. The temporal variations in the responses of both rotors are shown in Fig. 3(a). For clarity only the run up is shown in Fig. 3(a), which clearly shows that the system establishes contact at nearly the same instant on both rotors. Figure 3(b) shows the variation of the normalised clearance between the drillstring and borehole, with a value of one implying contact between the drillstring and borehole. Both the run up and run down is shown in Fig. 3(b) and it can be observed that the drillstring is continuously in contact with the borehole during run down. A value less than one indicates a bouncing behaviour since intermittent contact is observed. A waterfall spectrum may be used to study the key response frequencies. A waterfall spectrum is a three dimensional frequency response spectrum corresponding to various rotational speeds shown in Fig. 4(a). The frequency response was averaged by sampling across sections of the time series. The frequency corresponding to the maximum response for each rotor speed was extracted and a two dimensional version of the waterfall plot was generated, as shown in Fig. 4(b). It can be observed from Fig. 4(b) that the maximum response frequency jumps when the rotor speed is near the lateral natural frequency of the system. The subsequent peaks are observed at Rδ ω where ω is the rotor speed which corresponds to the backward whirl frequency with the drillstring rolling along the borehole surface. A hysteresis in the frequency content near the jump frequency was also observed. During run down the backward whirl frequency was persistent beyond the lateral frequency. Next the initial position of the eccentric mass on the second rotor was altered to be 180 degree out of phase to that on the first rotor. This excites the second lateral mode and thus delays the establishment of continuous

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contact between the drillstring and borehole for the same rotor speed sweep as shown in Fig. 5(a). The variation in the normalised clearance shown in Fig. 5(b) and is similar to Fig. 3(b) for the reverse sweep. However during run up there are instants where the system is undergoing forward whirl with no contact within the same rotor speed sweeping range since the first mode is not excited. Figure 6 confirms that the system establishes continuous contact and the jump in frequency content occurs when the rotor speed approaches the second lateral natural frequency of the system. For the study carried out until now the modes were well separated. In order to understand the influence of natural frequencies of the system the lateral coupling stiffness was reduced to 0.1 N/m. This reduces the spacing between the two lateral natural frequencies of the system [15, 16, 17]. The second out of phase lateral mode is now reduced to 0.17 Hz. The temporal variation in the response of the two rotors shown in Fig. 7 and indicates that the continuous contact is established at different instants on the two rotors. The waterfall plot shows similar behaviour with the natural frequency dominating the frequency content before contact and once continuous contact is established the backward whirl frequency Rδ ω dominates the response. The influence of natural frequency was analysed further by reducing the mass and the position of the eccentric mass on the second rotor to 0.1 kg and 0.01 m respectively. This reduces the magnitude of forced excitation on the second disk. It can be observed from Fig. 8 that the response of the two disks are more asymmetric with disk 1 undergoing continuous contact and disk 2 is rotating within the constraint with no contact during both the forward and reverse sweep. Therefore the response is localized to a single disk.

4

Conclusion

The backwhirl behaviour of a two disk rotor was analysed and different steady state regimes were identified. It was observed that the mode excited by the system can be altered by varying the initial phase difference between the eccentric masses of the two disks. This controlled the initiation of backward whirl or continuous contact between the drillstring and the borehole. The frequency content of the rotor response during run up and run down was analysed using the waterfall spectrum by extracting the frequency which produced the peak in the averaged lateral response. It was observed that the natural frequency dominates the response when the drillstring undergoes forward whirl with no contact with the borehole. However with continuous contact a jump in the frequency contents was observed. The jump in frequency content is observed when the rotor speed is near the lateral frequency. The frequency contents is then dominated by the backward whirl frequency. A hysteresis in the jump behaviour of the system was observed with the backward whirl frequency dominating the response even at rotor speeds below the lateral resonance during run down. The influence of natural frequency and excitation within the non-linear system was controlled by altering the lateral coupling stiffness and the position and mass of the eccentric mass. This altered the frequency spacing between the two lateral frequencies of the system as well as the unbalance excitation force acting on the rotor. It was observed that depending on the parameters the backward whirl could be localized to a single disk. Future work will analyse the behaviour on a MDOF system and a bifurcation analysis will be carried out to identify various steady state regimes by varying system parameters within the system.

Acknowledgments The authors acknowledge the support of the Engineering and Physical Sciences Research Council through grant number EP/K003836.

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References [1] S. Jardine, L. Malone, and M. Sheppard. Putting a damper on drilling’s bad vibrations. Oil Field Review, pages 15–20, 1994. [2] A. Muszynska, D. E. Bently, W. D. Franklin, J. W. Grant, and P. Goldman. Applications of sweep frequency rotating force perturbation methodology in rotating machinery for dynamic stiffness identification. Journal of Engineering for Gas Turbines and Power-Transactions of the ASME, 115(2):266–271, 1993. [3] P. Goldman and A. Muszynska. Chaotic behavior of rotor-stator systems with rubs. Journal of Engineering for Gas Turbines and Power-Transactions of the ASME, 116(3):692–701, 1994. [4] A. Muszynska and P. Goldman. Chaotic response of unbalanced rotor-bearing stator system with looseness or rubs. Chaos Solitons & Fractals, 5(9):1683–1704, 1995. [5] Agnieszka Muszyska. Rotordynamics. Taylor and Francis, 2005. [6] Edwards, S., Lees, A.W., and Friswell, M.I. The influence of torsion on rotor-stator contact in rotating machinery. Journal of Sound and Vibration, Vol. 225, No. 4:pp. 767–778, August 1999. [7] F. K. Choy and J. Padovan. Nonlinear transient analysis of rotor-casing rub events. Journal of Sound and Vibration, 113(3):529–545, 1987. [8] Giancarlo Genta. Dynamics of rotating systems. Springer, 2004. [9] Friswell, M.I., Penny, J.E.T., Garvey, S.D., and Lees, A.W. Dynamics of Rotating Machines. Cambridge University Press, 2010. [10] H. H. Jeffcott. The lateral vibrations of loaded shafts in the neighbourhood of a whirling speed - the effect of want of balance. Philosophical Magazine, 37:304–314, 1919. [11] J. D. Jansen. Nonlinear rotor dynamics as applied to oilwell drillstring vibrations. Journal of Sound and Vibration, 147(1):115–135, 1991. [12] F. Chu and Z. Zhang. Bifurcation and chaos in a rub-impact jeffcott rotor system. Journal of Sound and Vibration, 210(1):1–18, 1998. [13] N. Vlajic, X. Liu, H. Karkic, and B. Balachandrana. Torsion oscillations of a rotor with continuous stator contact. International Journal of Mechanical Sciences, Vol. 83:pp.65–75, 2014. [14] Chien Min Liao, Nicholas Vlajic, Hamad Karki, and Balakumar Balachandran. Parametric studies on drill-string motions. International Journal of Mechanical Sciences, 54(1):260 – 268, 2012. [15] K. Vijayan and J. Woodhouse. Shock transmission in a coupled beam system. Journal of Sound and Vibration, 332(16):3681 – 3695, 2013. [16] K. Vijayan and J. Woodhouse. Shock amplification, curve veering and the role of damping. Journal of Sound and Vibration, 333(5):1379 – 1389, 2014. [17] K. Vijayan. Vibration and shock amplification of drilling tools. PhD thesis, University of Cambridge, 2012.

E NGINEERING NON - LINEARITIES

M1 m1 e1 J1 k1 kt1 c1 ct1 R1 δ1 kc ktc

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1 0.05 0.05 0.0025 1 0.5 0.02 1.83E-04 0.05 0.01 1 0.5

M2 m2 e2 J2 k2 kt2 c2 ct2 R2 δ2

1 0.05 0.05 0.0025 1 0 0.02 1.83E-04 0.05 0.01

Units kg kg m kg m2 N/m N/m Ns/m Ns/m m m N/m N/m

Table 1: System parameters corresponding to the two disk model.

Natural frequency (Hz) x1 x2 y1 y2 θ1 θ2

ω1 0.16 -0.69 -0.69 0.00 0.00 0.00 0.00

ω2 0.16 0.00 0.00 -0.69 -0.69 0.00 0.00

ω3 0.27 0.00 0.00 -0.69 0.69 0.00 0.00

ω4 0.27 -0.69 0.69 0.00 0.00 0.00 0.00

ω5 1.36 0.00 0.00 0.00 0.00 -0.38 -0.92

ω6 3.42 0.00 0.00 0.00 0.00 -0.92 0.38

Table 2: Natural frequencies and mode shapes corresponding to the system parameters, without contact.

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Figure 1: Schematic of the two disk rotor model

Figure 2: Model of the two disk system including the contact model.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

0

500

1000

1500

2000 time(s) x2(m)

2500

3000

3500

4000

0

500

1000

1500

2000 time(s)

2500

3000

3500

4000

0.02

x2(m)

0.01 0 −0.01 −0.02

(a) Run up behaviour of the system time response.

(b) Variation in the normalised clearance during run up (red) and run down (green).

Figure 3: Temporal variation in the lateral response of the two rotors when the eccentric masses are in phase.

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Avg. x1 (dB)

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0 −100 −200 0.4

0.3

0.2

0.1

Avg. x2 (dB)

ωe (Hz)

0

0

2

4

6

8

10

8

10

Frequency (Hz)

0 −100 −200 0.4

0.3

0.2

0.1

ωe (Hz)

0

0

2

4

6

Frequency (Hz)

(a) Waterfall spectrum during run up (blue) and run down (green). Subplot 1: Rotor 1 Frequency (Hz)

2 1.5 1 0.5 0 0.05

0.1

0.15

0.2 0.25 Rotor speed (Hz) Subplot 2: Rotor 2

0.3

0.35

0.4

0.1

0.15

0.2 0.25 Rotor speed (Hz)

0.3

0.35

0.4

Frequency (Hz)

2 1.5 1 0.5 0 0.05

(b) The variation of the frequency of maximum response with rotor speed during run up (red) and run down (green). A jump in frequency is observed near the first lateral mode.

Figure 4: The variation in frequency content of the response with change in rotor speed.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

0

500

1000

1500

2000 time(s) x2(m)

2500

3000

3500

4000

0

500

1000

1500

2000 time(s)

2500

3000

3500

4000

0.02

x2(m)

0.01 0 −0.01 −0.02

(a) Run up behaviour of the system time response.

(b) Variation in the normalised clearance during run up (red) and run down (green).

Figure 5: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase.

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Avg. x1 (dB)

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0 −100 −200 0.4

0.3

0.2

0.1

Avg. x2 (dB)

ωe (Hz)

0

0

2

4

6

8

10

8

10

Frequency (Hz)

0 −100 −200 0.4

0.3

0.2

0.1

ωe (Hz)

0

0

2

4

6

Frequency (Hz)

(a) Waterfall spectrum during run up (blue) and run down (green). Subplot 1: Rotor 1 Frequency (Hz)

2 1.5 1 0.5 0 0.05

0.1

0.15

0.2 0.25 Rotor speed (Hz) Subplot 2: Rotor 2

0.3

0.35

0.4

0.1

0.15

0.2 0.25 Rotor speed (Hz)

0.3

0.35

0.4

Frequency (Hz)

2 1.5 1 0.5 0 0.05

(b) The variation of the frequency of maximum response with rotor speed during run up (red) and run down (green). A jump in frequency is observed near the second lateral mode.

Figure 6: The variation in frequency content of the response with change in rotor speed.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

0

500

1000

1500

2000 time(s) x2(m)

2500

3000

3500

4000

0

500

1000

1500

2000 time(s)

2500

3000

3500

4000

0.02

x2(m)

0.01 0 −0.01 −0.02

(a) Run up behaviour of the system time response.

(b) Variation in normalised clearance of the system in the Run up(red) and Run down (green).

Figure 7: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase and the lateral coupling stiffness is reduced.

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x1(m) 0.02

x1(m)

0.01 0 −0.01 −0.02

0

500

1000

1500

2000 time(s) x2(m)

2500

3000

3500

4000

0

500

1000

1500

2000 time(s)

2500

3000

3500

4000

0.01

x2(m)

0.005 0 −0.005 −0.01

(a) Forward sweep behaviour of the system time response.

(b) Variation in normalised clearance of the system in the forward sweep(red) and reverse sweep (green).

Figure 8: Temporal variation in the lateral response of the two rotors when the eccentric masses are out of phase and by reducing the eccentric mass and its position for the second rotor and also reducing the lateral coupling stiffness.