PAMM · Proc. Appl. Math. Mech. 10, 509 – 510 (2010) / DOI 10.1002/pamm.201010247

A Wave-Based Approach to Adaptively Control Self-Excited Vibrations

in Drill-Strings

Edwin Kreuzer1 and Michael Steidl1,∗

1 Institut für Mechanik und Meerestechnik, Technische Universität Hamburg-Harburg, 21071 Hamburg

Due to their small diameter-to-length-ratio, drill-strings are vulnerable to torsional vibrations. Moreover, the string is exposed

to unknown or uncertain time-variant and nonlinear loads (e.g. friction with falling friction characteristics, contact with

the borehole, differential sticking), which can result in severe torsional vibrations and stick-slip. The control law for the

boundary controller at the top drive of the string needs to adapt to those unknown loads in order to stabilize the vibrations.

The torsional vibrations of a drill-string are governed by the wave equation. Analytical solutions and control laws are often

based on a separation of the dynamics into a time- and a space-dependent part (modal representation). Here, we decompose the

vibrations into two traveling waves according to the D’Alembert solution, using only few measurements along the string. The

wave which travels up the string is then compensated by the actuator at the top drive. With this compensation, the upward-

traveling wave is no longer reflected back into the string and vibration energy is absorbed, thus stabilizing the torsional

vibrations.

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Control of Systems with Stick-Slip

In order to study the effect of stick-slip, simple spring-mass models on a conveyor belt have been studied for example by [1].

The existence of stick-slip limit cycles is due to the difference between the higher sticking friction force and the lower sliding

friction force in the nonlinear friction characteristic F (x) between the mass and the conveyor belt (Fig. 1). Independent of

the stick-slip limit cycle, a stable equilibrium point exists within this limit-cycle, if the friction characteristic linearized at the

belt velocity vB ,dF (x)

dx

∣

∣

∣

vB

, is positive. IfdF (x)

dx

∣

∣

∣

vB

is negative, this stable equilibrium point does not exist. In case of the

co-existence of a stable equilibrium point and the stick-slip limit cycle, a basin of attraction around the stable equilibrium point

exists in the two dimensional phase space. All initial values within this basin will converge asymptotically to the equilibrium

point, while all other initial values are attracted by the limit cycle. To stabilize the system when stick-slip vibrations occur,

the controller needs to dissipate enough energy from the system within one cycle to reach the basin of attraction. Otherwise,

the system will be in the sticking period again and stick-slip will prevail. Things become more complicated when a system

is considered which consists of a chain of masses and springs (Fig. 2). Due to the excitation of the last body, the whole

chain vibrates, whereas the frequencies and modes describing this vibration will depend on the excitation. In such a system,

the co-existence of a stable solution and a stick-slip solution may also be possible. The stable solution will exist if either

the slope of the linearized friction characteristic is positive or if some dissipation, e.g. due to a control law at the first body,

is introduced into the system. To stabilize such a system (transition from the stick-slip solution to the stable solution), it is

not enough to steer the last body of the chain into the basin of attraction. All other bodies have to be steered close to their

equilibrium position in phase space, as any disturbance within the chain will propagate through the chain and may eventually

push the last body out of the basin of attraction, which will lead to the new onset of stick-slip vibrations. We show how a

suitable feed forward control, applied to the first body of the chain, can be found, which dissipates a large amount of energy

from the vibrating chain within one cycle. With this feed forward control, the transition of the chain from stick-slip motion to

the basin of attraction for the stable motion can be achieved.

−5 0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5x 10

4

angular velocity [rad/s]

torq

ue [N

m]

Fig. 1 Nonlinear friction characteristic

mmm m

Fig. 2 Generic system with stick-slip

∗ Corresponding author: e-mail: steidl@tuhh.de, Phone: +0049 40 42878 2308 Fax: +0049 40 42878 2028

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

510 Section 11: Waves and Acoustics

Fig. 3 Vibrations along the string

40 50 60 70 80 90 100 1100

5

10

15

20

25

time [s]

angula

r velo

city [ra

d/s

]

Fig. 4 Velocity of top-drive (red) and bit (blue)

2 Feed-Forward Control Based on Modal Description

By measuring the dynamics of all or some of the bodies of the chain, the dynamics u(x, t) of the whole chain can be described

as a superposition of modes u(x, t) =∑n

i=1 ψi(x)ai(t). The evolution in time of each space-dependent mode ψi(x) is given

by ai(t), which is evaluated by projecting the measurements upon the spatial modes. The time-dependent modes may be

empirical modes retrieved from system data or the first n relevant eigenmodes which capture a large amount of the system

dynamics. The use of empirical modes may be advantageous, as the total number of relevant modes to describe the system

dynamics is usually lower. The idea of the feed forward control is to separate the observed vibrations into two waves traveling

to the left and the right of the chain. The wave traveling to the left is then absorbed by the controller, thus preventing reflection

of the wave into the system and dissipating the energy of the wave. Therefore, the observed vibrations have to be converted

into the form of the D’Alembert solution of the wave equation,δ2φ(x,t)

δt2= c2

δ2φ(x,t)δx2 . This is achieved by developing each

modal representation into series of sines, ψiai ≈∑m

j=1 aij sin(bijx)∑l

k=1 αijk sin(ωijkt), and then using the following sine

theorem: sinα sinβ = 12 [cos(α−β)− cos(α+β)]. Thus, the modal decomposition can be re-written as two traveling waves:

sin(bijx) sin(ωijkt) =1

2[cos(bijx− ωijkt) − cos(bijx+ ωijkt)] =

1

2[sin(bijx− ωijkt+

π

2) − sin(bijx+ ωijkt+

π

2)]. (1)

The feed-forward control is now determined by summing up over all terms which represent waves traveling to the left and

evaluating at x = 0.

3 Application to Drill-String Model

The feed-forward control is applied to a simple lumped-mass model of a drill-string of the form Mx + Dx + Kx = g(x),values are partly taken from [2]; g(x) contains a simple PI-control law which feeds back the velocity at the top of the string

and the nonlinear friction characteristic between the bit and the borehole (Fig.1), which may lead to strong torsional stick-slip

vibrations. As the linearized friction characteristic is negative at the operating velocity, the existence of the stable equilibrium

solution is due to the dissipation of energy in the controller at the top of the string. When the string is started from its ’natural’

initial conditions, zero twist and zero velocity, it shows strong stick-slip vibrations. However, with the feed-forward control

it is possible to reach the basin of attraction for the stable equilibrium solution. Figures 3 and 4 show the transition from

stick-slip vibrations to the stable solution. The time the feed-forward control is applied is marked by the two black bars in

Fig. 4. Figure 3 shows that the string is untwisted close to its stable equilibrium. Note that the velocity controller at the top

of the string is identical before and after application of the feed forward control. The feed forward control is only valid for a

short period of time, as it is calculated from the system dynamics before it is applied to the system. In case of the drill string

model, the wave speed is approx. 3100 m/s. For the modeled string with a length of 2000 m, the wave takes about 1.3 seconds

to travel from the top drive to the bottom of the string and back up, therefore the feed forward control is valid for 1.3 seconds.

References

[1] M. Oestreich, N. Hinrichs, K. Popp, Bifurcation and stability analysis for a non-smooth friction oscillator, Archive of Applied Me-

chanics 66. Berlin/Heidelberg: Springer-Verlag 1996[2] O. Kust, Selbsterregte Drehschwingungen in schlanken Torsionssträngen, Fortschr.-Ber. Reihe 11, Nr. 270. Düsseldorf: VDI-Verlag,

1998.

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