An Experimental and Theoretical Study of a Coupling Mechanism Between Longitudinal and Torsional Drillstring Vibrations at the Bit Ihor Viggo Aarrestad, Rogaland Research Inst.

ge Kyllingstad, SPE, Rogaland Research Inst.

Summary. A mechanism that couples longitudinal and torsional drillstring vibrations at the bit was studied. Torsional vibrations are associated with dynamic variations of the rotational bit speed. When a roller bit runs over a multilobed pattern, these speed variations have been shown to affect the input of longitudinal vibrations. The theory for this coupling mechanism is verified experimentally by high-rate data of near-bit accelerations and torque recorded in a IOOO-m [3,280-ft] -deep well.

Introduction Several publications 1-6 have dealt with vibrations in drillstrings. As discussed in the literature, vibrations influence the drilling sit­uation and can result in a lower rate of penetration or may cause fatigue failures. On the other hand, a thorough understanding of the vibrations can enable predictions of downhole conditions on the basis of top measurements.

The theory presented here is an attempt to explain some phenom­ena seen in experiments performed at Ullrigg, a full-scale offshore­type research drilling rig. These experiments comprise both real drilling and "drilling" with an exciter system, which induces a longitudinal, sinusoidal vibration with a constant amplitude at the bit. The experimental setup with data transmission system is thoroughly described in Ref. 1.

Frequency spectra on drilling data gathered at Ullrigg often con­tain small side lobes near the most dominant frequency components. The sources for these side lobes are also discussed here.

Theory It is assumed here that the axial motion ofthe bit can be described by an elevation function, s(so), where So is the angular displace­ment of the bit. Assuming also that the shape and orientation of the downhole patterns do not change or at least change very slowly with time implies that

s(so) =s(so + n27r) , ................................. (I)

where n is an integer. The axial force through the bit, or weight on bit (WOB), can

formally be written as

F=F+P, ........................................ (2)

where F denotes the mean force and P denotes the dynamic or fluc­tuating force.

The torque on bit (TOB) may also be separated into a mean torque, r, and a dynamic part that is represented by

ds i=F- . ....................................... (3)

dso Because of this dynamic torqu..e, the rotation rate of the bit will fluctuate about a mean value 0:

dso __ 0=-=0+0, .................................. (4)

dt

Copyright 1988 Society of Petroleum Engineers

12

where 0 =O(i) may be regarded as a dynamic torsional response to the input torque i .

Now, turning to the special case where the downhole patterns are pure sinusoidal three-lobed patterns with amplitude s I ' the axial bit motion can be written as

S=SI sin(3so) ..................................... (5)

Note that Eq. 5 is correct only with a three-cone roller bit and if the roller radius is much smaller than the pattern curvature radius. This requirement is reasonably well satisfied for the exciter 7 sys­tem, where sl =2 mm [0.08 in.].

In the first linear approximation, it is assumed that the dynamic variations of force and bit speed are negligibly small-i.e.,

iFl ~ IFI ......................................... (6)

and

101 ~O. . ........................................ (7)

The input for axial and torsional vibrations is then decoupled and may be written as

s =s I sin(30t) .................................... (8)

and

Te= -3Fs l cos(30t) ............................... (9)

The negative sign appears because the torque input-i.e., the torque transfer from the formation to the bit-is minus the TOB. Hence, the dynamic torque input is positive when the bit is running down­ward on a negative slope: ds/dso<O.

Eq. 8 has been used in several earlier publications on drillstring vibrations. 1-3,5,6 Experiments have revealed, however, that the as­sumptions for this first linear approximation are poorly satisfied­i.e., the dynamic force, P, and the dynamic speed, 0, are not negligibly small. Hence the displacement,

S=SI sin(3so), ................................... (10)

and the torsional excitation moment,

Te=-3(F+P)SI cos(3so), ......................... (11)

SPE Drilling Engineering. March 1988

w

"" ::> 1-.

A 0.5 _J 0... ~

"'" A 0.4 >-:z u.J ::;; w A 0.3 u

"'" -' 0... A O. 2 U)

"" -' A O. 1

"'" ><

"'" A - o. 0

~ ~--------~------~--------~--------~ 10 20 30 40

NON-DI MENSI ONAL TI ME

Fig. 1-Theoretical axial displacement of bit vs. time for six different normalized bit-speed amplitudes, A. II = o.

contain higher harmonics of the fundamental angular frequency,

WI =30. . ....................................... (12)

In the following, the dynamic force amplitude is assumed to be small compared with the mean part; i.e., iFl ~ IFI is assumed to be valid. The angular bit speed, however, is assumed to have a finite dynamic component:

n=-AOcos(wlt+(3) . ............................ (13)

This term can be regarded as a response to the torque input given by Eq. 9. The normalized speed amplitu<!.e, A, denotes the relative speed variations of the bit. If A=0.3 and 0=411"/3 rad/s [2400/sec], corresponding to 40 rev/min, then the true bit speed will oscillate between 28 and 52 rev/min. Phase (3 is the phase between the response and the driving torque, 7 e . As an example, (3 = 7r/2 cor­responds to the situation when the bit speed is 1.57 rad [90°] behind the input torque-i.e., maximum speed occurs in the lobe minima. In general, A and (3 will be functions of the frequency WI and of the drill string geometry.

Integrating the instantaneous angular bit speed,

o=o+n =O[1-A COS(Wlt+(3)], .................... (14)

the angular displacement function is given by

So =Ot- 'hA sin(wl t+(3), .......................... (15)

where the integration constant is already chosen to be equal to zero. The axial bit displacement now becomes

S=SI sin[wlt-A sin(wlt+(3)] . ...................... (16)

This function is periodic with a period T=27r/WI, and represents a distorted sine function. The axial acceleration may be written as

where So is given by Eq. is and 1/1 is an abbreviation for wlt+(3. The axial acceleration will be a relatively more distorted sine func­tion than the displacement function. This is shown in the following.

SPE Drilling Engineering. March 1988

w

"" ::> >-

A 0.5 -' 0... ~

"'" A 0.4 >-:z w ~ A 0.3 w u

"'" _..J

0... A 0.2 U)

"" -' A O. 1 "'" ><

"'" A 0.0

~ L-________ L-______ ~L-______ ~ ________ ~

10 20 30 40

NON-DIMENSIONAL TIME

Fig. 2-Theoretical axial displacement of bit vs. time for six different normalized bit-speed amplitudes, A. II = n12.

Because s(t) is periodic, it can formally be written as

00

S(t)=SI L: Kn sin(nwlt+O'n)' .................... (18) n=1

It follows that the acceleration is

Here Kn represents normalized Fourier coefficients for the axial displacement s(t), while n 2 Kn represents the normalized Fourier coefficients for the axial acceleration. Both Kn and O'n are functions of A and (3.

The dynamic part of the TOB becomes, still neglecting the dy­namic force,

T = -7e=3Fs I cos[wlt-A sin(wlt+(3)], • ............. (20)

which can also be written formally as

00

T "",3Fs I L: K~ cos(nwlt+O'~) . ................... (21) n=1

It can be shown that K~(A,(3)=Kn(A,(3-7r/2), showing that higher harmonics of the torque are less pronounced than those of the ac­celeration. Note, however, that higher harmonics can also rise from the product 3Fs I cos[wlt-A sin(wlt+(3)], the dynamic part of the force.

The normalized functions S(t)/SI and -(d2s/dt2 )/(WI 2SI) are plotted in Figs. I through 5 for various amplitudes A and phases (3. By comparing the shape of experimental time traces with these curves, we can estimate both A and (3.

In Figs. I through 6, theoretical curves for bit excitations and accelerations for various response amplitudes and phases are given. Figs. 1 and 2 give axial-bit-elevation curves as a function of time for different values of the normalized bit speed amplitude, A. The curves have been plotted as a function of the nondimensional time, 7=wlt. Phase (3 is 0 and 11"/2, respectively. A realistic figure for A is 0.3, as will be seen later. For (3=0, the input and response are in phase, as seen for high values of A, because climbing the

13

on UJ N

CI => .... -'

Q A O. 5 N

CL. :::;; «:

A O. 4 z on 0

I- A O. 3 «: 0::: UJ -' UJ A O. 2 (,,)

(,,)

«:

-' A O. 1

«:

>< A O. 0 «:

~ ~ ________ L-________ ~ ______ ~~ ______ ~

10 20 30 40

NON-DIMENSIONAL TIME

Fig. 3-Theoretical normalized axial acceleration of bit vs. time for six different normalized bit-speed amplitudes, A. p = O.

lobe leads to a smaller absolute value of the slope than running down the lobe.

Turning to Figs. 3 and 4, the axial acceleration is given for the same A and /3. These curves are, as expected, even more distorted by the torsional response amplitude and phase.

In Fig. 5, the normalized speed amplitude. A, has been fixed to 0.3, and the response phase, /3, is varied from -7r/2 to 7r/2 in steps of 7r/4. It is seen that by adding 7r to the response phase, /3, the acceleration equals minus the original acceleration signal plus a phase shift of 7r.

The Fourier coefficients, K 1, 4K2 , 9K3, and 16K4 (see Eq. 22), are given as functions of the response amplitude, A, in Fig. 6. These coefficients are only slightly affected by Phase /3, whereas Phases o!,' 0!2, 0!3' .. are strongly affected by /3. The result presented in

Q

u.J N

CI => .... -'

, 1';/2 CL. :::;; «:

z , 1';/4 0

.... «: 0::: , 0 UJ

-' UJ (,,)

(,,) , - 1';/4 <C

-'

"'" >< , - 1';/2 "'"

~ L-________ L-______ ~L-______ ~ ________ _J

10 20 30 40

NON--D! MENSI ONAl TI ME

Fig. 5-Theoretical normalized axial acceleration of bit vs. time for five different phase angles, p. The normalized bit­speed amplitude, A = 0.3.

14

on UJ N

CI => I-

Q A = 0.5 -' N CL. :::;; «:

A 0.4 Z 0

I- A 0.3 «: 0::: UJ

-' UJ A 0.2 (,,)

(,,)

«:

-' A O. 1

«:

>< «: A 0.0

~ L-________ L-______ ~L-______ ~ ________ _J

10 20 30 40

NON-D! MENSI ONAl TI ME

Fig. 4-Theoretical normalized axial acceleration of bit vs. time for six different normalized bit-speed amplitudes, A. p = n/2.

Fig. 6 indicates that a second harmonic component in the accelera­tion spectrum will appear even when the relative bit speed ampli­tude, A, is as small as 0.1.

Low-Frequency Perturbation of the Speed So far, it has been assumed that the dynamic perturbation of the angular bit speed is harmonic with a cycle frequency equal to the main vibration frequency, w,, In a real drilling situation, and also when the downhole exciter is used, the frictional interaction be­tween the drillstring and the wellbore wall will give rise to torque transfer to the drillstring of a more stochastic nature. It is reasonable to assume that this input is composed of many short torque spikes corresponding to a wide-band input spectrum. The response to this spectrum is very frequency-dependent and closely related to the

Q

c.n 11. ----.... HARMONIC :z UJ ~ , (,,) GO ~ .,; , lL.

~ lL. ___ P = ± IT /2 UJ 0 (,,)

0::: .,; _--,-P =0. IT

UJ

0::: => 0 lL.

.,; CI UJ N

'V -' «: .,; :::e 0::: 0 :z

'" '" 0.0 O. 1 0.2 0.3 0.4 0.5

RESPONSE AMPll TUDE

Fig. 6-Theoretical normalized acceleration Fourier coeffi­cients, n2Kn, as function of the normalized bit-speed ampli­tude, A. P=O and n/2.

SPE Drilling Engineering, March 1988

* '"

......... E

z: 0

z: 0 I >-

""" '" N

uJ , 4

UJ

'-' TI ME I N SECONDS '-'

"'" Fig. 7-Theoretical and measured time plots of near-bit acceleration with 130-kN nominal woe and a nominal rotary speed of 40 rev/min. The exciter system was used in the experiment.

torsional drillstring resonances. 8 Experience has shown that the lowest torsional resonance frequency is often dominating the dy­namic bit motion. This mode can be regarded as a pendulum mode because the collar section and the pipe section act as a torsional mass/spring system. The angular frequency for this mode is denoted by wo0 In analogy to the previous example, the instantaneous bit speed may be written as

O=O+AOO cos(wot), ............................. (22)

where Ao is the relative speed modulation. The angular bit dis­placement now becomes

_ 0 so=Ot+Ao-sin(wot), ........................... (23)

Wo

where Phase (3 of the perturbation is considered equal to zero, for convenience.

The axial displacement is now, still assuming that the bit is run­ning on a sinusoidal three-lobed pattern,

~

~ 0

'" * '"

.., ~

0 ......... E

N

:z ,,; i--:::I -:::>

'" ,,; I--

>-'-'

0 w I "- ,,; Vl 0

~

~ 0

'" ! * '"

..,

I

10

In the frequency modulation theory of signals, the function Aow l/wO is called the modulation index. 9 If the modulation index Aowl/wo~ 1, then the spectrum has two side bands with frequen­cies WI ±wo, in addition to the main component at the "carrier" frequency WI' At a larger mQdulation index-i.e., when the dy­namic velocity amplitude AoO increases-more spikes in the frc­quency spectrum will appear at WI ±2wo, WI ±3wo, etc.

The previous example, in which an angular speed perturbation with the main frequency w I was assumed, is in fact a special case where the modulation frequency equals the carrier frequency. Ac­cording to the previous theory, it is not surprising that higher har­monics at 2wI and 3wI are seen.

Experiments The experiments were performed in the IOOO-m [3,280-ft) -deep, nearly vertical well at the full-scale research drilling rig, Ullrigg. A hard-wire measurement-while-drilling tool used in the experi­ments gives high-rate data on many downhole parameters, including WOB, TOB, and axial acceleration. This setup gives a very good opportunity to study effects of the bit/formation interactions.

Details of the experiments can be found in other publications. I.?

In some of the experiments, a specially designed exciter system for constant-axial-displacement amplitude excitation was used. Real drilling with and without a shock absorber has also been performed

I

-

-

I

15 20

FREQUENCY. HZ

-

-

15 20

FREQ~ENCY. HZ

Fig. 8-Theoretical and measured frequency spectra of near-bit acceleration with 130-kN nomi­nal woe and a nominal rotary speed of 40 rev/min. The exciter system was used in the experiment.

SPE Drilling Engineering, March 1988 15

E :z

z -

w => 0

a 0< o I- .. THEORETI CAL

MEASURED ~~ ______ -L ________ ~ ______ ~ ________ ~ ______ ~

:;~

~~Jt~ I

10

4 5

TI ME I N SECONDS

I

15 20

FREQUENCY, HZ

Fig. 9-Theoretical and measured time plots and measured frequency spectrum of near·bit tQrque with 130-kN nominal WOB and a nominal rotary speed of 40 rev/min. The exciter system was used in the experiment.

during the experimental series. The real drilling tests were per­formed in a hard formation, augen gneiss, often producing a sinusoi­dal three-lobed pattern by the bit/formation interactions-Le., a dominating frequency of three times the rotational frequency was often seen.

Theory vs. Experimental Results Some of the experimental results obtained on the research rig are presented in Figs. 7 through 11. In Fig. 7, a time plot of the axial acceleration with corresponding theoretical results is presented. In Fig. 8, frequency spectra of the theoretical and measured signals are given. The nominal rotary speed is 40 rev/min, and the WOB is about 130 kN [29,225 lbf]. The data were gathered when the exciter system 7 was used with a three-lobed steel cam of elevation amplitude sl =2 mm [0.08 in.], an idealized experimental drilling situation. Two dominating frequencies in the spectrum are 2 and 4 Hz [2 and 4 cycles/sec]. These correspond to the first and second

5

z 0 i=w ~o.., w~ ..... .J_, ~~E ~~ ....

-5 0

t~ Z:>.., wo:: ..... :> ... ,

8~E 0::0.. .... lL.1Il

0

harmonic component of the expected vibration frequencies at a ro­tary speed of 40 rev/min. 1,2 The theoretical curves in Figs. 7 and 8 have been produced with an amplitude A",,0.3 and a response phase fJ = - 7r/2. When the experimental and theoretical signals in Fig. 7 are compared, the similarity of the curves is striking. Also, for the frequency spectra, the theoretical and measured results have the same form and magnitude. Note that the only signal processing has been to subtract the sensor offset caused mainly by gravitational acceleration from the measurements-Le., only the dynamic part is retained. The similarity in both time-trace and spectra indicates that this theory may explain the measured elevation acceleration within reasonable accuracy. Note that the second harmonic is not a resonance frequency of the drillstring, but a coupling effect be­tween torsional and longitudinal bit vibrations. Note also that an even better correspondence between the measured and theoretical results could have been obtained by a more accurate selection of the response amplitude, A.

TIME [5 ] 5

0 FREQUENCY [Hz] 20

16

Fig. 10-Time plot and frequency spectrum of measured near-bit acceleration with 130-kN nominal WOB and a nominal rotary speed of 80 rev/min. Real drilling was performed.

SPE Drilling Engineering. March 1988

2

z 0 i=w ~o,.., w~"1n ...J_, ~~E ~~ ....

-2 0 TIME [s] 5

0.2

>-U::::E Z:::>,.., wQ:::"1n :::> 1-, 8hlE Q:::Ll. .... LL.fJl

0.0 0 FREQUENCY [Hz] 20

Fig. 11-Time plot and frequency spectrum of measured near-bit acceleration with 4S-kN nominal WOB and a nominal rotary speed of 40 rev/min. Real drilling was performed.

0.4

-0.4 o TIME [s] 5

Fig. 12-Filtered time plot of measured near-bit acceleration with 4S-kN nominal WOB and nominal rotary speed of 40 rev/min. Real drilling was performed.

The TOB, as displayed in Fig. 9 and measured simultaneously with the parameters presented in Fig. 7, seems to have one dominat­ing frequency of 2 Hz [2 cycles/sec], corresponding to the three­lobed pattern. However, there are two sideband frequencies to this main frequency, as seen in the frequency spectrum. These side­band frequencies somewhat disturb the time-plot curve, as seen in the upper part of the figure. The frequency shift from the main fre­quency to the sideband frequencies corresponds to the lowest tor­sional normal mode of vibration. 8 In addition, a weak second harmonic frequency peak can be seen at 4 Hz [4 cycles/sec]. The theoretical torque, as given by the presented theory, has an addi­tional mean torque added to have direct comparison between the signals. Again, the form of the theoretical curve is the same as that for the measured signal. The dynamic amplitude of the signals is also comparable. The additional parts of the measured signals not explained by the present theory may be explained by the sideband frequencies and possibly the dynamic force effect not included.

Figs. 7 through 9 have been produced from data gathered with the exciter system in the drillstring. Real drilling has also been per­formed, and a time plot with a frequency spectrum for axial ac­celeration is displayed in Fig. 10. The rotary speed was held constant at 80 rev/min, and the WOB was about 130 kN [29,225 Ibfl. Again, the main frequency is seen to match the excitation frequency ex­actly for a three-lobed pattern, 4 Hz [4 cycles/sec] at 80 rev/min. The time plot is more complex here than for the idealized,drilling result. It is possible, however, to see the second harmonic at 8 Hz [8 cycles/sec], in addition to the sideband frequencies near these two main frequencies. Returning to the theoretical predictions, a torsional response amplitude of A "" 0.3, but with a response phase of (3 "" 7r/2, seems to give similarity between the experimental and theoretical signals. The phase shift of 7r from 40-rev/min idealized drilling to 80-rev/min real drilling is not really understood. Further research on the matter will be done.

SPE Drilling Engineering, March 1988

The last experimental results to be discussed here are presented in Figs. 11 and 12. The rotary speed was 40 rev/min, and a WOB of 45 kN [10, 1161bfl was used. The time plot and frequency spec­trum are not easy to interpret, mostly because of the lump of fre­quencies seen at around 20 Hz [20 cycles/sec] in the frequency spectrum. The source of these frequencies is unknown at the present time, but by taking a Fourier transform of the signal, multiplying the transformed signal with a digital filter that allows only frequen­cies between 1 and 7.5 Hz [I and 7.5 cycles7sec] to pass, and then taking the inverse Fourier transform of the transformed multiplied signal, a result as shown in Fig. 12 is obtained. The result is still not completely clear, but it is now also possible to interpret some of the theoretical results given here for these experimental data. The shape of the curve now aligns somewhat with the shape of the curve in Fig. 7, idealized drilling at 40 rev/min and 130-kN [29,225-lbfl WOB, and the theoretical time-trace curve in the same figure. It can therefore be concluded that the main frequency of the signal in Fig. 11 is induced by a three-lobed pattern in the for­mation. The phase of the signal in Fig. 11 (or 12) is the same as for the idealized experimental results presented in Fig. 7-i.e., the phases, (3, between the driving torque and the response are equal for the two situations. Further development of the theory is need­ed, however, to understand the sources producing the higher com­ponents in the frequency spectrum.

Conclusions I. Frequency spectra of such downhole parameters as axial ac­

celeration, WOB, or TOB often contain a dominating vibration fre­quency equal to three times the rotational frequency. This frequency, which is also seen in the top acceleration, clearly indicates that the bit has formed a three-lobed pattern in the formation.

2. The spectra also very often contain higher harmonics of this fundamental frequency. An explanation for this is presented in this

17

paper-a nonlinear coupling between torsional and axial vibrations. 3. Further, the frequency spectra often contain small side lobes

near the most dominant frequency components. The frequency spac­ing between these side lobes and the center frequency corresponds very well to the calculated and measured frequency for torsional pendulum oscillations of the drill-collar section.

4. The theoretical predictions given here seem to reproduce the experimental data gathered on the research rig, within reasonable accuracy. The theory will be further developed, however, to in­clude more effects to obtain improved analysis of experimental data.

Nomenclature A = normalized bit-speed amplitude, dimensionless

Ao = low-frequency bit-speed amplitude, dimensionless F = axial force through bit, N [Ibf] F = mean part of axial force through bit, N [Ibf] P = dynamic part of axial force through bit, N [Ibf]

Kn = normalized Fourier coefficient number, n, axial displacement

n = integer S = axial displacement at the bit, m [ft]

sl = amplitude of axial displacement at bit, m [ft] se = angular displacement of bit

t = time, seconds IX = phase of n harmonic in formal Fourier series for

axial ~isplacement fJ = phase of dynamic bit speed T = mean TOB, N'm [lbf-ft] T = dynamic TOB, N· m [Ibf-ft] T e = dynamic torque input at the bit, N' m [Ibf-ft] if; = phase used in Eq. 18

Wo = angular frequency for torsional pendulum mode, rad/s [degrees/sec]

WI = fundamental angul;;lr frequency=30, rad/s [degrees/ sec] .

{} = rotational bit speed, rad/s [degrees/sec] o = mean rotational bit speed, rad/s [degrees/sec] o = dynamic rotational bit speed, rad/s [degrees/sec]

Subscripts 1,2,3,4 = numbering of Fourier coefficients and phases

Superscript , = torque Fourier coefficients and phases

18

Acknowledgments We thank A/S Norske Shell and Den Norske Stats Oljeselskap A/S for funding and building the full-scale research drilling rig and for initiating and funding the projects concerning drillstring dynamic behavior. Furthermore, the funding of the preparation of this paper is appreciated. We also thank Inst. Franc;ais du Petrole for lending its Televigile system to the research projects and for assisting in the production of the accelerometer sub.

References I. Aarrestad, T.V., Tt'nnesen, H.A., and Kyllingstad, A.: "Drillstring

Vibrations: Comparison Between Theory and Experiments on a Full­Scale Research Drilling Rig," paper SPE 14760 presented at the 1986 IADC/SPE Drilling Conference, Dallas, Feb. 10-12.

2. Dareing, D.W. and Livesay, 8.J.: "Longitudinal and Angular Drill String Vibrations with Damping," J. Eng. Ind. (Nov. 1968) 671-79.

3. Kreisle, L.F. and Vance, J.M.: "Mathematical Analysis of the Effect of a Shock Sub on the Longitudinal Vibrations of an Oil well Drill String," SPEJ (Dec. 1970) 349-56. .

4. Eronini, I.G., Somerton, W.H., and Auslander, D.M.: "A DynamiC Model for Rotary Rock Drilling," J. Energy Resources Technol. (June 1982) 104, 108-20.

5. Alltner, U.: "Modellanalyse zur Wirkung von Schwingungsdiimpfern fur die Reduktion von Axialschwingungen in Bohrstriingen," PhD dis­sertation, Technical U., Clausthal, West Germany (1981).

6. Dareing, D.W.: "Vibrations Increase Available Power at the Bit," Oil & Gas J. (March 1984) 91-98.

7. Skaugen, E. and Kyllingstad, A.: "Performance Testing of Shock Ab­sorbers," paper SPE 15561 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, Oct. 5-8.

8. Halsey, G.W. eta!': "Torsional Drillstring Vibrations: Compari~on Between Theory and Experiments on a Full Scale Research Dnllmg Rig," paper SPE 15564 presented at the 1986 SPE Annual Technical Conference and Exhibition, New Orleans, Oct. 5-8.

9. Schwartz, M.: Information Transmission, Modulation and Noise. McGraw-Hili Book Co. Inc., New York City (1970) 228-50.

51 Metric Conversion Factors cycles/sec x 1.0* E+OO Hz

ft x 3.048* E-OI m Ibf x 4.448222 E+OO N

Ibf-ft x 1.355 818 E+OO N'm

* Conversion factor is exact. SPEDE

Original SPE manuscript received for review Oct 5, 1986. Paper accepted for publication July 16, 1987. Revised manuscript received Sept 30, 1987. Paper (SPE 15563) first present­ed at the 1986 SPE Annual Technical Conference and Exhibition held In New Orleans, Oct. 5--8.

SPE Drilling Engineering. March 1988