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Title:
DRILL STRING AXIAL VIBRATION ATTENUATOR
Document Type and Number:
WIPO Patent Application WO/2016/007689
Kind Code:
A1
Abstract:
The drill string axial vibration attenuator (10) is installed in the bottom hole assembly (BHA) of a drill string to attenuate axial and torsional vibrations of the drill string. The vibration attenuator (10) includes a massive elongate stabilizer (14) installed in a sealed chamber (12) that is, in turn, rigidly installed within the BHA of the drill string. Clearance between the inner wall of the chamber (12) and the stabilizer (14) is provided to preclude frictional interference therebetween. The stabilizer mass (14) is supported from below by a compression spring (18) and a shock absorber or damper (24) at the top to allow movement of the mass (14). The stabilizer (14) substantially reduces or eliminates axial and coupled torsional vibration of the drill string when the mass of the stabilizer (14), the rate of the spring, and the damper stiffness are properly configured.

Inventors:
SAEED AHMED (US)
PALAZZOLO ALAN B (US)
AHMED SHEHAB (QA)
Application Number:
PCT/US2015/039643
Publication Date:
January 14, 2016
Filing Date:
July 09, 2015
Export Citation:
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Assignee:
QATAR FOUNDATION FOR EDUCATION SCIENCE AND COMMUNITY DEV (US)
SAEED AHMED (US)
PALAZZOLO ALAN B (US)
AHMED SHEHAB (QA)
International Classes:
E21B17/07; E21B7/24
Foreign References:
US20140151122A12014-06-05
US20100139977A12010-06-10
US20100181111A12010-07-22
US20100224410A12010-09-09
US20080315471A12008-12-25
Attorney, Agent or Firm:
FORDE, Remmon R. et al. (P.A.8951 Center Stree, Manassas Virginia, US)
Download PDF:
Claims:
CLAIMS

We claim:

1. A drill string axial vibration attenuator, comprising:

an elongate sealed chamber adapted for being immovably affixed within a pipe of a drill string; and

an elongate stabilizer mass resiliently supported within the sealed chamber.

2. The drill string axial vibration attenuator according to claim 1, wherein:

the chamber is adapted for being disposed concentrically within one of the pipes of the drill string, the chamber having an outer surface and an inner surface, the chamber outer surface and the inner surface of the pipe defining a toroidal passage therebetween; and

the stabilizer mass is disposed concentrically within the sealed chamber, the stabilizer mass having an outer surface, the stabilizer mass outer surface and the chamber inner surface defining a toroidal stabilizer mass clearance volume therebetween, the stabilizer mass being free of contact with the inner surface of the chamber.

3. The drill string axial vibration attenuator according to claim 1, wherein the chamber has a lower end and an upper end opposite the lower end and the stabilizer mass has a lower end and an upper end opposite the lower end, the attenuator further comprising: a compression spring disposed between the lower end of the chamber and the lower end of the stabilizer mass; and

a damper disposed between the upper end of the chamber and the upper end of the stabilizer mass.

4. The drill string axial vibration attenuator according to claim 3, wherein the compression spring is selected from the group consisting of springs having linear spring rates and springs having nonlinear spring rates.

5. The drill string axial vibration attenuator according to claim 3, wherein the damper is selected from the group consisting of dampers having linear damper rates and dampers having nonlinear damper rates.

6. The drill string axial vibration attenuator according to claim 1, wherein the stabilizer mass is rotationally stationary relative to the chamber.

7. The drill string axial vibration attenuator according to claim 1, further comprising a motor communicating with the stabilizer mass, the motor selectively rotating the stabilizer mass.

8. A drill string axial vibration attenuator, comprising:

an elongate sealed chamber adapted for being disposed concentrically within a pipe of a drill string, the chamber having an outer surface and an inner surface, the chamber outer surface and the inner surface of the pipe defining a toroidal passage therebetween, the chamber having an upper end and a lower end;

a first resilient member extending downward within the chamber from the upper end; a second resilient member extending upward within the chamber from the lower end; and

an elongate stabilizer mass disposed concentrically within the sealed chamber, the stabilizer mass being attached to and suspended between the first and second resilient members, the stabilizer mass having an outer surface, the stabilizer mass outer surface and the chamber inner surface defining a toroidal stabilizer clearance volume therebetween, the stabilizer being free of contact with the inner surface of the chamber.

9. The drill string axial vibration attenuator according to claim 8, wherein:

the chamber is immovably affixed within one of the pipes of the drill string; and the stabilizer mass is resiliently supported within the sealed chamber.

10. The drill string axial vibration attenuator according to claim 8, wherein the stabilizer mass has a lower end and an upper end opposite the lower end, the attenuator further comprising;

a compression spring disposed between the lower end of the chamber and the lower end of the stabilizer mass; and

a damper disposed between the upper end of the chamber and the upper end of the stabilizer mass.

11. The drill string axial vibration attenuator according to claim 10, wherein the compression spring is selected from the group consisting of springs having linear spring rates and springs having nonlinear spring rates.

12. The drill string axial vibration attenuator according to claim 10, wherein the damper is selected from the group consisting of dampers having linear damper rates and dampers having nonlinear damper rates.

13. The drill string axial vibration attenuator according to claim 8, wherein the stabilizer mass is rotationally stationary relative to the chamber.

14. The drill string axial vibration attenuator according to claim 8, further comprising a motor communicating with the stabilizer mass, the motor selectively rotating the stabilizer mass.

15. A drill string with axial vibration attenuation, comprising:

a plurality of elongate pipes connected end-to-end to define a drill string, the drill string having an upper end and a lower end;

a drill bit mounted on the lower end of the drill string;

a bottom hole assembly located in the lower end of the drill string; and

at least one attenuator mounted in the bottom hole assembly, the attenuator having: an elongate sealed chamber mounted coaxially within the bottom hole assembly, the chamber being annularly spaced from the bottom hole assembly, the chamber having an upper end and a lower end;

a vibration damper mounted at the upper end of the chamber;

a spring mounted at the lower end of the chamber; and

an elongate mass having an upper end attached to the damper and a lower end attached to the spring, the elongate mass being annularly spaced from the chamber and resiliently move upward and downward to counteract axial forces occurring during drilling operations, thereby preventing and reducing axial vibration of the drill string.

16. The drill string with axial vibration attenuation according to claim 15, wherein the chamber is immovably affixed within one of the pipes of the drill string.

17. The drill string with axial vibration attenuation according to claim 15, wherein:

the chamber has an outer surface and an inner surface, the chamber outer surface and the inner surface of the pipe defining a toroidal passage therebetween; and

the elongate mass is disposed concentrically within the chamber, the elongate mass having an outer surface, the elongate mass outer surface and the chamber inner surface defining a toroidal stabilizer clearance volume therebetween, the elongate mass being free of contact with the inner surface of the chamber.

18. The drill string with axial vibration attenuation according to claim 15, wherein:

the spring is selected from the group consisting of springs having linear spring rates and spring having nonlinear spring rates; and the vibration damper is selected from the group consisting of vibration dampers having linear damper rates and vibration dampers having nonlinear damper rates.

19. The drill string with axial vibration attenuation according to claim 15, wherein the elongate mass is rotationally stationary relative to the chamber.

20. The drill string with axial vibration attenuation according to claim 15, further comprising a motor communicating with the elongate mass, the motor selectively rotating the elongate mass.

Description:
DRILL STRING AXIAL VIBRATION ATTENUATOR

TECHNICAL FIELD

The present invention relates generally to earth boring and drilling equipment, and particularly to a drill string axial vibration attenuator for damping or attenuating undesired axial motion in a drill string during drilling operations.

BACKGROUND ART

The problem of drill string vibration has been recognized as one of the prime causes of deterioration in drilling performance. These vibrations may be lateral, torsional, or axial. Field observations have indicated that drill strings can exhibit severe vibrations that may become even more severe at the bottom-hole assembly (BHA), which comprises the drill collars, stabilizers, and the drill bit, and may also include other logging tools and instruments.

This application is directed to axial and torsional drill stem vibrations, and means for reducing or eliminating such axial and torsional vibrations. As the drill bit is penetrating the underlying formation during the drilling operation, the normal reaction force or weight-on-bit (WOB) may become excessive and fluctuate, resulting in axial vibration in the drill string. This is known as "bit-bounce." Excessive axial vibration or bit-bounce may result in reduction of the rate of penetration (ROP) of the drill bit and/or damage to the drill bit, adverse effects upon telemetry tools and data conveyed to the surface, and fatigue of the drill pipes that form the drill stem. All of these factors result in decreased efficiency in the drilling process and increased costs of operation due to the need to replace various components more frequently than would be the case without such axial drill string vibrations.

Excessive torsional vibrations may eventually result in limit cycles where the BHA rotary speeds are bounded between zero and two or possibly even three times the designated rotary table (rotary drive for the drill stem) speed. At its extreme, this phenomenon is known as "stick-slip" where the rotational velocity of the drill bit is momentarily decreased to zero as it sticks at the bottom of the down hole, then slips and accelerates beyond the prescribed rotary table speed. This "stick-slip" phenomenon is also detrimental to the drill pipes and string, the drill bit, logging tools, and to the entire drilling operation.

Thus, a drill string axial vibration attenuator solving the aforementioned problems is desired.

DISCLOSURE OF INVENTION The drill string axial vibration attenuator comprises a massive, elongate stabilizer installed concentrically within a sealed internal chamber of the drill collar (DC) or BHA of a drill string, the chamber being rigidly secured concentrically within the DC or BHA. Drilling fluid or "mud" is routed around the sealed stabilizer chamber between the outer wall of the chamber and the inner wall of the drill string pipe. The stabilizer is supported within its chamber by a compression spring at its lower end and by a shock absorber or damper extending between the upper end of the stabilizer and the drill string structure. Clearance is provided around the stabilizer to preclude contact with the surrounding wall of the sealed chamber.

The mass of the suspended stabilizer, the spring rate of the supporting compression spring, and/or the stiffness of the damper may be adjusted or configured according to the needs of the system. The spring and damper rates may be linear or nonlinear. The stiffness, mass, and/or damping values for the assembly are selected after considering the drill bit type, type of geological formation, DC and/or BHA mass, and/or other drill string parameters. Axial vibrations of the drill string are dissipated due to the damping forces that arise from the relative velocity between the stabilizer mass and the DC and/or the BHA of the drill string. This provides two beneficial effects, namely, (1) resonant and off-resonance forced vibrations are attenuated; and (2) vibration instabilities caused by modulation of cutting force amplitudes are suppressed.

Torsional vibrations may also be attenuated by rotating or spinning the stabilizer within its chamber. A mathematical analysis of the various relevant parameters is also provided herein.

These and other features of the present invention will become readily apparent upon further review of the following specification and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a schematic elevation view of a drill string axial vibration attenuator according to the present invention.

Fig. 2 is a schematic elevation view showing the drill string axial vibration attenuator of Fig. 1 installed in the bottom hole assembly (BHA) of the drill string.

Fig. 3A is a schematic diagram of a drill string and drive table, illustrating various parameters related to axial and torsional forces exerted upon the drill string. Fig. 3B is a schematic diagram of the drill string and drive table of Fig. 3A, including a diagrammatic equivalent structure representation of a drill string axial vibration attenuator according to the present invention.

Fig. 4A is another schematic diagram of the drill string of Figs. 3 A and 3B, showing additional parameters affected by operation of a drill string axial vibration attenuator according to the present invention.

Fig. 4B is a schematic diagram of a polycrystalline-diamond-compact (PDC) drill bit, illustrating various additional parameters of concern when the drill string axial vibration attenuator according to the present invention is used therewith.

Fig. 5 A is a graph showing the minimum added mass required to suppress bit-bounce using the drill string axial vibration attenuator according to the present invention, wherein the rock stress is 160 MPa and the formation stiffness is 134 MPa.

Fig. 5B is a three-dimensional graph showing stable mass values for the drill string axial vibration attenuator according to the present invention.

Fig. 6 A is a graph of axial BHA velocity without the drill string axial vibration attenuator according to the present invention when rock stress is 160 MPa and formation stiffness is 134 MPa.

Fig. 6B is a graph of rotational BHA velocity without the drill string axial vibration attenuator according to the present invention when rock stress is 160 MPa and formation stiffness is 134 MPa.

Fig. 7A is a graph of axial BHA velocity with the drill string axial vibration attenuator according to the present invention installed in the BHA when rock stress is 160 MPa and formation stiffness is 134 MPa.

Fig. 7B is a graph of rotational BHA velocity with the drill string axial vibration attenuator according to the present invention installed in the BHA when rock stress is 160 MPa and formation stiffness is 134 MPa.

Fig. 8 is a three-dimensional plot of stable values for the drill string axial vibration attenuator according to the present invention, as installed and operating within the BHA.

Fig. 9 is a graph of axial velocity oscillations of a conventional drill string without the installation of the drill string axial vibration attenuator according to the present invention therein.

Fig. 10A is a plot of Eigenvalues in a conventional drill string without the installation of the drill string axial vibration attenuator according to the present invention. Fig. 10B is a plot of Eigenvalues for a drill string having the drill string axial vibration attenuator according to the present invention installed therein.

Fig. 11 is a coarse mesh stability chart for the drill string axial vibration attenuator according to the present invention.

Fig. 12A is a graph showing the axial velocity of the BHA of the drill string incorporating the drill string axial vibration attenuator according to the present invention at a top drive speed of 78 RPM and formation stiffness of 53 MPa.

Fig. 12B is a graph showing the axial velocity of the BHA of the drill string incorporating the drill string axial vibration attenuator according to the present invention at a top drive speed of 78 RPM and formation stiffness of 55 MPa.

Fig. 13A is a graph showing the drill bit rotational velocity of a drill string incorporating the drill string axial vibration attenuator according to the present invention at a top drive speed of 105 RPM and formation stiffness of 100 MPa.

Fig. 13B is a graph showing the axial velocity of the BHA of the drill string incorporating the drill string axial vibration attenuator according to the present invention at a top drive speed of 135 RPM and formation stiffness of 154 MPa.

Fig. 13C is a graph showing the drill bit rotational velocity of a drill string incorporating the drill string axial vibration attenuator according to the present invention at a top drive speed of 140 RPM and formation stiffness of 158 MPa.

Fig. 14 is an operating point stability chart showing various characteristics for a drill string with and without the drill string axial vibration attenuator according to the present invention.

Fig. 15A is a graph showing the axial velocity of the lower or distal portion of a drill stem adjacent to the drill bit prior to the installation of the drill string axial vibration attenuator according to the present invention.

Fig. 15B is a graph showing the torsional velocity of the lower or distal portion of a drill stem adjacent to the drill bit prior to the installation of the drill string axial vibration attenuator according to the present invention.

Fig. 15C is a graph showing the drill bit reaction force or weight on bit (WOB) prior to the installation of the drill string axial vibration attenuator according to the present invention. Fig. 15D is a graph showing the drill bit reaction torque, or torque on bit (TOB), prior to the installation of the drill string axial vibration attenuator according to the present invention.

Fig. 16A is a graph showing the axial velocity of the lower or distal portion of a drill stem adjacent to the drill bit after installation of the drill string axial vibration attenuator according to the present invention.

Fig. 16B is a graph showing the torsional velocity of the lower or distal portion of a drill stem adjacent to the drill bit after installation of the drill string axial vibration attenuator according to the present invention.

Fig. 16C is a graph showing the drill bit reaction force or weight on bit (WOB) after installation of the drill string axial vibration attenuator according to the present invention.

Fig. 16D is a graph showing the drill bit reaction torque, or torque on bit (TOB), after installation of the drill string axial vibration attenuator according to the present invention.

Fig. 17 A is a graph showing the universal performance of the drill string axial vibration attenuator according to the present invention with a polycrystalline-diamond- compact (PDC) drill bit over a range of rotational speeds and a formation stiffness of 80 MPa.

Fig. 17B is a graph showing the universal performance of the drill string axial vibration attenuator according to the present invention with a roller cone drill bit over a range of rotational speeds and a formation stiffness of 80 MPa.

Fig. 18A is a graph comparing the effects of sprung and unsprung masses to the BHA having the drill string axial vibration attenuator according to the present invention installed therein, using a PDC drill bit with a formation stiffness of 80 MPa.

Fig. 18B is a graph comparing the effects of sprung and unsprung masses to the BHA having the drill string axial vibration attenuator according to the present invention installed therein, using a roller cone drill bit with a formation stiffness of 80 MPa.

Fig. 18C is a graph comparing the effects of sprung and unsprung masses to the BHA having the drill string axial vibration attenuator according to the present invention installed therein, using a PDC drill bit with variable formation stiffness and a top drive rotational speed of 80 RPM.

Fig. 18D is a graph comparing the effects of sprung and unsprung masses to the BHA having the drill string axial vibration attenuator according to the present invention installed therein, using a roller cone drill bit with variable formation stiffness and a top drive rotational speed of 80 RPM.

Fig. 19A is a graph comparing the axial velocities of a drill string incorporating the drill string axial vibration attenuator according to the present invention with various shock absorber or damper characteristics in a relatively soft rock formation.

Fig. 19B is a graph comparing the torsional velocities of a drill string incorporating the drill string axial vibration attenuator according to the present invention with various shock absorber or damper characteristics in a relatively soft rock formation.

Fig. 20A is a graph comparing the axial velocities of a drill string incorporating the drill string axial vibration attenuator according to the present invention with and without a shock absorber or damper in a harder rock formation.

Fig. 20B is a graph comparing the torsional velocities of a drill string incorporating the drill string axial vibration attenuator according to the present invention with and without a shock absorber or damper in a harder rock formation.

Fig. 21 is a graph showing the relationship between the mass of the stabilizer and the stable top drive rotational speed of a drill string incorporating the drill string axial vibration attenuator according to the present invention.

Fig. 22 is a graph showing the axial velocity at the BHA at 76 RPM in a conventional drill string without the installation of the drill string axial vibration attenuator according to the present invention.

Fig. 23 is a graph showing the axial velocity at the BHA at 76 RPM in a drill string incorporating the drill string axial vibration attenuator according to the present invention when the attenuator has a mass of 20 percent of the BHA, a natural frequency of 30 rad/sec, and a damping ratio of 0.3.

Fig. 24 is a graph showing the axial velocity with various top drive rotational speeds of a BHA having a mass of 87 tons in a drill string using a PDC drill bit and incorporating the drill string axial vibration attenuator according to the present invention when the actuator has a natural frequency of 30 rad/sec and a damping ratio of 0.3 and the formation stiffness is 134 MPa.

Similar reference characters denote corresponding features consistently throughout the attached drawings.

BEST MODES FOR CARRYING OUT THE INVENTION The drill string axial vibration attenuator, also referred to herein as the attenuator, is installed within a length of drill pipe in a drill string to greatly reduce or eliminate vibrations in and along the drill string during drilling operations. The attenuator is particularly configured to dampen or eliminate axial vibrations, but may be used for the reduction or elimination of torsional vibrations as well.

Fig. 1 of the drawings provides a schematic elevation view of the drill string axial vibration attenuator 10. The attenuator 10 includes a hollow elongate chamber 12 that is sealed from the external environment. The chamber 12 is preferably in the form of an elongate cylinder, but other non-cylindrical shapes may be used, if desired. An elongate stabilizer mass 14 is installed concentrically within the chamber 12, the chamber 12 being immovably affixed within the drill pipe DP, as shown in Fig. 2. The stabilizer mass 14 is also preferably in the form of an elongate cylinder, but other non-cylindrical shapes may be used, if desired.

The stabilizer mass 14 is resiliently supported at its lower end 16 by a compression spring 18 that extends upward from within the lower end 20 of the chamber 12. The opposite, upper end 22 of the stabilizer mass 14 is connected to a shock absorber or damper 24 that extends downward from the upper end 26 of the chamber 12. The spring 18 may have a linear or constant spring rate, or alternatively, may have a nonlinear rate, e.g., having stiffer coils forming a portion of its length and lighter coils forming the remainder of the length. Similarly, the damper 24 may have a linear or nonlinear damping rate.

The chamber 12 has an inner wall or surface 28 defining an internal span or diameter that is larger than that of the stabilizer mass 14. Thus, the outer wall or surface 30 of the stabilizer mass 14 and the inner wall or surface 28 of the chamber 12 define a toroidal stabilizer mass clearance volume 32 therebetween. Thus, the stabilizer mass 14 is free of contact with the inner surface 28 of the chamber 12. The above-described structure enables the stabilizer mass 14 to move vertically, relative to the chamber 12, within the limits imposed by the spring 18 and damper 24. By selecting appropriate masses, spring rates, and damper rates according to the mass of the drill stem, or more appropriately, to the mass of the drill pipe or bottom hole assembly in which the attenuator 10 is installed, the attenuator 10 serves to reduce or eliminate vertical vibrations along the drill stem between the drill bit and the top drive at the surface.

However, the drill stem is also subject to torsional vibrations due to frictional drag at the drill bit and the relatively constant torque of the top drive. In extreme cases, the rotational velocity of the drill bit may be reduced to zero as the bit sticks in the substrate in which it is working, and then accelerate to two or three times the rotational speed of the top drive when the bit is released due to the energy stored in the drill string when the bit sticks and the inherent torsional elasticity of the elongate drill string. This phenomenon is known as "stick-slip" in the drilling industry.

The additional mass of the stabilizer 14 disposed within the drill pipe can serve to reduce or eliminate such torsional vibrations, even if the stabilizer mass 14 is restricted from rotational movement within the chamber 12, e.g., by the damper 24 serving as a rotationally rigid link between the stabilizer mass 14 and the chamber 12. However, it may be desired to provide for rotational movement of the stabilizer mass 14 relative to the chamber 12, adjusting the direction and speed of the rotation in accordance with any torsional vibration that may occur along the drill stem. This may be accomplished by installing a drive motor 34 at some convenient location between the structure of the chamber 12 and the stabilizer mass 14, e.g., affixing the motor 34 to the upper portion 26 of the chamber 12 and connecting the motor 34 to the damper 24 to selectively rotate the damper 24. Alternatively, the motor 34 might be combined structurally with the damper 24. In any event, appropriate power input to the motor 34 rotates or torsionally oscillates the stabilizer mass 14 accordingly to dampen or eliminate torsional vibrations during drilling operations.

Fig. 2 of the drawings provides a schematic illustration of the installation of the attenuator 10 of Fig. 1 within a drill string, or more specifically within the bottom hole assembly (BHA) of such a drill string. The BHA is conventionally that portion of the drill string at the lower end or bottom of the drill string. The drill bit, designated DB in the drawing, is attached to the lower end of the BHA. The BHA comprises a hollow length of drill pipe (designated DP in the drawing), and may have additional electronic or other devices installed therein to transmit information to the surface. The hollow drill pipe DP has an inner surface S defining its open core C. The attenuator 10 may be installed concentrically within the drill pipe DP. The attenuator 10 has a smaller span or diameter than that of the open core C, so that the outer surface 36 of the attenuator 10 and the inner surface S of the drill pipe DP define a toroidal passage P or attenuator clearance volume therebetween. Thus, the attenuator 10 remains free of contact with the inner surface S of the drill pipe DP. The attenuator 10 is preferably immovably affixed within the drill pipe DP, i.e., restricted from rotation relative to the drill pipe DP. Fig. 3A of the drawings is a schematic diagram of a drill string and drive table illustrating various parameters affected by the installation and performance of the attenuator 10, while Fig. 3B is a similar diagram illustrating parameters associated with the attenuator 10. Fig. 4A is another schematic diagram illustrating the foregoing parameters affected dynamically by use of the attenuator 10. Fig. 4B shows a diagrammatic representation of a polycrystalline-diamond-compact (PDC) drill bit, illustrating various parameters of concern when the attenuator 10 is used with a PDC bit.

Before proceeding further with a description of the attenuator 10, and particularly the various graphs and charts provided to illustrate the various parameters involved and their effects on the efficiency of the attenuator 10, it is appropriate to provide tables listing the various parameters, terms and symbols used herein and their definitions. The values shown in the right hand columns are purely exemplary, and may be adjusted as required.

Table 1 : Parameter Description and Exemplary Values

Parameter Definition Exemplary Value

BHA Bottom Hole Assembly N/A

DDE Delay Differential Equation N/A

PDC Polycrystalline-Diamond-Compact bit N/A

Rate of Penetration (of bit into

ROP N/A

substrate)

Torque on Bit during drilling

TOB N/A

operations

Weight on Bit during drilling

WOB N/A

operations

/ Drill Collar Inertia 415 kg-m 2

m d Drill Collar Mass 87000 kg

Equivalent Drill Pipe Torsional

K f 600 N-m/rad

Stiffness

Added Drill Collar Torsional

C T 500 N-s 2 /rad

Damping

Wo Static Load 100 kN

A Drill Bit Radius 0.15 m

I Wearflat Length 5 mm

σ Average Normal Stress 112 MPa

E Intrinsic Specific Energy 160 MPa

Inclination of Cutting Force on

ξ 0.7

Cutting Face

Γ Spatial Orientation of Wearflats 1.2

μο Coefficient of Friction 0.06

n Number of Blades 8

ntf Attenuator Mass Variable

Cf Attenuator Damping Variable

kf Attenuator Stiffness Variable

ld Rotary Speed of Rotary Table Variable Table 2: Simulation Parameters (Roller-Cone Drill Bits) and Exemplary Values

Parameter Definition Exemplary Value

k c Rock Formation Stiffness 67 MN/m s 0 Formation Elevation Amplitude 1 mm

B Formation Surface Function Constant 1

C] Penetration Constant 1.35e-8 c2 Penetration Constant -1.9e-4

Fig. 3A is a schematic diagram of a drill string, illustrating various parameters relevant to vibration of the drill string. In Fig. 3A, the BHA is represented by a lower cylinder having closely spaced horizontal lines thereacross, and the drill pipe and its elasticity are shown schematically by the spring symbol between the top of the BHA and the top drive of the assembly. The equivalent drill pipe torsional stiffness and added drill collar torsional damping of the drill pipe are indicated by the symbols K t and C t , respectively, as shown in Table 1 further above. The Weight on Bit (WOB) is shown by a vertical arrow at the bottom of the BHA, and the Torque on Bit (TOB) is shown by an elliptical representation of the circular path of the drill assembly during operation. The symbols φ and Ω represent torsional factors arising during the drilling process, and ¾ represents the depth of the hole from the surface to the bottom of the hole. The top drive is the prime mover that applies the torque to rotate the drill stem and its BHA and DB in the hole. The symbol W 0 represents the static load on the system, and the symbols <¾ represents the rotational drive speed of the rotary top drive table. T d and co represent additional factors.

Prior to the installation of the attenuator 10 in the BHA of the drill string, the torsional equation of motion is given as:

]φ + 0 τ φ + Κ τ φ = T d - TOB (1) where

T d = K T - a> d - 1. (2) The axial equation of motion is given as: m DC x d = W 0 - WOB. (3) The depth of cut per revolution per blade is given as: dn(t) = X d (t -X d t-t n ). (4)

The total depth of cut is given as:

d = n - d (5) where t n is the instantaneous time delay obtained by solving the equation:

0(t) - (t - t n ) = 2π/ η . (6)

The WOB and the TOB both have cutting and friction components:

WOB = W c + W f ; (7)

TOB = T C + T j (8)

The expressions for the cutting components are given as:

W c = a ξ ε d; (9)

T r =— ε d. (10)

The friction components are given as:

W f = a I σ; (11) a"

T f =— γ I σ μ. (12)

When the drill bit loses contact with the formation during bit bounce, the depth of cut, d, is negative and the friction components vanish.

By substituting Equations (10) and (12) into equation (1), we have:

]φ + 0 τ φ + Κ τ ψ +— ε η x d (t)

(13)

= T d -Tf +— -£-n-x d {t- t n ). By substituting Equation (9) and (11) into Equation (3) we have: m d x d + α·ξ·ε·η· x d t)

(14)

= W 0 — Wf + a ξ ε n x d t With reference to equation (14), the coefficient of x d represents the stiffness of the formation that can be expressed as: k c = α · ζ ε n. (15)

Fig. 3B is a schematic diagram including parameters that describe vibration of the drill string after installation of the drill string axial vibration attenuator 10. The various symbols provided in Fig. 3A are also provided in Fig. 3B, and have identical meanings in the two Figures. However, Fig. 3B also includes a representation of the sprung mass ntf, shown as stabilizer mass 14 in Figs. 1 and 2, and the depth below the surface x/ of the lower end of the sprung mass ntf or stabilizer mass 14, along with the attenuator stiffness and attenuator damping Cf. After adding the attenuator 10 of Figs. 1 and 2 to the assembly represented in Fig. 3 A, the equation of motion for the sprung mass ntf or stabilizer mass 14 is given as: m f - x f + Cf ixf - x DC ) + k f (x f - x d ) = 0 (16)

Hence the axial equation of motion of the BHA after adding the stabilizer mass is given as: m d x d + α · ξ · ε · η · x d t)

= F 0 - W f + a ^ - £ - n - x d (t - t n ) (17)

+ Cf {x f - x d ) + k f {x f - x d ).

The modeling can be enhanced into a finite element model (FEM) by considering both the torsional and axial flexibility of the BHA and the drill pipe. If the displacement vector of the element is denoted by W, = [x; φι X; + 1 φι+ι] τ , and if the moduli of elasticity and rigidity of the element are denoted by E; and G;, respectively, the element flexibility matrix Ki is given as:

where Ai and /j are the element cross-sectional area and moment area of inertia, respectively.

As an example, if a 200-meter long BHA is discretized into 20 elements equal in length and if a 1000-meter drill pipe is added to the FEM, 10 additional elements (also equal in length) are appended, providing a drill string having 62 degrees of freedom. Fig. 4A is another schematic drawing of the drill string illustrating additional parameters. This model includes coupling between the torsional and axial motions through the form of drill bit-formation interface, torque and axial force. Lumped inertias are included in the model for the axial and torsional motions of the BHA-DC and the absorber. Also included are spring and damping forces between the absorber and BHA. The absorber design variables are the absorber' s mass ntf, damping c , and stiffness kf. The BHA mass and inertia are ntd and /, respectively. The drill pipe torsional stiffness and damping are K r and C r , respectively. The WOB is shown by an axial vector arrow with the TOB shown as an elliptical representation of the rotary torque vector, as in Fig. 3B. The symbols φ and Ω represent torsional factors arising during the drilling process, Xf represents the depth below the surface of the lower end of the sprung mass m ; and ¾ represents the depth of the hole from the surface to the bottom of the hole. The top drive is the prime mover that applies the torque to rotate the drill stem, the BHA, and the DB in the hole. The symbol W 0 represents the static load on the system. The symbol <¾ represents the rotational drive speed of the rotary top drive table. T d and a>dt represent additional factors, as indicated in Fig. 3B.

Fig. 4B is a diagrammatic representation of a polycrystalline-diamond-compact (PDC) drill bit, illustrating various parameters of concern when the drill string axial vibration attenuator 10 is used therewith. The numbers 1 and 2 represent two blades or elements of the drill bit. /„ represents the wearflat length multiplied by the number of blades, while dn(l) represents the cutting depth multiplied by the number of blades, he variables Xd(l) and Xd(l-l n ) represent the vertical depth of the bit and the depth of thickness of the blades. The bit rotates about the vertical axis z. The arcuate span of each blade is represented by the term 2π/η. The variables Ω, φ(1-1„), and φ(1) represent various rotational factors involved in the bit operation.

Many drilling operations utilize a roller cone drill bit that has different dynamics than its PDC counterpart, but nevertheless the coupling between the axial and torsional modes still exists. The WOB in the case of a roller cone drill bit is given by:

WOB = c (¾ - s) if ¾ ≥ s

( 0 if x d < s

The formation surface elevation, s, is given as: s = s 0 - sin b(p. (20) The torque-on-bit (TOB) is given as: where a Stribeck friction model is expressed as:

2 - φ

μ = μ 0 ( tanh φ + + 0.010 (22)

1 + φ :

In this case, the depth of cut, d, is given as:

2 ■ ■ ROP

(23)

The rate of penetration (ROP) is defined as:

ROP = Cl F 0 Ιω ά + c 2 (24)

The description and exemplary values of the parameters for the PDC and roller cone drill bit cases are listed in tables 1 and 2, respectively, further above.

Equations 13 and 14 (or 16 and 17, after adding the attenuator) represent a system of delay differential equations (DDEs) that can be expressed in a state-space model with a time delay τ either constant or variable as: x(t) = A 0 x t) + Α 1 · {ί - τ) + Β · u t). (25)

The characteristic equation is a quasi-polynomial in the form of: i - ^o - ^ - e " " ! (26)

The presence of the term e ~T's leads to a theoretical infinite number of complex solutions for a continuous system. Here, the Chebyshev Spectral Method is presented as a numerical method to solve DDEs of a discrete system that will have a finite number of roots, since a closed form solution is virtually impossible to obtain. In the present system, the quasi- polynomial is solved using the approach developed by Breda et al. to discretize the solution operator. The solution operator is the operator transforming an initial condition φ onto the solution segment at a later timepoint specified by a parameter h, in the following sense. We define the solution operator of the DDE in equation (25) to be the operator transforming an initial condition φ to the solution segment at timepoint h. This operator is denoted by T(h) : C([-T, 0], ¾ N ) → C([-T, 0], ¾ N ). The solution operator applied to φ, i.e., (Τ(Κ)φ) (θ) =: ψ(θ), is the solution segment of (2.1) with initial condition ψ = φ at timepoint h. More precisely, ψ(θ) := (Τ(Κ)φ)(θ) = x(h + θ), Θ £ [—τ, 0J , where x(t) is the solution of Equation (25) with initial condition ψ = φ.

Every DDE can be rewritten as a partial differential equation (PDE) by introducing an additional memory-dimension. If the original DDE is represented as: x (t) = A 0 x(t) + A 1 x t - τ) t≥ 0

(27) x(t) = cp(t) t e [-τ, 0] ' then the equivalent PDE can be written as a boundary value problem as:

for the unknown u G C ([0,∞] x [-τ, 0] , ¾ n ). Let φ G C ([-T, 0] , ¾ n ) be given. Then if x(t) is the solution to equation (26), and if u(t, Θ) is a solution to equation (28), then: u(t, Θ) = x(t + θ), θε[-τ, 0], ί≥ 0. (29)

Let A correspond to the differentiation operator in 0-direction with the domain of functions fulfilling the boundary conditions in equation (28), that is:

(Αφ) (θ) := (θ), φ'(.0) = A^(-T) + Α 0 φ(0). (30)

Hence the problem is reduced to an abstract Cauchy-problem in the form:

^- x 1 = Ax 1 . (31) dt 1 1

The differentiation operator, A, of the abstract Cauchy-problem is expressed in terms of the solution operator of the DDE, T, as:

1

Αφ := Urn - (Τ(ί)φ - φ). (32) The eigenvalues of the operator A are the eigenvalues of the DDE. It is now required to discretize A and compute the eigenvalues of the corresponding finite-dimensional linear operator A N that represents the eigenvalues of the system with the time-delay incorporated.

For a given natural number, N, the Chebyshev nodes over the interval [— τ, 0] are defined as:

The Chebyshev differentiation matrix, D N , is obtained utilizing the Chebyshev nodes as:

Thus, AN can now be evaluated as: where m is the number of degrees of freedom of the original state space of the continuous DDE.

At a certain operating point there exists a minimum stability speed of the top drive of the rotary table, below which the system becomes unstable due to increased time delay due to the cutting action of the blades of the PDC drill bit. It is well known that time delay resembles negative damping that destabilizes a system. Hence, the BHA will be more susceptible to bit bounce when the speed of the top drive is lowered. In drilling operations, there are several reasons as to why the top drive speed is decreased. Logging-while-drilling (LWD) or measurement-while-drilling (MWD) operations usually require lower penetration rates in order to obtain accurate data of the well. Another reason is the power limitation on the motor driving the rotary table, since the torque load on the driving motor increases as the friction at the drill bit increases and puts a limit on the maximum rotary speed. Even during normal operation, with increasing torque on bit (TOB), the rotary speed of the drill bit would be oscillating around the mean value of the top drive speed, and when its speed falls significantly, the time delay between the cutting actions of the individual blades will increase. Moreover, the addition of new sections of drill pipe requires that the drilling action be temporarily paused, bringing the rotary table to a complete stop and then restarting and passing through low RPM ranges that are susceptible to bit bounce.

One of the methods for stabilizing the system is by adding additional mass to the BHA in order to suppress bit bounce. This can be done by either increasing the BHA length, which is referred to as the "unsprung mass" case, or by attaching attenuator units comprising a mass sprung on one or more springs and dampers, generally as shown schematically in Figs. 1 and 2. Fig. 5A is a graph showing the minimum mass ratio required to suppress bit bounce over a top drive speed ranging from 65 to 130 RPM at a rock stress of 160 MPa and formation stiffness of 134 MPa. At speeds of 130 RPM or higher the time delay is small enough and the system is stable, so no added mass is needed. It can be clearly seen that the sprung added mass needed is much smaller than its unsprung counterpart. A rigorous and controlled search using computerized numerical integration was carried out to arrive at the respective stiffness and damping values of the spring and damper that yielded the minimum and optimum sprung mass required to suppress bit bounce. Fig. 5B is a three dimensional plot illustrating the corresponding sprung mass values. The optimum mass values are shown in bold face. The optimum mass was chosen to be the minimum value at each rotational speed. It is to be noted that the entire search elapsed over twenty hours of computational time, resulting in 120,000 possible search points.

Consider a case where the top drive speed decreases to 105 RPM. Bit bounce vibrations will be seen, as shown clearly in the graph of Fig. 6A, and there also will be large fluctuations of the BHA rotary velocity, as shown in the corresponding graph of Fig. 6B. In these two graphs the intrinsic specific energy ε is 160 MPa, and the formation stiffness is 134 MPa. In this case, bit bounce suppression requires that the BHA mass be increased by 49 tons if the mass is fixed to the BHA, i.e., as unsprung mass. However, the added mass need only be increased by 17 tons if the additional mass is attached to the BHA by a spring 18 having appropriately selected stiffness, e.g., 42.4 MN/m, and a damper 24, e.g., 0.72 MN.s/m, to provide the sprung mass. The same result is achieved when the sprung mass is about three times less than its unsprung counterpart. Figs. 7A and 7B are graphs respectively illustrating the axial and torsional velocities of the BHA after adding the attenuator, with ε = 160 MPa and formation stiffness = 134 MPa, as in other examples above.

Since numerical integration of the dynamic equations of motion is very time- consuming, and thus a relatively small number of possible solutions can be scanned through trial and error, the Chebyshev-based discretization spectral method was implemented in order to find values of the attenuator mass, spring stiffness, and damping that achieve overall system stability. By applying equation (35) above one can obtain a characteristic matrix A^, in which the eigenvalues determine the stability. The system is stable if, and only if, all eigenvalues have negative real values. Hence, the computational time is reduced, and thousands of possible solutions can be scanned in just a matter of minutes. Fig. 8 is a three- dimensional plot of solutions obtained using the Chebyshev-based discretization spectral method. The number of search points was approximately ten million, yet the elapsed search time was less than four hours. If this search were to be conducted manually using numerical integration, the search time would be 58 days. If the drill string model were finite element- based, then at least one thousand days would be required to conduct this search.

The validity of the Chebyshev method is verified by numerical integration of the equations of motion under the same conditions. Consider a case when the equivalent soil stiffness is 67 MPa and the top drive speed is 80 RPM. Without the attenuator 10, the system is unstable, as can be seen in Fig. 9, where there exist oscillations of axial velocity. The Chebyshev method predicts this instability through the evaluation of eigenvalues, and it can be seen in Fig. 10A that there are two conjugate and complex poles on the imaginary axis. Note that in Fig. 10A the rightmost conjugate poles have a frequency of 31 rad/s. The periodic time from Fig. 9 is approximately 0.2 seconds, which corresponds to an oscillation frequency of 31 rad/s, which is consistent with the frequency directly evaluated from the eigenvalue plot. Upon installing an attenuator that is 15% in mass of the original BHA, and which has a natural frequency and damping ratio of 30 rad/s and 0.3, respectively, the system becomes stable. This is shown by the graph of Fig. 10B, showing the eigenvalue plot obtained by the Chebyshev method and indicating that the previously unstable poles have migrated into the left-hand plane, further validating the spectral method implemented here.

Another advantage of the Chebyshev method is that it can be implemented to obtain or predict an operating point stability chart. Fig. 11 is such a chart, showing a coarse mesh of stable operating points that indicate stable top drive spin speeds over a range of values of rock formation stiffness. The larger open ovals shown on the chart of Fig. 11 are on the stability boundary line, in which a small decrease in top drive spin speed or increase in formation stiffness value will result in bit bounce in the system. This is verified by numerical integration of the equations of motion by perturbing about the three different boundary line stable points shown by the larger open ovals. Fig. 12A shows the axial velocity of the drill bit at the first point of investigation, where the top drive spin speed is 78 RPM and the formation stiffness is 53 MPa. In this case, the axial velocity reaches a steady-state value with the system being stable, as the Fig. 12A graph indicates. Either a reduction of top drive rotational speed or an increase in formation stiffness would be destabilizing, as noted further above. In the graph of Fig. 12B the rotational velocity of the top drive, i.e., 78 RPM, is the same as that of the graph of Fig. 12A. However, the rock formation stiffness in Fig. 12B is 55PMa, slightly higher than the 53 MPa of Fig. 12A. This is sufficient to cause some destabilization, as indicated in Fig. 12B.

Figs. 13A and 13B are additional graphs showing the destabilizing effects of increases in rock formation stiffness. In Fig. 13A, the drill bit spin velocity has been increased to 105 RPM, but this is insufficient to offset the destabilizing influence of the increase in rock formation stiffness to 100 MPa, as opposed to the 53 to 55 MPa formation stiffnesses of the graphs of Figs. 12A and 12B. The graph of Fig. 13B shows the results of operation at a rotational velocity of 135 RPM, normally a stable speed in softer formations, but operating in a rock formation having a stiffness of 154 MPa. This combination results in axial instability or bit bounce, as indicated by the Fig. 13B graph. Similarly, the graph of Fig. 13C shows the resulting instability at a top drive rotational velocity of 140 RPM and a rock formation stiffness of 158 MPa.

Fig. 14 illustrates a very fine mesh stability chart showing top drive speed v. rock formation stiffness for different attenuator mass values. The chart was divided into 1.2 million search points, and the elapsed time to evaluate this chart was only forty-five minutes. In the chart of Fig. 14, the attenuator mass values (indicated as PPMD, or Passive Proof Mass Damper, in the legend of Fig. 14) increase from zero in the lowermost band up to an increase of 40% in the larger shaded area of the upper portion of the chart. The shading of the various bands indicating different percentages of mass increase correspond to the shadings and corresponding mass increases shown in the small circular areas of the legend.

Figs. 15A through 15D are graphs showing the system performance before adding the attenuator 10, where the system stiffness is 100 MPa. These graphs are based upon a lumped model and finite element modeling (FEM). Fig. 15A is a graph of the axial velocity near the drill bit, Fig. 15B is a graph of the torsional velocity near the drill bit, Fig. 15C is a graph of the drill bit axial reaction force (WOB), and Fig. 15D is a graph of the drill bit torsional reaction force (TOB). In each of these drawings, the lumped model is shown by the heavier continuous line, the FEM for the drill string is shown by the lighter continuous line, and the FEM for the BHA is shown by the broken line. The continuous instability that continues completely across each of these graphs is apparent.

In contrast, Figs. 16A through 16D are graphs showing the system performance after adding the attenuator 10, where the system stiffness is identical to that of Figs. 15A through 15D at 100 MPa. The mass of the added attenuator is 15% of the original mass of the BHA in these examples, and has a natural frequency and damping ratio of 30 rad/s and 0.3, respectively. The graphs of Figs. 16A through 16D are also based upon a lumped model and finite element modeling (FEM). Fig. 16A is a graph of the axial velocity near the drill bit, Fig. 16B is a graph of the torsional velocity near the drill bit, Fig. 16C is a graph of the drill bit axial reaction force (WOB), and Fig. 16D is a graph of the drill bit torsional reaction force (TOB). In each of these Figs., the lumped model is shown by the heavier continuous line, the FEM for the drill string is shown by the lighter continuous line, and the FEM for the BHA is shown by the broken line. While there is not a significant difference between the FEM of the complete drill string and the FEM of only the BHA in any of Figs. 15 A through 16D, it will be seen that the addition of the attenuator 10 rapidly attenuates the vibrational frequencies as time progresses, as shown by the broken lines representing the FEM BHA in each of Figs. 16A through 16D. The onset instability speed of approximately 95 RPM is nearly the same in all three cases, i.e., lumped model, FEM drill string, and FEM BHA, which indicates that the lowest modes of vibration are excited as described further above.

In order to demonstrate robustness, consider a universal attenuator with a natural frequency of 30 rad/s and a damping ratio of 0.3 to be fitted upon any BHA that utilizes either a PDC or roller cone drill bit. Fig. 17A is a graph illustrating the decrease in axial vibration amplitudes with increasing attenuator mass for different operating top drive spin velocities when a PDC drill bit is used. Fig. 17B is similar, but shows the results when a roller cone drill bit is used. Figs. 18 A and 18B are additional graphs illustrating a comparison between vibration amplitude values for added sprung and unsprung masses at a top drive spin speed of 80 RPM, with Fig. 18A being for a PDC bit and Fig. 18B being for a roller cone bit.

Figs. 18C and 18D are graphs illustrating vibration amplitude values v. rock formation stiffness when an attenuator having 15% total sprung and unsprung mass is added to the original exemplary 87-ton BHA and the top drive spinning at 80 RPM. Fig. 18C is for a PDC drill bit, while Fig. 18D is for a roller cone bit. To demonstrate the effectiveness and advantages of the attenuator over standard or conventional shock absorbers, consider a case where the soil formation is not sufficiently hard to induce bit bounce vibrations at a given top drive spin speed, and hence no axial vibration suppression device is needed. Equation 36 below represents a simplified model of a standard shock absorber with stiffness k a bs and damping c a bs- Four simulations are conducted, including (i), original BHA, (ii) tuned absorber, (iii) mistuned absorber, and (iv) with the attenuator 10. The equivalent rock formation stiffness is 150 MN/m, and the top drive speed is 150 RPM, at which there is no bit bounce at steady-state speed. Adding an attenuator of mass ratio 0.15, natural frequency of 30 rad/s, and damping ratio of 0.3, or a tuned shock absorber with stiffness value of 2e4 N/m and damping value of 30e3 N.s/m, does not affect the stability of the system, as shown in Figs. 19 A and 19B. However, if the stiffness of the shock absorber is increased to 2e5 N/m, the system becomes unstable, and in addition to bit bounce, stick-slip occurs, since there is a large fluctuation of both torsional and axial velocities. Thus, this demonstration confirms that adding shock absorbers may exacerbate the vibration, rather than mitigating such vibration.

abs ' X DC c abs ' X DC ½ > 0

Now consider the case when drilling into a harder formation of equivalent rock formation stiffness of 170 MN/m with the same top drive spin speed of 150 RPM. In this case, severe bit bounce and torsional vibrations occur that nearly force the system into stick- slip. As shown in Figs. 20A and 20B, adding the attenuator (passive proof mass damper, or PPMD) suppresses the bit bounce and torsional vibrations, and using the conventional "tuned" shock absorber worsens the vibration, since the system experiences stick-slip as the drill bit velocity slows to zero RPM. Thus, the robustness of using the attenuator 10 is further validated by comparison with a conventional shock absorber operating under identical conditions.

Figs. 21 through 24 are graphs illustrating further variations of the attenuator, drill string, and BHA assembly. Another approach to the attenuator configuration is to increase the attenuator mass while reducing the BHA and/or DC mass in order to maintain a constant total mass of the assembly. The graph of Fig. 21 illustrates the amount of attenuator mass (as a percentage of the exemplary 87-ton total mass) required to suppress bit bounce vs. top drive spin speed. This Fig. 21 graph corresponds to the use of a bladed PDC bit, but the attenuator is also effective in suppressing axial BHA vibrations when a roller cone bit is used, as noted further above.

Fig. 22 is a graph illustrating the undamped axial vibrations that occur in a formation having a rock formation stiffness of 67 MN/m and an undamped 87-ton BHA, i.e., the same total weight as the attenuator and BHA combination used to form the graph of Fig. 21, and a top drive rotational speed of 76 RPM. As can be seen in Fig. 22, the axial vibrations remain substantially constant over a given period of time with no damping whatsoever. By attaching an attenuator weighing 20% of the original weight of the BHA and having a natural frequency of 30 rad/s and damping ratio of 0.3, the vibrations are mitigated over a period of time, as indicated in the graph of Fig. 23.

Finally, consider an absorber with a natural frequency of 30 rad/s and a damping ratio of 0.3 that may be fitted to any BHA using either a PDC or roller cone drill bit. Fig. 24 shows the decrease of axial vibration amplitudes with increasing absorber mass for different operating top drive spin velocities when a PDC drill bit is used. This is similar to the results when using a roller cone drill bit, as shown in the graph of Fig. 17B further above.

In conclusion, a passive proof mass damper or attenuator serves to mitigate or fully suppress harmful axial vibrations in a drill string BHA. Simulation studies indicate that the sprung mass of the attenuator effectively mitigates (and can in some cases completely suppress) bit bounce, and that the value of the sprung mass is much less than the corresponding unsprung mass required to yield the same effect. This permits the driller to operate the top drive with minimal spin speed, thus conserving power and operational costs. Moreover, the spring and damper of the attenuator can be fine-tuned in order to minimize the sprung mass. The robustness of the attenuator is illustrated herein for a wide range of operating conditions for both PDC and roller cone drill bit models, as well as in comparison to a standard shock absorber conventionally used for such purposes. One, two, or more such attenuators may be installed in the drill string, particularly in the BHA of the drill string. Furthermore, the Chebyshev-based discretization spectral method has been applied in order to predict system stability by estimating eigenvalues and to estimate stabilizing attenuator parameters in a very time efficient manner, instead of the time-consuming trial and error process performed using numerical integration. Verification of the Chebyshev method DDE solver has been confirmed by comparison with the numerical integration of the equations of motion utilizing both a lumped model and FEM. It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.