Method for determining critical rotating speed of drill string with rotary steering tool

Abstract

The invention discloses a critical rotation speed determination method for a drill string with a rotary steering tool. Comprising the following steps that firstly, the combined structure of the drilling tool, the type of a pushing-type steering tool PowerDrive, the size rated pushing force, the sizes and weights of all parts of a drill column, the elasticity modulus of the drill column and the type of a drill bit are determined; 2, confirming the working mode of the rotary steering tool, and determining the excitation acting force of the rotary steering tool according to the working mode of the rotary steering tool; 3, according to the working mode of the rotary steering tool, the inherent frequency and the inherent vibration mode of the drill column with the rotary steering tool are calculated; 4, calculating the steady-state response of the drill stem with the rotary steering tool at different rotating speeds; and 5, comparing the steady-state response amplitudes of the rotary steering tool at different rotating speeds, and determining the rotating speed corresponding to the maximum steady-state response so as to determine the critical rotating speed. The method has the beneficial effects that the critical rotating speed can be determined, and resonance of a drill column with a rotary steering tool during underground drilling is avoided.

Description

Technical Field

[0001] The invention relates to the technical field of drilling tools, and in particular to a method for determining a critical rotation speed of a drill string with a rotary guide tool. Background Art

[0002] As deep oil and gas resources become a new area of ​​exploration and development, deep well drilling technology has developed rapidly. However, in deep, complex and difficult-to-drill formations, slow mechanical drilling speed and difficulty in controlling wellbore trajectory have always been important problems restricting the development of ultra-deep wells. Rotary steerable drilling tools can flexibly adjust the size and direction of the lateral force of the drill bit, greatly improve the control accuracy of the wellbore trajectory, and significantly improve drilling efficiency. Therefore, rotary steerable drilling technology is considered to be the mainstream development direction of future drilling technology.

[0003] However, with the increasing application of rotary steerable drilling tools, a series of problems such as poor steering ability, unsatisfactory trajectory control effect, and drill failure frequently occur. When using rotary steerable tools, the influence of rotation speed on mechanical drilling speed is very obvious. When the rotation speed used is lower than the critical speed, the drilling efficiency will be reduced. When the rotation speed used reaches a certain value, it will cause resonance of the drill string, which not only affects the working efficiency of the rotary steerable tool, but also brings risks to the safety of the drill string. At present, when using rotary steerable tools for drilling, the rotation speed is mostly selected based on experience, especially without considering the influence of the rotary steerable tool direction on the critical speed.

[0004] Therefore, in order to ensure the safety of the drill string and the efficient use of the rotary steerable tool, the present invention provides a method for determining the critical speed of the drill string with a rotary steerable tool, which can effectively determine the critical speed of the drill string with a rotary steerable tool during drilling, and guide the mine to select a reasonable speed when using the rotary steerable tool. Summary of the invention

[0005] The object of the present invention is to provide a method for determining the critical speed of a drill string with a rotary steering tool, which is based on the working mode and excitation force of the rotary steering tool, the steady-state response of the drill string with the rotary steering tool at different speeds, and the comparison of the steady-state response amplitudes of the rotary steering tool at different speeds to determine the speed corresponding to the maximum steady-state response. The critical speed is determined in this way to avoid resonance of the drill string with the rotary steering tool when drilling downhole, thereby ensuring the safety of the drill string with the rotary steering tool.

[0006] The present invention is implemented as follows: a method for determining the critical speed of a drill string with a rotary steering tool comprises the following steps:

[0007] Step 1: Determine the combined structure of the drilling tool, the type and size of the PowerDrive, the rated push force of the push-type steerable tool, the size and weight of each component of the drill string, the elastic modulus of the drill string and the type of drill bit;

[0008] Step 2: confirming the working mode of the rotary steerable tool, and determining the excitation force of the rotary steerable tool according to the working mode of the rotary steerable tool;

[0009] Step 3: Calculate the natural frequency and natural vibration mode of the drill string with the rotary steerable tool according to the working mode of the rotary steerable tool;

[0010] Step 4: Calculate the steady-state response of the drill string with the rotary steering tool at different rotation speeds according to the obtained natural frequency and natural vibration mode;

[0011] Step 5: Compare the steady-state response amplitudes of the rotary guide tool at different speeds, determine the speed corresponding to the maximum steady-state response, and thereby determine the critical speed.

[0012] Preferably, in step 2, the working mode of the rotary steering tool is a full-power steering mode or a non-full-power steering mode. When the deviation between the actual controlled inclination and the designed well inclination is greater than 1°, it is in the full-power steering mode, otherwise the non-full-power steering mode can be used.

[0013] Preferably, in full-power steering mode, the magnitude and orientation of the bias force at the guide ribs of the PowerDrive system remain unchanged. When the drill string rotates one circle, the three guide ribs of the PowerDrive system extend in turn to "slap" the well wall, which will apply a periodic excitation force to the drill string, and the frequency is three times the frequency corresponding to the rotation speed; the excitation force is;

[0014]

[0015] Where F is the excitation force, N; θ V is the initial phase of the excitation force, rad; F V is the maximum value of the “beating force”, N; Ω is the drill string rotation speed, rad / s; t is the time, s.

[0016] Preferably, in the non-full force steering mode, the upper disc valve of the PowerDrive system swings, and in the first half cycle of the upper disc valve swing, the rotation speed of the drill string and the rotation speed of the upper disc valve are in the same direction, and the time interval between two adjacent "flaps" of its steering rib increases, and the excitation force is;

[0017]

[0018] Where F is the excitation force, N; ω d is the swing speed of the upper valve, rad / s; θ V is the initial phase of the excitation force, rad; F V is the maximum value of the “beating force”, N; Ω is the drill string speed, rad / s; t is the time, s; the range of t is:

[0019]

[0020] In the formula, θ d is the swing angle range of the upper valve of the PowerDrive system, rad; ω d is the swing speed of the upper valve, rad / s; k is the number of flapping times; the "flapping" direction is no longer fixed, and the eccentric azimuth of the "flapping" is:

[0021] δ P =θ d0 +ω d t (4)

[0022] In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s;

[0023] In the second half of the swing cycle of the upper valve, the rotation speed of the drill string and the upper valve are opposite, and the time interval between two adjacent "beats" decreases. Then:

[0024]

[0025] Where, F is the excitation force, N; F V is the maximum value of the “beating force”, N; Ω is the drill string speed, rad / s; ω d is the swing speed of the upper valve, rad / s; t is the time, s; θ V2 is the initial phase of the excitation force in the second half of the cycle, rad;

[0026] The range of t is:

[0027]

[0028] The eccentric azimuth of the "flap" is:

[0029] δ P =θ d0 +θ d -ω d t (7)

[0030] In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s.

[0031] Preferably, in step three, the formula for calculating the natural frequency and natural vibration mode of the drill string with the rotary steering tool is:

[0032] (K-ω 2 M)ψ=0 (8)

[0033] In the formula, M is the mass matrix; K is the stiffness matrix. According to formula (8), the natural frequency ω can be solved: i , Hz, and the natural vibration modes ψ corresponding to different natural frequencies i ;

[0034] First, the entire drill string is divided into 9m long units, and the Power-Drive is a separate unit. The mass matrix and stiffness matrix of each unit are:

[0035]

[0036]

[0037] Where E is the elastic modulus, Pa; l is the unit length, m; A is the unit cross-sectional area, m 2 ; is the section inertia moment. For the whole drill string, the unit mass matrix and stiffness matrix can be superimposed according to the node position to form the overall mass matrix and stiffness matrix of the whole drill string;

[0038] Then, the matrix is ​​inverted to directly obtain the different natural frequencies ω corresponding to equation (8): i , and the natural vibration modes ψ corresponding to different natural frequencies i .

[0039] Preferably, in step 4, the steady-state response of the drill string with the rotary steerable tool at different rotation speeds is calculated according to the obtained natural frequency and natural vibration mode:

[0040]

[0041] Where U(t) is the steady-state response of the drill string; n is the order; r i is the frequency ratio, r i =γ / ω i ,γ is the drill bit excitation frequency;ζ i is the damping ratio, α, β are structural damping coefficients. For ultra-slender drill strings in drilling fluid, α = 0.1, β = 0.01 are generally used; θ i Phase angle at different frequencies, θ i =arctan[2ζ i r i / (1-r i 2 )];ψ i is the natural vibration mode; i represents different natural vibration modes; Ω is the drill string speed, rad / s; t is the time, s.

[0042] The beneficial effects of the present invention are as follows: based on the working mode and excitation force of the rotary steering tool, the steady-state response of the drill string with the rotary steering tool at different rotational speeds is compared, the steady-state response amplitude of the drill string with the rotary steering tool at different rotational speeds is compared, and the rotational speed corresponding to the maximum steady-state response is determined, thereby determining the critical speed, avoiding resonance of the drill string with the rotary steering tool when drilling downhole, and ensuring the safety of the drill string with the rotary steering tool. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 The excitation force of the rotary guide tool in Example 1 of the present invention when fully guiding.

[0044] Figure 2 The first five natural frequencies of the drill string with the rotary steering tool in Example 1 of the present invention.

[0045] Figure 3 The first five vibration modes of the drill string with a rotary steering tool in Example 1 of the present invention.

[0046] Figure 4 A curve showing the dynamic displacement response of a drill string with a rotary steering tool as a function of rotation speed in Example 1 of the present invention.

[0047] Figure 5 Projection diagram of the dynamic displacement response of the drill string with a rotary steering tool in Example 1 of the present invention.

[0048] Figure 6 The excitation force of the rotary guide tool in embodiment 2 of the present invention when the guide tool is not fully guided.

[0049] Figure 7 A curve showing the dynamic displacement response of a drill string with a rotary steering tool as a function of rotation speed in Example 2 of the present invention.

[0050] Figure 8 Projection diagram of the dynamic displacement response of the drill string with a rotary steering tool in Example 2 of the present invention. DETAILED DESCRIPTION

[0051] The present invention is described in further detail below so that those skilled in the art can implement it according to the description.

[0052] It should be understood that terms such as “having”, “including” and “comprising” used herein do not exclude the existence or addition of one or other elements or combinations thereof.

[0053] Example 1

[0054] A method for determining a critical speed of a drill string with a rotary steering tool according to the present invention comprises:

[0055] Step 1: Determine the combined structure of the drilling tool, the type and size of the PowerDrive, the rated push force of the push-type steerable tool, the size and weight of each component of the drill string, the elastic modulus of the drill string and the type of drill bit;

[0056] The combined structure of the drilling tool: Φ215.9.mm drill bit + Φ172.0mm Power V + Φ172.0mm drill collar + Φ213.0mm stabilizer + Φ172.0mm short drill collar + Φ127.0mm flexible short section + Φ172.0mm short drill collar + Φ213.0mm stabilizer + Φ172.0mm drill collar. The inner diameter of the drill collar used is 71.4mm.

[0057] Push-to-steer tool PowerDrive Type: Dynamic push-to-steer vertical drilling tool PowerV;

[0058] Elastic modulus: 210GPa; drill bit type: 6-blade PDC drill bit; weight: drill bit weight is 343N, PowerV weight is 5.87kN, stabilizer weight is 2.94kN, and drill collar weight per unit length is 1.52kN / m.

[0059] Step 2: When the deviation between the actual well control and the designed well inclination is greater than 1°, it is the full-power steering mode. The magnitude and direction of the bias force at the steering rib of the PowerDrive system remain unchanged. When the drill string rotates one circle, the three steering ribs of the PowerDrive system extend out in turn to "slap" the well wall, which will apply a periodic excitation force to the drill string, and the frequency is three times the frequency corresponding to the rotation speed; the excitation force is;

[0060]

[0061] Where F is the excitation force, N; θ V is the initial phase of the excitation force, rad; F V is the maximum value of the “beating force”, N; Ω is the drill string speed, rad / s; t is the time, s. When the speed is equal to 60rpm, F V Take 30kN, θ V =0rad, the excitation force of the PowerDrive in the full force steering mode can be calculated.

[0062] Step 3: Calculate the natural frequency and natural vibration mode of the drill string with the rotary steerable tool according to the working mode of the rotary steerable tool;

[0063] (K-ω 2 M)ψ=0 (8)

[0064] In the formula, M is the mass matrix; K is the stiffness matrix. According to formula (8), the natural frequency ω can be solved: i , Hz, and the natural vibration modes ψ corresponding to different natural frequenciesi ; First, divide the entire drill string into 9m long units, Power-Drive is a separate unit, and the mass matrix and stiffness matrix of each unit are:

[0065]

[0066] Where E is the elastic modulus, Pa; l is the unit length, m; A is the unit cross-sectional area, m 2 ; is the section inertia moment. For the whole drill string, the unit mass matrix and stiffness matrix can be superimposed according to the node position to form the overall mass matrix and stiffness matrix of the whole drill string;

[0067] Then, the matrix is ​​inverted to directly obtain the different natural frequencies ω corresponding to equation (8): i , and the natural vibration modes ψ corresponding to different natural frequencies i .

[0068] Step 4: Calculate the steady-state response of the drill string with the rotary steering tool at different rotation speeds according to the obtained natural frequency and natural vibration mode;

[0069]

[0070] Where U(t) is the steady-state response of the drill string; n is the order; r i is the frequency ratio, r i =γ / ω i ,γ is the drill bit excitation frequency;ζ i is the damping ratio, α, β are structural damping coefficients. For ultra-slender drill strings in drilling fluid, α = 0.1, β = 0.01 are generally used; θ i Phase angle at different frequencies, θ i =arctan[2ξ i r i / (1-r i 2 )];ψ i is the natural vibration mode; i represents different natural vibration modes; Ω is the drill string speed, rad / s; t is the time, s.

[0071] Step 5: Compare the steady-state response amplitudes of the rotary guide tool at different speeds, determine the speed corresponding to the maximum steady-state response, and thus determine the critical speed. Figure 4 and Figure 5 It can be seen that when the speed is 90rpm, the vibration amplitude is the largest. This speed is the zero-level speed and should be avoided in field applications.

[0072] Example 2

[0073] A method for determining the critical speed of a drill string with a rotary steering tool comprises the following steps:

[0074] Step 1: Determine the combined structure of the drilling tool, the type and size of the PowerDrive, the rated push force of the push-type steerable tool, the size and weight of each component of the drill string, the elastic modulus of the drill string and the type of drill bit;

[0075] The combined structure of the drilling tool: Φ215.9.mm drill bit + Φ172.0mm Power V + Φ172.0mm drill collar + Φ213.0mm stabilizer + Φ172.0mm short drill collar + Φ127.0mm flexible short section + Φ172.0mm short drill collar + Φ213.0mm stabilizer + Φ172.0mm drill collar. The inner diameter of the drill collar used is 71.4mm.

[0076] Push-to-steer tool PowerDrive Type: Dynamic push-to-steer vertical drilling tool PowerV;

[0077] Elastic modulus: 210GPa; drill bit type: 6-blade PDC drill bit; weight: drill bit weight is 343N, PowerV weight is 5.87kN, stabilizer weight is 2.94kN, and drill collar weight per unit length is 1.52kN / m.

[0078] Step 2: When the deviation between the actual well control and the designed well inclination is less than or equal to 1°, it is a non-full-force steering mode. In the non-full-force steering mode, the upper valve of the PowerDrive system swings. In the first half cycle of the swing of the upper valve, the drill string rotation speed and the upper valve rotation speed are in the same direction, and the time interval between two adjacent "flaps" of the steering rib increases. The excitation force is;

[0079]

[0080] Where F is the excitation force, N; ω d is the swing speed of the upper valve, rad / s; θ V is the initial phase of the excitation force, rad; F V is the maximum value of the “beating force”, N; Ω is the drill string speed, rad / s; t is the time, s; the range of t is:

[0081]

[0082] In the formula, θ d is the swing angle range of the upper valve of the PowerDrive system, rad; ω d is the swing speed of the upper valve, rad / s; k is the number of flapping times; the "flapping" direction is no longer fixed, and the eccentric azimuth of the "flapping" is:

[0083] δ P=θ d0 +ω d t (4)

[0084] In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s;

[0085] In the second half of the swing cycle of the upper valve, the rotation speed of the drill string and the upper valve are opposite, and the time interval between two adjacent "beats" decreases. Then:

[0086]

[0087] Where, F is the excitation force, N; F V is the maximum value of the “beating force”, N; Ω is the drill string speed, rad / s; ω d is the swing speed of the upper valve, rad / s; t is the time, s; θ V2 is the initial phase of the excitation force in the second half of the cycle, rad;

[0088] The range of t is:

[0089]

[0090] The eccentric azimuth of the "flap" is:

[0091] δ P =θ d0 +θ d -ω d t (7)

[0092] In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s.

[0093] Step 3: Calculate the natural frequency and natural vibration mode of the drill string with the rotary steerable tool according to the working mode of the rotary steerable tool;

[0094] (K-ω 2 M)ψ=0 (8)

[0095] In the formula, M is the mass matrix; K is the stiffness matrix. According to formula (8), the natural frequency ω can be solved: i , Hz, and the natural vibration modes ψ corresponding to different natural frequencies i ;

[0096] First, the entire drill string is divided into 9m long units, and the Power-Drive is a separate unit. The mass matrix and stiffness matrix of each unit are:

[0097]

[0098]

[0099] Where E is the elastic modulus, Pa; l is the unit length, m; A is the unit cross-sectional area, m 2 ; is the section inertia moment. For the whole drill string, the unit mass matrix and stiffness matrix can be superimposed according to the node position to form the overall mass matrix and stiffness matrix of the whole drill string;

[0100] Then, the matrix is ​​inverted to directly obtain the different natural frequencies ω corresponding to equation (8): i , and the natural vibration modes ψ corresponding to different natural frequencies i .

[0101] Step 4: Calculate the steady-state response of the drill string with the rotary steering tool at different rotation speeds according to the obtained natural frequency and natural vibration mode;

[0102]

[0103] Where U(t) is the steady-state response of the drill string; n is the order; r i is the frequency ratio, r i =γ / ω i ,γ is the drill bit excitation frequency;ζ i is the damping ratio, α, β are structural damping coefficients. For ultra-slender drill strings in drilling fluid, α = 0.1, β = 0.01 are generally used; θ i Phase angle at different frequencies, θ i =arctan[2ζ i r i / (1-r i 2 )];ψ i is the natural vibration mode; i represents different natural vibration modes; Ω is the drill string speed, rad / s; t is the time, s.

[0104] Step 5: Compare the steady-state response amplitudes of the rotary guide tool at different speeds to determine the speed corresponding to the maximum steady-state response. Figure 7 and Figure 8 It can be seen that the critical speeds in this example are 60 rpm and 90 rpm.

[0105] In summary,.

[0106] Although the embodiments of the present invention have been disclosed as above, they are not limited to the applications listed in the specification and the implementation modes, and they can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to the specific details and the illustrations shown and described herein.

Claims

1. A method for determining the critical speed of a drill string with a rotary steering tool, characterized in that: The following steps are involved: Step 1: Determine the combined structure of the drilling tool, the type and size of the PowerDrive, the rated push force of the push-type steerable tool, the size and weight of each component of the drill string, the elastic modulus of the drill string and the type of drill bit; Step 2: confirming the working mode of the rotary steerable tool, and determining the excitation force of the rotary steerable tool according to the working mode of the rotary steerable tool; Step 3: Calculate the natural frequency and natural vibration mode of the drill string with the rotary steerable tool according to the working mode of the rotary steerable tool; Step 4: Calculate the steady-state response of the drill string with the rotary steering tool at different rotation speeds according to the obtained natural frequency and natural vibration mode; Step 5: Compare the steady-state response amplitudes of the rotary guide tool at different speeds, determine the speed corresponding to the maximum steady-state response, and thereby determine the critical speed.

2. The method for determining the critical speed of a drill string with a rotary steering tool according to claim 1, characterized in that: In step 2, the working mode of the rotary steering tool is full-power steering mode or non-full-power steering mode. When the deviation between the actual controlled inclination and the designed well inclination is greater than 1°, it is in full-power steering mode, otherwise, non-full-power steering mode can be used.

3. The method for determining the critical speed of a drill string with a rotary steering tool according to claim 2, characterized in that: In full-power steering mode, the magnitude and direction of the bias force at the steering ribs of the PowerDrive system remain unchanged. When the drill string rotates one circle, the three steering ribs of the PowerDrive system extend in turn to "slap" the well wall, which will apply a periodic excitation force to the drill string, and the frequency is three times the frequency corresponding to the rotation speed; the excitation force is; Where, F is the excitation force, N; θ V is the initial phase of the excitation force, rad; F V is the maximum value of the "beating force", N; Ω is the drill string rotation speed, rad / s; t is the time, s.

4. The method for determining the critical speed of a drill string with a rotary steering tool according to claim 2, characterized in that: In non-full-power steering mode, the upper valve of the PowerDrive system swings. In the first half of the swing cycle of the upper valve, the rotation speed of the drill string and the rotation speed of the upper valve are in the same direction, and the time interval between two adjacent "flaps" of the steering rib increases. The excitation force is; Where, F is the excitation force, N; ω d is the swing speed of the upper valve, rad / s; θ V is the initial phase of the excitation force, rad; F V is the maximum value of "beating force", N; Ω is the drill string speed, rad / s; t is time, s; the range of t is: In the formula, θ d is the swing angle range of the upper valve of the PowerDrive system, rad; ω d is the swing speed of the upper valve, rad / s; k is the number of flapping times; the flapping direction is no longer fixed, and the eccentric azimuth of the flapping is: d P =θ d0 +oh d t (4) In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s; In the second half of the swing cycle of the upper valve, the rotation speed of the drill string and the upper valve are opposite, and the time interval between two adjacent "beats" decreases. Then: Where, F is the excitation force, N; F V is the maximum value of "beating force", N; Ω is the drill string speed, rad / s; ω d is the swing speed of the upper valve, rad / s; t is the time, s; θ V2 is the initial phase of the excitation force in the second half of the cycle, rad; The range of t is: The eccentric azimuth of "flapping" is: d P =θ d0 +θ d -oh d t (7) In the formula, δ p is the eccentric azimuth; θ d0 is the minimum swing angle of the upper valve, rad; ω d is the swing speed of the upper valve, rad / s; t is the time, s.

5. The method for determining the critical speed of a drill string with a rotary steering tool according to claim 3 or 4, characterized in that: In step 3, the formula for calculating the natural frequency and natural vibration mode of the drill string with a rotary steering tool is: (K-oh 2 M)ψ=0 (8) In the formula, M is the mass matrix; K is the stiffness matrix. According to formula (8), the natural frequency ω can be solved: i , Hz, and the natural vibration modes ψ corresponding to different natural frequencies i ; There are infinite solutions to equation (8), so i represents different solutions from small to large. First, the entire drill string is divided into 9m long units, and the Power-Drive is a separate unit. The mass matrix and stiffness matrix of each unit are: Where E is the elastic modulus, Pa; l is the unit length, m; A is the unit cross-sectional area, m 2 ; is the section inertia moment; for the whole drill string, the unit mass matrix and stiffness matrix can be superimposed according to the node position to form the overall mass matrix and stiffness matrix of the whole drill string; Then, the matrix is ​​inverted to directly obtain the different natural frequencies ω corresponding to equation (8): i , and the natural vibration modes ψ corresponding to different natural frequencies i .

6. The method for determining the critical speed of a drill string with a rotary steering tool according to claim 5, characterized in that: In step 4, the steady-state response of the drill string with a rotary steerable tool at different rotation speeds is calculated based on the obtained natural frequencies and natural vibration modes: Where U(t) is the steady-state response of the drill string; n is the order; r i is the frequency ratio, r i =γ / ω i ,γ is the drill bit excitation frequency;ζ i is the damping ratio, α, β are structural damping coefficients. For ultra-slender drill strings in drilling fluid, α = 0.1, β = 0.01 are generally used; θ i Phase angle at different frequencies, θ i =arctan[2ζ i r i / (1-r i 2 )];ψ i is the natural vibration mode; i represents different natural vibration modes; Ω is the drill string speed, rad / s; t is the time, s.

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