Regular Article
Research on kinematics analysis of spherical singlecone PDC compound bit and rock breaking simulation verification
^{1}
School of Mechanical Engineering, Xihua University, Chengdu, Sichuan 610039, PR China
^{2}
School of Mechatronic Engineering, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China
^{*} Corresponding author: kongcy@mail.xhu.edu.cn
Received:
16
April
2021
Accepted:
2
June
2021
The singlecone bit has become the first choice for slim hole sidetracking and deep well drilling with its unique rock breaking method and high ROP (Rate Of Penetration), with its main failure mode being of early excessive wear of the cutting teeth. In order to improve the adaptability of singlecone bits to hard and highly abrasive formations, a spherical singlecone Polycrystalline Diamond Compact (PDC) compound bit is designed. According to the characteristics of the tooth profile, the way of tooth arrangement and the way of contact between the cutting teeth and the rock, the acceleration equation to the cutting teeth of the spherical singlecone PDC compound bit is established. The acceleration of the singlecone bit is verified by numerical simulation experiment of rockbreaking. The shaft inclination angle of the cone, the position and height of the PDC teeth, the radius of the PDC teeth, the lateral rotation angle and the front inclination angle on the acceleration are studied. The results show that as the shaft inclination angle increases, the bit transmission ratio gradually increases, and the harder the rock formation, the larger the transmission ratio of the singlecone bit; the shaft inclination angle and the position of the PDC tooth have a greater influence on the acceleration of the PDC tooth, and the radius, lateral rotation angle and front inclination angle of the PDC tooth have a small influence on the acceleration of the PDC tooth; rock properties have an impact on the acceleration of the cutting teeth, with the acceleration of the cutting teeth in hard rock formations being higher than that in soft rock formations; near the top of the cone, the absolute acceleration of the cutting teeth will fluctuate sharply and cause severe wear of the cutting teeth, so the tooth distribution in this area should be strengthened; on the premise that the bearing life of the singlecone bit is allowed, the value of the shaft inclination angle β can be approached to 70°. The relative error between the theoretical analysis results of the acceleration of the PDC cutter and the rockbreaking simulation experiment results is between −0.95% and −2.24%. This research lays a theoretical foundation for the dynamic research of spherical singlecone PDC compound bit.
© C. Kong et al., published by IFP Energies nouvelles, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nomenclature
ρ : Radius vector, as shown in Figure 6, mm;
L : Distance from the center of the cone to the back cone plane of the cone, mm;
h _{C} : Position height of the center point C of the PDC cutting tooth surface, mm;
r _{C} : Radius of the characteristic circle where point C is located, mm;
α : Rotation angle of the cone relative to the cone shaft, degree;
α _{0} : Initial rotation angle of the cone relative to the cone shaft, degree;
x_{1}, y_{1}, z_{1}: Coordinate value of point Q in the coordinate system O_{1}X_{1}Y_{1}Z_{1}, mm;
r _{Q} : Radius of PDC cutter, mm;
β : Shaft inclination angle of singlecone bit, degree;
θ : Position angle of the drill bit within 0 ~ t, the position change caused by the rotation of the drill bit, degree;
θ _{0} : Extreme position angle of the origin O′ of the moving coordinate system, degree;
Z _{O′} : Vertical height of the moving coordinate origin O′, mm;
δ : Front inclination angle of PDC cutter, degree;
η : Lateral rotation angle of PDC cutter, degree;
γ : Angle between the baseline of the PDC tooth and the axis of the drill bit, degree;
σ _{0} : Uniaxial compressive strength of limestone, and σ_{0} = 124 MPa;
σ : Uniaxial compressive strength of any rock, MPa.
1 Introduction
The main drill bits currently used in drilling engineering are PDC bits and tricone bits. The singlecone bit is a cutting type drill bit between the tricone bit and the Polycrystalline Diamond Compact (PDC) bit [1], and the cutting teeth on the cone crush the rock by rolling, scraping, and twisting. With its unique rockbreaking method and high ROP (Rate Of Penetration), it has won the favor of the drilling industry and has become the preferred drill bit for slim hole sidetracking and deep well drilling [2–6]. Although the singlecone bit is a good choice for slim hole drilling because of its good formation adaptability, strong guiding ability and easy to be smallsized [1, 7], according to the field application tracking investigation and statistical analysis, the singlecone bit also has a fatal weakness, that is, the wear resistance of the cutters is seriously insufficient [1, 8, 9]. The spherical singlecone bits currently used in the oil field (Fig. 1) all use cemented carbide inserts (such as cone button teeth, wedge teeth, etc.), which can only be used in soft formations; For hard formations with high abrasiveness, the early wear is severe and the life span is short [10, 11]. This is because the singlecone bit will produce relatively large slip at the bottom of the well when it is working (Fig. 2), resulting in severe wear of the carbide cutting teeth. Therefore, the wear resistance of the cutting teeth will directly determine the working life of the singlecone bit. Once the cutters are dull, the drilling speed will decrease sharply. The main reason why the singlecone bit is less and less used is that the teeth are not wearresistant, which leads to the ROP decline and short service life of the bit. It is of great practical significance and development prospect to research and design singlecone bits with strong abrasion resistance and wider lithological adaptability [12, 13].
Fig. 1 Failed singlecone bit (a) worn cutting teeth (b) broken cutting teeth. 
Fig. 2 Bottom hole pattern of singlecone bit. 
Ma [14] founded the geometry and kinematics of modern roller cone bits. Wang et al. [15] and others [16] designed a specialshaped singlecone bit to increase the working life of the bit, but the specialshaped singlecone bit is not a fullhole drill, and the rockbreaking efficiency is still not high. Li [17] researched and designed a new type of impact pressin singlecone bit. After field application, it was found that the bit was worn seriously. Yu and Ding [18], Yu [19] and Deng et al. [20] studied the geometry, kinematics, and dynamics of the spherical singlecone bit, and discussed the design method of the tooth surface structure of the singlecone bit. Yu et al. [7] studied the influence of different geometric parameters and motion parameters for spherical singlecone bits on rockbreaking. However, many singlecone bit researchers have not conducted indepth and detailed studies on the acceleration of PDC cutting teeth for the special motion trajectory of singlecone bits.
In recent years, researchers have done a lot of research on bit technology. New bit technology is emerging, especially the improvement of bit performance brought by new bit structures. Baker Hughes Company has developed a new hybrid bit product by combining fixed PDC and unfixed cone. The special structure enables the bit to achieve good drilling results in some special formations and drilling requirements [21–25]. Smith bits of Schlumberger has successively launched 360° rotary PDC bit [26, 27] and conical diamond element Bits [28–30]. These two kinds of bit have good performance in some special difficult to drill formation. Chen et al. proposed a new technology of singlecone bit, which is composed of fixed and unfixed [31]. The research team led by Professor Yang of Southwest Petroleum University proposed a diamond bit with unfixed cutting structure through structural innovation [32, 33], and many useful research results have been obtained [34–36]. Compared with the existing cone bit and PDC bit, the new bit has obvious advantages in rock breaking efficiency and service life. The bit is expected to form an efficient rock breaking tool in the near future. The structure of the bit has a decisive influence on the adaptability and performance of the bit. With the structural innovation, new research on new drill bits and new tools will have new breakthroughs [37]. Sometimes, the new bit structure can play a role of surprise and drive the development of drilling technology.
In order to overcome the shortcomings of the prior art (Fig. 1), improve the wear resistance of the singlecone bit and prolong the service life of the bit, based on an indepth analysis of the structural characteristics and working principles of the singlecone bit, this paper adopts the design concept of the spherical singlecone bit + PDC bit and the principle that the PDC cutting teeth work slowly and alternately on the cone, with the design concept of spherical singlecone PDC compound bit proposed (Fig. 3). In order to make this new type of bit better serve the needs of oil drilling production, we need to conduct an indepth study on the kinematics and related parameters for the PDC teeth on the singlecone bit. However, due to the changes in the tooth profile characteristics and tooth arrangement method for the new drill bit, the characteristic points on the PDC cutting tooth edge can no longer be simplified to any point on the cone sphere. Moreover, PDC teeth have cutting angles such as front inclination angle and lateral rotation angles during the rockbreaking process. It is necessary to reestablish the kinematics equations of the cutting teeth of the spherical singlecone PDC compound bit to thoroughly grasp the acceleration characteristics of the new type of bit, which provides basic support and reference for the dynamic research, subsequent design, development and application of this kind of bit.
Fig. 3 Spherical singlecone PDC compound bit. 
2 Establishment coordinate system
As shown in Figure 4, when the spherical singlecone PDC compound bit is working, the movement of the cone includes the revolution of the bit body and cone around the center line of the bit, the movement along the direction of the center line of the bit, and the rotation of the cone around the cone axis. Any studies on the basic motions of cutters on cone shall be based on deep understanding of the features of bit in the coordinate system, basic plane and structure.
Fig. 4 The motion of spherical singlecone PDC compound bit. 
2.1 Static cylindrical coordinate system
Use the central axis of a singlecone bit as the coordinate axis OZ to establish a static/fixed cylindrical coordinate system (static coordinate system) [20], set the polar axis on a horizontal reference plane and in a certain direction. Thereafter, a point Q within the space can be represented by Q (ρ, θ, Z), as shown in Figure 5.
Fig. 5 Cylindrical coordinate system for bit. 
2.2 Dynamic cylindrical coordinate system
Set up a dynamic cylindrical coordinate system for a singlecone [19], with the origin O′ set at the bottom center point of cone, the vertical axis O′H′ coinciding with the axis of cone, and polar axes O′X′ and O′H′ perpendicular with each other. The plane defined by H′O′X′ is called as a polar axis plane of cone.
Figure 6 shows the projection relationships of a complex cylindrical coordinate system, including: (a) view from direction A for the back cone plane of cone; (b) front view of singlecone; and (c) top view in addition to (b). Figure 6 indicates the parameters for bit structure as (L, R, β), parameters for bit and cone positions as (θ_{0}, Z_{O′}, α_{0}), parameters for PDC cutter’s position on cone as (h_{C}, r_{C}, γ, δ, η), and parameters of point Q’s position on PDC cutter blade profile as (r_{Q}, φ).
Fig. 6 Geometry of spherical singlecone PDC compound bit. 
Set up four rectangular coordinate systems for PDC cutters and respectively from the center points of four cutters [38]. Set up a coordinate system O_{1}X_{1}Y_{1}Z_{1} for the position of PDC cutters, to make O_{1}Z_{1}//OZ and pointing to the same direction as OZ, and O_{1}X_{1} in a horizontal direction and perpendicular to OZ. Rotate O_{1}X_{1}Y_{1}Z_{1} around O_{1}Y_{1} by an angle γ (the included angle between the reference line of cutter and the axis of bit) in the clockwise direction to obtain a coordinate system O_{1}X_{2}Y_{2}Z_{2}, to make O_{1}X_{2} collinear with the reference line of cutter and pointing to the external of bit. Rotate O_{1}X_{2}Y_{2}Z_{2} around O_{1}X_{2} by η in the clockwise direction to obtain a coordinate system O_{1}X_{3}Y_{3}Z_{3}, to make O_{1}X_{3} collinear with the reference line of cutter and pointing to the direction against from the center line of bit (coinciding with O_{1}X_{3}). Rotate O_{1}X_{3}Y_{3}Z_{3} around O_{1}Z_{3} by δ in the counterclockwise direction to obtain a coordinate system O_{1}X_{4}Y_{4}Z_{4}, to make O_{1}Z_{4} perpendicular to the directional reference line of PDC cutter (coinciding with O_{1}Z_{3}).
In consideration of expressing all geometrical features of a cutter, such as points, lines and planes, define the parameter equations for the blade curves of the cutter as follows:
According to the principles for transformation of rectangular coordinate system, obtain the equations for the position coordinates for the blade profile of PDC cutter as follows:
3 Geometrical equations for describing motion of PDC cutters
During operation, the motion of a singlecone bit consists of rotation and displacement, which are for the convenience of study analyzed in separation before combination of them.
3.1 Rotation of cone
3.1.1 Relative rotation of cone around cone shaft
Assume that the bit does not rotate, and the cone rotates around the cone shaft. Define the center point on a surface of PDC cutter as the characteristic point C (ρ, θ, Z) and the initial position as α_{0}. After a time lapse t, point C moves to point C′ and arrives at a position angle α_{0} + α. Each view in Figure 6 includes the point C′ and its projection.
The true length is indicated as the distance between O and C′ to represent the radius vector of point C′ in the top view. OC′ constitutes the oblique side of the right triangle ON′C′, of which the right angle sides ON′ and N′C′ are as follows:
wherein:
Subject to the time lapse t, calculate the radius vector ρ_{Q} of a point Q on the blade profile of cutter can be expressed as:
3.1.2 Rotation of cone and cone shaft around the center axis of bit
Under the assumption that only the bit rotates, and the cone doesn’t rotate around the cone shaft, point C′ conducts fixedaxis rotation around the center axis of bit, and, after the time lapse t, point C′ moves to point C′′ after travelling over the angle θ, as shown in Figure 6. The true value is reflected in the top view for the polar angle θ_{C″} of the characteristic point C″, and θ_{C′′} is the sum of the polar angle of point O′ and ∠N′OC″, wherein:
Accordingly, subject to the time lapse t, calculate the polar angle θ_{Q} of a point Q on the blade profile of cutter can be expressed as:
3.2 Displacement of cone
During operation, the cone moves along the center axis of bit in addition to rotation. The true length is reflected in Figure 6b for the net vertical height Z_{C′} of the characteristic point C after the time t. Accordingly, subject to the time lapse t, calculate the vertical height Z_{Q} of a point Q on the blade profile of cutter can be expressed as:
Equations (6), (9) and (10) are kinematic geometric equations of the spherical singlecone PDC compound bit.
4 Acceleration equation of PDC tooth on singlecone
In order to facilitate the study of the acceleration of the PDC teeth on the cone, the rotation and movement of the cone are studied separately and combined.
The angular velocity of the bit body is ω_{1}, and the angular acceleration is ε_{1}.
The angular velocity of the cone around the cone axis is ω_{2}, and its angular acceleration is ε_{2}.
The up and down position of the bit body is Z_{O′}, and the acceleration is a_{Z}
ε_{1}, ε_{2} and a_{Z} are all random variables, and through experimental measurements, it can be seen how their instantaneous real values change with time.
The acceleration of a characteristic point on a singlecone PDC tooth, especially the acceleration when the cutting tooth impacts with the bottom of the hole, is an important parameter for rockbreaking and cutting tooth strength. However, the actual measurement of the acceleration and dynamic load of the cutting teeth is very troublesome and cannot be implemented on a large scale. It is obviously very useful if the mathematical relationship between the acceleration of the PDC teeth of the singlecone bit and the abovementioned measurable bit acceleration ε_{1}, ε_{2} and a_{Z} can be established.
In order to facilitate the study of the acceleration of the singlecone bit of the PDC tooth, the absolute movement of a certain characteristic point on the singlecone PDC tooth is regarded as the combination of the following three kinds of movements. That is, for the longitudinal implicated motion of point Q, its position variable is Z_{Q}, and its acceleration is ; for the tangentially implicated movement of point Q, its position variable is the polar angle θ_{Q}, and the acceleration is a_{Qτ}; for the radial relative movement of point Q, the position variable is the vector radius ρ_{Q}, and the acceleration is a_{Qρ}; the three moving directions are 90° with each other. Starting from the geometrical equation of motion of the singlecone bit, the acceleration equation of any point on the PDC tooth is established.
(1) Relative acceleration a _{ r }
The acceleration caused by the radial movement is the relative acceleration, with the relative acceleration being the derivative of the radial velocity with respect to time t.
Let
Then there is
(2) Tangentially implicated acceleration a _{ eτ }
Let
Then there is
(3) Implicated centripetal acceleration a _{ eρ }
The radially implicated centripetal acceleration caused by the rotation of the drill bit is:
(4) Longitudinal implicated acceleration a _{ eZ }
(5) Coriolis acceleration a _{ k }
The Coriolis acceleration caused by the change in the direction of the relative speed caused by the implicated rotation and the change in the magnitude of the implicated speed caused by the relative speed, expressed by the vector a_{k}. By the definition of Coriolis acceleration, its magnitude is:
wherein, Ψ is the angle between the relative speed and the angular velocity of the implicated rotation, Ψ = 90°; the direction of a_{k} is perpendicular to the plane defined by ω_{Q} and v_{Qρ}, pointing to the tangential direction.
Thus, the radial acceleration a_{Qρ}, the tangential acceleration a_{Qτ}, and the longitudinal acceleration a_{QZ} at point Q on the cutting tooth are obtained:
the radial acceleration a _{ Qρ }
the tangential acceleration a _{ Qτ }
the longitudinal acceleration a_{QZ}
so, the absolute acceleration a_{Q} at point Q on the cutting tooth is:
5 Calculation examples and analysis
5.1 Relationship between the transmission ratio of the singlecone bit and the rock
The relationship between the roller speed and the bit rotation speed is the transmission ratio. Through analyzing and summarizing the experimental data of the transmission ratio of the spherical singlecone bit in the literature [39], the empirical formula of the transmission ratio i can be obtained.
Three common rocks are selected: sandstone (σ = 33.44 MPa), limestone (σ = 124 MPa) and granite (σ = 236 MPa). According to the empirical formula of transmission ratio, the change trend of transmission ratio i of singlecone bit in three kinds of rocks with shaft inclination angle β can be obtained, as shown in Figure 7. It can be seen from the curve in the figure that as β increases, the bit transmission ratio i also gradually increases, and that the transmission ratio of a singlecone bit in granite is higher than that in sandstone, that is, the harder the rock, the higher the transmission ratio of the bit. Of course, the transmission ratio of a singlecone bit is not only related to the shaft inclination angle and rock characteristics, but also to the bit tooth surface structure. However, according to previous experimental studies, it can be known that the shaft inclination angle and rock characteristics are the main factors affecting the transmission ratio, so the empirical formula for transmission ratio has certain applicability to singlecone bits. The influence of the new tooth surface structure on the transmission ratio of the spherical singlecone PDC compound bit needs to be supplemented and improved in the followup experimental research.
Fig. 7 Change of transmission ratio at shaft inclination angle. 
5.2 Analysis of factors affecting acceleration
In order to further clarify the influence of each parameter of the PDC tooth singlecone bit on the acceleration, the acceleration of the PDC tooth singlecone bit is quantitatively analyzed and studied in combination with some parameters given in reference [20], with the given parameters being L = 30 mm, R = 50 mm, h_{C} = 40 mm, β = 60°, α = 30°, η = 5°, δ = 38°, r_{Q} = 6.72 mm, ω_{1} = 60 r/min, and i = 0.748 (limestone). The diameter of the spherical singlecone PDC compound bit is 118 mm. The front inclination angle δ, the radius r_{Q} of the PDC tooth, the lateral rotation angle η, the shaft inclination angle β, the position height h_{C} of the PDC tooth, and the rock strength are selected as independent variables. The above basic parameters are substituted into equations (27)–(30) for numerical calculation.
The acceleration simulation calculation result of the spherical singlecone PDC compound bit is shown in Figures 8–12. It can be seen from Figure 8 that the tangential acceleration a_{Qτ} and the absolute acceleration a_{Q} increase with the increase of the front inclination angle δ, and that the radial acceleration a_{Qρ} decreases slightly with the increase of δ; the longitudinal acceleration a_{QZ} has nothing to do with δ. It can be seen from Figure 9 that a_{Qρ}, a_{Qτ} and a_{Q} all increase slightly with the increase of the radius r_{Q} of the PDC tooth, but the amplitude of the change is small; a_{QZ} has nothing to do with r_{Q}. It can be seen from Figure 10 that a_{Qρ}, a_{Qτ} and a_{Q} all decrease slowly with the increase of the lateral rotation angle η; a_{QZ} has nothing to do with η. It can be seen from Figure 11 that the shaft inclination angle β has a relatively large impact on acceleration, where a_{Qρ} decreases as β increases, and that both a_{Qτ} and a_{QZ} first increase and then decrease with β increase; as β increases, a_{Q} first decreases, then remains unchanged, and finally decreases. It can be seen from Figure 12 that the position height h_{C} of the PDC tooth also has a relatively large impact on the acceleration, and between 0 and 65 mm, a_{Qρ}, a_{Qτ} and a_{Q} all increase as h_{C} increases; between 65 and 80 mm, a_{Qτ} decreases sharply with the increase of h_{C} and approaches zero, a_{Qρ} and a_{Q} increase greatly with the increase of h_{C}, and decrease sharply when they are close to the top of the cone; a_{QZ} has nothing to do with h_{C}.
Fig. 8 Effect of front inclination angle on acceleration. 
Fig. 9 Effect of PDC cutter radius on acceleration. 
Fig. 10 Effect of lateral rotation angle on acceleration. 
Fig. 11 Effect of shaft inclination angle on acceleration. 
Fig. 12 Effect of PDC cutter position on acceleration. 
Sandstone (σ = 33.44 MPa), limestone (σ = 124 MPa) and granite (σ = 236 MPa), and combine equation (31) are selected to quantitatively analyze and calculate the absolute acceleration of the PDC teeth of the singlecone bit, with the results shown in Figures 13–15. The analysis results show that because rock formations of different hardness will cause changes in the bit transmission ratio, the rock formations also have a certain influence on the acceleration of the cutting teeth, and that the acceleration of the cutting teeth in sandstone is slightly lower than that in granite; the absolute acceleration of the cutting teeth at most positions on the cone in different rock formations tends to be gentle with the increase of β. However, near the top of the cone, the absolute acceleration of the cutting teeth will fluctuate violently and cause severe wear of the cutting teeth; the greater β, the closer the absolute acceleration fluctuation position is to the top of the cone, and the smaller the fluctuation amplitude. Therefore, the tooth placement in the area to the right of the dotted line in Figures 13–15 require special attention from singlecone bit researchers. At the same time, the larger β, the smaller the absolute acceleration, indicating that the inertial force of the drill bit is smaller and that it is more beneficial to the strength of the PDC tooth; however, considering the wear of the roller bearing [40, 41], the shaft inclination angle β of the singlecone bit cannot be designed too large.
Fig. 13 Relationship between the absolute acceleration and h_{C} (sandstone). 
Fig. 14 Relationship between the absolute acceleration and h_{C} (limestone). 
Fig. 15 Relationship between the absolute acceleration and h_{C} (granite). 
The bit design should try to achieve the goal of equal life, that is, the life of each component of the cone bit is almost the same. For a singlecone bit, it mainly means that the roller bearing and the bit cutting teeth reach the same life. Therefore, under the premise that the bearing life of the singlecone bit is allowed, the shaft inclination angle β can be approached to 70°.
5.3 Multifactor composite analysis of acceleration
According to Figures 13–15, it is known that among the various factors affecting the acceleration of the cutting teeth of the spherical singlecone PDC compound bit, the shaft inclination angle and the position height of the PDC teeth have the greatest impact on the acceleration. Therefore, in order to comprehensively investigate the influence trend of β and h_{C} of the PDC tooth on the acceleration, β and h_{C} are used as independent variables at the same time in analyzing and calculating the radial acceleration a_{Qρ}, tangential acceleration a_{Qτ}, longitudinal acceleration a_{QZ} and absolute acceleration a_{Q} in the limestone with these two independent variables, as shown in Figures 16–19.
Fig. 16 Change of radial acceleration along with h_{C} and β. 
Fig. 17 Change of tangential acceleration along with h_{C} and β. 
Fig. 18 Change of longitudinal acceleration along with h_{C} and β. 
Fig. 19 Change of absolute acceleration along with h_{C} and β. 
It can be seen from Figures 16–19 that when the position height h_{C} of the PDC teeth and the shaft inclination angle β increase at the same time, the radial acceleration a_{Qρ} first decreases and then tends to be gentle. When β = 40°~60°, the radial acceleration a_{Qρ} change law tends to be stable as a whole, and slowly increases with the increase of h_{C}. The tangential acceleration a_{Qτ} first increases and then slowly decreases with the increase of β and h_{C}. When β = 30°~40°, the tangential acceleration a_{Qτ} reaches the maximum value, and the maximum value is close to the top area of the cone. The longitudinal acceleration a_{QZ} first increases and then decreases with the increase of β and h_{C}. When β = 40°, the longitudinal acceleration a_{QZ} of the PDC teeth in the middle of the cone reaches the maximum. The absolute acceleration a_{Q} decreases first and then stabilizes with the increase of β and h_{C}. When β and h_{C} take the minimum at the same time, the absolute acceleration a_{Q} reaches the maximum; when β takes the maximum, the absolute acceleration a_{Q} of the PDC tooth at the end of the cone reaches the minimum.
In order to verify the theoretical analysis results of the acceleration of the spherical singlecone PDC compound bit in this paper, the nonlinear dynamical model is established to simulate the dynamic rock breaking process of the spherical singlecone PDC compound bit on basis of which the acceleration state of the bit element during the rock breaking process is simulated. It provides a theoretical basis for the application and design of the spherical singlecone PDC compound bit.
6 Rockbreaking simulation experiment
In the actual drilling process, there are many factors that affect the rockbreaking of the drill bit. In the process of numerical simulation of the rockbreaking of the drill bit using the finite element method, it is impossible to consider the influence of all factors on the simulation results. The specific problems and research goals should be addressed. The simulation model is appropriately simplified and necessary assumptions are made. To facilitate the analysis the drilling process of the spherical singlecone PDC compound bit, the following assumptions are taken:
The palm and cone of the drill bit are regarded as rigid bodies.

The influences of temperature, confining pressure, and drilling fluid are neglected.
The bottom of the well is kept clean at all times, that is, the rock grid unit will be automatically deleted after failure, ignoring its impact on the subsequent cutting.
The rock is continuous isotropic medium, ignoring the effects of initial cracks and internal pressure.
The axis of the drill bit coincides with the axis of the borehole, ignoring the possible radial displacement of the drill bit.
6.1 FiniteElement model of the drill bitrock system
The PDC bit exhibits highly nonlinear characteristics during the rockbreaking process, including geometric nonlinearity, material nonlinearity and contact nonlinearity. By adopting the finite element method, treat the spatial domain of the cutterrock contacting system at time t as Ω, and the body force, boundary stress and Cauchy’s stress respectively as b, r, r_{C} and σ_{C}, then the contacting issue could be represented as [42, 43]:
wherein, Γ_{f} is the border for a given boundary force, Γ_{c} is the contact boundary, δ_{u} is the virtual displacement, δ_{e} is the virtual strain, ρ is the density, a_{1} represents the acceleration. By discretizing the spatial domain Ω with finite element method, one can obtain the following equation:
wherein, M is the mass matrix, ϋ is the acceleration vector, t is the time variable, p is the external force vector, c is the contact force and friction force vector, h is the internal stress vector, u is the object displacement, ξ is the variable associated with contact surface characteristics, and λ represents the variable associated with constitutive relation of materials.
Considering that a proper plastic constitutive model is the key of accurately simulating the yielding, hardening and damaging process, and that bottomhole rock is a kind of granular material so that the rock element will be expanded when suffering shear force, DruckerPrager model [36, 44] is adopted in this paper.
According to the above theory, combined with the bit structure designed in Figure 3, with the given structural parameters being L = 30 mm, R = 50 mm, β = 70°, η = 5°, δ = 38°, r_{Q} = 6.72 mm, the nonlinear dynamic finite element model of drill bitrock system was established in ABAQUS program, as shown in Figure 20a. According to the SaintVenant principle, the size of the rock is 220 mm in diameter and 100 mm in height. To reduce the time of bottom hole formation and make all the cutter interact with the rock earlier, a precontact bottom hole is set on the surface of the rock model according to the size of the spherical singlecone bit, as shown in Figure 20b. The 8node reduced integration element C3D8R with high accuracy, robustness and hourglass control was employed to discrete the rock model, and the rock’s area nearby the cutter was finely meshed, the grid size is about 2 mm. The rockbreaking simulation model was totally divided into 912 396 elements. And the system of unit (mmNs) was applied in the numerical model, while for convenience, results were displayed according to the engineering conventions. The mechanical parameters of the material of the drill bit are listed in Table 1. The mechanical parameters of rock material are listed in Table 2.
Fig. 20 Finite element model of the drill bitrock system (a) Overall model, (b) Rock model. 
Mechanical parameters of each part of the drill bit.
As shown in Figure 21, the overall coordinate system of the spherical singlecone PDC compound bit is in the same direction as the global coordinate system. The zaxis of the singlecone bit coincides with the rock center axis, and the positive direction of the zaxis is the longitudinal movement direction of the singlecone bit, the xaxis is horizontal and perpendicular to the zaxis, that is, the xoz plane is parallel to the surface of the rock model.
Fig. 21 Boundary setting of rockbreaking simulation. 
The rockbreaking simulation adopts ABAQUS/Explicit analysis method. Nonreflecting boundary and fixed constraint were applied to the rock surfaces except the top one. The contact type between the cutter and rock was eroding surface to surface. Considering the friction between the cutting surface and rock, the friction coefficient of contact surfaces was set to 0.4 [45]. The simulation loading process is divided into two steps: In the first step, the Weight On Bit (WOB) is 100 kg, which is applied to the upper surface of the bit leg to keep the cutter in contact with the rock. In the second step, the WOB is 2000 kg, which is applied to the upper surface of the bit leg (Fig. 21); at the same time, the rotation speed applied to the center axis of the bit is 60 r/min, so that the singlecone bit rotates at a uniform speed around the center axis of the rock model. Set the simulation time to 6 seconds. The singlecone bit designed in Figure 3 has 25 PDC cutters distributed. Due to the large number of cutting teeth, according to the distribution of PDC cutter on the surface of the cone, seven representative PDC cutters are selected for research, and their distribution positions on the cone are shown in Figure 22. The position height h_{C} of the seven PDC cutters are shown in Table 3.
Fig. 22 Number of cutters. 
Position parameters of PDC cutter.
6.2 Results and discussion
The equivalent plastic strain cloud diagram of the rock in the simulation of rockbreaking with a spherical singlecone PDC compound bit within 1–6 s is shown in Figure 23. It can be seen from Figure 23 that as the working time of the spherical singlecone PDC compound bit increases, a complete spherical bottom hole is gradually formed in the center of the rock model.
Fig. 23 Equivalent plastic strain in rockbreaking process. 
During the rockbreaking process, the overall longitudinal velocity and acceleration of the drill bit (in the direction of the borehole axis) vary with the drilling time as shown in Figures 24 and 25. It can be seen from Figure 24 that the longitudinal movement speed of the bit fluctuates in the range of −5 to 25 mm/s. Figure 25 shows that during the rockbreaking simulation of a singlecone bit, the acceleration of the bit fluctuates in the range of −2500~2500 mm/s^{2}, and the acceleration of most of the time is between −1000~1000 mm/s^{2}. The fluctuation of the speed and acceleration is due to the interaction between the bit and the rock during the rockbreaking process of the singlecone bit, which causes the bit to be accompanied by axial vibration during the drilling process. Therefore, the acceleration of the drill bit has a large extreme value at a certain moment, which is related to the unique rockbreaking method of the singlecone bit.
Fig. 24 Longitudinal moving speed of bit. 
Fig. 25 Longitudinal moving acceleration of bit. 
The acceleration simulation data of these seven PDC cutters is shown in Figure 26. It can be seen from Figure 26 that as the drilling time increases, the acceleration of the PDC cutter at each position on the surface of the cone gradually increases. After 1 s, the acceleration value suddenly increases, and the acceleration of the seven PDC cutters reaches the maximum when approaching 1.5 s. After 1.5 s, the acceleration of each PDC tooth is in a small fluctuation, but the overall trend is stable. The reason for the acceleration fluctuation in the simulation experiment is that the cone rotates around the cone axis during the drilling process of the singlecone bit, and generates rolling rockbreaking, which in turn causes the bit to vibrate, and ultimately causes the acceleration of the PDC cutter to fluctuate. The acceleration of the 4# tooth is greater than that of the 1# tooth, which is consistent with the analysis results of Figures 12 and 14. According to the acceleration fluctuation curve after 1.5 s in Figure 26, the average curve of each PDC tooth is made, and the theoretical calculation value in Figure 14 is compared with the simulation experiment average value in Figure 26 (Tab. 4). Table 4 shows that the acceleration values of the PDC cutter obtained through the simulation experiment are all greater than the theoretical calculation values, and the fractional error between the theoretical analysis results of the acceleration of the PDC cutter and the simulation experiment results is between −0.95% and −2.24%. This shows that this paper is advisable to study the acceleration theory of the spherical singlecone PDC compound bit.
Fig. 26 Absolute acceleration of PDC cutters. 
Theoretical and simulated values of PDC cutter acceleration.
7 Conclusion
In this paper, the acceleration equation of the spherical singlecone PDC compound bit is established; sandstone, limestone and granite are selected respectively, and the acceleration of the spherical singlecone PDC compound bit is quantitatively analyzed and calculated; the acceleration of the singlecone bit is verified by numerical simulation experiment of rockbreaking, with the following conclusions drawn.
The bit transmission ratio i increases as the shaft inclination angle β increases, and the harder the rock formation, the larger the singlecone bit transmission ratio i; the rock formation has an influence on the acceleration of the cutting teeth, and the acceleration of the cutting teeth in the hard rock layer is higher than that in the soft rock layer.
The shaft inclination angle and the position height of the PDC teeth have a greater impact on the acceleration of the PDC teeth, and close to the top of the cone, the absolute acceleration of the cutting teeth will fluctuate sharply and cause severe wear of the cutting teeth; the greater β, the closer the absolute acceleration fluctuation position is to the top of the cone, and the smaller the fluctuation amplitude; the absolute acceleration decreases as β increases, which means that the inertial force of the drill bit is smaller; however, considering the wear of the cone bearing, the shaft inclination angle β of the singlecone bit cannot be designed too large; on the premise that the bearing life of the singlecone bit is allowed, the value of β can be approached to 70°.
The relative error between the theoretical analysis results of the acceleration of the PDC cutter and the rockbreaking simulation experiment results is between −0.95% and −2.24%. This shows that this paper is advisable to study the acceleration theory of the spherical singlecone PDC compound bit.
Acknowledgments
This research work was supported by the Key Scientific Research Fund of Xihua University (No. Z171190303); Research Project of Key Laboratory Machninery and Power Machinery (Xihua University), Ministry of Education.
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All Tables
All Figures
Fig. 1 Failed singlecone bit (a) worn cutting teeth (b) broken cutting teeth. 

In the text 
Fig. 2 Bottom hole pattern of singlecone bit. 

In the text 
Fig. 3 Spherical singlecone PDC compound bit. 

In the text 
Fig. 4 The motion of spherical singlecone PDC compound bit. 

In the text 
Fig. 5 Cylindrical coordinate system for bit. 

In the text 
Fig. 6 Geometry of spherical singlecone PDC compound bit. 

In the text 
Fig. 7 Change of transmission ratio at shaft inclination angle. 

In the text 
Fig. 8 Effect of front inclination angle on acceleration. 

In the text 
Fig. 9 Effect of PDC cutter radius on acceleration. 

In the text 
Fig. 10 Effect of lateral rotation angle on acceleration. 

In the text 
Fig. 11 Effect of shaft inclination angle on acceleration. 

In the text 
Fig. 12 Effect of PDC cutter position on acceleration. 

In the text 
Fig. 13 Relationship between the absolute acceleration and h_{C} (sandstone). 

In the text 
Fig. 14 Relationship between the absolute acceleration and h_{C} (limestone). 

In the text 
Fig. 15 Relationship between the absolute acceleration and h_{C} (granite). 

In the text 
Fig. 16 Change of radial acceleration along with h_{C} and β. 

In the text 
Fig. 17 Change of tangential acceleration along with h_{C} and β. 

In the text 
Fig. 18 Change of longitudinal acceleration along with h_{C} and β. 

In the text 
Fig. 19 Change of absolute acceleration along with h_{C} and β. 

In the text 
Fig. 20 Finite element model of the drill bitrock system (a) Overall model, (b) Rock model. 

In the text 
Fig. 21 Boundary setting of rockbreaking simulation. 

In the text 
Fig. 22 Number of cutters. 

In the text 
Fig. 23 Equivalent plastic strain in rockbreaking process. 

In the text 
Fig. 24 Longitudinal moving speed of bit. 

In the text 
Fig. 25 Longitudinal moving acceleration of bit. 

In the text 
Fig. 26 Absolute acceleration of PDC cutters. 

In the text 