Study on kinematic characteristics of the vertical-wheel on hybrid PDC bit

Abstract

Aiming at improving the ROP of PDC bits in hard-to-drill formation, a new hybrid PDC bit with vertical-wheel is proposed in this paper, furtherly, the kinematic characteristics of the vertical-wheel are researched. Firstly, the coordinate system of the vertical-wheel is established, on basis of which the motion equation, the speed equation and the acceleration equation of the vertical-wheel is derived. Secondly, a series of unit rock-breaking experiments on the vertical-wheel with variable parameters are carried out, so that the average speed of vertical-wheel under different conditions is obtained and the theoretical model of average speed is established, furthermore, the influence of diameter and normal angle of the wheel on the average speed is analyzed. Thirdly, the speed of vertical-wheel under different conditions is measured on basis of the speed model with compensation coefficient, specifically, the calculated average speed of the Ø18 mm vertical-wheel according to the instantaneous speed model is 4.532 r/s, while the average speed obtained in experiment is 4.491 r/s, with an error of 0.9%, proving that the measuring method and modeling of the instantaneous speed are credible. Lastly, the theoretical conditions for continuous rotation of the wheel are obtained. The research results are of great significance to study the working mechanism of the vertical-wheel and provide technical support for the application of the hybrid PDC bit with vertical-wheels.

Introduction to the vertical-wheel hybrid PDC bit

Since most undeveloped oil and gas resources in China are buried in deep strata,^1–3 and steering drilling (including horizontal well drilling) technology can greatly improve oil and gas production and reduce the comprehensive cost of oil and gas development, deep and ultra-deep well drilling and directional well drilling are the key technologies for oil and gas resource development in the future. In deep or ultra-deep formation, the effects of overburden pressure and fluid pressure increase the rock strength and make the bottom-hole rock hard to drill. In addition, due to insufficient hydraulic energy and long tripping time, low drilling efficiency and high drilling cost become problems that need to be solved urgently.^4–8 In steering drilling, the rock-breaking behavior of the bit in the bottom-hole is much more complex than that of conventional drilling.^9–12

PDC (Polycrystalline Diamond Compact) bit is the most widely used drill bit in oil and gas drilling, and its drilling footage has reached more than 94% of the total drilling footage in the world. ¹³ Various and complex drilling conditions has posed more and severe challenges to PDC bit technology. In spite of improving the performance of PDC cutters, reducing and stabilizing the torque of the bit is also necessary, which including improving the anti-stick-slip performance and developing steering drilling performance of the bit. Moreover, it should be noticed that buffer protection for PDC cutters can greatly improve the service life of cutters, and that auxiliary cutting structure for pre-rock-breaking can prolong the service life of cutters.

On basis of these ideas, Baker Hughes has developed the Kymera^TM hybrid PDC bits, ¹⁴ including the Kymera FSR series, as shown in Figure 1 (left), and the Kymera XTreme series, as shown in Figure 1 (right). Each Kymera hybrid PDC bit consists several fixed wing-blades and a plurality of rotatable cones. Field application shows that the combination of these two rock-breaking structures can restrain the stick-slip behavior and develop the steering drilling performance for the bit in complex drilling conditions. On the other hand, Halliburton Company has invented the Cruzer^TM PDC bits, ¹⁵ as shown in Figure 2, each Cruzer PDC bit is configured with several small rollers behind the PDC cutters. Although the supporting structure and the bearing system of the roller tend to break easily, these rollers can theoretically improve the steering drilling performance of the bit and protect cutters by restricting the cutting depth. The application results shows that the hybrid structure has improved the drilling speed of the bit and prolonged the service life of the cutters thereon.

Figure 1.

Kymera FSR (left) and Xtreme (right) hybrid PDC bits of Baker Hughes.

Figure 2.

Cruzer hybrid PDC bit of Halliburton.

These researches have innovated the structures of the PDC bit. The vertical wheel, as mentioned above, is generally a cylindrical rock-breaking structure configured with a circle of teeth in the crown and with integrated shafts thereunder. By setting a series of vertical wheels on the wing-blades of the PDC bit, a new vertical-wheel hybrid PDC bit is formed, as shown in Figure 3. The vertical wheel (hereinafter referred to as wheel), on one hand, rotates around its own axis, and simultaneously revolves around the bit axis, thus forming a compound motion which is a little bit similar to the cone in a tri-cone bit, but the actual motion behavior of the wheel is quite different from the cone. The vertical wheel is disposed on the drill bit as a multifunction rock-breaking structure. First, it is an auxiliary rock-breaking structure that pre-fracture bottom-hole rock for the PDC cutters to improve penetrating capacity of the cutters; Second, friction and cutting resistance from the rock may result in fluctuation on the instantaneous speed, but if the bearing system and other geometric parameters (size, location and direction etc.) are well designed, the rotation of the wheel will not be stopped, thus lowering the torque response and reduces the stick-slip behavior of the bit, which is beneficial to the working capacity of PDC bit in steering drilling; Third, it is also a buffer for PDC cutters that protecting the cutters from impact load and WOB fluctuation, thus prolonging service life of the cutters.

Figure 3.

Vertical wheel hybrid PDC bit.

The vertical-wheel hybrid PDC bit is an organic combination of two sets of cutting structures with different motion modes, that is, the fixed cutting structure (the wing-blades and the PDC cutters fixed thereon) and the vertical-wheel. The kinematic mechanism of fixed cutting structure can be obtained through the existing literature.^16–19 Since the vertical-wheel is a new structure, it is necessary to study the kinematic characteristics of it so that the rock-breaking mechanism of the hybrid PDC bit can be revealed.

Kinematics theory of the vertical-wheel

Coordinates of the vertical-wheel

Since the wheel rotates along its own axis when the bit body revolves, the motion of the wheel is a compound motion, so that the coordinate system of the wheel should be established as a compound system, as shown in Figure 4, wherein, Figure 4(d) briefly shows the relationship of spatial positions between the bit and the wheel, Figure 4(a) is a detailed upward view of Figure 4(d), so that the bit rotates clockwise in this view. Figure 4(c) is aligned with Figure 4(a) and shows the local rotating coordinate system of the wheel together with Figure 4(b). In order to clarify the coordinate systems, points, angles, and line segments in these figures are marked in blue, the rotating axis of the wheel is marked in red and the other elements are black.

Figure 4.

Coordinate systems of the bit body and the wheel.

First is the coordinate system of the bit, which is defined as a cylindrical-coordinate system. As shown in Figure 4(d), point O is a certain point on the revolving axis of the bit, it is taken as the origin of the coordinate system, and the plane passes through the point and simultaneously perpendicular to the bit axis is defined as the reference plane of the bit. By taking axis OH (which is aligned with the bit revolving axis) as the longitudinal axis, OX (which is a certain radial axis perpendicular to the longitudinal axis) as the polar axis and $\theta$ as the azimuth, the cylindrical-coordinate system of the bit is established.

On the other hand, since the wheel will rotate along its own axis when the bit revolves, it is necessary to establish a rotating system for the points on it. As shown in Figure 4(b) and (c), take the vertex point Q( $r_Q,{\psi }_Q,h_Q$ ) of a certain tooth as a feature point of the wheel, obviously, the vertex points of all the teeth on the wheel determine a certain circle in a certain plane. The circle, of which the center point is O_Q, is called the teeth-disposing circle, and the plane is accordingly called the teeth-disposing plane. If take point O_W as the center point of the shaft root (i.e. the back end of the wheel shaft, the subscript _W is short for Wheel), then line O_WO_Q will be the rotating axis of the wheel and the line will obviously be perpendicular to the teeth-disposing plane. As shown in Figure 4(a) and (d), if the vertical wheel is designed with offset s, line O_WO_Q and axis OH will be skew lines. If a plane passes through line O_WO_Q and is parallel to axis OH, the plane is called the polar plane of the wheel (hereinafter referred to as the wheel polar plane). In the wheel polar plane, by taking O_W as the origin of the rotating system, O_WH_W (which is aligned with the wheel rotating axis) as the longitudinal axis, O_WX_W (which is perpendicular to the longitudinal axis on the wheel polar plane) as the polar axis and ${\psi }_Q$ as the azimuth, the cylindrical-coordinate system of the wheel is established.

Additionally, in the rotating coordinate system of the wheel, point O′ is the intersection point of line O_WO_Q and the reference plane, and axis O’H′ is parallel to the axis OH, the angle between O_WO_Q and O′H′ is the normal angle of the wheel, as shown in Figure 4(c). Considering that point O can move along the bit axis, if the distance between O and O′ equals to the offset s, axis O′H′ must be a vertical projection of axis OH on the wheel polar plane. As shown in Figure 4(a) and (d), line OX₀ is a line parallel to the wheel polar plane, if the plane X₀OH is taken as the feature plane of bit revolving, then the angle ${\theta }_0$ can be used to represent the angular displacement of the bit in the bit coordinate system. Furtherly, the coordinates of point O_W is used to describe the position of the wheel in the bit coordinate system, which comprises the radius c (i.e. the distance between point O and point O_W in the reference plane), the azimuth ${\theta }_W$ and the height h_W, as shown in Figure 4(a) and (c). Similarly, the coordinates of point O_Q in the wheel coordinate system can be represented as $r_Q$ , ${\psi }_Q$ , and $h_Q$ , as shown in Figure 4(b) and (c). Technically, the distance c is called the arm of the shaft-root center, and c₀ is the vertical projection of it on the wheel polar plane. The definition of $c_Q$ is similar to $c_0$ .

According to Professor Ma Dekun’s theory, ²⁰ the position or coordinates of point Q in the bit coordinate system can be expressed as:

$$\{\begin{array}{c}{\rho }_Q=\sqrt{{(c_Q+h_Q\cos \alpha \cos {\psi }_Q)}^2+{(r_Q\sin {\psi }_Q-s)}^2}\hfill \\ \multicolumn{1}{c}{{\theta }_Q={\theta }_W-\arctan \left(\frac{s}{c_0}\right)-\arctan \left(\frac{r_Q\sin {\psi }_Q-s}{c_Q+h_Q\cos \alpha \cos {\psi }_Q}\right)}\\ \multicolumn{1}{c}{H_Q=h_W-h_Q\sin \alpha \cos {\psi }_Q-h_Q\cos \alpha }\end{array}$$

(1)

Wherein, $H_Q$ is the vertical height of point in the bit coordinate system, which should be distinguished from $h_Q$ , and

$$c_Q=c_0-h_Q\sin \alpha$$

$${\theta }_W={\theta }_0-\frac{2\pi (i-1)}{3}+\arctan \left(\frac{s}{c_0}\right)$$

Speed equation of vertical wheel

According to the radius vector ${\rho }_Q$ , azimuth ${\theta }_Q$ and the vertical height $H_Q$ in equation (1), the position of the certain point Q on the wheel can be obtained. On the other hand, the motion speed of a certain point Q can be represented by three independent partial velocities, namely, the radial, tangential and axial speed components:

$$\overrightarrow{V_Q}=\overrightarrow{V_{Q\rho }}+\overrightarrow{V_{\mathrm{Qt}}}+\overrightarrow{V_{\mathrm{QH}}}$$

(2)

Apparently, the derivative of radius vector of point Q with respect to time is the radial component $\overrightarrow{V_{Q\rho }}$ , radius vector and the derivative of polar angle ${\theta }_Q$ with respect to time represent the tangential component $\overrightarrow{V_{\mathrm{Qt}}}$ , and the derivative of vertical height $H_Q$ to time is the axial component $\overrightarrow{V_{\mathrm{QH}}}$ . Thus, the speed equation of point Q can be expressed as:

$$\{\begin{array}{c}{V}_{Q\rho }=\frac{d{\rho }_Q}{\mathrm{dt}}={\lambda }_{Q\rho }{r}_Q{\omega }_W\hfill \\ \multicolumn{1}{c}{V_{\mathrm{Qt}}={\rho }_Q\frac{d{\theta }_Q}{\mathrm{dt}}={\rho }_Q{\omega }_b-{\lambda }_{\mathrm{Qt}}{r}_Q{\omega }_W}\\ \multicolumn{1}{c}{V_{\mathrm{QH}}=\frac{d{H}_Q}{\mathrm{dt}}=V_{\mathrm{bH}}+r_Q{\omega }_W\sin \alpha \sin {\psi }_Q}\end{array}$$

(3)

$$V_Q=\sqrt{{V_{Q\rho }}^2+{V_{\mathrm{Qt}}}^2+{V_{\mathrm{QH}}}^2}$$

(4)

Wherein,

$${\lambda }_{Q\rho }=\frac{0.5r_Q{\sin }^2\alpha \sin 2{\psi }_Q-(c_0-h_Q\sin \alpha )\cos \alpha \sin {\psi }_Q-s\cos {\psi }_Q}{\sqrt{{(r_Q\sin {\psi }_Q-S)}^2+{(c_0-h_Q\sin \alpha +r_Q\cos \alpha \cos {\psi }_Q)}^2}}$$

$${\lambda }_{\mathrm{Qt}}=\frac{(c_0-h_Q\sin \alpha )\cos {\mu }_Q+r_Q\cos \alpha -s\cos \alpha \sin {\psi }_Q}{\sqrt{{(r_Q\sin {\psi }_Q-s)}^2+{(c_0-h_Q\sin \alpha +r_Q\cos \alpha \cos {\psi }_Q)}^2}}$$

Wherein, $V_{\mathrm{bH}}$ is the axial feeding speed of the bit, namely, the ROP (rate of penetration) of the bit, mm/s; ${\omega }_W$ is the rotation speed of the wheel, rad/s; ${\omega }_b$ is the revolution speed of the bit, rad/s.

Acceleration equation of vertical wheel

The speed components of V_Q obtained above is perpendicular to each other in the axial, radial and tangential directions, and their respective acceleration and Coriolis acceleration are homogeneous in these three directions. Therefore, the acceleration of point Q can be divided into three acceleration components: axial $a_{\mathrm{QH}}$ , radial $a_{Q\rho }$ , and tangential $a_{\mathrm{Qt}}$ .

The axial acceleration $a_{\mathrm{QH}}$ of point Q has only one term, that is, the derivative of axial speed component $\overrightarrow{V_{\mathrm{QH}}}$ to time:

$$\begin{array}{c}\multicolumn{1}{c}{a_{\mathrm{QH}}=\frac{d{V}_{\mathrm{QH}}}{\mathrm{dt}}}\\ =\frac{d^2{\rho }_Q}{d{t}^2}=a_b+r_Q\sin \alpha (\sin {\psi }_Q{ϵ}_b+\cos {\psi }_Q{({\omega }_W)}^2)\hfill \end{array}$$

(5)

Wherein, $a_b$ is the axial acceleration of the bit, ${ϵ}_b$ is the revolution acceleration of the bit. which can be represented as:

$${ϵ}_b=\frac{d{\omega }_b}{\mathrm{dt}}=\frac{d^2\theta }{d{t}^2}$$

(6)

The radial acceleration $a_{Q\rho }$ includes two parts, one is the acceleration $a_{\rho }$ caused by the radial movement of point Q, which is the derivative of radial speed component to time, and the other is the centripetal acceleration $a_r$ caused by the rotation of point Q:

$$a_{\rho }=\frac{d{V}_{Q\rho }}{\mathrm{dt}}=\frac{d^2{\rho }_Q}{d{t}^2}$$

(7)

$$a_r=-{\rho }_Q{\left(\frac{d^2{\rho }_Q}{d{t}^2}\right)}^2$$

(8)

Then the radial acceleration $a_{Q\rho }$ can be expressed as:

$$a_{Q\rho }=\frac{A{r}_Q}{{\rho }_Q}-B{\rho }_Q$$

(9)

Where

$$A=(E-\mathrm{Scos}{\psi }_Q){ϵ}_W+F({\omega }_W{)}^2+{\frac{{({\omega }_W)}^2(E-\mathrm{Ssin}{\psi }_Q)}{{{\rho }_Q}^2}}^2$$

$$B=({ϵ}_W-G{ϵ}_b-G^{\prime }({\omega }_W{)}^2{)}^2$$

And,

$$E=\frac{r_Q}{2}\mathrm{co}{s}^2\alpha \sin 2{\psi }_Q-(c-h_Q\sin \alpha )\cos \alpha \sin {\psi }_Q$$

$$F=r_Q\mathrm{si}{n}^2\alpha \sin 2{\psi }_Q-(c-h_Q\sin \alpha )\cos \alpha \sin {\psi }_Q+s{\psi }_Q$$

$$G=\frac{r_Q((c-h_Q\sin \alpha )\cos {\psi }_Q+r_Q\cos \alpha -\mathrm{scos}\alpha \sin {\psi }_Q}{{(c-h_Q\sin \alpha +r_Q\cos \alpha \sin {\psi }_Q)}^2+{(r_Q\cos \alpha -s)}^2}$$

$${ϵ}_W=\frac{d{\omega }_W}{\mathrm{dt}}=\frac{d^2{\theta }_W}{d{t}^2}$$

Wherein, G’ is the derivative of G, ${ϵ}_W$ is the angular acceleration of the wheel, rad/s ² .

The tangential acceleration also includes two parts, one is the tangential acceleration $a_{\mathrm{Qt}}$ caused by the revolution of point Q around the center axis of the bit, the other is the Coriolis acceleration $a_k$ caused by the direction change of the radial speed of point Q caused by the rotation and axial movement of point Q and the change of the revolution speed of the bit caused by the radial movement of point Q:

$$a_{\mathrm{Qt}}={\rho }_Q\frac{d^2{\theta }_Q}{d{t}^2}$$

(10)

$$a_k=2\left(\frac{d{\theta }_Q}{\mathrm{dt}}\right)\left(\frac{d{\rho }_Q}{\mathrm{dt}}\right)$$

(11)

Thus, the tangential acceleration $a_{\mathrm{Qt}}$ can be expressed as:

$$a_{\mathrm{Qt}}={\rho }_Q({ϵ}_W-G{ϵ}_b-G^{\prime }{({\omega }_W)}^2)+\frac{2V_{Q_t}{V}_{Q_{\rho }}}{{\rho }_Q}$$

(12)

Where $G^{\prime }$ and G are the same as the expression in equation (9).

Therefore, the acceleration expression of point Q is:

$$a_Q=\sqrt{{a_{\mathrm{QH}}}^2+{a_{Q\rho }}^2+{a_{\mathrm{Qt}}}^2}$$

(13)

Trajectory equation of vertical wheel

Assuming that the feature point is the vertex of a certain tooth on the wheel, then the coordinates of the wheel tooth $T(x_T,y_T,z_T)$ in the rectangular coordinate system of the wheel should be represented as:

$$\{\begin{array}{c}{x}_T=r\cos w\hfill \\ y_T=r\cos w\hfill \\ z_T=0\hfill \end{array}$$

(14)

Where $x_T$ , $y_T$ , and $z_T$ are respectively the abscissa, ordinate and applicate of point T in the rectangular coordinate system of the wheel, and $w$ is the angular displacement of the wheel in its cylindrical coordinate system, and 0 $\le w\le$ 2 $\pi$ .

Through coordinates conversion, $T(x_T,y_T,z_T)$ in the rectangular coordinate system of the wheel can be converted to the rectangular coordinate system of the drill bit, which is,

$$\{\begin{array}{c}{X}_T=-(y_T+S)\sin {\theta }_0+(z_T·\cos \alpha -x_T·\sin \alpha +c)\cos {\theta }_0\hfill \\ \multicolumn{1}{c}{X_T=-(y_T+S)\cos {\theta }_0+(z_T·\cos \alpha -x_T·\sin \alpha +c)\sin {\theta }_0}\\ \multicolumn{1}{c}{Z_T=z_T\sin \alpha +x_T\cos \alpha +h_W}\end{array}$$

(15)

Assuming that $T(x_T,y_T,z_T)$ is a moving point and has got an angular displacement of w along its own axis, and that the wheel/bit speed ratio is i, then the angular displacement of the bit should be $w/i$ . Taking the footage of the bit per revolving circle (i.e. the angular displacement is 2 π) is d, and then,

$$\{\begin{array}{c}{X}_T=[-(y_T+s)\sin {\theta }_0+(z_T·\cos \alpha -x_T·\sin \alpha +c)\cos {\theta }_0]\cos \frac{w}{i}\hfill \\ \multicolumn{1}{c}{+[-(y_T+s)\cos {\theta }_0-(z_T·\cos \alpha -x_T·\sin \alpha +c)\sin {\theta }_0]\sin \frac{w}{i}}\\ \multicolumn{1}{c}{X_T=[(y_T+s)\sin {\theta }_0-(z_T·\cos \alpha -x_T·\sin \alpha +c)\cos {\theta }_0]\sin \frac{w}{i}}\\ \multicolumn{1}{c}{+[-(y_T+s)\cos {\theta }_0-(z_T·\cos \alpha -x_T·\sin \alpha +c)\sin {\theta }_0]\cos \frac{w}{i}}\\ \multicolumn{1}{c}{Z_T=z_T\sin \alpha +x_T\cos \alpha +h_W+\left(\frac{w}{2\pi ·i}\right)d}\end{array}$$

(16)

Where $X_T$ , $Y_T$ , and $Z_T$ are respectively the abscissa, ordinate and applicate of point T in the rectangular coordinate system of the bit.

Calculation of wheel/bit speed ratio

The ratio between the rotation speed of the wheel and revolution speed of the bit body is called the wheel/bit speed ratio, it is very important for kinematics analyzing, bearing system designing, and anti-wear performance of the bit.

On the feature plane of the bit, as the plane X₀OH shown in Figure 4, the wheel can be vertically projected on the polar plane of the bit body. In the projection view, the teeth-disposing circle should be a line segment, as the dashed and solid line segments shown in Figure 5 (left). If profile of the wheel (hereinafter referred to as wheel profile) perfectly matches with the cutting profile of the bit, the normal angle is defined as the original normal angle ${\alpha }_o$ , however, in order to motivate the wheel to rotate, the actual normal angle $\alpha$ is usually designed with a proper deviation angle $\mathrm{\Delta }\gamma$ on basis of the original normal angle, apparently, $\alpha$ is the sum of ${\alpha }_o$ and $\mathrm{\Delta }\gamma$ . Furtherly, if the lowest contacting point I between the wheel and the bit is determined, the teeth-disposing circle should rotate with the angle $\mathrm{\Delta }\gamma$ to ensure the normal angle $\alpha$ , accordingly, the center of the circle will be moved. As shown in Figure 5, the point O_Q1 represents the center of the original teeth disposing circle, while O_Q2 represents the updated one. Assuming that the wheel is a round disk and the friction force between the disk and the bottom-hole rock is sufficient, the rotation of the wheel will be pure rolling with a speed $\overrightarrow{V_I}$ . As shown in Figure 5 (right), it is a top view along the drilling direction, so the bit should revolve counterclockwise rather than clockwise as that shown in Figure 4.

$$V_I={\omega }_b·\sqrt{{(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}^2+s^2}$$

(17)

Wherein, $V_I$ is the magnitude of $\overrightarrow{V_I}$ . Theoretically, the speed ${\overrightarrow{V}}_I$ can be decomposed into a component ${\overrightarrow{V}}_{IC}$ parallel to the polar plane of the wheel and a component ${\overrightarrow{V}}_{IT}$ in the tangential direction. The component ${\overrightarrow{V}}_{IC}$ is the main reason that the wheel slips on the bottom-hole rock. Since the point I is the contacting point but not an actual point of the wheel, as point I moves forward, the actual rotation speed (along the wheel axis and in the cylindrical coordinate system of the wheel) of the corresponding point on the wheel will be equal to the component ${\overrightarrow{V}}_{IT}$ in magnitude but opposite to its direction, that is, the linear speed ${\overrightarrow{V}}_W$ of the wheel (with the radius being r) is equal to $-\overrightarrow{V_{IT}}$ . Take $V_{\mathrm{IT}}$ as the magnitude of $\overrightarrow{V_{IT}}$ and $V_W$ as the magnitude of ${\overrightarrow{V}}_W$ , then,

$$\begin{array}{c}{V}_W=V_{\mathrm{IT}}=V_I·\cos (\overrightarrow{V_{\mathrm{IT}}},\overrightarrow{V_I})\hfill \\ ={\omega }_b·\sqrt{{(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}^2+s^2}·\hfill \\ \frac{c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma )}{\sqrt{{(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}^2+s^2}}\hfill \\ ={\omega }_b·(c-\mathrm{rsin}(\alpha +\mathrm{\Delta }\gamma )\hfill \end{array}$$

(18)

So that the wheel/ bit speed ratio should be:

$$i=\frac{{\omega }_W}{{\omega }_b}=\frac{\frac{V_{\mathrm{IT}}}{r}}{{\omega }_b}=\frac{\frac{{\omega }_b·(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}{r}}{{\omega }_b}=\frac{c}{r}-\cos (\alpha +\mathrm{\Delta }\gamma )$$

(19)

It should be noted that the deviation angle $\mathrm{\Delta }\gamma$ in Figure 5 (right) is defined as positive because it makes the normal angle larger, otherwise, it should be negative. If it is negative, then the wheel/bit speed ratio should be:

$$i=\frac{c}{r}-\cos (\alpha -\mathrm{\Delta }\gamma )$$

(20)

Figure 5.

Normal angle (left) and speed decomposition of the wheel (right).

Measuring and modeling for average speed of the vertical wheel

Measuring and modeling for average speed of the vertical wheel

The average speed of the vertical wheel is measured during the unit rock-breaking experiment on the wheel. As shown in Figure 6, the experiment is carried out on a planning machine which is specially developed to be suitable for unit rock-breaking experiment. The experiment equipment mainly includes the planning machine, feeding speed adjusting module, penetrating depth adjusting module, tri-axial load transducer, rock sample fixture, data acquisition system, and high-speed camera. Limited by the dimensions of the rock sample fixture, the rock sample used in this experiment is a sandstone in the volume of 280 mm × 280 mm × 160 mm. Specifically, the wheel is rotatably mounted on the wheel seat through an integrated shaft, as shown in Figure 9 (right) in the following content, both the wheel shaft and the wheel seat are manufactured with grooves for rollers, and a plurality of rollers are put in the grooves through a hole on the seat, so that the wheel is enabled to rotate and is prevented to fall from the seat. Besides, a thrust bearing is configured at the back end of the wheel shaft to reduce friction. The fitting depth of the wheel can be adjusted by applying different thrust bearings and wheels, and the normal angle of the wheel can be adjusted by changing the mounting angle between the transducer and the wheel seat.

Figure 6.

Experiment equipment (left) and the wheels (right).

For the unit rock-breaking experiment, the experimental principle is that when the cutting teeth of the wheel contact with and start to penetrate in the rock, there will be an impact load between the teeth and the rock. On one hand, the impact load on the rock will break the rock, on the other, the cutting force (tangential force) on the wheel will increase and the angular speed will decrease sharply, which can be indicated by the force peak in the cutting force curve. Accordingly, a valley bottom will appear on the wheel angular speed curve. Therefore, in the whole rock-breaking process, the instantaneous angular speed of the wheel changes periodically as the wheel rotates, and the number of peaks (or valleys) on the curve is equal to the impacting times of the teeth penetrating in the rock. As shown in Figure 7, the curve shows the variation of tangential force of the wheel with its diameter being Ø30 mm, particularly, it is mounted with conical teeth and the normal angle is 15°. It can be seen from the curve that 16 peaks are produced totally, respectively, the total number of the rock-breaking tracks (or scratches) generated by the wheel are 16, as illustrated in the red box in Figure 8.

Figure 7.

Variation of tangential force on the wheel with respect of time.

Figure 8.

Scratches formed by the vertical wheel.

The experimental average speed $V_{\mathrm{We}}$ of the wheel can be converted according to the number of scratches:

$$V_{\mathrm{We}}=\frac{k_e}{\mathrm{mt}}$$

(21)

Where, $k_e$ is the number of times the wheel teeth contact the rock, $t$ is the rock-breaking time of wheel, and $m$ is the number of wheel teeth.

This experiment is carried out with variable parameters, which including different wheel diameters, normal angles, back rake angles, tooth profiles, height-diameter ratio and different cutting depth. Furthermore, each experiment is repeated for three times. Particularly,

1. Wheel diameter D: $Ø18$  mm, $Ø30$  mm;

2. Normal deviation angle $\mathrm{\Delta }\gamma$ of wheel: 0°, 5°, 10°, and 15°, as shown in Figure 9 (left). With a certain locating radius, when the wheel profile totally coincides with the cutting profile of drill bit, the wheel has a specific axis inclination, in this condition, the deviation angle $\mathrm{\Delta }\gamma$ is defined as zero. And when the axis of wheel deflects counterclockwise or clockwise so that only one side of wheel coincide with the cutting profile of drill bit, there will be an angle between the present wheel axis and the axis when $\mathrm{\Delta }\gamma =0$ , and the deviation angle $\mathrm{\Delta }\gamma$ is defined as the normal deviation angle;

3. The back rake angle of the wheel: 0°, 5°, 10°, and 15°, the definition of back rake angle of the wheel is similar to the normal deviation angle, the difference is that its deviation direction is perpendicular to the paper inwardly or outwardly at the position shown in Figure 9 (left);

4. Tooth profile: conical tooth, wedge tooth, circular section tooth and conical section tooth;

5. Both the normal deviation angle and back rake angle exist: the normal deviation angle is 15°, and the back rake angle is 10°;

6. Height-diameter ratio H_W/D of the wheel: 3, 2, 1, as shown in Figure 9 (right), wherein, H_W is the exposing height of the wheel;

7. Depth-diameter ratio (S / D): 1, 1/2, 1/3, wherein, S is the fitting depth of the wheel shaft;

8. Cutting depth of wheel: 1 mm, 1.5 mm, 2 mm.

According to the experiment, the experimental average velocities of the wheels are obtained, as shown in Table 1.

Figure 9.

Definition of normal deviation angle (left) and matching relationship between wheel and wheel seat (right).

Table 1.

Average speed of vertical wheels with variable parameters.

Diameter (mm)	Tooth profile	Shaft fitting depth (gear)	Normal deviation angle (º)	Back rake angle (º)	Cutting depth (mm)	Number of scratches (PCs)	Average speed (r/s)
18	Conical tooth	0	5	0	1	–	–
18	Conical tooth	0	10	0	1	12	4.497
18	Conical tooth	0	15	0	1	12	4.532
18	Conical tooth	1	5	0	1	–	–
18	Conical tooth	1	10	0	1	10	3.587
18	Conical tooth	1	15	0	1	10	3.759
18	Conical tooth	2	5	0	1	–	–
18	Conical tooth	2	10	0	1	9	3.266
18	Conical tooth	2	15	0	1	9	3.389
18	Wedge tooth	0	5	0	1	–	–
18	Wedge tooth	0	10	0	1	–	–
18	Wedge tooth	0	15	0	1	9	3.545
18	Conical section steel teeth	0	15	0	1	17	4.368
18	Circular section steel teeth	0	15	0	1	17	4.288
30	Conical tooth	0	5	0	1	13.5	2.319
30	Conical tooth	0	10	0	1	16	3.479
30	Conical tooth	0	15	0	1	16	3.484
30	Conical tooth	0	15	10	1	13	3.013
30	Conical tooth	0	15	0	1.5	15	3.280
30	Conical tooth	0	15	0	2	14.5	3.114
30	Wedge tooth	0	5	0	1	–	–
30	Wedge tooth	0	10	0	1	5	1.208
30	Wedge tooth	0	15	0	1	11.5	2.891

By analyzing and fitting the rotating speed of different wheels in the experiment, the fitting result can be concluded that the rotating speed $V_{\mathrm{Wf}}$ of the wheel conforms to the following equation:

$$V_{\mathrm{Wf}}=\frac{k}{\sqrt{r}}+\sin (\mathrm{\Delta }\mathrm{\gamma })$$

(22)

Where: $V_{\mathrm{Wf}}$ is the fitting average speed of wheel, r/s; r is the wheel radius (the distance from the tooth vertex to the wheel axis), mm; $\mathrm{\Delta }\gamma$ is the normal deviation angle of wheel, °; $k$ is a constant related to the tooth profile and rock properties. In this experiment, $k$ is taken as 12.8.

Analysis on factors affecting the average speed of vertical wheel

Influence of wheel size

Firstly, the wheel itself has two main dimensions, one is the radius r of the wheel, which is the distance from the tooth edge to the central axis of the wheel, and the other is the radius R of the teeth-disposing circle.

According to the average speed equation of the wheel, the speed ratio i of the wheel body is inversely proportional to $\sqrt{r}$ , that is, the larger the radius r of the wheel, the smaller the average speed of the wheel body, which is more beneficial to improving the service life of the wheel bearing. However, since the space in the bit is limited, the radius of the wheel should not be too large. The larger the space occupied by the wheel, the weaker the PDC wing-blade and the narrower the fluid channel. Too large a wheel will weaken the strength of the PDC bit and may decreasing the rock-debris removing capacity of the bit and finally causing bit balling in bottom-hole.

Under the condition of the same wheel radius r (i.e. $\overline{\mathrm{AB}}=\overline{\mathrm{AC}}=\overline{\mathrm{AD}}$ ), as shown in Figure 10, where $O_1$ , $O_2$ , and $O_3$ are respectively the center of the wheel profile, when teeth disposing radius R is different (i.e. R₃ = 2R₂ = 4R₁ in the figure), the relationship between the arm length of the shaft root of the wheel is $C_{Q1}<C_{Q2}<C_{Q3}$ , and the relationship between the angle is ${\alpha }_1+\mathrm{\Delta }{\gamma }_1>{\alpha }_2+\mathrm{\Delta }{\gamma }_2>{\alpha }_3+\mathrm{\Delta }{\gamma }_3$ , the relationship between the wheel/bit speed ratio is $i_3<i_1<i_2$ (where i₂ is 4.46% larger than i₁ and 9.34% larger than i₃). The difference of wheel/bit speed ratio indicates that the wheel/bit speed ratio is not sensitive to the teeth disposing radius. With the same wheel radius r, the smaller the teeth disposing radius R, the farther one side of the wheel profile deviates from the cutting profile of the bit, and the fewer teeth on the wheel will contact the rock at the same time, so that the shorter the interaction time between a certain tooth and the rock, and the shorter and narrower the scratch tracks formed on the rock.

Figure 10.

Wheel profiles with different teeth disposing radii and normal deviation angles.

Apparently, the teeth disposing radius of the wheel 3# shown in Figure 10 is larger than the radius of the cutting profile of the drill bit. In this case, if the wheel has no normal deviation angle, at least one side of the wheel will be exposed higher than the cutting profile of the drill bit, so that the cutting teeth on the wheel will no longer cut the rock alternatively, instead, all the teeth will contact with the rock continuously.

Influence of normal deviation angle of the wheel

According to the average speed equation of the wheel, the variation relationship between the average speed and the normal deviation angle is a sinusoidal function. Since the normal deviation angle of the wheel is generally set in the range of 10°–30°, it can be considered that the greater the normal deviation angle, the greater the average speed of the wheel. Then, in order to improve the service life of the wheel bearing, the normal deviation angle of the wheel should be set smaller; however, when the normal deviation angle decreases, especially when it approaches zero, the teeth will slide on the rock more obviously, so that the interaction time between the contacting tooth and the rock will be prolonged, thus, two or more teeth will cut the rock at the same time, which will accelerate the wear of the teeth. Therefore, the normal deviation angle should be reasonably designed according to specific application requirements for the bit.

Instantaneous speed and speed compensation for the vertical wheel

Compensation method for the rotating speed

Compared with the instantaneous speed, it is relatively easier to measure the average speed of the wheel, because the average speed can be obtained by counting the number of the teeth on the wheel and the scratches formed on the rock. The average speed can be used to calculate the wheel/bit speed ratio, and furtherly calculate the drilling speed and tooth wear of the bit. However, in fact, the rotation speed of the wheel is non-uniform. Additionally, due to the flexibility of the drill string, the revolution speed of the bit body in actual drilling condition is also not uniform. To study the dynamic load of the wheel in rock-breaking process, the instantaneous speed of the wheel is required.

Since the instantaneous speed of the planning machine is not uniform, before measuring the instantaneous speed of the wheel, the instantaneous speed of the planning machine must be corrected. By fixing a ruler on the rock sample, as shown in Figure 11 (left), the position of the wheel along the moving direction can be directly taken by the high-speed camera. In order to measure the instantaneous speed of the wheel, the cylindrical surface of the wheel is divided by marked degrees thereon, and the speed of the wheel is approximately calculated by the position taken in each picture collected by the high-speed camera. As shown in Figure 11 (right), each interval represents 2º. Thus, the approximate instantaneous speed can be measured by the angle variation of wheel on timeline.

Figure 11.

Planning machine with ruler (left) and wheels with measuring intervals (right).

Speed compensation coefficient and instantaneous speed model

Through data processing on the collected position of the planning machine, the moving speed of the machine can be calculated. It should be noted that random rock breakage usually occurs at the start and end point during the contacting process, the actual contacting length is shorter than the rock length (280 mm). Under the moving speed of the planning machine, the effective contacting time between the wheel and the rock is around 0.7 s, as a result, the time range of effective data acquisition is 0–0.7 s. During the experiment process, the position of the planning machine is recorded by the high-speed camera, so that the moving distance of the machine in each time interval can be achieved by comparing the position and time point in recorded frames. Through dividing the distance by time interval in adjacent frames, the instantaneous speed can be calculated. As shown in Figure 12, each blue box represents an instantaneous speed point on the time line. Considering the data variation trend, polynomial fitting method is applied to fit the instantaneous speed model. As the blue curve shown in Figure 12, the speed equation of the planning machine is derived as a quadratic polynomial, for the fitted speed $V_m$ of the machine, the equation of the planning machine is:

$$V_m=-0.2387t^2-0.0288t+0.3133$$

(23)

When calculating the rotating speed of the wheel, the influence of the planning machine speed should be taken into consideration. It can be observed in Figure 12 that the instantaneous speed of the planning machine gradually decreases form the start point, accordingly, if the speed of the planning machine is a constant value, the wheel should rotate faster than it is observed. As a result, the instantaneous speed compensation coefficient $k_m$ is introduced into the calculation of the instantaneous speed of the wheel. Take the planning machine speed at the start point as the reference value, that is, the compensation coefficient at this point is 1, then the rest coefficient points can be derived from the instantaneous speed of the planning machine, which is:

$$k_{\mathrm{mn}}=-V_{\mathrm{mn}}+V_{m1}+1$$

(24)

Where $k_{\mathrm{mn}}$ is the nth coefficient on the timeline, $V_{\mathrm{mn}}$ is the nth speed value, and $V_{m1}$ is the start speed of the planning machine. As shown in Figure 13, each green box represents a coefficient value. Similar as the derivation of equation (23), polynomial fitting method is applied to fit the compensation coefficient. As the green curve shown in Figure 13, the coefficient model is derived as a quadratic polynomial. According to numerical fitting result, the compensation coefficient $k_m$ conforms to the following equation:

$$k_m=1.5431t^2-0.1586t+0.9708$$

(25)

By recording the rock-breaking process of the wheel and comparing the angular position of the wheel frame by frame, the instantaneous rotation speed can be calculated. And furtherly, with the speed compensation coefficient, the instantaneous rotation speed is corrected, as shown in Figure 14, it can be found that the instantaneous speed variation is substantially a sinusoidal function in timeline. Among all the tested wheels, the rotating speed of the wheel with a diameter of 18 mm is unstable and fluctuates widely, while the rotating speed of the wheel with a diameter of 30 mm is relatively stable. The average speed of φ18 mm wheel calculated according to equation (18) and corrected according to equation (25) is 4.532 r/s, while the average speed obtained according to the collected pictures is 4.491 r/s, with an error of 0.9%, indicating that the compensation for the speed model is reliable.

Figure 12.

Speed fitting of planning machine.

Figure 13.

Numerical fitting for the speed compensation coefficient.

Figure 14.

Instantaneous and average speed of the φ18 mm wheel.

For the wheel with a diameter of 18 mm, considering the compensation coefficient, the theoretical model of instantaneous rotating speed $V_{\mathrm{Wi}}$ is derived as:

$$V_{\mathrm{Wi}}=4.44+1.18\sin \left(\frac{(t+0.065)\mathrm{\pi }}{0.014}\right)$$

(26)

Conditions for continuous rotation of vertical wheel

In the actual drilling process, the interaction between the teeth and the rock is a dynamic process. With a certain tooth spacing, the wheel can keep rotating as long as the normal deviation angle is large enough. Ideally, the friction on the shaft is small enough so that it can be ignored when the wheel rotates. In this condition, the tooth arrangement should satisfy the requirement that at least one tooth contact with the rock at the same time, that is, if the former tooth leaves the rock, at least one tooth is still in contact with or is contacting the rock. Therefore, the maximum teeth spacing can be represented as:

$$L_Z=2\sqrt{r^2-{(r-\frac{H-h}{\sin \left(\mathrm{\Delta }\gamma \right)})}^2}$$

(27)

Where, $L_Z$ is the center maximum distance of teeth, representing the maximum spacing between two adjacent teeth, as shown in Figure 15(b); h is the cutting depth; H is the exposing height of the teeth.

Figure 15.

Force diagram on the wheel and its A-direction view.

In the ideal condition, as long as the actual teeth spacing L on the wheel is smaller than $L_Z$ and the deviation angle $\mathrm{\Delta }\gamma$ of the wheel is more than 0 (as shown in Figure 5 (left)), the wheel can keep rotating continuously. In other words, if $L<L_Z$ and $\mathrm{\Delta }\gamma >0$ , the adjacent subsequent tooth (assumed as point I₂) will successively penetrate in the rock when the previous tooth (assumed as point I₁) just leaves the rock. This requirement is defined as the minimum teeth spacing condition of the teeth, but it is just the basic requirement for continuous rotation.

In the actual condition, taking the wheel with only one teeth ring as an example, as shown in Figure 15(a) which is a force diagram of the wheel shown in Figure 5, there are three force components on the contacting point I in the bit coordinate system, namely the axial force $F_N$ , the tangential force $F_T$ (perpendicular to the paper outwardly) and the radial force $F_R$ . Assuming that there is no sliding friction between the wheel and rock, all these three force components are resolved from the anti-force (including the normal pressure and the static friction force on the interface around the contacting point) from the bottom-hole rock. On the other hand, the above three forces can be furtherly decomposed in the wheel coordinate system, as shown in Figure 15(c) which is the force-component diagram of Figure 15(a). Specifically, if the offset equals to zero, the axial force $F_N$ can be decomposed as the force $N_1$ in the radial direction and $N_3$ in the axial direction of the wheel, similarly, the radial force $F_R$ can be decomposed as $N_2$ in the radial direction and $N_4$ in the axial direction of the wheel. As for the tangential force $F_T$ , it is a force in the circumferential direction of the wheel that motivate the rotation of the wheel, so it is the driving force for the rotation. Apparently, the resultant force $N_b$ in the radial direction of the wheel is the sum of $N_1$ and $N_2$ , and resultant force $N_c$ in the axial direction of the wheel is the sum of $N_3$ and $N_4$ .

Furtherly, when the wheel rotates, the existence of $N_b$ will cause a sliding friction force $F_b$ between the wheel shaft and the shaft sleeve, while the force $N_c$ will generate a friction force $F_c$ between the teeth and rock. As shown in Figure 15(d), which is a simplified diagram illustrating the force relationship and rotation mechanism of the wheel. In this figure, the bottom-hole rock is simplified as an L shaped groove with a horizonal floor and vertical wall, the teeth are simplified as a wheel rolling in the groove. The normal force $N_b$ exerted by the floor furtherly acts on the wheel shaft and finally causes the friction $F_b$ , while the normal force $N_c$ exerted by the wall acts on the wheel end and finally causes the friction $F_c$ . Apparently, relative sliding exists between the wheel shaft and the sleeve, as well as between the wheel end and the vertical wall. Therefore, both the friction forces $F_b$ and $F_c$ are sliding friction. On the other hand, since rotation of the wheel is assumed as pure rolling on the floor, the driving force $F_T$ can be simplified as the static friction between the wheel and rock. Obviously, the torque $T_T$ generated by force $F_T$ drives the wheel to rotate while the torques $T_b$ and $T_c$ generated by the friction forces are the resistance for the rotation. It should be noted that both $F_T$ and $F_c$ are the friction forces between the teeth and rock, but $F_T$ is a force component of the anti-force (including normal force and static friction), while $F_c$ is an additional sliding friction resulted from the normal force. As the simplified force model shown in Figure 15(d), static friction and sliding friction can be generated simultaneously at different area even the rotation is pure rolling.

Apparently, whether the wheel can rotate depends on if the driving force $F_T$ can overcome the additional friction forces $F_b$ and $F_c$ . ²¹ In a limiting case, there is only one tooth on the wheel contacting the rock, so the tangential force on the wheel can be derived according to equation (12), which is

$$\begin{array}{c}\multicolumn{1}{c}{F_T=m_W{a}_{\mathrm{Qt}}}\\ =m_W{\rho }_Q({ϵ}_W-G{ϵ}_b-G^{\prime }{({\omega }_W)}^2)+\frac{2V_{\mathrm{Qt}}{V}_{Q\rho }}{{\rho }_Q}\hfill \end{array}$$

(28)

Where $m_W$ is the mass of the wheel, ${ϵ}_W$ , ${ϵ}_b$ , $G$ , $G^{\prime }$ , $V_{\mathrm{Qt}}$ , and $V_{Q\rho }$ and are the same as the above mentioned.

If there are n teeth ring on the wheel and only one tooth per ring contact with the rock at the same time, then there will be n cutting teeth contact with the rock at the same time, so the tangential force of the wheel shall be the sum of the tangential forces of all contacting teeth, that is:

$$\begin{array}{c}\multicolumn{1}{c}{F_T=m_W{a}_{\mathrm{Qt}}}\\ =m_W\sum _{i=1}^n[{\rho }_i({ϵ}_W-G{ϵ}_b-G^{\prime }{({\omega }_W)}^2)+2V_{\mathrm{it}}{V}_{i\rho }/{\rho }_i]\hfill \end{array}$$

(29)

Where, $a_{\mathrm{Qt}}$ is the tangential acceleration of the wheel.

On the other hand, the friction resistance on the shaft is related to the friction coefficient $\mu$ _b and the normal pressure N_b, and the resistance force F_c comes from the friction coefficient $\mu$ _c between rock and teeth and normal pressure N_c. Assuming that the $\mathrm{\Delta }\gamma$ increasing the normal angle of wheel shaft is positive, vice versa, and that the axial direction of the wheel inclining to the outside of the bit is positive, and vice versa. Thus,

$$\{\begin{array}{c}{F}_b={\mu }_b(N_1+N_2)={\mu }_b(F_N\cos (\mathrm{\Delta }\gamma +\alpha )-F_R\sin (\mathrm{\Delta }\gamma +\alpha ))\hfill \\ \multicolumn{1}{c}{F_c={\mu }_c(N_3+N_4)={\mu }_c(F_N\sin (\mathrm{\Delta }\gamma +\alpha )+F_R\cos (\mathrm{\Delta }\gamma +\alpha ))}\end{array}$$

(30)

$$\begin{array}{c}{F}_N=m_c{a}_{\mathrm{QH}}\hfill \\ =n{m}_c{a}_b+m_c\sum _{i=1}^n(r_j\sin (\mathrm{\Delta }\gamma +\alpha )(\sin {\mu }_j{ϵ}_b+\cos {\psi }_j({\omega }^{\prime }{)}^2))\hfill \end{array}$$

(31)

$$F_R=m_c{a}_{Q\rho }=m_c\sum _{i=1}^n(\frac{A{r}_i}{{\rho }_i}-B{\rho }_i)$$

(32)

In the equation, the expressions of A and B are consistent with the above.

If $F_T>F_b+F_c$ , the following requirements should be satisfied:

$$\begin{array}{c}\sum _{i=1}^n({\rho }_i({ϵ}_W-G{ϵ}_b-G^{\prime }{({\omega }_W)}^2)+\frac{2V_{\mathrm{it}}{V}_{i\rho }}{{\rho }_i})>({\mu }_c\sin (\mathrm{\Delta }\gamma +\alpha )\hfill \\ +{\mu }_b\cos (\mathrm{\Delta }\gamma +\alpha ))\hfill \\ +(n{a}_b\sum _{i=1}^n{r}_j\sin (\mathrm{\Delta }\gamma +\alpha )(\sin {\psi }_j{ϵ}_b+\cos {\mu }_j({\omega }_W{)}^2)\hfill \\ +({\mu }_c\cos (\mathrm{\Delta }\gamma +\alpha )-{\mu }_b\sin (\mathrm{\Delta }\gamma +\alpha ))\sum _{i=1}^n(\frac{A{r}_i}{{\rho }_i}-B{\rho }_i))\hfill \end{array}$$

(33)

If the above equation is satisfied, the wheel can keep rotating continuously. In actual drilling process, the force F_c is usually small enough so that it can be ignored, besides, teeth are usually arranged as tightly as possible so that the teeth spacing is small enough. Actually, cemented carbide teeth is not the only option to be used on the wheel, in order to reduce the teeth spacing, steel teeth with smaller sizes or cemented carbide wheel with teeth integrated thereon can also be applied, so that the teeth can be disposed more tightly and the teeth spacing can be smaller. With smaller teeth spacing, when the teeth contact the rock alternatively, the impact vibration caused by subsequent teeth is smaller, and the scratch continuity caused by the teeth is better.

Conclusion

• (1) A new type of vertical-wheel hybrid PDC bit is proposed in this paper. The vertical wheel rotates along its vertical axis when the bit drills in bottom-hole, which is beneficial to the drilling performance of the bit. Firstly, it is an auxiliary cutting structure sharing rock-breaking work for fixed wing-blades. Secondly, the rotation of the wheel changes part of the sliding friction of the bit into rolling friction, so as to reduce the torsion and stabilize the torsion fluctuation on the bit. Thirdly, the wheel itself is a buffer for protecting the bit from impact load. Fourthly, the wear rate of the teeth can be reduced by penetrating the rock alternatively.

• (2) To study the rock-breaking behavior of the vertical-wheel hybrid PDC bit, a coordinate system for the vertical wheel is established, on basis of which the equations of speed and acceleration of the vertical wheel are derived. Specifically, the average rotation speed of the wheel is:

$$\begin{array}{c}\multicolumn{1}{c}{V_W={\omega }_b·\sqrt{{(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}^2+s^2}·}\\ \frac{c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma )}{\sqrt{{(c-\mathrm{rcos}(\alpha +\mathrm{\Delta }\gamma ))}^2+s^2}}\hfill \end{array}$$

According to the speed equation, the wheel/bit speed ratio is furtherly derived, which is:

$$i=\frac{{\omega }_W}{{\omega }_b}=\frac{c}{r}-\cos (\alpha +\mathrm{\Delta }\gamma )$$

• (3) According to the experiment, the average speed of the wheel is most sensitive to the changes of the wheel diameter and the normal deviation angle. Specifically, the fitted average speed is inversely proportional to the quadratic of diameter and varies with the normal deviation angle in a sinusoidal function:

$$V_{\mathrm{Wf}}=\frac{k}{\sqrt{r}}+\sin (\mathrm{\Delta }\mathrm{\gamma })$$

And analysis on the instantaneous rotation speed shows that the speed changes as a sinusoidal function with respect of time, which is:

$$V_{\mathrm{Wi}}=4.44+1.18\sin \left(\frac{(t+0.065)\mathrm{\pi }}{0.014}\right)$$

Accordingly, the calculated average speed of the Ø18 mm vertical-wheel according to the instantaneous speed model is 4.532 r/s, while the average speed obtained in experiment is 4.491 r/s, with an error of 0.9%, proving that the measuring method and modeling of the instantaneous speed are credible.

• (4) If the wheel shall rotate continuously, the driving force $F_T$ must be greater than the additional sliding friction F_c on the wheel and the friction resistance $F_b$ on the wheel shaft:

$$F_T>F_b+F_c$$

Besides, the teeth spacing should be less than the maximum value, which is:

$$L_Z=2\sqrt{r^2-{(r-\frac{H-h}{\sin \left(\mathrm{\Delta }\gamma \right)})}^2}$$