## STUDY ON SPINDLE OFFSET FORCE AND DEFLECTION PERFORMANCE OF POINT-BIT STEERABLE TOOL1)

TANG Bo,2), WANG Peng, ZHANG Hong, FENG Ding,3)

College of Mechanical Engineering, Yangtze University, Jingzhou 434023, Hubei, China

Hubei Engineering Research Center for Oil ＆ Gas Drilling and Completion Tools, Jingzhou 434023, Hubei, China

 基金资助: 1)湖北省技术创新专项(重大项目)资助(2019010)

2)唐博,硕士研究生,研究方向为油气装备及井下工具。E-mail:1878375392@qq.com

Abstract

In order to study the influence of the installation position of weight on bit (WOB) transfer device on the spindle offset force and deflection performance of point-bit steerable tool, a mechanical model of spindle considering axial load is developed based on the mechanical model of transverse bending beam and the bending beam with combined axial and lateral load. The beam element assembly method is used to solve the string mechanics by using the general solution of differential equation. The geometric equation and mechanical equation of spindle are derived. And the effects of the position of WOB transfer device on the spindle offset fore and deflection performance parameters such as bit side force and bit inclination are investigated. The results show that when increasing the position of the WOB transfer device from the bit, the offset force required for tool spindle deflect will be reduced, and the bit-side force and bit inclination will be increased.

Keywords： point-bit; shaft; offset force; deflection performance

TANG Bo, WANG Peng, ZHANG Hong, FENG Ding. STUDY ON SPINDLE OFFSET FORCE AND DEFLECTION PERFORMANCE OF POINT-BIT STEERABLE TOOL1). Mechanics in Engineering, 2022, 44(2): 310-316 DOI:10.6052/1000-0879-21-363

## 1 指向式导向工具主轴力学模型建立

### 图1

Fig.1   Simplified diagram of mechanical model

$Q-(Q+{\rm d}Q)-q{\rm d}x=0$

$Q{\rm d}x-{\rm d}M-P{\rm d}y-\frac{1}{2}q{\rm d}x{\rm d}x=0$

$M=-EI\frac{{\rm d}^{2}y}{{\rm d}x^{2}}$

$\frac{{\rm d}^{4}y}{{\rm d}x^{4}}-\frac{P}{EI}\frac{{\rm d}^{2}y}{{\rm d}x^{2}}=\frac{q}{EI}$

$\begin{array}{c}\omega(x)=C_{1} \sin \left(\sqrt{-\frac{P}{E I}} x\right)+C_{2} \cos \left(\sqrt{-\frac{P}{E I}} x\right)+ \\C_{3} x+C_{4}-\frac{q}{2 P} x^{2}\end{array}$

$\omega (x)=C_{1} \frac{x^{3}}{6}+C_{2} \frac{x^{2}}{2}+C_{3} x+C_{4}+\frac{q}{24EI}x^{4}$

## 2 柔性主轴管柱单元组合

### 2.1 管柱单元几何与力学属性

$\left[\begin{array}{c}\omega_{i} \\\theta_{i} \\M_{i} \\Q_{i}\end{array}\right]=\left[\begin{array}{c}\omega_{i} \\\omega_{i}^{\prime} \\-E_{i} I_{i} \omega_{i}^{\prime \prime} \\-E_{i} I_{i} \omega_{i}^{\prime \prime \prime}\end{array}\right]$

\begin{aligned}{\left[\begin{array}{c}\omega_{i}(s) \\\theta_{i}(s) \\M_{i}(s) \\Q_{i}(s)\end{array}\right] } &=\left[\begin{array}{cccc}\sin \left(k_{i} x\right) & \cos \left(k_{i} x\right) & x & 1 \\k_{i} \cos \left(k_{i} x\right) & -k_{i} \sin \left(k_{i} x\right) & 1 & 0 \\E_{i} I_{i}\left[k_{i}^{2} \sin \left(k_{i} x\right)\right. & k_{i}^{2} \cos \left(k_{i} x\right) & 0 & 0 \\E_{i} I_{i}\left[k_{i}^{3} \cos \left(k_{i} x\right)\right. & -k_{i}^{3} \sin \left(k_{i} x\right) & 0 & 0\end{array}\right] \\{\left[\begin{array}{c}C_{i}^{1} \\C_{i}^{2} \\C_{i}^{3} \\C_{i}^{4}\end{array}\right]-\left[\begin{array}{c}\frac{q_{i}}{2 P_{i}} x^{2} \\\frac{q_{i}}{P_{i}} x \\-\frac{E_{i} I_{i} q_{i}}{P_{i}} \\0\end{array}\right] }\end{aligned}

$\begin{array}{l}{\left[\begin{array}{l}\omega_{i}(s) \\\theta_{i}(s) \\M_{i}(s) \\Q_{i}(s)\end{array}\right]=\left[\begin{array}{cccc}\frac{x^{3}}{6} & \frac{x^{2}}{2} & x & 1 \\\frac{x^{2}}{2} & x & 1 & 0 \\-E_{i} I_{i} x & -E_{i} I_{i} & 0 & 0 \\-E_{i} I_{i} & 0 & 0 & 0\end{array}\right]\left[\begin{array}{c}C_{i}^{1} \\C_{i}^{2} \\C_{i}^{3} \\C_{i}^{4}\end{array}\right]+} \\{\left[\begin{array}{c}\frac{q_{i} x^{4}}{24 E_{i} I_{i}} \\\frac{q_{i} x^{3}}{6 E_{i} I_{i}} \\-\frac{q_{i} x^{2}}{2} \\-q_{i} x\end{array}\right]}\end{array}$

### 2.2 单元节点连接条件

$\left.\begin{array}{l} \omega_{i} (x\vert_{x=0\mbox{\scriptsize或}x=L_{i}})=e_{\omega_{i-1} } \\ M_{i} (x\vert_{x=0\mbox{\scriptsize或}x=L_{i}})=0 \\ \end{array} \right\}$

### 图2

Fig.2   Spindle element node attribute diagram

$\left. {\begin{array}{l} \omega_{i} (x\vert_{x=L_{i} } )-\omega_{i+1} (x\vert_{x=0} )=0 \\ \theta_{i} (x\vert_{x=L_{i} } )-\theta_{i+1} (x\vert_{x=0} )=0 \\ M_{i} (x\vert_{x=L_{i} } )-M_{i+1} (x\vert_{x=0} )=0 \\ Q_{i} (x\vert_{x=L_{i} } )-Q_{i+1} (x\vert_{x=0} )=0 \\ \end{array}} \right\}$

$\left. {\begin{array}{l} \omega_{i} (x\vert_{x=L_{i} } )-\omega_{i+1} (x\vert_{x=0} )=0 \\ \theta_{i} (x\vert_{x=L_{i} } )-\theta_{i+1} (x\vert_{x=0} )=0 \\ M_{i} (x\vert_{x=L_{i} } )-M_{i+1} (x\vert_{x=0} )=0 \\ \omega_{i+1} (x\vert_{x=0} )=e_{\omega_{i} } \\ \end{array}} \right\}$

$\left. {\begin{array}{l} \omega_{i} (x\vert_{x=0\mbox{\scriptsize或}x=L_{i}})=e_{\omega_{i-1} } \\ \theta_{i} (x\vert_{x=0\mbox{\scriptsize或}x=L_{i}})=0 \\ \end{array}} \right\}$

## 3 主轴偏置力与造斜能力分析

### 图3

Fig.3   Tool spindle structure parameter drawing

$\theta_{0} =C_{1}^{1} \cdot k+C_{1}^{3}$

$R_{0} =E_{1} I_{1} k_{1} C_{1}^{1}$

$R_{3} =E_{4} I_{4} C_{4}^{1} -E_{3} I_{3} C_{3}^{1} -qL_{3}$

### 图4

Fig.4   Diagram of bit inclination changing with thrust bearing position

### 图5

Fig.5   Diagram of bit lateral force changing with thrust bearing position

### 图6

Fig.6   Diagram of offset force changing with thrust bearing position

$K=\frac{R_{3} }{y_{\max } }$

### 图7

Fig.7   Diagram of spindle rigidity changing with thrust bearing position

## 4 结论

(1) 在结合横向弯曲梁和纵横弯曲梁力学模型的基础上,建立了一种考虑轴向载荷的指向式导向钻具组合的力学理论模型,对工具主轴进行力学分析,得到平面管柱力学微分方程,进而推导出主轴管柱各单元挠度、转角、弯矩及剪力的计算公式。

(2) 分析了钻压传递机构在主轴上的安装相对位置对偏置力的影响规律,得出结论,在主轴长度和偏心机构偏置量一定的情况下,随着钻压传递机构安装位置靠近调心组合轴承,主轴造斜所需的偏置力将减小,由23.897 kN减小至23.528 kN。

(3) 分析了钻压传递机构在主轴上的安装相对位置对主轴造斜性能的影响规律,随着钻压传递机构安装位置靠近调心组合轴承,导向工具的钻头侧向力与钻头倾角将增大,钻头倾角由0.223$^\circ$增大至0.236$^\circ$,钻头侧向力由5.9 kN增大至6.51 kN。

(4) 分析了钻压传递机构在主轴上的安装相对位置对主轴刚度的影响规律,得出结论,随着钻压传递机构安装位置靠近调心组合轴承,主轴刚度将减小,由3.37 kN/mm减小至3.33 kN/mm,钻压传递机构安装位置的变化对主轴的刚度影响较弱。

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