# Optimization of the Structural Parameters and the Teeth Shape of Slip in Drill Rig

Authors: Hua Zhang; Qianwei Zhu; Bin Gao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In order to improve the performance of slip and reduce the extrusion damage of the drill pipe in the drill rig, the optimization of structural parameters and teeth shape of the slip while clamping the drill pipe had been researched in this article. On the macroscale, the structural parameters of the slip had been optimized with response surface method (RSM) and Multiobjective Genetic Algorithm (MOGA). The optimized result showed that the single weight of the slip had been reduced from 3.99 kg to 2.91 kg and the maximum deformation of the drill pipe was reduced from 3.75 mm on both sides to 2.56 mm on both sides. On the microscale, a mathematical model for the single slip teeth while clamping the drill pipe had been established to give a detailed description to calculate the equivalent coefficient of friction and provide the relationship between the frictional torque with the allowable compression strength. In addition, the finite element model that had been set up by ABAQUS was used to verify the mathematical model, and the comparison of results had shown the accuracy of the mathematical model of the slip teeth while clamping the drill pipe. According to the mathematical model of the slip teeth in the drill rig, while clamping the drill pipe, the optimal shape of the slip teeth in the drill rig was achieved under the following condition: the slope of the slip teethθ is 60°, the top width of the slip teeth wh is 1.5 mm, and the depth of the slip teeth clamping the drill pipe d is 0.5 mm. The equivalent coefficient of friction fv can be increased from 1.73 to 2.06, and the optimal result fv increases 11.3%.


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## Body

## 1. Introduction

As a consequence of sustained interest in the development of underground space, tunnels played a more and more important role in human production activities and life. Hence, drill rigs had been widely used in various construction occasions because of the high automatic and working efficiency [1, 2].In the recent years, with the rapid development of anchoring construction and engineering construction, the crushing of the drill pipe has occurred frequently due to the more and more heavy and complex dynamic loadings. To solve this problem, a lot of research studies had been carried to analyze on the phenomenon and position of drill pipe failure mainly from the following four aspects: surface irregularities [3–5], upset area [6, 7], corrosion pitting [8, 9], and threaded connections [10–12].In order to avoid slip crushing, this study used the response surface method (RSM) and multiobjective optimization genetic algorithm (MOGA) to optimize the structure of the slip; in addition, this study also proposed a mathematical model to describe the interaction of the slip teeth and drill pipe in a drill rig. The physical model drill rig is shown in Figure1. The clamp was the most important module of the drill rig. It was always fixed in front of the propulsive frame and used to load or unload the drill pipe. The slip is also a key part of the clamp; there is a protuberant teeth which is called the slip teeth on the surface of slip. Three-dimensional schematic of the slip and the slip teeth is shown in Figure 1.Figure 1
The installation location of the clamp and slip.On the macroscale, we expected to reduce the slip crushing by optimizing the structural parameters of slip in the drill rig but the optimization is a nonlinear problem. Thus, it is difficult to establish an accurate mathematical model to evaluate and optimize the performance. Thanks to the research on progress of optimization methods. In [13], a new multilevel optimization strategy for efficient multiobjective optimization was used for optimizing the design of interior permanent magnet synchronous motors (IPMSMs). Then, the Kriging model was employed to approximate the finite element analysis for the multiobjective optimization in each level. In [14], a novel multiobjective system level optimization method for SRM drive systems had been presented. A SRM drive system with a segmented-rotor switched reluctance motor (SSRM) and the angle position control method were investigated to verify the effectiveness of the proposed method. In addition, the optimization of structural parameters was another research hotspot [15, 16]. The design optimization research studies of the engineering machine had been developed rapidly. The most popular method for optimizing engineering machine was performed using finite element analysis (FEA) coupled with optimization algorithms such as genetic algorithm (GA) [17], multiobjective genetic algorithm [18,19], and nondominated sorting genetic algorithm-II (NSGA-II) [20, 21].On the microscale, we wanted to establish a mathematical model for the single slip teeth, while clamping the drill pipe to reduce the slip crushing. The analysis of drill pipe failure due to the slip teeth could be traced back to the work by Reinhold and Spiri [22] in 1959. By regarding the slip system as an immoveable wedge, the ratio of the transverse loading to the axial loading can be derived and the average radial pressure acting on the outer wall of the drill pipe can be estimated. In this way, a formula for calculating the crushing load can be obtained by combining the axial stress and the tangential stress. Vreeland [23] verified the Reinhold–Spiri formula by carrying out eight groups of tests on 5 in drill pipes. The results had shown a satisfactorily fitting of the Reinhold–Spiri formula. Then, Hayatdavoudi [24] and Sathuvalli et al. [25] had performed research studies on reducing the probability of drill pipe yielding. Tang et al. [26] and Qian [27] used the FE model to study how the slip-insert geometry affects the mechanical behavior of the drill pipe.This study work included the following three contents. Firstly, the structure of the slip could be optimized to reduce the deformation of the drill pipe by the method of RSM and MOGA. Secondly, the mathematical model was used to calculate the equivalent coefficient of friction and provide the relationship between the torque with the allowable compression strength. Finally, the mathematical model was verified by the finite element model which was established by ABAQUS, and the optimization parameters of the slip teeth had been presented.


## 2. Optimization of the Structure of the Slip by RSM and MOGA

### 2.1. Determination of Optimized Parameters and DOE Experiment Design

The slip is usually machined from a square piece by NC milling. There are four vertical sides at the top, bottom, left, and right. In front of the slip, there are two symmetrical slip teeth surfaces. The two slip teeth surfaces are at a certain angle with the horizontal direction. The specific structure is shown in Figure2. As shown in Figure 2, the length and the width of the slip are represented by P1 and P2, respectively. The angle between the teeth surface and horizontal direction is represented by P3. P1, P2, and P3 are selected as the optimal design parameters.Figure 2
The diagram of the slip structure.According to the position between specific structures interference combined with design skills that include the diameter of the drill pipe, the size restriction of the roughneck guide device, and the width/ thickness ratio of the slip, the optimization range of design parameters could be concluded as shown in Table1.Table 1
Initial value and optimization range of design parameters.
Design parametersInitial valueOptimization range of design parametersP1 (mm)12096～144P2 (mm)12096～144P3 (degree)7056～84According to the optimization design object slip, slip mass (P4) and drill pipe deformation (P5) were taken as output parameters. Box Behnken Design (BBD) was used to design the experiment; meanwhile, the combination of design parameters was obtained to construct the response surface.


### 2.2. Establishment and Analysis of Standard Second-Order RSM

The standard regression equation was used to establish the mathematical model, which can be established as shown in the following equation:(1)y^=a+∑j=1mbjxj+∑j=1mbjjxj2+∑j<kbjkxjxk+ε,where y^ is the regression prediction value of output parameters, xj is, respectively, corresponding to the values of input parameter Pj, a is the constant coefficient value, bj, bjj, and bij are the coefficient values of the corresponding input parameter, and ε is the statistical error value. This is a FE simulation analysis, and the value of ε is taken as 0.A model for the standard quadratic response surface was established based on all quadratic terms, primary terms, and the interaction between various factors. With the calculation of software, the regression mathematical model of slip mass and drill pipe deformation with three design parameters can be obtained:(2)y^4=1.61+0.07X1+0.13X2+0.14X3−0.01X12+0.005X22+0.008X32+0.002X1X2+0.04X1X3+0.005X2X3,y^5=10.76−0.63X2+10.17X3+0.01X12+3.57X32+0.21X1X3,where y^4andy^5 are, respectively, regression prediction values of the parameter of slip mass and drill pipe deformation and X1, X2, and X3 are, respectively, corresponding to input parameters P1, P2, and P3 after Yeo–Johnson transformation [28].The output results ofP4 and P5 from RSM and the design parameters P1, P2, and P3 are shown in Figure 3. When the length of the slip is between 100–110 mm, the drill pipe deformation will have a great drop and then gradually becomes stable, as shown in Figure 3(a). It can be seen from Figure 3(b) that the deformation of the drill pipe decreases with the increase of the width of the slip. When the slip surfaces’ angle is about 60°, the drill pipe deformation achieves the minimum value within the optimized range, as shown in Figure 3(c).Figure 3
Output results ofP4 and P5 by the response surface method. (a) Output results with different lengths of the slip. (b) Output results with different widths of the slip. (c) Output results with different slip angles.
(a)(b)(c)


### 2.3. The Multiobjective Optimization Model of the Slip Structure

By establishing the mathematical regression model of output parameters, the method of MOGA was used to optimize the structural parameters of the slip. The optimization constraints were set as the minimum slip mass and the minimum deformation of drill pipe while clamping; the upper limit value of design variablex¯=144,144,70T and the lower limit value of design variable x = [96, 96, 56]T.Table2 had shown the eight Pareto optimal solutions obtained by importing software for MOGA calculation. According to the results given in Table 2, the optimal solution of width of the slip and the surface angle of the slip has a steady trend around a fixed value besides the length of slip changes in a large range. These observations implied that, with the increase of the length of the slip, although the mass of the slip was slightly increased, there was a significant decrease in the deformation of the drill pipe.Table 2
Eight groups of optimal solutions under the MOGA calculation.
P1 (mm)P2 (mm)P3 (°)First group111.8797.3259.05Second group120.7399.4458.53Third group130.897.1359.04Forth group123.2899.659.02Fifth group130.6597.3959.22Sixth group130.6797.4659.23Seventh group130.6797.4659.23Eighth group130.3297.4459.25By analyzing the eight optimization results listed in Table2 and combining with the processing technology at the same time, three groups of MOGA calculations were round-up. The final optimized results are shown in Table 3.Table 3
Three groups of the most optimal calculations.
First groupSecond groupThird groupP1 (mm)110120130P2 (mm)100100100P3 (°)606060P4 (kg)2.762.842.91P5 (mm)1.311.311.28Considering the main optimization object as the deformation of the drill pipe, the third group was preferred. Under the same working circumstance, the single weight of the slip is reduced from 3.99 kg to 2.91 kg and the maximum deformation of the drill pipe is reduced from 3.75 mm on both sides to 2.56 mm on both sides.


## 2.1. Determination of Optimized Parameters and DOE Experiment Design

The slip is usually machined from a square piece by NC milling. There are four vertical sides at the top, bottom, left, and right. In front of the slip, there are two symmetrical slip teeth surfaces. The two slip teeth surfaces are at a certain angle with the horizontal direction. The specific structure is shown in Figure2. As shown in Figure 2, the length and the width of the slip are represented by P1 and P2, respectively. The angle between the teeth surface and horizontal direction is represented by P3. P1, P2, and P3 are selected as the optimal design parameters.Figure 2
The diagram of the slip structure.According to the position between specific structures interference combined with design skills that include the diameter of the drill pipe, the size restriction of the roughneck guide device, and the width/ thickness ratio of the slip, the optimization range of design parameters could be concluded as shown in Table1.Table 1
Initial value and optimization range of design parameters.
Design parametersInitial valueOptimization range of design parametersP1 (mm)12096～144P2 (mm)12096～144P3 (degree)7056～84According to the optimization design object slip, slip mass (P4) and drill pipe deformation (P5) were taken as output parameters. Box Behnken Design (BBD) was used to design the experiment; meanwhile, the combination of design parameters was obtained to construct the response surface.


## 2.2. Establishment and Analysis of Standard Second-Order RSM

The standard regression equation was used to establish the mathematical model, which can be established as shown in the following equation:(1)y^=a+∑j=1mbjxj+∑j=1mbjjxj2+∑j<kbjkxjxk+ε,where y^ is the regression prediction value of output parameters, xj is, respectively, corresponding to the values of input parameter Pj, a is the constant coefficient value, bj, bjj, and bij are the coefficient values of the corresponding input parameter, and ε is the statistical error value. This is a FE simulation analysis, and the value of ε is taken as 0.A model for the standard quadratic response surface was established based on all quadratic terms, primary terms, and the interaction between various factors. With the calculation of software, the regression mathematical model of slip mass and drill pipe deformation with three design parameters can be obtained:(2)y^4=1.61+0.07X1+0.13X2+0.14X3−0.01X12+0.005X22+0.008X32+0.002X1X2+0.04X1X3+0.005X2X3,y^5=10.76−0.63X2+10.17X3+0.01X12+3.57X32+0.21X1X3,where y^4andy^5 are, respectively, regression prediction values of the parameter of slip mass and drill pipe deformation and X1, X2, and X3 are, respectively, corresponding to input parameters P1, P2, and P3 after Yeo–Johnson transformation [28].The output results ofP4 and P5 from RSM and the design parameters P1, P2, and P3 are shown in Figure 3. When the length of the slip is between 100–110 mm, the drill pipe deformation will have a great drop and then gradually becomes stable, as shown in Figure 3(a). It can be seen from Figure 3(b) that the deformation of the drill pipe decreases with the increase of the width of the slip. When the slip surfaces’ angle is about 60°, the drill pipe deformation achieves the minimum value within the optimized range, as shown in Figure 3(c).Figure 3
Output results ofP4 and P5 by the response surface method. (a) Output results with different lengths of the slip. (b) Output results with different widths of the slip. (c) Output results with different slip angles.
(a)(b)(c)


## 2.3. The Multiobjective Optimization Model of the Slip Structure

By establishing the mathematical regression model of output parameters, the method of MOGA was used to optimize the structural parameters of the slip. The optimization constraints were set as the minimum slip mass and the minimum deformation of drill pipe while clamping; the upper limit value of design variablex¯=144,144,70T and the lower limit value of design variable x = [96, 96, 56]T.Table2 had shown the eight Pareto optimal solutions obtained by importing software for MOGA calculation. According to the results given in Table 2, the optimal solution of width of the slip and the surface angle of the slip has a steady trend around a fixed value besides the length of slip changes in a large range. These observations implied that, with the increase of the length of the slip, although the mass of the slip was slightly increased, there was a significant decrease in the deformation of the drill pipe.Table 2
Eight groups of optimal solutions under the MOGA calculation.
P1 (mm)P2 (mm)P3 (°)First group111.8797.3259.05Second group120.7399.4458.53Third group130.897.1359.04Forth group123.2899.659.02Fifth group130.6597.3959.22Sixth group130.6797.4659.23Seventh group130.6797.4659.23Eighth group130.3297.4459.25By analyzing the eight optimization results listed in Table2 and combining with the processing technology at the same time, three groups of MOGA calculations were round-up. The final optimized results are shown in Table 3.Table 3
Three groups of the most optimal calculations.
First groupSecond groupThird groupP1 (mm)110120130P2 (mm)100100100P3 (°)606060P4 (kg)2.762.842.91P5 (mm)1.311.311.28Considering the main optimization object as the deformation of the drill pipe, the third group was preferred. Under the same working circumstance, the single weight of the slip is reduced from 3.99 kg to 2.91 kg and the maximum deformation of the drill pipe is reduced from 3.75 mm on both sides to 2.56 mm on both sides.


## 3. Mathematical Model of Conclusion for the Slip Teeth and Drill Pipe

### 3.1. Special Considerations

Considering the complex phenomenon of the multiple slip teeth clamp drill pipe at the same time, the following hypotheses [29, 30] were made before the governing equations had been given:(1)
Considering only one slip teeth clamps the drill pipe(2)
The slip teeth is a rigid body(3)
The drill pipe is a linear elastic deformable body


### 3.2. The Antislide Model

Figure4 shows the model of the single-slip teeth clamp drill pipe. In the model, the slip teeth is clamping into the drill pipe under the pressure of hydraulic pressure. To make it easy to explain the mathematical model, the slip teeth in Figure 4 scaled five times.Figure 4
The model of the single-slip teeth clamping into the drill pipe.As shown in Figure4, the tangential force of the slip teeth can be stated as(3)Fc=TR−d,where T is the drill pipe torque, Fc is the tangential force caused by torque moment, R is the outer radius of the drill pipe, and d is the depth of the slip teeth clamping into the drill pipe.Considering that the torque moment always squeezes the one side of the slip teeth, the squeeze side will achieve great frictional force, respectively. The friction force can be obtained by orthogonal decomposing of the tangential force (Fc) caused by torque moment.As shown in Figure4, the friction force can be stated as(4)Ff=Fn⋅f,where f is the coefficient of friction between the slip teeth and drill pipe, Ff is the friction force, and Fn is the normal component of the tangential force caused by torque moment. The pressing force can be described as(5)Fn=Fccos∠OAB.According to the angle relationship in ∆OAB, the ∠OAB and OB¯ can, respectively, be calculated as follows:(6)OB¯=R−d−wh2cotθ,(7)Rsin∠OBA=OB¯sin∠OAB,(8)sin∠OAB=R−d−wh/2cotθRsinθ,where wh is the top width of the slip teeth and θ is the slope of the slip teeth.Substituting (8) into (5), it can be expressed as(9)Fn=Fc1−R−d−wh/2cotθRsinθ2.Considering that the slip teeth is acting slowly while working, the force balance formula can be expressed as static analysis:(10)Ft=Ffcosθ+Fnsin∠OAB,(11)Fncos∠OAB=Ffsinθ,where Ft is the total force of preventing the slide between the slip teeth and the drill pipe:(12)FtR−dfv=T.According to the above equation, the result of the antislide model can be calculated as(13)Ft≥TR−dfv.The slip teeth and the drill pipe are not allowing to slide each other. Therefore, the equivalent friction torque generated when the slip teeth is stuck into the outer wall of the drill pipe should be greater than the resistance torque, and it can be expressed as(14)fv=1fcosθ1−R−d−wh/2cotθ/Rsinθ2+R−d−wh/2cotθ/Rsinθ1−R−d−wh/2cotθ/Rsinθ2,where fv is the equivalent coefficient of friction between the slip teeth and the drill pipe.Once slip occurs between the slip teeth and the outer wall of the drill pipe, both the slip teeth and the drill pipe will be broken fast. If the resistance torque does not fit the relationships in equation (11), it will not only affect the normal working state of the slip teeth but also accelerate the rate of breaking of both the slip teeth and drill pipe.


### 3.3. The Compressive Stress Model

Considering that the compressive stress needs to be calculated, the area of the vertical component of tangential force caused by torque moment must be provided. According to equation (10), the normal component of the tangential force caused by torque moment Fn has been calculated.As shown in Figure4, the area of the vertical component of tangential force caused by torque moment can be expressed as(15)An=ltsecθ2.Combining equation (9) with (15), the compression stress of the drill pipe can be expressed as(16)σn=T1−R−d−wh/2cotθ/Rsinθ2R−dltsecθ/2.In the above formula, equation (13) can also be expressed as inequality:(17)T≤R−dltsecθ/21−R−d−wh/2cotθ/Rsinθ2σn,where [σn] is the allowable compression strength of the drill pipe. If the drill pipe torque is more than the allowable compression strength of the drill pipe, the drill pipe will undergo plastic deformation. Since the plastic deformation cannot resume, the plastic deformation of the drill pipe must be avoided.


## 3.1. Special Considerations

Considering the complex phenomenon of the multiple slip teeth clamp drill pipe at the same time, the following hypotheses [29, 30] were made before the governing equations had been given:(1)
Considering only one slip teeth clamps the drill pipe(2)
The slip teeth is a rigid body(3)
The drill pipe is a linear elastic deformable body


## 3.2. The Antislide Model

Figure4 shows the model of the single-slip teeth clamp drill pipe. In the model, the slip teeth is clamping into the drill pipe under the pressure of hydraulic pressure. To make it easy to explain the mathematical model, the slip teeth in Figure 4 scaled five times.Figure 4
The model of the single-slip teeth clamping into the drill pipe.As shown in Figure4, the tangential force of the slip teeth can be stated as(3)Fc=TR−d,where T is the drill pipe torque, Fc is the tangential force caused by torque moment, R is the outer radius of the drill pipe, and d is the depth of the slip teeth clamping into the drill pipe.Considering that the torque moment always squeezes the one side of the slip teeth, the squeeze side will achieve great frictional force, respectively. The friction force can be obtained by orthogonal decomposing of the tangential force (Fc) caused by torque moment.As shown in Figure4, the friction force can be stated as(4)Ff=Fn⋅f,where f is the coefficient of friction between the slip teeth and drill pipe, Ff is the friction force, and Fn is the normal component of the tangential force caused by torque moment. The pressing force can be described as(5)Fn=Fccos∠OAB.According to the angle relationship in ∆OAB, the ∠OAB and OB¯ can, respectively, be calculated as follows:(6)OB¯=R−d−wh2cotθ,(7)Rsin∠OBA=OB¯sin∠OAB,(8)sin∠OAB=R−d−wh/2cotθRsinθ,where wh is the top width of the slip teeth and θ is the slope of the slip teeth.Substituting (8) into (5), it can be expressed as(9)Fn=Fc1−R−d−wh/2cotθRsinθ2.Considering that the slip teeth is acting slowly while working, the force balance formula can be expressed as static analysis:(10)Ft=Ffcosθ+Fnsin∠OAB,(11)Fncos∠OAB=Ffsinθ,where Ft is the total force of preventing the slide between the slip teeth and the drill pipe:(12)FtR−dfv=T.According to the above equation, the result of the antislide model can be calculated as(13)Ft≥TR−dfv.The slip teeth and the drill pipe are not allowing to slide each other. Therefore, the equivalent friction torque generated when the slip teeth is stuck into the outer wall of the drill pipe should be greater than the resistance torque, and it can be expressed as(14)fv=1fcosθ1−R−d−wh/2cotθ/Rsinθ2+R−d−wh/2cotθ/Rsinθ1−R−d−wh/2cotθ/Rsinθ2,where fv is the equivalent coefficient of friction between the slip teeth and the drill pipe.Once slip occurs between the slip teeth and the outer wall of the drill pipe, both the slip teeth and the drill pipe will be broken fast. If the resistance torque does not fit the relationships in equation (11), it will not only affect the normal working state of the slip teeth but also accelerate the rate of breaking of both the slip teeth and drill pipe.


## 3.3. The Compressive Stress Model

Considering that the compressive stress needs to be calculated, the area of the vertical component of tangential force caused by torque moment must be provided. According to equation (10), the normal component of the tangential force caused by torque moment Fn has been calculated.As shown in Figure4, the area of the vertical component of tangential force caused by torque moment can be expressed as(15)An=ltsecθ2.Combining equation (9) with (15), the compression stress of the drill pipe can be expressed as(16)σn=T1−R−d−wh/2cotθ/Rsinθ2R−dltsecθ/2.In the above formula, equation (13) can also be expressed as inequality:(17)T≤R−dltsecθ/21−R−d−wh/2cotθ/Rsinθ2σn,where [σn] is the allowable compression strength of the drill pipe. If the drill pipe torque is more than the allowable compression strength of the drill pipe, the drill pipe will undergo plastic deformation. Since the plastic deformation cannot resume, the plastic deformation of the drill pipe must be avoided.


## 4. The Finite Element Model Verification

### 4.1. The Establishment of the Finite Element Model

The model of the slip teeth and the drill pipe is shown in Figure5. R1 and R2 are the external diameter and the inner diameter of the drill pipe; θ, ζ, and wh, respectively, are the thread angle, the height of the slip teeth, and the top width of the slip teeth; d is clamp depth. The values are shown in Table 4.Figure 5
Diagram of the model of the slip teeth and drill pipe. (a) Diagram of the model of the drill pipe. (b) Diagram of the model of the slip teeth.
(a)(b)Table 4
The parameters of the slip teeth and the drill pipe.
ParametersValueR1 (mm)65R2 (mm)73Θ (°)30°ζ (mm)2wh (mm)2d (mm)1Considering that the strain of the drill pipe must be controlled in the elastic scope, the material of the drill pipe can be given a general property of the linear elastic steel which has density of 7.85e-9 ton/mm3, Young’s modulus of 2.1e5 MPa, and Poisson’s ratio of 0.3.The model of the slip teeth clamping the drill pipe can be substituted as the 2D model, as shown in Figure6(a). The finite element analysis of the 2D model of the slip teeth and the drill pipe was implemented by using the finite element analysis ABAQUS software. The pinball of the mesh where the slip teeth clamping the drill pipe should be more fine for the accurate results is shown in Figure 6(b). The elements’ type of the drill pipe is CPS4R which has 4 nodes and 1 Gauss point; the number of elements of the drill pipe is 1168.Figure 6
Finite element model of the slip teeth and the drill pipe. (a) Plane model. (b) Grid model.
(a)(b)The contact between the slip teeth and the drill pipe can be set as surface-to-surface contact. The surface of the slip teeth is the master surface, and the surface of the drill pipe is the slave surface.


### 4.2. The Boundary and Solver Control of the Finite Element Model

The boundary of the finite element model is shown in Figure7. As shown in Figure 7, the edge A is the inner boundary of the drill pipe. Reference A is the circle center of the drill pipe. Reference B is the middle point of the slip teeth.Figure 7
The boundary of the finite element model.There are two solver steps in this analysis. In the first solver step, the edge A is coupling to reference point A and the slip teeth is coupling to the reference point B; then, the fixed constraint inx and y directions is applied on reference point A, and 2 mm displacement at y direction is applied on reference point B. In the second solver step, 30,000 N/mm torque in rotation z is applied on reference point A and the boundary of the reference point B is preserved.Under the great torque, the deformation of the drill pipe is so large that the elements have great change in FEM analysis. Therefore, the quasi-static [31] analysis by the explicit solver should be used to substitute to the implicit analysis by the standard solver.


### 4.3. The Verification of the Finite Element Model

According to equations (3) and (6), the normal component of the tangential force caused by torque moment can be calculated:(18)Fn=TR−d1−R−d−wh/2cotθRsinθ2.Based on the parameters of Table4 and 30,000 N/mm torque, Fn can be calculated by equation (18).To get the reaction force normal component of the tangential force caused by torque moment, the reaction force of nodes at the boundary must be all counted. As shown in Figure8, the master surface is the boundary of the teeth clamp with the solid edge and the slave surface is the boundary of the drill pipe with the dash edge. The contact area is between the solid edge with the dash edge. The nodes from N1 to N13 are the nodes of the slip teeth during the contact area. The value of the node force at the contact surface can be extracted by ABAQUS, and the reaction force caused by torque moment is equal to the sum of the node force at the contact surface.Figure 8
Diagram of the contact node area of the slip teeth and the drill pipe.As shown in figures9(a)–9(c), the value of Fn which, respectively, calculated by the finite element model and mathematical model had been compared under the different parameters of d, wh, and θ.Figure 9
The FE model compared with the mathematical model under different parameters. (a) The normal component of the tangential force caused by torque momentFn about d. (b) The normal component of the tangential force caused by torque moment Fn about wh. (c) The normal component of the tangential force caused by torque moment Fn about θ.
(a)(b)(c)According to figures9(a)–9(c), the tangential force caused by torque moment Fn can also be extracted by the finite element model. By comparing the result of the mathematical model with the result of the finite element model, the result of the mathematical model is all close to the result of the finite element model. It proves that the mathematical model can get a good confirmation to the finite element model.


## 4.1. The Establishment of the Finite Element Model

The model of the slip teeth and the drill pipe is shown in Figure5. R1 and R2 are the external diameter and the inner diameter of the drill pipe; θ, ζ, and wh, respectively, are the thread angle, the height of the slip teeth, and the top width of the slip teeth; d is clamp depth. The values are shown in Table 4.Figure 5
Diagram of the model of the slip teeth and drill pipe. (a) Diagram of the model of the drill pipe. (b) Diagram of the model of the slip teeth.
(a)(b)Table 4
The parameters of the slip teeth and the drill pipe.
ParametersValueR1 (mm)65R2 (mm)73Θ (°)30°ζ (mm)2wh (mm)2d (mm)1Considering that the strain of the drill pipe must be controlled in the elastic scope, the material of the drill pipe can be given a general property of the linear elastic steel which has density of 7.85e-9 ton/mm3, Young’s modulus of 2.1e5 MPa, and Poisson’s ratio of 0.3.The model of the slip teeth clamping the drill pipe can be substituted as the 2D model, as shown in Figure6(a). The finite element analysis of the 2D model of the slip teeth and the drill pipe was implemented by using the finite element analysis ABAQUS software. The pinball of the mesh where the slip teeth clamping the drill pipe should be more fine for the accurate results is shown in Figure 6(b). The elements’ type of the drill pipe is CPS4R which has 4 nodes and 1 Gauss point; the number of elements of the drill pipe is 1168.Figure 6
Finite element model of the slip teeth and the drill pipe. (a) Plane model. (b) Grid model.
(a)(b)The contact between the slip teeth and the drill pipe can be set as surface-to-surface contact. The surface of the slip teeth is the master surface, and the surface of the drill pipe is the slave surface.


## 4.2. The Boundary and Solver Control of the Finite Element Model

The boundary of the finite element model is shown in Figure7. As shown in Figure 7, the edge A is the inner boundary of the drill pipe. Reference A is the circle center of the drill pipe. Reference B is the middle point of the slip teeth.Figure 7
The boundary of the finite element model.There are two solver steps in this analysis. In the first solver step, the edge A is coupling to reference point A and the slip teeth is coupling to the reference point B; then, the fixed constraint inx and y directions is applied on reference point A, and 2 mm displacement at y direction is applied on reference point B. In the second solver step, 30,000 N/mm torque in rotation z is applied on reference point A and the boundary of the reference point B is preserved.Under the great torque, the deformation of the drill pipe is so large that the elements have great change in FEM analysis. Therefore, the quasi-static [31] analysis by the explicit solver should be used to substitute to the implicit analysis by the standard solver.


## 4.3. The Verification of the Finite Element Model

According to equations (3) and (6), the normal component of the tangential force caused by torque moment can be calculated:(18)Fn=TR−d1−R−d−wh/2cotθRsinθ2.Based on the parameters of Table4 and 30,000 N/mm torque, Fn can be calculated by equation (18).To get the reaction force normal component of the tangential force caused by torque moment, the reaction force of nodes at the boundary must be all counted. As shown in Figure8, the master surface is the boundary of the teeth clamp with the solid edge and the slave surface is the boundary of the drill pipe with the dash edge. The contact area is between the solid edge with the dash edge. The nodes from N1 to N13 are the nodes of the slip teeth during the contact area. The value of the node force at the contact surface can be extracted by ABAQUS, and the reaction force caused by torque moment is equal to the sum of the node force at the contact surface.Figure 8
Diagram of the contact node area of the slip teeth and the drill pipe.As shown in figures9(a)–9(c), the value of Fn which, respectively, calculated by the finite element model and mathematical model had been compared under the different parameters of d, wh, and θ.Figure 9
The FE model compared with the mathematical model under different parameters. (a) The normal component of the tangential force caused by torque momentFn about d. (b) The normal component of the tangential force caused by torque moment Fn about wh. (c) The normal component of the tangential force caused by torque moment Fn about θ.
(a)(b)(c)According to figures9(a)–9(c), the tangential force caused by torque moment Fn can also be extracted by the finite element model. By comparing the result of the mathematical model with the result of the finite element model, the result of the mathematical model is all close to the result of the finite element model. It proves that the mathematical model can get a good confirmation to the finite element model.


## 5. The Optimization and Analysis of the Slip Teeth

According to equation (14), the depth of the slip teeth clamping the drill pipe d, the top width of the slip teeth wh, and the slope of the slip teeth θ had been chosen as the key parameters. The equivalent coefficient of friction between the slip teeth and the drill pipe fv had been chosen as the optimized target. The boundary constraint of design variables for the slip teeth structure can be set, as shown in Table 5.Table 5
The boundary constraint of design variables for the slip teeth structure.
Design variablesInitial valueVariable ranged (mm)0.75(0.5, 1)wh (mm)1.5(1, 2)θ (°)45(30, 60)According to the established mathematical model and combining the upper and lower limits of design variables, the different parameters about the trend chart of the equivalent coefficient of friction are shown in Figure10. As shown in Figure 10(a), it is apparent that the slope of the slip teeth θ has large influence on the equivalent coefficient of friction fv. When the slope of the slip teeth θ is set as 45°, the equivalent coefficient of friction fv can achieve the minimum value. As shown in Figures 10(b) and 10(c), the depth of the clamp teeth clamping the drill pipe d and the top width of the clamp teeth wh are nearly having no influence on the equivalent coefficient of friction fv. According to Figure 10, it is obviously shown that fv. is mainly related to θ; in addition, fv has little relationship with other two parameters separately and interaction between two parameters.Figure 10
Different parameters about the trend chart of the equivalent coefficient of friction. (a) The trend chart aboutθ with fv. (b) The trend chart about wh with fv. (c) The trend chart about d with fv.
(a)(b)(c)The optimization parameters of slip teeth are set as follows: the slope of slip teeth θ is 60°, the top width of slip teethwh is 1.5 mm, and the depth of slip teeth d is 0.75 mm. As shown in Figure 10, the parameters of θ and wh have nearly no influence on fv; then, the values of parameters which are convenient for machining are preferred. The top width of the slip teeth wh is chosen as 1.5 mm and the depth of slip teeth d is chosen as 0.75 mm, respectively. Under this circumstance, the equivalent coefficient of friction fv can be increased from 1.73 to 2.06, and the optimal result fv increases by 11.3%.In the previous studies [30], Wei had optimized the slope of the triangle slip teeth θ by the orthogonal optimization method. Comparing the type of slip teeth in this research, the optimized result of the slope of triangle slip teeth θ was 55° ,which is similarity to 60°. By analyzing different profiles between the two slip teeth, the slip teeth in this research is a trapezoid slip tooth, while the slip teeth in the research of Wei’s study was a triangle slip teeth with a filleted corner at the top of the triangle slip teeth. For the above mentioned two types of the slip teeth, the optimization results had both shown that the slope of the slip teeth θ had achieved the upper of the optimization value. On the one hand, this phenomenon shows that the equivalent friction coefficient fv has little effect on the profile shape of the slip teeth. On the other hand, it also shows that the larger the slope of the slip teeth θ is, the larger the theoretical equivalent friction coefficient is. Moreover, the comparison between the two types of slip teeth can also prove the accuracy of the established mathematical model in this study.


## 6. Conclusions

In this study, in order to prevent drill pipe damage because of slip, the optimization works include two contents. On the macroscale, the structural parameters of the slip had been optimized with RSM and MOGA. On the microscale, a mathematical model for the single-slip teeth clamping the drill pipe was established, and a finite element model was set up by ABAQUS to verify the mathematical model.The following are the key conclusions drawn from this study.(1)
By the optimization of the slip based on the method of RSM, it is apparently shown that when the slip lengthP1 reaches 110 mm, the minimum deformation of the drill pipe is obtained; then, the minimum value does not decrease again as P1 increases. The deformation of the drill pipe is steadily decreasing as the slip width P2 increases. When the slip surfaces’ angle is about 60°, the drill pipe deformation achieves the minimum value. With the method of MOGA, the optimal values can be calculated as P1 is 130 mm, P2 is 100 mm, and P3 is 60°. The weight of the single slip had been reduced from 3.99 kg to 2.91 kg with a reduction rate of 27%, and the maximum deformation of the drill pipe had been reduced from 3.75 mm on both sides to 2.56 mm with a reduction of 31.9%.(2)
By simplifying the complex phenomenon of the multiple slip teeth clamp drill pipe at the same time, this study derived the mathematical model of the single-slip teeth clamp drill pipe. The equivalent coefficient of friction between the slip teeth and the drill pipefv could be set a relationship with the top width of the slip teeth wh, the slope of the slip teeth θ, and the depth of the slip teeth clamping into the drill pipe d.(3)
The mathematical model was verified by the finite element model established by ABAQUS by extracting the value of the node force at the contact surface. Then, the FE model has a good fitting with the mathematical model of the single-slip teeth clamp drill pipe.(4)
The optimization parameters of the slip teeth had been given by analyzing the mathematical model. Through the single-factor comparison and cross-factor analysis between the three parameters, it was apparently shown that the slope of the slip teethθ was taken as the primary factor. When the slope of the slip teeth θ was 60°, the top of slip teeth’s width wh was 1.5 mm and the depth of the slip teeth while clamping the drill pipe d was 0.5 mm. The equivalent coefficient of friction fv could be increased from 1.73 to 2.06, and the optimal result fv increased by 11.3%.(5)
Besides the optimization of the slip and slip teeth parameters, the topological shape of the slip and slip teeth will be the hot topic in the future research.



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*Source: 9930182-2021-05-07.xml*