Abstract
In the process of drilling coal, the kinematics of drillrod is quite complicated. The drillrod not only vibrates in longitudinal, transverse and torsional direction, but also random impacts and contacts coal wall. Considering the drilling load of drillbit and coal, contact impact of the drillrod and coal wall, the drillrods are dispersed into a number of finite elements. At the same time, the nonlinear dynamic model of drillrod system coupling longitudinal, transverse and torsional vibration is set up. The simulations of the dynamic model are researched under the conditions of different hardness coal (2.7, 3.7, 4.65). In order to decrease the vibration of auger drill, the stabilizer is added onto the drilling mechanism. And the underground experiments are done at 2404 working face of XieZhang Coal Mine in ShanDong Province of China. The results indicate that the transverse vibration radius, the longitudinal vibration frequency and amplitude all decrease with the rock hardness. The maximum transverse vibration radius shows an exponential relation with the drilling depth under the condition of the same rock. Under the same condition, the drilling depth of auger drill with stabilizer is 1.39 times that with no stabilizer, and the drilling pressure decreases about 2/3.
1. Introduction
The drilling system often deviates from the set direction, which caused by the vibration in drilling. Drilling system vibration of auger drill results in a deteriorated drilling performance. Field experience reveals that it is crucial to understand the complex vibrational mechanisms of a drilling system in order to better control its functional operation and improve its performance. There is little material or literature about the auger drill for it’s a new machine used in extremely thin coal seam. But a lot of researches are done about the vibration characteristics of drilling system of other machines similar with auger drill, which can be as a reference for this research.
The vibration coupling the longitudinal and transverse of drillrod based on the Hertz contact theory was researched and the coupling vibration dynamics model was built [13]. A nonlinear dynamic inversion control design method was raised to suppress the transverse and the torsional vibrations of a nonlinear drillrod, which reduced the vibration significantly [4]. A dynamic model of the drillrod including both drillpipe and drillcollars was formulated in which considered the gyroscopic effect, the torsional/bending inertia coupling, and the effect of the gravitational force field [56]. Through multivariate linear regression analysis and sensitivity analysis, a model was developed to determine which factor had the greatest effect on surface heave development with horizontal directional drilling, the result indicated that the relationship between surface heave and drilling practice was not a linear relationship, but a dynamic system where changed in factors produce nonlinear responses [7]. The geometrical stiffening of the drillrod using a nonlinear finite element approximation was analyzed, in which large rotations and nonlinear straindisplacements were taken into account [8]. The operation recommendations and the detection of safe drilling parameters in a conventional oil well drillrod was presented by using dynamical analysis tools, which took into account the fact that the drillrod length increased as drilling operation and analyzed the reasons of the sliding motion giving rise to selfexcited drillbit stickslip oscillations and drillbit sticking phenomena at the bottomhole assembly [9]. A stochastic computational model was proposed to model uncertainties in the bitrock interaction system and the nonlinear dynamical equations were discretized by means of the finite element method [10]. The drillrod vibrations with differential quadrature method were analyzed, which indicated the method was efficiency and accuracy in dealing with drillrod vibration problems [11]. Using the perturbation techniques, the nonlinear model for a drillrod system in deviated well with axially moving motion and axial loading was developed, and the effects of rotating speed, axial compression load, imbalance mass and nonlinear fluid force on the drillrod responses were investigated in detail, nonlinear natural frequencies and their corresponding mode shapes were presented [12]. The problems on investigation of the dynamic bending of elongated drillrod tubes was considered, which indicated that the free bending vibrations of unbounded rods could be realized only in the modes of regular dextral and sinistral spiral waves propagating with different velocities in different directions [13]. A method for controlling the torsional and stickslip vibrations by exactly decomposing the drill string dynamics into two traveling waves traveling in the direction of the top drive and in the direction of the drillbit was presented [14]. A new method named “time history shape function method” was developed to study the dynamic boundary condition between the revolving drillrod and borehole wall, which proved the method was accurate and reasonable [15]. The nonlinear dynamic model of drilling system coupling longitudinal, transverse and torsional vibration was built to study the numerical method of high nonlinear dynamic model [1617].
The above researches provide the references for this study. But the vibration of the CoalDrillrodDrillbit interacted system under the conditions of different hardness coal and different rotary speeds has not been researched. In order to master the vibration actions of the drilling system and improve the drilling efficiency, the coupling vibration of drilling system is researched in this paper. For the complexity of coupling system, the coupling system is simplified and assumed, and the interacted dynamics models between drillrod and coal, drillbit and coal are built respectively. And then, the models are solved according to the contact collision condition between drillrod and coal. The coupling vibration simulations of drilling system are done basing on the experiment data, which will provide technical support for improving drilling performance and predicting motion trail of drilling system.
2. Coupling nonlinear dynamics model of interacted system
2.1. Basic hypothesis
The coal drilling broken is a dynamic process, and the drilling system will be influenced by the gravity, feed resistance, resistance torque, damping, impacting force and frictional resistance between drillrod and coal wall. Under these external forces, there is not only longitudinal, transverse and torsional vibrations, but also coupling vibration existing in the collision between drillrod and coal wall. The collision force is a random variable changing with the time. The drilling system is taken as the researched object, shown in Fig. 1. Where ${V}_{y}$ is axial feed speed of drillbit, $M$ is the resistance torque acted on drillbit, $F\left(t\right)$ is longitudinal force of drillbit, $n$ is rotary speed of drillbit.
The basis hypothesis as follows:
(a) The crosssection of drilling hole is round, and the diameter of drilling hole is not changed with the time.
(b) The drillbit is rigid, and the drillrod is elastomeric. The original axis of drillrod is coincident with the axis of drilled hole, and the drillrod axis can deviate from the original axis when drilling coal.
(c) The collision happened between drillrod and coal wall is multidirectional and random. The drilling pressure is a constant along the drilling hole axis.
(d) The transverse vibration is researched through decomposing two component $x$ and $z$ on the crosssection.
(e) The friction force between the drillrod and the broken coal is not considered, and the connection between drillrod and driver motor is simplified as a spring with the stiffness ${K}_{t}$.
Fig. 1Discretization of drilling system
2.2. Coupling nonlinear dynamics model
The drillrod is highly nonlinear, and there is friction between drillrod and coal wall. So the analytic solution of the continuous drillrod system cannot be acquired. The drillrod is dispersed into a number of elements, and transferred to a multidegree system. The drillrod is connected together by a spring and damping. The rotational inertia of drillbit is ${J}_{A}$. And the rotational inertia of each drillrod is ${J}_{i}\left(i=1,2,3,\dots ,40\right)$. The parameters of drillrod are given in Table 1.
Table 1Parameters of drilling system
Parameter  Value 
Length $l$  48.2 m 
Rotational inertia ${J}_{A}$  10.79 kg·m^{2} 
Damping $c$  0.001 N·s 
Polar moment ${I}_{P}$  1.884·10^{−}^{5} m^{4} 
Shear modulus $G$  7.96·10^{9} N/m^{2} 
Density $\rho $  7801 kg/m^{3} 
The vibration equation of drilling system as follows [14]:
where $\mathbf{M}$ is mass matrix of drilling system, $\mathbf{c}$ is damping matrix of drilling system, $\mathbf{k}$ is stiffness matrix of drilling system, $\ddot{\mathbf{u}}\left(t\right)$, $\dot{\mathbf{u}}\left(t\right)$ and $\mathbf{u}\left(t\right)$ are acceleration, speed and displacement vector of drilling system, $\mathbf{F}\left(t\right)$ is equivalent external load vector, $\mathbf{M}$, $\mathbf{c}$ and $\mathbf{k}\in {\text{R}}^{41\times 41}$.
The expressions of the rotational inertia ${J}_{i}$, damping ${c}_{i}$ and connecting stiffness ${k}_{i}$ are shown as follows:
Taking the interacted load between drillbit and coal as the boundary condition, the coupling nonlinear dynamics model can be acquired according to the mass matrix, damping matrix and stiffness matrix of drilling system.
2.3. Interacted load model of drillbit and coal
In the drilling process, the drillbit is influenced by the feed resistance, transverse force and resistance torque. The longitudinal load of drillbit is composed of the drilling pressure, the reacting force of the coal and the dynamic load of drillbit. And all of these forces should be balanced at any time when drilling. So the longitudinal mechanical model of drillbit and coal is shown in equation (8):
where $W\left(t\right)$ is the drilling pressure of drillbit at any time, ${m}_{A}$ is mass of drillbit, $a\left(t\right)$ is acceleration of drillbit at any time, ${F}_{y}\left(t\right)$ is longitudinal force of drillbit at any time.
The transverse force of drillbit endured is a resultant, which is computed through decomposing into two forces along $x$coordinate and $z$coordinate:
where ${f}_{j}\left(t\right)$ is transverse force of drillbit, ${\alpha}_{j}\left(t\right)$ is the drilling angle of the $j$th pick at time $t$.
In order to ensure the stability of drilling, the resistance torque of drillbit should be equivalent with the drive torque. If the coordinate of $j$th pick at time $t$ is $\left(x\left(j\right),y\left(j\right),z\left(j\right)\right)$, the resistance torque acted on drillbit is:
where ${l}_{xj}\left(t\right)$ is the distance between the pick tip and the $x$ axis, ${l}_{zj}\left(t\right)$ is the distance between the pick tip and the $z$ axis in the coordinate system of drillbit, ${f}_{x}\left(t\right)$ and ${f}_{z}\left(t\right)$ are components of the transverse forces decomposed into the $x$ and $z$ axis respectively.
2.4. Contact collision conditions and dynamic model solution
2.4.1. Contact collision conditions of drillrod and coal wall
The energy of drillrod will be lost because of the friction between drillrod and coal wall in drilling process. According to the energy conservation law and ignoring the lost of thermal energy, the lost energy of drillrod will be transferred to the elastic deformation energy and the kinetic energy of coal wall. Due to the randomness and multidirectionality of the collision between drillrod and coal wall, it is difficult to describe the motion law of drillrod exactly. So the dynamic gap element is introduced between the discretized drillrod and coal wall, which is a virtual element. The inner boundary of dynamic gap element connects with the outside surface of drillrod, and outside boundary of dynamic gap element is coincident with the coal wall (Fig. 2a). The motion of drillrod is not influenced when there is no collision between the drillrod and the coal wall (Fig. 2b), and the motion of drillrod will be prevented when the collision happens (Fig. 2c). A certain deformation energy will be stored after the dynamic gap element deforms, which can be used to describe the energy lost in colliding process [18].
Fig. 2Collision between drillrod and coal wall
If the initial gap between drillrod and coal wall is $\mathrm{\Delta}{C}_{0}$, the normal relative displacement between drillrod along the direction ${n}_{t}$ and the normal of coal wall is $\mathrm{\Delta}C\left(t\right)$ at time $t$, the contact point tangential speed of drillrod and coal wall is $V\left(t\right)$, the maximum elastic deformation of coal wall is $\mathrm{\Delta}M$, the judging conditions of collision between drillrod and coal wall are shown as follows.
Freedom state:
Collision state:
where $V\left(t\right)=0$ indicates that the friction along the circumference of drillrod is rolling friction, $V\left(t\right)\ne 0$ indicates that the friction is sliding friction. But the friction along the axis of drillrod is always sliding friction.
The elastic modulus of coalwall is related with the hole depth, coal characteristics, drilling speed, drilling acceleration and time. Due to the inaccuracy of the variables, the transversal displacement of drillrod will exceed the drilled hole wall in simulation. So the elastic modulus of dynamic gap element needs to be revised until satisfy the conditions of convergence, shown as follows.
Freedom state:
Collision state:
where $\delta $ and ${F}_{c0}$ are infinitely small quantities greater than zero, ${F}_{cn}\left(t\right)$ is normal reaction force on dynamic gap element at time $t$.
The contact collision state between drillrod and coal wall is divided into two kinds according to contact time: continuous contact collision and intermittent contact collision. Continuous contact collision easily happens between the elements far to the end of drillrod and coal wall, and intermittent contact collision easily happens between the elements near to the end of drillrod and coal wall. The elements happened intermittent contact collision with coal wall rebound, which makes the conditions of collision between drillrod and coal wall not satisfied. And the elements will stay in the freedom state until that the conditions of collision between drillrod and coal wall was satisfied. It might also be noted that the discription of contact collision process will be more fine through decreasing the time step when adopting the dynamic gap element method.
2.4.2. Dynamic model solution
In order to acquire the solution of Eq. (1), the Houbolt method is utilized. Supposing the thirdorder approximate solution can be used to express the value of $\mathbf{u}$ in the scope of $\mathbf{u}(t2\Delta t)~\mathbf{u}(t+\Delta t)$, the value of $\ddot{\mathbf{u}}$ can be obtained by twice differential, and the values of $\dot{\mathbf{u}}(t+\Delta t)$ and $\mathbf{u}(t+\Delta t)$ can be got respectively through one integral and twice integral from $t$ to $t+\Delta t$. The solution of Eq. (1) is shown in Eq. (15):
3. Coupling vibration simulation
According to the previous experiment data and field experience, the compression strength of the artificial coal is 2.5 times the natural coal [19]. Because the artificial coal is homogeneous while there are bedding and joint in natural coal. The compression strengths of artificial coal in this experiment are 18.6 MPa, 14.775 MPa and 10.725 MPa. So they are equivalent to the natural coal with compression strength of 46.5 MPa ($f=$ 4.65), 36.9375 MPa ($f=$ 3.7) and 26.8125 MPa ($f=$ 2.7). Where $f$ is the Proctor coefficient of coal, which is also known as the rock solid coefficient and fastening coefficient and its value is 1/100 of the limit of the rock uniaxial compressive strength. The experiments are done on the rock & coal cutting test bed [20], the drillbit rotary speed is 60 rev/min, and the drilling feed speed is 1 m/min. The drilling torque and resistance force are shown in Fig. 3 and Fig. 4.
According to the theory above, the interacted system coupling dynamic simulations of the drilling mechanism and coal are done by ADAMS. Simulation conditions: drilling hole depth is 40 m, rotary speed of drillbit is 60 rev/min, diameter of drillbit is 550 mm, length of drillbit is 770 mm, diameter of drillrod is 450 mm, length of drillrod is 48.2 m. The constraints of drilling mechanism are set in ADAMS, and the drillrod is replaced with a flexibility body. The contact between the flexible drillrod and coal wall is set, too. The rotary driver and line driver are added on the end of drillrod, and the torque resistance and feed resistance acquired from the experiments are added on the drillbit. The dynamic model of interacted system is shown in Fig. 5. Under the conditions of different coal compression strength and different rotary speed, the coupling dynamic simulation is done. The simulation time is 10 seconds, and the simulation steps are 600.
Fig. 3Drilling torque of different coal
Fig. 4Feed resistance force of different coal
Fig. 5Dynamics simulation model of interacted system
3.1. Vibration of drillrod when $\mathit{f}=$ 4.65
The motion trails of measure points (1, 2, 3, 4, 5) are shown in Fig. 6a~Fig. 6e. The distance between the measure points and the end of drillrod are 4 m, 8 m, 12 m, 16 m and 20 m. In order to compare the motion trails of each measure point on the crosssection, the simulation data is input to statistical analysis. The vibration circle takes the initial position of measure points as the center of circle, and takes the maximum vibration displacement of measure points as radius. Then the mean and the variance of the vibration amplitude are acquired, which are shown in Figs. 6 and 7.
Fig. 6Motion trails of different measure points on xz crosssection
According to the Figs. 6 and 7, the maximum radius shows an exponential relation with the changing of drilling depth. The transverse vibration amplitude of measure point 1 is the minimum. It will enhance if the drilling depth increases, which even resulting in the drillrod colliding the coal wall. And the variance of measure point 3 is the maximum, which indicates that the collision between drillrod and coal wall is serious.
It can be seen from Fig. 8 that the axial vibration tendency of measure points is consistent. The vibration amplitude increases with the drilling depth, and the biggest vibration amplitude is 6.474 mm/s.
Fig. 7Radius statistics of measure points
Fig. 8Axial vibration velocity of measure points
3.2. Vibration of drillrod when $\mathit{f}=$ 3.7
According to the Fig. 9, the relationship between the biggest vibration radius and the drilling depth is approximately exponential. The transverse vibration amplitude of measure point 1 is the minimum. The change of the mean and variance of vibration radius is relatively slow, which indicates that the collision between drillrod and coal wall is little. Comparing the Fig. 10 with the Fig. 8, it can be known that the axis vibration frequency is decreased, and the biggest vibration amplitude reduces to 5.243 mm/s.
Fig. 9Radius statistics of measure points
Fig. 10Axial vibration velocity of measure points
3.3. Vibration of drillrod when $\mathit{f}=$ 2.7
It can be seen from the Fig. 11, the biggest vibration radius increases exponentially with the drilling depth. And the growth rate of vibration radius is faster, which indicates that the transverse vibration amplitude of measure points increases rapidly, and the serious collision happens at point 5.
The Fig. 12 shows the axial vibration velocity of the drillrod. Comparing with the Fig. 8 and the Fig. 10, the axial vibration frequency is the smallest, and the maximum amplitude of axial velocity reduces to 3.785 mm/s.
Fig. 11Radius statistics of measure points
Fig. 12Axial vibration velocity of measure points
From Fig. 7 and Fig. 9, when $f$ are 4.65 and 3.7, the mutation of Measure 2 maximum radius comparing Measure 1 is great. And the mutation value is 11.5 mm. But there is not large difference in maximum radius between Measure 3, 4, 5 and Measure 2. Moreover, the mutation of Measure 1 under $f=$ 4.65 is much bigger than that under $f=$ 3.7. Thus, stabilizers shoud be added on the drillrods per 4 meters and 8 meters under $f=$ 4.65 and $f=$ 3.7 repectively to avoid great vibration. From Fig. 11, when $f$ is 2.7, it is the maximum radius of Measure 3, not Measure 2, is 1 mm more than Measure 1. Thus, stabilizers shoud be adopt on the drillrods per 12 meters under $f=$ 2.7. In brief, stabilizers added under large hardness should be much more than that under small hardness.
4. Underground experiment of auger drill
According to the above analysis, the vibration of drillrod increases with the drilling depth. In order to decrease the vibration of drillrod, the method of adding stabilizer is proposed shown in Fig. 13 and Fig. 14.
Fig. 13Structure of auger drill
Fig. 14Structure of stabilizer
From Fig 14, the stabilizer is mainly consisting of stabilizer frame and stabilizer connector, and the stabilizer frame is rigidly connected at the air duck through stabilizer connector. The diameter of stabilizer frame is 20~30 mm bigger than the diameter of drillrod. The width of stabilizer frame is about 0.6 times the diameter of stabilizer frame.
The drilling experiment is done on the 2404 working face of XieZhuang Coal Mine in ShanDong Province of China. Where the incline angle of coal seam is 8º, the Proctor coefficient of coal seam is 2.5, thickness of coal seam is 0.60.8 m. And the mechanical parameters of auger drill are shown in Table 2. The structure and working principle of the experiment system are shown in Fig. 15. The experiment system comprises an experiment propulsion hydraulic system and a drilling system. The composition and working principle of each subsystem are as follows:
(1) The experiment propulsion hydraulic system which is developed by the authors mainly consists of oil tank, high pressure pump: 16 MPa, rated flow: 100 L/min, made in China, used for providing the pressure source for propulsion system, relief valve, pressure gauge, magnetic valve, hydraulic lock, speed control valve, and hydraulic cylinder and so on. The hydraulic oil arrives to the hydraulic cylinder through the magnetic valve, the hydraulic lock and the speed control valve, which provides a propulsion power for the drilling system. Where the hydraulic lock is used to prevent the hydraulic cylinder declining under the feed resistance, the speed control valve is used to regulate the speed of the hydraulic cylinder, and the relief valve is used to produce a backpressure and prevent system overload.
(2) The drilling system mainly includes transmission framework, drive system (drive motors: 2×110 kW, made in China, used for providing power for the mechanical drilling, gear reducer and hydraulic coupler), drillrod, drillbit (mechanical parameters are shown in Table 2), stabilizer and gearbox and so on. When the system starts working, the drive motors driving drillbits cutting the coal wall through the drillrods and gearbox, and the hydraulic cylinder push forward the drilling mechanism to complete drilling the coal wall.
Table 2Mechanical parameters of auger drill
Drilling power  2×110 kW 
Diameter of drillbit  550 mm, 425 mm 
Numbers of drillbit  3 
Diameter of drillrod  480 mm 
Length of drillrod  1500 mm 
Rated operational voltage  Three phases AC 660 V 
Feed speed  0~1 m/min 
Rotary speed  60 rev/min 
Fig. 15Structure and working principle of the experiment system
Before the experiments, the roadway roof must be supported stably and the auger drill must be fastened. The drilling depth is measured through the numbers of drillrods, and the drilling pressure of auger drill is taken as the criterion to judge the deflection of drillrods from the pressure gauge in Fig. 14. The experiments are divided into four groups, in the first three groups, the stabilizers are added on the drillrods per 4 meters, 6 meters, and 8 meters, and the fourth group works without stabilizer. Four groups of drilling experiments are conducted, and the drilling depth and the drilling pressure obtained are shown in Fig. 16.
Fig. 16Drilling pressure of drillrod with stabilizer and no stabilizer
It can be seen from Fig. 16 the drilling pressure increases with the number of drillrod, the upward trends of the first three groups cruves are consistent, and the fourth group cruve increases greatly after 14 drillrods, and the drilling pressure of the fourth group is the largest and the drilling depth is the smallest. For the further analysis, the drilling depth and drilling pressure increment of the experiment are obtained, shown in Table 3. It can be seen that the drilling depth reduces from 37.5 m to 27 m, and the drilling pressure increment increases from 2.5 MPa to 7 MPa. Field experience reveals that the deflection is large if the drilling pressure increases quickly, conversely the deflection is small. The experiment shows that the drilling depth of auger drill with stabilizer is 1.39 times that with no stabilizer and the drilling pressure increment decreases 2/3, and the closer of stabilizer interval, the better of effect, but for better economic, the stabilizers are added on the drillrods per 4 meters in the actual drilling process. The experiment confirms that the method of adding stabilizer is correct and effective.
Table 3Drilling depth and drilling pressure increment of the experiment
Item  Group 1  Group 2  Group 3  Group 4 
Drilling depth  37.5 m  34.5 m  30 m  27 m 
Drilling pressure increment  2.5 MPa  4 MPa  5 MPa  7 MPa 
5. Conclusions
(1) Taking the system of Coal – Drillrod – Drillbit as the researched object, considering the collision conditions of drillrod and rock, the interacted system coupling dynamics model of drilling mechanism and rock is built. Combining with the experimental data, the dynamics model is solved by ADAMS, and the vibration coupled the longitudinal, transverse and torsional of the drillrod is acquired under different compression strength rock and different rotary speeds of drill.
(2) The deflection of drillrod is caused by the complexity of coal, the nonhomogeneity of interacted force between drillbit and coal. With the decrease of coal compression strength, the transverse vibration radius is reduced; the longitudinal vibration frequency and amplitude are all reduced, too. For the same coal, the biggest vibration radius of drillrod shows exponentially growth with the drilling depth.
(3) The method of adding stabilizer is proposed. At the same condition, the drilled depth of auger drill with stabilizer is 1.39 times than that with no stabilizer. The drilling pressure increment decreases 2/3, and the closer of stabilizer interval, the better of effect, which provides an effective method to prevent deflection of auger drill.
References

Yigit A. S., Christoforou A. P. Coupled axial and transverse vibrations of oilwell drillstrings. Journal of Sound and Vibration, Vol. 195, Issue 4, 1996, p. 617627.

Yigit A. S., Christoforou A. P. Coupled torsional and bending vibrations of drillstrings subject to impact with friction. Journal of Sound and Vibration, Vol. 215, Issue 1, 1998, p. 167181.

Christoforou A. P., Yigit A. S. Fully coupled vibrations of actively controlled drillstrings. Journal of Sound and Vibration, Vol. 267, 2003, p. 10291045.

AlHiddabi S. A., Samanta B., Seibi A. Nonlinear control of torsional and bending vibrations of oilwell drillstrings. Journal of Sound and Vibration, Vol. 265, 2003, p. 401415.

Khulief Y. A., AlNaserb H. Finite element dynamic analysis of drillstrings. Finite Elements in Analysis and Design, Vol. 41, 2005, p. 12701288.

Khulief Y. A., AlSulaiman F. A., Bashmal S. Vibration analysis of drillstrings with selfexcited stick–slip oscillations. Journal of Sound and Vibration, Vol. 299, 2007, p. 540558.

Lueke Jason S., Ariaratnam Samuel T. Numerical characterization of surface heave associated with horizontal directional drilling. Tunnelling and Underground Space Technology, Vol. 21, 2006, p. 106117.

Sampaio R., Piovan M. T., VeneroLozano G. Coupled axial/torsional vibrations of drillstrings by means of nonlinear model. Mechanics Research Communications, Vol. 34, 2007, p. 497502.

NavarroLopez Eva M., Cortes Domingo. Avoiding harmful oscillations in a drillstring through dynamical analysis. Journal of Sound and Vibration, Vol. 307, 2007, p. 152171.

Rittoa T. G., Soizeb C., Sampaioa R. Nonlinear dynamics of a drillstring with uncertain model of the bit–rock interaction. International Journal of NonLinear Mechanics, Vol. 44, 2009, p. 865876.

Hakimi H., Moradi S. Drillstring vibration analysis using differential quadrature method. Journal of Petroleum Science and Engineering, Vol. 70, 2010, p. 235242.

Sahebkar S. M., Ghazavi M. R., Khadem S. E., Ghayesh M. H. Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well. Mechanism and Machine Theory, Vol. 46, 2011, p. 743760.

Gulyayev V. I., Borshch O. I. Free vibrations of drill strings in hyper deep vertical borewells. Journal of Petroleum Science and Engineering, Vol. 78, 2011, p. 759764.

Kreuzer E., Steidl M. Controlling torsional vibrations of drill strings via decomposition of traveling waves. Arhive of Applied Mechanics, Vol. 82, 2012, p. 515731.

Zhu Xiaohua, Tong Hua, Liu Qingyou, Feng Linxian. Research on the dynamic boundary condition between revolving drill string and borehole wall. China Mechanic Engineering, Vol. 18, Issue 15, 2007, p. 18331837, (in Chinese).

Zhu Caichao, Feng Daihui, Lu Bo, Yang Yingxin. Nonlinear study on dynamic action of integrated drill stringwell rock system. Chinese Journal of Mechanical Engineering, Vol. 43, Issue 5, 2007, p. 145149, (in Chinese).

Zhu Caichao, Song Chaosheng, Wang Qingfeng. Nonlinear stability analysis for helical buckling of drill string in wellbore. Journal of Southwest Petroleum University (Science & Technology Edition), Vol. 32, Issue 2, 2010, p. 11591163, (in Chinese).

Su Hua, Wang Guangyuan, Zhang Xuehong. Dynamic gap element method for contactimpact problem between slender rod and wall of a round hole. Earthquake Engineering and Engineering Vibration, Vol. 16, Issue 1, 1996, p. 7986, (in Chinese).

Xu Xiaohe. Rock Crushing Theory. Beijing, Coal Industry Press, 1984, (in Chinese).

Liu Songyong, Du Changlong, Cui Xinxia. Research on the cutting force of a pick. Mining Science and Technology, Vol. 19, Issue 4, 2009, p. 514517.
About this article
This Project is supported by Technology Research and Development Program of China (863 Program) (No. 2012AA062102), the Fundamental Research Funds for the Central Universities (Project No. 2012QNA22) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.