Analysis of the Longitudinal-Bending-Torsional Coupled Vibration Mechanism of the Drilling of a Roof Bolter for Mine Support System

The condition of the drilling of a roof bolter for mine support system is complex, and the existing analytical approaches for the vibration of the drilling for oil wells cannot be fully applicable to it. In order to reveal the vibration mechanism of bolt drilling, some models for longitudinal, bending, and torsional coupled vibration were established based on the energy method and finite element method. A global matrix was assembled in the order of units. Taking the drilling condition as the boundary conditions, a reduced-order model was constructed. The simulations for drilling mudstone show that the order of mode shape is positively correlated with the corresponding natural frequency, and the low-order mode shape plays a decisive role in the vibration characteristics of a drill string. The order of causing drill string displacement from large to small is as follows: longitudinal vibration, torsional vibration, and bending vibration. Compared with the bit, the displacements of longitudinal and torsional vibration of the tail of the drill string are decreased by 93.2% and 84.7%, respectively. The perturbed force and reaction torque are the main reasons for inducing the coupled vibration of it. This research provides a theoretical basis for analysing the vibration characteristics of the drilling of roof bolters for mine support systems and developing their absorbers.


Introduction
Roof bolters are key mechanical equipment for roadway supports for coal mines, which can realize fragmentizing coal rock and drilling by the high-speed impact and rotation of drill string. ere is a complex vibration in drill string in operation under the combined excitation of di erent geotechnical parameters, axial thrust, and torque. e main vibration forms including bending vibration, longitudinal vibration, and torsional vibration, and various vibrations are nonlinear coupled, which possibly lead to the stick-slip and bit-bounce of drill string [1]. ese are the main reasons for the premature failure of drill string and a ecting the support e ect.
Currently, there are transfer matrix method [2], lumped mass method [3], generalized coordinate method [4], hypothetical mode method [5], nite element method [6,7], elastic line method [8], and other methods to analyse the vibration mechanism of drill string. For a drill string system with multiple pipe sections, Han et al. calculated the total vibration transfer matrix by multiplying all individual matrices; each is obtained for an individual pipe section [2].
is method simpli es the modelling process and vibration analysis. However, its essence is to solve higher-order algebraic equations. It is di cult to obtain accurate solutions while dealing with multi-DOF vibration systems. Chen et al. concentrated the masses along the drill string and proposed a kinematic coupled approach to correlate the axial displacement of the bit to the rotation angle of the bit [3]. However, this approach only discusses torsional and longitudinal coupled vibration, and the natural frequency error increases with the increase of the order of the mode shape.
Cheng et al. discretized the drill string into some absolute nodal coordinate beam units and established its kinetic models of coupled vibration, which can describe the coupled vibration phenomenon of the drill string [4]. e key to this approach lies in the physical coordinates of the drill string. Moreover, and the selected mode shape must meet all the boundary conditions given by the system, which is difficult to operate and harsh to use. Jeong and Yoo proposed a hypothetical mode method to conduct the static or dynamic analysis of a flexible beam undergoing large deflection, and simplify the expressions of strain energy and geometric constraints among deformation variables [5]. However, this method uses a finite number of hypothetical modal vibrations to approximately simulate it, which requires many nodal coordinates and cumbersome calculation. Ren et al. presented a coupled model for axial/torsional/lateral vibrations of the drill string, in which the nonlinear dynamics and qualitative analysis method are employed to find out the key factors and sensitive zone for coupled vibration [6]. Deng et al. obtained the time domain response of the drill string based on the shape and strength of the geotechnical surface [9]. Ritto et al. proposed a stochastic computational model by the nonlinear Timoshenko beam theory to model uncertainties in the bit-rock interaction model [7]. However, the 3-order elastic matrix in 2D plane is used will reduce the accuracy of the whole drill string model while calculating the strain energy of it. Tucker and Wang analysed the stability of a drill string under torsional, axial, and bending coupled disturbances by the elastic line method and transformed an exact motion equation into an ordinary differential-difference equation [8]. e space does not need to be discretized by this method; thus, it is suitable for higher frequency vibration.
Most of the research objects in the above work are the drill pipes for oil wells, which are essentially different from the drill strings for mine support systems, which are mainly reflected as follows: (1) there are large differences in structure and size between them. A drill pipe system for oil wells is composed of a drilling rig, a top drive system, drill pipes, a drill collar, and a drill bit. e drill pipes are connected by bolts and have a length of hundreds of meters. However, the drill string of the roof bolter is mostly a whole steel pipe wound with spiral blades, with a length of 0.5 to 3 m. (2) eir stress conditions are very different. e force loading on the drill string of the roof bolter during construction is much less than that of another. e existing analytical methods for the drilling vibration of the drill pipes for oil wells are not completely suitable for the roof bolter for the mine support system. In this paper, a dynamic model for 6 degree of freedom (DOF) bolt drilling will be established; a set of mass, stiffness, damping, and force loading matrixes will be assembled; and the order of the finite element models will be reduced. ese are aimed at revealing the longitudinal-bending-torsional coupled vibration mechanism of the bolt drilling. rough the modal analysis and vibration response analysis for drilling mudstone, the law and change trend of the coupled vibration of the drill string will be explored. It is expected to provide a theoretical basis for analysing the vibration characteristics of drill strings and developing absorbers.

Underlying Assumption.
For the nonlinear coupled of the 3 types of vibration forms of a drill string, there are 6 assumptions as follows: (1) e drill string is a thin straight rod with equal cross section, and the cross section is circular; (2) e drill bit is regarded as a rigid body, and the body of the drill string as an elastomer with small deformation; (3) Ignoring the influence of temperature rise caused by friction between drill pipe and coal rock during the process of fragmentizing coal rock [10,11]; (4) e direction of axial thrust is along the axis of the drill string; (5) e displacement of the bending vibration of the drill string is decomposed into x and y axes; (6) e drill string is regarded as an Euler-Bernoulli beam, namely, the effects of shear deformation and rotational inertia are ignored.

Kinetic Energy Equation.
As shown in Figure 1, the drill string system is discretized into 1 to n units, and each unit has 6-DOF. e displacement of longitudinal vibration is u; that of bending vibration are v and w; the rotation angle around x, y, and z axis is θ x , θ y , and θ z , respectively. e translational kinetic energy of the unit of drill string (UDS) is expressed as [12] T p � 1 2 l 0 ρAs T s dz. (1) According to Figure 1, the translational velocity of the unit is Substitute (2) into (1), As shown in Figure 2, the rotation velocity of the unit is expressed by Euler angle. Firstly, rotating the global coordinate system o − xyz around the y-axis by an angle ψ to get the coordinate system o − x 1 y 1 z 1 ; secondly, rotating it again around the z 1 -axis by an angle θ to get the coordinate system o − x 2 y 2 z 2 ; finally, rotating it once more around the x 2 -axis by an angle φ to get the coordinate system o − x 3 y 3 z 3 .
Using Euler kinematics equation [13], the rotational velocity is shown as, e rotational kinetic energy of the UDS is expressed as [12] T z 1 2 R s · J · R s .
Substitute (4) into (5), Due to the rotation angle of the drill string restricted by coal rock, there is an assumption as Substitute (7) into (6), According to simultaneous (3) and (8), the total kinetic energy of the UDS is shown as

Strain Energy Equation.
e strain energy of the UDS is described as follows [14]: Due to the length of the drill string is much larger than its diameter, the stress and strain are mainly concentrated in the normal direction of the cross-section of the drill string. Its components can be simplified as Substituting (11) into (10), the strain energy of the UDS is where λ � (2μ − 1)(μ − 1)/(μ + 1). e strains of the UDS are as follows [15]: According to the assumption of Euler-Bernoulli beam [16], compared with bending deformation, shear deformation can be ignored, thus, Substituting (14) into (13), and then substituting the result into (12), the linear strain energy of the UDS is as follows: e nonlinear strain energy of the UDS is generated by longitudinal vibration, longitudinal-torsional coupled vibration, longitudinal-bending coupled vibration, and bending-torsional coupled vibration.
e expressions of them are as follows: According to simultaneous (16) and (19), the total strain energy of the UDS is shown as

Mass Matrix.
Taking any unit on the drill string to analyse, the upper and lower nodes of it are represented by j and i, respectively. e unit coordinate systems j-xyz and ixyz are established respectively, as shown in Figure 3. ere are 6-DOF for each node of the discrete model of the UDS. e displacement matrix of the unit is as follows.
According to the displacement shape function [17], each displacement matrix of the UDS is as follows: where (9) by Hamiltonian principle [18,19], ignoring infinitesimal quantity, the mass matrix [M e ] of the UDS is obtained as (23).

Stiffness Matrix.
e linear strain energy of the UDS is shown in (15). In the same way as solving the unit mass matrix, substituting (22) into (15), the linear stiffness matrix of the UDS is shown in (24). Substituting (22) into equations (16) to (19), the nonlinear stiffness of the UDS is shown in equations (25) to (28). e stiffness matrix of the UDS, combined with the linear and nonlinear ones of it, is shown in (29).

Damping Matrix.
According to the Rayleigh damping [20], a unit damping matrix [21] is built by combing the mass matrix and linear sti ness matrix of the UDS with a certain proportion coe cient.
where α 2ξω a ω b /ω a + ω b and β 2ξ/ω a + ω b . e damping matrix changes with the natural frequency of the drill string. e frequency band that is easy to induce vibration can be obtained according to the kinetic characteristics of the drill string structure and the excitation of force loading.

Force Loading Matrix.
ere is mainly gravity (F g ), axial thrust (F a ), perturbed force (F r ), impact force (F c ), frictional force (F m ), reaction torque (M o ), and torque (M t ) on the drill string. According to the displacement matrix shown in (21), the directions of gravity and axial thrust are along the axis of the drill string, and their equivalent nodal forces are as equations (31) and (32). e torque rotates along the axial direction of the UDS, and its equivalent nodal force is shown in (33). If the lateral displacement is greater than the distance between the drill string and coal rock, it is judged that an impact collision occurs. e impact forces in the x-axis and y-axis are, respectively, as equations (34) and (35) [22], and their equivalent nodal force is shown in (36). e frictional force between the drill string and coal rock is shown in (37), and its equivalent nodal force is shown in (38) [23].
3.5. Assembly of Unit Matrixes. According to the order of units, the global force loading matrix is assembled, as shown in equation (39).
Since the local coordinate system of each unit is consistent with the global one, coordinate transformation is not required. Equations (23) and (29) are assembled according to the order of the units by using the direct stiffness method [24,25]. In this paper, the number n of the units is 7 (8 nodes). e characteristics of the global mass and stiffness matrix of the drill string are obtained by MATLAB, as shown in Figure 4.
As shown in Figure  ere is obvious symmetry in Figure 4(a), while asymmetric in Figure 4(b). Consequently, the global stiffness matrix is singular and irreversible.
After the UDS matrix is assembled into a global matrix, the kinetic equation of the drill string is shown as

Boundary Condition
Working conditions of the drill string are as follows: (1) the axial thrust and torque are applied to the tail of the drill string by the roof bolter; (2) the drill string drill along its axis. Consequently, the boundary conditions of the drill string system are as follows: (1) all DOF of the first node of the drill string are constrained, in which the node displacements of the element are all zero. (2) the lateral displacements of the 8 th node is constrained, namely, in which the lateral displacements in x and y axes are eliminated. e boundary conditions include global mass, stiffness, damping, and force loading and can be expressed as Equation (41) is applied to (40) to eliminate the DOF of the displacement of the corresponding node so as to eliminate the singularity of the global sti ness matrix of the drill string.
A drilling model is constructed according to the above drill string, mudstone parameters, and boundary conditions. As shown in Figure 5, point "RP" is xed, and the drilling direction is along the shaft of the drill string. A drilling simulation is implemented by ABAQUS [26], and then the perturbed force and reaction torque of mudstone on the drill string are obtained. e rst six modal shapes of the drill string are obtained by ABAQUS used for the modal analysis of it, as shown in Figure 6. e rst to third order mode shapes of the drill string are oscillating; and the fourth to sixth ones are secondorder bending. e natural frequencies of the rst six modal shapes are 0.051, 0.224, 0.314, 0.715, 0.870, and 1.094 Hz, respectively. e order of the mode shape is positively correlated with the corresponding natural frequency. e higher the natural frequency, the smaller the vibration period and the faster the attenuation, and vice versa. It means that the low-order mode shape plays a decisive role in the vibration characteristics of the drill string.
Consequently, the natural frequencies of the rst and second mode shapes are employed, and their corresponding angular frequencies are as follows: ω a 0.320 rad/s and ω b 0.224 rad/s. Taking the ξ 0.0048, substituting the parameters above into (30), the scale coe cients are obtained as follows: α 1.26 × 10 − 3 and β 1.76 × 10 − 2 .

Interpolation of Perturbed Force and Reaction Torque.
To substitute the perturbed force and reaction torque into (39), it is necessary to interpolate them. e perturbed force and reaction torque of the mudstone on the drill string are imported into the "ppval" function of MATLAB for cubic interpolation. As shown in Figure 7, " * " indicates data points; the sampling interval is 0.001 s; and the sampling time t is set to 3 s. It can be seen from Figure 7(a)    Mathematical Problems in Engineering  Figure 5: Drilling model. Step: Step-  Mathematical Problems in Engineering string rebounds. As can be seen from Figure 7 .87], the reaction torque on the drill string is zero. At this time, the drill string is stuck and stops suddenly, and the actual revolving speed is zero. [27] in MATLAB, the responses to the coupled vibration on the drill pipe are output to 7 oscilloscopes, respectively, as shown in Figure 8. Substitute (41) into (40), the modi ed global matrix, including mass, sti ness, damping, and force load, are substituted into the " nite element model." e calculation process can be described as follows. e coupled vibration displacement responses to the drilling of roof bolter are solved by using the "ode23tb" decoder in Simulink, as shown in Figure 9.

Displacement of Unit Vibration. Employing Simulink
As shown in Figure 9, u n , v n , w n , and θ n x are the responses to the longitudinal, x-axis, y-axis, and torsional vibration displacement of the UDS, respectively. e positive or negative values of the vibration displacement curves only represent their directions. e variation intervals of u 8 to u 2 shown in Figures 9(a)      , respectively. eir vibration responses curves are similar, and the amplitudes of them decrease gradually. Compared with the bit, the longitudinal vibration displacement of the tail of the drill string is decreased by 93.2%. ere is zero in the curves of the longitudinal vibration displacements during this period, which is due to the bouncing movement of the drill string that is consistent with the rebound time of the drill string in Figure 7(a). Compared with Figures 9(a) to 9(f ), the amplitude of u 8 increased signi cantly, which shows that during process of fragmentizing coal rock, the longitudinal vibration is rst transmitted to the drill bit, and then attenuated to each unit of the drill string in turn. It means that the perturbed force is the main cause of inducing the longitudinal vibration.
e v 8 ∼ v 2 and w 8 ∼ w 2 curves in Figures 9(a) to 9(f ) are shown as mirror symmetry, but the amplitudes of them is not consistent, which shows that the characteristics of the impact forces caused by the bending vibration are similar, but their magnitudes are di erent.  Figures 9(b) and 9(a) to 9(g), it is found that the curve of reaction torque is linearly related to the curve of the torsional vibration displacement of the drill string. eir vibration displacements are intermittently zero, indicating that the drill strings are stuck, which is consistent with the sticking time in Figure 7(b).
is means that the uneven change of the reaction torque is the main cause of the torsional vibration.

Conclusions
(1) Based on the energy method and nite element method, combined with the boundary conditions of the cantilever beam, and drilling along the axis of the roof bolter, the reduced-order kinetic models of the bending, longitudinal, and torsional coupled vibrations of the drill string are constructed. e coupled vibration displacement responses of each unit of the drill pipe are obtained, in which the coupled vibration mechanism of the drilling of a roof bolter for a mine support system has been revealed. (2) According to the simulated analysis of mudstone drilling, the order of causing drill string displacement from large to small is as follows: longitudinal vibration, torsional vibration, and bending vibration. e curve of reaction torque is linearly related to the curve of the torsional vibration displacement of the drill string. e displacements of the longitudinal vibration and torsional vibration decrease gradually from the bit to the tail of the drill string. In the instance of this paper, the longitudinal and torsional vibration displacements of the tail are decreased by 93.2% and 84.7%, respectively, compared with those of the tip. e dynamic characteristics of the tail of the drill string under coupled vibration can be predicted according to the longitudinal and torsional vibration displacement of the bit.
(3) e bending vibration displacement curves along the x and y-axes are shown as mirror symmetry, but the amplitudes of them are not consistent. Moreover, they are intermittently zero, indicating that the contact collision between the drill string and the mudstone does not occur continuously. e perturbed force and reaction torque are the main causes of inducing the coupled vibration. To reduce the vibration amplitude of drill string and avoid tripping and sticking, the axial thrust and torque of drill string can be reduced, which can reduce its disturbing force and reaction torque.