Anti-Vibration Method for the Near-Bit Measurement While Drilling of Pneumatic Down-the-Hole Hammer Drilling

Abstract

Pneumatic down-the-hole (DTH) hammer drilling technology has been used extensively in the fields of heat reservoir exploitation and geological exploration owing to its advantages of high efficiency and low pollution. However, the vibration near the bit is up to 40 g while DTH hammer drilling, which significantly affects the performance and longevity of the near-bit measurement while drilling (MWD). To enhance the environmental adaptability of the near-bit MWD in pneumatic DTH operations, a design method for a vibration-damping system based on the parameter optimization of a non-dominated sorting genetic algorithm II (NSGA-II) is proposed in this study. First, the whole structure of the near-bit MWD is designed, including the MWD sub-shell, sensors, measurement circuits, batteries, and connecting structures (the circuit unit). Secondly, this study analyzes the vibration characteristics of the pneumatic DTH hammer near the bit. According to the damping structure, the vibration response model for the circuit unit and the damping model are established. Thirdly, NSGA-II is employed to optimize the parameters of the damping model in terms of the low-frequency, high-intensity vibration characteristics near the bit in pneumatic DTH operations, thereby devising a damping scheme tailored to the unique conditions of DTH hammer drilling. Finally, vibration experiments were conducted to verify the effectiveness of the vibration-damping device. The experimental results indicate that within the vibration frequency range of 5–20 Hz and vibration level of 10–40 g, the peak attenuation rate of the circuit unit is more than 86.446%, and the improvement rate of the vibration stability of the system is more than 75.214%; the anti-vibration performance of the near-bit MWD system in DTH hammer drilling is improved remarkably. This study provides strong technical support for the stability of MWD equipment under such special working conditions. It has broad engineering application prospects.

1. Introduction

Pneumatic down-the-hole (DTH) hammer drilling technology has been widely used in the fields of thermal storage mining and geological exploration [1,2]. Its principle is that the piston inside the DTH hammer is mainly driven by compressed air to carry out high-speed reciprocating motion to break the rock [3], which has the advantages of high drilling efficiency and small pollution to the formation compared with traditional rotary drilling [4,5]. In the process of pneumatic DTH hammer drilling, the near-bit measurement while drilling (MWD) is crucial. It is a device that is used to monitor various important drilling parameters relating to the near drill bit in real time. By accurately measuring data, the real-time drilling status can be monitored, and the drilling efficiency can be optimized. However, the vibration at the drill bit during pneumatic DTH hammer drilling is as high as 40 g, which poses a serious challenge to the performance of the near-bit MWD. The existing MWD tools are mainly designed for oil drilling and geological drilling. Its vibration level is only 20 g. Adapting to the high-intensity and low-frequency vibrations in pneumatic DTH hammer drilling is difficult. Therefore, for the special vibration environment of pneumatic DTH hammer drilling, it is important to research and develop a new type of anti-vibration MWD to improve drilling efficiency and ensure operational safety.

Since the vibration response of a circuit unit is caused by the vibration of the drill string near the drill bit, studying the vibration characteristics of the drill string is crucial for establishing a vibration model. Many scholars have carried out research on the vibration mechanics and vibration control of drill strings. For example, Rey O F [6] used the finite difference method to analyze the effect of damping, mass, and other factors on vibration by simplifying the drilling system. Aarrestand [7] and Wei et al. [8] focused on the vibration dissipation process of a drill string by using the linear elasticity theory and experimental methods. They pointed out that damping plays an important role in the vibration of a drill string. Zhang et al. [9,10] investigated the effect of the damper’s installation position on the longitudinal vibration of a drill string by establishing a mathematical model and using a computer simulation. They determined the optimal installation position of the damper. Hakimi H [11] calculated the intrinsic frequencies of a drill string’s longitudinal, transverse, and torsional vibration by using the differential quadrature method (DQM). Bu [12] investigated the longitudinal forced vibration of a drill string under the action of cyclic impacts of the drilling cycle of a DTH hammer in hard rock. Dong [13] explored vibration and impact-suppression techniques for drill strings. They proposed three control methods: passive control, active control, and semi-active control. Further, they comparatively analyzed the characteristics of the various methods, equipment, and application scopes. Tian [14] analyzed the effects of stiffness and damping of the damper and the installation position on the vibrations of downhole drilling tools. Riane et al. [15] eliminated the severe stick–slip vibrations that appear along the drill string of a rotary drilling system according to the LQG observer-based controller approach. The above studies revealed the vibration mechanisms of a drill pipe and quantified its vibration characteristics. However, there is still a gap in the research related to the forced vibration response of the internal unit of the near-bit MWD during DTH hammer drilling and its damping technology.

In other fields, the research and development of vibration-damping methods is more mature. Du [16] proposed the principle of elastic vibration isolation and the frequency design criterion. She established a dynamic model of an engine’s elastic system and determined the dynamic characteristic parameters. Wang [17] addressed the problem of the vibration resistance of MEMS IMUs in high-g shock environments. He embedded gyroscopes and accelerometers into a symmetric hexahedral frame and utilized a viscoelastic-damping damper to achieve internal vibration damping. Zhou et al. [18,19,20,21] addressed the problem of damper–damping matching in a vehicle’s steel plate spring suspension system. They established mathematical models of the suspension system for the optimal damping ratio, and the root mean square values of dynamic deflection and vibration velocity. The optimal damping ratio of the damper in the suspension system was determined. Li [22] proposed four new passive vibration control damping forms based on inertia–spring–damping (ISD) structures. Through the establishment of dynamic equations and the acquisition of a steady-state amplitude amplification factor, the optimal damping ratio and stiffness ratio of the system were obtained. Shatskyi and Velychkovych [23] presented a new dry friction shell shock absorber design. Such shell shock absorbers are projected to be used in the mining, oil and gas industries. The development of the above damping methods is more mature, and they have good damping effects in related fields. However, these methods are mostly applied to vibration suppression in precision instruments and have not yet been utilized in the more severe environment of low-frequency, high-intensity impact vibrations.

Accordingly, this study proposes a vibration-damping system, which is based on non-dominated sorting genetic algorithm (NSGA II) parameter optimization. This method is designed to solve the problems relating to the significant longitudinal vibrations in pneumatic DTH hammer drilling processes. First, the forced vibration model of a short-section circuit unit under complex drilling conditions was constructed and analyzed. Then, the vibration-damping model of the circuit unit was established via model simplification, the separation of variables, the superposition of vibration shapes, etc. For the problem relating to the complexity of the model parameter solution, the forced vibration full response function of the circuit unit was used for analysis. This study proposes two key parameters, the logarithmic attenuation rate of the amplitude and the displacement amplification factor, with both affecting the performance of the vibration-damping system. NSGA II is used to find the optimal stiffness-damping parameters of the system. The optimal stiffness and damping parameters of the system are derived using NSGA II(Surrogates and Evolutionary Algorithms Laboratory, Indian Institute of Technology, Surat, India). Finally, a vibration-damping scheme adapted to the drilling conditions of a DTH hammer was designed. It effectively suppress the vibration amplitude of the circuit unit in a violent vibration environment. This enhances the efficiency and safety of the DTH hammer drilling operations.

2. Overall Design for the Near-Bit MWD

The MWD is installed near the drill bit for the real-time measurement of downhole pressure, temperature, and other parameters. The main structure includes the measuring sub-shell, sensor, circuit, battery, and connecting structure between the upper and lower drill pipes. Among them, the sensors are mounted on the measuring sub-shell and the circuit unit, which are used to measure the real-time downhole data; the circuit and battery are mounted on the circuit unit, which are used for signal acquisition, signal processing, and power supply. The vibration-damping device support frame is mounted between the vibration-damping device and the circuit unit, which is used to provide space for the sensor to be mounted. The vibration-damping devices at both ends are mounted tightly on the inside of the measuring sub-shell by the reducer union. The reducer union connects the MWD to the upper and lower drill pipes. The pressure detection hole is opened on the wall of the shell, which is used to measure the pressure of the annulus. According to the actual project background, the preliminary design of the MWD is 127 mm in diameter and 750 mm in length. The internal installation structure of the MWD is shown in Figure 1.

In pneumatic DTH hammer drilling operation processes, a drill string will produce very complex coupled vibrations, of which longitudinal vibrations are the most intense. Although the focus of this study is on longitudinal vibrations, it is worth noting that transverse and torsional vibrations in the drilling string system can similarly affect the performance of a system. Although these forms of vibrations were not explored in depth in this study, their presence cannot be ignored. Future research efforts will be directed towards the development of more comprehensive vibration-damping solutions to cover all types of vibration. This will further improve the reliability and measurement accuracy of the system.

To protect the circuit unit, this study adopts spring-damper elements at both ends of the circuit unit. These elements are flexibly connected to the upper and lower reducer unions. The elastic deformation of the spring element and the energy dissipation of the damping element can isolate the vibration. However, the structure and parameters of the stiffness damper directly affect the vibration-damping performance. Therefore, it is necessary to optimize the parameters of the damper.

3. The Vibration Model of Circuit Unit

To solve and design the structure and parameters of the stiffness damper, and to investigate the damping performance of the spring-damping system, it is necessary to study the vibration response characteristics of the circuit unit.

3.1. Vibration Response Model of Circuit Unit

A DTH hammer piston-bit-rock-drill string system is a complex and highly nonlinear coupled vibration system. During rock-drilling processes, there is not only the impact vibration generated by the impactor but also the reverse force generated by the rock on the drill pipe. These forces jointly provoke the complex vibration response of the drill string system, which, in turn, will directly act on the near-bit MWD [24,25]. To facilitate analysis and research, the following assumptions are made [26,27]:

(i) The drill string is in the state of linear elastic deformation, within the range of elastic deformation; (ii) the effect of shear force on the deformation of the drill string is not considered; (iii) the effect of temperature is omitted; (iv) both ends of the drill string are free, and only the horizontal displacement is limited; (v) transverse and torsional vibrations of the drill string are not considered, and only the longitudinal vibration characteristics of the drill string are investigated; and (vi) the vibration-damper is regarded as a stiff and damped damper, and the mass of the damper is on the circuit unit.

For the drill string, the impact force generated by the drill bit is an external excitation, and the drill string system is a forced vibration. For the circuit unit, the vibration response of the drill string provides excitation, and the circuit unit is forced vibration. Therefore, it is necessary to establish a dynamic model of the drill string system, including the DTH hammer impactor, drill string, circuit unit impactor, etc. The analytical model is shown in Figure 2.

3.2. Dynamic Equation of Circuit Unit

Since only the longitudinal vibration characteristics of the circuit unit are investigated, the system can be regarded as a damped single-degree-of-freedom system under external excitation [28]. Based on the mechanical structure of the circuit unit mounting described above, the circuit unit is simplified to a mass, and the mass is represented by m. The vibration-damping unit is simplified as a set of spring-damper systems, where stiffness and damping are k and c.

Due to the influence of gravity, the static equilibrium position of the damping system is not the center point. The static equilibrium position is taken as the origin of the coordinate system to establish a coordinate system. The downward direction is specified as the positive direction. Given the role of damping in energy consumption, vibration analysis is often used in the principle of energy equivalence. Moreover, the circuit unit quality does not change with movement. In this system, the elastic force of the system can be simplified to linear, and the damping can be simplified to linear viscous damping. As the deformations of the two groups of springs are consistent, they can be regarded as a set of springs in parallel with equivalent stiffness: $k=k_1+k_2$. Damping is the same, with equivalent damping $c=c_1+c_2$. Therefore, the model is further simplified, as shown in Figure 3.

The force analysis of the circuit unit is carried out, and the mass m is taken as the separator. According to Newton’s second law and D’Alembert principle, the vibration equation can be obtained.

$$m\ddot{x}\left(t\right)=-c\left[\dot{x}\right(t)-\dot{y}(t\left)\right]-k\left[x\right(t)-y(t\left)\right]$$

(1)

This is expressed as follows:

$$m\ddot{x}\left(t\right)+c\dot{x}\left(t\right)+kx\left(t\right)=c\dot{y}\left(t\right)+ky\left(t\right)$$

(2)

The above equation is the forced vibration response of the circuit unit under external excitation, where m is the mass of the circuit unit in kg, k is the stiffness of the damping mechanism, c is the damping coefficient of the damping mechanism, $\ddot{x}\left(t\right)$ is the acceleration of the measuring mechanism in m/s², $\dot{x}\left(t\right)$ is the velocity of the measuring mechanism in m/s, $\dot{y}\left(t\right)$ is the speed of the drill string in m/s, $x\left(t\right)$ is the displacement of the measuring mechanism in m, and $y\left(t\right)$ is the displacement of the drill string in m.

4. Solving the Vibration Response of Circuit Unit

4.1. Decomposition of the Vibration Response

The forced total vibration response under simple harmonic force is similar to the superposition of a general solution of the damped free vibration response and a special solution of simple harmonic vibration. Equation (2) is a second-order linear non-homogeneous ordinary differential equation, whose solution should be a superposition of the general solution of the homogeneous equation and a special solution of the non-homogeneous equation. The general solution of the homogeneous equation is the solution of the damped free vibration response of the system under no external excitation. Over time, the amplitude of the part of the general solution gradually decays. After a while, it decays to zero. This part is the transient response of the circuit unit. The special solution part of the response amplitude does not change with time; the frequency and excitation frequency are consistent, whereas the amplitude and phase depend on the excitation amplitude and system parameters, independent of the initial conditions. This part is the steady-state response of the circuit unit. The forced vibration of the whole system is the superposition of transient vibration and steady-state vibration. The full response of the vibration system is the superposition of the steady-state response and transient response of the circuit unit.

4.1.1. Transient Response

The homogeneous part of $m\ddot{x}\left(t\right)+c\dot{x}\left(t\right)+kx\left(t\right)=F\left(t\right)$ is written as follows:

$$m\ddot{x}\left(t\right)+c\dot{x}\left(t\right)+kx\left(t\right)=0$$

(3)

The damping coefficient and stiffness are known to be $c=\xi \times 2\sqrt{mk}$, $k=m{{\omega }_n}^2$, which are obtained by substituting it into Equation (3):

$$\ddot{x}\left(t\right)+2\xi {\omega }_n\dot{x}\left(t\right)+{{\omega }_n}^{2}x\left(t\right)=0$$

(4)

Let the general solution be $x\left(t\right)=X{e}^{st}$, and substitute it into Equation (4) to obtain the general solution of the equation.

$$x(t)=e^{-\xi {\omega }_{n}t}(X_1{e}^{{\omega }_n\sqrt{{\xi }^2-1t}}+X_2{e}^{-{\omega }_n\sqrt{{\xi }^2-1t}})$$

(5)

When $\xi \ge 1$, it is an over-damped case in which the circuit unit will gradually return to the equilibrium position after only one time at most, with no periodic change in the factor and no oscillation characteristics. In this case, the energy input to the system by the initial excitation is quickly consumed by the damping. The system is too late to generate reciprocating vibration and is not conducive to the damping of continuous excitation.

When $0<\xi <1$, it is an under-damped case. The transient response of the circuit unit is expanded and collated, using Euler’s formula to obtain:

$$x(t)=e^{-\xi {\omega }_{n}t}[(X_1+X_2)\mathit{cos}({\omega }_{d}t)+i(X_1-X_2)\mathit{sin}({\omega }_{d}t)]$$

(6)

Denoted by

$$x(t)=X{e}^{-\xi {\omega }_{n}t}\mathit{cos}({\omega }_{d}t-\phi )$$

(7)

where $X=\sqrt{{x_0}^2+{\displaystyle \frac{(v_0+\zeta {\omega }_n{x}_0{)}^2}{{{\omega }_d}^2}}}$ is the vibration amplitude of the transient response and $\phi =\mathit{arctan}({\displaystyle \frac{v_0+\zeta {\omega }_n{x}_0}{x_0{\omega }_d}})$ is the phase of the transient response.

This is an under-damped condition in which the circuit unit vibrates back and forth near the system’s equilibrium position. Its amplitude decreases with the exponential law $X{e}^{-\xi {\omega }_{n}t}$. The logarithmic decay rate of the amplitude is introduced here to describe the speed of amplitude decay.

$$\delta =\mathit{ln}{\displaystyle \frac{e^{-\xi {\omega }_{n}t}}{e^{-\xi {\omega }_n(t+T_d)}}}={\displaystyle \frac{2\pi \xi }{\sqrt{1-{\xi }^2}}}$$

(8)

Equation (8) is transformed into an expression of the stiffness and damping coefficients. Denoted as $\delta (k,c)$:

$$\delta ={\displaystyle \frac{2\pi c}{\sqrt{4mk-c^2}}}$$

(9)

For this system, the larger the damping rate, the faster the amplitude decay of the transient response and the shorter the time to reach the steady state. However, too large a damping rate can also result in the excitation of the system by the drill string being applied directly to the circuit unit. This will cause the circuit unit to be subjected to excessive transient forces. In engineering applications, it is generally taken that $0<\xi <0.2$. That is

$$0<{\displaystyle \frac{c}{2\sqrt{mk}}}<0.2$$

(10)

4.1.2. Steady-State Response

The periodic excitation $F\left(t\right)$ can be decomposed into the sum of infinitely many harmonic components using a Fourier expansion. That is

$$F(t)={\displaystyle \frac{a_0}{2}}+\sum _{j=1}^{\infty }{c}_j\mathit{sin}(j\omega t+{\varphi }_j)$$

(11)

where $\begin{array}{ccc}{c}_j=\sqrt{{a_j}^2+{b_j}^2}& ,& {\varphi }_j=\mathit{arctan}({\displaystyle \frac{a_j}{b_j}})\end{array}$.

For each harmonic component, a non-homogeneous special solution is obtained. According to the principle of the superposition of linear systems, the steady-state characteristics of the system can be obtained. The second-order non-homogeneous differential equation of arbitrary harmonics is written as follows:

$$\ddot{x}\left(t\right)+2\xi {\omega }_n\dot{x}\left(t\right)+{{\omega }_n}^{2}x\left(t\right)={{\omega }_n}^{2}A\mathit{cos}(\omega t)$$

(12)

Let the special solution of the above equation be $x\left(t\right)=X\mathit{cos}(\omega t-\phi )$, and substituting it into the differential equation:

$$X\left[\right({{\omega }_n}^2-{\omega }^2)\mathit{cos}\phi +2\xi {\omega }_n\omega \mathit{sin}\phi ]\mathit{cos}(\omega t)+X[({{\omega }_n}^2-{\omega }^2)\mathit{sin}\phi +2\xi {\omega }_n\omega \mathit{cos}\phi ]\mathit{sin}(\omega t)={{\omega }_n}^{2}A\mathit{cos}(\omega t)$$

(13)

The above equation can hold for any time, so the coefficients of the terms $\mathit{cos}(\omega t)$ and $\mathit{sin}(\omega t)$ on both sides of the equal sign must be equal. That is

$$\left\{\begin{array}{c}X\left[\right({{\omega }_n}^2-{\omega }^2)\mathit{cos}\phi +2\xi {\omega }_n\omega \mathit{sin}\phi ]={{\omega }_n}^{2}A\\ X\left[\right({{\omega }_n}^2-{\omega }^2)\mathit{sin}\phi +2\xi {\omega }_n\omega \mathit{cos}\phi ]=0\hfill \end{array}\right.$$

(14)

The solution is $X=A\left|H\left(\omega \right)\right|$, $\phi =\mathit{arctan}\left[{\displaystyle \frac{2\xi \omega /{\omega }_n}{1-(\omega /{\omega }_n{)}^2}}\right]$, where $\left|H\left(\omega \right)\right|$ is the amplification factor:

$$\left|H\left(\omega \right)\right|={\displaystyle \frac{1}{\sqrt{[1-(\omega /{\omega }_n{)}^2{]}^2+(2\xi \omega /{\omega }_n{)}^2}}}$$

(15)

Equation (14) is transformed into an expression of the stiffness and damping coefficients. Denoted as $\beta (k,c)$:

$$\beta (k,c)={\displaystyle \frac{k}{\sqrt{(k-m\omega {)}^2+c^2{\omega }^2}}}$$

(16)

$\beta (k,c)$ indicates the magnification of the static displacement amplitude under dynamic vibration. The smaller the amplification factor, the better the damping effect. Furthermore, only when the amplification factor is greater than 0 but less than 1 will the amplitude of the vibration will be attenuated. Therefore, in the parameter design of the vibration-suppression system for the MWD, the amplification factor should be ensured to be small enough. That is

$$0<\beta <1$$

(17)

At the same time, the vibration isolation effect of the damping system is achieved when the frequency ratio is greater than $\sqrt{2}$. However, to ensure the stability and reliability of the system, it is usually no greater than 5. That is

$$\sqrt{2}<{\displaystyle \frac{\omega }{{\omega }_n}}<5$$

(18)

4.2. Solving for Optimal Stiffness and Damping Coefficient of the Damper

Through the decomposition of the vibration response of the above circuit unit, two main parameters, the logarithmic attenuation rate of the amplitude and the displacement amplification factor, are obtained. They are both functions of damping and stiffness, and the two parameters will affect each other when seeking their optimum values. Therefore, the two functions are used as the objective functions, which are transformed into a multi-objective optimization problem. The optimal stiffness and damping parameters of the vibration system can be obtained by seeking the optimal solution through a suitable algorithm.

The multi-objective optimization problem is more than a single-objective optimization problem, multiple objective functions may conflict. Therefore, the result is an optimal solution set. In this study, NSGA-II was chosen to solve the optimal stiffness damping parameters of the vibration system, which has the advantages of fast convergence, high computational efficiency, and good stability compared with other multi-objective optimization algorithms.

Based on the previous section, Equations (9) and (16) are used as the objective functions of this multi-objective optimization problem, and Equations (10), (17) and (18) are used as the boundary conditions of this function. This is sorted out as follows:

$$\left\{\begin{array}{c}f(k,c)\left\{\begin{array}{c}\mathit{min}\beta (k,c)={\displaystyle \frac{k}{\sqrt{(k-m\omega {)}^2+c^2{\omega }^2}}}\\ \mathit{max}\delta (k,c)={\displaystyle \frac{2\pi c}{\sqrt{4mk-c^2}}}\hfill \end{array}\right.\\ s.t.\left\{\begin{array}{c}0<\beta <1\hfill \\ 0<{\displaystyle \frac{c}{2\sqrt{mk}}}<0.2\hfill \\ \sqrt{2}<{\displaystyle \frac{\omega }{{\omega }_n}}=\omega \sqrt{{\displaystyle \frac{m}{k}}}<5\hfill \end{array}\right.\hfill \end{array}\right.$$

(19)

A non-dominated sorting genetic algorithm was developed based on the above objective function and boundary conditions. To optimize programming and calculation, take the reciprocal of the displacement amplification factor and find the maximum of both objective functions simultaneously using the algorithm.

An initial set of populations is first generated randomly when the algorithm is executed. Each spring-damping parameter represents a set of solutions and ensures that these initial values satisfy the boundary conditions in Equation (19). Then, a hierarchy is performed based on the dominance of the spring-damped parameter over the set of multi-objective functions. This can ensure that each individual parameter solved maintains the property of being a non-inferior solution on at least one objective. Next, a crowding calculation is performed for each spring-damped parameter to quantify the intensity of the solution. Thus, this promotes the diversity and equilibrium of the solution set. After that, the spring-damped parameter with a higher grade is preferred. According to the congestion index, the spring-damped parameter with a sparse distribution is selected in each layer. Then, it is added to the population of the subsequent evolution. Then, new populations are obtained through the use of two genetic operations, crossover, and mutation, to increase the diversity of solutions within the population. After the newly generated individuals are merged with the existing population through non-dominated sorting and crowding evaluation, the individuals with higher priority are again filtered to form the new generation of the population. This is carried out until the maximum number of iterations is reached. Eventually, the algorithm will converge to a set of high-quality optimal solution sets, resulting in the optimal stiffness and damping coefficients that are applicable to the model. The optimization process is shown in Figure 4.

The known circuit unit prototype is 8 kg, and the DTH hammer (China University of Geosciences, Beijing, China) used in the mining of geothermal resources has an operating frequency range of 5–15 Hz. Let the impact frequency be 10 Hz, and then substitute it into Equation (19). A set of optimal stiffness and damping coefficients are obtained using MATLAB R2023a (MathWorks, Natick, MA, USA), which are rounded to $k=128$, $c=12$, respectively. The optimal displacement amplification factor and amplitude attenuation rate of $\beta =0.2570$ and $\delta =1.1994$ are obtained, respectively. The parameter that measures the damping effect of the system is the absolute motion conductivity—$T_d=\sqrt{{\displaystyle \frac{1+(2\zeta \lambda {)}^2}{(1-{\lambda }^2{)}^2+(2\zeta \lambda {)}^2}}}=0.0922$—which damps up to 90.78%.

5. Vibration Simulation Experiment of DTH Hammer

5.1. The Experiment Platform

5.1.1. Prototype for MWD

The parameters of the spring-damped vibration-damping system are calculated as $k=128$ and $c=12$. Two sets of vibration-damping systems consisting of ten spring-damper elements connected in parallel are designed. Their structure is shown in Figure 5. Since the calculated parameters of each spring-damper element are non-standard parts, they need to be customized independently. To reduce the cost, this study chooses the standard parts with stiffness and damping coefficients of $k=6.25$ and $c=0.5$, respectively. Its total stiffness and damping coefficient are $k=125$ and $c=10$, respectively.

They are installed between the two reducer unions and the circuit unit, respectively. By tightening the threads between the reducer union and the shell, the dampers at both ends are compressed to achieve preloading. The components of the prototype are shown in Figure 6.

5.1.2. Vibration Table

A DC4000-40 electrodynamic vibration table (Sushi, Suzhou, China) was chosen to perform the simulation experiment. The principle of this experiment is to input different spectral density functions into the vibration table to simulate the longitudinal vibration of the drill pipe near the bit when a DTH hammer strikes the rock. In this way, the vibration characteristics and parameters of the MWD and the circuit unit are studied.

In the simulation experiment, the vibration table will apply the excitation force to the MWD. The applied excitation force approximates the excitation force of the drill bit to the upper drill pipe in a real drilling operation. It is used as the external excitation of the vibration-damping system. To isolate the vibration of the drill pipe, the damper is mounted between the drill pipe and the circuit unit. The accelerometer mounted inside the circuit unit is used to measure the vibration signal after the vibration damper has damped the vibration. The accelerometer mounted on the surface of the vibration table is used to measure the actual vibration signal of the MWD. By comparing the measured signals after damping with the signals before damping, we can investigate the vibration attenuation effect of the damper.

5.2. Experimental Content

5.2.1. Investigating the Vibration-Damping Performance at Different Frequencies

The frequency of the pneumatic DTH hammer during the mining of geothermal resources is 5 Hz–15 Hz, mostly around 9 Hz. Therefore, to investigate the vibration damping of the DTH hammer drilling rock with different frequencies at the same vibration level, the vibration level is controlled to 10 g. The damping effect of the damping device is investigated at frequencies of 5 Hz, 7 Hz, 9 Hz, 12 Hz, 15 Hz, and 20 Hz, respectively.

In the experiment, the MWD is mounted on the surface of the vibration table using the mounting base. One of the sensors is installed at the bottom of the measuring sub-shell, as shown in Figure 7. The direction is upward in the direction of the vibration. The other sensor is installed in the internal circuit unit, as shown in Figure 6, and it is not in contact with the measuring sub-shell. The direction is upward in the direction of the vibration. Then, the vibration table controller is configured. The vibration mode is that of a classical impact. The output level is 10 g. The control of different frequencies is achieved by adjusting the pulse interval. Multiple sets of experiments are performed for each frequency. Finally, using the host computer to collect the data for each group of frequency experiments, five pulses were selected to draw images for comprehensive analysis. The data are as follows:

The green curve presented in Figure 8 shows the vibration level of the measuring sub-shell as the vibration level before vibration damping. The red curve shows the vibration level of the circuit unit as the vibration level after vibration damping. The above six groups of experimental curves verified that the vibration-damping device can effectively reduce the harmful vibrations of the circuit unit in the frequency range of 5 Hz–20 Hz.

After completing six groups of experiments with different frequencies at the same level. The damping effect of the damping device was quantitatively analyzed with the help of four parameters: peak difference, the peak attenuation rate, fluctuation rate, and the system stability enhancement rate. These parameters were chosen based on their generality and validity in terms of vibration analysis. The peak difference and attenuation rate visualize the changes before and after vibration damping, while the fluctuation rate and system stability enhancement rate further help us to understand the stability and reliability of the device. Therefore, the combination of these parameters can provide powerful data to support the design of vibration-damping devices. The calculated collated results are shown in

Table 1. Comparison of the data before and after damping in six groups of experiments with different frequencies at the same level.

Frequency	Peak Difference	Peak Attenuation Rate	Fluctuation Rate		System Stability Enhancement Rate
			Before	After
5 Hz	9.086	88.247%	1.012	0.127	87.513%
7 Hz	9.174	87.756%	0.866	0.124	85.626%
9 Hz	9.450	87.825%	1.377	0.229	83.303%
12 Hz	10.474	90.029%	1.423	0.212	85.095%
15 Hz	8.719	88.072%	0.856	0.201	76.600%
20 Hz	9.947	86.443%	1.113	0.276	75.214%

below.

By comparing the fluctuation rates before and after vibration damping in each group of experiments, the efficiency of the damping device in improving the stability of the system in each group of experiments can be obtained. As shown in Table 1, the effect of the vibration-damping device on the stability of the system in terms of enhancing the efficiency of the system is greater than 75.214% in the case where a vibration level of 10 g and a frequency range of 5–20 Hz are used. The enhancement effect is significant. The amplitude attenuation rate of the vibration-damping device on the system is more than 86.443%, which is similar to the theoretical analysis result of 90.78% presented in the previous section. The vibration-damping performance is good.

5.2.2. Investigating the Vibration-Damping Performance at Different Levels

Compared with other drilling processes, pneumatic DTH hammer drilling has more intense shock vibration, up to 40 g. Since the common frequency of DTH hammer drilling is about 9 Hz, in this experiment, five levels of 10 g, 15 g, 20 g, 30 g, and 40 g were explored at an excitation frequency of 9 Hz. This research focuses on the damping effect of the damping device under different vibration levels.

The device required for this section of the experiment is the same as that used in the previous section. The frequency is set to 9 Hz. The control of different experiments is achieved by adjusting the output level. Then, several groups of experiments were carried out separately for each vibration level. A pulse was selected for each group of experiments to be analyzed after the data were collected using the host computer. The data are as follows (Figure 9):

To more intuitively understand the damping performance of the damping device at different levels, the four parameters used in the previous section are still used to describe the damping effect. The calculated collated results are shown in

Table 2. Comparison of data before and after damping in five groups of experiments with different levels at the same frequency.

Level	Peak Difference	Peak Attenuation Rate	Fluctuation Rate		System Stability Enhancement Rate
			Before	After
10 g	8.988	94.710%	1.345	0.085	93.660%
15 g	13.906	91.978%	2.707	0.247	90.862%
20 g	20.598	94.816%	2.346	0.374	84.053%
30 g	26.938	86.549%	5.932	0.704	88.132%
40 g	34.750	90.938%	6.975	0.738	89.413%

below.

The data show that the designed vibration-damping device has a good damping effect at different levels of excitation under the same vibration frequency. By calculating the peak attenuation rate of the vibration signal before and after damping, it can be seen that with the increase in excitation level, the attenuation rate of the damping system on the input peak value decreases, the stability of the system also gradually reduced, and the damping effect decreases. The peak attenuation rate and the enhancement effect on the stability of the system in the vibration range with a frequency of 9 Hz and vibration level of 10–40 g can be guaranteed to be above 86.549% and 84.053%. The vibration-damping performance is good.

6. Discussion

In this study, an innovative method for the vibration damping of the near-bit MWD is proposed for use in the field of pneumatic DTH hammer drilling. The vibration-damping system designed using this method is highly stable. It can achieve an optimal vibration-damping effect concerning the mass and excitation frequency of the MWD, which significantly improves the accuracy and service life of the MWD in the drilling process. Meanwhile, the parameter optimization method based on NSGA-II is used to solve the optimal damping parameters. The method not only significantly improves the computational efficiency of the damping system design but also enhances the variety of parameter choices and the accuracy of the optimization results. Thus, this study provides a highly efficient and reliable vibration-damping solution for pneumatic DTH hammer drilling projects.

However, there are some shortcomings in this research process:

• The lower impact of transverse and torsional vibrations was neglected: In the investigation of the vibration characteristics of the circuit unit, this study mainly considered the impact of longitudinal vibrations and simplified or neglected torsional and transverse vibrations, as well as the three kinds of vibration coupling effects. These vibrations will have a certain impact on the vibration-damping performance of the vibration-damping device. Despite the fact that the simulation experiment process shows a good vibration-damping effect, the effect of this coupled vibration on the vibration-damping device has not yet been fully investigated in relation to the practical applications concerning impact. Further studies are needed to investigate the interference of the torsional vibrations of the drill string and its coupling with transverse vibrations on the damping device.
• Differences between the simplified model and the actual working conditions: This study establishes a forced vibration response model of a circuit unit by studying the vibration characteristics of a drill string near a drill bit and finds the optimal damping parameters of a damping system. However, many uncertainties concerning the actual drilling process may have an impact on the performance of the vibration-damping system, such as formation changes, fluctuations in drilling fluid performance, etc. These factors are not fully reflected in the model. Future research work will address these shortcomings and consider more practical working condition factors to further improve and optimize the model in order to achieve more accurate and reliable vibration damping.

7. Conclusions

Based on the actual engineering background of the pneumatic DTH hammer drilling, this study researched the vibration-damping method for the internal circuit unit of the near-bit MWD. The following conclusions can be drawn:

• Aiming at the high-intensity and low-frequency vibration problems faced by a near-bit MWD in a pneumatic DTH hammer drilling technology, a vibration-damping system design method was innovatively proposed. The method has a certain adaptive ability. It can achieve an optimal vibration-damping effect throughout the drilling process.
• Through the modeling analysis and parameter optimization of the vibration characteristics of a drill string near a drill bit, the vibration-damping device based on the parallel connection of multiple spring-damper elements was innovatively designed. Its effectiveness was verified through the use of vibration simulation experiments involving a DTH hammer. The experimental results show that, in the vibration frequency range of 5–20 Hz and a vibration level range of 10–40 g, the vibration-damping device can achieve a peak vibration attenuation rate of more than 86.446% for the circuit unit and an improvement rate of more than 75.214% in terms of system vibration stability. The device significantly improves the vibration-damping performance in the drilling process.
• The experimental results show that the vibration-damping device attenuates the peak vibration of the circuit unit by 94.8165% and improves the system vibration stability by 84.053% under normal drilling operations. Under extreme conditions, the vibration peak attenuation rate is 90.9384%, and the system vibration stability enhancement rate is 89.413%.

In summary, this study provides an effective vibration-damping solution for a near-bit MWD in a pneumatic DTH hammer drilling and develops a vibration-damping system design method. This has a wide range of engineering applications. It is of great significance for improving the stability and service life of MWD.