Optimal Design Parameters of a Percussive Drilling System for Efficiency Improvement

0is paper aims to determine the optimal design parameters for percussive drilling systems considering the bit-rock interaction. First, the motion dynamics of a bit impacted by a dropped piston are modeled by impact stress propagation and a rock-breaking mechanism representing the penetration resistance coefficient and unloading constant. Next, the damping vibration behavior of the bit is investigated considering the impact duration and the rock loading/unloading condition. In addition, the proposed dynamics are simplified by adopting two dimensionless parameters representing the bit-piston mass ratio and the rock-piston stiffness ratio. Finally, the drilling efficiency, defined by the energy transmitted from the piston to the rock, is presented in terms of the proposed parameters. 0e use of optimal design parameters for percussive drilling systems improved the drilling efficiency. 0ese results are applicable to the design and performance estimation of down-the-hole and top-hammer systems.


Introduction
Drilling equipment is the collective term used for machines that apply impact and rotation forces to drill (for the most part) surfaces and blastholes, and it is classified as tophammer drilling (THD), down-the-hole (DTH) drilling, and rotary drilling (RD) rigs, depending on the operating method (Figure 1).In general, THD is used mostly for mining and civil blasting works, where the rig drills into the earth usually to a depth of 1-20 m and at most 40 m; DTH is used mainly for groundwater development and can create holes to a maximum depth of 4000 m; and RD bores the deepest holes, most commonly for petroleum gas extraction and geothermal development, being propelled by its own weight to reach depths of up to 10,000 m [1].
Simply put, a percussive drilling mechanism utilizes the percussive energy resulting from the repeated impact of the drifter (THD rigs) or the DTH hammer (DTH rigs) and the feed force and rotation force that are transmitted to the drill bit through the drill rod.e energy generated from the repeated impacts is then converted into wave energy, which is transmitted to the rock via the drill bit.Finally, the drill bit, now with enough impact energy for drilling, cuts into and crushes the rock.e rate at which the impact-generated energy in a percussive drilling system is transmitted (i.e., the drilling efficiency) is determined by complex effects such as drilling rod, coupling sleeve, the compressive strength of the rock, and interactions between the drill bit and the rock.erefore, certain drilling methods are highly efficient, with high rates of penetration, when drilling soft rock (uniaxial compressive strength, UCS, < 20 MPa) or medium hard rock (UCS 50-120 MPa), but the efficiency decreases when drilling very hard rock (UCS > 200 MPa) [2].
ere are numerous previous studies regarding drill bit, rock drilling, the transmission of impact energy, and drilling efficiency.Hustrulid and Fairhurst [3][4][5][6] investigated energy transmission between the drill steel and rock and measured the specific energy resulting from the impact force.
Chiang and Elias [7] used the finite element method (FEM) to study factors such as energy transmission, interactions between the drill bit and rock, and the rock penetration process, which are important in the research and evaluation of drill equipment.Song et al. [1] proposed optimal design factors and range for top-hammer drill bit based on multiphase ow simulation and response surface methodology (RSM).Also, Song et al. [8] designed a labscale testing system to test drilling performance of a drill bit and investigated the e ect of bit design factors on the performance of drilling.Kwon et al. [9] validated the e ect of drill bit button arrangement on drilling e ciency through a piston percussion test using piston drop testing system.Lundberg and Okrouhlik [10] conducted a study of the 3D e ects on the e ciency of three representative methods for percussive rock drilling (hammer drilling, DTH drilling, and churn drilling) in which the e ciency of various drilling methods was compared via 1D analysis and 3D FEM.Lundberg and Collet [11] investigated the optimal incident wave that maximizes the conversion e ciency of the wave energy in the THD method.All these prior studies essentially considered the transmission of impact energy from the piston and drill bit.However, they failed to examine the e ect of the dynamic properties of the drill bit resulting from the piston impact.Li et al. [2] analyzed the piston reaction force characteristics of percussive hammers using the stress wave propagation theory and the law of conservation of energy and also studied the energy transmission e ciency of a DTH hammer tool [12].e studies by Li et al. [2,12] considered the dynamics of the drill bit resulting from the piston impact.However, this consideration only addressed the underdamped modes of the system response characteristics that result from the rotational movement of the drill bit and the e ect of the rock.e work; therefore, neglected the dynamics of the drill bit generated under various other damped modes.
e study was further limited in that it employed an excessively large range of rock-to-piston sti ness ratios (β) in the transmission e ciency of impact energy.Overall, while many studies have investigated rock drilling and the transmission of impact energy, few have addressed the damping characteristics generated from the penetration process, the rock e ect, and the movement of the drill parts or investigated the related drilling e ciency.e analysis of behavior of drill bit should be performed for the design and manufacturing of a percussive drilling system with high drilling e ciency because the drilling e ciency and the range of rock fracturing depend on movement characteristics of drill bit during percussion and because drill bit also percusses the rock directly, meaning the close relationship with the delivery of percussion energy and drilling e ciency.
is study aimed to identify design parameters that could optimize the drilling e ciency in a percussive drilling system, while taking dynamic drill bit-rock interactions resulting from the piston impact into consideration.Drilling dynamic models (i.e., dynamic models of the drill bit) were established by introducing drill bit-rock interactions to determine the rock-penetrating properties of the bit. is was done using impact stress wave propagation theory, the penetration resistance coe cient, and the unloading constant in relation to the impacting piston and drill bit.e drilling dynamics was established considering the movement of the drill bit induced by piston impacts and the loading and unloading conditions resulting from the e ects of the rock.
e drilling dynamics proposed here employ two dimensionless parameters: the rst (α) is related to the pistonto-drill bit mass ratio and β (de ned above) and the second is β itself.is introduction is followed by a discussion of the rate at which the impact energy is transmitted from the piston to the rock during percussion (i.e., the drilling eciency).e drilling e ciency is the work completed by the drill bit relative to the incident-stress wave that causes it to crush the rock.We also analyzed the dynamics of the drill bit with regard to dimensionless parameters and damping ratios and considered impact-energy transmission rates given m r (the piston-to-drill bit mass ratio) and β.Combinations of parameters to maximize the e ciency of a percussive drilling system are proposed for various rock strengths.
To that end, Section 2 of this paper proposes a drilling dynamic model for a percussive drilling system that takes into account the dynamic properties of the drill bit, and with consideration to the theory of stress-wave propagation via piston impact.Drill bit behaviors and response characteristics observed during the percussion process (i.e., over, critical, and underdamped modes) involving the piston, the bit, and the rock via the parameters α and β are discussed.e physical meaning was analyzed by vertical development of dimensionless parameters (i.e., α and β).e system responses of drill bit to the piston percussion were extended to six conditions depending on the presence of incident stress wave and loading or unloading condition.e typical motion of drill bits involves four conditions.However, this study expanded them into six conditions.e response characteristics resulting from the rock and the movement of the drill bit were also analyzed to investigate the e ciency or the transfer rate of the percussive energy from the piston to rock. 2 Advances in Materials Science and Engineering e energy transmission e ciency via the impact of the percussive drilling system is examined in Section 3. Section 4 discusses the combinations of parameters that might optimize the drilling e ciency given the dynamic drill bit characteristics and rock sti ness (i.e., β).Conclusions are presented in Section 5. e rock powder is discharged to the outside by compressed air supplied through a ushing duct located inside the drill bit, thus enabling the drill to proceed further [1].

Bit Motion and Stress
Wave Propagation.Figure 2 illustrates the principles of impact and rock penetration by a percussive drilling system.e piston mass, length, cross-sectional area, descending speed, density, and longitudinal wave velocity are designated as m p , L p , A p , v, ρ, and c, respectively.e underlying assumptions were that the drill bit remains immobile on the surface and that the diameters of the drill bit and the piston are equal.Also, the e ects of buttons embedded in the bit were ignored.
e following equation expresses the motion of the drill bit resulting from the force applied to it by the piston [12]: where u is the penetration depth (displacement) of the drill bit, the second term is the force applied to the drill bit when the impact is exerted upon it, F is the force generated at the rock-bit interface (i.e., the penetration force), and A is the cross-sectional area of the drill bit and piston.
According to the relation between the dynamic stress and velocity of a mass in an elastic body system, the velocity of the contacting end of the bit can be expressed as follows [2]: du dt here σ i and σ r are the incident and re ected stress waves, respectively.e e ect of the secondary incident stress wave was not considered.For analytical simpli cation, the piston and the drill bit were assumed to be of the same diameter and material; both were assumed to have a rectangular pulse with a duration of incident stress wave τ and an amplitude of 0.5 ρcv.
Typically, nearly all the percussion impact energy is converted into wave energy.e piston hitting the drill bit generates an incident stress wave (σ i ), at which point a portion of the impact energy is transmitted to the rock via the drill bit, crushing the rock [11].e remaining energy is converted into a re ected stress wave (σ r ) and is used to increase the speed of the piston ascending from the drill bit top to speed v e (Figure 2) [2].e re ected stress wave can in turn be transmitted to the piston and drill bit in the form of a secondary incident stress wave at the piston-bit interface and at the drill bit-rock interface.
e secondary incident stress wave in a percussive drilling system has been reported not to contribute to the crushing of rocks [11]; therefore, it would not signi cantly a ect the energy transmission e ciency.Furthermore, a study of drilling speed found the incident stress wave to have an insigni cant e ect on the impact force and penetration characteristics [14].
e interaction between the drill bit and rock (the force-penetration relationship, Figure 3) is expressed in (3) [2], which separately considers the loading and unloading conditions: k is the rock penetration coe cient index, c is the unloading constant, and τ is the duration of the incident stress wave: e duration of the incident stress wave τ generated by the impact of the piston can be expressed with respect to time t in two scenarios, as de ned in (4) [2].
e rst scenario involves transmission of an incident stress wave, whereas the second does not: For the loading condition, we obtain the following equations [2]: where the initial condition is t 0 and F 0.
For the unloading condition, the equations can be expressed as follows [2]: where the initial condition is t τ and F(t) lim t → τ F(t).e response characteristics of a drill bit (i.e., the damped characteristics) can be under, critical, or overdamped given its motion equation, that is, (1).e movement of the drill bit under loading and unloading conditions was examined.e damping characteristics of a percussive drilling system are categorized depending on the damping ratio (ζ) and the dimensionless parameter α.
When loading, the equation of motion for the drill bit is de ned as follows: where the response-characteristics determinant ζ 1 is When unloading, the drill-bit equation of motion is de ned by (9).e ζ 2 term is given in (10): e dimensionless parameters α and β de ned in ( 8) and ( 10) can be expressed as follows (extended from ( 2), ( 12)): As explained earlier, the typical motion of drill bits involves four conditions.However, this study considers six conditions, as listed in Table 1 (Appendix A), and includes the loading/unloading condition and the presence or absence of an incident stress wave.To investigate the rate at which the impact energy is transmitted from the piston to the rock during the percussion process (i.e., the drilling e ciency), this study rst considered the response characteristics resulting from the motion of the drill bit during the percussion process and the e ect of the rock.e aim was to nd the response characteristics that lead to the rock's maximum crushing displacement, u m .
e study also de ned the motion of percussive drill bits resulting from the impact of the piston and analyzed their behavior given the system responses.e drill bit-rock interaction was considered here, including the force-penetration relationship for the rock.
Table 1 presents the relationships between loading and unloading and the duration of stress wave τ in percussive drilling systems.In Table 1, τ is the duration of the incident stress wave being transmitted to the drill bit via the piston during loading; it is referenced relative to the time t. e loading condition can be de ned in two ways: 0 < t ≤ τ or τ < t ≤ t m , where the latter expression means that τ arrived sooner than t; that is, the transmission of τ was complete, but the loading continued owing to the inertial e ect of the drill bit.If τ > t m , the drill bit arrived at u m before the transmission of τ to the bit had completed and indicates the time when the system switches to the unloading condition.
In summary, in Cases 1 to 5 in Table 1, the motion of the drill bit following the piston impact continues until the loading condition applying external force to the rock reaches τ < t ≤ t m .After that time, the stress wave arriving at the drill bit is converted into unloading at τ > t m , provided that the rock displacement occurs sooner than t m , that is, the maximum time of arrival.However, the τ in Case 6 represents a stress wave that is still progressing while the rock displacement owing to the drill bit has already reached its maximum; that is, unloading occurs during τ. 4 Advances in Materials Science and Engineering

Results of Bit Motion Analysis.
Numerical simulations on a dynamic model of drill bit properties were run for the six cases and used the dynamic drill bit properties during induced percussion (Section 2), the drill bit-rock interaction model, and the dimensionless parameters α and β.
Figure 4 shows the simulation results (ζ) for different conditions (α, Table 1) and the effect of loading and unloading.e mass ratio, m r , was 1, and the same conditions were applied to the piston mass and drill bit mass.Figure 4(a) shows an overview of the response characteristics for all six cases.e initial black part of each curve refers to the sustained effect of τ; the red parts indicate the loading condition that marks the effect of drill bit inertia, and the blue parts represent the transition to unloading.Figure 4(b) shows Case 1, where a single-incident stress wave indicates the drill bit displacement at u(τ) and the drill bit arrival at u m as an overdamped motion due to the loading continuing owing to external forces and inertia.Later, in the transition to unloading, characteristic overdamped motion appears, and there is no convergence to u f (Figure 3).
Case 2 (Figure 4(c)) appears similar to Case 1, but the bit reaches u m sooner owing to τ and the effect of drill bit inertia.As the system transitions to unloading and converges to u f , it shows critical damping.Case 3 (Figure 4(d)) shows a longer τ compared with Case 2 because of its quicker arrival at u m .During unloading, it converges to u f as underdamped motion.Case 4 (Figure 4(e)) has the drill bit displaced at u(τ) due to τ, and it arrives at u m as critically damped motion owing to continued loading resulting from the inertia of the drill bit.e subsequent unloading phase converges to u f as underdamped motion.
Case 5 (Figure 4(f)) has a longer τ than the preceding four cases and also reaches u m sooner.Furthermore, all its zones showed underdamped motion.Finally, in Case 6 (Figure 4 (g)), the rock displacement has already reached its maximum while τ is still in progress.is means that unloading begins during τ.Underdamped motion was observed in all zones.
Overall, the longer the duration of τ, the shorter the loading time and the sooner the u m is reached, owing to drill bit inertia.Table 2 summarizes the damping characteristics for the loading and unloading conditions and presents the associated damping mode that arrives at u m .Cases 1-3 show overdamping, Case 4 critical damping, and Cases 5 and 6 underdamping.
e underdamped mode showed the highest responsiveness, indicating the fastest transmission time of τ. e results show that increasing the τ-transmission time decreased the loading effect resulting from drill bit inertia and led to faster responses.e drill bit inertial effect is inversely proportional to τ.

Energy Transmission in Percussive Drilling System
In Section 2, we examined the response characteristics of drill bits that can arrive at u m , which is the maximum drilling displacement of rocks.Based on the examined characteristics in Section 3, we calculate the efficiency of percussive drilling systems and discuss the combinations of parameters that could maximize it.

Energy Transmission Efficiency.
In a percussive drilling system, the piston directly hits the drill bit at speed v (Figure 2).For a transmitted impact energy, E i , at the time of the piston descending and hitting the drill bit, the final speed of the piston immediately preceding the impact, v, can be expressed as follows [2,10]: Table 1: Expansion conditions for dynamic drill-bit properties for loading, unloading, and stress-wave duration, τ.
Case no.

Initial conditions
Advances in Materials Science and Engineering   Advances in Materials Science and Engineering e initial height h 0 is v 2 /2g, and the rebound, e h , can be de ned as the ratio between the initial height and the piston's postimpact rebound height, h.
ese are expressed as follows [2]: e e ciency, η, of impact energy transmission is dened as the ratio between the kinetic energy generated from the impact of the piston and the energy transmitted to the rock [12] and is given as Which can be expressed as follows through dimensional analysis: e value u m during loading can be obtained from ( 7), the equation of motion for the drill bit.e obtained u m can be expressed as in (16) for each damping mode, where the initial conditions are 0 < t ≤ τ, F ku i , u 0 0, and _ u 0 0:

Advances in Materials Science and Engineering
Moreover, for t m < τ, u m is expressed as follows: e initial conditions are 0 < t ≤ τ, F � ku i , u i � 0, and _ u i � 0. Next, the displacement value u f during unloading can be obtained via the equation of motion for a drillbit (9), as shown in (18).e initial conditions during unloading are t > τ, F � 0, u 0 � u(τ), and 8 Advances in Materials Science and Engineering e e ciency of the percussive drilling systems for given β can be calculated using (14)(15)(16)(17).
e factor u m , which is important to the e ciency calculations, is discussed in depth.

Results of Energy Transmission E ciency.
e results of simulations on the drill bit-rock interaction model shown in Figure 5 indicate the damped modes that can arrive at u m (for varying values of β) as the response by the drill bit at the time of piston impact.e damped mode is determined by ζ and α, which indictate the system responses.Overdamped mode has α < 1 and can be expressed as follows: e critical damped mode has α 1 and is expressed as follows: e underdamped mode has α > 1 and is expressed as where m b /m h m r .
e special underdamped mode refers to the case where τ is faster than t, indicating that the drill bit can arrive at u m only via an incident stress wave.In this case, the bit reaches its maximum displacement and transitions to the unloading state while the transmission of τ is underway.
e damped mode of each m r shown in Figure 5 reveals that the underdamped mode is dominant of the e ects.Also, overdamped and critical damped were observed, where β was low. is is possibly because of the e ects of the internal energy (i.e., compression strength) of rock and the percussive energy.e overdamped and critical damped were also observed.is is possibly because of internal energy (i.e., compression strength) of rock and energy caused by percussion.
Figure 6 shows dimensionless time, t m t/τ, capable of reaching U m for β and τ.In Section A of t > τ, the bit reaches U m after termination of τ, and in Section B of t < τ, the bit reaches U m before the termination of τ. e lower the m r and β, the faster the termination of τ, and increase in these tended to make the duration of τ delivered to rock through bit to increase proportionally to β. is suggests that the higher rock strength requires longer duration of τ for drilling the rock.
Figure 7 shows dimensionless U m state de ned as ratio u m , a displacement u m by β, and u ∞ caused by incident stress wave and loading condition.e prediction of displacement of drill bit, drilling displacement, by characteristic of rock was limited.
is study, accordingly, judged the dimensionless drilling state, U m , only by using ratio u ∞ , a displacement during loading and u m considering inertia caused during percussion (22): where the u ∞ only by loading is de ned as follows: e e ect of initial condition on u ∞ decreased with the increase in β. e u m is determined by considering loading condition and inertia e ect, is a ected by overresponse, and decreases as β increases.
Section A refers to u m < u ∞ , with only τ and loading e ect present (i.e., the condition for the normal state in the initial mode): u ∞ means only the τ e ect resulting from loading exists.Section B refers to u m > u ∞ , a state in which u m can be reached through τ and loading is exceeded.is section is where the drill bit inertia and underdamped motion appear.
e gradient of the curve for maximum displacement of the bit varies with β, indicating that as the rock strength increases, the displacement allowing fracture decreases.Near value 2, β changes its gradient for maximum displacement.is means that, in Section A, the rock is less sti than the piston owing to the rock-sti ness e ect, and in Section B, the rock is sti er than the piston.β is de ned as the sti ness ratio between the rock and the piston, as in (11).Hence, when β < 2, a condition is established for rock fracture to occur, in that the piston sti ness exceeds the rock sti ness.
In examining u m for piston-bit mass ratios of m r < 2, the rock strength that can be defeated by τ occurring in the initial loading tends to increase.is tendency is suspected to result from the inertial e ect of the mass of the bit.For β > 4, the u m gradient tended to decrease with increasing mass of the bit.Hence, the e ciency is considered to decrease, which is attributed to the e ect of the rock's compressive strength.e uniaxial compressive strength of very hard rock is typically at least 200 MPa, while the tensile strength of the H13 tool steels that are used mostly for pistons (ASTM A681 and DIN EN ISO 4957) is known to be at least 359-1170 MPa [15].
Given the steels generally used to make pistons, the e ective range of β is considered to be 0.34 < β < 1.12, which was calculated considering the strength ratio between the hard rock and an H13 steel tool.Hence, the e ective range of β could be no greater than 2. Rocks with a sti ness ratio of β > 2 might cause failure through the piston's plastic deformation and fracture.is study examined the e ective interval of β and investigated the parameter combinations giving high drilling e ciency in this interval.
Figure 8 illustrates the motion of the bit generated during the percussion process and the energy transmission rate (i.e., e ciency) for the response characteristics, τ, and percussion impact.Analysis of m r found the maximum drilling e ciency in interval 1 < β < 2, with a tendency efciency decreasing after m r increased above a certain value.is result was attributed to the tendency whereby increasing m r converts the system's dynamic response to τ due to the piston from underdamped to overdamped.Furthermore, increasing the mass of the bit can increase its internal energy (i.e., friction energy) to above the piston transmitted energy (i.e., kinetic and potential energy).Hence, the bit's dynamic response would be overdamped and could compromise drilling e ciency.In other words, the heavier the bit, the more delayed the response to percussion.
In Figure 8, the left side of red line (β 1.8) represents the e ect of only incident stress wave, meaning that the higher the value in graph the higher the drilling e ciency in incident stress wave.e right side of red line, representing unloading behavior after reaching maximum e ciency in incident stress wave, is a ected by both brittleness characteristics of rock and inertia e ect of drill bit.
e piston should have twice the mass of the drill bit (i.e., m r 0.5) for the most e cient drilling for β ≤ 2, which is established to be the e ective interval of β and corresponds to both soft and hard rocks.Drilling sti er rocks (2 < β < 4) might be less e cient, but increasing m r to 1 would help improve drilling e ciency.

Results and Discussion
is study defined equations for the motion of a drill bit when struck by a piston and expanded the dynamic properties of the drill bit based on the conditions ζ and α.It also analyzed the relationship between the bit dynamics and the duration of the incident stress wave (τ).Underdamped drill bits displayed the greatest responsiveness and the fastest transmission of τ.Increasing the transmission of τ decreased the loading effect due to the drill bit's inertia but led to a fast response.e inertial effect of the drill bit was confirmed to be inversely proportional to the transmission of τ.
A percussive drilling system accomplishes its work via the percussion impact of a piston.In the systems studied here, τ ended more quickly with lower m r and β values; as these values increased, the duration of τ transmitted to the rock via the drill bit increased in proportion to β. e results also showed that the maximum bit displacement (u m ) decreased as β increased.Analysis of the drilling dynamics confirmed the effective intervals of β.With reference to the rock strengths suggested by the International Society for Rock Mechanics (ISRM), the effective interval of β is considered to be no greater than 2.
e main purpose of this study was to examine the impact energy transmission rate and drilling efficiency in a percussive drilling system (Figure 8).e results established a m r value of 0.5 as the most efficient for rocks whose strength corresponds to the interval β < 2. e improved drilling efficiency would lead to advantages such as reduced bit production costs.Rocks stiffer than 2 < β < 4 can be most efficiently drilled by selecting m r � 1.
e movement and response characteristics of bit during percussion process, τ, and transfer rate of energy caused by percussion, an efficiency, depend on m r and β, where increase in m r leads to overdamping of bit movement characteristics due to τ by piston and reduction in drilling efficiency due to internal energy effect by mass of drill bit.As the increase in mass of drill bit, for the section of 4 < β < 6, the drilling efficiency of percussive drilling system decrease, and for the very hard rock (UCS > 200 MPa) section of 4 < β < 6, it is considered that the reasonable applied m r of drilling tool is 1 or 2.
is study neglected the effects of the secondary incident stress wave and also the effect of buttons embedded in the drill bit.Additionally, for analytical simplification, the piston and drill bit were assumed to be of the same diameter and material, and a rectangular pulse with an incident stress wave of duration τ and amplitude 0.5ρcv was assumed.e effect of the flexural stress wave depending on the configuration was not considered.ese limitations of this paper should be addressed in further investigations that take into account the configuration of the piston and drill bit, the effect of buttons, and the drilling efficiency given differing shapes of the incident stress wave.

Conclusion
is paper aimed to identify the optimal design parameters for percussive drilling systems by introducing a drill bit-rock interaction model that could verify the bit's motion during percussion and the resulting damping characteristics.e study analyzed drilling efficiency and drew the following conclusions.
Percussive drilling systems have six dynamic drill bit properties that can be expandable.is paper discussed the physical meaning of the dimensionless parameters α and β.
eir values determined the damping characteristics that can lead to the rock's maximum fracture displacement.e fastest response of τ was observed for underdamped drill-bit motion.
Drilling was most efficient in the interval 1 < β < 4, where increasing m r at a given β decreased drilling efficiency.e value of m r for efficient drilling was determined by the rock strength (i.e., β).
e results indicate that application of a piston-to-drill bit mass ratio of 0.5 (i.e., a piston mass twice that of the drill bit mass) to the rocks whose stiffness corresponds to β ≥ 2 would likely be most efficient and also reduce drill bit production costs.Furthermore, a selection of m r � 1 would be valid when developing drill tools for boring complex rocks (1 < β < 4).At β > 4, the best efficiency could be achieved when the bit mass equals or exceeds the piston mass.
where bit displacement u � u f is used to indicate the rock's fracture displacement.Furthermore, considering the bit's damping ratio when unloading, its behavior can be expressed as Damped mode � Second, under critical damping where ζ 1 � 1 and α � 1, τ is transmitted to the drill bit, indicating loading.Equation ( 7) can be expressed as in (A.1).e initial conditions are 0 < t ≤ τ, σ i � 0.5ρcv, F � ku, u 0 � 0, and _ u 0 � 0. In this case, assuming the normal state, u can be calculated using (A.2): e following is the case where τ is complete and the loading effect is sustained owing to external forces and drill bit inertia.Applying the conditions of τ < t ≤ t m , σ i � 0, F � ku, u 0 � u(τ), and _ u 0 � _ u(τ) to (A.1) give (A.3)where u � 0. Last, we examine the unloading condition where t m < t, σ i � 0, F � cku − cku f , u 0 � u m , and _ u 0 � 0. Equation (9) when unloading can be expressed as (A.4).
In this case, bit displacement u � u f is used to indicate the rock's fracture displacement.Furthermore, considering the bit's damping ratio when unloading, its behavior follows Damped mode � e second condition of critical damping is defined as α � 1.When unloading drill bits, c refers to the rock's property effect and is always less than 1.Hence, when unloading, no other cases but the underdamped mode are allowed.In other words, when unloading in the critically damped mode owing to the rock properties, the bit is always underdamped.e third condition is underdamping with ζ 1 < 1 and α > 1.Here, τ is transmitted to the drill bit, meaning loading.Equation ( 7) can be expressed by (A.1).e initial conditions are 0 < t ≤ τ, σ i � 0.5ρcv, F � ku, u 0 � 0, and _ u 0 � 0. Where, assuming the normal state, u can be calculated as (A.2).
e following is the case where τ is complete, and the loading effect is sustained owing to external forces and drill bit inertia.Applying the conditions of τ < t ≤ t m , σ i � 0, F � ku, u 0 � u(τ), and _ u 0 � _ u(τ) to (A.1) results in (A.3), where, u � 0. Last, we examine the unloading condition where t m < t, σ i � 0, F � cku − cku f , u 0 � u m , and _ u 0 � 0. Equation ( 9) when unloading can be expressed as (A.4).
In this case, the drill bit displacement u � u f is used to indicate the rock's fracture displacement.Furthermore, considering the damping ratio in the drill bit's unloading, the bit's behavior can be expressed as Damped mode � ).
e third condition is the critically damped mode defined as α > 1.During drill bit unloading, c refers to the rock property effect and is always less than 1.Hence, when unloading, no other cases but underdamping are allowed.
Next, the underdamped condition is where ζ 1 < 1 and α > 1, and τ is slower than t m (the time at which the rock's maximum fracture displacement is reached), that is, t m < τ.In this case, τ is transmitted to the drill bit, meaning loading.Equation ( 7) can be expressed by (A.1).e initial conditions are 0 < t ≤ t m , σ i � 0.5ρcv, F � ku, u 0 � 0, and _ u 0 � 0. Here, assuming the normal state, u can be calculated as (A.2). e following case is where conversion is made to unloading after the drill bit reaches its maximum displacement, but the incident stress wave is sustained.Applying the conditions of t m < t ≤ τ, σ i � 0.5ρcv, F � cku − cku f , u 0 � u m , and _ u 0 � 0 to (9) results in where the definition of u � 1/c(ρcA/k)v + u f is possible.
Last, this unloading condition is where the incident stress wave is resolved.When unloading the drill bit, using the conditions t < τ, σ i � 0, F � cku − cku f , u 0 � u(τ), and _ u 0 � _ u(τ) allows ( 9) to be expressed as (A.4).In this case, drill bit displacement u � u f is used to indicate the rock's fracture displacement.e analysis of drill bit dynamics showed that when loading transitions to unloading in the underdamped condition, the dynamic properties of the bit exist only in the underdamping mode.

Figure 1 :
Figure 1: Drilling mechanisms of two types of percussive drilling system: (a) top-hammer drilling and (b) down-the-hole drilling (modi ed from Song et al. [1]).

Figure 3 :
Figure 3: Force versus penetration relationship representing the bit-rock interaction curve.

2 Figure 7 : 10 Figure 8 :
Figure 7: Dimensionless time capable of reaching U m for an incident stress-wave τ acting on the bit and β.

Table 2 :
Dynamic drill-bit properties for loading, unloading, and τ, and the damping modes capable of reaching u m .
* Damping modes capable of reaching u m .