Laboratory-Scale Limestone Rock Linear Cutting Tests with a Conical Pick: Predicting Optimal Cutting Conditions from Tool Forces

Abstract

This study introduces a simplified method for predicting the optimal cutting conditions to maximize excavation efficiency based on tool forces. A laboratory-scale linear rock-cutting test was conducted using a conical pick on Finike limestone. The tool forces and their ratios were analyzed in relation to cutting parameters such as penetration depth and spacing. While the cutting force (FC) and normal force (FN) increased with the penetration depth and spacing, this relationship could not predict the optimal cutting conditions. The ratio of the mean normal force to the mean cutting force (FN_m/FC_m) increased with the penetration depth and the ratio of spacing to penetration depth (s/d). However, even while including this relationship, predicting optimal cutting conditions remained challenging. The ratio of the peak cutting force to the mean cutting force (FC_p/FC_m) reached a maximum value at a specific s/d, which is similar to the relationship between the specific energy (SE) and s/d. The optimal s/d obtained through the SE methodology was found to be between 3 and 5, and FC_p/FC_m reached a maximum at s/d. The error between the optimal s/d and the s/d in which FC_p/FC_m was maximized was less than 5%. Therefore, it was confirmed that the optimal cutting conditions could be predicted through the relationship between FC_p/FC_m and s/d. Additionally, by using the results from previous studies, the optimal cutting conditions obtained from the SE methodology and the proposed methodology were found to agree within a margin of error of 20%. The proposed methodology can be beneficial for the design of cutter heads and the operation of excavation machines.

1. Introduction

In mining and civil engineering, two primary methods are commonly employed for rock excavation: mechanical excavation and drilling and blasting. Mechanical excavation is preferred over drilling and blasting owing to its productivity, cost-effectiveness, reliability, and safety [1]. Consequently, mechanical excavation has been increasingly applied in rock excavation in the fields of mining and civil engineering in recent decades [2,3].

The efficiency of the mechanical excavation processes was evaluated using specific energy (SE). SE refers to the energy consumed to excavate a unit volume and is a property of the rock-cutting process, not a rock property [4]. A high SE indicates that more energy is required to remove a given volume of rock, potentially leading to increased civil and mining project costs [5]. The SE in MJ/m³ was calculated using Equation (1), as follows:

$$SE={\displaystyle \frac{W}{V}}={\displaystyle \frac{{FC}_m \times L \times \mathsf{\rho }}{M}}$$

(1)

where W is the work or energy required to cut the rock (MJ), V is the cutting volume (m³), FC_m is the mean cutting force (kN), L is the cutting length (mm), ρ is the density of rock (g/cm³), and M is the mass of the rock fragments (g).

SE is more affected by spacing than by penetration depth [6]. If the spacing is too small, the rock is overcrushed, resulting in decreased excavation efficiency. Conversely, if the spacing is too wide, tensile fractures from adjacent cuts cannot form a rock chip, which also leads to reduced excavation efficiency. The minimum SE was obtained with the optimal spacing (Figure 1) [3]. Based on this principle, SE methodology has been widely employed to determine the optimal conditions for rock cutting. Under optimal cutting conditions, the excavation process is accelerated, which directly affects project schedules and cost planning. Therefore, various methods have been proposed to determine the optimal cutting conditions.

Full-scale linear cutting machine (LCM) testing is the most reliable method for predicting the performance and design of excavation machines [7]. This test is commonly employed to determine the optimal cutting conditions that can maximize excavation efficiency, as highlighted in various studies [2,6,8,9,10,11,12,13,14,15,16]. Nevertheless, full-scale LCM tests present inherent challenges because of their high cost, time consumption, and the difficulty or impossibility of obtaining full-sized rock samples [2,17,18]. Small-scale LCM tests can solve these shortcomings [19,20,21,22]. However, a fundamental issue affecting the accuracy of the LCM test lies in the volume measurement. Volume measurements in an LCM test are typically derived from the weight of the rock fragments produced during the cutting process. This reliance on rock fragment weight for volume measurement introduces challenges: larger rock chips may spill beyond the measurement area, and small rock chips and dust can scatter outside the measurement zone, potentially compromising the accuracy of the volume measurements. To address these challenges, Cho et al. [17] utilized the ShapeMetrix3D photogrammetric measurement system to obtain accurate volume measurements and predict the optimal conditions. Similarly, Choi et al. [23] and Kim [24] measured the cutting volume using 3D scanning. Ayawah et al. [5] measured the resulting underbreaks and overbreaks in excavated rocks using laser profilometry. They confirmed that the volume of overbreaks decreased up to optimal cutting conditions, suggesting a new method for determining optimal cutting conditions. Despite these improvements in volume measurement accuracy, the additional processes increase the complexity of the process and require significant time.

Numerical simulations have been employed as an alternative method to determine the optimal cutting conditions, as they can avoid the drawbacks inherent in physical tests. Cho et al. [25] utilized AUTODYN-3D, a finite element method (FEM) numerical simulation, to obtain the optimal spacing of tunnel boring machine (TBM) disc cutters for eight different types of Korean rock. Moon and Oh [26] investigated the optimal cutting conditions for hard rocks using the discrete element method (PFC2D). They reported that the brittleness of the rock was directly proportional to the optimal s/d ratio. Zhou et al. [27] conducted a linear cutting test using an FEM simulation and found that the optimal cutter spacing decreased as the rock strength increased. Additionally, various studies have employed numerical simulations to derive optimal cutting conditions [28,29,30,31,32]. However, numerical methods display some shortcomings, such as the inability to accurately replicate the crack propagation in rocks or the need for complex parameter calibration owing to the involvement of microscopic parameters [33].

Therefore, this study proposes a simplified approach for predicting optimal cutting conditions by analyzing the tool forces measured in LCM tests. Rock-cutting tests were performed at various penetration depths and spacings for Finike limestone using a conical pick. The tool forces measured during rock cutting were analyzed in relation to penetration depth and spacing. Additionally, an analysis was conducted to examine the relationship between the ratio of tool forces and the cutting conditions. The proposed approach verified the accuracy of predicting the optimal cutting conditions compared to results obtained using the existing SE methodology.

2. Rock Linear Cutting Test

2.1. Laboratory-Scale Linear Cutting Machine

The laboratory-scale LCM (LLCM) used for rock cutting in this study is shown in Figure 2. The LLCM was electronically driven. The rock box could accommodate samples measuring up to 470 mm × 500 mm × 450 mm, and it was repositioned using x-axis and y-axis servomotors. The x-axis servomotor, which was responsible for the movement in the spacing direction, had a stroke of 500 mm. The y-axis servomotor, which is responsible for the movement in the cutting direction, can achieve a maximum speed of 100 mm/s and has a stroke of 700 mm. The penetration depth was controlled by adjusting the z-axis servomotor, which moved the cutting tool up and down (Figure 2a,c).

The tool forces generated in the three axis directions were measured by installing load cells in all direction, each with a data accumulation capacity of 10 Hz. The x-axis load cell, with a capacity of 15 tons (150 kN), measures the side force (FS), whereas the y- and z-axis load cells, both with capacities of 25 tons (250 kN), measure the cutting force (FC) and normal force (FN), respectively (Figure 2b).

The cutting parameters, such as penetration depth, spacing, and cutting speed, were set in the control box. Real-time tool forces were visualized on the display and simultaneously stored in the CSV (comma-separated values) format.

2.2. Conical Pick and Rock Sample

The LLCM can be considered a small-scale LCM. In small-scale LCM tests, observing the interactions between adjacent cutting grooves becomes challenging when the rock sample is insufficiently large. In addition, when using a downsized cutting tool, it is impossible to estimate the cutting performance directly or to obtain the design parameters of the cutter head [20,34]. Therefore, in this study, a linear rock-cutting test was performed using a full-sized conical pick and a sufficiently large rock sample to ensure interaction between cutting grooves.

Model SM06, a conical pick manufactured by Kennametal Inc. (Pittsburgh, PA, USA), was used for the linear rock-cutting tests in this study. The tip angle of the conical pick was 90°, and the pick gauge and shank lengths were 45.25 mm and 41.91 mm, respectively. Detailed specifications of the conical pick are shown in Figure 3a. The attack angle was set to 45° and was applied equally in all tests (Figure 3b).

The rock selected for this study was Finike limestone. This medium-strength rock is a calcareous sedimentary rock obtained from the Finike district in southern Turkey. The rock samples selected for the cutting tests contained no bedding planes and or and were cut into dimensions of 350 mm × 350 mm × 250 mm. Subsequently, to expand the measurement range and the usable portion for the experiment, the sample was placed in a mold with dimensions of 400 mm × 400 mm × 300 mm, and the surrounding space was filled with concrete. The mechanical properties of Finike limestone are listed in

Table 1. Mechanical properties of Finike limestone.

Rock Name	ρ (g/cm3)	E (GPa)	ν	σc (MPa)	σt (MPa)
Finke limestone	2.22	21.1	0.13	49	5

. The uniaxial compressive strength (σ_c), elastic modulus (E), and Poisson’s ratio (ν) were determined based on ASTM D 7012-14 [35]. Tensile strength (σ_t) and density (ρ) were determined using the testing methods recommended in ASTM D3967-08 [36] and ASTM C97/C97M-15 [37], respectively.

2.3. Cutting Parameters and Data Processing

In all cutting tests, the attack angle was set to 45°, as mentioned earlier, whereas the skew angle and tilt angle were both set to 0°. The penetration depths were set to 3, 6, and 9 mm, and the spacing was adjusted to maintain a ratio of spacing to penetration (s/d) between 1 and 6. This adjustment is based on the findings of Bilgin et al. [3], who reported that the optimal s/d for a conical pick is between 2 and 5. Compared with other parameters, such as the penetration depth, the cutting speed does not significantly affect the specific energy [38]. Consequently, the effect of cutting speed on rock cutting was ignored, and cutting was conducted at a slow speed of 4 mm/s to minimize the loss of rock chips used for volume measurement during the cutting process. Cutting was performed at least three times under identical conditions.

The tool forces measured during a single cutting in the rock-cutting test were recorded, as illustrated in Figure 4. The red line represents the FC, whereas the blue and black lines correspond to the FN and FS, respectively. The mean tool forces were obtained using the values recorded in the data window (represented by a yellow box) to mitigate the impact of cutting the concrete and excessive crushing at the beginning and end of the rock sample. For example, in Figure 4, FC_m is an illustration of the mean tool force. The peak tool forces, such as the peak cutting force (FC_p) shown in Figure 4, were calculated as the average of the peak values within the data window. The FC is directly related to the torque requirement and the specific energy of the excavation machine. In contrast, the FN is used to estimate the effective capacity and thrust of a machine. Meanwhile, the FS is related to the vibration of the machine and has relatively little effect on the excavation process [9]. Therefore, in this study, the analysis was performed only for FC and FN. The results of the linear rock-cutting tests are summarized in

Table 2. Results of linear rock-cutting test with LLCM.

d (mm)	s (mm)	s/d	FCm (kN)	FNm (kN)	FCp (kN)	FNp (kN)	FCp/FCm	FNp/FNm	FNm/FCm	FNp/FCp	SE (MJ/m3)
3	3	1	1.567	0.448	3.183	0.925	2.031	2.066	0.286	0.291	178.02
	6	2	1.650	0.594	4.139	1.284	2.508	2.164	0.360	0.310	96.91
	9	3	1.994	0.727	5.204	1.965	2.610	2.701	0.365	0.378	78.22
	12	4	2.187	0.834	5.294	1.796	2.419	2.154	0.381	0.339	72.89
	15	5	2.533	0.937	6.114	2.062	2.425	2.200	0.370	0.336	80.75
	18	6	2.733	1.037	6.752	2.440	2.470	2.352	0.380	0.361	95.80
6	6	1	3.325	0.900	6.933	1.369	2.085	1.521	0.271	0.197	108.33
	9	1.5	3.506	0.961	7.718	1.699	2.201	1.768	0.274	0.220	63.04
	12	2	3.820	1.093	8.830	2.135	2.312	1.953	0.286	0.242	54.57
	18	3	4.594	1.376	11.055	4.135	2.406	3.005	0.299	0.374	47.37
	24	4	4.937	1.575	13.618	4.783	2.758	3.036	0.319	0.351	45.95
	36	6	5.656	1.921	13.346	3.786	2.360	1.971	0.340	0.284	57.71
9	9	1	4.868	0.777	11.342	2.182	2.330	2.809	0.160	0.192	66.61
	12	1.33	5.125	0.878	12.144	1.972	2.370	2.247	0.171	0.163	54.26
	18	2	6.066	1.425	14.188	3.699	2.328	2.597	0.235	0.262	39.21
	24	2.67	6.570	1.818	16.483	3.774	2.509	2.076	0.277	0.229	32.89
	36	4	7.157	2.271	18.141	4.840	2.535	2.131	0.317	0.267	33.46
	48	5.33	8.135	2.644	19.646	5.615	2.415	2.123	0.325	0.286	45.87

Note: d is the penetration depth, and s is the spacing; FC_m and FN_m are the mean cutting and normal forces, respectively; FC_p and FN_p are the peak cutting and normal forces, respectively.

.

3. Results and Discussions

3.1. Analysis of Tool Forces

3.1.1. Correlation between Tool Forces and Cutting Conditions

The penetration depth is a fundamental parameter that defines the cutting conditions. In rock cutting, an increase in penetration depth correlates with higher production; however, an escalation in tool forces accompanies it. Although deep penetration is advantageous for enhancing production, it has the drawbacks of accelerating tool wear and reducing tool lifespan.

The correlation between the penetration depth and the tool forces was analyzed for all data that underwent cutting, with penetration depths of 3 mm, 6 mm, and 9 mm (Table 2). Figure 5 illustrates the variations in FC_m and FC_p with respect to penetration depth. Both FC_m and FC_p increased linearly as the penetration depth increased (R² = 0.680–0.804, F = 27.613–65.789, p = 0.000). In Figure 6, the variations in FN_m and FN_p with respect to penetration depth are depicted. While both FN_m and FN_p demonstrated a linear increase with penetration depth, the correlation coefficient remained lower than 0.371 (R² = 0.195–0.371, F = 3.386–9.425, p = 0.007–0.087). This implies that the FC is more affected by the penetration depth. Moreover, FC_p and FN_p increased significantly compared to FC_m and FN_m. Wang et al. [39] confirmed that FC and FN have a repetitive pattern of building up to a peak and then dropping as cutting is performed. As the penetration depth increased, the peak value rose, and the pattern period extended, indicating the generation of larger rock chips at greater depths. Therefore, it can be expected that FC_p and FN_p increase significantly more than FC_m and FN_m because the period of the pattern becomes longer. The expected equations for the tool forces, depending on the penetration depth, are summarized in

Table 3. Regression equations for tool forces based on penetration depth.

Regression Equation	R2	F-Value	p-Value
${FC}_m=0.701d + 0.037$	0.804	65.789	0.000
${FC}_p=1.304d + 1.615$	0.680	27.613	0.000
${FN}_m=0.361d + 0.145$	0.371	9.425	0.007
${FN}_p=1.316d + 0.211$	0.195	3.386	0.087

.

Tool spacing is a critical parameter in cutter head design, directly influencing rock chip size and excavation efficiency [40,41]. Wide spacing results in unrelieved cutting, in which cutting grooves do not interact, increasing tool forces and reducing excavation efficiency. Conversely, a close spacing results in decreased tool forces but causes over-crushing and excavation occurring deeper than the penetration depth, ultimately diminishing excavation efficiency.

The correlation between spacing and tool forces was analyzed using the cutting data obtained at spacings ranging from 3 to 48 mm. Accordingly, the s/d ratio is between 1 and 6, and it is noted that it is a relieved mode in which interactions occur between cutting grooves. As illustrated in Figure 7 and Figure 8, the tool forces exhibited an increase following a power–function relationship, with an increase in spacing.

Table 4. Regression equations of tool forces based on spacing.

Regression Equation	R2	F-Value	p-Value
${FC}_m=0.916s^{0.554}$	0.676	137.844	0.000
${FC}_p=1.992s^{0.590}$	0.723	151.232	0.000
${FN}_m=0.175s^{0.695}$	0.932	632.802	0.000
${FN}_p=0.461s^{0.649}$	0.843	257.610	0.000

provides the regression equation and statistical parameters detailing the relationship between the tool forces and the spacing. Because all F-values were greater than 31, and all p-values were less than 0.05, all regression curves were considered statistically significant at a confidence level of 99%. Moreover, the correlation coefficient ranged from 0.676 to 0.923, which was stronger than the correlation with the penetration depth (R² = 0.195–0.804). Therefore, it can be seen that both FC and FN are more affected by the spacing than by the penetration depth. This correlation was notably more robust for FN_m and FN_p (R² = 0.676–0.723) than for FC_m and FC_p (R² = 0.843–0.932), suggesting that FN was more sensitive to spacing variations than was FC.

In summary, it was confirmed that increasing the penetration depth and spacing increases the tool forces.

Table 5. Multiple regression models and statistical parameters of tool forces by penetration depth and spacing.

Regression Equation	R2	F-Value	p-Value	Coefficient
				d		s		Constant
				t	p-Value	t	p-Value	t	p-Value
${FC}_m=0.507d + 0.081s-0.229$	0.995	1454.468	0.000	30.820	0.000	23.698	0.000	−2.442	0.027
${FC}_p=1.176d + 0.088s-0.750$	0.977	320.838	0.000	13.412	0.000	12.259	0.000	−1.498	0.155
${FN}_m=0.043d + 0.044s + 0.208$	0.961	183.104	0.000	3.099	0.007	14.996	0.000	2.606	0.020
${FN}_p=0.103d+ 0.094s + 0.538$	0.839	39.189	0.000	1.557	0.140	6.859	0.000	1.429	0.173

summarizes the multiple regression model and the statistical parameters of the tool forces for the penetration depth and spacing. However, this relationship did not confirm the ability to predict optimal cutting conditions.

3.1.2. Correlation between Ratio of Normal to Cutting Force and Cutting Condition

The FN/FC ratio is an essential parameter for determining the efficiency of excavation machines. As the tool wears, FN increases rapidly, leading to a corresponding increase in FN/FC [42]. Moreover, FN/FC is related to the excavation efficiency. A high FN/FC ratio causes excessive tool wear, increased crushing, less chipping, and low penetration rates [43]. However, a lower FN/FC indicates a more efficient cutting tool performance [39]. Therefore, the FN/FC is expected to be associated with the optimal cutting conditions for achieving maximum efficiency.

Figure 9 and Figure 10 illustrate the variations in FN_m/FC_m according to the penetration depth and spacing, respectively. In this study, FN_m/FC_m was between 0.160 and 0.381 (Table 2). It was confirmed that the FN_m/FC_m decreased linearly as the penetration depth increased. In this relationship, the correlation coefficient was 0.502, and the F-value and p-value were 16.134 and 0.001, respectively, which were statistically significant at the 99% confidence level. However, FN_m/FC_m did not show any correlation with spacing. This is partly inconsistent with the results of Wang et al. [39], where FN_m/FC_m increased linearly as the penetration depth increased and decreased within the power function as the spacing increased. However, as shown in Figure 11, this study confirmed that FN_m/FC_m was significantly correlated with s/d (R² = 0.529, F = 397.166, and p = 0.000). FN_m/FC_m increased within the power–function relationship as s/d increased. The regression equation and statistical parameters of the correlation between FN_m/FC_m and the cutting parameters are summarized in

Table 6. Regression equations for ratios between tool forces considering penetration depth and s/d.

Regression Equation	R2	F-Value	p-Value
${FN}_m/{FC}_m=-0.018d+ 0.410$	0.502	16.134	0.001
${FN}_m/{FC}_m= 0.234{s/d}^{0.253}$	0.529	397.166	0.000
${FN}_p/{FN}_m=-0.064{s/d}^2+ 0.461s/d+ 1.645$	0.154	1.367	0.285
${FC}_p/{FC}_m=-0.043{s/d}^2+ 0.343s/d+ 1.867$	0.614	11.935	0.000

. The multiple regression model and parameters of FNm/FCm considering the penetration depth and spacing are listed in

Table 7. Multiple regression models and statistical parameters of ratios between tool forces by penetration depth and spacing.

Regression Equation	R2	F-Value	p-Value	Coefficient
				d		s		Constant
				t	p-Value	t	p-Value	t	p-Value
${FN}_m/{FC}_m =-0.027d+0.004s+0.397$	0.892	62.024	0.000	−10.882	0.000	7.364	0.000	27.853	0.000
${FN}_p/{FC}_p =-0.024d+0.003s+0.374$	0.665	14.898	0.000	−5.410	0.000	3.264	0.005	14.641	0.000
${FC}_p/{FC}_m=-0.016d+0.007s+2.364$	0.195	1.818	0.196	−0.892	0.386	1.906	0.076	22.626	0.000
${FN}_p/{FN}_m=-0.009d+0.001s+2.212$	0.004	0.027	0.974	0.179	0.860	0.042	0.967	8.056	0.000

. Nevertheless, predicting the optimal cutting conditions remains challenging, both in the correlation between FN_m/FC_m and the penetration depth and the correlation between FN_m/FC_m and s/d.

3.1.3. Correlation between Ratio of Peak-to-Mean Tool Force and Cutting Condition

The peak-to-mean tool force ratio is an essential factor in design and practice. This ratio plays a significant role in influencing the vibration of the cutter head and the potential breakdown of mechanical parts [9]. The peak-to-average tool force ratio is also related to the excavation efficiency during rock cutting. Bilgin et al. [3] found that, in relieved cutting under optimal conditions, this ratio was higher than that in unrelieved cutting, which was attributed to the larger rock chips obtained in the relieved mode. Additionally, the FC_p/FC_m ratio represents the ease of cutting the rock, as a rock with a high FC_p/FC_m ratio is harder to cut than a rock with the same SE but a lower FC_p/FC_m ratio [44]. Kim et al. [21] suggested that FC_p/FC_m may be related to the optimal cutting conditions. Barker [45] and Copur et al. [9] found that the ratio of the peak-to-mean tool force did not correlate with the penetration depth and spacing, which was verified by the results of the multiple regressions performed in this study (Table 7). Therefore, in this study, an analysis was performed on the relationship between the ratio of peak to mean tool forces and s/d.

In this study, FN_p/FN_m was determined to be 2.271 ± 0.411, while FC_p/FC_m was found to be 2.393 ± 0.174 (Table 2). Figure 12 and Figure 13 illustrate the relationships between s/d and FN_p/FN_m, as well as FC_p/FC_m, and the regression equations and statistical parameters are listed in Table 6. Both FN_p/FN_m and FC_p/FC_m tended to reach their maximum values when s/d was between 3 and 4. It is noticeable that this resembles the concept of the SE methodology, where the optimal cutting conditions are determined by the theory that the SE reaches a minimum at a particular s/d. Nevertheless, FC_p/FC_m and s/d were statistically significant at the 99% confidence level (R² = 0.614, F = 11.935, and p = 0.000). In contrast, FN_p/FN_m and s/d were not statistically significant, with correlation coefficients, F-values, and p-values of 0.154, 1.367, and 0.285, respectively. Therefore, in this study, it was assumed that the optimal cutting conditions could be predicted based on the relationship between FC_p/FC_m and s/d.

3.2. Simplified Approach to Predicting Optimal Cutting Conditions

3.2.1. Optimal Cutting Conditions Using SE Methodology

In this study, the SE was calculated using Equation (1) by collecting the rock fragments generated by cutting. The SE was found to be vary from 72.89 to 178.02 MJ/m³ for cutting with a penetration depth of 3 mm, from 45.95 to 108.33 MJ/m³ for cutting with a 6 mm depth, and from 32.89 to 66.61 MJ/m³ for cutting with 9 mm depth (Table 2). This indicates an increase in SE with increasing penetration depth. The SE was the highest when s/d was 1 compared to when s/d was greater than 5 at all penetration depths, indicating that cutting was performed exclusively in the relieved mode.

Figure 14 shows the relationship between SE and s/d according to the penetration depth. At all penetration depths, a significant correlation between SE and s/d was observed, with a 90% confidence level (R² = 0.791–0.970, F = 5.689–48.449, p = 0.005–0.095). The corresponding regression equations and statistical parameters are listed in Table 6. The regression curves indicated that the minimum SE was 65.65 MJ/m³ at a 3 mm penetration depth, decreasing to 37.83 MJ/m³ and 29.19 MJ/m³ at depths of 6 mm and 9 mm, respectively. Additionally, the optimal s/d ratio for minimizing SE, irrespective of the penetration depth, was found to be between 3 and 5. This is consistent with the results of Bilgin et al. [3], who found that the optimal s/d for a conical pick was between 2 and 5.

Table 8. Regression equations for SE based on s/d.

d (mm)	Regression Equation	R2	F-Value	p-Value
3	$SE=10.482{s/d}^2-86.657s/d+244.754$	0.935	21.430	0.017
6	$SE=6.007{s/d}^2-48.700s/d+136.539$	0.791	5.689	0.095
9	$SE=5.482{s/d}^2-38.637s/d+97.271$	0.970	48.449	0.005

lists the optimal s/d ratios derived from the regression curve, which illustrates the SE and s/d relationships.

3.2.2. Comparison with SE Methodology

The relationship between FC_p/FC_m and s/d is shown in Figure 15, according to penetration depth. As confirmed in Section 3.1.2, a parabolic relationship was observed, indicating that FC_p/FC_m reaches its maximum value at a specific s/d (s/d_FC). The s/d_FC was found to be between 3 and 5, which corresponds to the optimal s/d range. Therefore, it may be seen that s/d_FC can serve as a substitute for the optimal s/d determined by the SE methodology. Figure 16 illustrates the proposed methodology, based on these findings.

To validate the proposed methodology, the optimal s/d and s/d_FC values were compared for each penetration depth. Figure 17 presents a comparison of s/d_FC, derived from the FC_p/FC_m and s/d relationship, with the optimal s/d ascertained through SE methodology. The discrepancy between the optimal s/d and s/d_FC was found to be less than 5%, demonstrating high precision. Consequently, the simplified approach introduced in this study has been demonstrated to be highly accurate in determining optimal cutting conditions.

3.2.3. Assessment of Proposed Methodology

To further verify the methodology proposed in this study, the data obtained from the LCM test, using drag-type tools performed in four studies, were analyzed.

Table 9. Summary of LCM tests obtained from the references and this study.

No.	References	Cutting Tools	Rock Types	σc (MPa)	d (mm)	Optimal s/d	s/d FC
1	This study	Conical pick	Finike limestone	49.00	3	4.13	4.09
					6	4.05	4.11
					9	3.52	3.69
2	Kim et al. [21]	Chisel pick	Cement mortar	18.00	3	3.34	2.88
					6	3.26	3.22
					9	3.38	3.46
				29.30	9	3.21	2.52
				42.00	3	3.46	3.11
					6	2.99	2.86
					9	3.23	2.59
				51.80	6	3.11	3.08
					9	3.14	2.77
3	Wang et al. [39]	Conical pick	Sandstone	17.91	3	5.06	5.30
					6	2.48	2.30
					9	2.57	2.88
					12	3.02	3.64
				79.20	3	2.82	2.92
					6	3.44	4.16
					9	3.84	3.01
4	Park et al. [46]	Conical pick	Cement mortar	21.00	4	3.51	3.17
						3.11	3.58
					6	3.27	3.03
				41.00	4	3.64	3.02
					6	2.82	2.92
				57.00	4	3.44	4.16
						2.39	2.20
					6	2.87	2.78
						2.83	2.69
5	Yasar [47]	Conical pick	Red andesite	72.85	9	4.02	3.72
			Grey andesite	99.92	9	5.94	5.16
			Green tuff	51.65	9	5.92	5.12
			Grey tuff	62.63	9	6.26	5.64
			Brown vitric tuff	88.15	9	6.08	5.02
			Yellow vitric tuff	62.48	9	3.68	2.59
			Metasiltstone	1.89	9	5.57	4.50
			Crystal tuff	2.44	9	4.00	4.26
			Volcanic sandstone	12.34	9	6.40	6.37

shows a summary of the LCM tests from the references and this study, including the cutting tool used, rock type, penetration depth, and the resulting optimal s/d and s/d_FC. It is noteworthy that the optimal s/d ratio was determined by the regression curve. Additionally, some cases were excluded from the analysis because either the correlation between SE and s/d or the correlation between FC_p/FC_m and s/d was not clear, or the optimal s/d or s/d_FC could not be determined.

Figure 18 compares s/d_FC with the optimal s/d in Table 9. In all cases, the error of s/d_FC for the optimal s/d was less than approximately 20%, and the coefficient of determination was 0.774. This demonstrates the potential generalizability of the proposed simplified approach for drag-type tools.

4. Conclusions

This study introduces a simplified approach for predicting optimal cutting conditions based on the tool forces measured during rock cutting. A laboratory-scale linear rock-cutting test was conducted on Finike limestone using a conical pick. Tool forces and the ratios between tool forces were analyzed in relation to cutting parameters such as penetration depth and spacing. The proposed prediction methodology was validated by comparing it with the SE methodology. The main conclusions are summarized as follows:

• FC and FN exhibited linear and power function relationships, respectively, with increasing penetration depth and spacing. Spacing more significantly influenced both FC and FN than did penetration depth. However, the optimal cutting conditions could not be predicted based on the relationship between the tool forces and cutting parameters.
• FN_m/FC_m exhibited a linear decrease as the penetration depth increased, but it did not show any correlation with the spacing. Additionally, FN_m/FC_m demonstrated the strongest correlation with s/d, increasing within the power function relationship as s/d increased. However, the optimal cutting conditions could not be predicted, even for the relationship between FN_m/FC_m and the cutting parameters.
• FN_p/FN_m and FC_p/FC_m exhibited a parabolic relationship with s/d. They increased as s/d increased up to a certain point, after which they decreased, behavior similar to that observed for the relationship between SE and s/d. However, the relationship between FN_p/FN_m and s/d was not significant. Therefore, it was assumed that the optimal cutting conditions could be predicted based on the relationship between FC_p/FC_m and s/d.
• The optimal s/d for the conical pick in Finike limestone was between 3 and 5, and FC_p/FC_m reached its maximum value when s/d was between 3 and 5. The difference between the optimal s/d and s/d_FC was less than 5%. Furthermore, the proposed methodology showed reasonable agreement with the results of standard techniques, with a margin of error within 20%, when compared with the results of the LCM test performed in previous studies that employed drag-type tools.

It was found that the optimal cutting conditions could be predicted from the relationship between FC_p/FC_m and s/d. The proposed methodology provides valuable information for cutterhead design and ensures accelerated excavation during operation. However, this study considered only drag-type tools and limited rock types. Therefore, further research using other types of cutting tools and rocks is required to test the generalizability of the proposed methodology.