RCSA-Based Analysis of Stability Lobes in Milling Incorporating Tool Clamping Errors

Abstract

This study proposes a methodology for selecting robust stable cutting conditions from a Receptance Coupling Substructure Analysis (RCSA)-based Stability Lobe Diagram (SLD) by considering tool clamping errors that may occur during operator tool setup. However, most existing RCSA studies have been conducted under the assumption of a constant tool clamping length and thus do not sufficiently reflect the clamping length variation observed in practical machining environments. Since the tool tip dynamic characteristics can be sensitive even to small variations in clamping length, operator-induced tool clamping errors in actual processes can introduce such variations and consequently degrade the prediction accuracy of the SLD. Moreover, uncertainty studies in milling stability have largely focused on variations in model parameters, such as cutting coefficients, damping, and modal parameters, whereas experimental quantification of operator-induced clamping length variability and its direct integration into RCSA-based tool tip Frequency Response Function (FRF) and SLD prediction has been relatively limited. Therefore, this study quantifies the distribution of tool clamping errors through clamping experiments and incorporates it into RCSA to derive an SLD band that accounts for tool clamping errors. The width of the SLD band is defined as a physical variation induced by clamping uncertainty, and the corresponding uncertainty range is set as an avoidance region. Robust cutting conditions are then selected from the remaining stable region while considering the physical variation width. The physical variation width was quantified as 60 rpm (minor axis) and 1.62 mm (major axis), representing the dispersion of the stability limit in the spindle speed and axial depth directions caused by clamping errors. As a result, stable cutting conditions that do not cross the stability limit can be determined even in the presence of process variations and disturbances.

1. Introduction

In the manufacturing industry, High-Speed Machining (HSM) in milling requires chatter avoidance design to improve productivity and machining quality. Chatter is a self-excited vibration that occurs between the cutting tool and the workpiece, which deteriorates surface quality, reduces machining precision, and shortens tool life [1,2,3]. Accordingly, various studies have been conducted to predict and analyze chatter and to derive stable cutting conditions [4,5]. In HSM, achieving a high material removal rate can be limited by regenerative chatter, where surface waviness left by the previous cut modulates chip thickness in the subsequent cut and amplifies vibrations [6]. Chang et al. [7] proposed a chatter analysis and Stability Lobe Diagram (SLD) prediction system using real-time monitoring and compensation techniques based on the zero-order and envelope methods. Yamato et al. [8] developed an automatic chatter suppression system for parallel milling by combining observer-based monitoring with real-time spindle speed control. Wan et al. [9] presented an effective approach to suppress milling chatter by applying an Active Magnetic Bearing (AMB) and a projection-based robust adaptive controller.

The SLD is a widely adopted tool for chatter avoidance. The SLD represents stable and unstable cutting regions as a function of spindle speed and axial depth of cut, providing an essential standard for selecting stable cutting conditions. Furthermore, Urbikain et al. reported that the SLD can be used to evaluate the maximum achievable productivity under chatter-free conditions, even at the machine design stage before the machine is built [10]. To further enhance the efficiency and accuracy of chatter stability prediction, various advanced numerical and analytical methods have been developed. For instance, Compean et al. [11] introduced a homotopy-based method to solve the transcendental equations of milling stability, providing a robust approach for identifying stability boundaries. Similarly, Urbikain et al. [12] employed Chebyshev polynomials to discretize delay differential equations in milling, particularly for tools with specific geometries like rectangular end mills, effectively capturing the complex dynamics of the process. Furthermore, recent studies by Sanz-Calle et al. [13] have explored sophisticated modeling techniques to improve the reliability of Stability Lobe Diagrams (SLDs) under diverse machining conditions. While these numerical methods offer high precision in solving the governing equations, generating the SLD requires Frequency Response Function (FRF) data at the tool tip [14,15]. Ealo et al. reported that, in machine tools, joints can influence the tool tip dynamic characteristics; therefore, accurate identification of joint properties is important [16].

However, performing vibration tests for a wide range of cutting tools each time is highly inefficient in terms of time and cost. To address this inefficiency, Schmitz and Donaldson [17] first proposed Receptance Coupling Substructure Analysis (RCSA), which predicts the receptance of an assembled system by coupling the receptances of individual components. The reliability of this method was demonstrated by Schmitz et al. [18]. Subsequently, Schmitz and Ducan [19] advanced the approach into a second-generation RCSA by modeling the spindle holder cutting tool assembly as three substructures. Erturk et al. [20] established a mathematical model for RCSA by applying Timoshenko beam theory. Park et al. [21] and Cheng et al. [22] integrated rotational degrees of freedom into the RCSA model and demonstrated that rotational degrees of freedom have a significant impact on the accuracy of RCSA. Albertelli et al. [23] improved the prediction accuracy of the rotational FRF by using RCSA that combines experimentally identified machine tool dynamic characteristics with a Finite Element Analysis (FEA) model of the cutting tool. Ozsahin et al. [24], Park et al. [25], and Mostaghimi et al. [26] proposed new RCSA approaches by introducing Artificial Neural Networks (ANNs) and machine learning.

Although the importance of rotational FRF has been highlighted in efforts to improve the prediction accuracy of RCSA, there are research limitations in estimating it with high precision. The method of Durate et al. [27] introduces errors in the antiresonance frequency range. The technique of Albertelli et al. [23] requires FRF measurements under multiple excitations, which can lead to noise amplification. Kim et al. [28] estimated rotational FRF using a compensation strategy. However, limitations remain depending on boundary conditions. To overcome the limitations in estimating rotational FRF, Kim et al. [29] proposed a new RCSA methodology that improves accuracy in the antiresonance region and eliminates unnecessary additional experiments. In their study, the cutting tool was divided into the flute and shank. Using Euler–Bernoulli beam theory, they estimated characteristic curves. Rotational FRF was then estimated by curve fitting between the experimentally measured tool FRF and the estimated characteristic curves.

Although this method showed improved predictive performance compared with previous approaches, it did not consider uncertainties in real machining environments, such as tool clamping errors. In real machining processes, the overhang length can vary depending on operational requirements, which can significantly alter the dynamic characteristics of the tool–holder system [30]. These overhang variations modify the tool tip dynamic characteristics, thereby shifting the stability boundaries and significantly affecting overall machining stability [31,32]. Consequently, the quantification of variations caused by the uncertainty in tool clamping is imperative to ensure the reliability of machining processes at industrial sites.

This study presents an SLD analysis accounting for tool clamping error uncertainty based on the RCSA methodology proposed by Kim et al. [29]. The tool clamping error distribution obtained from operator clamping experiments was incorporated into the RCSA-based SLD analysis. For the same clamping condition, the RCSA-based SLD and the experimental SLD were compared using Intersection over Union (IoU) to demonstrate high reliability. However, in real machining environments, operator-induced tool clamping errors occur and can change the tool tip dynamic characteristics, thereby reducing the prediction accuracy of the SLD. Therefore, this study generates an SLD band that reflects tool clamping errors and defines the width of the SLD band as a physical variation induced by uncertainty. In addition, the SLD band caused by tool clamping errors was defined as an avoidance region and excluded from candidate cutting conditions. Cutting conditions were then determined within the remaining stable region while accounting for physical variation. This study proposes a methodology for identifying stable cutting conditions that remain stable even when unpredictable disturbances occur during machining. Therefore, the distinct novelty of this work lies in establishing a framework for robustness under uncertainty. Rather than aiming solely for precise point prediction, this methodology defines a robust stability region that remains invariant to setup errors, thereby bridging the gap between theoretical SLDs and the stochastic nature of manual industrial environments.

2. Tool Tip Dynamic Prediction and Stability Lobe Diagram Analysis Incorporating Tool Clamping Errors

This study presents, in Figure 1, a schematic of the procedure for predicting the tool tip dynamic characteristics of the overall system without repeated experiments by using the RCSA methodology and for constructing the SLD based on the predicted dynamics. RCSA is a methodology that predicts the receptance of an assembly by coupling the receptance of individual substructures. Through this approach, the FRF can be efficiently predicted. Based on the predicted dynamic characteristics, the axial depth of cut as a function of spindle speed is calculated to construct the SLD. In addition, because the stability boundary can vary with small changes in clamping length, this study derives an SLD band that accounts for tool clamping error-induced variations in clamping length.

2.1. Receptance Coupling Substructure Analysis

Receptance is a type of FRF that represents the dynamic characteristics of a structure. It is a physical quantity that expresses the displacement response to an applied excitation force in the frequency domain. RCSA is a method for predicting the dynamic characteristics of an assembly using the dynamic characteristics of its substructures. The notation for displacement, rotation, force, and moment used in RCSA is shown in Figure 2. The full receptance matrix of the assembly is given in Equation (1)

$$G_{ij}=\left[\begin{array}{cc}{\displaystyle \frac{X_i}{F_j}}& {\displaystyle \frac{X_i}{M_j}}\\ {\displaystyle \frac{{\Theta }_i}{F_j}}& {\displaystyle \frac{{\Theta }_i}{M_j}}\end{array}\right]=\left[\begin{array}{cc}{H}_{ij}& L_{ij}\\ N_{ij}& P_{ij}\end{array}\right]$$

(1)

where $H_{ij}$, $L_{ij}$, $N_{ij}$, and $P_{ij}$ denote displacement to force, displacement to moment, rotation to force, and rotation to moment receptances, respectively. $X$ and $\Theta$ represent the displacement and rotation at the response point, respectively. $F$ and $M$ denote the excitation force and moment applied at the excitation point. The subscripts $i$ and $j$ indicate the response point and the excitation point, respectively. When $i=j$, the term is referred to as direct receptance; otherwise, it is referred to as cross-receptance. The full receptance matrix of the substructure is given in Equation (2).

$$R_{ij}\left(\omega \right)=\left[\begin{array}{cc}{\displaystyle \frac{x_i}{f_j}}& {\displaystyle \frac{x_i}{m_j}}\\ {\displaystyle \frac{{\theta }_i}{f_j}}& {\displaystyle \frac{{\theta }_i}{m_j}}\end{array}\right]=\left[\begin{array}{cc}{h}_{ij}& l_{ij}\\ n_{ij}& p_{ij}\end{array}\right]$$

(2)

where $h_{ij}$, $l_{ij}$, $n_{ij}$, and $p_{ij}$ denote displacement to force, displacement to moment, rotation to force, and rotation to moment receptances, respectively. $x$ and $\theta$ represent the displacement and rotation at the response point, respectively. $f$ and $m$ denote the force and moment applied at the excitation point, respectively.

To derive the SLD, the dynamic characteristics at the tool tip are essentially required. Since the cutting tool exhibits different geometric and material properties in the shank and flute sections, simplifying it as a single structure may fail to sufficiently reflect these differences, thereby reducing the prediction accuracy of the dynamic characteristics. Therefore, to improve the prediction accuracy, this study divides the cutting tool into two substructures, as proposed by Kim et al. [29], and performs RCSA. The receptance of each substructure is obtained using Euler–Bernoulli beam theory. After assuming a rigid connection at the coupling interface, the tool subassembly is constructed as shown in Figure 3 by applying displacement compatibility and force equilibrium conditions. The full receptance matrix of the resulting tool subassembly is defined as ${GS}_{ij}$, and the corresponding relationships are given in Equations (3)–(6).

$${GS}_{11}\left(\omega \right)=R_{11}\left(\omega \right)-R_{12a}\left(\omega \right){\left[R_{2a2a}\left(\omega \right)+R_{2b2b}\left(\omega \right)\right]}^{-1}{R}_{2a1}\left(\omega \right)$$

(3)

$${GS}_{13a}\left(\omega \right)=R_{13a}\left(\omega \right){\left[R_{2a2a}\left(\omega \right)+R_{2b2b}\left(\omega \right)\right]}^{-1}{R}_{2b3a}\left(\omega \right)$$

(4)

$${GS}_{3a1}\left(\omega \right)=R_{3a2b}\left(\mathsf{\omega }\right){\left[R_{2a2a}\left(\mathsf{\omega }\right)+R_{2b2b}\left(\mathsf{\omega }\right)\right]}^{-1}{R}_{2a1}\left(\mathsf{\omega }\right)$$

(5)

$${GS}_{3a3a}\left(\omega \right)=R_{3aa3}\left(\omega \right)-R_{3a2b}\left(\omega \right){\left[R_{2a2a}\left(\omega \right)+R_{2b2b}\left(\omega \right)\right]}^{-1}{R}_{2b3a}\left(\omega \right)$$

(6)

where $u_1$, $u_{2a}$, $u_{2b}$ and $u_{3a}$ denote the node at the flute tip, the interface node on the flute side, the interface node on the shank side, and the node at which the shank is coupled to the holder, respectively. $R_{ij}$ and ${GS}_{ij}$ denote the full receptance matrix of a substructure and the full receptance matrix of the cutting tool subassembly obtained from the shank flute RCSA, respectively. Using RCSA with the obtained cutting tool ${GS}_{ij}$ and the holder receptance $R_{3b3b}$, the full receptance matrix at the tool tip of the overall system, $G_{11}$ is defined in Equation (7).

$$G_{11}={GS}_{11}\left(\mathsf{\omega }\right)-{GS}_{13a}\left(\mathsf{\omega }\right){\left[{GS}_{3a3a}\left(\mathsf{\omega }\right)+R_{3b3b}\left(\mathsf{\omega }\right)\right]}^{-1}{GS}_{3a1}\left(\omega \right)$$

(7)

where the holder’s full receptance matrix $R_{3b3b}$ is determined as expressed in Equation (8) through Inverse Receptance Coupling Substructure Analysis (IRCSA), which involves removing the cutting tool from the assembled system.

$$R_{3b3b}={GS}_{3a1}\left(\omega \right)\left[{GS}_{11}\left(\omega \right)-G_{11}\left(\omega \right)\right]{GS}_{13a}\left(\omega \right)-{GS}_{3a3a}\left(\omega \right)$$

(8)

2.2. Stability Lobe Diagram

The SLD is defined as a relationship between spindle speed and axial depth of cut, where the region below the stability boundary indicates stable cutting and the region above it indicates cutting. The axial depth of cut in the SLD is given in Equation (9) [33].

$$b_{lim}={\displaystyle \frac{-1}{2K_{s}Re\left[{FRF}_{Orient}\right]N}}$$

(9)

where $K_s$, $N$, and orient FRF denote the cutting coefficient, the average number of teeth engaged in the cut, and the oriented FRF, respectively. In milling processes, ${FRF}_{orient}$ is used because the dynamics depend on the two-directional tool tip dynamic characteristics, as given in Equation (10) [33].

$${FRF}_{Orient}={\mu }_x{FRF}_x+{\mu }_y{FRF}_y$$

(10)

where ${FRF}_x$, ${FRF}_y$, ${\mu }_x$ and ${\mu }_y$ denote the tool tip FRF in the $x$ direction, the tool tip FRF in the $y$ direction, the $x$ directional coefficient, and the $y$ directional coefficient, respectively. Therefore, tool clamping errors arising during the operator’s clamping process introduce uncertainty in the tool tip FRF, which ultimately leads to variations in the SLD and makes it difficult to ensure prediction reliability. While various factors contribute to machining instability, this study specifically explicitly limits its claims to the robustness against operator-dependent setup errors. Consequently, other stochastic variables, including runout and tool wear, are excluded from the current uncertainty bands to strictly evaluate the sensitivity of the stability boundary to clamping length variation.

2.3. Analysis of Tool Clamping Errors

Tool clamping errors occurring in industrial milling environments constitute one of the uncertainties that induce variations in the dynamic characteristics of the machining system. Lee et al. [34] showed that variations in operating conditions during the milling process change the tool tip dynamic characteristics and that variations in SLD affect the reliability of stability prediction. Specifically, under manual clamping, slight variations in tool overhang length can arise due to human-induced factors and environmental factors, and such uncertainty decreases the accuracy of predicting the tool tip dynamic characteristics. In this study, to quantify the variability introduced during the clamping process, repeated clamping experiments were conducted with four operators (10 trials per operator, 40 trials in total). While the sample size is primarily intended for methodological validation rather than a universal population study, it effectively captures the representative range of physical variation in industrial tool setup. By applying Kernel Density Estimation (KDE) to these 40 trials, the 90% confidence interval of the clamping error was confirmed to be between −2.52 mm and 2.08 mm. However, we intentionally extended the analysis range to ±4 mm to encompass the remaining 10% of data points that fall outside this confidence interval. This extension was specifically intended to account for the worst-case conditions among all possible error ranges that can arise from manual operation.

This physical variation during clamping directly induces changes in the tool tip FRF, thereby increasing the uncertainty in the SLD. The distribution of these variations is illustrated in Figure 4, which shows the tool clamping error distribution characterized by kernel density estimation (KDE). The results indicate that the clamping error follows a probabilistic distribution centered around the nominal position. Based on these results, the overhang length variation is expressed as Equation (11)

$$L=L_0+∆L \hspace{1em}\hspace{1em}∆L\in \left[-4, 4\right]\mathrm{m}\mathrm{m}$$

(11)

where $L$ denotes the actual clamped overhang length used in the milling setup, $L_0$ is the standard overhang length defined as the nominal position, and $∆L$ represents the clamping error induced by the clamping process, as statistically represented in Figure 4.

3. Experimental Methodology

This study proposes a methodology for estimating stable cutting conditions in milling by predicting tool tip dynamic characteristics while accounting for operator-related tool clamping errors. To ensure high-precision dynamic measurements, Frequency Response Functions (FRFs) were obtained using an impulse hammer (PCB 086E80, PCB Piezotronics, Inc., Depew, NY, USA) and an ultra-miniature accelerometer (PCB 352A21, PCB Piezotronics, Inc., Depew, NY, USA). The data acquisition and signal processing were performed using the Simcenter SCADAS Mobile hardware and Simcenter Testlab software (https://plm.sw.siemens.com/en-US/simcenter/physical-testing/testlab/, accessed on 2 January 2026, Siemens Digital Industries Software, Plano, TX, USA). Notably, the accelerometer possesses a minimal mass of only 0.6 g, which is sufficiently light to ensure that the mass loading effect on the tool tip dynamics remains negligible, thereby securing the reliability of the measured data. Based on these predicted dynamics, the Stability Lobe Diagram (SLD) is derived. Figure 5 presents the overall workflow for analyzing the RCSA-based SLD incorporating clamping errors, building on the RCSA proposed by Kim et al. [29], and for determining optimal cutting conditions.

• As shown in Figure 6, the beam FRF was measured via an impact test. Divided into substructures 1 and 2, the uniform beam was analyzed using Euler–Bernoulli beam theory for each substructure, and receptance coupling was performed to obtain the characteristic curves of the uniform beam. The obtained characteristic curves and the FRF measured from vibration experiments were used for modal fitting to estimate the material properties of each substructure, such as the elastic modulus, density, and damping ratio. Using the estimated material properties, the full receptance matrix of the entire uniform beam was calculated. In addition, to perform Inverse Receptance Coupling Substructure Analysis (IRCSA), the full receptance matrix of the overhang uniform beam was calculated.
• As shown in Figure 7, to perform IRCSA, two accelerometers were attached to the uniform beam assembled with a BT40 holder, and an impact excitation was applied at the beam tip to measure the FRF. The full receptance matrix was then estimated from the measured FRF using Schmitz’s method.
• Using IRCSA, the full receptance matrix of the holder was obtained from the full receptance matrix of the uniform beam assembled with the holder estimated in the previous step and the full receptance matrix of the overhang uniform beam. The estimated displacement to force receptance of the holder, $R_{3b3b},$ is shown in Figure 8. This enables the overall system dynamic characteristics to be predicted in subsequent cases by measuring the FRF only for an additional cutting tool to be coupled and performing RCSA with the obtained holder receptance.
• As shown in Figure 9, the cutting tool FRF was measured via an impact test. Divided into substructures 1 and 2 to account for the heat-treated region and geometric differences, the cutting tool was analyzed using Euler–Bernoulli beam theory for each substructure, and receptance coupling was performed to obtain the characteristic curves of the cutting tool. The obtained characteristic curves and the FRF measured from vibration experiments were used for modal fitting to estimate the material properties of each substructure, such as the elastic modulus, density, and damping ratio. Using the estimated material properties, the full receptance matrix of the cutting tool was calculated. In addition, the full receptance matrix of the overhang cutting tool incorporating tool clamping errors, ${{GS}_{ij}}^{o-tool}$, was calculated. The displacement-to-force receptance ${h_{11}}^{o-tool}$ is shown in Figure 10, the displacement-to-moment (=rotation-to-force) ${l_{11}}^{o-tool}\left(={n_{11}}^{o-tool}\right)$ is shown in Figure 11, and the rotation-to-moment receptance ${p_{11}}^{o-tool}$ is shown in Figure 12.
• As shown in Figure 13, the tool tip FRFs predicted by RCSA for the 70 mm overhang condition incorporating a tool clamping error of $\pm$4 mm were obtained. As summarized in

  Table 1. Comparison of experimental and RCSA prediction natural frequencies for the 70 mm overhang condition.

  Overhang (mm)	Mode	Experiment (Hz)	Prediction (Hz)	Error Rate (%)
  70	$1_{st}$	368.75	368.80	0.01
  	$2_{nd}$	1609.38	1614.12	0.29

  , the prediction errors of the first and second natural frequencies were both within 1%, confirming the prediction reliability. As shown in Figure 14, the cutting tool FRF was measured via an impact test with the cutting tool assembled with a BT40 holder, and this measurement was used for comparison with the predicted FRFs.
• To derive the RCSA-based SLD incorporating tool clamping errors, as shown in Figure 15, the modal parameters at the tool tip are required. In this study, modal parameters such as the natural frequency and damping ratio were estimated using a peak-picking method, and a Single Degree of Freedom (SDOF) FRF was constructed based on the estimated parameters to calculate the SLD [35].

The SLD was obtained under the down-milling and 50% radial immersion conditions shown in Figure 16, and the cutting conditions used for the calculation are summarized in

Table 2. Cutting parameters for SLD down milling at 50% radial immersion.

Parameters	Value
Cutting coefficient $\left(K_s\right)$	850 $\left(\mathrm{N}/{\mathrm{m}\mathrm{m}}^2\right)$
Force angle $\left(\beta \right)$	${50}^{°}$
No. of teeth $\left(N_t\right)$	0.5
Workpiece	Al 7075-T6

.

Where $\beta$ denotes the angle between the normal vector $n$ and the tangential vector $F$. In addition, for an overhang of 70 mm, the RCSA-based SLD was compared with the experimental SLD using the IoU method to validate the prediction reliability, and the SLD band incorporating tool clamping errors was then presented based on the validated procedure.

4. Result and Discussion

Stable cutting conditions for chatter avoidance in milling environments were determined using an RCSA-based SLD band incorporating tool clamping errors.

4.1. Validation of the Reliability of the RCSA-Based SLD

The RCSA-based SLD was calculated using the tool tip FRF predicted by RCSA. Figure 17 presents an IoU comparison between the RCSA-based SLD and experimental SLD, resulting in an IoU of 97.92%. This result indicates high reliability of the proposed RCSA-based SLD.

4.2. Analysis of Milling Cutting Conditions Using an SLD Incorporating Tool Clamping Errors

The RCSA-based SLD was analyzed for the overhang range of 70 ± 4 mm, incorporating tool clamping errors. As shown in Figure 18, the RCSA-based SLD incorporating tool clamping errors was obtained as an SLD band, which represents the range of stability boundaries. From this band, the dispersion ranges for the minor and major axes were derived as 60 RPM and 1.62 mm, respectively, representing the variations in spindle speed and axial depth of cut caused by clamping uncertainty.

When selecting machining conditions, the uncertainty region expressed by the SLD band was defined as an exclusion region and removed from candidate conditions, and cutting conditions were selected within the remaining stable cutting region. Based on this, the physical variation was incorporated into the remaining stable region to define a robust, high-efficiency cutting condition range that does not cross the stability boundary even under uncertain changes in spindle speed and axial depth of cut unforeseen process disturbances.

5. Conclusions

This study proposes a methodology for selecting robust stable cutting conditions from an RCSA-based SLD by considering tool clamping errors that may occur during operator tool setup in industrial milling environments.

• Repeated clamping experiments with four operators (10 trials per operator, 40 trials in total) were conducted to obtain clamping length samples, and KDE confirmed that the tool clamping error forms a distribution of approximately $\pm$4 mm.
• The $\pm$4 mm distribution was incorporated into RCSA to predict the tool tip dynamic characteristics and to derive an SLD band that accounts for tool clamping error-induced variations in clamping length.
• For the 70 mm overhang condition, the RCSA-based SLD and the experimental SLD were compared using IoU to quantify the overlap of stable regions, and the IoU comparison resulted in 97.92%, indicating high reliability of the proposed RCSA-based SLD.
• The width of the SLD band was defined as a physical variation caused by clamping uncertainty, and the minor axis and major axis of the physical variation were determined to be 60 RPM and 1.62 mm, respectively, representing the range over which spindle speed and axial depth of cut are dispersed due to tool clamping errors. When selecting machining conditions, the uncertainty region expressed by the SLD band was defined as an exclusion region and removed from candidate conditions, and cutting conditions were selected within the remaining stable cutting region while accounting for the physical variation, thereby enabling stable cutting conditions that do not cross the stability boundary even under uncertain changes in spindle speed and axial depth of cut and unforeseen process disturbances. In conclusion, this paper presents a methodology to secure robustness under uncertainty by converting the physical clamping error distribution into a stability avoidance zone. While the model is currently limited to clamping variations, it provides a critical foundation for robust process planning, ensuring that selected cutting conditions remain valid even in the presence of unpredictable manual setup errors.