The Chaotic Property of BTA Deep-Hole Machining System under the Effect of Inner Cutting Fluid

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Introduction
With the development of science and technology, mankind needs to face the constant challenges of aerospace, deep-sea, deep-earth, and other extreme environments; at the same time, in these extreme environments, there are many parts of scientifc exploration equipment which are also facing the challenge of a high, precision, sharp limit bottleneck. Te nonlinear characteristics of these high-precision parts during service have become a major factor afecting the overall dynamic stability of extreme equipment [1][2][3], among which the impact and application of high-precision deep-hole components with the large length-diameter ratio is particularly prominent in the abovementioned felds [4][5][6][7].
Te Boring Trepanning Association (BTA) deep-hole machining can be classifed into three categories: BTA system drilling, BTA trepanning drilling, and BTA boring. Among them, the former two are commonly used for processing solid components with low machining accuracy that require secondary precision machining, while the latter is often employed to expand holes in components with high machining accuracy and capability of achieving precision machining [8]. BTA deep-hole processing technology is an important means of processing deep-hole parts. Its system is a typical complex nonlinear process system that involves the coexistence and interaction of mechanical, electrical, and hydraulic felds. Te complexity of its motion state lies in the simultaneous rotation and axial feed of the boring bar, accompanied by the infow and outfow of cutting fuid. Te analysis of the chaotic characteristics of the system is crucial in elucidating the impact of parameter changes on system stability, which is essential for overcoming the quality bottleneck in deep-hole parts processing [9][10][11][12]. At present, there is no general analytical paradigm for analyzing a complex nonlinear dynamic system [13][14][15][16], which also brings great challenges to the dynamic stability analysis of complex BTA deep-hole machining system with mechanical, electrical, and hydraulic multifeld coupling.
Chin et al. [17][18][19] studied transverse vibration frequency characteristics of BTA deep-hole boring bar under internal cutting fuid and axial pressure using the Euler-Bernoulli beam and Timoshenko beam as models, respectively. It is found that the natural frequencies of the same order modes under the two models are quite diferent, and the diference becomes more and more obvious with the change in boring bar speed. At the same time, the change of external force can also change the natural frequency of the system. Matsuzaki et al. [20,21], on the premise of ignoring the infuence of torsional vibration, axial force, and cutting fuid, established the motion equation of the bending boring bar and found that the natural frequency and natural mode of the boring bar will change complicated with the length of boring bar entering the workpiece. Tis means that when the bending vibration frequency of the boring bar is close to or lower than the natural frequency of the system, there will be rifing marks. To suppress the vibration of the system, Raabe et al. [22,23] studied the futter disturbance and spiral regeneration efect in the deep-hole machining system and simulated the correlation between the stability and instability of the system on the futter and spiral through modeling. On this basis, Messaoud et al. [24][25][26] further proposed an online futter monitoring strategy based on control chart through dynamic modeling of the nonlinear time series of the deep-hole machining system, and investigated the infuence of the position of the adjusted guide block and drilling depth on the system chatter. Steininger and Bleicher [27] adopted a continuous multidimensional sensor system to monitor the dynamic disturbance of diferent parameter changes on the deep-hole machining system, and adjusted the parameter changes to deal with the system futter and rotary vibration, which proved that variable speed cutting plays a role in improving the system stability. Te actual production system for BTA deep hole machining involves the rotation of the boring bar, which is accompanied by the infow and outfow of cutting fuid. Tis creates a typical fuid-structure coupling system with multi-energy feld nonlinear action between the boring bar and fuid. Tis further increases the complexity of the system analysis. Hu and Miao [28] derived the nonlinear expression of the cutting fuid reaction force acting on the rotating boring bar, established the basic motion equation of the rotating boring bar under the fuid-structure coupling, and obtained the basic criterion of the halffrequency vortices and instability of the boring bar caused by the cutting fuid, which laid a foundation for the subsequent in-depth analysis of the nonlinear characteristics of the fuid-structure coupling system in BTA deep-hole machining. In the nonlinear study of the fuid-structure coupling system, Utsumi [29] studied the nonlinear vibration characteristics of the simply supported cylindrical rotor under the fuid-structure coupling efect by the semianalytical method. Te results show that increasing the fuid volume will reduce the excitation efect of nonlinear pressure gradient on the fuid velocity, and make the stability of the whole system stronger. Wang et al. [30,31] analyzed the stability of a fexible rotor based on a Bernoulli-Euler beam. It is found that the unstable region of the system will gradually move to the low-speed region with the increase of the ratio of fuid mass to rotor mass. Firouz-Abadi et al. [32][33][34] applied the frst-order shear deformation shell theory and quasi-dimensional linear Navier-Stokes theory to analyze the stability of rotating cylindrical shells under fuidstructure coupling and found that the stability of the system increases with the increase in the ratio of fuid mass to rotor mass. Zhao et al. [5,[35][36][37] combined the rod beam theory and the fuid-structure coupling theory and applied the system dynamics analysis method to establish the lateral nonlinear vibration model of the deep-hole boring bar containing the cutting fuid disturbance; analyzed the infuence of the cutting fuid free surface, dynamic viscosity, system damping ratio and system cubic stifness, and other parameters on the system vibration characteristics; and clarifed the mechanism of the cutting fuid disturbance on the nonlinear vibration of the BTA deep-hole machining system. At the same time, the research on the external cutting fuid in the process of BTA deep-hole machining can be compared to the fuid motion between the slender ring gaps in the liquid-flled state, which has been thoroughly studied in fuid dynamics. While the inner cutting fuid is the fuid movement in the slender cylindrical cavity with an incomplete flling, its fuid state is complex [38][39][40][41][42] and its infuence mechanism on the stability of the BTA deep-hole machining system is not clear. Terefore, it is necessary to further study the mechanism of the fuid-structure coupling efect formed by the internal cutting fuid on the BTA deep-hole machining system.
To explore the chaotic property of the BTA deep-hole machining system under the fuid efect of internal cutting fuid, this paper frstly established the system's equation of motion by considering the dynamic characteristics of the system under the fuid efect of the internal cutting fuid, and then based on the Hamiltonian function and Melnikov function under the near-Hamilton plane system, the critical conditions for the chaos of the system are deduced. Secondly, digital simulation was used to study the mechanism of the efect of the liquid flling ratio of the internal cutting fuid, the fow rate of the internal cutting fuid, and the frequency ratio on the critical unstable surface of the system in the processing system is studied. Finally, through the change of the flling ratio and the frequency ratio, the joint action relationship between the torque coefcient and the resultant force of the fuid force and the cutting force is explored, and the theoretical analysis conclusion is checked and verifed with the physical experimental results.

Chaos of Lateral Vibration in BTA Deep-Hole System under the Effect of Inner Cutting Fluid
To describe the motion state of the boring bar, the coordinate system of the BTA deep-hole machining system, as shown in Figure 1, is established according to the actual working conditions of the BTA deep-hole machining system under the consideration of the fuid efect of internal cutting fuid. In Figure 1, o-xyz is the coordinate system of the boring bar body, o-x a y a z a is the boring bar axis coordinate system, o-x c y c z c is the coordinate system of boring bar velocity, o-ξηζ is the rotating coordinate system of the boring bar, α is the nutation angle, β is the precession angle, c is the angle of rotation, u and v are the radial and tangential disturbance velocities of fuid in the cylindrical coordinate system, respectively, r and θ are polar coordinates in the cylindrical coordinate system, o ′ is the axis of the boring bar under ideal condition, Ω is the speed at which the boring bar rotates around its axis of symmetry (z c ) in the velocity coordinate system (o-x c y c z c ), F xa and F ya are the components of the combined force of fuid force and cutting force in the direction of x a and y a , respectively, and P 1 is the total axial pressure (P 1 � P xa + P ya , P xa , and P ya are the components of the additional axial force of boring bar in the direction of x a and y a , respectively).

Motion Equation of BTA Deep-Hole Machining System under Internal Cutting Fluid Efect.
When only considering the fuid efect of the internal cutting fuid, the motion state of the cutting fuid at any position in the fow feld can be represented by polar coordinates (as shown in Figure 1), and the relative disturbance velocity of the cutting fuid ( v → r ) can be expressed as follows: Tus, the linearized motion equation and continuity equation of ideal incompressible fuid can be obtained as follows: where is the radial disturbance pressure feld, ρ f is the cutting fuid density, P 3 is the radial fuid pressure of the cutting fuid, which is a function of (r, θ), ω 2 is the disturbance frequency of cutting fuid, b is the radius of the free liquid surface, and a → e is the implicated acceleration at any point.
Substitute equations (3) into (2) and take div to get where ∆ is the Laplace operator. Te boundary conditions of the fow feld are determined as follows: where d 1 is the inner diameter of the boring bar.
In the BTA deep-hole processing system, r � b + τ(θ, t), where τ is a frst-order small quantity and the radial disturbance velocity on the free surface is u| r�b � zτ/zt. Considering that the relative pressure on the free surface in the boring bar is zero, the radial disturbed pressure feld is P 2 | r�b � −ρ f ω 2 2 bτ, and the boundary conditions on the free surface can be fnally determined as follows: Assuming that the x a , y a , and z axes coincide with the central inertia axis of the system to the center of mass, K is the polar moment of inertia, N is the lateral moment of inertia, and the moment of momentum of the system can be expressed as follows: where ω xa � − _ β sin α, ω ya � _ α, ω z � _ c + ω za , and ω za � _ β cos α. Te projections of the relative disturbance velocity v r of the cutting fuid on the x a , y a , and z axes are v r1 , v r2 , and v r3 , respectively; Q 2xa , Q 2ya , and Q 2z are, respectively, represented as follows: Ten, the components M xa , M ya , and M z of the momentum moment equation on the x a , y a , and z axes are follows:

Shock and Vibration
Equation (9) is the global motion model of the BTA deep-hole machining system considering various factors in the working process. Te analysis of this equation needs to be combined with equations (2) and (3), taking into account the boundary conditions of cutting fuid motion, that is, equations (5) and (6).
In actual production processes, precision deep hole processing typically employs a post-guidance mode in technology. Tis involves installing the guide key behind the tool with the feed direction of the tool as its front. When considering the fuid efect, M xa � M ya � M z � 0. It is also assumed that the viscosity of cutting fuid μ � 0. In the generalized coordinates α, β, c, x a , y a , and z, the kinetic energy of the system can be written as follows: According to the Lagrange principle, β and c are cyclic coordinates. After cyclic integration, we can get Te reaction torque of the internal cutting fuid on the system is as follows: where m f is the mass of cutting fuid, F LRM � −F LSM is the gyro torque coefcient, and F LIM is the phase plane torque coefcient. All these can be expressed as functional relations related to the vibration frequency of cutting fuid, boring bar size, liquid flling ratio, viscosity coefcient of cutting fuid, and other parameters, which can refect the relationship between the fuid characteristics of internal cutting fuid and torque coefcient in the internal structure of the boring bar.
Under the general working condition, the damping moment generated by considering the deviation of boring bar axis is as follows: where k y is the characteristic number of the static moment, k yy is the characteristic number of damping torque, and V 0 is the fow velocity of the inner cutting fuid. Combined equation (11) can be deduced as follows: Substituting it into f equations (11), (12), and (9), we get the motion equation of the boring bar defection angle (α) as follows:  (15) near zero point exists as follows: Performing a Taylor expansion on equation (15), and utilizing the approximation sinα � α -α 3 /6 + o(α) 5 , equation (15) can be simplifed to: where 32 , where K L is the polar moment of inertia of the internal cutting fuid. According to the universality condition of the system operation, there exists ε � 1/V 0 ≪ 1, and by substituting z 1 � α, z 2 � _ α, and υ � ω 2 t + υ 0 as follows, equation (17) can be transformed into Determining ε � 0, there is a Hamiltonian function for (z 1 , z 2 ) components: Tus, equation (20) has hyperbolic periodic orbits in z 1 -z 2 -υ phase space: and connected by a pair of the heteroclinic orbits: when z 2 > 0, "+" is taken in equation (22); when z 2 < 0, "−" is taken in equation (22); and the components of (z 1 , z 2 ) can be obtained by the plane curve of H � η 2 ω 2 0 /(4ϕ).
When ε ≠ 0, the hyperbolic periodic orbit also exists in equation (19), denoted as X ε , in which the stable manifold is denoted as R s (X ε ), and the unstable manifold is denoted as R u (X ε ). By the Melnikov function method, we can get Te condition for the intersection of R s (X ε ) and R u (X ε ) is as follows: From the intersection of stable and unstable manifolds of hyperbolic fxed points on the cross section [43], it can be seen that, when σ p F/D f > R 0 (ω 2 ), two manifold cross sections intersect; when σ p F/D f < R 0 (ω 2 ), the cross sections of the two manifolds never intersect; and when σ p F � D f R 0 (ω 2 ), the quadratic heterologous bifurcation occurs. So, the critical condition for chaos in BTA deephole machining system is σ p F/D f < R 0 (ω 2 ). Substitute equations (18) and (24) into this critical condition to obtain Let h � 2b/d 1 and ω � ω 2 /ω 0 , equation (54) can be converted into 6 Shock and Vibration , then the critical condition for chaos in the BTA deep-hole machining system is as follows: ����������
Under diferent cutting fuid fow velocities, the efects of the flling ratio and frequency ratio on the chaotic characteristics of the system are shown in Figures 2(a) Tis means that the region where chaos occurs gradually becomes larger and the instability of the system is intensifed. Terefore, the increase of cutting fuid velocity in the boring bar will accelerate the transition process from laminar fow to turbulent fow, which makes the fuid disturbance enhance and aggravate the instability of the system.
Te physical meaning of the positive and negative frequencies in the complex frequency is the frequency when the rotation factor is rotated counter clockwise, it is a positive frequency, and the frequency when it is rotated clockwise, it is a negative frequency. In Figure 2(a), the infuence of frequency ratio ω change on chaotic characteristics of the system is as follows: when the absolute value of frequency ratio ω decreases monotonically within the interval −6 ≤ ω ≤ − 3, the torque coefcient ���������� F 2 LSM + F 2 LIM value represented by surface f decreases. Tis shows that the decrease of frequency ratio ω in this range will promote the stability of the system. When the absolute value of frequency ratio ω decreases monotonically within the interval −3 < ω ≤ − 1, the torque coefcient ���������� F 2 LSM + F 2 LIM value represented by the surface f increases, which means that the chaotic region of the system becomes larger. It shows that the decrease in the frequency ratio value in this region will intensify the instability and chaos of the system. When the frequency ratio ω continues to decrease in the absolute value of interval −1 < ω ≤ − 0.6, the value of the surface f decreases, that is to say, the chaotic region of the system becomes smaller. Tis indicates that the reduction of frequency ratio ω in the interval will promote the stability of the system. In the interval −0.6 < ω ≤ 0, the surface f value increases with the decrease of the frequency ratio ω value, which indicates that the chaotic region of the system becomes larger and the instability of the system is intensifed. In the interval 0 < ω ≤ 1.4, the value of the surface f decreases with the increase of the frequency ratio ω value, which means that the chaotic region of the system becomes smaller. Tis indicates that the increase of frequency ratio ω in the interval is benefcial to the stability of the system. In the interval 1.4 < ω ≤ 6, the surface f value increases with the increase of frequency ratio ω, which indicates that the chaotic region of the system becomes larger and the instability of the system is intensifed. Similarly, in Figures 2(b) and 2(c), the infuence trend of frequency ratio ω changes considering diferent cutting fuid fow velocities on chaotic characteristics of the system is consistent with the abovementioned law.
Te efects of cutting fuid fow velocity and frequency ratio on the chaotic characteristics of the system under diferent flling ratios are shown in Figure 3. According to the comprehensive analysis of Figures 2 and 3, the following conclusions can be drawn from the physical mechanism. Te increase of the liquid flling ratio reduces the chaotic characteristics of the system. Te increase in cutting fuid velocity changes the fuid motion pattern, intensifes the chaotic characteristics of the system, and makes the stability of the system deteriorate. When the frequency ratio changes in the small value interval, it is not easy to trigger the frequency doubling relationship between the disturbance frequency and the system frequency, which reduces the probability of resonance of the system, so the stability of the system is better; when the frequency ratio changes in a large value range, the frequency doubling relationship between the disturbance frequency and the system frequency is easy to trigger, and the probability of resonance of the system increases, so the stability of the system is poor.
To further clarify the sensitivity of the chaotic characteristics of the system to the frequency ratio, the liquid flling ratio, and the cutting fuid fow velocity, a sensitivity analysis of equation (26) is carried out, as shown in Figure 4: In Figure 4(a), to further clarify the boundary region, the expression of the critical curve can be obtained by data ftting the intersecting lines of two sensitive surfaces as follows: f(V 0 , h) � V 0 − 0.2631h −1.239 . Furthermore, it can be seen that in Figure 4(a), when coordinate (V 0 , h) makes function f(V 0 , h) > 0, the sensitivity of chaotic characteristics of the system to liquid flling ratio h is greater than that of cutting fuid fow velocity V 0 . When the coordinate (V 0 , h) makes the function f(V 0 , h) ≤0, the sensitivity of the chaotic characteristics of the system to the liquid flling ratio h is less Shock and Vibration 7 than or equal to the sensitivity of the cutting fuid fow velocity V 0 . In Figure 4(b), the sensitivity of the chaotic characteristics of the system to the frequency ratio ω is greater than that of the cutting fuid fow velocity V 0 in most regions. Only in the frequency ratio ω when the value is taken near 1.5 or the cutting fuid fow velocity V 0 is taken in the range of 0 ≤ V 0 ≤ 0.67, the chaotic characteristics of the system are more sensitive to the frequency ratio ω than to the velocity of cutting fuid V 0 . To further clear the boundary area, the critical curve is obtained by data ftting expression as follows: can be seen that in Figure 4(b), when coordinate (V 0 , ω) makes function f(V 0 , ω) > 0, the sensitivity of chaotic characteristics of the system to the frequency ratio ω is greater than that of cutting fuid fow velocity V 0 . When the coordinate (V 0 , ω) makes the function f(V 0 , ω) ≤ 0, the sensitivity of the chaotic characteristics of the system to the frequency ratio ω is less than or equal to the sensitivity of the cutting fuid fow velocity V 0 .
In Figure 4(c), the sensitivity of chaotic characteristics of the system to the frequency ratio ω and cutting fuid fow velocity V 0 is bounded by the liquid flling ratio h � 0.58. When the liquid flling ratio 0 ≤ h ≤ 0.58, the sensitivity of chaotic characteristics of the system to the frequency ratio ω  When the liquid flling ratio is 0.58 < h ≤ 1, the sensitivity of chaotic characteristics to the frequency ratio ω is less than that to the liquid flling ratio h in most regions. To further clear the boundary area, the critical curve is obtained by data ftting expression as follows: when 0 < ω < 1.5, f(h, ω) � h − 0.0399ω 2 + 0.3872ω − 0.3482; when 6 ≥ ω ≥ 1.5, f(h, ω) � h + 0.209ω 2 + 0.4008ω − 0.9541. When coordinate (h, ω) makes function f(h, ω) > 0, the sensitivity of chaotic characteristics of the system to the frequency ratio ω is greater than that of the liquid flling ratio h. When the coordinate (h, ω) makes the function f(h, ω) ≤ 0, the sensitivity of the chaotic characteristics of the system to the frequency ratio ω is less than or equal to the sensitivity of the liquid flling ratio h.
To sum up, in most regions, the sensitivity of the chaotic characteristics of the system to liquid flling ratio h is greater than that of cutting fuid fow velocity V 0 , and the sensitivity of the chaotic characteristics of the system to the frequency ratio ω is greater than the sensitivity to the cutting fuid fow velocity V 0 , while the sensitivity of the chaotic characteristics of the system to the liquid-flled ratio V 0 , and the frequency ratio ω needs to be judged based on the position of the coordinate point in the function. Given a set of parameters, the sensitivity order of the chaotic characteristics of the system to the liquid flling ratio h, cutting fuid fow velocity V 0 , and frequency ratio ω can be quickly determined by the abovementioned method. From the perspective of qualitative analysis, it can be considered that in most of the value regions, the chaotic characteristics of the system are sensitive to frequency ratio, liquid flling ratio, and cutting fuid fow velocity in the following order: when 0 ≤ h ≤ 0.58, the sensitivity of chaotic characteristics of the system to each parameter from strong to weak is frequency ratio ω, flling ratio h, and fow rate of cutting fuid V 0 ; when 0.58 < h ≤ 1, the sensitivity of chaotic characteristics of the system to each parameter from strong to weak is flling ratio h, frequency ratio ω, and cutting fuid fow velocity V 0 .

Experimental Equipment and Scheme.
Select a boring bar made of 30CrMnSi and make a precision boring process for a deep hole with an inner diameter of ∅50 mm (d 2 � ∅50 mm), in which the boring bar size is D 1 � ∅46 mm, d 1 � ∅30 mm, and L � 9 m. Te workpiece material is 35CrNiMoV, where the elastic modulus E � 214 × 10 3 MPa, shear modulus G � 82.9 × 10 3 MPa, and Poisson's ratio μ � 0.3. By changing the test condition, the infuence of parameter change on the dynamic stability of the system is studied.

Experimental Equipment.
Te test equipment is composed of a deep-hole processing machine tool, power module, signal acquisition module, and signal analysis module. Te test platform built is shown in Figure 5.
In Figure 5, the tool system employs a hard alloy boring tool with machine clip. Te tool angles are as follows: rake angle of 5°, relief angle of 8°, end relief angle of 10°, tool cutting edge angle of 45°, tool minor cutting edge angle of 3°, and inclination angle of 0°. Te workpiece bore diameter is 50 mm and the drill length is 3 m. Te cutting speed of the tool during the machining process is 10 meters per minute, with a depth of cut of 0.1 millimeters and a feed rate of 0.1 millimeters per revolution.

Experimental Scheme.
First, set the environment parameters of the system to be the same as those of the numerical simulation, and then collect the single variable signal with the flling ratio (h) and cutting fuid fow velocity (V 0 ) as the variables, respectively. Finally, determine the infuence of each parameter change on the system stability through power spectrum analysis. Te selection of variable parameters is shown in Table 1.

Experimental Data Analysis.
In this paper, the infuence of parameter changes on the stability of the system is studied through the power spectrum characteristics of the transverse vibration amplitude. Te full name of the power spectrum is the power spectrum density function, which is defned as the signal power in the unit frequency band. It can transform the vibration description in the time domain into the vibration description in the frequency domain, so it can be used as an efective tool for the analysis of system vibration characteristics [44,45].
According to data analysis, when the liquid-flled ratio h changes, the power spectrum of transverse vibration amplitude is shown in Figure 6: It can be seen from the power spectra in Figure 6 that with the increase of the liquid flling ratio h, although the frequencies of each order of resonance in the system remain unchanged, the spectral peaks of each order of frequency gradually decrease. Tis indicates that the vibration characteristics of the system are weakened and the stability of the system is enhanced with the increase of liquid flling ratio h. Tis is consistent with the efect of liquid-flled ratio h change on system stability in Figures 2 and 3.
When cutting fuid velocity changes V 0 , the power spectrum of transverse vibration amplitude is shown in Figure 7.
From Figure 7, with the increase of the cutting fuid fow velocity, although the resonance point of the system is unchanged, the system's discrete spectrum peak gradually increased, and the peak value is getting larger. Tis shows that the increase of the cutting fuid velocity aggravates the chaotic characteristics of the system, which is consistent with the numerical simulation results in Figures 2(b) and 2(c).

Conclusion
In this paper, the nonlinear lateral vibration motion equation of the BTA deep-hole machining system is established by analyzing the internal cutting fuid efect. On this basis, the chaotic characteristics of the system are studied systematically; and by combining numerical simulation and physical experiment, the mechanism of the dynamic stability of the system under the changes of flling ratio, cutting fuid fow velocity, and frequency ratio was preliminarily obtained.
(1) In precision boring, the increase of the liquid flling ratio improves the system quality, reduces the system frequency, and causes the resonance frequency of the system to decrease, which reduces the resonance area and weakens the chaotic characteristics of the system. Terefore, in the actual machining process, increasing the flling ratio is one of the measures to improve the stability of the machining system. (2) In precision boring, the increase of cutting fuid velocity will change the movement of fuid and aggravate the chaotic efect of the system. Terefore, under the condition of meeting the production demand, the cutting fuid velocity should be reduced as much as possible to improve the stability of the system. (3) When the frequency ratio changes in a small range, the frequency doubling relationship between the disturbance frequency and the system frequency is    not easy to trigger, the probability of resonance of the system is reduced, and the system stability is good. When the frequency ratio changes in a large numerical range, the frequency doubling relationship between the disturbance frequency and the system frequency is easy to trigger, which increases the probability of resonance of the system, and the system stability is poor. Terefore, in the actual processing process, the resonance region can be avoided by adjusting the frequency ratio value range to improve the stability of the system.
In summary, the research conclusions of this paper on the dynamic stability of the BTA deep-hole machining system under the consideration of the internal cutting fuid efect can lay a certain theoretical foundation for the analysis, control, and optimization of its complex mechanical behavior in engineering practice.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.