A Method for Predicting the Boring Topography Error of Shaft Hole on a Thin-Walled Box Based on a Proposed Dynamic Model

Traditional boring dynamic model ignores the effect of the uneven radial stiffness of the shaft hole of the thin-walled box on the boring vibration characteristic, which will lead to the deviation in the prediction for both the cutting force and the surface topography of the machining hole. To address this shortcoming, a new calculation model of the equivalent radial stiffness of the shaft hole is proposed and then validated by the finite element (FE) method. On this basis, a boring dynamic model of the shaft hole is developed, and an experimental study on the dynamic characteristics of the shaft hole boring is carried out to verify the correctness and effectiveness of the proposed dynamic model. Moreover, a prediction method for the surface topography error of the machined shaft hole is presented based on the proposed dynamic model. Finally, the comparison investigation on the hole machined surface topography predicted by the proposed method and measured in practice is carried out. The results show that the proposed theoretical models and method can achieve accurate predictions for both the boring cutting force and the machined surface topography error of the shaft hole on the thin-walled box. This study can provide a theoretical basis for machining error control and process optimization of the fine boring of the thin-walled transmission box.


I. INTRODUCTION
High-power density transmission box system has the advantages of compact structure, and precise and efficient transmission, which is widely used in the transmission equipment of vehicles, ships and aviation [1], [2], [3]. However, it is easy to deform or vibrate in processing due to the box-shaped part structure with lower stiffness. Thus, the machining accuracy of the transmission box shaft hole is difficult to control [4]. As a key feature of the transmission box, the machined surface topography error of the shaft hole directly determines the assembly accuracy of the transmission components and is closely related to the dynamic service performance of the whole box. The boring dynamic model is an important theoretical means to study the dynamic behavior of shaft hole The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso . machining and to develop prediction methods for the shaft hole machined surface topography error. To establish the boring dynamic model is the benefit to understand the forming mechanism of machining error, and be timely controlled to avoid unexpected phenomena occurrence, but how to improve its calculation accuracy is a prerequisite for predicting the cutting force and the surface topography of the shaft hole [5].
The fluctuation of cutting force can lead to the vibration of the cutter and workpiece, and then affect the machining precision of the workpiece in process. Accurately modelling cutting force has become an important theoretical means for scholars to establish the prediction of machining surface topography error and evaluate the cutting stability [6], [7], [8]. For example, Sabberwal et al. [9] were the first to demonstrate that the cutting force is proportional to the chip load area, and the proportionality coefficient depends on the workpiece and cutter material characteristics and cutting VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ conditions. Subramani et al. [10] proposed a simple analytical model for calculating the cutting force of boring. In their model, the cutting force components were obtained by using an equivalent chip flow angle. Based on the work in [10], Kuster et al. [11] carried out a cutting dynamic model by considering the regenerative chatter. Then, Atabey et al. [12] developed two models to predict the cutting force during the boring process. Young et al. [13] proposed a model for predicting the cutting force, in which the equivalent cutting edge was defined and used. Later, Wang et al. [14] improved their model by considering the influences of inclination and rake angles. The cutting force depends heavily on the dynamic parameters of the cutting system and can be obtained from the dynamic model of the machining system. Establishing the dynamic model of the machining system is a basic theoretical method to evaluate the mechanical properties of machining and predict the precision of the machined surface topography, which plays an important role in industrial practices [15]. In recent years, scholars have paid more and more attention to the dynamic modelling of machining systems [16]. For example, Lazoglu et al. [16] proposed a time-domain mathematical model for solving the dynamics in the boring process and then studied the influence of boring parameters on cutting force. Ozlu et al. [17] developed a discretization method of the trapezoidal differential element to calculate the cutting area and cutting force, and then to observe the impact of tool nose radius on machining stability. Moetakef-Imani et al. [18] used the Euler-Bernoulli beam theory to conduct dynamic modelling of the boring bar. Miguelez et al. [19] established a dynamic model to study the flutter suppression performance of a boring bar. Ghorbani et al. [20] evaluated the vibration characteristics of the boring bar by changing its dynamic parameters. Chockalingam et al. [21] established a model to study the damping capacity and flutter performance of the boring tool. Fallah et al. [22] proposed a chatter vibration mechanical model of the boring bar to simulate the nonlinear dynamic characteristics of the forward path.
In the current boring dynamic modelling, the influence of the radial stiffness of the cutter rod on the dynamic behavior of the boring system is focused. However, the thin wall with the shaft hole attached is prone to a large deformation in the boring process, which will lead to a large difference in the radial stiffness of the shaft hole in different directions, and then affect the dynamic characteristics of the boring system. While this effect can not be considered in the traditional dynamic model of the boring system, which is bound to lead to a large deviation in the prediction for both the boring dynamic behavior and machining topography error.
Therefore, given the problems mentioned above, the main contributions and highlights of this paper are summarized as follows: • A new analytical calculation model of the radial stiffness of the shaft hole on the thin-walled box is established, and its accuracy is validated by the FE method.
• A dynamic model of the boring system is proposed by considering the supporting deformation of the shaft hole, and its correctness and effectiveness are validated by conducting an experimental study.
• Combined with the established dynamic model, a prediction method for the surface topography error of the boring shaft hole is developed, and its prediction effect is proved by actual measured results. The following structure of this paper is arranged as follows: The problem that needs to be addressed in this paper is introduced in Section II. The proposed shaft hole radial stiffness calculation model and boring dynamic model are demonstrated in Section III. The established shaft hole radial stiffness verification FE model and the verification experiment of the boring dynamic characteristics are presented in Section IV. Then the proposed prediction method for the surface topography error of the machined shaft hole is described in Section V. In addition, the results obtained from the proposed models and method are analyzed and discussed in Section VI. Finally, some main conclusions are drawn in Section VII.

II. PROBLEM FORMULATION
The force analysis of the boring system is shown in Fig. 1 (a). The complex time-varying cutting force is generated due to the extrusion motion of the insert and the hole wall. The cutting force is decomposed into three mutually perpendicular components, namely F f , F r and F t , along the axial, radial and tangential directions of the boring rod. Fig. 1 (b) shows the error generation process analysis during the boring process. The point O N is the fixed reference point of the relative machine coordinate system {G}. On the plane XYO N , the boring cutter is deflected by the radial force F cr and tangential force F ct , which makes the center point O C of the boring bar deviate from the position O N and form the displacement e c . According to the third law of Newton mechanics, there are equivalent inverse forces F hr and F ht acting on the hole wall to cause additional deformation of the box wall, which makes the center point O H of the shaft hole away from the position O N and then the offset e h occurs. The offset e c and e h together constrain the change of instantaneous chip thickness in the boring process.
Combining with Fig. 1 (a) and (b), the transformation relation of cutting component forces in two coordinate forms that describe box wall deformation by cartesian coordinates and boring bar deflection by polar coordinates, is established, namely: Fig. 2 shows the simplified cantilever beam model of the shaft hole on the thin-walled box. In Fig. 2, points A and B denote ectopic two points on the circumference of the shaft hole and point C refers to the endpoint of the shaft hole attached to the thin wall. Since the shaft hole is contained in the suspended beam structure, it is assumed that the component of the cutting force in the X direction is always collinear with point C, thus the bending moment of this component on point C equals zero. Therefore, when the cutting force acts at points A and B, the bending moment of point C is, Assuming that the bending strength of the box wall at point C is The angle of the wall-attached suspension beam caused by M C A and M C B refers to φ C A and φ C B , respectively. Thus there is φ C A < φ C B . In these two cases, the offset of the hole axis point O H can be expressed as, The equivalent directional stiffness of points A and B relative to the hole center O H can be expressed as, where i =A or B, x C and y C are the directional offsets at point C of the box thin wall caused by the cutting force.
Due to x Aφ = x Bφ and y Aφ = y Bφ , there is k Ae hx = k Be hx and k Ae hy = k Be hy . Therefore, through the above derivation, it can be seen that due to the influence of the shaft hole's own structure and the deformation of the thin wall, the directional stiffness of the shaft hole changes with the variation of the machining position. However, this effect has been ignored in the existing boring dynamic models, which will inevitably lead to deviation of the prediction in both the dynamic responses of the boring system and the machined shaft hole surface topography error.

III. BORING DYNAMIC MODELLING OF SHAFT HOLE ON THIN-WALLED BOX
In this section, the analytical expression of the cutting force of the boring system is established, and the calculation model of the directional stiffness of the shaft hole is proposed. On this basis, the boring dynamic model of the shaft hole of the thinwalled box is finally developed.

A. ANALYTICAL EXPRESSION OF CUTTING FORCE
The analysis of cutting forces expressed in different coordinate systems for a single-insert boring is presented in Fig. 3. The rotating coordinate system (r, t, f ) is fixed to the boring rod and rotates with the machine tool spindle. The coordinate axis symbols r, t and f successively represent the radial, tangential and feed direction of the boring rod, in which the tangential coordinate axis t is parallel to the cutting speed direction V but opposite. The three-dimensional cutting force in boring is simplified as a concentrated force and acted on the primary cutting edge. The cutting force decomposition reference frame (u, v, w) of the inclined cutting force model is introduced to describe the state of cutting force decomposition. The three-dimensional (3D) Cartesian coordinate system (u, v, w) is defined as: fixed on the front cutting VOLUME 11, 2023 surface, the origin falls on the primary cutting edge, the v axis is parallel to the cutting speed V direction, the w axis is perpendicular to the plane formed by the straight line between the v axis and the primary cutting edge, and the u axis is perpendicular to the v axis and the w axis respectively. The coordinate axis direction follows the right-hand rule. Sabberwal [9] pointed out that the cutting force is proportional to the cross-sectional area of the chip load. According to the boring experiment in [23], it is found that the cutting force is related to the cutting speed, and there is a certain critical value. Therefore, to improve the prediction accuracy of boring cutting force, a quadratic polynomial cutting force coefficient correction term based on the change of cutting speed is carried out, and the cutting force calculation formula of a single-insert boring cutter can be expressed as [17], of which, K uc , K vc and K wc represent the cutting force coefficients along u, v and w directions respectively, and their values depend on the boring insert geometry parameters and the material properties of the workpiece and the cutter [24]. A c refers to the load area of the chip in contact with the rake face without deformation. a ik denotes the quadratic polynomial coefficient of cutting force modified based on the cutting speed, and its subscript symbols i = u, v, w and k = 0,1, 2.
The boring insert geometry parameters, such as the cutting edge inclination of the primary cutting edge λ s , the rake angle of the tool γ 0 , the normal rake angle γ n and the cutting edge angle κ r can be expressed as, Since the reference coordinate system (u, v, w ) describing the cutting force and the rotating coordinate system (r, t, f ) of the boring cutter movement is fixed to the cutter, the transformation relationship of the cutting force in the two coordinate systems can be established, namely, Substitute Eq. (7) into Eq. (1), and the expression of the cutting force in the coordinate system can be obtained, According to Eqs. (5) and (8), the cutting force is related to A c , namely the area of the undeformed chip in contact with the rake face. Fig. 4 shows the formation analysis of instantaneous chip load area A c when the single-insert boring is adopted. Boring process is a periodic rotary movement, when the cutting vibration is slight, it can be considered that the instantaneous cutting area of the current cycle is jointly constrained by the actual cutting contour of the current and previous cutting cycles, which is represented in the blue shadow in Fig. 4.
Since the boring tool tip has a geometric circle shape, the fluctuation of cutting depth h θ in the adjacent feeding period will cause the nonlinear change of the load area of the instantaneous chip in the current period. To calculate the dynamic change of cutting force accurately, the effect of the cutting rotation period delay should be considered. The instantaneous chip load area A c can be expressed as, In Fig. 4, xO 1 y is the established plane local coordinate system. The origin of the coordinate system coincides with the center of the tool tip at time t-T, and the X -axis and Y -axis directions are parallel to the axial and radial directions of the boring bar respectively. Suppose the angle between the line from point O 1 as the starting point to any point on the edge envelope curve DABE at time t-T and the positive direction of the X -axis is ϕ 1 , the sign shows a positive value when angleϕ 1 is in the counterclockwise direction. According to the value range of ϕ 1 , the equation g 1 (x 1 , y 1 ) of the envelope curve DABE of the cutting edge can be expressed as, According to Eq. (10), the corresponding azimuth angle of points B, E and S in Fig. 4 can be obtained separately: ϕ B 1 = arccos(f /2r 0 ), ϕ E 1 = π/2+C e and ϕ S 1 = C s . The cutting point of the insert at the circular contour of the shaft hole is point D, and the parameter p D = h θ (t-T)-r 0 (1-sinC s ) is used to judge whether point D is on the primary cutting edge of the boring tool, According to the principle of coordinate translation, the coordinate system xO 1 y is translated by a feed f in the positive direction of X -axis, and then another local plane coordinate system xO 2 y can be established, where the equation g 2 (x 2 , y 2 ) of the envelope curve CB of the cutting edge can be written as, The azimuth angle of points B and C in the coordinate system xO 2 y can then be deduced as, Using the curve equation in Eq. (12), the arc length of BC can be obtained as, Similarly, the coordinate system xO 2 y is translated by h = h θ (t-T)-h θ (t) along the y-axis direction to construct the local coordinate system xO 3 y, then the equation g 3 (x 3 , y 3 ) of the boring tool edge envelope curve C' MAB' in the current cutting cycle can be given as, According to Eq. (15), the corresponding azimuth angles of points B and C in coordinate system xO 3 y can be obtained: The azimuth of M is judged VOLUME 11, 2023 according to p M = h θ (t)-r 0 (1-sinC s ), that is, MO 3 x can be expressed as, Through simultaneous Eqs. (10) and (15), the intersection point A of the envelope curve of the cutting edge at time t-T of the last cycle and the time t of the current cycle can be solved. The azimuth angle [ϕ A 1 , ϕ A 3 ] T corresponding to coordinate system xO 1 y and xO 3 y can be calculated according to the positive and negative signs of the radial cutting depth fluctuation h in the current boring machining cycle: x When h≥0, that is, the cutting depth h θ (t) in the current cutting period is less than h θ (t-T) in the last cutting period. The parameter p A = h-r 0 (1-sinC s ) is constructed to determine whether point A is on the primary cutting edge of the tool tip circle. Then, the piecewise functions applied in Eqs. (10) and (15) can be selected by the positive and negative signs of y When h<0, the discriminant parameters are set as follows, Here, P As denotes the discriminant parameter to judge whether the cutting point of the insert is on the tool nose when point A is on the cutting edge of the boring tool pair, and its value is negative.
From the above derivation, when the fixed boring cutter type and cutting feed f are given, A c can be expressed as a function of the time variable cutting depth h θ (t) and h θ (t-T) of the adjacent cutting periods, namely,

B. MODELLING OF RADIAL STIFFNESS OF SHAFT HOLE ON THE THIN-WALLED BOX
From Fig. 2 we can know that the directional stiffness of the shaft hole changes with the variation of the machining position, which will affect the dynamic characteristics of the boring system. To more accurately express the directional stiffness of the shaft hole, Eq. (4) can be written as, where, k θ e hx and k θ e hy denote the equivalent stiffness of the shaft hole along the X and Y axes at position angle θ, respectively. F θ hx and k θ hy refer to the cutting forces acting on the cutter site at position angle θ. The direction offsets x C and y C at the point C shown in Fig. 2 can be expressed as, Here, k C hx and k C hy represent the directional stiffness at point C, respectively.
Considering that the box wall structure is usually characterized by asymmetry, uneven wall thickness and irregular distribution, a parameter φ C Y 0 that is only related to the structural form of the box wall is proposed and added into the shaft hole radial stiffness model and makes it generally representative.
Here, φ C Y 0 represents the unit angle of line segment CO H caused by the unit force acting on the Y direction of point C in Fig. 2.
Combining with Eq. (2), the bending deformation angle φ C =M C θ /I C can be rewritten as, Usually, φ C is a small rotational angle, thus sin(φ C ) ≈ φ C and cos(φ C ) ≈1. Eq. (27) can then be simplified as, Since (φ C ) 2 is a higher order minimum, we can consider x θφ ≈0. Substituting Eqs. (26), (28) and (29) into (25), the calculation formula of the radial equivalent stiffness of the shaft hole on the thin-walled box can be finally obtained as, C. PROPOSED DYNAMIC MODEL OF BORING SYSTEM Fig. 5 displays the proposed spring-damping system model for boring the shaft hole. In this model, the boring cutter and the shaft hole are assumed to be the concentrated mass objects with two degrees of freedom respectively. Under the excitation of the cutting force fluctuation, the boring cutter and the shaft hole will produce dynamic displacement responses along the X and Y axis respectively, which will cause periodic fluctuation of radial instantaneous cutting depth h θ (t). According to Newton's law, the differential equation of cutting vibration of the shaft hole and boring cutter can be expressed as follows, of which, In Eqs (31) and (32), m h and m c mean equivalent mass of the shaft hole and boring cutter, respectively. c hx , c hy , c cx and c cy represent the equivalent damping of the shaft hole and boring cutter along the X and Y directions separately. k θe hx , k θ e hy , k cx and k cy denote the equivalent directional stiffness of the shaft hole and boring cutter along the X and Y directions respectively. F hx , F hy , F cx and F cy respectively represent the cutting forces acting on the shaft hole and boring cutter along X and Y directions.
In Fig.1 (b), the displacement vector e c and e h of the center of the boring cutter and shaft hole can be calculated as, The azimuth angles 1 φ cθ and 1 φ hθ of e c and e h projected on the XOY plane in coordinate system {W} are expressed as, VOLUME 11, 2023 The relation function between the radial cutting depth h(θ) and e c , e h has been given in Eq. (9). To facilitate the unified description of variables in the time domain, it can be rewritten as in (37), shown at the bottom of the previous page, of which, Substituting Eq. (37) into Eq. (24), the instantaneous cutting force can be obtained by parallel establishing Eqs. (5) and (8). In this study, the dynamic parameters of the boring bar are assumed to be the same, namely, k cx = k cy = k cr = k c and c cx = c cy = c cr = c c . Finally, the dynamic model of the boring system in the form of the matrix can be described as,

IV. VERIFICATION FOR THE PROPOSED DYNAMIC MODEL
To verify the correctness of the proposed boring dynamic model, the FE model of the directional stiffness of the shaft hole on the thin-walled box is firstly conducted. Then the experiment set for validating the dynamic characteristics of the boring system is described.

A. FE VERIFICATION MODEL FOR THE DIRECTIONAL STIFFNESS OF THE SHAFT HOLE
The FE is a universal method to identify the dynamic parameters of complex components [25], [26], [27]. In this section, the radial deformation of the shaft hole under static action is obtained through FE statics analysis, and then the radial stiffness of the shaft hole in different directions can be calculated.
The proposed stiffness calculation model, namely Eq. (30), can be verified by comparing with the results obtained from the FE model. The FE calculation process of the equivalent directional stiffness of the shaft hole on the thin-walled box is displayed in Fig. 6. The specific steps are as follows, 1) Establish the 3D model of the thin-walled box, and define the parameters such as material properties and element type, then generate the FE model of the box. In this simulation, the main parameters are set as: the box material is aluminium alloy 6061-T6, its density ρ =2770 kg/m 3 , elastic modulus E = 70 GPa and Poisson's ratio µ =0.33.

B. EXPERIMENT SETTING FOR VALIDATING THE PROPOSED BORING DYNAMIC MODEL
The experiment system for verifying the proposed boring dynamic model is shown in Fig. 7, which mainly includes the boring processing system and signal acquisition system. For the boring processing system, the boring tool used in the experiment is WNC (Winock) four-axis horizontal boring and milling machine. The repeated positioning accuracy of the machine tool is ±0.005 mm, and the effective working space range is 1200 mm×1500 mm×1000 mm. The parameters of the boring tool are presented in Table 1.
For the data acquisition system, the cutting force was acquired using the dynamometer, its type is 9257B, and its measuring range and sensitivity is ±5 KN and 7.5 pC/N, respectively. The spatial coordinates of the characteristic points on the boring machining surface are measured by using FARO manipulator three-coordinate measuring instrument. The sampling frequency of cutting force is 25000 HZ.

V. PROPOSED SURFACE TOPOGRAPHY ERROR PREDICTION METHOD
As described in Section II, the dynamic behavior of the boring system has an important influence on the surface topography of the machined shaft hole. In this section, a cutting scanning surface trajectory model of the single-insert boring cutter is proposed to realize the visualization of the machined surface topography shaft hole. On this basis, a surface topography error prediction method for the boring shaft hole is proposed.

A. MODELLING FOR CUTTING SCANNING SURFACE TRAJECTORY OF THE SINGLE-INSERT BORING CUTTER
To establish the analytical expression of the cutting scanning surface trajectory of the single-insert boring cutter, the layout of each local reference coordinate system of the boring system is established as shown in Fig. 8. Five reference coordinate systems are set in the figure to describe the relative motion relationship between the boring cutter and the shaft hole. In Fig. 8 VOLUME 11, 2023 For the nominal feature point p i on the hole-forming surface, its two-parameter motion coordinate system can be described as, To establish the relation between coordinate system {W } and {F pi }, an orthogonal coordinate network is established on the hole-forming surface. H(u,v) is carried out to represent the radial diameter on the surface, and the orthogonal unit vectors η i , ξ i and ζ i are selected as the coordinate axis directions of {F pi }, namely, where sα i = sin α i , cα i = cos α i , and other parameters are analogous. From the given rotation matrix (43), find the equivalent rotation angles α i , β i and γ i around x-y-z, let, Introduce the bivariate inverse tangent function Atan2(y,x). According to the value of β i , the values of α i , β i and γ i can be described as following three cases, 1) When 2) When β i = π/2 The homogeneous transformation matrix of {W } relative to {F pi } is, Taking into account the deformation of the box wall in the boring process, it can be, Here, f= (f x , f y , f z ) T is the feed velocity vector of the cutter center point, and O w s = (x w os , y w os , z w os ) T refers to the initial position vector of the origin O S of {S} in {W }. 0 e h = (x h0 , y h0 , z h0 ) T denotes the coordinate of the deviation vector of the shaft hole position in the initial clamping state. e h = (x h , y h , z h ) T represents the instantaneous eccentric vector coordinate of the origin of {F pi } deviating from its nominal position, which is related to the vibration of the box wall.
To obtain the relationship between the cutting point of the boring cutter and the nominal feature point to be processed on the hole-forming surface, the position relationship of the cutting point in the coordinate system {C} of the boring insert, that is, the position relationship of any cutting point E i in {C} is established and shown in Fig. 9. In Fig. 9, O r is the center of the tool nose circle of the boring insert and its radius is r 0 . κ r denotes the cutting edge angle of the boring insert, κ r means the trail edge angle, R represents the nominal radius of the finished hole and h 0 refers to the nominal depth of cut.
The position coordinates of point E i in coordinate system {C} are described as, where ϕ i represents the angle between unit vector -Z c and vector O r E r . Combining with Eq. (10), the length value |O r E r | can be expressed as, The homogeneous coordinate transformation matrix of {C} relative to {W } is, Usually, the axis of the lathe spindle is parallel to the directrix of the shaft hole cylinder when fine boring. Therefore, the pose unit vector of the directrix of the shaft hole cylinder with respect to {W} is assumed to be (0, 0, 1) T . Then Eq. (40) can be further described as, Taking the external normal direction of the shaft hole surface as positive, and substituting Eq. (55) into Eq. (42), then combining with Eq. (44), we can get, In the actual boring process, the edge point E i and the hole surface characteristic point p i should have the same position azimuth angle in the corresponding time domain, and the hole radius value R is much larger than e h and e c , then we can set u = π/2+θ. Considering that the axial stiffness of the cutter rod and shaft hole is relatively large, all small quantities in the Z direction can be omitted, then Eq. (54) can be further simplified as, To obtain the position vector (x

B. PROPOSED SURFACE TOPOGRAPHY PREDICTION METHOD FOR BORING SHAFT HOLE
Due to the vibration of the cutter and shaft hole in the boring operation, the end cutting edge near the tool nose may be cut to the machined contour of the last cycle and then form a new shape contour. Therefore, it is necessary to take the minimum cutting residual height scanned by the insert as the contour of the final forming surface. Based on Eq. (58), the problem of solving the local residual height of the boring cutting-formed surface can be transformed into a nonlinear programming optimization problem, namely,

A. VERIFICATION RESULT ANALYSIS OF THE PROPOSED HOLE DIRECTIONAL STIFFNESS MODEL
The comparison between the analytical stiffness model and the FE model is displayed in Fig. 10. As can be seen that the curve variation of the predicted stiffness value obtained from the proposed model converges with the FE simulation value. It can be calculated that the prediction deviation rates of k θe hx and k θ e hy within the period (0, 2π) are within 7% and 11%, respectively. The comparison results verify the correctness of  Table 2 shows the equivalent dynamic parameters of the boring system used in the simulation and test. Based on the data in Table 2, the proposed boring dynamic model is solved to obtain the dynamic cutting force and displacement response. Then the results are compared with the experimental results to verify the effectiveness of the proposed dynamic model. Stable cutting condition is the desired boring state. However, when the selected combination of cutting feed and cutting speed parameters is distributed in the unstable cutting region of the Lobe diagram of the boring shaft hole, the chatter will occur and further affect the machining quality of the hole surface. Therefore, to verify the proposed dynamic model universality, both cutting states are considered in this comparison study, which are shown in Case #1 (chatter) and Case #2 (stable) in Table 3. The comparison results of cutting forces in the stable and chatter cutting cases obtained by the proposed model and the test are shown in Fig. 11. It can be seen that the changing trend of the cutting force calculated by the proposed dynamic model fits well with that of the measured value in both the stable and chatter cutting cases, which indicates the effectiveness and universality of the proposed boring dynamic model.
To validate the influence of the radial stiffness of the shaft hole on the boring cutting characteristics, the cutting force obtained from the traditional model in [18] without considering the stiffness, the dynamic model proposed in this paper and experiment are compared and presented in Fig. 12. The prediction results of the traditional model show that the cutting force will appear chatter phenomenon, while the proposed model and test results present that the cutting force are both in a stable cutting state. Due to that considering the flexible support of the shaft hole will produce a vibration absorption effect when boring, thus the results obtained from the proposed model are more consistent with the actual situation.

C. ANALYSIS OF MACHINING TOPOGRAPHY ERROR PREDICTION RESULTS
On the basis of verifying the proposed dynamic model, the surface topography of the shaft hole to be machined is predicted based on the proposed topography prediction method, and then the predicted results are compared with the measured results to verify the correctness of the proposed models and method.
The comparison between the measured results and the predicted results based on the proposed method is displayed in Fig. 13. In the surface measurement experiment, the VHX-7000 ultra depth of field 3D micro-graph (see Fig. 13(b)) is used to measure the surface topography of the shaft hole, which can observe and obtain the point cloud data of the measured surface, and then the topography reconstruction of the measured surface can be realized by using 3D software.
It can be seen in Fig. 13 (a), the 3D topography and contour of the hole surface predicted based on the proposed model have high similarity with the measured results. From the comparison of contour residual height results in the stable cutting case (pictured in Fig. 13 (d1)), it can be seen that the residual peak-valley difference values of the hole profile at the same azimuth angle θ =45 are roughly similar. The predicted maximum peak-valley difference is 5.088 µm while the measured one is 4.232 µm, and it can be calculated that the measured peak-valley value is 16.82% smaller than the predicted one. Fig. 13 (c) and (d2) show the comparison results between the measured and predicted topography and the residual peak heights. It can be observed that the predicted and measured hole surface topography both agree well and the topography amplitude of the peak-valley difference of the residual profile under this condition both increase exponentially compared with that under the stable cutting condition. The predicted maximum peak-valley difference is 16.351 µm and the measured one is 13.501 µm, their difference reaches 17.27%.
On the whole, the measured and predicted residual heights of the hole profile fluctuate around the mean residual heights under both the stable and chatter cutting conditions, which indicates the correctness of the proposed surface topography error prediction method for the boring shaft hole.

VII. CONCLUSION
The traditional boring dynamic model ignores the influence of the radial deformation of the shaft hole of the thin-walled box on the boring dynamic characteristics, which will lead to a deviation in the prediction of the machining topography error of the shaft hole. To solve this problem, a new radial stiffness calculation model of the shaft hole on the thin-walled box is established and verified by the FE method. Moreover, a dynamic model of the boring system is developed, and the experimental verification of the presented dynamic model is also carried out. Combining with the proposed dynamic model, a prediction method of machining topography error of the shaft hole is proposed, and the prediction and measurement analysis of the shaft hole topography error are finally carried out. Through this research, some main conclusions can be drawn as follows: • The proposed radial stiffness calculation model of the shaft hole on the thin-walled box can achieve high calculation accuracy, the FE verification shows that the calculation errors of the hole radial stiffness in the X and Y directions of the proposed model are within 7% and 11%, respectively.
• The experimental results of boring dynamic characteristics show that the proposed boring dynamic model can accurately predict the real cutting force fluctuation state of the boring system, and the influence of the radial stiffness of the shaft hole on the boring dynamic behavior can not be ignored.
• The proposed surface topography error prediction method shows good performance by comparing with the measured surface topography error. His research interests include signal processing, noise analysis and pattern recognition, equipment fault diagnosis, intelligent monitoring and residual life prediction technology, precision transmission, and system integration technology. VOLUME 11, 2023