Corner milling force prediction and improvement method of aviation thin-walled structural parts

Corner is one of the typical characteristics of aerospace monolithic components. In order to achieve a stable processing environment and reduce the mutation of milling force in the process of machining, the manufacturer has been searching a method to control the milling force of corner machining. Based on the cutting contact relationship and the instantaneous chip thickness, the milling force prediction model of corner machining is established. By analyzing different types of corner contact relationship, the influence of cutting contact angle and cutting arc length on milling force is established. The corner feed rate iterative calculation method is used to improve the corner milling force mutation. The corner milling experiment proves that the proposed method can not only effectively control the corner milling force, but also improve the surface quality of the bottom surface of the workpiece. The proposed method is suitable for automatic programming of multi-corner pocket machining and provides theoretical support for stable machining of frame part.


Introduction
The progress in the aerospace field is accompanied by the development of manufacturing industry. In order to improve the overall performance of the aircraft and control its overall quality, the thin-walled design of the overall structure has become an obvious feature. The aerospace monolithic components have the advantages of light weight, high specific strength, and compact space structure, which have wide applications in the new generation of aircraft. The cavities of such structures are complex in shape and have pockets and thin-wall features. In the process of milling cavity, the corner position is prone to lack of cutting and chatter, so manual grinding is needed to eliminate defects. This phenomenon not only reduces the tool life, but also affects the geometric size and machining efficiency of the workpiece. Therefore, the research on corner milling is necessary to improve the machining efficiency of structural parts.
Corner processing is one of the important tasks of pocket processing. According to the structural characteristics of the workpiece, the corner types are also diversified, with the demand of modern aircraft for the high geometric accuracy and high surface quality of the overall structural components. To obtain stable processing state and high-quality products is the hope of the manufacturer. But in the pocket processing, corner processing breaks the state of stable cutting. Due to the constraint of pocket geometry, the stable line path will change, and the tool-part contact relationship will change, resulting in a mutation in the milling force. This sudden change of force is caused by the change of radial milling width when the end mill enters and moves out of the pocket corner. The abrupt change of cutting parameters will lead to the deflection of the tool, poor surface quality, poor geometric accuracy, and aggravate tool wear in the machining process.
As an important factor to describe the machining process, the correct prediction of milling force is the basic problem for analyzing the machining process. Therefore, scholars in various countries have conducted a lot of research on the milling force prediction model. Martellotti [1] first proposed the cycloid trajectory of planar milling and the instantaneous chip thickness formula. Subsequently, Altıntaş and Lee [2] proposed the cutting slip line method, and based on the assumed stress acting on the material, the establishment of the orthogonal slip line was completed from the maximum direction of the shear. Tlusty and MacNeil [3] and Budak and Altintas [4] expressed the unit cutting force coefficient as an exponential function of instantaneous undeformed chip thickness. Wan and Zhang [5] introduced the cutter/ workpiece deflections and cutting contact angle into the calculation of milling force when calculating instantaneous chip thickness. Kang et al. [6] extended the average force model of Kline and established the average cutting force model under different cutting parameters. The cutting force coefficient was expressed as a polynomial form of spindle speed, cutting width, and cutting depth per tooth feed. Song et al. [7] considered the influence of tool angle and cutting speed on cutting force and introduced the polynomial product of tool angle and cutting speed into the cutting force coefficient. Chen et al. [8] introduced the deformation in the process of thin-walled milling into the instantaneous chip thickness and established the coupling model between force and deformation.
In the pocket machining process, the prediction of corner milling force and the stable machining state are the issues that many scholars and manufacturers have always paid attention to. Based on the cutting force models of different machining conditions [9][10][11], the cutting force prediction method for corner machining is proposed. Zhang and Zheng [12] discreted the corner milling into steady-state milling process and introduced the radial milling depth at the instantaneous position of the tool into each stage to predict the corner milling force. Li and Liu [13] established the milling force prediction model of inner and outer circular corners through the time-varying intersection point of tool and workpiece when tool path moves. Based on the infinitesimal cutting force model, Ding et al. [14] regarded the discrete element of the ball-end cutter as the equivalent end mill, calculated the cutting contact area, and established the milling force model for the angular machining of the ball-end cutter. Li et al. [15] established a general model of corner milling force by analyzing the variation of milling width.
A series of research results have been obtained in the process of improving the corner processing of pockets. At present, the optimization methods mainly focus on tool path planning and machining process optimization. The former research mainly focuses on the tool path planning and optimization path of the corner structure. Choy and Chan [16] appended the bow cutter path to the corner position of the basic cutter path, controlled the cutting contact length by adding the number of cutter paths, and realized the optimized angle machining. The spiral tool path generation method proposed by Held and Spielberger [17] can realize the efficient processing of single connected two-dimensional pockets formed by straight lines and arcs. Zhao et al. [18] inserted a double circular arc transition section in the parallel tool path to remove the corner residual material, reduce the processing time, and improve the processing efficiency. Ferreira and Ochoa [19] proposed the method of generating two-dimensional cycloidal tool path by using the central axis transformation. In the latter study, the corner milling parameters were optimized based on dynamic modeling and simulation. Dotcheva and Millward [20] proposed the mapping relationship between tool deviation and corner milling force, established the calculation method of maximum chip thickness, and realized the high-precision corner machining process. Wei et al. [21] proposed a prediction method of milling force considering the change of feed direction and tool contact area, which improved the prediction accuracy of milling force in corner machining. Liu et al. [22] established the feed optimization method based on load control, corrected the milling force model by calculating the instantaneous chip thickness of arc trajectory, and controlled the maximum cutting load by modifying the feed rate. Suzuki et al. [23] considered the nonlinear fluctuation law of undeformed chip thickness and the process damping caused by back tool contact and improved the machined surface roughness by optimizing the feed rate. Qiong et al. [24] optimized the milling parameters by changing the cutter center coordinates and analyzing the geometric contact relationship of cutting with constant cutting force as the goal. Godwin [25] took the minimum pocket processing time as the goal and considered the axial and radial cutting depths to achieve efficient pocket processing.
In the production process, it is efficient and easy to achieve by optimizing the process parameters. Based on the relationship model between the processing results and the input process parameters, also with wide applications, the ideal combination of processing parameters was obtained by using the optimization algorithm. At present, genetic algorithm [26] and particle swarm algorithm [27] are widely used in machining process. Li et al. [28] combined the Taguchi method and particle swarm optimization algorithm to optimize the process parameters with processing time and energy as the research objectives. A trade-off point is obtained between energy and low processing time through experiments. Kumar [29] obtained the combination of machining parameters with the best surface quality and the highest machining efficiency by using genetic algorithm. Meng et al. [30] established a mean particle swarm optimization algorithm to obtain the results of angular processing using low spindle speed and selected the optimal processing parameters from the Pareto set. In the actual machining process, multi-pass processing completes the final pocket geometry, and because of the diversity of the workpiece geometry, the complex optimization algorithm is not universal, and it is not suitable for batch production and automatic programming.
In this paper, aiming at the corner structure characteristics of pockets in aviation structural parts, a corner milling force prediction method and corner machining improvement method based on common information such as corner structure are proposed. The method is mainly used in semi-finishing and finishing processes of pocket. By improving the corner feed speed, the mutation of milling force is reduced and the surface quality of the workpiece is improved. Firstly, through the different contact relationships of tool-part, the accurate model of chip thickness in the process of linear and curved cutting is established. Then, the influence of the cutting contact relationship of different types of corner machining on the change of milling force is analyzed, and the milling force of corner machining is predicted through experiments. Finally, taking the constant milling force as the goal, an iterative calculation method of angular machining feed speed is proposed, and the improved feed speed is obtained by mean square error calculation. The experimental results verify the effectiveness of the improvement method.

Model definition
Milling force is an important factor to measure the cutting quality, and the change of milling force will lead to defects in the processed parts. Therefore, cutting force is the most basic research object in cutting phenomenon. Cutting parameters, tool geometry, physical properties of tool and workpiece materials, friction effect, and plastic strain all affect the stability of cutting force. In the machining process of structural parts, the cutting force of the corner is complex, and the milling force of the corner should be calculated based on the suitable cutting force model.

Modeling of cutting forces
The cutting edge of the end mill is the geometric structure of the spatial spiral, so the contact form between the tool and the workpiece is unevenly distributed. The cutting edge involved in cutting is discreted into equal height (j) layer along the Z-axis direction by the tool, and the cutting force is calculated in the form of oblique cutting with each disk [11]. Each disk describes the differential tangential force (dF t ), radial force (dF r ), and axial force (dF a ) in the oblique cutting process [2].
where ϕ is the angular position; K tc,i,j , K rc,i,j , and K ac,i,j are coefficients of shear force (K tc,i,j = 796.1, K rc,i,j = 168.8, K ac,i,j = 222); K te,i,j , K re,i,j , and K ae,i,j are coefficients of (1) plowing forces (K te,i,j = 27.7, K re,i,j = 30.8, K ae,i,j = 1.4); and h i, j (ϕ) is the nominal undeformed chip thickness. Due to the existence of tool helix angle (β), the cutting contact relationship between tool and workpiece changes with tool rotation. Radial force and tangential force also change with tool rotation, as shown in Fig. 1.
The tangential force (dF t ), radial force (dF r ), and axial force (dF a ) in Eq. (1) are transferred to X, Y, and Z directions by Cartesian coordinates.
Therefore, when the tool rotates to the angle (ϕ), the total cutting force in X, Y, and Z directions is expressed as follows:

Tool-part contact model
In the milling process of frame parts, the tool path consists of straight line and curve, as shown in Fig. 2. In the process of linear path machining, the contact relationship between the tool and the workpiece does not change. When the cutting parameters are the same, the stable contact relationship has little effect on the change of cutting force. In the curve path, the contact between tool and workpiece is divided into convex and concave contact. In curve path machining, the contact relationship between the tool and the workpiece is different from that of the linear path. Under the same cutting parameters (cutting width (a e ), cutting depth (a p ), and cutting feed rate (v f )), the change of the tool path will have a significant influence on the cutting force [24].
Three different tool paths have different effects on cutting force. Through the cutting force model in Section 2.1, it can be found that the contact relationship between different tools and workpiece changes the instantaneous chip thickness. Therefore, establishing accurate instantaneous chip thickness model according to different contact relationships is the basis for calculating cutting force. In the side milling process, the simplified equation is used to calculate the instantaneous chip thickness, which is to improve the calculation efficiency. The change of cutting force is obvious in the moving path of the curve, so it is necessary to accurately calculate the chip thickness. The chip thickness in straight and curve paths is composed of part 1 and part 2 (the process of cutting edge entering and removing the workpiece), as shown in Fig. 3.
In the calculation of chip thickness, P and Q points are established to judge the current disk is located in part 1 or part 2. P is the intersection of the ray from the tool center to the tip of the cutting disk and the unprocessed surface. When the disk cuts into the workpiece to the maximum cutting thickness A point, the P point can be expressed as Q is the intersection point between the rays from the tip of the cutting disk and the machined surface. The disk moves from point A to point M, and the Q point can be expressed as In the cutting process of linear path, the chip thickness of part 1 and part 2 is expressed by h L1 and h L2 , which can be expressed as where r is the tool radius, f is the feed per tooth of tool movement, ϕ en is the cutting disk entry angle, ϕ ex is the cutting disk exit angle, and ϕ max is the tool rotation angle at the maximum chip thickness.
In the cutting process of curve path, the cutting area of curve path is different from that of straight path, so the chip thickness of part 1 and part 2 is expressed by h C1 and h C2 , and the expression is In the process of curve path cutting, there are two paths of convex curve and concave curve. When calculating h C1 , the position of surface center and tool center at cutting time should be analyzed. In the convex curve path, the cutter center and the surface center are located on both sides of the cutting zone. In the concave curve path, the tool center and the surface center are on one side of the cutting zone.
The contact relationship of cutting is also different under three different paths. Therefore, the values of cutting contact angle (θ) and cutting arc length (L) in the cutting process are also different. The cutting edge contact parameters in twodimensional expansion are defined, as shown in Fig. 4. The expressions of contact angle (θ L ) and cutting arc (L L ) for milling in straight path are Cutting contact angle (θ cave ) and cutting arc length (L cave ) in convex curve path can be expressed as Cutting contact angle (θ cave ) and cutting arc length (L cave ) in concave curve path can be expressed as

Corner milling
Milling force is one of the decisive factors that reflect the excellent milling process. The defect of workpiece processing reflects the instability of milling force. Therefore, milling force is regarded as the basis of studying milling process. Due to the geometric structure characteristics of the corner, the change of cutting force at this position is complex. It is necessary to accurately model the structural characteristics of the corner and the tool-workpiece contact relationship.

Type of corner
In the corner contour machining of frame parts, the inner corner machining is regarded as the cylindrical surface left by the previous tool path. The curvature of the cylindrical surface is a constant. The structure of the corner position is arced by two side walls to be machined and the inner contour to be machined [31]. Set the angle γ between the machined surfaces of the two-side wall theory. According to the use and design requirements of frame structure components, the internal corner model can be divided into sharp angle, blunt angle, and right angle according to the value of γ, as shown in Table 1.
In the corner milling process, the tool-part contact relationship is composed of two contact modes: the radius of the corner is equal to the tool radius (type 1), and the radius of the corner is greater than the tool radius (type 2). Take the right-angle corner processing as an example, as shown in Fig. 5.

Tool-part contact area model
When analyzing the tool-part contact relationship, the tool is discretized along the tool axis direction, and the discrete cutting unit is taken as the research object of the two-dimensional tool-part contact relationship [24]. In the corner milling process, the whole milling process is divided into five stages according to the moving trajectory of the tool: the linear path milling process, the process of the tool straight trajectory entering the corner arc, the cutting process of the arc position, the process of cutting out the arc, and the linear path milling process, as shown in Fig. 6.
In the above tool path, the trajectory is divided into five processes by A, B, C, and D points: the first process (the linear path milling process, A), the second process (A, B), the third process (B, C), the fourth process (C, D), and the fifth process (D, the linear path milling process). It is worth noting that the first and second process and the fourth and fifth process are opposite.
1. When the tool moves from the distal end to point A and the radial cutting depth remains unchanged, the contact angle and contact arc length of tool-part can be expressed by Eq. (10) and Eq. (11). Similarly, the cal- culation process is the same when the tool moves from D point to another remote point. 2. In the corner milling process, when the tool moves from A to B, the tool path is still straight, but the cutting edge has entered the arc L T1 in the previous machining contour. This cutting process is a radial cutting width change process.
Through the contour equation of arc (L T1 ), the expressions of tool center trajectory and tool rotation equation are obtained: where (x B , y B ) is the trajectory of the tool center point, which is found by numerical control (NC) system program.
The coordinates of the cur rent entr y point (B en (x B, en , y B, en )) are obtained by Eq. (16), and the expressions of the tool contact angle (θ B ) and the arc (L B ) at the current position can be obtained  3. The tool moves from B to C, and its trajectory is curve path. In this milling process, the radial milling width is constant, and it is concave curve machining. Therefore, the contact angle (θ L ) and arc length (L L ) are calculated by Eq. (14) and Eq. (15).

Analysis of corner milling force
In the pocket processing, the intersection of two adjacent thin walls must appear corner. In the corner milling process, the trajectory of tool feed direction changes, which makes the contact relationship between tool and workpiece mutation. The change of cutting contact relationship induces the mutation of cutting force. In Fig. 7, when the mill enters the corner, the contact angle between the tool and the workpiece suddenly increases, that is, the milling contact arc length increases. Figure 7 is the corner milling contact relationship of type 1. The milling arc length (L) of the tool at any contact position at the corner is calculated by the tool trajectory and the workpiece geometric contour. In the inner angle milling process, the correlation between the milling arc length (L) and the radial cutting depth (a e ) can be calculated by Eq. (15). After the tool enters the corner, the radial cutting width (a e ) increases, and the cutting force also increases. The change trend of corner milling force forming cone is shown in Fig. 8.
The milling force increases when the tool enters the corner from A, and reaches the maximum value after B. B to D is the process of tool moving out of the corner. The increasing trend of milling force from A to B presents a parabolic shape. When the tool moves to B, the contact angle between the tool and the workpiece reaches the maximum, the radial cutting depth reaches the maximum, and the cutting force reaches the maximum. In the process from B to C, the tool center coincides with the tip of the trajectory, and the cutting force decreases from the maximum to the minimum. In this process, the contact angle of the tool and the workpiece decreases linearly, and the decrease rate is fast. After reaching D, the tool begins to enter the linear motion trajectory again, and the contact angle between the tool and the workpiece enters a constant state. In corner milling type 2, the trajectory of the tool center point at the corner will be an arc trajectory, as shown in Fig. 9. The calculation method of cutting force is similar to that of type 1.
In Fig. 9, in the process from A to B, the tool trajectory is a straight line. The cutting tool begins to enter the corner, and the cutting contact arc length begins to increase gradually. The cutting force also increases with the increase of radial cutting width. The contact angle (θ B ) and contact arc length (L B ) are calculated by Eq. (17) and Eq. (18). According to Section 3.2, in the process of C to D, the contact relationship between tool and workpiece is opposite to A to B.
The tool-workpiece contact relationship of type 2 is different from that of type 1, so the contour shape of cutting force is also different. The stage of BC is a stable concave curve cutting process, so the cutting force is in a relatively stable stage, as shown in Fig. 10. Figure 11 illustrates the experimental set-up. The spindle speed of VDL-1000E three-axis CNC machine tool used as experimental equipment is 45~8000 rpm. Kistler 5236B rotary dynamometer by force measurement, using a matching 5238B charge amplifier, is used to collect cutting force signals during machining (the sampling ratio is set as 10 kHz).

Set-up
GM-4E-D10.0-75 cemented carbide end mill with four edges is used in the experiment. The diameter is 10 mm, the helix angle is 45°, and the extension length is 45 mm. The workpiece material is 7075 aluminum alloy, and the geometric size of the blank is 100 mm × 100 mm × 50 mm. The shape of the workpiece after processing is shown in Fig. 12.

Data analysis
In the machining process of corner structure, the milling force is obviously distorted due to the change of the contact relationship between the tool and the workpiece. The change of cutting force has significant influence on machining stability, workpiece geometric accuracy, and tool wear. Therefore, it is necessary to accurately predict corner milling force and improve corner machining environment.
Firstly, milling experiments are carried out on six different types of corners of right angle, blunt angle, and sharp angle to verify the accuracy of the prediction model. The machining trajectory is automatically generated according to the workpiece geometry without special processing in the cutting trajectory. Milling experiments are performed using the machining parameters in Table 2.
Through the above processing parameters, the milling force of the straight path and the milling force of the corner are analyzed, respectively. For corner machining, the milling force of different corner types is given in Table 3 in the Y direction of the cutting speed 4000 r/min and feed speed 400 mm/min. It can be observed that the milling force increases significantly when the tool enters the corner position. The increase of milling force is caused by the increase of cutting width (a e ) and material removal rate after the tool enters the corner area. From the results in Table 3, it can be observed that the experimental results of type 1 and type 2 are consistent with the analysis results in Section 3.3. The maximum force comparison between line segment processing and corner processing of six types of corners shows X-force as shown in Fig. 13. In three cases, the line segment cutting process decreases with the decrease of feed rate, the unit removal material volume decreases, and the force decreases. Similarly, the force at the corner decreases with the decrease of feed rate. Among the six types of corners, acute-angle type 1 has the largest force increase of 164.6%.
Right-angle type 2 has the smallest proportion of increase, with a value of 40.1%.

Improved method
In the actual manufacturing process, the operator realizes the stable cutting of the corner by changing the feed speed. Experience shows that reducing the feed rate of angular position can reduce the cutting force. After the corner machining is completed, the original feed rate is restored to improve the milling efficiency. This method is mostly modified by processing experience and experimental method. In this process, an inappropriate small feed rate is easy to cause chatter, tool deflection, and transition cutting in the angular processing, which will increase subsequent repair work and affect product quality.
When any cutting edge (i) of the tool moves to point A, the corner processing begins. When the cutting edge i + 1 enters the corner at the initial feed rate (f), the increase of milling force is due to the increase of cutting width (a e ). If the feed rate of cutting edge i + 1 is reduced, the material  removal can be reduced to reduce the milling force. At present, the contact angle (θ 1 ) and arc length (L vex-1 ) between the tool and the workpiece are recalculated by reducing n × df (n = 1, 2, 3, … ) and the new chip thickness (h) and milling force (F S-corner-1 ) (S = X, Y, Z) are calculated. When F S-corner-1 is equal to F S-line , the output is the cutter location point (x 1 , y 1 ) and f corner (x 1 ,y 1 ). Then, when the cutting edge i + 2 enters the workpiece, it will move with f corner (x 1 ,y 1 ). At this moment, the above process reducing the feed rate is repeated, and the output F S-corner-2 and F S-line are compared here. When the values are equal, the cutter location point (x 2 , y 2 ) and f corner (x 2 ,y 2 ) at this moment are output. The feed rate f corner (x m ,y m ) of all cutter points at the corner is calculated according to the above steps. The calculation process is shown in Fig. 14.
In the machining process, NC machine tools give each cutter location by identifying NC code. The nominal cutter location calculated through the above process will deviate from the cutter location given by NC programming. The nominal feed (f) cannot be accurately assigned to the actual tool position, which makes the milling force deviate from the predicted value. Moreover, the frequent changes in feed speed put forward a test for the response speed and motion inertia control of machine tool motion.
In the actual machining process, the formation of the corner is the result of multi-pass milling. If the calculated Acute -angle machining Type 1 T ype 2 feed rate f corner (x m ,y m ) is assigned to each cutter location of the corner, it will be a time-consuming and error-prone work. Therefore, this paper proposes an approximate principle calculation method of feed rate suitable for actual processing. The feed rate f corner (x m ,y m ) obtained by the above iterative calculation is used to achieve the final result through Eq. (19).
where C(f) is the mean square deviation of feed rate, E(f) is the expected value of feed rate, and f c − m (x m , y m ) is the feed rate of each cutter location point. After obtaining C(f) and E(f), the final feed rate at the corner position is

Experimental verification
Through the calculation method of Section 4.3, the feed speed in the corner machining process is obtained, and the updated feed speed (f c ) is the input to the corner machining trajectory position in the programming process. Taking the parameters of 4000 r/min and feed speed 400 mm/min as examples, Table 4 shows the improved milling force in Y direction.
From the results of Table 4, it can be observed that the maximum corner milling force after improvement increases by 11.7%, 8.89%, 17.52%, 23.56%, 76.92%, and 38.9%, respectively, compared with the maximum milling force in the line stage. After improvement, the corner milling force decreases obviously. In acute-angle machining, the maximum milling force of type 1 improved method is 22.2% lower than that before the improvement. The sudden force change is still present, which is determined by the geometry of acute angle. If according to the feed rate of equal force processing, feed per tooth becomes smaller, there will be a friction phenomenon between the flank face and the workpiece, reducing the processing efficiency. During corner milling, due to the change of tool-workpiece contact relationship, the milling force in X, Y, and Z directions increases. The increase in the milling force at the corner position makes the tool have a slight deflection. Tool deflection induces poor surface quality at the bottom of the workpiece. The surface roughness of the workpiece bottom surface before improvement and after improvement is measured by Fig. 15.
Measured values at different locations are averaged and displayed in Table 5. It can be observed from the measurement results in Table 5 that the surface roughness of the corner before improvement is 1.98-2.63 times that of the line position roughness. It can be observed from the measurement results in Table 5 that the surface roughness of the corner before improvement is 1.98-2.63 times that of the line position roughness, and the surface roughness of the improved corner is 1.01-1.32 times that of the line position roughness. The improved corner roughness decreased by 41.07-56.65%.
The experimental results show that the improved cutting parameters not only affect the corner milling force, but also affect the surface quality of the bottom corner. The results show that reducing the feed rate of corner cutting reasonably not only effectively reduces the mutation of milling force, but also improves the surface quality of workpiece. Finally, this method is applied to the pocket machining, as shown in Fig. 16. The method of improving corner feed speed is applied to the cutting trajectory of semi-finishing and finishing in pocket machining. It can be seen from the results that the surface roughness of corner and line position conforms to the variation law of the above method.

Conclusions
In this paper, the cutting contact relationship of multiple types of corner is analyzed, and the corner milling force prediction model is established. According to the change rule of milling force and cutting contact relationship, the feed rate of corner machining is improved. Finally, the effectiveness of the method is verified by experiments. The research contents of this paper are as follows: 1. Based on the instantaneous milling force model, the cutting contact relationship models of linear, concave, and convex side milling are established. The calculation method of instantaneous chip thickness for different contact relationships is proposed, and the expressions of contact angle (θ) and the length of cutting arc (L) for different contact relationships are obtained. 2. Combined with the pocket machining characteristics of frame parts, the corner types are divided into six categories, and the contact relationship between different corner types in milling process is analyzed. The cutting contact relationship at each stage of corner machining is analyzed. Based on the cutting contact angle (θ) and   force, an iterative calculation and improvement method of corner milling feed rate is proposed. The mutation of corner milling force is reduced by improving the feed rate in the corner machining process. The average milling force of the improved corner decreased by 19.6-23.8%. The tool deflection is reduced, and the surface quality of the bottom surface of the workpiece is improved. After improvement, the bottom roughness of the corner is reduced by 41.07-56.65%. The effectiveness of the method is verified by pocket processing experiments.
The established prediction model of corner milling force is one of the important factors in the analysis of the stable machining process of frame and beam structures. Reasonable machining strategy is an effective means to control machining geometric accuracy, chatter, and tool wear. In addition, the method avoids the input of discrete point feed rate one by one, which is suitable for multi-corner machining environment in pocket, and conforms to the automatic programming strategy and the actual production process, which provides help for the stable machining of frame and beam parts.
Availability of data and material The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability Not applicable
Author contribution Zhitao Chen: methodology, investigation, experiments, writing of original draft, writing including review and editing, and visualization; Caixu Yue: project administration, investigation, conceptualization, and supervision; Xianli Liu: project administration, validation, and supervision; Steven Y. Liang: writing including review and editing, supervision, and validation.

Declarations
Ethics approval and consent to participate The content studied in this article belongs to the field of metal processing and does not involve humans and animals. This article strictly follows the accepted principles of ethical and professional conduct. The authors would like to opt in to In Review.

Consent for publication
The authors agree with the Copyright Transfer Statement.