Research on the Dynamic Flexible Support Machining Method for Propeller Blades

Research on the Dynamic Flexible Support Machining Method for Propeller Blades | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Research on the Dynamic Flexible Support Machining Method for Propeller Blades Songmo Li, Rui Wang, Yuhao Ge, Xiangyu Guo, Mengmeng Liu, Xiaohu Zhu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4016972/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 26 Jul, 2025 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted 5 You are reading this latest preprint version Abstract At present, most of the propeller machining adopts single-sided machining, and its machining vibration and deformation seriously affect the machining accuracy. To reduce the machining vibration, a dynamic flexible support machining method is proposed, i.e., while the tool is machining, the multi-point flexible support device supports the blade and counteracts the milling force to suppress the vibration and deformation. Due to the complex shape of the blade and the special structure of the support device, the blade is divided into different areas, and a support motion trajectory combining symmetric and asymmetric motions is planned, and then a set of post-processing systems is introduced. After obtaining the tool position points, the support points are solved cyclically by the ergodic method. Subsequently, the support points are interpolated, and the vectors are smoothed to obtain smooth and continuous support trajectories. Finally, the machining parameters are calculated, and the machining data applicable to the XYZ-3RPS hybrid machine are integrated. The feasibility of the proposed support trajectory and post-processing algorithm was ultimately demonstrated through practical machining experiments. propeller trajectory planning dynamic flexible support ergodic method post-processing Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 1 Introduction Propellers are the core propulsion components of ships and submarines, which work in high-speed and complex flow field environments. If the surface machining quality and accuracy of the propeller are low, it will reduce its propulsion efficiency, cause noise, and even affect the underwater performance [ 1 ]. Therefore, high-precision, high-efficiency, intelligent processing equipment is particularly important. In recent years, with the development of CNC machine tools, there have been many special propeller processing of heavy-duty horizontal multi-linkage CNC machine tools. However, the issues of vibration, interference, and secondary clamping during the machining process of large propellers have not been effectively solved, significantly impacting the machining precision and quality of the propeller blade surface [ 2 ]. The main reason for vibration is that during the machining of the propeller, it is in a cantilever state, and the tool generates a significant overturning moment on the part of the blade far from the hub. Currently, traditional machining methods often involve single-sided machining, where one side of the propeller blade surface is machined first, followed by flipping it for the second clamping process, which inevitably introduces positioning errors and reduces machining accuracy. To solve the above problems, many scholars have researched double-sided co-machining, which is machined by two tools on both sides of the part at the same time. Double-sided co-machining not only solves the problem of interference and secondary clamping but also improves the machining efficiency. However, double-sided co-machining cannot guarantee that the two tools are always opposing. The misalignment will produce local torque, so there is still the problem of vibration. In recent years, many expert teams have proposed mirror machining technology, which has achieved significant results in suppressing vibration. Mirror machining refers to the mirror-symmetric distribution of the tool and support device on both sides of the workpiece to be machined. While the tool is machining, the support device supports the workpiece. Mirror machining has two characteristics, one is the tool axis and support axis to be co-linear and coincide with the normal vector of the workpiece surface; the second is the workpiece requires equal wall thickness. In response to the problems of secondary clamping and machining vibration, this study intends to draw inspiration from the mentioned mirror machining technology. The propeller is vertically clamped, and the cutting tool and support device are arranged on both sides to achieve dynamic flexible support machining. The aim is to further reduce vibration and improve the precision of propeller machining. With the continuous development and improvement of CNC technology, many types of machining equipment have appeared, and propeller machining methods have become diversified and refined. Youn et al. applied the iterative B-spline surface modeling technique to improve the accuracy and efficiency of the model, generated tool trajectories with a good surface finish and proposed that the method of calculating the check vectors based on the tool dimensions and the tool position points along the tool path can be used to obtain interference-free tool paths [ 3 ]. Liu et al. designed a CAD/CAM system for the propeller machining process, using Creo to parameterize the original design data to build a 3D model of the propeller and generate a process for manufacturing propeller blades, which improved the design and manufacturing efficiency [ 4 ]. Korsmik used topology optimization to redesign the original model of the propeller. By comparing the stress state with the original propeller and evaluating the stress-strain state parameters, the structure of the propeller was optimized while ensuring the reliability characteristics of the solid type, and a control program for the LMD manufacturing method was developed [ 5 ]. Jiang of Huazhong University of Science and Technology obtained the three-dimensional flow field map of the propeller blade surface during the rotating work by CFD fluid analysis and proposed that making the machining path of the tool consistent with the direction of the blade surface fluid traces can reduce the impact of the propeller surface unevenness on the underwater performance [ 6 ]. Zhang et al. proposed a machining method combining dynamic adjustment of tool axis vector and optimization of machining parameters. The feasible machining domains of complex regions are obtained through area division and interference adjustment, then the optimal process parameters are obtained through AdvantEdgeFEM simulation, and finally, the effectiveness of this machining method is verified by five-axis machining experiments on propellers [ 7 ]. The traditional single-tool machining method not only has low machining efficiency but also causes precision loss in secondary clamping. For parts like propeller blades with double-sided features, double-tool double-sided machining can be used. Shamoto et al. used the speed difference method to suppress machining vibration. By controlling the milling cutters on both sides of the workpiece to process at different speeds, the regenerative chatter can be eliminated, the machining stability can be improved, thus realizing high-precision and high-efficiency simultaneous machining of both sides of the workpiece [ 8 , 9 ]. Mori et al. proposed a new method for synchronized milling of flexible thin-walled parts with double-sided single teeth, which can offset the milling force by synchronized two-flute milling cutters, and the final experiments showed that the flatness of the processed thin plate was increased by 2 times, and the machining efficiency was increased by 3 times [ 10 ]. Fu et al. used a double parallel machine tool (PKMS) to achieve synchronous and asynchronous machining and support of thin-walled parts, and the test results showed that the static and dynamic characteristics of the workpieces obtained by double PKMS machining were improved, the dimensional accuracy and surface quality were improved, and the material removal rate was doubled [ 11 ]. Mejbel et al. developed a double-sided milling cutter co-machining technology, two vertical face milling cutters at the same time machining two thin-walled surfaces, cutting forces offset each other, the test results show that the workpiece flatness and straightness are significantly increased [ 12 ]. Song et al. proposed a tool path smoothing algorithm and trajectory optimization algorithm based on geometric constraints, to ensure the synchronous movement of the dual tool while smoothing the machining trajectory of the tool. Finally, the results of simulation and machining experiments show that the method reduces the nonlinear error in the machining process, and the contour error of the machined surface meets the tolerance requirements of the process [ 13 , 14 ]. Ge et al. utilized a hybrid machine tool to complete double-cutter double-sided machining of propeller blades, using single-sided machining in the thin tip area, staggered machining in the edge area, and using the top-to-top machining in the thicker inner area, which effectively reduces the blade vibration. This method cannot be used in the tip area of double-sided symmetric machining, which needs to improve the processing effect through an auxiliary support device, so this program has more room for improvement [ 15 ]. At present, many scholars and organizations have begun to propose a new machining method - mirror machining, as shown in Fig. 1 . The advantage of this machining method is that the support device can offset the cutting force generated by the tool, so it can play a role in suppressing vibration, reducing deformation, the wall thickness of the effect of controllable, and thus improving the quality of machining and surface accuracy [ 16 ]. Regarding mirror image processing, many scholars have carried out research. Liu introduced an airflow-assisted support device and explored the impact of different pressure fields and velocity fields of air jet streams on the force and deformation of thin-walled parts. Finally, experiments demonstrated that this method enhanced machining accuracy by using appropriate parameters [ 17 ]. Bao makes the stiffness distribution of the workpiece in the machining process have the same trend with the change of cutting force to optimize the support position of the multi-point support machining. The research results show that after optimizing the support position, the contour error of the part is significantly reduced [ 18 , 19 ]. Hao established an integrated stiffness model of the workpiece and the support mechanism for the mirror-image support equipment and optimized the distribution of the integrated stiffness on the surface of the workpiece by increasing the redundancy drive. The experimental results proved that the average stiffness of the overall mechanism was doubled [ 20 ]. Muller et al. developed an adsorptive multipoint flexible support device based on a parallel robot, with a support head with vacuum suction cups that can adhere to the surface of a workpiece with changing curvature to achieve stable support [ 21 ]. Mahmud et al. simulated the milling force model of an annular face milling cutter to predict the milling force and derived the relationship between the cutting force and the support force generated by different tool angles [ 22 ]. Moradi proposed an optimization algorithm based on the modal method to design an adjustable damper to suppress the regenerative vibration in the milling process of cantilevered thin plates, and the optimal position of the damper was obtained by analyzing the nonlinear error presented by the cutting force [ 23 ]. Bolsunovsky et al. installed a special damper on a flexible workpiece to eliminate its vibration according to the rotational frequency of the machining spindle. The damper was applied to workpieces of any shape and frequency, and the results of the cutting test showed that the vibration of the workpiece was reduced by 20 times [ 24 ]. Based on the above research status of propeller machining, double-sided collaborative machining, and mirror machining, this paper proposes the following scheme: XYZ-3RPS hybrid mechanism is arranged on both sides of the propeller to be machined at the same time, and the tool and support device are installed respectively at the end of the tool to synchronously follow the tool to move and provide dynamic support force to reduce the machining vibration and deformation. The hybrid machining equipment used in this study has the advantages of both tandem and parallel equipment, which ensures a larger workspace and improves the stiffness-to-weight ratio and load capacity. At the same time, this solution can complete the machining task in one clamping, which can solve the problem of positioning error and accuracy degradation caused by secondary clamping. In addition, the following machining steps will be adopted in this study: rough machining with double-sided double-tool machining, semi-finishing machining with double-sided collaborative machining, and final finishing machining with dynamic flexible support machining, which can improve the machining efficiency of the propeller and ensure the machining accuracy. The so-called dynamic flexible support is that the support device follows the movement of the tool to realize the follower support and the support force provided by the support device can change with the change of the size of the milling force to realize the flexible support. The main research content of the article is as follows. Firstly, introducing the overall structure of the machining test prototype, i.e. hybrid machine tool, and designing a multi-point follower support device. Secondly, dividing the propeller surface into reasonable areas and planning the trajectory of the support device. Completing a series of post-processing from the conversion of the original tool position points to the machining parameters, including the processing of the tool position points, the acquisition of the support points, the interpolated smooth processing, and the calculation of machining parameters, etc. Finally, online simulation verification and actual machining test verification are carried out. 2. Structural analysis of hybrid machine tool and support device The experimental prototype model of the XYZ-3RPS hybrid machine tool used in this study is shown in Fig. 3 . The establishment method is to install a 3RPS parallel mechanism on the X -axis sliding rail of the XYZ series mechanism, with the same symmetrical structure on both sides. One side of the parallel mechanism is equipped with a cutting tool at the end, and the other side is equipped with a support device to perform machining and support work respectively. For the serial part, the direction perpendicular to the ground is defined as the Z-axis positive direction, and the orientation of the initial position of the tool or support device is defined as the X -axis positive direction. The Y -axis direction is determined through the right-hand rule of the spatial coordinate system. Each axis is driven by a screw module. The parallel mechanism is installed on the X -axis sliding platform of the series mechanism, which includes three driving rods, a rotating platform, and a translational platform. The two ends of the driving rods are connected to the rotating platform and the translational platform using ball joint pairs and rotating pairs, respectively. The length of the rod is also changed through a screw module. The cutting tool and support device are installed at the ends of the rotating platforms on both sides. The XYZ series mechanism is responsible for the overall large-scale movement of the machine tool, while the 3RPS realizes the attitude change of the end effector. The function of the support device is that it can support the workpiece while processing to reduce vibration and deformation. To achieve stable support for propeller blades, it is necessary to design a lightweight and flexible support device that can adaptively adjust the support force according to the shape of the components. As shown in Fig. 4 , the multi-point flexible support device mainly consists of 7 universal balls, 7 linear bearings, 7 springs, a copper bushing, and a thin cylinder [ 25 ]. Universal balls can withstand large forces in the axial direction and reduce friction and scratches on the surface of the parts caused by the support head moving with the tool. The center is supported by a large-sized universal ball as the main support, while the remaining 6 are evenly arranged on the circumference centered on the main support. During the machining process, multiple points are supported together, which increases the local stiffness of the components near the milling point while applying center point support. This device uses a single-action extrusion-type thin cylinder as the supporting element and relies on an internal spring to retract when not supplying air. 7 springs are installed inside the support body. During the machining process, the piston rod of the cylinder extends, and the universal ball contacts the part. Due to the presence of the spring, the supporting device can adaptively adjust the length of the push rod extension according to the surface curvature of the part, achieving a flexible support effect. In addition, due to the reciprocating motion between the push rod and the support body, as well as between the support body and the housing, this device uses copper bushing and 7 linear bearings to reduce friction and lubrication. The advantage of this support device is that even when processing curved thin-walled parts with complex curvatures, it can ensure that each universal ball always contacts the surface of the thin-walled component, playing a good supporting effect. In the actual machining process of propeller blades, semi-finishing machining adopts double-sided dual-tool collaborative machining. Therefore, before finishing machining, this support device needs to replace the cutting tool on one side of the hybrid machine tool to process one side of the propeller blade. After the machining is completed, the positions of the two are exchanged to complete the machining of the other side of the blade. This dynamic support machining method does not require flipping and reinstalling the blades, thus avoiding positioning errors caused by secondary clamping. 3. Overall planning of dynamic support machining When machining propeller blades, a torsional moment is generated between the tool contact point and the support point, and an overturning moment is generated between the tool contact point, support point, and blade, which together constitute a deformation moment. The deformation torque is an important factor that causes blade deformation and leads to a decrease in machining accuracy. To reduce the deformation torque, this section will introduce how to set a reasonable attitude relationship between the tool and the support device, plan the tool path, and support motion trajectory. 3.1 Original tool path generation and area division During the processing of propeller blades, defects (i.e. knife marks) are inevitably generated on the surface. If the defect direction is parallel to the fluid trace during blade operation, vibration and noise can be minimized to the greatest extent. Therefore, the tool path should overlap or approach the streamline on the blade surface to the maximum extent possible [ 6 ]. Generally, when the blade is rotating during operation, the direction of the streamline on its surface resembles the curve direction formed by the intersection of axisymmetric cylindrical surfaces at different radii with the blade. Therefore, the tool path used in this study is a zig-zag reciprocating cutting method, and the generation method is as follows: the radius of the trajectory circle swept by the blade tip of the propeller during operation is called the radius of the propeller, and a series of coaxial cylindrical surfaces are made around the axis of the propeller with a set radius \(\Delta R\) increment until the radius of the cylindrical surface reaches the propeller radius R . Intersecting the cylindrical surface with the surface of the blade to obtain a set of contour curves. Dividing the set of curves into two parts using the guide and trailing edges of the propeller blades, which serve as the machining tool paths on both sides. The tool path that matches the streamline direction in all tool paths is called the surface machining tool path, and the one that connects adjacent surface machining tool paths is called the transformation machining tool path, as shown in Fig. 5 . Based on the thickness and cutting depth of the rough workpiece after rough machining, a series of biases can be applied to the final finishing machining tool path to obtain multiple finishing machining tool paths. It should be noted that the last two finishing machining use dynamic support machining, while the previous processes all use double-sided double cutting. For dynamic support machining, the tool position points on the machining side adopt the original machining points, and the original machining points on the support side are used as the initial support points, which are then processed into the final support points. To avoid interference and collision between the support device and the blade, as well as between the support device and the tool, the surface of the blade needs to be divided based on the structural characteristics of the support device and the blade. Firstly, the area division parameters are introduced: support radius \({R_{\sup }}\) and shrinkage distance \({D_{{\text{shr}}}}\) . The support radius refers to the radius of the circle formed by the auxiliary support of the support device (with the main support as the center). Shrinkage distance refers to the distance at which the original tool path is shrunk in all directions along the Y and X axes (as shown in Fig. 6 ), where the shrink distance \({D_{{\text{shr}}}}={R_{\sup }}\cdot {P_{{\text{shr}}}}\) , \({P_{{\text{shr}}}}\) is the shrink ratio and is generally taken as 1-1.5. The specific division rules are as follows: (1) Performing shrinkage processing in the X direction and referring to all points cloud of the tool paths as {P} . Dividing {P} into N rows of tool paths \({P_i}(i=1,2,...,N)\) according to the reciprocating machining method and obtaining the points with the highest X -axis coordinate value among all points on each row of tool paths \(X_{{\hbox{max} }}^{i},i=1,2,...,N\) . (2) Calculating two height parameters, where: \({H_1}=X_{{\hbox{max} }}^{1} - {D_{shr}},{H_2}=X_{{\hbox{max} }}^{N}+{D_{shr}}\) . (3) Filtering the number of tool path rows j that meet the condition \({H_2}<X_{{\hbox{max} }}^{j}<{H_1}\) , \(j={n_1},...,{n_2},1<{n_1}<{n_2}<N\) . (4) The area formed by tool paths 1 to \({n_1}\) is called the tip area, and the area formed by tool paths \({n_2}\) to N is called the root area. So far, the first shrinkage is completed. (5) Similarly, shrinking the remaining j rows of tool paths in the Y direction, and the filtered tool paths on both sides are called the edge area. The final shrunken tool path is called the internal area, as shown in Fig. 6 . In the Figure, the shaded area is the trajectory area (i.e. internal area) where the main support head moves, while the areas that the main support head cannot reach are the tip area, edge area, and root area. In the internal area, the tool and the support device perform symmetrical motion, that is, the two move synchronously, and the tool position points correspond one-to-one with the support points (the point on the main support head motion trajectory is referred to as the support point in the text). In the edge area, the tool processes normally. To prevent interference between the support device and the blade, the support device stops waiting. For the tip and root areas, the tool and support device perform asymmetric motion. Finally, to maximize the internal area, the shrinkage distance \({D_{{\text{shr}}}}\) should be reasonably selected while ensuring safe machining, so that the tool and support device can perform symmetrical motion as much as possible. 3.2 Support motion trajectory planning To achieve stable support, it is necessary to obtain a reasonable support attitude based on the tool position point. The attitude scheme adopted in this study is to use the normal vector on the surface of the blade as the support axis and pass through the tool position point, which can counteract the local torque to the maximum extent generated by the cutting tool during processing, providing effective and stable support for the blade, as shown in Fig. 7 . Dynamic support machining needs to be described through six key points: the machining edge point of the n th trajectory \(P_{{{\text{edge}}}}^{n}\) , the supporting edge point \(S_{{{\text{edge}}}}^{n}\) and its corresponding machining point \(P_{{cor}}^{n}\) of the n th trajectory, the machining edge point \(P_{{{\text{edge}}}}^{{n+1}}\) of the ( n + 1)th trajectory, the supporting edge point \(S_{{{\text{edge}}}}^{{n+1}}\) and its corresponding machining point \(P_{{cor}}^{{n+1}}\) of the ( n + 1)th trajectory. The motion process of the cutting tool and support device is designed and planned as follows: (1) The cutting tool enters from the top of the blade tip area and performs reciprocating machining according to the established tool paths. At the same time, the support device moves back and forth on the first trajectory to support. Since the main support and the cutting tool do not correspond one-to-one according to the above attitude scheme, asymmetric motion is performed, and the cutting force is counteracted by auxiliary support currently. When the tool reaches the path corresponding to the first support trajectory, the two begin to move symmetrically. (2) Fig. 8 (a) shows the process of the tool and support device moving from the internal area to the edge area. When the support device reaches the edge point on the n th trajectory \(S_{{{\text{edge}}}}^{n}\) , the tool reaches the corresponding point \(P_{{cor}}^{n}\) , and the support device stops waiting. When the tool reaches the edge point \(P_{{{\text{edge}}}}^{n}\) , the two transition synchronously to the ( n + 1)th trajectory. (a) Moving from the internal area to the edge area (3) Fig. 8 (b) shows the process of the tool and support device moving from the edge area to the internal area. The support device stops waiting at the edge point \(S_{{{\text{edge}}}}^{{n+1}}\) on the ( n + 1)th trajectory, and the tool starts to process from the edge point \(P_{{{\text{edge}}}}^{{n+1}}\) . When the tool reaches the point of processing \(P_{{cor}}^{{n+1}}\) , the tool and support device begin to move symmetrically, and the subsequent motion trajectory is the same. (b) Moving from the edge area to the internal area Figure 8 The motion process of dynamic support machining (4) After the support device moves to the last trajectory, it will move back and forth to support the workpiece, while the tool is processing normally. Currently, auxiliary support is used to counteract the cutting force until the processing of the root area is completed. Finally, the tool and support device exit synchronously and return to zero point. In most areas, the tool and support device maintain symmetrical motion, and the lateral and radial distances between the two are very small, which can reduce torsional and overturning moments. In addition, throughout the entire machining process, the support device always adheres to the surface of the blade, providing a full support effect for the blade and effectively suppressing the vibration caused by tool machining. 4 The post-processing of dynamic support machining In CNC programming, the process of converting tool position data generated by CAM into the final machining program that can be read by the CNC machine tool according to the machine tool structure and its instruction format is called post-processing [ 26 ]. Similarly, the original tool position data of the propeller blade needs to be processed by corresponding post-processing programs before it can be converted into dynamic support machining data. 4.1 Preprocessing of tool position point cloud According to the theory of blade path generation mentioned above, with the center of the propeller axis as the center, a fixed radius increment is set to create a series of concentric circles, which are projected onto the surface of the blade to obtain the contour curve. It is used as a streamline for reciprocating machining of the blade path, and a double-sided blade path model is obtained. The original tool position file includes instruction information, machining point pose parameters, identifiers for different types of tool paths, etc. Different instructions guide different machining information. Therefore, by identifying the corresponding instructions and identifiers, the extraction of tool position information can be completed and become the machining points cloud. The tool path partition identifier is used to implement partition management of tool position information. Extracting the tool path of the final process of surface finishing into point cloud information, with one side as the processing points cloud {P} and the other side as the original support points cloud {S} . According to the area division method in section 3.1 , by setting an appropriate shrinkage distance, completing shrinkage processing on the machining points cloud, and dividing the blade surface into four areas. Temporarily discarding the machining points in the blade tip area, root area, and edge area, and retaining the machining points in the internal area, as shown in Fig. 9 , in preparation for obtaining support points in the following text. 4.2 The method of support point acquisition The support scheme adopted in this study is that the axis of the support device passes through the tool position point, and based on this scheme, the support point is solved iteratively through the ergodic method. Firstly, after completing the semi-finishing machining of the propeller blade, obtain the 3D points cloud information of the machining surface and support surface. Fitting the surface with the original support points cloud {S} , and then selecting support points within a certain area based on the tool position points. If none of them meet the distance condition, use the ergodic method to select different r values to solve for the support points \(S_{e}^{j}\) and support vectors \(n_{e}^{j}\) . The specific process is shown in Fig. 10 . (1) A fifth-degree polynomial is used to fit the surface S according to the 3D support points cloud. The surface equation is \({z_S}=f({x_S},{y_S})\) , and the surface fitting result is shown in Fig. 11 . (2) Based on the known tool position point \(P({x_P},{y_P},{z_P})\) , projecting along the Z -axis onto the support surface, the reference point \(B({x_B},{y_B},{z_B})\) of the support surface is obtained. The X -axis and Y -axis coordinate values of this point are then incremented and decremented in a fixed increment value to obtain a rectangular boundary \(D:\{ {x_B} - \Delta x \leqslant x \leqslant {x_B}+\Delta x,{y_B} - \Delta y \leqslant y \leqslant {y_B}+\Delta y\}\) . The support points within the boundary form a region \({S_T}\) , as shown in Fig. 12 . (3) Solving the normal of S at the support point \(S_{T}^{i}({x_T},{y_T},{z_T})(i=1,2,...,n)\) : $$n_{T}^{i}=\frac{{[f({x_S},{y_S})/\partial {x_S}{|_{{x_S}=x_{S}^{i}}},f({x_S},{y_S})/\partial {y_S}{|_{{y_S}=y_{S}^{i}}}, - 1]}}{{||[f({x_S},{y_S})/\partial {x_S}{|_{{x_S}=x_{S}^{i}}},f({x_S},{y_S})/\partial {y_S}{|_{{y_S}=y_{S}^{i}}}, - 1]||}}$$ 1 In the equation, \({z_S}=f({x_S},{y_S})\) represents the surface equation of the support surface S , and then calculating the spatial distance \(d_{T}^{i}\) between the normal \(n_{T}^{i}\) and the tool position point P : 2 In the formula, represents the vector pointing from the support point to the tool position point. Since the normal \(n_{T}^{i}\) is the unit vector, the result is the vertical projection of on the normal \(n_{T}^{i}\) . According to the Pythagorean theorem, the distance can be obtained. (4) If the minimum value of all distances obtained is smaller than the set threshold \(\hbox{min} \{ d_{T}^{i}\} <e\) , then the corresponding \(S_{T}^{i}\) is taken as the support point corresponding to the tool position point P , and the corresponding normal \(n_{T}^{i}\) is the support direction of that point. If \(\hbox{min} \{ d_{T}^{i}\} \geqslant e\) , taking the support point corresponding to the minimum distance and calling it \({S_e}({x_e},{y_e},{z_e})\) , then obtaining the new support point and support vector through ergodic method (i.e. numerical solution). (5) Taking \(({x_e},{y_e},{z_e})\) as the center of the circle and creating a circle based on the set initial radius r , taking points \((x_{e}^{j},y_{e}^{j},{z_e})\) on the circle at different angles \({\theta _j}\) , \(x_{e}^{j}={x_e}+r\cdot \cos {\theta _j},y_{e}^{j}={y_e}+r\cdot \cos {\theta _j}\) , and projecting them along the Z -axis onto the surface S to obtain the projection points \(S_{e}^{j}(x_{e}^{j},y_{e}^{j},z_{e}^{j})\) . (6) Solving the normal \(n_{e}^{j}\) of S at the projection point \(S_{e}^{j}(x_{e}^{j},y_{e}^{j},z_{e}^{j})\) and calculating the spatial distance \(d_{e}^{i}\) between the normal \(n_{e}^{j}\) and the tool position point P . The calculation method is the same as (4). (7) If the distance is less than the threshold \(d_{e}^{j}<e\) , take the projection point \(S_{e}^{j}\) as the support point corresponding to the tool position point P , and the corresponding normal \(n_{e}^{i}\) is the support direction of that point. If \(\hbox{min} \{ d_{T}^{i}\} \geqslant e\) , increasing the angle \(\theta\) by a fixed angle increment \(\Delta \theta\) until \(2\pi\) and taking the point on the circle. Repeating steps (5)–(6). If the threshold condition is not met, increase the radius uniformly in a fixed radius increment \(\Delta r\) and repeat the above process until the condition is satisfied, as shown in Fig. 13 . When the radius r increases to a certain value, further increase will enter a meaningless cycle, so the maximum radius \({r_{\hbox{max} }}\) needs to be set. If \(r>{r_{\hbox{max} }}\) , taking the support point \(S_{T}^{i}\) and support vector \(n_{T}^{i}\) in step (4). The above can serve as the process for obtaining the support point corresponding to the first tool position point. Repeating the above process to obtain the remaining support points one by one. The difference is that starting from the second point, in process (2), the previous support point can be directly used as the reference point, without the need for projection processing. The final support points cloud obtained through the ergodic method is shown in Fig. 14 . During the machining process, the axis of the cutting tool and the support device are both perpendicular to the surface of the blade, and the support axis passes through the tool position point, which can minimize the excitation caused by the cutting tool machining and the local torque generated by the cutting tool and support, providing stable dynamic support for the blade. 4.3 Support point interpolation and support vector smoothing processing As shown in Figs. 5 and 6 , the original machining point cloud and the original support point cloud both move following the surface reciprocating tool path along the Y -axis. For each trajectory, the spacing between adjacent points is similar, indicating good continuity and regularity. However, most of the support points obtained by the ergodic method have obvious “breakpoints” on their trajectories, mainly because this method is an approximate numerical solution, unlike analytical methods that can accurately solve problems. Through debugging, it is known that when looping to the j th point of the n th trajectory, the ergodic method can be used to obtain the new support point \(\{ {S_e}\} _{n}^{j}\) corresponding to the machining point \(P_{n}^{j}\) near the original support point \(S_{n}^{j}\) . When looping to the next machining point \(P_{n}^{{j+1}}\) , \(S_{n}^{{j+1}}\) nearby points do not meet the distance threshold condition, and \(S_{n}^{{j+2}}\) nearby points just meet the condition. So, there is a “gap” between the support points corresponding to the adjacent two machining points. To ensure a continuous and smooth support trajectory, interpolation needs to be performed at “gaps”, and the interpolation method is as follows. (1) Dividing the support points into M trajectories based on the Y -axis coordinate values of support points. (2) Performing first-order differential processing on the Y -axis coordinates of each point \({S_e}({x_e},{y_e},{z_e})\) on the trajectory, \(dif_{n}^{{j - 1}}=\{ {y_e}\} _{n}^{j} - \{ {y_e}\} _{n}^{{j - 1}},1 \leqslant n \leqslant M,2 \leqslant j \leqslant N\) , where N is the total number of support points. (3) For the n th trajectory, take the average of the difference results between its adjacent two trajectories and the current trajectory \({m_n}=(di{f_{n - 1}}+di{f_n}+di{f_{n+1}})/3\) , and calculate the distance threshold \({e_d}={m_n}\cdot {K_d}\) , where \({K_d}\) is the distance coefficient. (4) By filtering the difference results that meet the condition \(di{f_n} \geqslant {e_d}\) in the n th trajectory, it can be concluded that the midpoint between the support point \(S_{e}^{{j - 1}}\) and \(S_{e}^{j}\) is the “gap”. Performing linear interpolation at this position. Calculating the midpoint of the two support points \(S_{m}^{j}(x_{m}^{j},y_{m}^{j},z_{m}^{j})\) , then projecting \(S_{m}^{j}\) along the Z -axis onto the support surface S , \(z_{{\operatorname{int} er}}^{j}=f(x_{m}^{j},y_{m}^{j})\) , and \((x_{m}^{j},x_{m}^{j},z_{{\operatorname{int} er}}^{j})\) being the interpolation point. Finally, to ensure that the amount of points cloud on the machining side is consistent and one-to-one correspondence with the support side, the machining side needs to synchronously interpolate between the machining point \(P_{e}^{{j - 1}}\) and \(P_{e}^{j}\) corresponding to the support point \(S_{e}^{{j - 1}}\) and \(S_{e}^{j}\) , using the same method as (4). Repeating processes (3)–(4) to complete the interpolation processing of all “breakpoints”. If the support vectors are not smooth and continuous, the swinging angle of the support axis may change sharply during actual machining, which causes the parallel mechanism of the machine tool to shake and exacerbate the vibration of the blade, especially for propellers with large curvature. Therefore, to ensure the smooth motion of the machine tool, it is necessary to smooth the support vectors. For research on vector smoothing, most optimization methods currently focus on reducing the angles between adjacent tool axis vectors [ 27 ]. This study draws inspiration from this method, aiming to smooth the angles between adjacent support axis vectors as much as possible to make the support vectors smooth and continuous. Firstly, calculating the angles between adjacent support vectors on each trajectory: $${\theta _i}=\arccos \frac{{{V_i}\cdot {V_{i+1}}}}{{|{V_i}|\cdot |{V_{i+1}}|}}(1 \leqslant i \leqslant N - 1)$$ 3 In the equation, where N is the total number of support vectors, the angle \({\theta _i}\) reflects the smoothness of the support vectors. As shown in Fig. 15(a), it illustrates the angles between adjacent support vectors on the n th and ( n + 1) th support trajectory before smoothing. The curve generally shows an upward or downward trend, which is due to the asymmetric nature of the propeller blades. The curvature at both ends of a trajectory is different, which leads to varying degrees of vector change, resulting in this trend. However, the curve exhibits abrupt changes and discontinuities, where the angle undergoes sharp variations, causing vector discontinuity. Therefore, smoothing processing is needed. (a) Before smoothing (b) After smoothing Figure 15 Angles between adjacent support vectors before and after smoothing As indicated by Eq. ( 3 ), smoothing the support vectors implies smoothing the angles between vectors. Taking the n th and ( n + 1) th support trajectory and respectively using the first support point as the origin to establish a three-dimensional Cartesian coordinate system. Converting each support vector \({V_i}({I_i},{J_i},{K_i})\) into angles with the coordinate axes. As shown in Fig. 16 , the relationship between vector components and angles is given by the following: $${\alpha _i}=\arcsin (\frac{{{J_i}}}{{\sin {\beta _i}}})=\arccos (\frac{{{I_i}}}{{\sin {\beta _i}}}),{\beta _i}=\arccos ({K_i})$$ 4 The obtained angles \({\alpha _i},{\beta _i}\) are fitted with a sixth-degree polynomial to obtain curve equations \({F_\alpha },{F_\beta }\) , where the fitted results for the angle \({\alpha _i}\) are shown in Fig. 17 . Calculating the points on the fitted curve and using them as the angles after smoothing \(\alpha _{f}^{i}={F_\alpha }(i),\beta _{f}^{i}={F_\beta }(i)\) . According to Eq. ( 4 ), reverse-calculating the vector components after smoothing. $$I_{f}^{i}=\cos \alpha _{f}^{i}\cdot \sin \beta _{f}^{i},J_{f}^{i}=\sin \alpha _{f}^{i}\cdot \sin {\beta _i},K_{f}^{i}=\cos \beta _{f}^{i}$$ 5 The end \(V_{f}^{I}(I_{f}^{i},J_{f}^{i},K_{f}^{i})\) represents the vector after smoothing, and the angles between adjacent vectors after smoothing are shown in Fig. 15(b). Additionally, the vectors before smoothing are unit vectors, and subject to the constraint in Eq. ( 5 ), smoothing the angles \({\alpha _i},{\beta _i}\) separately ensures that the vectors after smoothing remain unit vectors. Strictly speaking, the vectors after smoothing no longer satisfy the support scheme passing through the tool position points. However, the difference between the angles and the smoothed sixth-degree polynomial fitting results is very small. The impact on the overall support scheme can be negligible, and it can still achieve a good support result. Therefore, this smoothing method is feasible. 4.4 Calculation of processing parameters The final step of the post-processing program is to calculate the machining parameters, which are the input files for the control system of the XYZ-3RPS hybrid machine tool prototype. These parameters include machining time, drive displacements of each axis, drive speed, and drive acceleration. Based on the inverse kinematic analysis theory of the XYZ-3RPS hybrid machine tool, the processed machining-side and support-side pose data are converted into the drive displacements of each axis of the machine tool through corresponding algorithms. Then, the machining time is reasonably allocated. Finally, based on the time-displacement sequence, cubic spline interpolation is performed to obtain cubic polynomial functions for each segment. The first and second derivatives are then calculated to obtain the required velocities and accelerations. In addition, both the machining-side and support-side trajectories consist of several streamlined trajectories. The connection between each streamlined trajectory is not smooth; instead, there are certain corners and forming turning points. Near these turning points, the direction of velocity and acceleration will change abruptly, thereby reducing the stability of the machine tool motion. Therefore, an improvement is made. Each turning point is used as a boundary to divide trajectories on both sides into different curve groups. Time-displacement interpolation is performed separately for each group, and the initial and final velocities of each group are set to zero. Before performing time-displacement interpolation, the machining points/support points are defined as nodes. At each node, machining time needs to be allocated, which depends on the drive displacements and maximum drive speeds of the 12 axes on the machining side and support side, as given in Eq. ( 6 ): $${t_n}=\hbox{max} \{ \frac{{{{\{ d_{m}^{P}\} }_n} - {{\{ d_{m}^{P}\} }_{n - 1}}}}{{{v_{\hbox{max} }}}},\frac{{{{\{ d_{m}^{S}\} }_n} - {{\{ d_{m}^{S}\} }_{n - 1}}}}{{{v_{\hbox{max} }}}}\} +{t_{n - 1}}$$ 6 In the equation, \({\{ d_{m}^{P}\} _n}/{\{ d_{m}^{S}\} _n}\) represents the drive displacement of the machining side/support side at the n th node; m is the number of drive axes, \(m=1,...,6\) ; \({t_n}\) is the machining time at the n th node, \(n=2,3...\) , where \({t_1}=0\) ; \({v_{\hbox{max} }}\) is the maximum speed limit of the drive axes. After obtaining the machining time, segment-wise cubic spline interpolation can be performed for each of the curve groups divided as described above. Taking one of the curve groups with k nodes as an example and considering the drive displacement along the X -axis on the machining side, the segment-wise cubic spline equation is as follows: $$x_{{}}^{P}(t)={A_i}+{B_i}(t - {t_i})+{C_i}{(t - {t_i})^2}+{D_i}{(t - {t_i})^3}$$ 7 In the equation, \(t \in [{t_i},{t_{i+1}}]\) , \(i=1,...,k - 1\) . Because the first and second derivatives of each adjacent pair of segments' equations are continuous at the nodes [ 28 ]. The following equation can be obtained: $${A_i}=x_{{}}^{P}({t_i})$$ 8 $${B_i}+{C_i}{T_i}+{D_i}{T_i}^{2}={V_i}$$ 9 $${B_i}+2{C_i}{T_i}+3{D_i}{T_i}^{2}={B_{i+1}}$$ 10 $${C_{i - 1}}{T_{i - 1}}+3{D_{i - 1}}{T_{i - 1}}={C_i}$$ 11 Where \({T_i}={t_{i+1}} - {t_i}\) , \({V_i}=(x_{{}}^{P}({t_{i+1}}) - x_{{}}^{P}({t_i}))/{T_i}\) . After simplification, it can be obtained as follows: $$\frac{{{B_{i - 1}}}}{{{T_{i - 1}}}}+2(\frac{1}{{{T_{i - 1}}}}+\frac{1}{{{T_i}}}){B_i}+\frac{{{B_{i+1}}}}{{{T_i}}}=3(\frac{{{V_{i - 1}}}}{{{T_{i - 1}}}}+\frac{{{V_i}}}{{{T_i}}})$$ 12 $${C_i}=\frac{1}{{{T_i}}}(3{V_i} - {B_{i+1}} - 2{B_i})$$ 13 $${D_i}=\frac{1}{{{T_i}^{2}}}({B_i}+{B_{i+1}} - 2{V_i})$$ 14 In Eq. ( 12 ), there are k − 1 unknown parameters \({B_1},{B_2},...,{B_{k - 1}}\) , but only k − 3 equations. Therefore, two additional boundary conditions are needed to solve all parameters, specifically, the first derivative values at the initial and final two points of this curve group. Here, we set the first derivatives at the initial and final two nodes equal to 0, i.e., and \({B_1},{B_{k - 1}}\) are both equal to 0. With this, all variable parameters in equations ( 12 )–( 14 ) can be solved, and by substituting them into Eq. ( 7 ), polynomial equations for each segment can be obtained. Successively solving for their first and second derivatives will yield polynomial equations for velocity and acceleration: $${v^P}(t)={B_i}+2{C_i}(t - {t_i})+3{D_i}{(t - {t_i})^2}$$ 15 $${a^P}(t)=2{C_i}+6{D_i}(t - {t_i})$$ 16 By substituting the time for each node into equations ( 15 ) and ( 16 ), the velocity \({v^P}({t_{i+1}})\) and acceleration \({a^P}({t_{i+1}})\) for each node can be solved. Finally, the time, displacement, velocity, and acceleration sequences are integrated into the input file for the hybrid machine tool. 5 Simulation verification and experimental verification 5.1 ADAMS simulation verification The original tool position file is transformed into machining data through the above preprocessing, interpolation, calculation of machining parameters, etc., and a portion of it is extracted. Table 1 shows the time, series drive displacement, and parallel drive displacement for the support side. Table 2 , on the other hand, presents the time, series drive velocity, and parallel drive velocity for the support side. Table 1 Partial time-displacement data for the support side \(t(ms)\) \(d_{x}^{S}(mm)\) \(d_{y}^{S}(mm)\) \(d_{z}^{S}(mm)\) \(d_{1}^{S}(mm)\) \(d_{2}^{S}(mm)\) \(d_{3}^{S}(mm)\) 110459 -36.2743 21.9991 85.5620 3.0331 -22.8370 23.3242 110565 -36.4132 23.0541 85.1803 2.9672 -22.7910 23.3310 110671 -36.5545 24.1125 84.7980 2.9006 -22.7439 23.3369 110778 -36.6982 25.1738 84.4153 2.8331 -22.6955 23.3421 …… …… …… …… …… …… …… 1603588 -131.2947 99.9463 61.9526 -1.3161 -12.2766 14.6180 1603799 -130.8497 97.9564 62.3266 -1.1369 -12.5214 14.7122 1604008 -130.4134 96.0902 62.6963 -0.9661 -12.7534 14.8014 1604216 -129.9568 94.2166 63.0697 -0.7855 -12.9920 14.8887 Table 2 Partial time-velocity data for the support side \(t(ms)\) \(v_{x}^{S}(mm/ms)\) \(v_{y}^{S}(mm/ms)\) \(v_{z}^{S}(mm/ms)\) \(v_{1}^{S}(mm/ms)\) \(v_{2}^{S}(mm/ms)\) \(v_{3}^{S}(mm/ms)\) 110459 0.0001 0.0099 -0.0035 -0.0003 0.0000 0.0002 110565 -0.0017 0.0100 -0.0036 -0.0007 0.0005 0.0000 110671 -0.0014 0.0099 -0.0036 -0.0006 0.0004 0.0001 110778 -0.0008 0.0099 -0.0035 -0.0006 0.0004 0.0000 …… …… …… …… …… …… …… 1603588 0.0022 -0.0091 0.0017 0.0009 -0.0012 0.0005 1603799 0.0021 -0.0093 0.0018 0.0008 -0.0011 0.0004 1604008 0.0022 -0.0089 0.0018 0.0008 -0.0011 0.0004 1604216 0.0021 -0.0090 0.0018 0.0008 -0.0011 0.0004 To validate the correctness and feasibility of the toolpath and support trajectory, ensuring the safe machining of the hybrid machine tool, all machining data is imported into the motion model of the hybrid machine tool created in the software ADAMS in proportion. The motion simulation process is shown in Fig. 18. (a) Motion simulation model of dynamic support machining (b) Toolpath on the machining side and motion trajectory on the support side Figure 18 Motion simulation dynamic support machining Finally, multiple sets of machining data are obtained from different original tool position data, and each set undergoes motion simulation verification. Neither the toolpath nor the support trajectory shows any abnormalities and is consistent with the planning, indicating that the processing procedure and results are correct. Further experimental analysis can be carried out using the prototype of the hybrid machine tool. 5.2 Experimental validation The experiment used the XYZ-3RPS hybrid machine tool described above, as shown in Fig. 19 . The cutting tool and support device are installed at the end center of the parallel mechanism, with the tool using a 3mm radius ball-end mill. The workpiece is a nylon material with dimensions of 200mm×50mm×350mm, vertically clamped to the worktable through six clamping support brackets. The machining steps include rough machining, semi-finishing, and finishing. Rough machining shapes the workpiece into the basic form of a propeller blade, then semi-finishing further reduces the thickness of the blade, and finishing is done using dynamic flexible support machining. Additionally, it should be noted how the machining side/support side engages and disengages. Taking the support side as an example, the support device moves from the zero point to the first support point with the attitude (i.e., normal) of the first support point, then retracts a certain safe distance from the last support point before returning to the zero point. The machining side follows a similar process. This approach helps to avoid interference collisions between the tool and support device with the blade during the tool engagement and disengagement processes. Finally, the integrated machining data is input into the control system of the machine tool to start the machining experiment. Figure 20 depicts the process of dynamic flexible support machining. In the internal area, the cutting tool and the support device move synchronously, and their attitudes conform to the expected attitude scheme. In the edge area, the cutting tool performs its machining task while the support device waits, then both the cutting tool and the support device move to the next trajectory at the same time, which is consistent with the planned trajectory. The above results indicate the accuracy of the post-processing algorithm proposed in this study, and the suggested support motion trajectory is proven to be correct and effective. (a) Internal area machining stage (b) Edge area machining stage Figure 20 Dynamic flexible support machining experiment Conclusion To reduce the vibration during the machining process of the propeller, a dynamic flexible support machining scheme is proposed in this paper, drawing inspiration from mirror machining technology. To counteract the cutting forces generated during machining as much as possible, a multi-point flexible support device is designed and installed at the end of the machine tool. Based on the structure of the blade and the support device, the surface of the propeller blade is divided into four areas, each employing a different motion method, and the support motion trajectory is planned accordingly. The paper introduces a post-processing method for dynamic support machining, realizing the conversion from the original CAM tool position file to the machining file recognized by the machine tool. Finally, the correctness and feasibility of the planned support motion trajectory and algorithm are verified through simulation and machining experiments. Throughout the entire machining process, the support device is in constant contact with the blade, providing support force to the blade that varies with the milling force. However, the precise calculation of the support force magnitude is not performed, and while the support effect can be improved, there is still room for improvement in this scheme. Therefore, in future work, it is necessary to establish a dynamic model for the interaction between the tool, blade, and support device, analyze the mapping relationship between the support force and milling force, and thereby precisely control the support force to further reduce machining deformation and vibration. Declarations Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant number [51975157]). Author Rui Wang has received this research support. Competing Interests: The authors have no relevant financial or non-financial interests to disclose. References Ji, Y.Y.; Tian, G.Z.; Zhou, H.G. Research progress in advanced manufacturing technology for marine propellers. Ship Science and Technology, 2015, 37, 9-15. Ren, Y.Y. Research on Structural Parameter Calibration of Six Axis Hybrid Horizontal Machining Device. Masters’ thesis, Harbin Institute of Technology, Harbin, China, 2014. Youn, J.W.; Jun, Y.; Park, S. Interference-free tool path generation in five-axis machining of a marine propeller. INT J PROD RES , 2003, 41, 4383-4402. Liu, X.; Liu, R. A Creo-Based Modeling and Toolpath Generating System for the NC Whirling Process of Propeller Blades. In Proceedings of the ICGG 2018-Proceedings of the 18th International Conference on Geometry and Graphics: 40th Anniversary-Milan, Italy, August 3-7, 2018, 18, 2019; pp. 1350-1357. 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Ph.D. Thesis, Northwestern Polytechnical University, Xi an, China, 2003. Cite Share Download PDF Status: Published Journal Publication published 26 Jul, 2025 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted Editorial decision: Major Revisions Needed 20 Aug, 2024 Reviewers agreed at journal 17 Apr, 2024 Reviewers invited by journal 08 Mar, 2024 Editor assigned by journal 07 Mar, 2024 First submitted to journal 05 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4016972","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":277572268,"identity":"54f5b978-0513-40c9-b5df-404ffa219c1f","order_by":0,"name":"Songmo Li","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Songmo","middleName":"","lastName":"Li","suffix":""},{"id":277572269,"identity":"335bffbb-29da-4670-87fd-4baeb7f16a47","order_by":1,"name":"Rui 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Ge","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Yuhao","middleName":"","lastName":"Ge","suffix":""},{"id":277572271,"identity":"6c0e3b9b-61d2-4679-acd0-f52266c7fd1c","order_by":3,"name":"Xiangyu Guo","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Xiangyu","middleName":"","lastName":"Guo","suffix":""},{"id":277572272,"identity":"d95500a1-012e-4f88-b37e-2a641c972113","order_by":4,"name":"Mengmeng Liu","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Mengmeng","middleName":"","lastName":"Liu","suffix":""},{"id":277572273,"identity":"2adb7ae1-ea19-4e16-93da-7e2c29375e68","order_by":5,"name":"Xiaohu Zhu","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Xiaohu","middleName":"","lastName":"Zhu","suffix":""}],"badges":[],"createdAt":"2024-03-05 12:10:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4016972/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4016972/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00170-025-16136-z","type":"published","date":"2025-07-26T15:57:02+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":52574128,"identity":"b7d3ec1d-570d-440e-8a46-dd5537ba22db","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":23331,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of mirror machining technology\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/83de54e884d2da2340f7e791.png"},{"id":52574129,"identity":"433cd9b1-d8b7-47d4-831a-e8543ce21587","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":10680,"visible":true,"origin":"","legend":"\u003cp\u003eMachining steps\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/e2d4de35c24a3298e0f7404f.png"},{"id":52574132,"identity":"656f51c1-78f3-41c2-bb8f-dc8d274a8f4b","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":226069,"visible":true,"origin":"","legend":"\u003cp\u003ePrototype model of hybrid machine tool\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/2401b4bf18db68a2dee1e749.png"},{"id":52574130,"identity":"0a8b6302-e378-4d32-9897-0edb206cdfc3","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":115387,"visible":true,"origin":"","legend":"\u003cp\u003eThe model diagram of the support device\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/50584ef59da3e5685e36bd27.png"},{"id":52574131,"identity":"96779ceb-b74d-4e0b-819a-f8de31d6d401","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":24265,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of machining tool path\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/0f8e301f618083742b0a4eeb.png"},{"id":52575101,"identity":"83d7571a-1ac7-49e9-a8aa-d97c3fc23b3f","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":27640,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of the division of areas\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/24824c4cfb84bd40a04b0fbb.png"},{"id":52575103,"identity":"45742dcc-35fe-481c-850b-4ec5f71177fd","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":14410,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of tool and support attitude scheme\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/51778936dd886312aba4c08a.png"},{"id":52574137,"identity":"6859de43-9e58-4e33-955e-916dd89f23ca","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":51906,"visible":true,"origin":"","legend":"\u003cp\u003eThe motion process of dynamic support machining\u003c/p\u003e\n\u003cp\u003e(a) Moving from the internal area to the edge area\u003c/p\u003e\n\u003cp\u003e(b) Moving from the edge area to the internal area\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/50a09a729fdd8b86dfafb616.png"},{"id":52575106,"identity":"ac516b81-9272-42e1-851a-fd27de033065","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":130130,"visible":true,"origin":"","legend":"\u003cp\u003eShrinkage processing of machining points\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/520bf2f4104ec243e2ee9efe.png"},{"id":52575102,"identity":"76052f7c-2d87-4559-ba9c-3fc767102f86","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":39076,"visible":true,"origin":"","legend":"\u003cp\u003eThe flowchart for support points acquisition\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/a228f4d9d589e4df0a0d176a.png"},{"id":52574135,"identity":"85b8495e-591a-4bab-9080-e23c2bc950b0","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":87185,"visible":true,"origin":"","legend":"\u003cp\u003eSurface fitting result\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/f0990b33c1618a6a0916447f.png"},{"id":52574143,"identity":"01cbffd5-f710-4b08-8812-302883c6c317","added_by":"auto","created_at":"2024-03-13 06:29:37","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":234966,"visible":true,"origin":"","legend":"\u003cp\u003eSelecting the support surface area based on the tool position point\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/86d2a3fc74cdb524f7d337dd.png"},{"id":52575104,"identity":"59e6f918-d912-4ce5-bd6e-6205d81cfdf8","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":125990,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of ergodic\u003cem\u003e θ \u003c/em\u003eand\u003cem\u003e \u003c/em\u003eergodic \u003cem\u003er\u003c/em\u003e\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/ae09f64dbe459efa81e34fd7.png"},{"id":52575107,"identity":"8c3d3b5a-6542-4875-a5ce-1c62c66ccdc2","added_by":"auto","created_at":"2024-03-13 06:37:37","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":167422,"visible":true,"origin":"","legend":"\u003cp\u003eSupport points obtained through the ergodic method\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/6236d5820b682c34ca067f86.png"},{"id":52574138,"identity":"7922c48a-316e-4c07-9384-f9f289715338","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":30717,"visible":true,"origin":"","legend":"\u003cp\u003eAngles between adjacent support vectors before and after smoothing\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/63204ea879b9654346d41243.png"},{"id":52574144,"identity":"24bf4468-4cfc-4521-acf9-cd7796f0a898","added_by":"auto","created_at":"2024-03-13 06:29:37","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":22676,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of converting vector to angles\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/5409ac9c6a135df52d29673d.png"},{"id":52575105,"identity":"6ade5b21-c195-44c9-ad63-43ac978e1b78","added_by":"auto","created_at":"2024-03-13 06:37:36","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":24415,"visible":true,"origin":"","legend":"\u003cp\u003eFitting result for the angle α\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/4c2ee0d2def8ae7433e1db01.png"},{"id":52574142,"identity":"18e49210-2d9b-43a3-9974-4aaf1821ee7b","added_by":"auto","created_at":"2024-03-13 06:29:36","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":91645,"visible":true,"origin":"","legend":"\u003cp\u003eMotion simulation dynamic support machining\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/459224cb46d69af87edb0421.png"},{"id":52574146,"identity":"d8a8bf5b-1af8-4637-8855-1ec283a79108","added_by":"auto","created_at":"2024-03-13 06:29:37","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":300799,"visible":true,"origin":"","legend":"\u003cp\u003eExperimental prototype of the hybrid machine tool\u003c/p\u003e","description":"","filename":"19.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/a64e7cbd58bfcc0b21eb73e0.png"},{"id":52574145,"identity":"a9d8bf4b-8208-475b-bc54-ae0db1a9889e","added_by":"auto","created_at":"2024-03-13 06:29:37","extension":"png","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":378727,"visible":true,"origin":"","legend":"\u003cp\u003eDynamic flexible support machining experiment\u003c/p\u003e","description":"","filename":"20.png","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/930e80d24cac3622a16a3051.png"},{"id":87756838,"identity":"12d21247-60f5-42fe-a99b-5506f1af1726","added_by":"auto","created_at":"2025-07-28 16:09:34","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2964112,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4016972/v1/b9ab6029-445b-4939-93a0-76d4dd8e3a2a.pdf"}],"financialInterests":"","formattedTitle":"Research on the Dynamic Flexible Support Machining Method for Propeller Blades","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003ePropellers are the core propulsion components of ships and submarines, which work in high-speed and complex flow field environments. If the surface machining quality and accuracy of the propeller are low, it will reduce its propulsion efficiency, cause noise, and even affect the underwater performance [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Therefore, high-precision, high-efficiency, intelligent processing equipment is particularly important. In recent years, with the development of CNC machine tools, there have been many special propeller processing of heavy-duty horizontal multi-linkage CNC machine tools. However, the issues of vibration, interference, and secondary clamping during the machining process of large propellers have not been effectively solved, significantly impacting the machining precision and quality of the propeller blade surface [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. The main reason for vibration is that during the machining of the propeller, it is in a cantilever state, and the tool generates a significant overturning moment on the part of the blade far from the hub. Currently, traditional machining methods often involve single-sided machining, where one side of the propeller blade surface is machined first, followed by flipping it for the second clamping process, which inevitably introduces positioning errors and reduces machining accuracy.\u003c/p\u003e \u003cp\u003eTo solve the above problems, many scholars have researched double-sided co-machining, which is machined by two tools on both sides of the part at the same time. Double-sided co-machining not only solves the problem of interference and secondary clamping but also improves the machining efficiency. However, double-sided co-machining cannot guarantee that the two tools are always opposing. The misalignment will produce local torque, so there is still the problem of vibration. In recent years, many expert teams have proposed mirror machining technology, which has achieved significant results in suppressing vibration. Mirror machining refers to the mirror-symmetric distribution of the tool and support device on both sides of the workpiece to be machined. While the tool is machining, the support device supports the workpiece. Mirror machining has two characteristics, one is the tool axis and support axis to be co-linear and coincide with the normal vector of the workpiece surface; the second is the workpiece requires equal wall thickness. In response to the problems of secondary clamping and machining vibration, this study intends to draw inspiration from the mentioned mirror machining technology. The propeller is vertically clamped, and the cutting tool and support device are arranged on both sides to achieve dynamic flexible support machining. The aim is to further reduce vibration and improve the precision of propeller machining.\u003c/p\u003e \u003cp\u003eWith the continuous development and improvement of CNC technology, many types of machining equipment have appeared, and propeller machining methods have become diversified and refined. Youn et al. applied the iterative B-spline surface modeling technique to improve the accuracy and efficiency of the model, generated tool trajectories with a good surface finish and proposed that the method of calculating the check vectors based on the tool dimensions and the tool position points along the tool path can be used to obtain interference-free tool paths [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Liu et al. designed a CAD/CAM system for the propeller machining process, using Creo to parameterize the original design data to build a 3D model of the propeller and generate a process for manufacturing propeller blades, which improved the design and manufacturing efficiency [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Korsmik used topology optimization to redesign the original model of the propeller. By comparing the stress state with the original propeller and evaluating the stress-strain state parameters, the structure of the propeller was optimized while ensuring the reliability characteristics of the solid type, and a control program for the LMD manufacturing method was developed [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. Jiang of Huazhong University of Science and Technology obtained the three-dimensional flow field map of the propeller blade surface during the rotating work by CFD fluid analysis and proposed that making the machining path of the tool consistent with the direction of the blade surface fluid traces can reduce the impact of the propeller surface unevenness on the underwater performance [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Zhang et al. proposed a machining method combining dynamic adjustment of tool axis vector and optimization of machining parameters. The feasible machining domains of complex regions are obtained through area division and interference adjustment, then the optimal process parameters are obtained through AdvantEdgeFEM simulation, and finally, the effectiveness of this machining method is verified by five-axis machining experiments on propellers [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe traditional single-tool machining method not only has low machining efficiency but also causes precision loss in secondary clamping. For parts like propeller blades with double-sided features, double-tool double-sided machining can be used. Shamoto et al. used the speed difference method to suppress machining vibration. By controlling the milling cutters on both sides of the workpiece to process at different speeds, the regenerative chatter can be eliminated, the machining stability can be improved, thus realizing high-precision and high-efficiency simultaneous machining of both sides of the workpiece [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Mori et al. proposed a new method for synchronized milling of flexible thin-walled parts with double-sided single teeth, which can offset the milling force by synchronized two-flute milling cutters, and the final experiments showed that the flatness of the processed thin plate was increased by 2 times, and the machining efficiency was increased by 3 times [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Fu et al. used a double parallel machine tool (PKMS) to achieve synchronous and asynchronous machining and support of thin-walled parts, and the test results showed that the static and dynamic characteristics of the workpieces obtained by double PKMS machining were improved, the dimensional accuracy and surface quality were improved, and the material removal rate was doubled [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. Mejbel et al. developed a double-sided milling cutter co-machining technology, two vertical face milling cutters at the same time machining two thin-walled surfaces, cutting forces offset each other, the test results show that the workpiece flatness and straightness are significantly increased [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Song et al. proposed a tool path smoothing algorithm and trajectory optimization algorithm based on geometric constraints, to ensure the synchronous movement of the dual tool while smoothing the machining trajectory of the tool. Finally, the results of simulation and machining experiments show that the method reduces the nonlinear error in the machining process, and the contour error of the machined surface meets the tolerance requirements of the process [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. Ge et al. utilized a hybrid machine tool to complete double-cutter double-sided machining of propeller blades, using single-sided machining in the thin tip area, staggered machining in the edge area, and using the top-to-top machining in the thicker inner area, which effectively reduces the blade vibration. This method cannot be used in the tip area of double-sided symmetric machining, which needs to improve the processing effect through an auxiliary support device, so this program has more room for improvement [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAt present, many scholars and organizations have begun to propose a new machining method - mirror machining, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The advantage of this machining method is that the support device can offset the cutting force generated by the tool, so it can play a role in suppressing vibration, reducing deformation, the wall thickness of the effect of controllable, and thus improving the quality of machining and surface accuracy [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Regarding mirror image processing, many scholars have carried out research.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eLiu introduced an airflow-assisted support device and explored the impact of different pressure fields and velocity fields of air jet streams on the force and deformation of thin-walled parts. Finally, experiments demonstrated that this method enhanced machining accuracy by using appropriate parameters [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. Bao makes the stiffness distribution of the workpiece in the machining process have the same trend with the change of cutting force to optimize the support position of the multi-point support machining. The research results show that after optimizing the support position, the contour error of the part is significantly reduced [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. Hao established an integrated stiffness model of the workpiece and the support mechanism for the mirror-image support equipment and optimized the distribution of the integrated stiffness on the surface of the workpiece by increasing the redundancy drive. The experimental results proved that the average stiffness of the overall mechanism was doubled [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. Muller et al. developed an adsorptive multipoint flexible support device based on a parallel robot, with a support head with vacuum suction cups that can adhere to the surface of a workpiece with changing curvature to achieve stable support [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e]. Mahmud et al. simulated the milling force model of an annular face milling cutter to predict the milling force and derived the relationship between the cutting force and the support force generated by different tool angles [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Moradi proposed an optimization algorithm based on the modal method to design an adjustable damper to suppress the regenerative vibration in the milling process of cantilevered thin plates, and the optimal position of the damper was obtained by analyzing the nonlinear error presented by the cutting force [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Bolsunovsky et al. installed a special damper on a flexible workpiece to eliminate its vibration according to the rotational frequency of the machining spindle. The damper was applied to workpieces of any shape and frequency, and the results of the cutting test showed that the vibration of the workpiece was reduced by 20 times [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eBased on the above research status of propeller machining, double-sided collaborative machining, and mirror machining, this paper proposes the following scheme: XYZ-3RPS hybrid mechanism is arranged on both sides of the propeller to be machined at the same time, and the tool and support device are installed respectively at the end of the tool to synchronously follow the tool to move and provide dynamic support force to reduce the machining vibration and deformation. The hybrid machining equipment used in this study has the advantages of both tandem and parallel equipment, which ensures a larger workspace and improves the stiffness-to-weight ratio and load capacity. At the same time, this solution can complete the machining task in one clamping, which can solve the problem of positioning error and accuracy degradation caused by secondary clamping. In addition, the following machining steps will be adopted in this study: rough machining with double-sided double-tool machining, semi-finishing machining with double-sided collaborative machining, and final finishing machining with dynamic flexible support machining, which can improve the machining efficiency of the propeller and ensure the machining accuracy.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe so-called dynamic flexible support is that the support device follows the movement of the tool to realize the follower support and the support force provided by the support device can change with the change of the size of the milling force to realize the flexible support. The main research content of the article is as follows. Firstly, introducing the overall structure of the machining test prototype, i.e. hybrid machine tool, and designing a multi-point follower support device. Secondly, dividing the propeller surface into reasonable areas and planning the trajectory of the support device. Completing a series of post-processing from the conversion of the original tool position points to the machining parameters, including the processing of the tool position points, the acquisition of the support points, the interpolated smooth processing, and the calculation of machining parameters, etc. Finally, online simulation verification and actual machining test verification are carried out.\u003c/p\u003e"},{"header":"2. Structural analysis of hybrid machine tool and support device","content":"\u003cp\u003eThe experimental prototype model of the XYZ-3RPS hybrid machine tool used in this study is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The establishment method is to install a 3RPS parallel mechanism on the \u003cem\u003eX\u003c/em\u003e-axis sliding rail of the XYZ series mechanism, with the same symmetrical structure on both sides. One side of the parallel mechanism is equipped with a cutting tool at the end, and the other side is equipped with a support device to perform machining and support work respectively.\u003c/p\u003e \u003cp\u003eFor the serial part, the direction perpendicular to the ground is defined as the Z-axis positive direction, and the orientation of the initial position of the tool or support device is defined as the \u003cem\u003eX\u003c/em\u003e-axis positive direction. The \u003cem\u003eY\u003c/em\u003e-axis direction is determined through the right-hand rule of the spatial coordinate system. Each axis is driven by a screw module. The parallel mechanism is installed on the \u003cem\u003eX\u003c/em\u003e-axis sliding platform of the series mechanism, which includes three driving rods, a rotating platform, and a translational platform. The two ends of the driving rods are connected to the rotating platform and the translational platform using ball joint pairs and rotating pairs, respectively. The length of the rod is also changed through a screw module. The cutting tool and support device are installed at the ends of the rotating platforms on both sides. The XYZ series mechanism is responsible for the overall large-scale movement of the machine tool, while the 3RPS realizes the attitude change of the end effector.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe function of the support device is that it can support the workpiece while processing to reduce vibration and deformation. To achieve stable support for propeller blades, it is necessary to design a lightweight and flexible support device that can adaptively adjust the support force according to the shape of the components.\u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, the multi-point flexible support device mainly consists of 7 universal balls, 7 linear bearings, 7 springs, a copper bushing, and a thin cylinder [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. Universal balls can withstand large forces in the axial direction and reduce friction and scratches on the surface of the parts caused by the support head moving with the tool. The center is supported by a large-sized universal ball as the main support, while the remaining 6 are evenly arranged on the circumference centered on the main support. During the machining process, multiple points are supported together, which increases the local stiffness of the components near the milling point while applying center point support.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThis device uses a single-action extrusion-type thin cylinder as the supporting element and relies on an internal spring to retract when not supplying air. 7 springs are installed inside the support body. During the machining process, the piston rod of the cylinder extends, and the universal ball contacts the part. Due to the presence of the spring, the supporting device can adaptively adjust the length of the push rod extension according to the surface curvature of the part, achieving a flexible support effect. In addition, due to the reciprocating motion between the push rod and the support body, as well as between the support body and the housing, this device uses copper bushing and 7 linear bearings to reduce friction and lubrication.\u003c/p\u003e \u003cp\u003eThe advantage of this support device is that even when processing curved thin-walled parts with complex curvatures, it can ensure that each universal ball always contacts the surface of the thin-walled component, playing a good supporting effect. In the actual machining process of propeller blades, semi-finishing machining adopts double-sided dual-tool collaborative machining. Therefore, before finishing machining, this support device needs to replace the cutting tool on one side of the hybrid machine tool to process one side of the propeller blade. After the machining is completed, the positions of the two are exchanged to complete the machining of the other side of the blade. This dynamic support machining method does not require flipping and reinstalling the blades, thus avoiding positioning errors caused by secondary clamping.\u003c/p\u003e"},{"header":"3. Overall planning of dynamic support machining","content":"\u003cp\u003eWhen machining propeller blades, a torsional moment is generated between the tool contact point and the support point, and an overturning moment is generated between the tool contact point, support point, and blade, which together constitute a deformation moment. The deformation torque is an important factor that causes blade deformation and leads to a decrease in machining accuracy. To reduce the deformation torque, this section will introduce how to set a reasonable attitude relationship between the tool and the support device, plan the tool path, and support motion trajectory.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Original tool path generation and area division\u003c/h2\u003e \u003cp\u003eDuring the processing of propeller blades, defects (i.e. knife marks) are inevitably generated on the surface. If the defect direction is parallel to the fluid trace during blade operation, vibration and noise can be minimized to the greatest extent. Therefore, the tool path should overlap or approach the streamline on the blade surface to the maximum extent possible [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. Generally, when the blade is rotating during operation, the direction of the streamline on its surface resembles the curve direction formed by the intersection of axisymmetric cylindrical surfaces at different radii with the blade. Therefore, the tool path used in this study is a zig-zag reciprocating cutting method, and the generation method is as follows: the radius of the trajectory circle swept by the blade tip of the propeller during operation is called the radius of the propeller, and a series of coaxial cylindrical surfaces are made around the axis of the propeller with a set radius\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Delta R\\)\u003c/span\u003e\u003c/span\u003eincrement until the radius of the cylindrical surface reaches the propeller radius \u003cem\u003eR\u003c/em\u003e. Intersecting the cylindrical surface with the surface of the blade to obtain a set of contour curves. Dividing the set of curves into two parts using the guide and trailing edges of the propeller blades, which serve as the machining tool paths on both sides. The tool path that matches the streamline direction in all tool paths is called the surface machining tool path, and the one that connects adjacent surface machining tool paths is called the transformation machining tool path, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBased on the thickness and cutting depth of the rough workpiece after rough machining, a series of biases can be applied to the final finishing machining tool path to obtain multiple finishing machining tool paths. It should be noted that the last two finishing machining use dynamic support machining, while the previous processes all use double-sided double cutting. For dynamic support machining, the tool position points on the machining side adopt the original machining points, and the original machining points on the support side are used as the initial support points, which are then processed into the final support points.\u003c/p\u003e \u003cp\u003eTo avoid interference and collision between the support device and the blade, as well as between the support device and the tool, the surface of the blade needs to be divided based on the structural characteristics of the support device and the blade. Firstly, the area division parameters are introduced: support radius \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R_{\\sup }}\\)\u003c/span\u003e\u003c/span\u003e and shrinkage distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D_{{\\text{shr}}}}\\)\u003c/span\u003e\u003c/span\u003e. The support radius refers to the radius of the circle formed by the auxiliary support of the support device (with the main support as the center). Shrinkage distance refers to the distance at which the original tool path is shrunk in all directions along the \u003cem\u003eY\u003c/em\u003e and \u003cem\u003eX\u003c/em\u003e axes (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), where the shrink distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D_{{\\text{shr}}}}={R_{\\sup }}\\cdot {P_{{\\text{shr}}}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_{{\\text{shr}}}}\\)\u003c/span\u003e\u003c/span\u003eis the shrink ratio and is generally taken as 1-1.5. The specific division rules are as follows:\u003c/p\u003e \u003cp\u003e(1) Performing shrinkage processing in the \u003cem\u003eX\u003c/em\u003e direction and referring to all points cloud of the tool paths as \u003cem\u003e{P}\u003c/em\u003e. Dividing \u003cem\u003e{P}\u003c/em\u003e into \u003cem\u003eN\u003c/em\u003e rows of tool paths \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P_i}(i=1,2,...,N)\\)\u003c/span\u003e\u003c/span\u003eaccording to the reciprocating machining method and obtaining the points with the highest \u003cem\u003eX\u003c/em\u003e-axis coordinate value among all points on each row of tool paths \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(X_{{\\hbox{max} }}^{i},i=1,2,...,N\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e(2) Calculating two height parameters, where: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({H_1}=X_{{\\hbox{max} }}^{1} - {D_{shr}},{H_2}=X_{{\\hbox{max} }}^{N}+{D_{shr}}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e(3) Filtering the number of tool path rows \u003cem\u003ej\u003c/em\u003e that meet the condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({H_2}\u0026lt;X_{{\\hbox{max} }}^{j}\u0026lt;{H_1}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(j={n_1},...,{n_2},1\u0026lt;{n_1}\u0026lt;{n_2}\u0026lt;N\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e(4) The area formed by tool paths 1 to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n_1}\\)\u003c/span\u003e\u003c/span\u003e is called the tip area, and the area formed by tool paths \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n_2}\\)\u003c/span\u003e\u003c/span\u003e to \u003cem\u003eN\u003c/em\u003e is called the root area. So far, the first shrinkage is completed.\u003c/p\u003e \u003cp\u003e(5) Similarly, shrinking the remaining \u003cem\u003ej\u003c/em\u003e rows of tool paths in the \u003cem\u003eY\u003c/em\u003e direction, and the filtered tool paths on both sides are called the edge area. The final shrunken tool path is called the internal area, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the Figure, the shaded area is the trajectory area (i.e. internal area) where the main support head moves, while the areas that the main support head cannot reach are the tip area, edge area, and root area. In the internal area, the tool and the support device perform symmetrical motion, that is, the two move synchronously, and the tool position points correspond one-to-one with the support points (the point on the main support head motion trajectory is referred to as the support point in the text). In the edge area, the tool processes normally. To prevent interference between the support device and the blade, the support device stops waiting. For the tip and root areas, the tool and support device perform asymmetric motion. Finally, to maximize the internal area, the shrinkage distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D_{{\\text{shr}}}}\\)\u003c/span\u003e\u003c/span\u003e should be reasonably selected while ensuring safe machining, so that the tool and support device can perform symmetrical motion as much as possible.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Support motion trajectory planning\u003c/h2\u003e \u003cp\u003eTo achieve stable support, it is necessary to obtain a reasonable support attitude based on the tool position point. The attitude scheme adopted in this study is to use the normal vector on the surface of the blade as the support axis and pass through the tool position point, which can counteract the local torque to the maximum extent generated by the cutting tool during processing, providing effective and stable support for the blade, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eDynamic support machining needs to be described through six key points: the machining edge point of the \u003cem\u003en\u003c/em\u003eth trajectory \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{{\\text{edge}}}}^{n}\\)\u003c/span\u003e\u003c/span\u003e, the supporting edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{{{\\text{edge}}}}^{n}\\)\u003c/span\u003e\u003c/span\u003e and its corresponding machining point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{cor}}^{n}\\)\u003c/span\u003e\u003c/span\u003e of the \u003cem\u003en\u003c/em\u003eth trajectory, the machining edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{{\\text{edge}}}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e of the (\u003cem\u003en\u0026thinsp;+\u003c/em\u003e\u0026thinsp;1)th trajectory, the supporting edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{{{\\text{edge}}}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e and its corresponding machining point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{cor}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e of the (\u003cem\u003en\u0026thinsp;+\u003c/em\u003e\u0026thinsp;1)th trajectory. The motion process of the cutting tool and support device is designed and planned as follows:\u003c/p\u003e \u003cp\u003e(1) The cutting tool enters from the top of the blade tip area and performs reciprocating machining according to the established tool paths. At the same time, the support device moves back and forth on the first trajectory to support. Since the main support and the cutting tool do not correspond one-to-one according to the above attitude scheme, asymmetric motion is performed, and the cutting force is counteracted by auxiliary support currently. When the tool reaches the path corresponding to the first support trajectory, the two begin to move symmetrically.\u003c/p\u003e \u003cp\u003e(2) Fig.\u0026nbsp;8 (a) shows the process of the tool and support device moving from the internal area to the edge area. When the support device reaches the edge point on the \u003cem\u003en\u003c/em\u003eth trajectory \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{{{\\text{edge}}}}^{n}\\)\u003c/span\u003e\u003c/span\u003e, the tool reaches the corresponding point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{cor}}^{n}\\)\u003c/span\u003e\u003c/span\u003e, and the support device stops waiting. When the tool reaches the edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{{\\text{edge}}}}^{n}\\)\u003c/span\u003e\u003c/span\u003e, the two transition synchronously to the (\u003cem\u003en\u0026thinsp;+\u003c/em\u003e\u0026thinsp;1)th trajectory.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(a) Moving from the internal area to the edge area\u003c/p\u003e \u003cp\u003e(3) Fig.\u0026nbsp;8 (b) shows the process of the tool and support device moving from the edge area to the internal area. The support device stops waiting at the edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{{{\\text{edge}}}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e on the (\u003cem\u003en\u0026thinsp;+\u003c/em\u003e\u0026thinsp;1)th trajectory, and the tool starts to process from the edge point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{{\\text{edge}}}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e. When the tool reaches the point of processing \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{{cor}}^{{n+1}}\\)\u003c/span\u003e\u003c/span\u003e, the tool and support device begin to move symmetrically, and the subsequent motion trajectory is the same.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(b) Moving from the edge area to the internal area\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;8 The motion process of dynamic support machining\u003c/p\u003e \u003cp\u003e(4) After the support device moves to the last trajectory, it will move back and forth to support the workpiece, while the tool is processing normally. Currently, auxiliary support is used to counteract the cutting force until the processing of the root area is completed. Finally, the tool and support device exit synchronously and return to zero point.\u003c/p\u003e \u003cp\u003eIn most areas, the tool and support device maintain symmetrical motion, and the lateral and radial distances between the two are very small, which can reduce torsional and overturning moments. In addition, throughout the entire machining process, the support device always adheres to the surface of the blade, providing a full support effect for the blade and effectively suppressing the vibration caused by tool machining.\u003c/p\u003e \u003c/div\u003e"},{"header":"4 The post-processing of dynamic support machining","content":"\u003cp\u003eIn CNC programming, the process of converting tool position data generated by CAM into the final machining program that can be read by the CNC machine tool according to the machine tool structure and its instruction format is called post-processing [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e]. Similarly, the original tool position data of the propeller blade needs to be processed by corresponding post-processing programs before it can be converted into dynamic support machining data.\u003c/p\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Preprocessing of tool position point cloud\u003c/h2\u003e \u003cp\u003eAccording to the theory of blade path generation mentioned above, with the center of the propeller axis as the center, a fixed radius increment is set to create a series of concentric circles, which are projected onto the surface of the blade to obtain the contour curve. It is used as a streamline for reciprocating machining of the blade path, and a double-sided blade path model is obtained. The original tool position file includes instruction information, machining point pose parameters, identifiers for different types of tool paths, etc. Different instructions guide different machining information. Therefore, by identifying the corresponding instructions and identifiers, the extraction of tool position information can be completed and become the machining points cloud. The tool path partition identifier is used to implement partition management of tool position information. Extracting the tool path of the final process of surface finishing into point cloud information, with one side as the processing points cloud \u003cem\u003e{P}\u003c/em\u003e and the other side as the original support points cloud \u003cem\u003e{S}\u003c/em\u003e. According to the area division method in section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e, by setting an appropriate shrinkage distance, completing shrinkage processing on the machining points cloud, and dividing the blade surface into four areas. Temporarily discarding the machining points in the blade tip area, root area, and edge area, and retaining the machining points in the internal area, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e9\u003c/span\u003e, in preparation for obtaining support points in the following text.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e4.2 The method of support point acquisition\u003c/h2\u003e \u003cp\u003eThe support scheme adopted in this study is that the axis of the support device passes through the tool position point, and based on this scheme, the support point is solved iteratively through the ergodic method. Firstly, after completing the semi-finishing machining of the propeller blade, obtain the 3D points cloud information of the machining surface and support surface. Fitting the surface with the original support points cloud \u003cem\u003e{S}\u003c/em\u003e, and then selecting support points within a certain area based on the tool position points. If none of them meet the distance condition, use the ergodic method to select different \u003cem\u003er\u003c/em\u003e values to solve for the support points \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e and support vectors \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e. The specific process is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(1) A fifth-degree polynomial is used to fit the surface S according to the 3D support points cloud. The surface equation is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z_S}=f({x_S},{y_S})\\)\u003c/span\u003e\u003c/span\u003e, and the surface fitting result is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e11\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(2) Based on the known tool position point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P({x_P},{y_P},{z_P})\\)\u003c/span\u003e\u003c/span\u003e, projecting along the \u003cem\u003eZ\u003c/em\u003e-axis onto the support surface, the reference point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(B({x_B},{y_B},{z_B})\\)\u003c/span\u003e\u003c/span\u003e of the support surface is obtained. The \u003cem\u003eX\u003c/em\u003e-axis and \u003cem\u003eY\u003c/em\u003e-axis coordinate values of this point are then incremented and decremented in a fixed increment value to obtain a rectangular boundary\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(D:\\{ {x_B} - \\Delta x \\leqslant x \\leqslant {x_B}+\\Delta x,{y_B} - \\Delta y \\leqslant y \\leqslant {y_B}+\\Delta y\\}\\)\u003c/span\u003e\u003c/span\u003e. The support points within the boundary form a region \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S_T}\\)\u003c/span\u003e\u003c/span\u003e, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e12\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(3) Solving the normal of \u003cem\u003eS\u003c/em\u003e at the support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{T}^{i}({x_T},{y_T},{z_T})(i=1,2,...,n)\\)\u003c/span\u003e\u003c/span\u003e:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$n_{T}^{i}=\\frac{{[f({x_S},{y_S})/\\partial {x_S}{|_{{x_S}=x_{S}^{i}}},f({x_S},{y_S})/\\partial {y_S}{|_{{y_S}=y_{S}^{i}}}, - 1]}}{{||[f({x_S},{y_S})/\\partial {x_S}{|_{{x_S}=x_{S}^{i}}},f({x_S},{y_S})/\\partial {y_S}{|_{{y_S}=y_{S}^{i}}}, - 1]||}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z_S}=f({x_S},{y_S})\\)\u003c/span\u003e\u003c/span\u003e represents the surface equation of the support surface \u003cem\u003eS\u003c/em\u003e, and then calculating the spatial distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e between the normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e and the tool position point \u003cem\u003eP\u003c/em\u003e:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the formula, \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e represents the vector pointing from the support point to the tool position point. Since the normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e is the unit vector, the \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e result is the vertical projection of \u003cspan class=\"InlineEquation\"\u003e\u003c/span\u003e on the normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e. According to the Pythagorean theorem, the distance can be obtained.\u003c/p\u003e \u003cp\u003e(4) If the minimum value of all distances obtained is smaller than the set threshold \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\hbox{min} \\{ d_{T}^{i}\\} \u0026lt;e\\)\u003c/span\u003e\u003c/span\u003e, then the corresponding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e is taken as the support point corresponding to the tool position point \u003cem\u003eP\u003c/em\u003e, and the corresponding normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e is the support direction of that point. If \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\hbox{min} \\{ d_{T}^{i}\\} \\geqslant e\\)\u003c/span\u003e\u003c/span\u003e, taking the support point corresponding to the minimum distance and calling it \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S_e}({x_e},{y_e},{z_e})\\)\u003c/span\u003e\u003c/span\u003e, then obtaining the new support point and support vector through ergodic method (i.e. numerical solution).\u003c/p\u003e \u003cp\u003e(5) Taking \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(({x_e},{y_e},{z_e})\\)\u003c/span\u003e\u003c/span\u003e as the center of the circle and creating a circle based on the set initial radius \u003cem\u003er\u003c/em\u003e, taking points \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((x_{e}^{j},y_{e}^{j},{z_e})\\)\u003c/span\u003e\u003c/span\u003e on the circle at different angles \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\theta _j}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x_{e}^{j}={x_e}+r\\cdot \\cos {\\theta _j},y_{e}^{j}={y_e}+r\\cdot \\cos {\\theta _j}\\)\u003c/span\u003e\u003c/span\u003e, and projecting them along the \u003cem\u003eZ\u003c/em\u003e-axis onto the surface \u003cem\u003eS\u003c/em\u003e to obtain the projection points \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}(x_{e}^{j},y_{e}^{j},z_{e}^{j})\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e(6) Solving the normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e of \u003cem\u003eS\u003c/em\u003e at the projection point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}(x_{e}^{j},y_{e}^{j},z_{e}^{j})\\)\u003c/span\u003e\u003c/span\u003e and calculating the spatial distance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{e}^{i}\\)\u003c/span\u003e\u003c/span\u003e between the normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e and the tool position point \u003cem\u003eP\u003c/em\u003e. The calculation method is the same as (4).\u003c/p\u003e \u003cp\u003e(7) If the distance is less than the threshold \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{e}^{j}\u0026lt;e\\)\u003c/span\u003e\u003c/span\u003e, take the projection point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e as the support point corresponding to the tool position point \u003cem\u003eP\u003c/em\u003e, and the corresponding normal \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{e}^{i}\\)\u003c/span\u003e\u003c/span\u003e is the support direction of that point. If \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\hbox{min} \\{ d_{T}^{i}\\} \\geqslant e\\)\u003c/span\u003e\u003c/span\u003e, increasing the angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e by a fixed angle increment \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Delta \\theta\\)\u003c/span\u003e\u003c/span\u003e until \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(2\\pi\\)\u003c/span\u003e\u003c/span\u003e and taking the point on the circle. Repeating steps (5)\u0026ndash;(6). If the threshold condition is not met, increase the radius uniformly in a fixed radius increment \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\Delta r\\)\u003c/span\u003e\u003c/span\u003e and repeat the above process until the condition is satisfied, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e13\u003c/span\u003e. When the radius \u003cem\u003er\u003c/em\u003e increases to a certain value, further increase will enter a meaningless cycle, so the maximum radius \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r_{\\hbox{max} }}\\)\u003c/span\u003e\u003c/span\u003e needs to be set. If \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(r\u0026gt;{r_{\\hbox{max} }}\\)\u003c/span\u003e\u003c/span\u003e, taking the support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e and support vector \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n_{T}^{i}\\)\u003c/span\u003e\u003c/span\u003e in step (4).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe above can serve as the process for obtaining the support point corresponding to the first tool position point. Repeating the above process to obtain the remaining support points one by one. The difference is that starting from the second point, in process (2), the previous support point can be directly used as the reference point, without the need for projection processing. The final support points cloud obtained through the ergodic method is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig13\" class=\"InternalRef\"\u003e14\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eDuring the machining process, the axis of the cutting tool and the support device are both perpendicular to the surface of the blade, and the support axis passes through the tool position point, which can minimize the excitation caused by the cutting tool machining and the local torque generated by the cutting tool and support, providing stable dynamic support for the blade.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Support point interpolation and support vector smoothing processing\u003c/h2\u003e \u003cp\u003eAs shown in Figs.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e and \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the original machining point cloud and the original support point cloud both move following the surface reciprocating tool path along the \u003cem\u003eY\u003c/em\u003e-axis. For each trajectory, the spacing between adjacent points is similar, indicating good continuity and regularity. However, most of the support points obtained by the ergodic method have obvious \u0026ldquo;breakpoints\u0026rdquo; on their trajectories, mainly because this method is an approximate numerical solution, unlike analytical methods that can accurately solve problems. Through debugging, it is known that when looping to the \u003cem\u003ej\u003c/em\u003eth point of the \u003cem\u003en\u003c/em\u003eth trajectory, the ergodic method can be used to obtain the new support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\{ {S_e}\\} _{n}^{j}\\)\u003c/span\u003e\u003c/span\u003e corresponding to the machining point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{n}^{j}\\)\u003c/span\u003e\u003c/span\u003e near the original support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{n}^{j}\\)\u003c/span\u003e\u003c/span\u003e. When looping to the next machining point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{n}^{{j+1}}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{n}^{{j+1}}\\)\u003c/span\u003e\u003c/span\u003e nearby points do not meet the distance threshold condition, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{n}^{{j+2}}\\)\u003c/span\u003e\u003c/span\u003e nearby points just meet the condition. So, there is a \u0026ldquo;gap\u0026rdquo; between the support points corresponding to the adjacent two machining points. To ensure a continuous and smooth support trajectory, interpolation needs to be performed at \u0026ldquo;gaps\u0026rdquo;, and the interpolation method is as follows.\u003c/p\u003e \u003cp\u003e(1) Dividing the support points into \u003cem\u003eM\u003c/em\u003e trajectories based on the \u003cem\u003eY\u003c/em\u003e-axis coordinate values of support points.\u003c/p\u003e \u003cp\u003e(2) Performing first-order differential processing on the \u003cem\u003eY\u003c/em\u003e-axis coordinates of each point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S_e}({x_e},{y_e},{z_e})\\)\u003c/span\u003e\u003c/span\u003e on the trajectory, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(dif_{n}^{{j - 1}}=\\{ {y_e}\\} _{n}^{j} - \\{ {y_e}\\} _{n}^{{j - 1}},1 \\leqslant n \\leqslant M,2 \\leqslant j \\leqslant N\\)\u003c/span\u003e\u003c/span\u003e, where \u003cem\u003eN\u003c/em\u003e is the total number of support points.\u003c/p\u003e \u003cp\u003e(3) For the \u003cem\u003en\u003c/em\u003eth trajectory, take the average of the difference results between its adjacent two trajectories and the current trajectory \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({m_n}=(di{f_{n - 1}}+di{f_n}+di{f_{n+1}})/3\\)\u003c/span\u003e\u003c/span\u003e, and calculate the distance threshold \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e_d}={m_n}\\cdot {K_d}\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K_d}\\)\u003c/span\u003e\u003c/span\u003e is the distance coefficient.\u003c/p\u003e \u003cp\u003e(4) By filtering the difference results that meet the condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(di{f_n} \\geqslant {e_d}\\)\u003c/span\u003e\u003c/span\u003e in the \u003cem\u003en\u003c/em\u003eth trajectory, it can be concluded that the midpoint between the support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{{j - 1}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e is the \u0026ldquo;gap\u0026rdquo;. Performing linear interpolation at this position. Calculating the midpoint of the two support points \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{m}^{j}(x_{m}^{j},y_{m}^{j},z_{m}^{j})\\)\u003c/span\u003e\u003c/span\u003e, then projecting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{m}^{j}\\)\u003c/span\u003e\u003c/span\u003e along the \u003cem\u003eZ\u003c/em\u003e-axis onto the support surface \u003cem\u003eS\u003c/em\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z_{{\\operatorname{int} er}}^{j}=f(x_{m}^{j},y_{m}^{j})\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((x_{m}^{j},x_{m}^{j},z_{{\\operatorname{int} er}}^{j})\\)\u003c/span\u003e\u003c/span\u003e being the interpolation point. Finally, to ensure that the amount of points cloud on the machining side is consistent and one-to-one correspondence with the support side, the machining side needs to synchronously interpolate between the machining point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{e}^{{j - 1}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(P_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e corresponding to the support point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{{j - 1}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(S_{e}^{j}\\)\u003c/span\u003e\u003c/span\u003e, using the same method as (4). Repeating processes (3)\u0026ndash;(4) to complete the interpolation processing of all \u0026ldquo;breakpoints\u0026rdquo;.\u003c/p\u003e \u003cp\u003eIf the support vectors are not smooth and continuous, the swinging angle of the support axis may change sharply during actual machining, which causes the parallel mechanism of the machine tool to shake and exacerbate the vibration of the blade, especially for propellers with large curvature. Therefore, to ensure the smooth motion of the machine tool, it is necessary to smooth the support vectors. For research on vector smoothing, most optimization methods currently focus on reducing the angles between adjacent tool axis vectors [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. This study draws inspiration from this method, aiming to smooth the angles between adjacent support axis vectors as much as possible to make the support vectors smooth and continuous. Firstly, calculating the angles between adjacent support vectors on each trajectory:\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${\\theta _i}=\\arccos \\frac{{{V_i}\\cdot {V_{i+1}}}}{{|{V_i}|\\cdot |{V_{i+1}}|}}(1 \\leqslant i \\leqslant N - 1)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the equation, where \u003cem\u003eN\u003c/em\u003e is the total number of support vectors, the angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\theta _i}\\)\u003c/span\u003e\u003c/span\u003e reflects the smoothness of the support vectors. As shown in Fig.\u0026nbsp;15(a), it illustrates the angles between adjacent support vectors on the \u003cem\u003en\u003c/em\u003eth and (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;1) th support trajectory before smoothing. The curve generally shows an upward or downward trend, which is due to the asymmetric nature of the propeller blades. The curvature at both ends of a trajectory is different, which leads to varying degrees of vector change, resulting in this trend. However, the curve exhibits abrupt changes and discontinuities, where the angle undergoes sharp variations, causing vector discontinuity. Therefore, smoothing processing is needed.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(a) Before smoothing (b) After smoothing\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;15 Angles between adjacent support vectors before and after smoothing\u003c/p\u003e \u003cp\u003eAs indicated by Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), smoothing the support vectors implies smoothing the angles between vectors. Taking the \u003cem\u003en\u003c/em\u003eth and (\u003cem\u003en\u003c/em\u003e\u0026thinsp;+\u0026thinsp;1) th support trajectory and respectively using the first support point as the origin to establish a three-dimensional Cartesian coordinate system. Converting each support vector \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V_i}({I_i},{J_i},{K_i})\\)\u003c/span\u003e\u003c/span\u003e into angles with the coordinate axes. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig14\" class=\"InternalRef\"\u003e16\u003c/span\u003e, the relationship between vector components and angles is given by the following:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\alpha _i}=\\arcsin (\\frac{{{J_i}}}{{\\sin {\\beta _i}}})=\\arccos (\\frac{{{I_i}}}{{\\sin {\\beta _i}}}),{\\beta _i}=\\arccos ({K_i})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe obtained angles \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha _i},{\\beta _i}\\)\u003c/span\u003e\u003c/span\u003e are fitted with a sixth-degree polynomial to obtain curve equations \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({F_\\alpha },{F_\\beta }\\)\u003c/span\u003e\u003c/span\u003e, where the fitted results for the angle \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha _i}\\)\u003c/span\u003e\u003c/span\u003e are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig15\" class=\"InternalRef\"\u003e17\u003c/span\u003e. Calculating the points on the fitted curve and using them as the angles after smoothing \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha _{f}^{i}={F_\\alpha }(i),\\beta _{f}^{i}={F_\\beta }(i)\\)\u003c/span\u003e\u003c/span\u003e. According to Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e), reverse-calculating the vector components after smoothing.\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$I_{f}^{i}=\\cos \\alpha _{f}^{i}\\cdot \\sin \\beta _{f}^{i},J_{f}^{i}=\\sin \\alpha _{f}^{i}\\cdot \\sin {\\beta _i},K_{f}^{i}=\\cos \\beta _{f}^{i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe end \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(V_{f}^{I}(I_{f}^{i},J_{f}^{i},K_{f}^{i})\\)\u003c/span\u003e\u003c/span\u003e represents the vector after smoothing, and the angles between adjacent vectors after smoothing are shown in Fig.\u0026nbsp;15(b). Additionally, the vectors before smoothing are unit vectors, and subject to the constraint in Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e5\u003c/span\u003e), smoothing the angles \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha _i},{\\beta _i}\\)\u003c/span\u003e\u003c/span\u003e separately ensures that the vectors after smoothing remain unit vectors. Strictly speaking, the vectors after smoothing no longer satisfy the support scheme passing through the tool position points. However, the difference between the angles and the smoothed sixth-degree polynomial fitting results is very small. The impact on the overall support scheme can be negligible, and it can still achieve a good support result. Therefore, this smoothing method is feasible.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Calculation of processing parameters\u003c/h2\u003e \u003cp\u003eThe final step of the post-processing program is to calculate the machining parameters, which are the input files for the control system of the XYZ-3RPS hybrid machine tool prototype. These parameters include machining time, drive displacements of each axis, drive speed, and drive acceleration. Based on the inverse kinematic analysis theory of the XYZ-3RPS hybrid machine tool, the processed machining-side and support-side pose data are converted into the drive displacements of each axis of the machine tool through corresponding algorithms. Then, the machining time is reasonably allocated. Finally, based on the time-displacement sequence, cubic spline interpolation is performed to obtain cubic polynomial functions for each segment. The first and second derivatives are then calculated to obtain the required velocities and accelerations.\u003c/p\u003e \u003cp\u003eIn addition, both the machining-side and support-side trajectories consist of several streamlined trajectories. The connection between each streamlined trajectory is not smooth; instead, there are certain corners and forming turning points. Near these turning points, the direction of velocity and acceleration will change abruptly, thereby reducing the stability of the machine tool motion. Therefore, an improvement is made. Each turning point is used as a boundary to divide trajectories on both sides into different curve groups. Time-displacement interpolation is performed separately for each group, and the initial and final velocities of each group are set to zero.\u003c/p\u003e \u003cp\u003eBefore performing time-displacement interpolation, the machining points/support points are defined as nodes. At each node, machining time needs to be allocated, which depends on the drive displacements and maximum drive speeds of the 12 axes on the machining side and support side, as given in Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e):\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${t_n}=\\hbox{max} \\{ \\frac{{{{\\{ d_{m}^{P}\\} }_n} - {{\\{ d_{m}^{P}\\} }_{n - 1}}}}{{{v_{\\hbox{max} }}}},\\frac{{{{\\{ d_{m}^{S}\\} }_n} - {{\\{ d_{m}^{S}\\} }_{n - 1}}}}{{{v_{\\hbox{max} }}}}\\} +{t_{n - 1}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\{ d_{m}^{P}\\} _n}/{\\{ d_{m}^{S}\\} _n}\\)\u003c/span\u003e\u003c/span\u003e represents the drive displacement of the machining side/support side at the \u003cem\u003en\u003c/em\u003eth node; \u003cem\u003em\u003c/em\u003e is the number of drive axes, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m=1,...,6\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t_n}\\)\u003c/span\u003e\u003c/span\u003eis the machining time at the \u003cem\u003en\u003c/em\u003eth node, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n=2,3...\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t_1}=0\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v_{\\hbox{max} }}\\)\u003c/span\u003e\u003c/span\u003e is the maximum speed limit of the drive axes.\u003c/p\u003e \u003cp\u003eAfter obtaining the machining time, segment-wise cubic spline interpolation can be performed for each of the curve groups divided as described above. Taking one of the curve groups with \u003cem\u003ek\u003c/em\u003e nodes as an example and considering the drive displacement along the \u003cem\u003eX\u003c/em\u003e-axis on the machining side, the segment-wise cubic spline equation is as follows:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$x_{{}}^{P}(t)={A_i}+{B_i}(t - {t_i})+{C_i}{(t - {t_i})^2}+{D_i}{(t - {t_i})^3}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn the equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t \\in [{t_i},{t_{i+1}}]\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(i=1,...,k - 1\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eBecause the first and second derivatives of each adjacent pair of segments' equations are continuous at the nodes [\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e28\u003c/span\u003e]. The following equation can be obtained:\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${A_i}=x_{{}}^{P}({t_i})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${B_i}+{C_i}{T_i}+{D_i}{T_i}^{2}={V_i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$${B_i}+2{C_i}{T_i}+3{D_i}{T_i}^{2}={B_{i+1}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${C_{i - 1}}{T_{i - 1}}+3{D_{i - 1}}{T_{i - 1}}={C_i}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T_i}={t_{i+1}} - {t_i}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V_i}=(x_{{}}^{P}({t_{i+1}}) - x_{{}}^{P}({t_i}))/{T_i}\\)\u003c/span\u003e\u003c/span\u003e. After simplification, it can be obtained as follows:\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\frac{{{B_{i - 1}}}}{{{T_{i - 1}}}}+2(\\frac{1}{{{T_{i - 1}}}}+\\frac{1}{{{T_i}}}){B_i}+\\frac{{{B_{i+1}}}}{{{T_i}}}=3(\\frac{{{V_{i - 1}}}}{{{T_{i - 1}}}}+\\frac{{{V_i}}}{{{T_i}}})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$${C_i}=\\frac{1}{{{T_i}}}(3{V_i} - {B_{i+1}} - 2{B_i})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$${D_i}=\\frac{1}{{{T_i}^{2}}}({B_i}+{B_{i+1}} - 2{V_i})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn Eq.\u0026nbsp;(\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e), there are \u003cem\u003ek\u003c/em\u003e\u0026thinsp;\u0026minus;\u0026thinsp;1 unknown parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B_1},{B_2},...,{B_{k - 1}}\\)\u003c/span\u003e\u003c/span\u003e, but only \u003cem\u003ek\u003c/em\u003e\u0026thinsp;\u0026minus;\u0026thinsp;3 equations. Therefore, two additional boundary conditions are needed to solve all parameters, specifically, the first derivative values at the initial and final two points of this curve group. Here, we set the first derivatives at the initial and final two nodes equal to 0, i.e., and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B_1},{B_{k - 1}}\\)\u003c/span\u003e\u003c/span\u003e are both equal to 0. With this, all variable parameters in equations (\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e)\u0026ndash;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e) can be solved, and by substituting them into Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e), polynomial equations for each segment can be obtained. Successively solving for their first and second derivatives will yield polynomial equations for velocity and acceleration:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$${v^P}(t)={B_i}+2{C_i}(t - {t_i})+3{D_i}{(t - {t_i})^2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$${a^P}(t)=2{C_i}+6{D_i}(t - {t_i})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eBy substituting the time for each node into equations (\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e15\u003c/span\u003e) and (\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e16\u003c/span\u003e), the velocity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v^P}({t_{i+1}})\\)\u003c/span\u003e\u003c/span\u003e and acceleration \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a^P}({t_{i+1}})\\)\u003c/span\u003e\u003c/span\u003e for each node can be solved. Finally, the time, displacement, velocity, and acceleration sequences are integrated into the input file for the hybrid machine tool.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Simulation verification and experimental verification","content":"\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e5.1 ADAMS simulation verification\u003c/h2\u003e \u003cp\u003eThe original tool position file is transformed into machining data through the above preprocessing, interpolation, calculation of machining parameters, etc., and a portion of it is extracted. Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e shows the time, series drive displacement, and parallel drive displacement for the support side. Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, on the other hand, presents the time, series drive velocity, and parallel drive velocity for the support side.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePartial time-displacement data for the support side\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t(ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{x}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{y}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{z}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{1}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{2}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(d_{3}^{S}(mm)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110459\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-36.2743\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e21.9991\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e85.5620\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.0331\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-22.8370\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e23.3242\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110565\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-36.4132\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e23.0541\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e85.1803\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.9672\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-22.7910\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e23.3310\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110671\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-36.5545\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e24.1125\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e84.7980\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.9006\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-22.7439\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e23.3369\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110778\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-36.6982\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e25.1738\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e84.4153\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.8331\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-22.6955\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e23.3421\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1603588\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-131.2947\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e99.9463\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e61.9526\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-1.3161\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-12.2766\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e14.6180\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1603799\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-130.8497\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e97.9564\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e62.3266\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-1.1369\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-12.5214\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e14.7122\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1604008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-130.4134\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e96.0902\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e62.6963\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.9661\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-12.7534\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e14.8014\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1604216\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-129.9568\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e94.2166\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e63.0697\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.7855\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-12.9920\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e14.8887\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePartial time-velocity data for the support side\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e\u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t(ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{x}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{y}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{z}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{1}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{2}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v_{3}^{S}(mm/ms)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110459\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0001\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0099\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-0.0035\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.0003\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.0000\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0002\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110565\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.0017\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0100\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-0.0036\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.0007\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.0005\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0000\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110671\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.0014\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0099\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-0.0036\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.0006\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.0004\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0001\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e110778\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-0.0008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.0099\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e-0.0035\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e-0.0006\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.0004\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0000\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e……\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1603588\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0022\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.0091\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0017\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0009\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-0.0012\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0005\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1603799\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0021\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.0093\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0018\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-0.0011\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0004\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1604008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0022\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.0089\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0018\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-0.0011\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0004\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1604216\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0021\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e-0.0090\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.0018\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0008\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e-0.0011\u003c/p\u003e \u003c/td\u003e\u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e0.0004\u003c/p\u003e \u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003eTo validate the correctness and feasibility of the toolpath and support trajectory, ensuring the safe machining of the hybrid machine tool, all machining data is imported into the motion model of the hybrid machine tool created in the software ADAMS in proportion. The motion simulation process is shown in Fig.\u0026nbsp;18.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(a) Motion simulation model of dynamic support machining\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(b) Toolpath on the machining side and motion trajectory on the support side\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;18 Motion simulation dynamic support machining\u003c/p\u003e \u003cp\u003eFinally, multiple sets of machining data are obtained from different original tool position data, and each set undergoes motion simulation verification. Neither the toolpath nor the support trajectory shows any abnormalities and is consistent with the planning, indicating that the processing procedure and results are correct. Further experimental analysis can be carried out using the prototype of the hybrid machine tool.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e5.2 Experimental validation\u003c/h2\u003e \u003cp\u003eThe experiment used the XYZ-3RPS hybrid machine tool described above, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig16\" class=\"InternalRef\"\u003e19\u003c/span\u003e. The cutting tool and support device are installed at the end center of the parallel mechanism, with the tool using a 3mm radius ball-end mill. The workpiece is a nylon material with dimensions of 200mm×50mm×350mm, vertically clamped to the worktable through six clamping support brackets. The machining steps include rough machining, semi-finishing, and finishing. Rough machining shapes the workpiece into the basic form of a propeller blade, then semi-finishing further reduces the thickness of the blade, and finishing is done using dynamic flexible support machining.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAdditionally, it should be noted how the machining side/support side engages and disengages. Taking the support side as an example, the support device moves from the zero point to the first support point with the attitude (i.e., normal) of the first support point, then retracts a certain safe distance from the last support point before returning to the zero point. The machining side follows a similar process. This approach helps to avoid interference collisions between the tool and support device with the blade during the tool engagement and disengagement processes. Finally, the integrated machining data is input into the control system of the machine tool to start the machining experiment.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;20 depicts the process of dynamic flexible support machining. In the internal area, the cutting tool and the support device move synchronously, and their attitudes conform to the expected attitude scheme. In the edge area, the cutting tool performs its machining task while the support device waits, then both the cutting tool and the support device move to the next trajectory at the same time, which is consistent with the planned trajectory. The above results indicate the accuracy of the post-processing algorithm proposed in this study, and the suggested support motion trajectory is proven to be correct and effective.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(a) Internal area machining stage\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e(b) Edge area machining stage\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;20 Dynamic flexible support machining experiment\u003c/p\u003e \u003c/div\u003e"},{"header":"Conclusion","content":"\u003cp\u003eTo reduce the vibration during the machining process of the propeller, a dynamic flexible support machining scheme is proposed in this paper, drawing inspiration from mirror machining technology. To counteract the cutting forces generated during machining as much as possible, a multi-point flexible support device is designed and installed at the end of the machine tool. Based on the structure of the blade and the support device, the surface of the propeller blade is divided into four areas, each employing a different motion method, and the support motion trajectory is planned accordingly. The paper introduces a post-processing method for dynamic support machining, realizing the conversion from the original CAM tool position file to the machining file recognized by the machine tool. Finally, the correctness and feasibility of the planned support motion trajectory and algorithm are verified through simulation and machining experiments.\u003c/p\u003e\u003cp\u003eThroughout the entire machining process, the support device is in constant contact with the blade, providing support force to the blade that varies with the milling force. However, the precise calculation of the support force magnitude is not performed, and while the support effect can be improved, there is still room for improvement in this scheme. Therefore, in future work, it is necessary to establish a dynamic model for the interaction between the tool, blade, and support device, analyze the mapping relationship between the support force and milling force, and thereby precisely control the support force to further reduce machining deformation and vibration.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eAcknowledgment\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Natural Science Foundation of China (Grant number [51975157]). Author Rui Wang has received this research support.\u003c/p\u003e\n\u003cp\u003eCompeting Interests: The authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eJi, Y.Y.; Tian, G.Z.; Zhou, H.G. 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Ph.D. Thesis, Northwestern Polytechnical University, Xi an, China, 2003.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"propeller, trajectory planning, dynamic flexible support, ergodic method, post-processing","lastPublishedDoi":"10.21203/rs.3.rs-4016972/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4016972/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAt present, most of the propeller machining adopts single-sided machining, and its machining vibration and deformation seriously affect the machining accuracy. To reduce the machining vibration, a dynamic flexible support machining method is proposed, i.e., while the tool is machining, the multi-point flexible support device supports the blade and counteracts the milling force to suppress the vibration and deformation. Due to the complex shape of the blade and the special structure of the support device, the blade is divided into different areas, and a support motion trajectory combining symmetric and asymmetric motions is planned, and then a set of post-processing systems is introduced. After obtaining the tool position points, the support points are solved cyclically by the ergodic method. Subsequently, the support points are interpolated, and the vectors are smoothed to obtain smooth and continuous support trajectories. Finally, the machining parameters are calculated, and the machining data applicable to the XYZ-3RPS hybrid machine are integrated. The feasibility of the proposed support trajectory and post-processing algorithm was ultimately demonstrated through practical machining experiments.\u003c/p\u003e","manuscriptTitle":"Research on the Dynamic Flexible Support Machining Method for Propeller Blades","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-13 06:29:31","doi":"10.21203/rs.3.rs-4016972/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2024-08-21T00:02:30+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2024-04-17T12:32:12+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-03-09T02:52:06+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-03-07T05:51:44+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2024-03-05T07:09:44+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"b8f7720a-8c81-4ef7-8d49-f3cb55536d18","owner":[],"postedDate":"March 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-07-28T16:05:33+00:00","versionOfRecord":{"articleIdentity":"rs-4016972","link":"https://doi.org/10.1007/s00170-025-16136-z","journal":{"identity":"the-international-journal-of-advanced-manufacturing-technology","isVorOnly":false,"title":"The International Journal of Advanced Manufacturing Technology"},"publishedOn":"2025-07-26 15:57:02","publishedOnDateReadable":"July 26th, 2025"},"versionCreatedAt":"2024-03-13 06:29:31","video":"","vorDoi":"10.1007/s00170-025-16136-z","vorDoiUrl":"https://doi.org/10.1007/s00170-025-16136-z","workflowStages":[]},"version":"v1","identity":"rs-4016972","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4016972","identity":"rs-4016972","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}