Dynamic characteristic analysis method for tool nose point in milling process of five-axis hybrid machine tool

Abstract

The invention discloses a tool nose point dynamic characteristic analysis method in the milling process of a five-axis hybrid machine tool, and belongs to the technical field of five-axis hybrid machine tools. The method comprises the steps that firstly, system dynamics modeling is conducted on the five-axis hybrid machine tool, and in the milling process of the five-axis hybrid machine tool, a dynamics equation of a cutting system composed of a cutter and the five-axis hybrid machine tool is established; 2, setting the kinetic parameters according to the situation that the kinetic parameters can change along with the pose of the hybrid machine; 3, setting a main shaft coordinate system, a tool coordinate system and a workpiece coordinate system to calculate three-way dynamic milling force borne by the tool in the milling process; and 4, according to the three-way dynamic milling force, track influences caused by radial runout and milling vibration of the cutter are determined and combined into a real motion track of the cutter. According to the method, the rigid change of the five-axis hybrid machine tool is calculated, the corresponding structural dynamics model is established, and the dynamic characteristic estimation of the machine tool under different postures is realized.

Description

Technical Field

[0001] This invention relates to the technical field of five-axis hybrid machine tools, and in particular to a method for analyzing the dynamic characteristics of the tool tip during the milling process of a five-axis hybrid machine tool. Background Technology

[0002] Five-axis hybrid machine tools, due to their high flexibility and large workspace, have become essential equipment for manufacturing large and complex workpieces. Compared to traditional rigid machine tools, their multi-link structure, while maintaining wide coverage, often comes with lower and more variable system rigidity. Changes in rigidity directly affect the magnitude of cutting forces and vibration responses during milling, thus limiting further improvements in machining accuracy and surface quality. Therefore, a thorough understanding of the rigidity characteristics of five-axis hybrid machine tools in different machining positions and their coupling effect on milling behavior is crucial for optimizing cutting parameters, predicting system stability, and achieving high-precision machining.

[0003] Currently, common methods for predicting robot stiffness can be divided into two main categories: experimental fitting methods and structural dynamics methods. Experimental fitting methods obtain parameters through modal testing and then use regression, convolutional neural networks, transfer learning, and other techniques to construct predictive models. These methods are intuitive and easy to verify, but require a large number of experiments, are costly, and struggle to cover frequency response function prediction across the entire workspace. Structural dynamics methods rely on physical constitutive equations to construct mass, damping, and stiffness matrices and predict dynamic responses. Commonly used methods include the finite element method and the substructure synthesis method. The former integrates CAD / CAE to achieve high-fidelity simulation, but due to the need for frequent mesh re-drilling and high computational cost caused by attitude changes, it is mostly used for theoretical verification and local analysis. The latter divides the linkage into beam elements and uses order reduction processing to assemble the overall dynamic equations, significantly improving efficiency while maintaining accuracy. It has been successfully applied to the stiffness and low-order modal analysis of various parallel and five-axis hybrid machine tools.

[0004] Currently, research on the rigidity prediction of five-axis hybrid machine tool bodies is quite mature. However, in the milling process, in addition to the influence of system stiffness, five-axis hybrid machine tools also experience nonlinear dynamic regeneration effects, which can cause complex milling vibration problems. Predicting milling dynamic characteristics requires establishing a cutting dynamics model. However, current dynamic models do not consider the rigidity changes of the robot body; they are mostly treated as constants and obtained through experimental modal analysis, failing to accurately reflect milling behavior. Summary of the Invention

[0005] The purpose of this invention is to provide a method for analyzing the dynamic characteristics of the tool tip during milling in a five-axis hybrid machine tool. Considering the rigidity changes of the robot body, a structural dynamic model of a novel five-axis hybrid machine tool is established, and the natural frequency of the machine tool and its vibration response under a unit step load are analyzed, enabling the prediction of the dynamic characteristics of the machine tool under different postures. Based on the milling regeneration effect and the dynamic characteristics of the machine tool, a dynamic model of the milling system is established by combining workpiece material removal simulation, enabling the prediction of milling force, milling vibration and form and position errors in the machining of complex curved surfaces.

[0006] To achieve the above objectives, this invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool, comprising the following steps: Step 1: Perform system dynamics modeling on the five-axis hybrid machine tool. During the milling process of the five-axis hybrid machine tool, establish the dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool. Step 2: Tune the dynamic parameters to account for changes in the pose of the hybrid machine; Step 3: Set up the principal axis coordinate system Tool coordinate system and workpiece coordinate system To calculate the three-dimensional dynamic milling force on the tool during the milling process; Step 4: Based on the three-dimensional dynamic milling force, determine the trajectory effects caused by the radial runout of the tool and milling vibration, and incorporate them into the actual motion trajectory of the tool.

[0007] Preferably, in step one, the dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool are given as follows: ; In the above formula, The mass matrix of the cutting system is represented. This represents the damping matrix of the cutting system. Represents the stiffness matrix of the cutting system. Represented as t Vibration response of the cutting system at a given time. Represented as t The cutting force excitation of the cutting system at any given time. Represented as The first derivative, Represented as The second derivative of .

[0008] Preferably, the specific process for tuning the dynamic parameters in step two is as follows: Based on the structural characteristics of the five-axis hybrid machine tool, it is equivalent to a spatial mechanical system composed of variable cross-section beam entities and revolute joints. The following assumptions are made during the dynamic modeling of the machine tool structure: ① It is assumed that each substructure can be equivalent to a variable cross-section beam entity. This is achieved by dividing the beam entity into a series of variable cross-section spatial beam elements and using the finite element method to construct the mass and stiffness matrices of each substructure; ② It is assumed that the sliding components of each branch are fixedly connected to the machine tool, and the five-axis hybrid machine tool in different poses is considered as a series of transient structures; ③ It is assumed that there is an ideal assembly relationship between each variable cross-section beam entity, ignoring the effects of installation clearance, damping, and friction in the revolute joints. The dynamic parameters of the five-axis hybrid machine tool include the mass matrix. Damping matrix and stiffness matrix Due to the structural characteristics of five-axis hybrid machine tools, their dynamic parameters will change with the change of the machine's position and posture. Therefore, the dynamic parameters of the five-axis hybrid machine tool need to be tuned. The tuning process is as follows: The moving platform system of a five-axis hybrid machine tool includes an upper moving platform, a lower moving platform, a tool-spindle module, and several branches. Each branch All of them include the same three substructures: sliding component, intermediate link and end link. The dynamic platform system of the five-axis hybrid machine tool is subdivided into six substructures. Based on the beam element theory, each substructure is analyzed and the corresponding mass and stiffness matrix is ​​established. Based on the structural characteristics of the upper and lower moving platforms, the upper platform is divided into... Each unit node divides the lower platform into There are 12 unit nodes, each with 6 degrees of freedom. Therefore, the stiffness matrix and mass matrix of the moving platform have 1 / 2 dimensions. The dimensions of the stiffness matrix and mass matrix of the lowering platform are both [missing information]. ; In the tool-spindle module, the tool is fixed to the machine tool spindle by a spring collet; therefore, the tool and the machine tool spindle are considered as a single unit, and the module is divided into... Each element node has a mass and stiffness matrix of dimension 1. Among them, the node corresponding to the connection between the tool and the spindle is The node corresponding to the connection between the tool-spindle module and the upper moving platform is ; The sliding assembly comprises multiple parts, which are connected by bolts or other means. Therefore, it is considered as a whole. Based on the structural characteristics of the sliding assembly, it is divided into... Each element node has a mass and stiffness matrix of dimension 1. ; Based on the structural characteristics of the intermediate and end links, they are divided into... and There are 1 unit nodes, therefore, the dimensions of the mass and stiffness matrices of the intermediate link are both 1 / 2. The dimensions of the mass and stiffness matrices of the end link are both [missing information]. ; Based on the above analysis, the stiffness matrices of the upper moving platform, lower moving platform, tool-spindle module, intermediate connecting rod, and end connecting rod are obtained. , , , and It is uniformly represented as: ; In the formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure Submatrices of the stiffness matrix of each beam element, with their subscripts pp The index is used to calculate the position of the submatrix within the beam element stiffness matrix; Mass matrix of upper moving platform, lower moving platform, tool-spindle module, intermediate link and end link , , , and The unified representation is as follows: ; In the formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure A submatrix of the mass matrix of each beam element, with its subscripts pp The index is used to calculate the position of the submatrix within the beam element mass matrix; Due to the sliding component beam element at the node Since the connection is in parallel, it can be considered as a beam entity composed of a series of beam elements connected in series and parallel. According to the static equilibrium equations, its stiffness matrix is... Revised to: ; Mass matrix of sliding component Revised to: ; Based on the structural properties of the five-axis hybrid machine tool, the mass and stiffness matrices of each substructure are assembled into the mass and stiffness matrices of the machine tool using the same joint assembly method. Since the mass and stiffness matrices of each substructure are consistent in both structural form and assembly method, the assembly process of the stiffness matrix of the five-axis hybrid machine tool is as follows: In each branch In the middle, the sliding component is connected to the intermediate link, and the intermediate link is connected to the end link via revolute joints. and Connect and rotate the pair and The corresponding turning angle is and In order to assemble the stiffness matrix of the branch system in the same reference coordinate system, the stiffness matrix of the intermediate link is... From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the intermediate link The calculation is as follows: ; In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​used to transform the stiffness matrix of the end link. From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the connecting rods at their ends Represented as: ; In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​as follows: ; In the above formula, α y To bypass Y The angle of rotation of the axis should be replaced with a specific angle value during calculation; The sliding assembly, intermediate link, and end link are all connected by revolute joints, and their rotation axes are all in the branch coordinate system. of With axes parallel, the stiffness matrix is ​​respectively based on the joint assembly method. , and Equivalent transformation to , and Then each branch stiffness matrix Represented as: ; In the above formula, , and These are the stiffness matrices. , and The assembly matrix is ​​as follows: ; In the above formula, and These are the identity matrix and the zero matrix, respectively, and their subscripts are... and Represent the number of rows and columns of the matrix, respectively; the stiffness matrix of the branch. From the branch coordinate system Transform to the moving platform system coordinate system In Chinese, the formula is as follows: ; In the above formula, Indicates the coordinate system of the moving platform system The lower branch i stiffness matrix, Indicates the coordinate system of the moving platform system Relative to the machine tool static coordinate system The transformation matrix, Representing the coordinate system The angle of rotation about the z-axis, Indicates the roll angle. Indicates the roll angle. coordinate system Around The coordinate transformation matrix for axis rotation is as follows: ; In the above formula, α z To bypass z The angle of rotation of the axis should be replaced with a specific angle value during calculation; Due to the nodes on the upper platform Node with the tool-spindle module The two are connected in parallel, therefore their stiffness matrices are... Represented as: ; In the above formula, the superscript This means first dividing the matrix Middle node With nodes Swap the corresponding column vectors, and then swap the nodes. to The corresponding column vector is shifted to the node. to Location; and They are respectively and The assembly matrix is ​​as follows: ; stiffness matrix Transform from the coordinate system of the upper moving platform to the coordinate system of the moving platform system to obtain The formula is as follows: ; ; ; In the above formula, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The transformation matrix, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The corner, the node of the moving platform Nodes of the lower platform The connection between them is a rotary link, and the rotation axis is the moving platform system. The stiffness matrix of the axis is respectively based on the joint assembly method. and Equivalent transformation to and The stiffness matrix of the moving platform system Represented as: ; In the formula, and These are the stiffness matrices. and The assembly matrix is ​​as follows: ; In the formula, Stiffness matrix The number of nodes, and ; Each branch is connected via a rotating joint. Connected to the moving platform system, and rotating pair and The axis of rotation is the coordinate system of axis, Rotary joint and The axis of rotation is the coordinate system of The stiffness matrix of the moving platform system is determined based on the joint assembly method. Stiffness matrix of each branch , , and Equivalent transformation to , , , and Then the stiffness matrix of the machine tool express: ; In the formula, , , , and These are the stiffness matrices. , , , and The assembly matrix is ​​as follows: ; In the formula, Stiffness matrix of the moving platform system Dimension and ; Stiffness matrix of the branched system Dimension and ; Using the same method as the stiffness matrix, the mass matrix of the machine tool is obtained based on the joint assembly method. Due to the nodes in the sliding component and nodes All connections between the nodes and the machine tool are fixed, and their corresponding generalized displacements are all zero. Therefore, the nodes... and nodes In the mass and stiffness matrices, the corresponding rows and columns are eliminated during the solution of the motion differential equations. Since the mass and stiffness matrices of the beam elements are expanded to zero vectors during the assembly of the revolute joint mass and stiffness matrices, when a row or column in the machine tool's mass and stiffness matrix is ​​a zero vector, that row and column are eliminated, ultimately yielding the mass matrix of the five-axis hybrid machine tool. and stiffness matrix Considering the influence of damping in the machine tool, its damping matrix is ​​as follows: ; In the above formula, and It is a constant for calculation and is greater than 0.

[0009] Preferably, in step three, the three-dimensional dynamic milling force on the tool during the milling process is determined, and the specific process is as follows: At any moment during the milling process t The radial runout of the tool will cause the tool tip to move relative to a given machining path. shaft and Dynamic offset distance in the axial direction and The calculation formula is as follows: ; In the above formula, The eccentricity distance representing the radial runout of the tool. This represents the rotation angle of the tool runout; considering the tool's rotation around the workpiece coordinate system during five-axis milling... shaft and Shaft tilt angle and lean angle The actual coordinates of the milling tool's actual machining trajectory in the workpiece coordinate system are: , and The formula is as follows: ; ; ; In the above formula, and These represent the cutting tool winding. shaft and The rotation matrix of the axis; , and These represent the coordinates of the given machining trajectory in the workpiece coordinate system and in the tool coordinate system, respectively. In the middle, for the serial number is The height is The blade, any point on its blade coordinates , and The formula is as follows: ; In the above formula, Indicates the effective milling radius of the cutting tool. The blade hysteresis angle is expressed by the following formula: ; ; In the calculation formula, Indicates the tool radius. The formula for calculating the cutting edge hysteresis angle indicates the tool radius. N Indicates the number of teeth. Indicates the helix angle of the cutting tool; For the serial number is The blade has an axial height of point Considering the combined effects of tool radial runout and milling vibration, the tool's performance under these conditions is calculated. axis, shaft and The true coordinate position of the axis , and as follows: ; In the above formula, Let be the rotation matrix about the Z-axis; The tool rotation angle is related to the spindle speed. related; , and The three-dimensional milling vibrations are respectively at the tool tip point; To determine the instantaneous undeformed cutting thickness corresponding to the tool edge micro-element using milling geometry simulation, the workpiece is first divided into a series of workpiece micro-elements, with the discrete directions being... Axis and spacing settings When discrete interval The workpiece element is small enough that it can be considered a spatial cylinder with a constant lateral geometry along discrete axes, and thus can be considered as a series of boundary points. The resulting continuous and closed boundary curves, while the tool tilt angle exists during milling. and lean angle This results in an angle between the axial rotation plane of the tool and the workpiece micro-element plane. Therefore, projecting the axial rotation plane of the tool onto the workpiece micro-element plane simplifies the spatial contact problem between the tool and the workpiece into a planar contact problem. Then, the planar cutting thickness of the tool micro-element can be solved numerically. ; Therefore, for any moment in the five-axis milling process Based on the geometric relationship between the axial rotation plane of the cutting edge and the projection plane, the dynamic chip thickness of the cutting edge micro-element is determined. The formula is as follows: ; In the above formula, i For the first i One blade edge point; intersection point The intersection point is the axial section of the tool and the central axis of the tool. The intersection of the horizontal cutting plane of the tool and the central axis of the tool. Intersection to the intersection Based on the infinitesimal milling force model, the tangential force of the infinitesimal cutting edge element is calculated. radial force and axial force as follows: ; In the above formula, , , These are the shear force coefficients for tangential force, radial force, and axial force, respectively. , , These are the plowing shear force coefficients for tangential force, radial force, and axial force, respectively; , These represent the axial cutting width and the cutting arc length of the cutting edge micro-element, respectively. The milling forces acting on all cutting edge micro-elements are transformed into the workpiece coordinate system, and the three-dimensional dynamic milling forces acting on the tool are obtained through numerical integration. , and They are as follows: ; ; In the above formula, This represents the immersion angle of the blade element.

[0010] Preferably, in step four, the radial runout of the tool and the trajectory influence caused by milling vibration are determined based on the three-dimensional dynamic milling force, and then incorporated into the actual motion trajectory of the tool. The specific process is as follows: use The numerical discretization method is used to solve the dynamic equations of the cutting system to obtain the vibration response of the system, defining the initial state. and All are zero, so select an appropriate time increment. Sum of numerical integration parameters φ Calculate the constant of the integration process. , and as follows: ; Then any time The formula is as follows: ; ; After the solution is completed, extract the reference point at the end of the tool. The elastic displacement vector is used as the milling vibration response of the cutting system, through the tool in... axis, shaft and The true coordinate position of the axis , and This is superimposed onto the actual movement trajectory of the cutting tool.

[0011] Therefore, the present invention employs the above-mentioned method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool, which has the following advantages: (1) In this invention, considering the rigidity change of the robot body, a structural dynamic model of a new five-axis hybrid machine tool was established, and the natural frequency of the machine tool and its vibration response under a unit step load were analyzed, so as to realize the prediction of the dynamic characteristics of the machine tool under different postures. (2) In this invention, based on the milling regeneration effect and the dynamic characteristics of the machine tool, a dynamic model of the milling system was established by combining the workpiece material removal simulation, and the prediction of milling force, milling vibration and form and position error in complex surface machining was realized.

[0012] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0013] Figure 1 This is the overall flowchart of the present invention; Figure 2 This is a schematic diagram illustrating the division of the upper and lower moving platform beam units in this invention; Figure 3 This is a schematic diagram of the beam unit division of the tool-spindle module in this invention; Figure 4 This is a schematic diagram of the beam element division of the sliding component in this invention; Figure 5 This is a structural diagram showing the division of the intermediate connecting rod and end connecting rod beam units in this invention; Figure 6 This is a simplified kinematic diagram of the five-axis hybrid machine tool in this invention; Figure 7 This is a comparison of experimental simulations of the tool tip dynamic characteristics analysis method in the milling process of a five-axis hybrid machine tool according to the present invention, when the tool tip is in the X direction, with a side tilt angle of 0 degrees and a front tilt angle of 0 degrees. Figure 8This invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. The experimental simulation comparison diagram shows the tool tip with a tilt angle of 0 degrees and a forward tilt angle of 0 degrees in the Y direction. Figure 9 This invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. The experimental simulation comparison shows the tool tip angles of 0 degrees and 20 degrees in the Y direction. Figure 10 This invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. The experimental simulation comparison shows the tilt angle of 20° and the forward tilt angle of 0° when the tool tip is in the Y direction. Figure 11 This invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. The experimental simulation comparison diagram shows the tilt angle of 40 degrees and the forward tilt angle when the tool tip is in the X direction. Figure 12 This invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. The experimental simulation comparison diagram shows the tool tip in the X direction with a tilt angle of 40 degrees and a forward tilt angle of 0 degrees. Detailed Implementation

[0014] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Specific model specifications need to be selected and determined according to the actual specifications of the device, etc. The specific selection calculation method adopts existing technology in the art, and therefore will not be described in detail.

[0015] Example like Figure 1 As shown, this invention provides a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool, comprising the following steps: Step 1: Perform system dynamics modeling on the five-axis hybrid machine tool. During the milling process of the five-axis hybrid machine tool, establish the dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool. The dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool are given below: ; In the above formula, The mass matrix of the cutting system is represented. This represents the damping matrix of the cutting system. Represents the stiffness matrix of the cutting system. Represented as t Vibration response of the cutting system at a given time. Represented ast The cutting force excitation of the cutting system at any given time. Represented as The first derivative, Represented as The second derivative of .

[0016] Step 2: Tune the dynamic parameters to account for changes in the pose of the hybrid machine; Based on the structural characteristics of the five-axis hybrid machine tool, it is equivalent to a spatial mechanical system composed of variable cross-section beam entities and revolute joints. The following assumptions are made during the dynamic modeling of the machine tool structure: ① It is assumed that each substructure can be equivalent to a variable cross-section beam entity. This is achieved by dividing the beam entity into a series of variable cross-section spatial beam elements and using the finite element method to construct the mass and stiffness matrices of each substructure; ② It is assumed that the sliding components of each branch are fixedly connected to the machine tool, and the five-axis hybrid machine tool in different poses is considered as a series of transient structures; ③ It is assumed that there is an ideal assembly relationship between each variable cross-section beam entity, ignoring the effects of installation clearance, damping, and friction in the revolute joints. The dynamic parameters of the five-axis hybrid machine tool include the mass matrix. Damping matrix and stiffness matrix Due to the structural characteristics of five-axis hybrid machine tools, their dynamic parameters will change with the change of the machine's position and posture. Therefore, the dynamic parameters of the five-axis hybrid machine tool need to be tuned. The tuning process is as follows: The moving platform system of a five-axis hybrid machine tool includes an upper moving platform, a lower moving platform, a tool-spindle module, and several branches. Each branch All of them include the same three substructures: sliding component, intermediate link, and end link. Therefore, the moving platform system of the five-axis hybrid machine tool is subdivided into six substructures, including upper moving platform, lower moving platform, tool-spindle module, sliding component, intermediate link, and end link. Based on beam element theory, each substructure is analyzed and the corresponding mass and stiffness matrix is ​​established. In this embodiment, the number of branches is set to 4. like Figure 2 Based on the structural characteristics of the upper and lower moving platforms, the upper platform is divided into... Each unit node divides the lower platform into There are 12 unit nodes, each with 6 degrees of freedom. Therefore, the stiffness matrix and mass matrix of the moving platform have 1 / 2 dimensions. The dimensions of the stiffness matrix and mass matrix of the lowering platform are both [missing information]. It is worth noting that the nodes on the platform... Nodes of the lower platform The connection between them is a rotary joint. .

[0017] like Figure 3 As shown, in the tool-spindle module, the tool is fixed to the machine tool spindle by a spring collet. Therefore, the tool and the machine tool spindle are considered as a whole, and the module is divided into... Each element node has a mass and stiffness matrix of dimension 1. Among them, the node corresponding to the connection between the tool and the spindle is The node corresponding to the connection between the tool-spindle module and the upper moving platform is ; The sliding assembly comprises multiple parts, which are connected by bolts or other means. Therefore, it is considered as a whole. Based on the structural characteristics of the sliding assembly, it is divided into... Each element node has a mass and stiffness matrix of dimension 1. ; specific examples Figure 4 As shown, node Node with intermediate link The two parts are revolute joints, corresponding to the revolute joints in a branched system. ;node With nodes These are boundary constraint points of the machine tool structure, and their corresponding generalized displacements are 0. Furthermore, at the nodes... At this point, the beam elements of the sliding component are connected in parallel.

[0018] Based on the structural characteristics of the intermediate and end links, they are divided into... and There are 1 unit nodes, therefore, the dimensions of the mass and stiffness matrices of the intermediate link are both 1 / 2. The dimensions of the mass and stiffness matrices of the end link are both [missing information]. For example Figure 5 As shown, the intermediate link node By rotating pair With end link node Connected; end link node With the upstream platform node and Downstream platform nodes and All connections are made via revolute joints, and each corresponds to a revolute joint. , , and .

[0019] Based on the above analysis, the stiffness matrices of the upper moving platform, lower moving platform, tool-spindle module, intermediate connecting rod, and end connecting rod are obtained. , , , and It is uniformly represented as: ; In the above formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure Submatrices of the stiffness matrix of each beam element, with their subscripts pp The index is used to calculate the position of the submatrix within the beam element stiffness matrix; Mass matrix of upper moving platform, lower moving platform, tool-spindle module, intermediate link and end link , , , and The unified representation is as follows: ; In the above formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure A submatrix of the mass matrix of each beam element, with its subscripts pp The index is used to calculate the position of the submatrix within the beam element mass matrix; Due to the sliding component beam element at the node Since the connection is in parallel, it can be considered as a beam entity composed of a series of beam elements connected in series and parallel. According to the static equilibrium equations, its stiffness matrix is... Revised to: ; Mass matrix of sliding component Revised to: ; Based on the structural properties of the five-axis hybrid machine tool, the mass and stiffness matrices of each substructure are assembled into the mass and stiffness matrices of the machine tool using the same joint assembly method. Since the mass and stiffness matrices of each substructure are consistent in both structural form and assembly method, the assembly process of the stiffness matrix of the five-axis hybrid machine tool is as follows: In each branch In the middle, the sliding component is connected to the intermediate link, and the intermediate link is connected to the end link via revolute joints. and Connect and rotate the pair and The corresponding turning angle is and Its joint coordinate system can be referenced. Figure 6 In order to assemble the stiffness matrix of the branch system in the same reference coordinate system, the stiffness matrix of the intermediate link is... From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the intermediate link The calculation is as follows: ; In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​used to transform the stiffness matrix of the end link. From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the connecting rods at their ends Represented as: ; In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​as follows: ; In the above formula, α y To bypass Y The angle of rotation of the axis should be replaced with a specific angle value during calculation; The sliding assembly, intermediate link, and end link are all connected by revolute joints, and their rotation axes are all in the branch coordinate system. of With axes parallel, the stiffness matrix is ​​respectively based on the joint assembly method. , and Equivalent transformation to , and Then each branch stiffness matrix Represented as: ; In the above formula, , and These are the stiffness matrices. , and The assembly matrix is ​​as follows: ; In the above formula, and These are the identity matrix and the zero matrix, respectively, and their subscripts are... and Represent the number of rows and columns of the matrix, respectively; the stiffness matrix of the branch. From the branch coordinate system Transform to the moving platform system coordinate system In Chinese, the formula is as follows: ; In the above formula, Indicates the coordinate system of the moving platform system The lower branch i stiffness matrix, Indicates the coordinate system of the moving platform system Relative to the machine tool static coordinate system The transformation matrix, Representing the coordinate system The angle of rotation about the z-axis, Indicates the roll angle. Indicates the roll angle. coordinate system Around The coordinate transformation matrix for axis rotation is as follows: ; In the above formula, α z To bypass z The angle of rotation of the axis should be replaced with a specific angle value during calculation; Due to the nodes on the upper platform Node with the tool-spindle module The two are connected in parallel, therefore their stiffness matrices are... Represented as: ; In the formula, superscript This means first dividing the matrix Middle node With nodes Swap the corresponding column vectors, and then swap the nodes. to The corresponding column vector is shifted to the node. to Location; and They are respectively and The assembly matrix is ​​as follows: ; stiffness matrix Transform from the coordinate system of the upper moving platform to the coordinate system of the moving platform system to obtain The formula is as follows: ; ; ; In the above formula, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The transformation matrix, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The corner, the node of the moving platform Nodes of the lower platform The connection between them is a rotary link, and the rotation axis is the moving platform system. The stiffness matrix of the axis is respectively based on the joint assembly method. and Equivalent transformation to and The stiffness matrix of the moving platform system Represented as: ; In the formula, and These are the stiffness matrices. and The assembly matrix is ​​as follows: ; In the formula, Stiffness matrix The number of nodes, and satisfying ; Each branch is connected via a rotating joint. Connected to the moving platform system, and rotating pair and The axis of rotation is the coordinate system of axis, Rotary joint and The axis of rotation is the coordinate system of The stiffness matrix of the moving platform system is determined based on the joint assembly method. Stiffness matrix of each branch , , and Equivalent transformation to , , , and Then the stiffness matrix of the machine tool express: ; In the formula, , , , and These are the stiffness matrices. , , , and The assembly matrix is ​​as follows: ; In the formula, Stiffness matrix of the moving platform system Dimension and ; Stiffness matrix of the branched system Dimension and ; Using the same method as the stiffness matrix, the mass matrix of the machine tool is obtained based on the joint assembly method. Due to the nodes in the sliding component and nodes All connections between the nodes and the machine tool are fixed, and their corresponding generalized displacements are all zero. Therefore, the nodes... and nodes In the mass and stiffness matrices, the corresponding rows and columns are eliminated during the solution of the motion differential equations. Since the mass and stiffness matrices of the beam elements are expanded to zero vectors during the assembly of the revolute joint mass and stiffness matrices, when a row or column in the machine tool's mass and stiffness matrix is ​​a zero vector, that row and column are eliminated, ultimately yielding the mass matrix of the five-axis hybrid machine tool. and stiffness matrix Considering the influence of damping in the machine tool, its damping matrix is ​​as follows: ; In the above formula, and It is a constant for calculation and is greater than 0.

[0020] Step 3: Set up the principal axis coordinate system Tool coordinate system and workpiece coordinate system The three-dimensional dynamic milling force on the tool during the milling process is calculated as follows: At any moment during the milling process t The radial runout of the tool will cause the tool tip to move relative to a given machining path. shaft and Dynamic offset distance in the axial direction and The calculation formula is as follows: ; In the above formula, The eccentricity distance representing the radial runout of the tool. This represents the rotation angle of the tool runout; considering the tool's rotation around the workpiece coordinate system during five-axis milling... shaft and Shaft tilt angle and lean angle The actual coordinates of the milling tool's actual machining trajectory in the workpiece coordinate system are: , and The formula is as follows: ; In the above formula, and These represent the cutting tool winding. shaft and The rotation matrix of the axis; , and These are the coordinates of the given machining trajectory in the workpiece coordinate system and in the tool coordinate system, respectively. In the middle, for the serial number is The height is The blade, any point on its blade coordinates , and The formula is as follows: ; In the above formula, Indicates the effective milling radius of the cutting tool. The blade hysteresis angle is expressed by the following formula: ; ; In the calculation formula, Indicates the tool radius. The formula for calculating the cutting edge hysteresis angle indicates the tool radius. N Indicates the number of teeth. Indicates the helix angle of the cutting tool; For the serial number is The blade has an axial height of point Considering the combined effects of tool radial runout and milling vibration, the tool's performance under these conditions is calculated. axis, shaft and The true coordinate position of the axis , and as follows: ; In the above formula, Let be the rotation matrix about the Z-axis; The tool rotation angle is related to the spindle speed. related; , and The three-dimensional milling vibrations are respectively at the tool tip point; To determine the instantaneous undeformed cutting thickness corresponding to the tool edge micro-element using milling geometry simulation, the workpiece is first divided into a series of workpiece micro-elements, with the discrete directions being... Axis and spacing settings When discrete interval The workpiece element is small enough that it can be considered a spatial cylinder with a constant lateral geometry along discrete axes, and thus can be considered as a series of boundary points. The resulting continuous and closed boundary curves, while the tool tilt angle exists during milling. and lean angle This results in an angle between the axial rotation plane of the tool and the workpiece micro-element plane. Therefore, projecting the axial rotation plane of the tool onto the workpiece micro-element plane simplifies the spatial contact problem between the tool and the workpiece into a planar contact problem. Then, the planar cutting thickness of the tool micro-element can be solved numerically. ; Therefore, for any moment in the five-axis milling process Based on the geometric relationship between the axial rotation plane of the cutting edge and the projection plane, the dynamic chip thickness of the cutting edge micro-element is determined. The formula is as follows: ; In the above formula, i For the first i One blade edge point; intersection point The intersection point is the axial section of the tool and the central axis of the tool. The intersection of the horizontal cutting plane of the tool and the central axis of the tool. Intersection to the intersection Based on the infinitesimal milling force model, the tangential force of the infinitesimal cutting edge element is calculated. radial force and axial force as follows: ; In the above formula, , , These are the shear force coefficients for tangential force, radial force, and axial force, respectively. , , These are the plowing shear force coefficients for tangential force, radial force, and axial force, respectively; , These represent the axial cutting width and the cutting arc length of the cutting edge micro-element, respectively. The milling forces acting on all cutting edge micro-elements are transformed into the workpiece coordinate system, and the three-dimensional dynamic milling forces acting on the tool are obtained through numerical integration. , and They are as follows: ; ; In the above formula, This represents the immersion angle of the blade element.

[0021] In step four, based on the three-dimensional dynamic milling force, the trajectory effects caused by the radial runout of the tool and milling vibration are determined and incorporated into the actual motion trajectory of the tool. The specific process is as follows: use The numerical discretization method is used to solve the dynamic equations of the cutting system to obtain the vibration response of the system, defining the initial state. and All are zero, so select an appropriate time increment. Sum of numerical integration parameters φ Calculate the constant of the integration process. , and as follows: ; Then any time The formula is as follows: ; ; After the solution is completed, extract the reference point at the end of the tool. The elastic displacement vector is used as the milling vibration response of the cutting system, through the tool in... axis, shaft and The true coordinate position of the axis , and This is then superimposed onto the actual motion trajectory of the cutting tool. Simulations and experiments are conducted to verify this. The experimental platform is as follows: Machine tool modal identification and milling vibration measurement experiments were conducted in a hybrid robot. For the machine tool modal identification experiment, the frequency response function was measured using a Siemens LMS SCADAS Mobile 05 data acquisition system. The hammer head was made of steel, and the accelerometers were all PCB 352C03. During the experiment, the hammer struck the tool tip, and the acceleration frequency response signals at the tool tip, the smooth part of the tool, and the double-moving platform were measured. For the milling vibration measurement experiment, the vibration signal was measured using an optoNCDT 2300 laser reflective displacement sensor. The measurement location was the smooth part of the tool, and the sampling frequency was set to 10000 Hz. To avoid interference from machine tool vibration on the measurement results, the sensor was mounted on a fixed support placed on the ground. Since the displacement sensor was fixed, the tool should remain stationary during the vibration response measurement process, i.e., the tool tilt angle during milling. and lean angle The values ​​are fixed, and due to limitations of the measuring equipment, only the horizontal milling vibration response can currently be measured. Since the vibration response of the tool tip cannot be measured using a laser displacement sensor during cutting, the vibration response at the smooth part of the tool is measured to indirectly reflect the vibration response of the tool tip. In the milling vibration measurement experiment, a supercarbide ball end mill with a diameter of 8 mm, a helix angle of 30°, and 2 cutting edges was used. The workpiece material was aluminum alloy 7075, with a thickness of 50 mm and a length and width of 170 mm.

[0022] First, the natural frequency was verified, and experimental simulation results were obtained. Figures 7-12 Excitation was applied at the tool tip, and acceleration frequency response signals were obtained at the tool tip, the smooth area of ​​the tool, and the double-moving platform for each image. The results show that the experimental and simulation results are in good agreement, verifying the accuracy of the method of this invention. The comparison results also indicate that the higher-order natural frequencies at the tool tip and the smooth area of ​​the tool remain essentially unchanged. In fact, these higher-order natural frequencies are directly related to the inherent properties of the tool and the spindle module themselves, and can be regarded as local vibrations of both, while the machine tool's pose has little influence on them.

[0023] Therefore, this invention employs a method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool. Considering the rigidity changes of the robot body, a structural dynamic model of a novel five-axis hybrid machine tool is established. The natural frequencies of the machine tool and its vibration response under a unit step load are analyzed, enabling the prediction of the machine tool's dynamic characteristics under different postures. Based on the milling regeneration effect and the dynamic characteristics of the machine tool, a dynamic model of the milling system is established by combining workpiece material removal simulation, enabling the prediction of milling force, milling vibration, and form and position errors in the machining of complex curved surfaces.

[0024] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool, characterized in that: Includes the following steps: Step 1: Perform system dynamics modeling on the five-axis hybrid machine tool. During the milling process of the five-axis hybrid machine tool, establish the dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool. Step 2: Tune the dynamic parameters to account for changes in the pose of the hybrid machine; Step 3: Set up the principal axis coordinate system Tool coordinate system and workpiece coordinate system To calculate the three-dimensional dynamic milling force on the tool during the milling process; Step 4: Based on the three-dimensional dynamic milling force, determine the trajectory effects caused by the radial runout of the tool and milling vibration, and incorporate them into the actual motion trajectory of the tool.

2. The method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool according to claim 1, characterized in that: In step one, the dynamic equations of the cutting system consisting of the cutting tool and the five-axis hybrid machine tool are given as follows: ； In the above formula, The mass matrix of the cutting system is represented. This represents the damping matrix of the cutting system. Represents the stiffness matrix of the cutting system. Represented as t Vibration response of the cutting system at a given time. Represented as t The cutting force excitation of the cutting system at any given time. Represented as The first derivative, Represented as The second derivative of .

3. The method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool according to claim 2, characterized in that: In step two, the specific process for tuning the dynamic parameters is as follows: Based on the structural characteristics of the five-axis hybrid machine tool, it is equivalent to a spatial mechanical system composed of variable cross-section beam entities and revolute joints. The following assumptions are made during the dynamic modeling of the machine tool structure: ① It is assumed that each substructure can be equivalent to a variable cross-section beam entity. This is achieved by dividing the beam entity into a series of variable cross-section spatial beam elements and using the finite element method to construct the mass and stiffness matrices of each substructure; ② It is assumed that the sliding components of each branch are fixedly connected to the machine tool, and the five-axis hybrid machine tool in different poses is considered as a series of transient structures; ③ It is assumed that there is an ideal assembly relationship between each variable cross-section beam entity, ignoring the effects of installation clearance, damping, and friction in the revolute joints. The dynamic parameters of the five-axis hybrid machine tool include the mass matrix. Damping matrix and stiffness matrix Due to the structural characteristics of five-axis hybrid machine tools, their dynamic parameters will change with the change of the machine's position and posture. Therefore, the dynamic parameters of the five-axis hybrid machine tool need to be tuned. The tuning process is as follows: The moving platform system of a five-axis hybrid machine tool includes an upper moving platform, a lower moving platform, a tool-spindle module, and several branches. Each branch All of them include the same three substructures: sliding component, intermediate link and end link. The dynamic platform system of the five-axis hybrid machine tool is subdivided into six substructures. Based on the beam element theory, each substructure is analyzed and the corresponding mass and stiffness matrix is ​​established. Based on the structural characteristics of the upper and lower moving platforms, the upper platform is divided into... Each unit node divides the lower platform into There are 12 unit nodes, each with 6 degrees of freedom. Therefore, the stiffness matrix and mass matrix of the moving platform have 1 / 2 dimensions. The dimensions of the stiffness matrix and mass matrix of the descent platform are both [missing information]. ; In the tool-spindle module, the tool is fixed to the machine tool spindle by a spring collet; therefore, the tool and the machine tool spindle are considered as a single unit, and the module is divided into... Each element node has a mass and stiffness matrix of dimension 1. Among them, the node corresponding to the connection between the tool and the spindle is The node corresponding to the connection between the tool-spindle module and the upper moving platform is ; The sliding assembly comprises multiple parts, which are connected by bolts or other means. Therefore, it is considered as a whole. Based on the structural characteristics of the sliding assembly, it is divided into... Each element node has a mass and stiffness matrix of dimension 1. ; Based on the structural characteristics of the intermediate and end links, they are divided into... and There are 1 unit nodes, therefore, the dimensions of the mass and stiffness matrices of the intermediate link are both 1 / 2. The dimensions of the mass and stiffness matrices of the end link are both [missing information]. ; Based on the above analysis, the stiffness matrices of the upper moving platform, lower moving platform, tool-spindle module, intermediate connecting rod, and end connecting rod are obtained. , , , and It is uniformly represented as: ； In the formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure Submatrices of the stiffness matrix of each beam element, with their subscripts pp The index is used to calculate the position of the submatrix within the beam element stiffness matrix; Mass matrix of upper moving platform, lower moving platform, tool-spindle module, intermediate link and end link , , , and The unified representation is as follows: ； In the formula, The corresponding number of beam elements They are respectively equal to , , , and ; Indicates the first substructure A submatrix of the mass matrix of each beam element, with its subscripts pp The index is used to calculate the position of the submatrix within the beam element mass matrix; Due to the sliding component beam element at the node Since the connection is in parallel, it can be considered as a beam entity composed of a series of beam elements connected in series and parallel. According to the static equilibrium equations, its stiffness matrix is... Revised to: ； Mass matrix of sliding component Revised to: ； Based on the structural properties of the five-axis hybrid machine tool, the mass and stiffness matrices of each substructure are assembled into the mass and stiffness matrices of the machine tool using the same joint assembly method. Since the mass and stiffness matrices of each substructure are consistent in both structural form and assembly method, the assembly process of the stiffness matrix of the five-axis hybrid machine tool is as follows: In each branch In the middle, the sliding component is connected to the intermediate link, and the intermediate link is connected to the end link via revolute joints. and Connect and rotate the pair and The corresponding turning angle is and In order to assemble the stiffness matrix of the branch system in the same reference coordinate system, the stiffness matrix of the intermediate link is... From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the intermediate link The calculation is as follows: ； In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​used to transform the stiffness matrix of the end link. From rotating joint Reference coordinate system Transform to branched coordinate system In the middle, for each branch The stiffness matrix of the connecting rods at their ends Represented as: ； In the above formula, for Around Axis rotation The coordinate transformation matrix is ​​as follows: ； In the above formula, α y To bypass Y The angle of rotation of the axis should be replaced with a specific angle value during calculation; The sliding assembly, intermediate link, and end link are all connected by revolute joints, and their rotation axes are all in the branch coordinate system. of With axes parallel, the stiffness matrix is ​​respectively based on the joint assembly method. , and Equivalent transformation to , and Then each branch stiffness matrix Represented as: ； In the above formula, , and These are the stiffness matrices. , and The assembly matrix is ​​as follows: ； In the above formula, and These are the identity matrix and the zero matrix, respectively, and their subscripts are... and Represent the number of rows and columns of the matrix, respectively; the stiffness matrix of the branch. From the branch coordinate system Transform to the moving platform system coordinate system In Chinese, the formula is as follows: ； In the above formula, Indicates the coordinate system of the moving platform system The lower branch i stiffness matrix, Indicates the coordinate system of the moving platform system Relative to the machine tool static coordinate system The transformation matrix, Representing the coordinate system The angle of rotation about the z-axis, Indicates the roll angle. Indicates the roll angle. coordinate system Around The coordinate transformation matrix for axis rotation is as follows: ； In the above formula, α z To bypass z The angle of rotation of the axis should be replaced with a specific angle value during calculation; Due to the nodes on the upper platform Node with the tool-spindle module The two are connected in parallel, therefore their stiffness matrices are... Represented as: ； In the above formula, the superscript This means first dividing the matrix Middle node With nodes Swap the corresponding column vectors, and then swap the nodes. to The corresponding column vector is shifted to the node. to Location; and They are respectively and The assembly matrix is ​​as follows: ； stiffness matrix Transform from the coordinate system of the upper moving platform to the coordinate system of the moving platform system to obtain The formula is as follows: ； ； ； In the above formula, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The transformation matrix, Indicates the coordinate system of the moving platform Relative to the coordinate system of the moving platform The corner, the node of the moving platform Nodes of the lower platform The connection between them is a rotary link, and the rotation axis is the moving platform system. The stiffness matrix of the axis is respectively based on the joint assembly method. and Equivalent transformation to and The stiffness matrix of the moving platform system Represented as: ； In the formula, and These are the stiffness matrices. and The assembly matrix is ​​as follows: ； In the formula, Stiffness matrix The number of nodes, and satisfying ; Each branch is connected via a rotating joint. Connected to the moving platform system, and rotating pair and The axis of rotation is the coordinate system of axis, Rotary joint and The axis of rotation is the coordinate system of The stiffness matrix of the moving platform system is determined based on the joint assembly method. Stiffness matrix of each branch , , and Equivalent transformation to , , , and Then the stiffness matrix of the machine tool express: ； In the formula, , , , and These are the stiffness matrices. , , , and The assembly matrix is ​​as follows: ； In the formula, Stiffness matrix of the moving platform system Dimension and ; Stiffness matrix of the branched system Dimension and ; Using the same method as the stiffness matrix, the mass matrix of the machine tool is obtained based on the joint assembly method. Due to the nodes in the sliding component and nodes All connections between the nodes and the machine tool are fixed, and their corresponding generalized displacements are all zero. Therefore, the nodes... and nodes In the mass and stiffness matrices, the corresponding rows and columns are eliminated during the solution of the motion differential equations. Since the mass and stiffness matrices of the beam elements are expanded to zero vectors during the assembly of the revolute joint mass and stiffness matrices, when a row or column in the machine tool's mass and stiffness matrix is ​​a zero vector, that row and column are eliminated, ultimately yielding the mass matrix of the five-axis hybrid machine tool. and stiffness matrix Considering the influence of damping in the machine tool, its damping matrix is ​​as follows: ； In the above formula, and It is a constant for calculation and is greater than 0.

4. The method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool according to claim 3, characterized in that: In step three, the three-dimensional dynamic milling force on the tool during the milling process is determined. The specific process is as follows: At any moment during the milling process t The radial runout of the tool will cause the tool tip to move relative to a given machining path. shaft and Dynamic offset distance in the axial direction and The calculation formula is as follows: ； In the above formula, The eccentricity distance representing the radial runout of the tool. Indicates the rotation angle of the tool runout; Considering the process of five-axis milling, the tool's position around the workpiece coordinate system shaft and Shaft roll angle and lean angle The actual coordinates of the milling tool's actual machining trajectory in the workpiece coordinate system are: , and The formula is as follows: ； ； ； In the above formula, and These represent the cutting tool winding. shaft and The rotation matrix of the axis; , and These represent the coordinates of the given machining trajectory in the workpiece coordinate system and in the tool coordinate system, respectively. In the middle, for the serial number is The height is The blade, any point on its blade coordinates , and The formula is as follows: ； In the above formula, Indicates the effective milling radius of the cutting tool. The blade hysteresis angle is expressed by the following formula: ； ； In the calculation formula, Indicates the tool radius. The formula for calculating the cutting edge hysteresis angle indicates the tool radius. N Indicates the number of teeth. Indicates the helix angle of the cutting tool; For the sequence number is The blade has an axial height of point Considering the combined effects of tool radial runout and milling vibration, the tool's performance under these conditions is calculated. axis, shaft and The true coordinate position of the axis , and as follows: ； In the above formula, Let be the rotation matrix about the Z-axis; The tool rotation angle is related to the spindle speed. related; , and The three-dimensional milling vibrations are respectively at the tool tip point; To determine the instantaneous undeformed cutting thickness corresponding to the tool edge micro-element using milling geometry simulation, the workpiece is first divided into a series of workpiece micro-elements, with the discrete directions being... Axis and spacing settings When discrete interval The workpiece element is small enough that it can be considered a spatial cylinder with a constant lateral geometry along discrete axes, and thus can be considered as a series of boundary points. The resulting continuous and closed boundary curves, while the tool tilt angle exists during milling. and lean angle This results in an angle between the axial rotation plane of the tool and the workpiece micro-element plane. Therefore, projecting the axial rotation plane of the tool onto the workpiece micro-element plane simplifies the spatial contact problem between the tool and the workpiece into a planar contact problem. Then, the planar cutting thickness of the tool micro-element can be solved numerically. ; Therefore, for any moment in the five-axis milling process Based on the geometric relationship between the axial rotation plane of the cutting edge and the projection plane, the dynamic chip thickness of the cutting edge micro-element is determined. The formula is as follows: ； In the above formula, i For the first i One blade edge point; intersection point The intersection point is the axial section of the tool and the central axis of the tool. The intersection of the horizontal cutting plane of the tool and the central axis of the tool. Intersection to the intersection Based on the infinitesimal milling force model, the tangential force of the infinitesimal cutting edge element is calculated. radial force and axial force as follows: ； In the above formula, , , These are the shear force coefficients for tangential force, radial force, and axial force, respectively. , , These are the plowing shear force coefficients for tangential force, radial force, and axial force, respectively; , These represent the axial cutting width and the cutting arc length of the cutting edge micro-element, respectively. The milling forces acting on all cutting edge micro-elements are transformed into the workpiece coordinate system, and the three-dimensional dynamic milling forces acting on the tool are obtained through numerical integration. , and They are as follows: ； ； In the above formula, This represents the immersion angle of the blade element.

5. The method for analyzing the dynamic characteristics of the tool tip during milling on a five-axis hybrid machine tool according to claim 4, characterized in that: In step four, based on the three-dimensional dynamic milling force, the trajectory effects caused by the radial runout of the tool and milling vibration are determined and incorporated into the actual motion trajectory of the tool. The specific process is as follows: use The numerical discretization method is used to solve the dynamic equations of the cutting system to obtain the vibration response of the system, defining the initial state. and All are zero, so select an appropriate time increment. Sum of numerical integration parameters φ Calculate the constant of the integration process. , and as follows: ； Then any time The formula is as follows: ； ； After the solution is completed, extract the reference point at the end of the tool. The elastic displacement vector is used as the milling vibration response of the cutting system, through the tool in... axis, shaft and The true coordinate position of the axis , and This is superimposed onto the actual movement trajectory of the cutting tool.

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