CN114895565A - Milling working condition-oriented dynamic characteristic real-time prediction method for double-turntable five-axis machine tool - Google Patents
Milling working condition-oriented dynamic characteristic real-time prediction method for double-turntable five-axis machine tool Download PDFInfo
- Publication number
- CN114895565A CN114895565A CN202210558844.2A CN202210558844A CN114895565A CN 114895565 A CN114895565 A CN 114895565A CN 202210558844 A CN202210558844 A CN 202210558844A CN 114895565 A CN114895565 A CN 114895565A
- Authority
- CN
- China
- Prior art keywords
- machine tool
- double
- axis
- turntable
- numerical control
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 238000003801 milling Methods 0.000 title claims abstract description 62
- 238000000034 method Methods 0.000 title claims abstract description 59
- 238000005096 rolling process Methods 0.000 claims description 46
- 239000011159 matrix material Substances 0.000 claims description 32
- 238000004458 analytical method Methods 0.000 claims description 19
- 230000009466 transformation Effects 0.000 claims description 17
- 230000033001 locomotion Effects 0.000 claims description 16
- 238000006073 displacement reaction Methods 0.000 claims description 11
- 238000006243 chemical reaction Methods 0.000 claims description 10
- 238000005381 potential energy Methods 0.000 claims description 10
- 230000009471 action Effects 0.000 claims description 6
- 230000006870 function Effects 0.000 claims description 5
- 230000005484 gravity Effects 0.000 claims description 5
- 238000005520 cutting process Methods 0.000 claims description 4
- 238000013519 translation Methods 0.000 claims description 4
- 230000005489 elastic deformation Effects 0.000 claims description 3
- 238000011276 addition treatment Methods 0.000 claims description 2
- 230000004323 axial length Effects 0.000 claims description 2
- 230000007850 degeneration Effects 0.000 claims description 2
- 238000009795 derivation Methods 0.000 claims description 2
- 230000004069 differentiation Effects 0.000 claims description 2
- 239000000463 material Substances 0.000 claims description 2
- 230000036316 preload Effects 0.000 claims description 2
- 239000007787 solid Substances 0.000 claims 2
- 230000009977 dual effect Effects 0.000 claims 1
- 238000004364 calculation method Methods 0.000 abstract description 13
- 230000003068 static effect Effects 0.000 abstract description 4
- 238000004519 manufacturing process Methods 0.000 abstract description 3
- 230000010354 integration Effects 0.000 abstract 1
- 238000003754 machining Methods 0.000 description 12
- 230000036544 posture Effects 0.000 description 8
- 238000005316 response function Methods 0.000 description 8
- 230000008859 change Effects 0.000 description 6
- 238000010586 diagram Methods 0.000 description 6
- 238000012545 processing Methods 0.000 description 5
- 238000011160 research Methods 0.000 description 4
- 238000012546 transfer Methods 0.000 description 4
- 238000005457 optimization Methods 0.000 description 3
- 238000002474 experimental method Methods 0.000 description 2
- 230000004044 response Effects 0.000 description 2
- 230000009286 beneficial effect Effects 0.000 description 1
- 230000005540 biological transmission Effects 0.000 description 1
- 238000005094 computer simulation Methods 0.000 description 1
- 230000008878 coupling Effects 0.000 description 1
- 238000010168 coupling process Methods 0.000 description 1
- 238000005859 coupling reaction Methods 0.000 description 1
- 230000006866 deterioration Effects 0.000 description 1
- 230000005284 excitation Effects 0.000 description 1
- 230000008569 process Effects 0.000 description 1
- 238000004088 simulation Methods 0.000 description 1
- 238000012916 structural analysis Methods 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
- G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
- G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
- G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
Landscapes
- Engineering & Computer Science (AREA)
- Health & Medical Sciences (AREA)
- Artificial Intelligence (AREA)
- Computer Vision & Pattern Recognition (AREA)
- Evolutionary Computation (AREA)
- Medical Informatics (AREA)
- Software Systems (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Numerical Control (AREA)
Abstract
The invention discloses a method for predicting dynamic characteristics of a double-turntable five-axis machine tool in real time under a milling working condition, and belongs to the technical field of machine tool manufacturing. The method comprises the following steps: describing a machine tool kinematic model by using a standard D-H method; establishing a machine tool kinetic equation by using a Lagrange method; establishing a dynamic model of a main shaft-tool handle, a main shaft-bearing, a guide rail sliding block and a ball screw joint part by adopting a Gicunx Kozaki integration method, a Hertz theory and elastic mechanics to obtain contact characteristic parameters of the joint part; and finally, establishing a kinetic model by combining a kinetic equation and the contact characteristics of the binding part. The dynamic characteristics of the machine tool are predicted in real time according to the milling working condition, the working position and the posture in the actual milling state of the machine tool, the problem that the dynamic characteristics of a static machine tool in the traditional method are different from those in the actual milling state is solved, the calculation efficiency is high, and the dynamic characteristics in the milling operation of the machine tool can be simulated, tracked and predicted by using a digital twin organism.
Description
Technical Field
The invention belongs to the technical field of machine tool manufacturing, and particularly relates to a dynamic characteristic prediction method of a double-turntable five-axis numerical control machine tool under a milling working condition.
Background
The multi-axis numerical control machine tool has a very deep influence on high-end fields such as national military, aviation, aerospace, scientific research, precise instruments, high-precision medical equipment and the like. High-level numerical control machine tools have the advantages of high speed, high precision, good reliability and the like, are applied to various branches of modern industries almost, and are starting points of most industrial manufacturing. In order to realize stable machining process, improve surface machining quality and avoid workpiece machining surface deterioration caused by chatter in the machining process of the numerical control machine tool, the dynamic characteristics of the numerical control machine tool are determined, and the optimization of the machining process is guided by the dynamic characteristics. Therefore, the method can efficiently and accurately simulate and predict the dynamic characteristics of the numerical control machine tool, and has important significance for ensuring the stability of the machining process and the machining quality of the surface of the workpiece.
In the actual machining process of the numerical control machine tool, the factors such as the operation working condition, the machining position, the posture, the contact between the combined parts, the friction and the like of the machine tool are in the process of dynamic change, the dynamic characteristic of the machine tool is changed dynamically at the moment, and the difference between the dynamic characteristic and the static state is obvious, so that the factors such as the milling working condition, the machining position, the posture, the contact between the combined parts, the friction and the like are considered to carry out high-efficiency and accurate numerical control machine tool dynamic characteristic prediction, and the method is particularly important for guiding process optimization and controlling the surface quality of a workpiece.
Currently, common dynamic modeling methods include lumped mass models, transfer matrix models, substructure models, and finite element analysis models.
1. The lumped mass model method is to segment the research objects and connect them by fixed mass limit points, each segment is regarded as rigid body or elastic body without fixed mass limit, so that the continuous system is converted into a system with limited freedom.
2. The transfer matrix model method is to regard an overall structure as a mechanical problem of connection and transfer of a plurality of substructures, establish a substructure transfer matrix, and analyze the overall structure through matrix multiplication, but when solving the dynamic problem of a large-scale complex rotor system, the problems of low calculation speed and unstable value are caused by the increase of trial calculation frequency.
3. The substructure model method is similar to the transmission matrix model method, and is different from the method that the kinetic characteristic parameters of each substructure are obtained through theoretical research, actual experiments and numerical calculation, and then the kinetic characteristics of the overall structure are obtained by integrating the substructure kinetic models.
4. The finite element model method is to regard the complete structure as that a plurality of units are connected at node positions, to regard the displacement of the connected nodes as unknown parameters to be expressed by a related displacement function, and to obtain the vibration balance equation of the complete structure by the energy variation principle. The method has high solving precision and mature relevant commercial software, but only one dynamic model can be established in single processing, when the working position, the posture and the milling working condition are changed, the model needs to be established again for solving, the method is not suitable for real-time tracking simulation of the dynamic characteristics of the machine tool under the milling working condition, the calculation speed is low, the consumption is long, and the modeling and analyzing efficiency is low.
The dynamic characteristics of the machine tool change along with the change of the milling working condition and the pose in the actual milling state, so that the high-efficiency prediction of the pose change in the whole working space of the machine tool under different milling working conditions is difficult to realize by the conventional dynamic modeling method. In addition, in the current research on the dynamic characteristics of the machine tool, excitation is applied to the machine tool body in a static state to obtain a response signal, and the modal parameters of the machine tool are obtained by processing the response signal, so that the method is greatly different from the actual operation condition and is not close to the actual condition.
Disclosure of Invention
The dynamic model of the double-rotary-table five-axis machine tool with low degree of freedom is established by considering the milling working condition, the machining position, the posture and the contact characteristic of a combined part, the dynamic characteristic of the machine tool is predicted in real time according to the Jacobian matrix of the transformation position and different milling working conditions of the machine tool, the dynamic model is closer to the actual milling state and has high calculation efficiency, and meanwhile, the dynamic model can be used as a digital twin body for simulating, tracking and predicting the dynamic characteristic of the machine tool in the milling operation, so that the difficult problems that the actual operation working condition of the machine tool is neglected when the dynamic characteristic of the machine tool in a static state is researched by the traditional method, the dynamic model is low in calculation speed, does not have real-time performance and is difficult to transform and analyze the posture in the whole working space of the machine tool are solved.
The invention aims to provide a method for predicting the dynamic characteristics of a double-turntable five-axis numerical control machine tool in real time under the milling working condition, which comprises the following steps:
the method comprises the following steps: establishing a motion chain of the double-turntable five-axis numerical control machine tool according to the structure and the motion state of the machine tool, establishing a parameter model by using a standard D-H method, and describing an integral complete kinematics model of the double-turntable five-axis numerical control machine tool through homogeneous coordinate transformation between coordinate systems;
step two: establishing a three-dimensional entity model of the double-turntable five-axis numerical control machine tool, acquiring the mass, the mass center of the double-turntable five-axis numerical control machine tool and the inertia tensor of the mass center by using the three-dimensional entity model, obtaining a kinetic energy and potential energy expression of the whole system of the double-turntable five-axis numerical control machine tool, and deducing a kinetic equation of the double-turntable five-axis numerical control machine tool by using a Lagrange method;
step three: establishing a dynamic model of a main shaft-tool handle combination part, a main shaft-bearing combination part, a guide rail sliding block combination part and a ball screw combination part under a milling working condition by adopting a Gimura Korea Zengxian integral method, a Hertz contact theory and elastic mechanics to obtain contact characteristic parameters of the main shaft-tool handle combination part, the main shaft-bearing combination part, the guide rail sliding block combination part and the ball screw combination part under the milling working condition;
step four: and (3) establishing a complete multi-body dynamic model of the double-rotary-table five-axis numerical control machine tool by combining a dynamic equation of the double-rotary-table five-axis numerical control machine tool and contact characteristic parameters of a combining part, and solving the model.
The invention provides a method for predicting the dynamic characteristics of a double-turntable five-axis numerical control machine tool in real time under the milling working condition, which can achieve the following beneficial effects:
1. the method can predict the dynamics of the double-turntable five-axis numerical control machine tool in real time under the milling working condition, and predict the dynamics characteristics aiming at different milling working conditions, different processing positions and postures, and compared with the existing method which does not consider the milling working condition and the processing position posture change, the prediction result is closer to the actual motion state of the machine tool and is more accurate, so that the dynamics characteristics of the machine tool are analyzed and evaluated for guiding the optimization of the milling process and improving the milling processing efficiency and the surface quality of the processed workpiece.
2. The dynamic model established by the invention considers the actual milling working condition of the machine tool, the pose of the machine tool can be changed only by changing the position Jacobian matrix parameter, the machine tool under the specific condition is not required to be repeatedly modeled, and the problem of low efficiency of performing dynamic characteristic analysis work pose transformation on the double-rotary-table five-axis numerical control machine tool is solved.
3. According to the method, the characteristic parameters (mass, self mass center and inertia tensor at the mass center) of each structural part are obtained by establishing a three-dimensional model of the double-turntable five-axis numerical control machine tool, meanwhile, the contact characteristic parameters of the joint parts are obtained by establishing contact models of each joint part (a main shaft-handle joint part, a main shaft-bearing joint part, a guide rail slide block joint part and a ball screw joint part) of the machine tool, the influences of milling load, centrifugal force and gyro moment in the milling process are particularly considered, an overall dynamics model is established by integrating the structural part and the joint part of the machine tool, the dynamics characteristics of the double-turntable five-axis numerical control machine tool obtained by the method are more in line with the actual conditions, the accuracy is higher, and the accuracy requirements of dynamics modeling and analysis of the machine tool are met.
4. The method has high calculation efficiency in establishing the dynamic model, meets the requirements of high efficiency and real-time performance of modeling and analysis, and can be used as a digital twin body for simulating the tracking and predicting dynamic characteristics of the machine tool in the milling operation.
Drawings
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
FIG. 1 is a structural analysis diagram of a double-turntable five-axis numerical control machine tool according to the invention;
FIG. 2 is a stress analysis diagram of a spindle system of a double-turntable five-axis numerical control machine tool according to the invention;
FIG. 3 is a mechanical model diagram of a guide rail combination part of a double-turntable five-axis numerical control machine tool under a milling load according to the invention;
FIG. 4 is a simplified model stress analysis diagram of a rolling linear guide rail of a double-turntable five-axis numerical control machine tool according to the invention;
FIG. 5 is a schematic diagram of the geometric relationship of rolling bodies of a ball screw kinematic pair of a double-turntable five-axis numerical control machine tool according to the invention;
FIG. 6 is a schematic diagram of a dynamic model analysis result of the double-turntable five-axis numerical control machine tool;
FIG. 7 is a frequency response function curve of the double-turntable five-axis numerical control machine tool in different postures;
FIG. 8 is a frequency response function curve of the double-turntable five-axis numerical control machine tool under different milling loads;
fig. 9 is a specific flow chart of the double-turntable five-axis numerical control machine tool multi-body dynamic characteristic real-time prediction method.
In the figure: 1. a machine tool frame; 2. an X-axis guide rail; 3. a Y-axis guide rail; 4. a shaft A is arranged on a platform; 5. a C-axis workbench; 6. a Z-axis guide rail; 7. a spindle cutter;
Detailed Description
The specific steps of the embodiments of the method of the present invention will be described more fully hereinafter with reference to the accompanying drawings.
S1: taking a double-turntable five-axis numerical control machine tool as an example, a motion chain model of the double-turntable five-axis numerical control machine tool is established according to the motion relation between the whole machine structure and each rotor structure of the double-turntable five-axis numerical control machine tool, as shown in fig. 1, coordinate system transformation among the seven rotor structures of the double-turntable five-axis numerical control machine tool is described by using a standard D-H parameter method, and a complete whole machine kinematics model of the double-turntable five-axis numerical control machine tool is established; the method specifically comprises the following steps:
s11: the double-turntable five-axis numerical control machine tool is provided with five linkage shafts, and an X-axis guide rail 2, a Y-axis guide rail 3, a Z-axis guide rail 6, an A-axis swing table 4 and a C-axis workbench 5 are respectively arranged on the linkage shafts. X-axis guide rail 2: the slide rail moves back and forth; y-axis guide rail 3: the slide rail is connected with the workbench to move left and right; the Z-axis guide rail 6: the slide rail moves up and down perpendicular to the ground; the A-axis swing table 4: the workbench swings around the Y axis; c-axis table 5: the worktable rotates 360 degrees around the Z axis.
The cutter is fixed on a Z axis and is connected with a part of a machine tool frame perpendicular to the ground through a slide rail, the part of the machine tool frame parallel to the ground is connected with an X-axis guide rail 2, the X-axis guide rail 2 is connected with a Y-axis guide rail 3, and the Y-axis guide rail 3 is connected with a C-axis workbench 5 through an A-axis swing table 4. When the machine tool works, a workpiece is fixed on the C-axis workbench 5, the position of the workpiece is adjusted through the rotation of the C-axis workbench 5, the swing of the A-axis swing table 4, the translation of the Y-axis guide rail 3 and the translation of the X-axis guide rail 2, and a cutter moves up and down along with the Z-axis guide rail 6 to machine the workpiece. Therefore, the kinematic chain relationship of the five-axis numerical control machine tool with the double rotary tables is as follows: the C-axis table 5 → the a-axis placing table 4 → the Y-axis guide rail 3 → the X-axis guide rail 2 → the machine frame 1 → the Z-axis guide rail 6.
S12: when the coordinate system transformation between the rotor structures of the double-turntable five-axis numerical control machine tool is described by using a standard D-H parameter method, a is used i 、α i 、d i And theta i Four parameters, wherein a i Refers to the common normal distance, alpha, of the two axes of the structural member i Meaning that the two axes of the structure are perpendicular to a i Angle in the plane, d i Indicating the distance, theta, of adjacent structural elements i Representing the corners between adjacent structural members, so that the relationship between two adjacent structural members is the coordinate system { x } 1 ,y 1 ,z 1 To a coordinate system x 0 ,y 0 ,z 0 And (4) performing homogeneous coordinate transformation, wherein the coordinate transformation relationship between two adjacent structural parts according to a standard D-H method is as follows:
wherein
Representing the position and orientation of the coordinate system i relative to the coordinate system i-1,
is shown along Z i-1 Axial movement d i ，
Is shown along Z i-1 Axis of rotation theta i ，
Is shown along X i Axial movement a i ，
Is shown along X i Rotation of the shaft alpha i 。
The D-H parameter table of the double-turntable five-axis numerical control machine tool is shown in table 1, and q is 1 Representing the X-axis, q-axis of the machine tool movement 2 Showing the machine tool movement axis Y, q 3 Indicating the Z-axis, q of the machine tool movement axis 4 Representing the axis A, q of the machine tool swing 5 Showing the machine tool axis of rotation C, i.e. the table.
TABLE 1 Standard D-H parameters of double-turntable five-axis numerical control machine tool
S13: establishing a machine tool coordinate system, and establishing a machine tool coordinate system OBED by taking the center of the bottom of the machine tool as an origin O, wherein an X-axis corresponds to a coordinate system OSX, a Y-axis corresponds to a coordinate system OSY, an A-axis corresponds to a coordinate system OSA, a workbench coordinate system OST, a Z-axis corresponds to a coordinate system OSZ and a tool location point coordinate system OTCP. Analyzing the kinematic chain transformation relation from the workbench to the whole cutter of the double-turntable five-axis numerical control machine tool to obtain the result that the transformation from the workbench coordinate system to the cutter coordinate system needs to undergo six times of coordinate transformation, namely OST → OSA → OSY → OSX → OBED → OSZ → OTCP, so that the whole complete kinematic model of the double-turntable five-axis numerical control machine tool uses a total coordinate transformation matrix
To show that:
wherein
Representing a rotation matrix of the structural member, p ═ p x p y p z ] T Representing a displacement matrix of the structure.
S2: establishing a complete machine entity model of the double-rotary-table five-axis numerical control machine, acquiring the mass, the mass center coordinate and the inertia tensor at the mass center of each part, obtaining a kinetic energy and potential energy expression of a complete machine system of the double-rotary-table five-axis numerical control machine, and deducing a kinetic equation of the double-rotary-table five-axis numerical control machine by using a Lagrange method; the method specifically comprises the following steps:
s21: establishing a complete machine three-dimensional entity model of the double-turntable five-axis numerical control machine tool by referring to a machine tool manual, and acquiring the mass m of each part by using the three-dimensional entity model i Centroid coordinate G i And its own centroid G i Tensor I of inertia i 。
S22: translation speed v of all components of double-turntable five-axis numerical control machine tool under same coordinate system of structural parts i And a rotational speed w i Calculated by time differentiation of a homogeneous transformation matrix, i.e. converted into the corresponding Jacobian matrix form:
the system kinetic energy of the double-turntable five-axis numerical control machine tool is divided into two parts, wherein one part is the kinetic energy E of a structural part l The other part being the kinetic energy E of the motor-driven rotor r . Let the kinetic energy of the i-th structural member relative to the inertial coordinate system be E li The kinetic energy of the differential mass dm on the structural part is recorded as dE li And then:
note I i An inertia tensor matrix for structure i:
mass of the structural member is m i The inertia tensor matrix is I i Then, the kinetic energy of the whole structure of the double-turntable five-axis numerical control machine tool is as follows:
the driving rotor of the motor on the ith structural member is recorded as I zi And the sum of the rotational kinetic energy of the n rotors of the double-turntable five-axis numerical control machine tool is as follows:
the overall kinetic energy of the double-turntable five-axis numerical control machine tool is obtained as the sum of the kinetic energy of the drive shaft rotor and the kinetic energy of the structural part:
s23: the only source of potential energy of the double-turntable five-axis numerical control machine tool is gravity, and the mass of an object is concentrated at the center of mass, so that the potential energy of the ith structural part is as follows:
P i ＝m i gC i (9)
wherein g is the gravity vector in the inertial coordinate system, C i Is the centroid coordinate of the structural member i. The total potential energy of the double-rotary table five-axis numerical control machine tool is as follows:
s24: and recording a Lagrange function L as the difference value of the kinetic energy and the potential energy of the double-turntable five-axis numerical control machine tool:
L＝E k -E p (11)
the Lagrange equation of the dynamics of the double-turntable five-axis numerical control machine tool can be obtained by carrying out derivation on the formula:
substituting the kinetic energy and the potential energy into the Lagrange function constructed by the step (12) to obtain the following result:
by substituting the above formulae (8) and (10), the following can be obtained:
the formula is simplified as follows:
where M (q) is a mass matrix for each structural member,
is a centrifugal force vector and a Coriolis force vector, G (q) is a structural member gravity vector matrix, F is an application force vector formed by a friction force and a cutting force generated in the cutting process, M (q),
Expressed as the associated jacobian matrix:
s3: establishing a rigidity model of a main shaft-cutter handle combination part by adopting a Ginku Xiaoxian integral method, obtaining a dynamic model of a main shaft-bearing combination part, a guide rail slide block combination part and a ball screw combination part by utilizing a Hertz contact theory and a mechanical balance principle, and establishing contact characteristic parameters of each combination part of the double-turntable five-axis numerical control machine tool under a milling working condition in a simultaneous manner; the method specifically comprises the following steps:
s31: the stress analysis is carried out on the main shaft system in the milling state, as shown in figure 2, the milling load F is applied to the main shaft-tool shank combination part x 、F y 、F z And influence of centrifugal force, main shaftAverage tool shank coupling normal force P n1 Comprises the following steps:
wherein F 0 Pull rod force for spindle-shank clamping, F n1 Is the normal force of the main shaft-tool shank combined part, S 1 Is the contact area of the spindle and the tool shank, phi 1 Is included angle of the knife handle, L 1 Is the axial length r of the main shaft-tool shank combination part 1 And r 2 A large radius and a small radius of the joint part, mu 1 Is the coefficient of friction.
The equivalent spring stiffness of the main shaft-handle combining part is obtained by a Ginchun xianxiao integral method:
wherein k (delta) can be obtained by the conversion of the filial piety integral curve of Ji Cunning 0 、β 0 As a contact characteristic parameter of the joint, δ rm For radial milling loads
The maximum deformation of the components in the combining part is realized under the action of the elastic deformation,
e is the elastic modulus, upsilon is the Poisson ratio, R is the radius of the joint, and delta j The radial clearance of the joint part is caused by centrifugal force, and when the rotating speed of the main shaft is n, the centrifugal force is
And b is the outer diameter of the main shaft.
S32: the milling load F is applied to the spindle-bearing joint in the milling state x 、F y 、F z Centrifugal force and gyroscopic moment, as shown in FIG. 3According to the Hertz theory, the stress balance equation of the tth rolling element of the bearing is as follows:
wherein Q it And Q ot For contact load of rolling elements with inner and outer races, α it And alpha ot Contact angle of rolling element with inner and outer races, M gt Is gyro moment, F ct D is the diameter of the rolling element.
Carrying out integral stress analysis on the bearing:
wherein R is i Is the radius of the center circle of curvature of the inner raceway, r i Is the inner raceway radius, F rx Radial load to the front and rear spindle-bearing joints, wherein the front spindle-bearing joint is subjected to radial load
Radial load experienced by the rear spindle-bearing interface
γ t The t-th rolling element is the azimuth angle and z is the number of rolling elements in the bearing.
Solving an equation set by applying a Newton-Raphson iterative method, obtaining stress balance types with different degrees of freedom of the bearing inner ring through coordinate transformation, accumulating contact forces of all rolling bodies and the inner ring and the outer ring, and deriving a displacement by the obtained resultant force to obtain a contact stiffness matrix of the main shaft-bearing joint part.
S33: the joint part of the rolling linear guide rail is subjected to milling load F in the milling state x 、F y 、F z Taking the joint of the Z-axis guide rail as an example, a mechanical model under the milling load is established, as shown in fig. 3, a mechanical balance equation is expressed as follows:
wherein M is x 、M y 、M z Are x, y, z three-way torque, P x 、P y For integral force components acting in the x, y directions of the rail pair, F qz The traction force applied to the ball screw pair, G 1 Weight of tool holder, G 2 The weight of the guide rail pair is the whole weight.
Since the z direction is the slider motion direction, there is no fixed constraint, so there is no reaction force. Dividing the combination part between the guide rail and the slide block into four areas, evenly distributing the decomposed load generated when each support reaction force and each reaction moment act independently to the combination part between the slide block and the guide rail, and obtaining the normal force and the tangential force acting on each combination part after the addition treatment according to the following formula:
wherein i and j are 1 and 2 respectively represent the number of each binding part, and P xij (n)、P yij (n) the component force is generated by each support reaction force or moment acting on the joint part alone.
Because the sizes and the number of the balls in the guide rail are the same, the guide rail, the sliding block and the balls can be correspondingly simplified, and the elastic deformation amount between the combined parts under the load is solved, as shown in fig. 4.
Only considering the action and the normal force on the guide rail, carrying out stress analysis on the rolling linear guide rail:
(F 1 -F 3 )nsinγ＝F y /2 (24)
according to the theory of mechanical superposition:
wherein n is the number of balls in a single-row track, F P For normal component of preload acting on the balls, gamma is the ball-race junctionThe included angle formed by the contact.
According to the Hertz contact theory, the contact degeneration quantities between the ball and the slide block and between the ball and the guide rail are respectively as follows:
wherein E 1 、E 2 、E 3 And mu 1 、μ 2 、μ 3 Respectively represents the elastic modulus and the Poisson ratio of the materials of the ball, the slide block and the rolling linear guide rail, wherein Σ ρ is the sum of the main curvatures at the internal contact point, and Σ ρ is ρ 1 +ρ 2 +ρ 3 +ρ 4 And ρ is 1 ＝ρ 2 ＝2/d b ，ρ 3 ＝-f/d b ，ρ 4 ＝0，d b Is the diameter of the ball, f is the ratio of the curvature radius of the raceway to the diameter of the ball, J/m a The value of (d) can be obtained by calculating a table lookup for the value of τ, and τ ═ ρ 3 -ρ 4 |/∑ρ。
The normal and tangential contact stiffness of the guide rail joint part are respectively as follows:
wherein alpha is the included angle between the normal acting force of the rolling body and the longitudinal axis.
S34: taking a Z-axis ball screw joint as an example for analysis, a mechanical model of the joint in a milling state is established, fig. 5 shows a geometric relationship of contact deformation of a rolling body and a screw raceway, the left side shows a geometric relationship of the rolling body and the upper side of the screw raceway, and the right side shows a geometric relationship of the rolling body and the lower side of the screw raceway, so that the following can be obtained:
wherein when the rolling bodies are in contact with the upper side of the screw raceway, the + delta is taken tz When the rolling bodies contact with the lower side of the screw raceway, take-delta tz ，α 0 、α t Respectively, the contact angle before deformation and the contact angle after deformation, A 0 Is the center distance between the curvature center of the screw raceway and the curvature center of the nut raceway, V tz 、V ty Respectively the axial distance and the radial distance of the curvature center of the screw rod raceway after contact deformation, delta tz 、δ ty Respectively axial displacement and radial displacement of the curvature center of the screw roller path.
Further analysis yields the displacement equilibrium equation:
and (3) analyzing the stress of the rolling body of the ball screw kinematic pair, wherein the stress balance equation of the rolling body is as follows:
wherein Q t For rolling element contact load, f tx 、f ty 、f tz For the force to be applied to the kth rolling element,
and the t-th rolling body azimuth angle is phi screw raceway spiral angle.
Under the action of milling load, the stress analysis is carried out between the rolling bodies and the screw roller path in the ball screw kinematic pair, and the mechanical balance equation between all the rolling bodies and the screw roller path is established as follows:
and accumulating the contact forces of all the rolling bodies through the stress balance of each rolling body, and deriving the displacement by the resultant force to obtain a contact stiffness matrix of the joint part of the ball screw kinematic pair in a milling state.
S4: and finally, establishing a complete multi-body dynamic model of the double-turntable five-axis numerical control machine tool by a dynamic equation of the simultaneous double-turntable five-axis numerical control machine tool and contact characteristic parameters of a main joint part, wherein the complete multi-body dynamic model is represented by the following differential equation:
k (q) is a main joint part contact stiffness matrix, a complete machine dynamics equation of the double-turntable five-axis numerical control machine tool under the milling working condition is analyzed and solved by combining contact characteristic parameters of each part and a joint surface of the double-turntable five-axis machine tool, a frequency response function curve is obtained through computer simulation calculation and calculation analysis and is shown in figure 6, wherein a solid line represents a frequency response function curve obtained through actual experiment calculation, and a dotted line represents a frequency response function curve obtained through calculation of the dynamics model established by the invention, so that the model established by the invention is accurate, the precision of the dynamics model is improved, and the dynamics characteristic of the machine tool under the actual operation working condition is closer to the dynamics characteristic of the machine tool under the actual operation working condition; the dynamic characteristics of the machine tool can be predicted in real time by changing the position Jacobian matrix and the milling load, a frequency response function curve of the double-turntable five-axis machine tool when the machining position changes is shown in figure 7, a frequency response function curve of the double-turntable five-axis machine tool when the milling load changes is shown in figure 8, the time required for analyzing all modal parameters of a dynamic model by using a computer language is less than 0.1ms, the calculation efficiency is high, and the requirement of real-time prediction is met; the specific flow implemented by the method is shown in fig. 9, so that the inherent frequency and the frequency response function of the double-turntable five-axis numerical control machine tool under different poses and different milling loads can be obtained, and the efficient analysis of the pose change of the machine tool under the milling working condition is facilitated.
The method for predicting the dynamic characteristics of the double-turntable five-axis machine tool in real time under the milling working condition is described in detail, the principle and the implementation mode of the method are explained by applying the double-turntable five-axis numerical control machine tool, and the description of the embodiment is only used for helping to understand the method and the core idea of the method; meanwhile, for a person skilled in the art, according to the idea of the present invention, there may be variations in the specific embodiments and the application scope, and in summary, the content of the present specification should not be construed as a limitation to the present invention.
Claims (2)
1. The method for predicting the dynamic characteristics of the double-turntable five-axis machine tool in real time under the milling working condition is characterized by comprising the following steps of: s1: analyzing the structure and the motion state of the double-rotary-table five-axis numerical control machine tool, and establishing a motion chain of the double-rotary-table five-axis numerical control machine tool; establishing a standard D-H parameter table of the double-turntable five-axis numerical control machine tool; describing a coordinate system transformation relation among all the substructure by using a standard D-H method, and describing a complete kinematic model of the double-turntable five-axis numerical control machine;
s2: establishing a complete machine three-dimensional solid model of the double-turntable five-axis numerical control machine tool, and acquiring the mass, the mass center coordinates and the inertia tensor of the mass center of each part by using the three-dimensional solid model; obtaining a kinetic energy and potential energy expression of the whole system of the double-turntable five-axis numerical control machine tool, and deducing a kinetic equation of the double-turntable five-axis numerical control machine tool by using a Lagrange method;
s3: establishing a dynamic model of a main shaft-handle combination part, a main shaft-bearing combination part, a guide rail sliding block combination part and a ball screw combination part under a milling working condition by adopting a Gimura Sampur Xiao integral method, a Hertz contact theory and elastic mechanics to obtain contact characteristic parameters of the main shaft-handle combination part, the main shaft-bearing combination part, the guide rail sliding block combination part and the ball screw combination part under the milling working condition;
s4: finally, a complete multi-body dynamic model of the double-turntable five-axis numerical control machine tool is established by combining a dynamic equation of the double-turntable five-axis numerical control machine tool and contact characteristic parameters of a combining part, and the model is solved;
2. the method for predicting the dynamic characteristics of the double-turntable five-axis numerical control machine tool under the milling working condition in real time according to claim 1, wherein the step S1 specifically comprises the following steps:
s11: establishing a machine tool motion chain of the double-turntable five-axis numerical control machine tool: c-axis workbench → A-axis placing table → Y-axis slide rail → X-axis slide rail → machine tool body → Z-axis slide rail;
s12: establishment of dual rotors using standard D-H parametric methodsTable five-axis numerical control machine tool parameter table, using a i 、α i 、d i And theta i The four parameters describe the coordinate system transformation relationship between adjacent structural members:
wherein
Representing the position and orientation of the coordinate system i relative to the coordinate system i-1,
is shown along Z i-1 Axial movement d i ，
Is shown along Z i-1 Axis of rotation theta i ，
Is shown along X i Axial movement a i ，
Is shown along X i Rotation of the shaft alpha i ；
S13: establishing a machine tool coordinate system, taking the center of the bottom of a machine tool body as an origin O, establishing a machine tool coordinate system OBED, an X-axis corresponding coordinate system OSX, a Y-axis corresponding coordinate system OSY, an A-axis corresponding coordinate system OSA, a workbench coordinate system OST, a Z-axis corresponding coordinate system OSZ and a tool location point coordinate system OTCP, and performing six times of coordinate conversion when converting from the workbench coordinate system to the tool coordinate system, namely OST → OSA → OSY → OSX → OBED → OSZ → OTCP, so that the overall complete kinematics model of the five-axis numerical control machine tool with the double turntables uses a total coordinate conversion matrix
To show that:
wherein
Representing a rotation matrix of the structural member, p ═ p x p y p z ] T A displacement matrix representing the structure;
the step S2 specifically includes:
s21: establishing a complete machine three-dimensional entity model of the double-rotary-table five-axis numerical control machine tool, and acquiring the mass m of each structural part by using the three-dimensional entity model i Centroid coordinate C i Its own centroid C i Tensor I of inertia i ；
S22: translation speed v of all mass centers under same coordinate system i And a rotational speed w i Calculated by the time differentiation of the homogeneous transformation matrix, the expression is as follows:
the kinetic energy of the ith structural member relative to the inertial coordinate system is E li The inertia tensor matrix I of the structure I i According to the kinetic energy of the differential mass on the structural part, the kinetic energy E of the complete machine structural part of the double-turntable five-axis numerical control machine tool is obtained l Comprises the following steps:
the driving rotor of the motor on the ith structural member is recorded as I zi And the sum of the rotational kinetic energy of the n rotors of the double-turntable five-axis numerical control machine tool is as follows:
the overall kinetic energy of the double-turntable five-axis numerical control machine tool is obtained as the sum of the kinetic energy of the drive shaft rotor and the kinetic energy of the structural part:
s23: the total potential energy of the double-turntable five-axis numerical control machine tool is as follows:
wherein G is the gravity vector in the inertial coordinate system, G i Is the centroid coordinate of the structural part i;
s24: constructing a Lagrange function L of the double-turntable five-axis numerical control machine tool:
L＝E k -E p (8)
the Lagrange equation of the dynamics of the double-turntable five-axis numerical control machine tool can be obtained by carrying out derivation on the formula:
and (3) substituting the kinetic energy and the potential energy into the Lagrange function constructed in the step (9), and finishing to obtain:
the formula is simplified as follows:
where M (q) is a mass matrix for each structural member,
is a centrifugal force vector and a Coriolis force vector, G (q) is a structural member gravity vector matrix, F is an application force vector formed by a friction force and a cutting force generated in the cutting process, M (q),
Expressed as the associated jacobian matrix:
the step S3 specifically includes:
s31: carrying out stress analysis on the spindle system in a milling state to obtain the average normal force P of the spindle-tool handle combination part n1 Comprises the following steps:
wherein F 0 Pull rod force for spindle-shank clamping, F n1 Is the normal force of the main shaft-tool shank combined part, S 1 Is the contact area of the spindle and the tool shank, phi 1 Is included angle of the knife handle, L 1 Is the axial length r of the main shaft-tool shank combination part 1 And r 2 A large radius and a small radius of the joint part, mu 1 Is the coefficient of friction;
the equivalent spring stiffness of the joint was obtained by the gemcunx xianxiao integral method:
wherein k (delta) can be obtained by the filial generation integral curve transformation of Ji Cunning, alpha 0 、β 0 Is a contact of the jointCharacteristic parameter, δ rm For radial milling loads
The maximum deformation of the components in the combining part is realized under the action of the elastic deformation,
e is the elastic modulus, upsilon is the Poisson ratio, R is the radius of the joint, and delta j The radial clearance of the joint part is caused by centrifugal force, and when the rotating speed of the main shaft is n, the centrifugal force is
b is the outer diameter of the main shaft;
s32, analyzing the stress of the spindle-bearing joint in the milling state, wherein according to the Hertz theory, the stress balance equation of the t-th rolling element of the bearing is as follows:
wherein Q it And Q ot For contact load of rolling elements with inner and outer races, α it And alpha ot Contact angle of rolling element with inner and outer races, M gt Is gyro moment, F ct Is the centrifugal force, D is the diameter of the rolling body;
carrying out integral stress analysis on the bearing:
wherein R is i Is the radius of the center circle of curvature of the inner raceway, r i Is the inner raceway radius, F rx Radial load to the front and rear spindle-bearing joints, wherein the front spindle-bearing joint is subjected to radial load
Received by the rear spindle-bearing jointRadial load
γ t The t-th rolling element is an azimuth angle, and z is the number of the rolling elements in the bearing;
solving an equation set by applying a Newton-Raphson iterative method to obtain a contact stiffness matrix of the main shaft-bearing joint part;
s33: establishing a mechanical balance equation of a rolling linear guide rail joint part in a milling state:
wherein M is x 、M y 、M z Are x, y, z three-way torque, P x 、P y For integral force components acting in the x, y directions of the rail pair, F qz The traction force applied to the ball screw pair, G 1 Weight of tool holder, G 2 The weight of the guide rail pair is the whole weight;
the combining part between the guide rail and the slide block is divided into four areas, the decomposed load generated when each support reaction force and each reaction moment act independently is evenly distributed to the combining part between the slide block and the guide rail, and the normal force and the tangential force acting on each combining part can be obtained after the addition treatment according to the following formula:
wherein i and j are 1 and 2 respectively represent the number of each binding part, and P xij (n)、P yij (n) each support reaction force or moment acts on the joint part independently to generate component force;
and (3) only considering the action and the normal force on the guide rail, and calculating the internal force of the rolling linear guide rail:
(F 1 -F 3 )n sinγ＝F y /2 (20)
wherein n is the number of balls in a single-row track, F P The normal component force of the preload acting on the ball is adopted, and gamma is an included angle formed by the contact of the ball and the roller path;
according to the Hertz contact theory, the contact degeneration quantities between the ball and the slide block and between the ball and the guide rail are respectively as follows:
wherein E 1 、E 2 、E 3 And mu 1 、μ 2 、μ 3 Respectively representing the elastic modulus and the Poisson ratio of materials of the ball, the sliding block and the rolling linear guide rail, wherein Sigma rho is the sum of main curvatures at an internal contact point, and Sigma rho is rho 1 +ρ 2 +ρ 3 +ρ 4 And ρ is 1 ＝ρ 2 ＝2/d b ，ρ 3 ＝-f/d b ，ρ 4 ＝0，d b Is the diameter of the ball, f is the ratio of the curvature radius of the raceway to the diameter of the ball, J/m a The value of (d) can be obtained by calculating a table lookup for the value of τ, and τ ═ ρ 3 -ρ 4 |/∑ρ；
The normal and tangential contact stiffness of the guide rail joint part are respectively as follows:
wherein alpha is an included angle between the normal acting force of the rolling body and a longitudinal axis;
s34: taking the Z-axis ball screw joint as an example for analysis, establishing a mechanical model of the joint in a milling state, and further analyzing to obtain a displacement balance equation:
and (3) analyzing the stress of the rolling body of the ball screw kinematic pair to obtain a stress balance equation of the rolling body:
wherein Q t For rolling element contact load, f tx 、f ty 、f tz For the force to be applied to the kth rolling element,
the t-th rolling body azimuth angle and the psi lead screw raceway spiral angle are obtained;
under the action of milling load, the stress analysis is carried out between the rolling bodies and the screw roller path in the ball screw kinematic pair, and the mechanical balance equation between all the rolling bodies and the screw roller path is established as follows:
accumulating all rolling body contact forces through the stress balance of each rolling body, and deriving the displacement by the resultant force to obtain a contact stiffness matrix of the ball screw kinematic pair joint part in a milling state;
the step S4 specifically includes:
establishing a complete multi-body dynamic model of the double-turntable five-axis numerical control machine tool by a simultaneous double-turntable five-axis numerical control machine tool dynamic equation and a main joint contact characteristic parameter, wherein the complete multi-body dynamic model is expressed by the following differential equation:
and K (q) is a joint contact stiffness matrix, and equations are solved by combining parameters of all parts of the double-turntable five-axis machine tool in simultaneous formulas (11), (15), (17), (23) and (26).
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210558844.2A CN114895565B (en) | 2022-05-21 | 2022-05-21 | Real-time prediction method for dynamics characteristics of double-turntable five-axis machine tool under milling working condition |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202210558844.2A CN114895565B (en) | 2022-05-21 | 2022-05-21 | Real-time prediction method for dynamics characteristics of double-turntable five-axis machine tool under milling working condition |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN114895565A true CN114895565A (en) | 2022-08-12 |
| CN114895565B CN114895565B (en) | 2024-09-10 |
Family
ID=82724311
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202210558844.2A Active CN114895565B (en) | 2022-05-21 | 2022-05-21 | Real-time prediction method for dynamics characteristics of double-turntable five-axis machine tool under milling working condition |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN114895565B (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117644431A (en) * | 2024-01-29 | 2024-03-05 | 南京航空航天大学 | CNC machine tool machining quality analysis method and system based on digital twin model |
Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107389340A (en) * | 2017-07-20 | 2017-11-24 | 哈尔滨理工大学 | High-speed spindle system dynamics contactless measuring device and method of testing |
| CN111291479A (en) * | 2020-01-21 | 2020-06-16 | 清华大学 | Method for predicting milling stability of series-parallel machine tool |
| WO2022007753A1 (en) * | 2020-07-06 | 2022-01-13 | 北京卫星制造厂有限公司 | Digital twin modeling method oriented to mobile robot milling processing |
| CN113985812A (en) * | 2021-10-19 | 2022-01-28 | 安徽科技学院 | Machining error forecasting method for multi-axis numerical control machine tool |
-
2022
- 2022-05-21 CN CN202210558844.2A patent/CN114895565B/en active Active
Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN107389340A (en) * | 2017-07-20 | 2017-11-24 | 哈尔滨理工大学 | High-speed spindle system dynamics contactless measuring device and method of testing |
| CN111291479A (en) * | 2020-01-21 | 2020-06-16 | 清华大学 | Method for predicting milling stability of series-parallel machine tool |
| WO2022007753A1 (en) * | 2020-07-06 | 2022-01-13 | 北京卫星制造厂有限公司 | Digital twin modeling method oriented to mobile robot milling processing |
| CN113985812A (en) * | 2021-10-19 | 2022-01-28 | 安徽科技学院 | Machining error forecasting method for multi-axis numerical control machine tool |
Non-Patent Citations (1)
| Title |
|---|
| 姜彦翠;刘献礼;吴石;李茂月;李荣义;: "考虑结合面和轴向力的主轴系统动力学特性", 机械工程学报, no. 19, 31 October 2015 (2015-10-31) * |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN117644431A (en) * | 2024-01-29 | 2024-03-05 | 南京航空航天大学 | CNC machine tool machining quality analysis method and system based on digital twin model |
| CN117644431B (en) * | 2024-01-29 | 2024-04-02 | 南京航空航天大学 | CNC machine tool machining quality analysis method and system based on digital twin model |
Also Published As
| Publication number | Publication date |
|---|---|
| CN114895565B (en) | 2024-09-10 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Zhang et al. | Kinetostatic-model-based stiffness analysis of Exechon PKM | |
| Li et al. | Dynamic performance comparison and counterweight optimization of two 3-DOF parallel manipulators for a new hybrid machine tool | |
| Klimchik | Enhanced stiffness modeling of serial and parallel manipulators for robotic-based processing of high performance materials | |
| Dong et al. | A screw theory-based semi-analytical approach for elastodynamics of the tricept robot | |
| CN110315396B (en) | Industrial robot constant-force grinding and polishing method based on big data | |
| Li et al. | Stiffness analysis of a Stewart platform-based parallel kinematic machine | |
| Aginaga et al. | Improving static stiffness of the 6-RUS parallel manipulator using inverse singularities | |
| Tsai et al. | Dynamic modeling and decentralized control of a 3 PRS parallel mechanism based on constrained robotic analysis | |
| CN102495550A (en) | Forward dynamic and inverse dynamic response analysis and control method of parallel robot | |
| CN109634111B (en) | Dynamic deformation calculation method for high-speed heavy-load robot | |
| Shan et al. | Structural error and friction compensation control of a 2 (3PUS+ S) parallel manipulator | |
| Wang et al. | Simplified strategy of the dynamic model of a 6-UPS parallel kinematic machine for real-time control | |
| CN114895565A (en) | Milling working condition-oriented dynamic characteristic real-time prediction method for double-turntable five-axis machine tool | |
| Russo et al. | A review of parallel kinematic machine tools: Design, modeling, and applications | |
| Zhang et al. | Elastodynamic Model-Based Vibration Characteristics Prediction of a Three Prismatic–Revolute–Spherical Parallel Kinematic Machine | |
| Le et al. | Dynamic modeling and control design for a parallel-mechanism-based meso-milling machine tool | |
| CN112001087B (en) | Nonlinear dynamics modeling analysis method for rotary joint type industrial robot | |
| Zhang et al. | Elastodynamic modeling and joint reaction prediction for 3-PRS PKM | |
| Luo et al. | A dynamic parameter identification method for the 5-DOF hybrid robot based on sensitivity analysis | |
| Liu et al. | Performance evaluation of a special 6-PUS type parallel manipulator | |
| Sun et al. | Comparative study of stiffness modeling methods for a novel industrial robotic arm with hybrid open-and closed-loop kinematic chains | |
| 汪满新 et al. | Stiffness analysis of a 4-DOF hybrid robot | |
| Liang et al. | Accuracy analysis of SCARA industrial robot based on screw theory | |
| CN116861596B (en) | Dynamics modeling method and system for 6-degree-of-freedom parallel robot | |
| Jiang et al. | Development of a Multi-DOFs Pneumatic Servo Polishing Robot |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |