CN107346356B - Method for predicting milling stability of box-type thin-wall part - Google Patents

Method for predicting milling stability of box-type thin-wall part Download PDF

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CN107346356B
CN107346356B CN201710500039.3A CN201710500039A CN107346356B CN 107346356 B CN107346356 B CN 107346356B CN 201710500039 A CN201710500039 A CN 201710500039A CN 107346356 B CN107346356 B CN 107346356B
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matrix
cutter
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position point
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张卫红
杨昀
万敏
马颖超
党学斌
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Northwestern Polytechnical University
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Abstract

The invention discloses a box-type thin-wall part milling stability prediction method which is used for solving the technical problem that the prediction precision of the existing thin-wall part milling stability prediction method is low. Firstly, measuring modal parameters of a cutter and a modal damping ratio of a workpiece by using a modal hammering experiment; then, calculating the dynamic characteristics of the workpiece considering the material removal effect by utilizing a substructure mode synthesis method; then extracting dynamic displacement of the workpiece at different cutter positions and different axial heights; and finally, establishing a multi-point contact milling dynamic model, and substituting the obtained dynamic characteristics of the workpiece into the model and solving the stability. The method simultaneously considers the change of the dynamic parameters of the workpiece caused by material removal, the change of the dynamic parameters at different cutter positions and the change of the dynamic parameters along the axial direction of the cutter, and improves the precision of the milling stability prediction of the box-type thin-wall part.

Description

Method for predicting milling stability of box-type thin-wall part
Technical Field
The invention relates to a method for predicting milling stability of a thin-wall part, in particular to a method for predicting milling stability of a box-type thin-wall part.
Background
Thin-wall parts are used in the aerospace field in a large amount, but efficient and high-precision machining of the thin-wall parts is a difficult problem in the manufacturing field. The part consists of box-shaped units with the wall thickness of 1.5-4mm, the length-thickness ratio of more than 100 and the height-thickness ratio of more than 10, and because the box-shaped units have thin walls, large span and deep cavities, unstable cutting is easy to occur in the milling process, the milling processing efficiency and the part quality of the thin-wall part are seriously influenced, and meanwhile, the cutter abrasion is accelerated, and even the cutter is damaged. Therefore, the method for accurately predicting the milling process stability of the thin-wall part has important significance for improving the processing efficiency and the processing quality of the thin-wall part. Research shows that different from the conventional milling process of a non-thin-wall part, in the milling process of the thin-wall part, the milling stability is greatly influenced by the change of the dynamic characteristics of a workpiece caused by different positions of contact areas of a cutter and the workpiece and the material removal effect. And the dynamic parameters of the box-type thin-wall part are obviously changed due to different positions and material removal, so that researchers develop a great deal of research work on the stability of the milling process of the box-type thin-wall part.
Document 1 "j.x.fei, b.lin, s.yan, x.f.zhang, j.lan, s.g.dai, Chatter prediction for milling of flexible socket-structure, International Journal of advanced manufacturing Technology 89(2017) 2721-2730" discloses a box-type thin-walled workpiece milling stability prediction method; the vibration of the cutter along the feeding direction, the vibration perpendicular to the feeding direction and the vibration of the workpiece along the cutter shaft direction are considered, the dynamic parameters of the workpiece at the center position are used for replacing the dynamic parameters of the workpiece at all positions, and the milling stability of the bottom of the box-type thin-wall part is predicted.
Document 2 "K.Ahmadi, finish strip modeling of the varying dynamics of the in-walled pocket structure reduction, International Journal of advanced manufacturing Technology 89(2017) 2691-; and (3) establishing a structural dynamic model of the thin-wall part by using a finite-strip method, integrating the structural dynamic model with a dynamic model of a milling process, and predicting the milling stability of the box-type thin-wall part.
The method for predicting the milling stability of the existing box-type thin-wall part has the main defects that the accuracy of acquiring the dynamic parameters of the workpiece is reduced and the accuracy of predicting the stability of the box-type thin-wall part is finally reduced without simultaneously considering the change of the dynamic parameters of the workpiece due to material removal, the change of the dynamic parameters of the workpiece at different cutter positions and the change of the dynamic parameters of the workpiece along the axial direction of the cutter.
Disclosure of Invention
The invention provides a box-type thin-wall part milling stability prediction method, aiming at overcoming the defect that the prediction precision of the existing thin-wall part milling stability prediction method is low. Firstly, measuring modal parameters of a cutter and a modal damping ratio of a workpiece by using a modal hammering experiment; then, calculating the dynamic characteristics of the workpiece considering the material removal effect by utilizing a substructure mode synthesis method; then extracting dynamic displacement of the workpiece at different cutter positions and different axial heights; and finally, establishing a multi-point contact milling dynamic model, and substituting the obtained dynamic characteristics of the workpiece into the model and solving the stability. The method simultaneously considers the change of the dynamic parameters of the workpiece caused by material removal, the change of the dynamic parameters at different cutter positions and the change of the dynamic parameters along the axial direction of the cutter, and improves the precision of the milling stability prediction of the box-type thin-wall part.
The technical scheme adopted by the invention for solving the technical problems is as follows: the method for predicting milling stability of the box-type thin-wall part is characterized by comprising the following steps of:
step one, clamping a milling cutter used in the milling process on a machine tool spindle, performing a modal hammering experiment on a cutter-handle-spindle system, measuring to obtain a frequency response function of a plurality of points of the cutter along the axial direction, performing experimental modal analysis on the cutter-handle-spindle system through the frequency response function of the plurality of points to obtain an inherent frequency matrix omega of the cuttertDamping ratio matrix ζtSum mode shape matrix
Figure BDA0001333457500000022
Step two, clamping the workpiece to be machined on a machine tool workbench, and carrying out a modal hammering experiment on the initial workpiece to be machined to obtain a damping ratio matrix zeta of the workpiecew；
Step three, regarding the initial workpiece to be processed as a processed workpiece and a removed material two substructure;
step four, establishing a finite element model for the substructure of the machined workpiece, wherein the undamped vibration dynamic equation of the machined workpiece is as follows:
Mmwümw(t)+Kmwumw(t)＝Fmw(t)
wherein t represents time, MmwAnd KmwRespectively mass matrix and rigidity matrix u of the machined workpiecemw(t) and Fmw(t) displacement and force vector of each node of the processed workpiece;
step five, writing the undamped vibration dynamics equation of the workpiece processed in the step four into
Figure BDA0001333457500000021
Wherein the index i indicates the inner nodes of the finite elements of the machined workpiece which are not in contact with the removed material, the index b indicates the boundary nodes in contact with the removed material, umw,i(t) and umw,b(t) displacement vectors representing inner nodes and boundary nodes, respectively, Fmw,b(t) represents the force vector at the boundary node, Mmw,ii，Mmw,ib，Mmw,biAnd Mmw,bbRespectively, the quality block matrix, K, corresponding to the inner node and the boundary nodemw,ii，Kmw,ib，Kmw,biAnd Kmw,bbRespectively are rigidity block matrixes corresponding to the inner nodes and the boundary nodes;
step six, reducing the finite element model of the processed workpiece by using a fixed boundary substructure modal synthesis method to obtain a reduced matrix
Figure BDA0001333457500000031
Wherein phimw,ikSet of reserved modes, phi, representing fixed boundariesmw,ibRepresenting a set of constrained modalities, Imw,bbRepresenting an identity matrix corresponding to the degree of freedom of the boundary node;
step seven, carrying out finite element modeling on the removed material substructure in the step three to obtain an undamped vibration kinetic equation of the removed material substructure before milling
Figure BDA0001333457500000032
Wherein the content of the first and second substances,
Figure BDA0001333457500000033
and
Figure BDA0001333457500000034
respectively mass matrix and stiffness matrix, u, of material removed before millingm(t) and
Figure BDA0001333457500000035
respectively removing displacement and force vectors of each node of the material before milling;
step eight, dispersing the tool track to obtain nCSA tool position point;
calculating a tool sweeping contour when the tool moves from the initial tool position point to the mth tool position point according to the tool track, and judging units contained in the removed material in the process;
step ten, multiplying the units contained in the removed material by-0.999999 to obtain a mass and rigidity change matrix, namely
Figure BDA0001333457500000036
And
Figure BDA0001333457500000037
eleven, obtaining a matrix of the undamped vibration dynamic equation of the removed material substructure before milling in the seventh step and the mass and rigidity change of the mth cutter position point, namely
Figure BDA0001333457500000038
And
Figure BDA0001333457500000039
combining to obtain an undamped vibration dynamic equation of the removed material substructure at the mth cutter position point
Figure BDA00013334575000000310
Wherein the content of the first and second substances,
Figure BDA00013334575000000311
the force vector of each node of the removed material at the mth cutter position point;
step twelve, removing undamped vibration of the material substructure according to the undamped vibration dynamic equation of the workpiece processed in the step four and the position point of the mth cutter in the step elevenAssembling a dynamic equation, and using the reduced matrix R obtained in the step sixmwAnd transforming the assembled equation set by the displacement coordination condition and the force balance condition of the two substructures to obtain a workpiece finite element reduction model at the mth cutter position point
Figure BDA00013334575000000312
Wherein the content of the first and second substances,
Figure BDA00013334575000000313
and
Figure BDA00013334575000000314
respectively the mass matrix and the stiffness matrix, p, of the finite element reduction model of the workpiece at the mth tool position pointwRepresenting the displacement vector of each generalized node of the workpiece at the mth cutter position point;
thirteen, carrying out calculation modal analysis on the twelve finite element reduction models to obtain the natural frequency matrix of the workpiece at the mth cutter position point
Figure BDA0001333457500000041
And modal transformation matrix
Figure BDA0001333457500000042
Fourteen, extracting dynamic displacement matrixes of a plurality of points in the cutting area of the cutter-workpiece according to the coordinates and the axial cutting depth of the mth cutter position point
Figure BDA0001333457500000043
Wherein
Figure BDA0001333457500000044
Is that
Figure BDA0001333457500000045
A sub-matrix of (a);
fifteen, utilizing the tool natural frequency matrix omega obtained in the first steptDamping ratio matrix ζtSum mode shape matrix
Figure BDA0001333457500000046
The damping ratio matrix zeta of the workpiece obtained in the second stepwThirteenth step, obtaining the natural frequency matrix of the workpiece in the machining process
Figure BDA0001333457500000047
Step fourteen, obtaining dynamic displacement matrix of workpiece in machining process
Figure BDA0001333457500000048
Establishing a dynamic equation of the milling process when the cutter moves to the mth cutter position point:
Figure BDA0001333457500000049
wherein, gamma ist(t)、
Figure BDA00013334575000000410
And
Figure BDA00013334575000000411
respectively displacement, speed and acceleration vectors of the modal coordinates of the tool,
Figure BDA00013334575000000412
Figure BDA00013334575000000413
Γw(t) displacement, velocity and acceleration vectors of the modal coordinates of the workpiece, respectively, and f (t) milling force vector acting on the tool-workpiece cutting area;
sixthly, judging the stability of the kinetic equation in the step fifteen by using the popularized semi-discrete time domain method, and drawing a stability lobe graph.
The invention has the beneficial effects that: firstly, measuring modal parameters of a cutter and a modal damping ratio of a workpiece by using a modal hammering experiment; then, calculating the dynamic characteristics of the workpiece considering the material removal effect by utilizing a substructure mode synthesis method; then extracting dynamic displacement of the workpiece at different cutter positions and different axial heights; and finally, establishing a multi-point contact milling dynamic model, and substituting the obtained dynamic characteristics of the workpiece into the model and solving the stability. The method simultaneously considers the change of the dynamic parameters of the workpiece caused by material removal, the change of the dynamic parameters at different cutter positions and the change of the dynamic parameters along the axial direction of the cutter, and improves the precision of the milling stability prediction of the box-type thin-wall part.
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
Drawings
FIG. 1 is a schematic view of a thin-walled box-shaped member having planar side walls in example 1 of the method of the present invention.
FIG. 2 is a graph comparing the predicted stability lobe plot with the experimental results at 5600rpm for the spindle speed in example 1 of the method of the present invention, in which the solid line represents the predicted value of this example, ○ represents the stable results of the experiment, and × represents the unstable results of the experiment.
FIG. 3 is a schematic view of a thin-walled box-shaped member with a curved side wall according to example 2 of the method of the present invention.
FIG. 4 is a graph of predicted stability lobes for a spindle speed of 7000rpm in example 2 of the method of the present invention.
FIG. 5 is a graph of the predicted stability lobe of the tool at 174.8mm from the start point along the tool path in example 2 of the method of the present invention.
Detailed Description
The following examples refer to fig. 1-5.
Embodiment 1, the prediction of the milling stability of the box-type thin-wall part with the planar side walls is carried out by adopting the method, wherein the box size is 200mm × 100mm × 27mm, the box bottom thickness is 3mm, the side wall before milling is 3mm, the radial cutting depth is 1mm, the workpiece material is aluminum alloy 7050, and the workpiece is clamped on a machine tool workbench through a pressing plate.
(1) Clamping a 3-tooth flat-head end mill with the diameter of 16mm on a main shaft of a machine tool through a spring chuck tool shank, wherein the extension length of a tool is 50mm, and the tool is made of hard alloy; tool for cutting toolPerforming a modal hammering experiment on the handle-spindle system, measuring to obtain a frequency response function of 3 points of the cutter along the axial direction, performing experimental modal analysis on the cutter-handle-spindle system through the 3-point frequency response function to obtain an inherent frequency matrix omega of the cuttertDamping ratio matrix ζtSum mode shape matrix
Figure BDA0001333457500000052
(2) Clamping a workpiece to be processed on a machine tool workbench, and carrying out a modal hammering experiment on an initial workpiece to be processed to obtain a damping ratio matrix zeta of the workpiecew；
(3) Regarding an initial workpiece to be processed as a processed workpiece and a material-removing substructure;
(4) establishing a finite element model for the substructure of the machined workpiece, wherein the undamped vibration dynamic equation of the machined workpiece is as follows:
Mmwümw(t)+Kmwumw(t)＝Fmw(t)
wherein t represents time, MmwAnd KmwRespectively mass matrix and rigidity matrix u of the machined workpiecemw(t) and Fmw(t) displacement and force vector of each node of the processed workpiece;
(5) writing the undamped vibration dynamic equation of the workpiece processed in the step (4) as
Figure BDA0001333457500000051
Wherein the subscript "i" denotes an inner node of the nodes of the finite element of the machined workpiece which is not in contact with the removed material, the subscript "b" denotes a boundary node in contact with the removed material, and umw,i(t) and umw,b(t) displacement vectors representing inner nodes and boundary nodes, respectively, Fmw,b(t) represents the force vector at the boundary node, Mmw,ii，Mmw,ib，Mmw,biAnd Mmw,bbRespectively, the quality block matrix, K, corresponding to the inner node and the boundary nodemw,ii，Kmw,ib，Kmw,biAnd Kmw,bbRespectively are rigidity block matrixes corresponding to the inner nodes and the boundary nodes;
(6) reducing the finite element model of the processed workpiece by using a fixed boundary substructure modal synthesis method to obtain a reduced matrix
Figure BDA0001333457500000061
Wherein phimw,ikSet of reserved modes, phi, representing fixed boundariesmw,ibRepresenting a set of constrained modalities, Imw,bbRepresenting an identity matrix corresponding to the degree of freedom of the boundary node;
(7) finite element modeling is carried out on the removed material substructure in the step (3), and an undamped vibration dynamic equation of the removed material substructure before milling is obtained
Figure BDA0001333457500000062
Wherein the content of the first and second substances,
Figure BDA0001333457500000063
and
Figure BDA0001333457500000064
respectively mass matrix and stiffness matrix, u, of material removed before millingm(t) and
Figure BDA0001333457500000065
respectively removing displacement and force vectors of each node of the material before milling;
(8) the tool track is dispersed to obtain nCSA tool position point;
(9) calculating a tool sweeping contour when the tool moves from the initial tool position point to the mth tool position point according to the tool track, and judging a unit contained in the removed material in the process;
(10) the mass and stiffness variation matrix, i.e., the mass and stiffness variation matrix, is obtained by multiplying the elements contained in the removed material by-0.999999
Figure BDA0001333457500000066
And
Figure BDA0001333457500000067
(11) by using the undamped vibration dynamic equation of the removed material sub-structure before milling in the step (7) and the mass and rigidity change matrix of the mth cutter position point, namely
Figure BDA0001333457500000068
And
Figure BDA0001333457500000069
combining to obtain the undamped vibration dynamic equation of the removed material substructure at the mth cutter position point
Figure BDA00013334575000000610
Wherein the content of the first and second substances,
Figure BDA00013334575000000611
the force vector of each node of the removed material at the mth cutter position point;
(12) assembling the undamped vibration dynamic equation of the processed workpiece in the step (4) and the undamped vibration dynamic equation of the material substructure removed in the mth cutter position point in the step (11), and using the reduced matrix R obtained in the step (6)mwAnd transforming the assembled equation set by the displacement coordination condition and the force balance condition of the two substructures to obtain a workpiece finite element reduction model at the mth cutter position point
Figure BDA00013334575000000612
Wherein the content of the first and second substances,
Figure BDA00013334575000000613
and
Figure BDA00013334575000000614
respectively the mass matrix and the stiffness matrix, p, of the finite element reduction model of the workpiece at the mth tool position pointwRepresenting the displacement vector of each generalized node of the workpiece at the mth cutter position point;
(13) performing calculation mode analysis on the finite element reduction model in the step (12), and obtaining the natural frequency matrix of the workpiece at the mth cutter position point
Figure BDA0001333457500000071
And modal transformation matrix
Figure BDA0001333457500000072
(14) Extracting a dynamic displacement matrix of a plurality of points in a cutting area of the cutter-workpiece according to the coordinates and the axial cutting depth of the mth cutter position point
Figure BDA0001333457500000073
Wherein
Figure BDA0001333457500000074
Is that
Figure BDA0001333457500000075
A sub-matrix of (a);
(15) utilizing the inherent frequency matrix omega of the tool obtained in the step (1)tDamping ratio matrix ζtSum mode shape matrix
Figure BDA0001333457500000076
Damping ratio matrix zeta of the workpiece obtained in the step (2)wThe natural frequency matrix of the in-process workpiece obtained in step (13)
Figure BDA0001333457500000077
The dynamic displacement matrix of the workpiece in the machining process obtained in the step (14)
Figure BDA0001333457500000078
Establishing milling when tool moves to mth tool location pointKinetic equation of process:
Figure BDA0001333457500000079
wherein, gamma ist(t)、
Figure BDA00013334575000000710
And
Figure BDA00013334575000000711
respectively displacement, speed and acceleration vectors of the modal coordinates of the tool,
Figure BDA00013334575000000712
Figure BDA00013334575000000713
Γw(t) displacement, velocity and acceleration vectors of the modal coordinates of the workpiece, respectively, and f (t) milling force vector acting on the tool-workpiece cutting area;
(16) and (5) judging the stability of the kinetic equation in the step (15) by using a generalized semi-discrete time domain method, and drawing a stability lobe graph.
Through the steps, the peripheral milling stability lobe graph of the box-type thin-wall part with the planar side walls can be predicted, and as can be seen from fig. 2, the prediction result of the method is consistent with the experiment, and the accuracy of the method is proved.
Example 2: the method is adopted to predict the peripheral milling stability of the box-shaped thin-walled part with the curved surface on the side wall, one side wall of the box-shaped part is the curved surface, and the equation of the curved surface is
x(u,v)＝RWcos(θW+(u-1)(2θW-π))
y(u,v)＝RWsin(θW+(u-1)(2θW-π))+0.155-RWu,v∈[0,1]
z(u,v)＝28v
Wherein, thetaW＝1.164705892287963rad，RW0.43035714285714m, wherein x, y and z are m. The other 3 sidewalls are planar. The box size is 340mm× 155mm, × 28mm, the thickness of the box bottom is 2mm, the wall thickness before milling is 2.8mm, the radial cutting depth is 1mm, the workpiece material is aluminum alloy 7050, and the workpiece is clamped on a workbench of a machine tool through a pressing plate.
(1) Clamping a 3-tooth flat-head end mill with the diameter of 16mm on a main shaft of a machine tool through a spring chuck tool shank, wherein the extension length of a tool is 50mm, and the tool is made of hard alloy; performing modal hammering experiment on the tool-tool handle-main shaft system, measuring to obtain a frequency response function of 3 points of the tool along the axial direction, performing experimental modal analysis on the tool-tool handle-main shaft system through the 3-point frequency response function to obtain an inherent frequency matrix omega of the tooltDamping ratio matrix ζtSum mode shape matrix
Figure BDA0001333457500000081
(2) Clamping a workpiece to be processed on a machine tool workbench, and carrying out a modal hammering experiment on an initial workpiece to be processed to obtain a damping ratio matrix zeta of the workpiecew；
(3) Regarding an initial workpiece to be processed as a processed workpiece and a material-removing substructure;
(4) establishing a finite element model for the substructure of the machined workpiece, wherein the undamped vibration dynamic equation of the machined workpiece is as follows:
Mmwümw(t)+Kmwumw(t)＝Fmw(t)
wherein t represents time, MmwAnd KmwRespectively mass matrix and rigidity matrix u of the machined workpiecemw(t) and Fmw(t) displacement and force vector of each node of the processed workpiece;
(5) writing the undamped vibration dynamic equation of the workpiece processed in the step (4) as
Figure BDA0001333457500000082
Wherein subscript "i" denotes an inner node of the nodes of the finite element of the machined workpiece which is not in contact with the removed material, and subscript "b" denotes a boundary node in contact with the removed material，umw,i(t) and umw,b(t) displacement vectors representing inner nodes and boundary nodes, respectively, Fmw,b(t) represents the force vector at the boundary node, Mmw,ii，Mmw,ib，Mmw,biAnd Mmw,bbRespectively, the quality block matrix, K, corresponding to the inner node and the boundary nodemw,ii，Kmw,ib，Kmw,biAnd Kmw,bbRespectively are rigidity block matrixes corresponding to the inner nodes and the boundary nodes;
(6) reducing the finite element model of the processed workpiece by using a fixed boundary substructure modal synthesis method to obtain a reduced matrix
Figure BDA0001333457500000083
Wherein phimw,ikSet of reserved modes, phi, representing fixed boundariesmw,ibRepresenting a set of constrained modalities, Imw,bbRepresenting an identity matrix corresponding to the degree of freedom of the boundary node;
(7) finite element modeling is carried out on the removed material substructure in the step (3), and an undamped vibration dynamic equation of the removed material substructure before milling is obtained
Figure BDA0001333457500000084
Wherein the content of the first and second substances,
Figure BDA0001333457500000091
and
Figure BDA0001333457500000092
respectively mass matrix and stiffness matrix, u, of material removed before millingm(t) and
Figure BDA0001333457500000093
respectively removing displacement and force vectors of each node of the material before milling;
(8) the tool track is dispersed to obtain nCSA tool position point;
(9) calculating a tool sweeping contour when the tool moves from the initial tool position point to the mth tool position point according to the tool track, and judging a unit contained in the removed material in the process;
(10) the mass and stiffness variation matrix, i.e., the mass and stiffness variation matrix, is obtained by multiplying the elements contained in the removed material by-0.999999
Figure BDA0001333457500000094
And
Figure BDA0001333457500000095
(11) by using the undamped vibration dynamic equation of the removed material sub-structure before milling in the step (7) and the mass and rigidity change matrix of the mth cutter position point, namely
Figure BDA0001333457500000096
And
Figure BDA0001333457500000097
combining to obtain the undamped vibration dynamic equation of the removed material substructure at the mth cutter position point
Figure BDA0001333457500000098
Wherein the content of the first and second substances,
Figure BDA0001333457500000099
the force vector of each node of the removed material at the mth cutter position point;
(12) assembling the undamped vibration dynamic equation of the processed workpiece in the step (4) and the undamped vibration dynamic equation of the material substructure removed in the mth cutter position point in the step (11), and using the reduced matrix R obtained in the step (6)mwAnd transforming the assembled equation set by the displacement coordination condition and the force balance condition of the two substructures to obtain a workpiece finite element reduction model at the mth cutter position point
Figure BDA00013334575000000910
Wherein the content of the first and second substances,
Figure BDA00013334575000000911
and
Figure BDA00013334575000000912
respectively the mass matrix and the stiffness matrix, p, of the finite element reduction model of the workpiece at the mth tool position pointwRepresenting the displacement vector of each generalized node of the workpiece at the mth cutter position point;
(13) performing calculation mode analysis on the finite element reduction model in the step (12), and obtaining the natural frequency matrix of the workpiece at the mth cutter position point
Figure BDA00013334575000000913
And modal transformation matrix
Figure BDA00013334575000000914
(14) Extracting a dynamic displacement matrix of a plurality of points in a cutting area of the cutter-workpiece according to the coordinates and the axial cutting depth of the mth cutter position point
Figure BDA00013334575000000915
Wherein
Figure BDA00013334575000000916
Is that
Figure BDA00013334575000000917
A sub-matrix of (a);
(15) utilizing the inherent frequency matrix omega of the tool obtained in the step (1)tDamping ratio matrix ζtSum mode shape matrix
Figure BDA00013334575000000918
Damping ratio matrix zeta of the workpiece obtained in the step (2)wThe natural frequency matrix of the in-process workpiece obtained in step (13)
Figure BDA00013334575000000919
The dynamic displacement matrix of the workpiece in the machining process obtained in the step (14)
Figure BDA00013334575000000920
Establishing a dynamic equation of the milling process when the cutter moves to the mth cutter position point:
Figure BDA0001333457500000101
wherein, gamma ist(t)、
Figure BDA0001333457500000102
And
Figure BDA0001333457500000103
respectively displacement, speed and acceleration vectors of the modal coordinates of the tool,
Figure BDA0001333457500000104
Figure BDA0001333457500000105
Γw(t) displacement, velocity and acceleration vectors of the modal coordinates of the workpiece, respectively, and f (t) milling force vector acting on the tool-workpiece cutting area;
(16) and (5) judging the stability of the kinetic equation in the step (15) by using a generalized semi-discrete time domain method, and drawing a stability lobe graph.
Through the steps, the peripheral milling stability lobe graph of the box-type thin-walled part with the curved surface in fig. 3 can be predicted, and fig. 4 and 5 illustrate that the method is suitable for the box-type thin-walled part with the curved surface, and the milling stability of the part at different cutter positions and different rotating speeds can be predicted.

Claims (1)

1. The method for predicting milling stability of the box-type thin-wall part is characterized by comprising the following steps of:
step one, using in the milling processClamping the milling cutter on a machine tool spindle, carrying out modal hammering experiment on the cutter-handle-spindle system, measuring to obtain frequency response functions of a plurality of points of the cutter along the axial direction, carrying out experimental modal analysis on the cutter-handle-spindle system through the frequency response functions of the plurality of points to obtain an inherent frequency matrix omega of the cuttertDamping ratio matrix ζtSum mode shape matrix
Figure FDA0001333457490000011
Step two, clamping the workpiece to be machined on a machine tool workbench, and carrying out a modal hammering experiment on the initial workpiece to be machined to obtain a damping ratio matrix zeta of the workpiecew；
Step three, regarding the initial workpiece to be processed as a processed workpiece and a removed material two substructure;
step four, establishing a finite element model for the substructure of the machined workpiece, wherein the undamped vibration dynamic equation of the machined workpiece is as follows:
Figure FDA0001333457490000012
wherein t represents time, MmwAnd KmwRespectively mass matrix and rigidity matrix u of the machined workpiecemw(t) and Fmw(t) displacement and force vector of each node of the processed workpiece;
step five, writing the undamped vibration dynamics equation of the workpiece processed in the step four into
Figure FDA0001333457490000013
Wherein the index i indicates the inner nodes of the finite elements of the machined workpiece which are not in contact with the removed material, the index b indicates the boundary nodes in contact with the removed material, umw,i(t) and umw,b(t) displacement vectors representing inner nodes and boundary nodes, respectively, Fmw,b(t) represents the force vector at the boundary node, Mmw,ii，Mmw,ib，Mmw,biAnd Mmw,bbRespectively, the quality block matrix, K, corresponding to the inner node and the boundary nodemw,ii，Kmw,ib，Kmw,biAnd Kmw,bbRespectively are rigidity block matrixes corresponding to the inner nodes and the boundary nodes;
step six, reducing the finite element model of the processed workpiece by using a fixed boundary substructure modal synthesis method to obtain a reduced matrix
Figure FDA0001333457490000014
Wherein phimw,ikSet of reserved modes, phi, representing fixed boundariesmw,ibRepresenting a set of constrained modalities, Imw,bbRepresenting an identity matrix corresponding to the degree of freedom of the boundary node;
step seven, carrying out finite element modeling on the removed material substructure in the step three to obtain an undamped vibration kinetic equation of the removed material substructure before milling
Figure FDA0001333457490000021
Wherein the content of the first and second substances,
Figure FDA0001333457490000022
and
Figure FDA0001333457490000023
respectively mass matrix and stiffness matrix, u, of material removed before millingm(t) and
Figure FDA0001333457490000024
respectively removing displacement and force vectors of each node of the material before milling;
step eight, dispersing the tool track to obtain nCSA tool position point;
calculating a tool sweeping contour when the tool moves from the initial tool position point to the mth tool position point according to the tool track, and judging units contained in the removed material in the process;
step ten, multiplying the units contained in the removed material by-0.999999 to obtain a mass and rigidity change matrix, namely
Figure FDA0001333457490000025
And
Figure FDA0001333457490000026
eleven, obtaining a matrix of the undamped vibration dynamic equation of the removed material substructure before milling in the seventh step and the mass and rigidity change of the mth cutter position point, namely
Figure FDA0001333457490000027
And
Figure FDA0001333457490000028
combining to obtain an undamped vibration dynamic equation of the removed material substructure at the mth cutter position point
Figure FDA0001333457490000029
Wherein the content of the first and second substances,
Figure FDA00013334574900000210
the force vector of each node of the removed material at the mth cutter position point;
step twelve, assembling the undamped vibration dynamic equation of the workpiece processed in the step four and the undamped vibration dynamic equation of the material substructure removed in the mth cutter position point in the step eleven, and using the reduced matrix R obtained in the step sixmwAnd transforming the assembled equation set by the displacement coordination condition and the force balance condition of the two substructures to obtain a workpiece finite element reduction model at the mth cutter position point
Figure FDA00013334574900000211
Wherein the content of the first and second substances,
Figure FDA00013334574900000212
and
Figure FDA00013334574900000213
respectively the mass matrix and the stiffness matrix, p, of the finite element reduction model of the workpiece at the mth tool position pointwRepresenting the displacement vector of each generalized node of the workpiece at the mth cutter position point;
thirteen, carrying out calculation modal analysis on the twelve finite element reduction models to obtain the natural frequency matrix of the workpiece at the mth cutter position point
Figure FDA00013334574900000214
And modal transformation matrix
Figure FDA00013334574900000215
Fourteen, extracting dynamic displacement matrixes of a plurality of points in the cutting area of the cutter-workpiece according to the coordinates and the axial cutting depth of the mth cutter position point
Figure FDA00013334574900000216
Wherein
Figure FDA00013334574900000217
Is that
Figure FDA00013334574900000218
A sub-matrix of (a);
fifteen, utilizing the tool natural frequency matrix omega obtained in the first steptDamping ratio matrix ζtSum mode shape matrix
Figure FDA0001333457490000031
The damping ratio matrix zeta of the workpiece obtained in the second stepwThirteenth step, obtaining the natural frequency matrix of the workpiece in the machining process
Figure FDA0001333457490000032
Step fourteen, obtaining dynamic displacement matrix of workpiece in machining process
Figure FDA0001333457490000033
Establishing a dynamic equation of the milling process when the cutter moves to the mth cutter position point:
Figure FDA0001333457490000034
wherein, gamma ist(t)、
Figure FDA0001333457490000035
And
Figure FDA0001333457490000036
respectively displacement, speed and acceleration vectors of the modal coordinates of the tool,
Figure FDA0001333457490000037
Figure FDA0001333457490000038
Γw(t) displacement, velocity and acceleration vectors of the modal coordinates of the workpiece, respectively, and f (t) milling force vector acting on the tool-workpiece cutting area;
sixthly, judging the stability of the kinetic equation in the step fifteen by using the popularized semi-discrete time domain method, and drawing a stability lobe graph.
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