CN104680000A - Orthogonal polynomial-based milling stability prediction method - Google Patents
Orthogonal polynomial-based milling stability prediction method Download PDFInfo
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- CN104680000A CN104680000A CN201510068454.7A CN201510068454A CN104680000A CN 104680000 A CN104680000 A CN 104680000A CN 201510068454 A CN201510068454 A CN 201510068454A CN 104680000 A CN104680000 A CN 104680000A
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Abstract
The invention relates to the field of advanced manufacturing, in particular to an orthogonal polynomial-based milling stability prediction method. The orthogonal polynomial-based milling stability prediction method adopts orthogonal polynomials to approach a state term, a delay term and a periodical coefficient term in a kinetic equation, and adopts a plurality of known time points and responses thereof to fit the needed terms, so that the local errors of the calculation method are reduced, and thereby the precision of the prediction method is increased; meanwhile, in the process of obtaining a stability lobe diagram, an H matrix is introduced instead of directly substituting an F matrix for calculation, so that the number of iterations in the F matrix calculation process is reduced, consequently, the time of the calculation method is saved, and the efficiency of calculation is increased.
Description
Technical field
The present invention relates to advanced manufacture field, be specifically related to a kind of milling stability Forecasting Methodology based on orthogonal polynomial.
Background technology
Along with China constantly incorporates globalizing market, the competitive pressure that manufacturing industry faces is increasing, High-speed Machining Technology is arisen at the historic moment, and be widely used in the complex parts manufacture in the fields such as Aeronautics and Astronautics, boats and ships, mould and automobile, flutter in working angles has a strong impact on the surface quality of workpiece, reduce lathe and cutting-tool's used life, and likely cause the destruction of lathe.The flutter stability of milling process is predicted, selects suitable processing conditions, the generation of flutter can be controlled, improve working (machining) efficiency, cut down finished cost.If do not carry out stability prediction, need to test many times in actual process, just can determine optimum machined parameters, cause a large amount of wastes of the wearing and tearing of cutter and workpiece material, greatly increase manufacturing cost.Therefore carry out prediction to the flutter stability in milling process to have great importance.
A kind of milling parameter stability prediction method based on semi-discrete method is disclosed in " Updated semi-discretization method for periodic delay-differential equations with discrete delay " literary composition on famous International Periodicals " International Journal For Numerical Methods In Engineering " magazine the 61st volume in 2004, use the method can obtain the flutter stability limit, but weak point is, the counting yield of the method is lower, obtain the stability lobes diagram and need long time, and the computational accuracy of this algorithm awaits promoting.
Summary of the invention
The milling stability Forecasting Methodology based on orthogonal polynomial that technical matters to be solved by this invention is to provide a kind of Computationally efficient, high computational accuracy is applicable to multiple parts.
For solving the problems of the technologies described above, the present invention adopts following technical scheme: the present invention includes following steps:
1. the kinetics equation in milling cutter single-degree-of-freedom milling process is set up:
(1)
Wherein,
for constant coefficient matrix,
for the matrix of coefficients of mechanical periodicity in time,
represent that cutter exists
the condition responsive in moment,
represent the natural frequency of point of a knife point,
represent relative damping,
represent modal mass,
represent axial cutting depth,
represent time lag;
represent momentary-cutting-thick, its expression formula is:
(2) in formula (2),
represent the cutter tooth number of milling cutter,
with
be respectively Cutting Force Coefficient that is tangential and normal direction,
be
the angular displacement of cutter tooth, expression formula is
, window function
definition is:
(3)
In formula (3),
with
be respectively
the incision of cutter tooth and cut out angle, when adopting climb cutting,
; When adopting upmilling,
,
for radial direction immerses ratio, the i.e. ratio of radial cutting-in/tool diameter;
2. by the time lag item of the milling process kinetics equation (1) of single-degree-of-freedom
be equally divided into
individual minizone, then time step is
, wherein any one time minizone is expressed as
By equation (1) in time minizone
on carry out integration, obtain
(4)
3. build by structure the status items that orthogonal polynomial carrys out fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
, detailed process is as follows:
Orthogonal polynomial race
coefficient by
(6) determine;
Formula (6) be with
for power is about point set
orthogonal polynomial, wherein
(7)
(8)
By step 2. in the status items of equation (4)
be expressed as by orthonormal polynomial approximation:
(9)
Wherein
it is polynomial expression
coefficient, pass through
(10) determine, wherein
be
the condition responsive in moment;
Get in (6) formula
pass through time point
with its response
come fit procedure 2. in status items,
(7) formula of utilization calculates
, and substitute into
can obtain:
(11)
So
(12)
Obtained by (10) formula
(13)
(14)
(12), (13), (14) formula are substituted into (10) formula and obtain
(15)
Arrange
(16)
In like manner time lag item
can be expressed as:
(17)
(18)；
4. build Floquet transition matrix, (16), (17), (18) formula substituted into (4) formula, can obtain:
(19)
Wherein
(20a)
(20b)
(20c)
(21a)
(21b)
(21c)
(21d)
Equation (19) can be written as
(22)
Wherein
(23)
By equation (22), can obtain current cutter tooth and a upper cutter tooth respond between mapping relations, as follows by matrix representation:
(24)
Wherein
(25a)
(25b)
(25c)
The Floquet transition matrix of system can be expressed as
(26)
Wherein
(27)
5. Floquet transition matrix is calculated
eigenwert, by the stability of the mould decision-making system of eigenwert, concrete decision criteria is as follows:
.
Step of the present invention 3. in also can adopt status items based on building Quadratic Orthogonal polynomial method and come fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
; Detailed process is as follows:
In modus ponens (6)
with
, pass through time point
with the response of its correspondence
carry out matching status items
, wherein
(29a)
(29b)
(29c)
By (7) Shi Ke get
(30)
So
(31)
(32)
(33)
Order
, and substitute into can obtain (30), (31), (32), (33) formula obtain
(34)
(35)
(36)
(37)
(34), (35), (36), (37) formula are substituted into (29c) and obtain
(38)
(29a), (36) and (38) formula are substituted into (10) formula and obtain
So
Arrange
。
The present invention in order to accurately obtain step 1. in modal parameter, need bracing or strutting arrangement be adopted;
Described bracing or strutting arrangement comprises brace table, the first circuit, the second circuit, X to slideway, the cooperation button being provided with through hole, the first slideway, the second slideway, the 3rd slideway, the 4th slideway and the supporting mechanism for fixed support thin-wall part;
Described first circuit and the second circuit are with one heart and be successively set on brace table, described X also passes through the center of circle of the first circuit and the second circuit on brace table to installation of slide track, described first slideway and the second slideway are symmetricly set on X to slideway both sides, described 3rd slideway and the 4th slideway are symmetricly set on X to slideway both sides, and described cooperation button is located at the intersection of the first circuit, the second circuit and the first slideway, the second slideway, the 3rd slideway, the 4th slideway;
Described supporting mechanism comprises train wheel bridge, lower plate, spring and pole, and described lower plate is arranged on pole, and described train wheel bridge is connected with train wheel bridge by spring, and described pole lower end is provided with shoulder, and described pole is connect with coordinating to link by its lower end shoulder.
Good effect of the present invention is as follows: the present invention adopts orthogonal polynomial several to status items, time lag item and the periodic system of approaching in kinetics equation, multiple known time point and response thereof is adopted to carry out matching the required entries, reduce the local error of computing method, thus improve the precision of Forecasting Methodology; Simultaneously in the process obtaining the stability lobes diagram, introduce H matrix, instead of directly substitution F matrix calculates, and reduces the iterations in F matrix computation process, thus save the time of computing method, improve counting yield.
Accompanying drawing explanation
Fig. 1 is that the present invention's radial direction immerses than stability diagram when being 0.05;
Fig. 2 is that the present invention's radial direction immerses than stability diagram when being 0.5;
Fig. 3 is that the present invention's radial direction immerses than stability diagram when being 1;
Fig. 4 is support table structure schematic diagram of the present invention;
Fig. 5 is lower plate structural representation of the present invention;
In the drawings: 1 first circuit, 2 second circuits, 3 brace tables, 4 coordinate button, 5X to slideway, 6 through holes, 7-1 first slideway, 7-2 second slideway, 8-1 the 3rd slideway, 8-2 the 4th slideway, 9-1 train wheel bridge, 9-2 lower plate, 10 springs, 11 poles.
Embodiment
The present invention includes following steps:
1. the kinetics equation in milling cutter single-degree-of-freedom milling process is set up:
(1)
Wherein,
for constant coefficient matrix,
for the matrix of coefficients of mechanical periodicity in time,
represent that cutter exists
the condition responsive in moment,
represent the natural frequency of point of a knife point,
represent relative damping,
represent modal mass,
represent axial cutting depth,
represent time lag;
represent momentary-cutting-thick, its expression formula is:
(2) in formula (2),
represent the cutter tooth number of milling cutter,
with
be respectively Cutting Force Coefficient that is tangential and normal direction,
be
the angular displacement of cutter tooth, expression formula is
, window function
definition is:
(3)
In formula (3),
with
be respectively
the incision of cutter tooth and cut out angle, when adopting climb cutting,
; When adopting upmilling,
,
for radial direction immerses ratio, the i.e. ratio of radial cutting-in/tool diameter;
2. by the time lag item of the milling process kinetics equation (1) of single-degree-of-freedom
be equally divided into
individual minizone, then time step is
, wherein any one time minizone is expressed as
By equation (1) in time minizone
on carry out integration, obtain
(4)
3. build by structure the status items that orthogonal polynomial carrys out fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
, detailed process is as follows:
Orthogonal polynomial race
coefficient by
(6) determine;
Formula (6) be with
for power is about point set
orthogonal polynomial, wherein
(7)
(8)
By step 2. in the status items of equation (4)
be expressed as by orthonormal polynomial approximation:
(9)
Wherein
it is polynomial expression
coefficient, pass through
(10) determine, wherein
be
the condition responsive in moment;
Get in (6) formula
pass through time point
with its response
come fit procedure 2. in status items,
(7) formula of utilization calculates
, and substitute into
can obtain:
(11)
So
(12)
Obtained by (10) formula
(13)
(14)
(12), (13), (14) formula are substituted into (10) formula and obtain
(15)
Arrange
(16)
In like manner time lag item
can be expressed as:
(17)
(18)；
4. build Floquet transition matrix, (16), (17), (18) formula substituted into (4) formula, can obtain:
(19)
Wherein
(20a)
(20b)
(20c)
(21a)
(21b)
(21c)
(21d)
Equation (19) can be written as
(22)
Wherein
(23)
By equation (22), can obtain current cutter tooth and a upper cutter tooth respond between mapping relations, as follows by matrix representation:
(24)
Wherein
(25a)
(25b)
(25c)
The Floquet transition matrix of system can be expressed as
(26)
Wherein
(27)
5. Floquet transition matrix is calculated
eigenwert, by the stability of the mould decision-making system of eigenwert, concrete decision criteria is as follows:
.
the present inventiondescribed step 3. in also can adopt status items based on building Quadratic Orthogonal polynomial method and come fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
; Detailed process is as follows:
In modus ponens (6)
with
, pass through time point
with the response of its correspondence
carry out matching status items
, wherein
(29a)
(29b)
(29c)
By (7) Shi Ke get
(30)
So
(31)
(32)
(33)
Order
, and substitute into can obtain (30), (31), (32), (33) formula obtain
(34)
(35)
(36)
(37)
(34), (35), (36), (37) formula are substituted into (29c) and obtain
(38)
(29a), (36) and (38) formula are substituted into (10) formula and obtain
So
Arrange
。
The present invention in order to accurately obtain step 1. in modal parameter, bracing or strutting arrangement need be adopted, as shown in Figure 4,5, described bracing or strutting arrangement comprises brace table 3, first circuit 1, second circuit 2, X to slideway 5, cooperation button 4, the first slideway 7-1, the second slideway 7-2 that are provided with through hole 6, the 3rd slideway 8-1, the 4th slideway 8-2 and the supporting mechanism for fixed support thin-wall part;
Described first circuit 1 and the second circuit 2 are with one heart and be successively set on brace table 3, described X to be arranged on brace table 3 and by the center of circle of the first circuit 1 and the second circuit 2 to slideway 5, described first slideway 7-1 and the second slideway 7-2 is symmetricly set on X to slideway 5 both sides, described 3rd slideway 8-1 and the 4th slideway 8-2 is symmetricly set on X to slideway 5 both sides, and described cooperation button 4 is located at the intersection of the first circuit 1, second circuit 2 and the first slideway 7-1, the second slideway 7-2, the 3rd slideway 8-1, the 4th slideway 8-2;
Described supporting mechanism comprises train wheel bridge 9-1, lower plate 9-2, spring 10 and pole 11, described lower plate 9-2 is arranged on pole 11, described train wheel bridge 9-1 is connected with train wheel bridge 9-1 by spring 10, described pole 11 lower end is provided with shoulder, and described pole 11 is connected with cooperation button 4 by its lower end shoulder.
The present invention coordinates the support attitude that can complete different spaces shape on brace table 3 by coordinating button 4 with circuit, slideway, effectively can complete and the support of thin-wall part is fixed, utilize bracing or strutting arrangement to carry out clamping and can obtain more accurate modal parameter thus make by utilizing the limit of stability of orthogonal polynomial prediction more accurate.
When given relevant parameter: milling cutter tooth number is 2, and tangential force coefficient and normal force coefficient are respectively
with
, the first natural frequency of point of a knife point is
, modal damping is 0.011, and modal mass is 0.03993kg, climb cutting.By time lag
be divided into 40 minizones, the plane be made up of the speed of mainshaft and the radial cutting degree of depth is divided into
grid.
Above-mentioned steps and parameter are carried out programming by Matlab software and draws stability Lobe figure, the stability in milling process is predicted by stability diagram, choosing different radial directions and immerse ratio, obtaining stability diagram for not getting 0.05,0.5,1, as shown in Figure 1, 2, 3.
The above embodiment is only the preferred embodiments of the present invention, and and the feasible enforcement of non-invention exhaustive.For persons skilled in the art, to any apparent change done by it under the prerequisite not deviating from the principle of the invention and spirit, all should be contemplated as falling with within claims of the present invention.
Claims (3)
1., based on a milling stability Forecasting Methodology for orthogonal polynomial, it is characterized in that it comprises the following steps:
1. the kinetics equation in milling cutter single-degree-of-freedom milling process is set up:
(1)
Wherein,
for constant coefficient matrix,
for the matrix of coefficients of mechanical periodicity in time,
represent that cutter exists
the condition responsive in moment,
represent the natural frequency of point of a knife point,
represent relative damping,
represent modal mass,
represent axial cutting depth,
represent time lag;
represent momentary-cutting-thick, its expression formula is:
(2) in formula (2),
represent the cutter tooth number of milling cutter,
with
be respectively Cutting Force Coefficient that is tangential and normal direction,
be
the angular displacement of cutter tooth, expression formula is
, window function
definition is:
(3)
In formula (3),
with
be respectively
the incision of cutter tooth and cut out angle, when adopting climb cutting,
; When adopting upmilling,
,
for radial direction immerses ratio, the i.e. ratio of radial cutting-in/tool diameter;
2. by the time lag item of the milling process kinetics equation (1) of single-degree-of-freedom
be equally divided into
individual minizone, then time step is
, wherein any one time minizone is expressed as
By equation (1) in time minizone
on carry out integration, obtain
(4)
3. build by structure the status items that orthogonal polynomial carrys out fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
, detailed process is as follows:
Orthogonal polynomial race
coefficient by
(6) determine;
Formula (6) be with
for power is about point set
orthogonal polynomial, wherein
(7)
(8)
By step 2. in the status items of equation (4)
be expressed as by orthonormal polynomial approximation:
(9)
Wherein
it is polynomial expression
coefficient, pass through
(10) determine, wherein
be
the condition responsive in moment;
Get in (6) formula
pass through time point
with its response
come fit procedure 2. in status items,
(7) formula of utilization calculates
, and substitute into
can obtain:
(11)
So
(12)
Obtained by (10) formula
(13)
(14)
(12), (13), (14) formula are substituted into (10) formula and obtain
(15)
Arrange
(16)
In like manner time lag item
can be expressed as:
(17)
(18)；
4. build Floquet transition matrix, (16), (17), (18) formula substituted into (4) formula, can obtain:
(19)
Wherein
(20a)
(20b)
(20c)
(21a)
(21b)
(21c)
(21d)
Equation (19) can be written as
(22)
Wherein
(23)
By equation (22), can obtain current cutter tooth and a upper cutter tooth respond between mapping relations, as follows by matrix representation:
(24)
Wherein
(25a)
(25b)
(25c)
The Floquet transition matrix of system can be expressed as
(26)
Wherein
(27)
5. Floquet transition matrix is calculated
eigenwert, by the stability of the mould decision-making system of eigenwert, concrete decision criteria is as follows:
.
2. a kind of milling stability Forecasting Methodology based on orthogonal polynomial according to claim 1, is characterized in that: described step 3. in also can adopt status items based on building Quadratic Orthogonal polynomial method and come fit procedure 2. Chinese style (4)
, hangover state item
several with time dependent periodic system
; Detailed process is as follows:
In modus ponens (6)
with
, pass through time point
with the response of its correspondence
carry out matching status items
, wherein
(29a)
(29b)
(29c)
By (7) Shi Ke get
(30)
So
(31)
(32)
(33)
Order
, and substitute into can obtain (30), (31), (32), (33) formula obtain
(34)
(35)
(36)
(37)
(34), (35), (36), (37) formula are substituted into (29c) and obtain
(38)
(29a), (36) and (38) formula are substituted into (10) formula and obtain
So
Arrange
。
3. a kind of milling stability Forecasting Methodology based on orthogonal polynomial according to claim 1, is characterized in that: in order to accurately obtain step 1. in modal parameter, need bracing or strutting arrangement be adopted;
Described bracing or strutting arrangement comprises brace table (3), the first circuit (1), the second circuit (2), X to slideway (5), the cooperation button (4) being provided with through hole (6), the first slideway (7-1), the second slideway (7-2), the 3rd slideway (8-1), the 4th slideway (8-2) and the supporting mechanism for fixed support thin-wall part;
Described first circuit (1) and the second circuit (2) are with one heart and be successively set on brace table (3), described X is arranged on brace table (3) to slideway (5) and goes up and the center of circle passing through the first circuit (1) and the second circuit (2), described first slideway (7-1) and the second slideway (7-2) are symmetricly set on X to slideway (5) both sides, described 3rd slideway (8-1) and the 4th slideway (8-2) are symmetricly set on X to slideway (5) both sides, described cooperation button (4) is located at the first circuit (1), second circuit (2) and the first slideway (7-1), second slideway (7-2), 3rd slideway (8-1), the intersection of the 4th slideway (8-2),
Described supporting mechanism comprises train wheel bridge (9-1), lower plate (9-2), spring (10) and pole (11), described lower plate (9-2) is arranged on pole (11), described train wheel bridge (9-1) is connected with train wheel bridge (9-1) by spring (10), described pole (11) lower end is provided with shoulder, and described pole (11) is detained (4) by its lower end shoulder with cooperation and is connected.
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| CN106774148A (en) * | 2017-01-12 | 2017-05-31 | 太原科技大学 | A kind of milling stability Forecasting Methodology based on Bull formula |
| CN110162733A (en) * | 2018-02-13 | 2019-08-23 | 中国商用飞机有限责任公司 | Milling stability analysis method based on whole discrete strategies |
| CN110750891A (en) * | 2019-10-12 | 2020-02-04 | 北京理工大学 | Parallel synchronous orthogonal turning and milling flutter stability lobe graph prediction method |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105184077A (en) * | 2015-09-06 | 2015-12-23 | 河南师范大学 | Excessively-close-range particle-swarm exponential method for optimizing efficiency of resonant electric energy transmitting system |
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| CN106774148A (en) * | 2017-01-12 | 2017-05-31 | 太原科技大学 | A kind of milling stability Forecasting Methodology based on Bull formula |
| CN106774148B (en) * | 2017-01-12 | 2018-11-13 | 太原科技大学 | A kind of milling stability prediction technique based on Bull formula |
| CN110162733A (en) * | 2018-02-13 | 2019-08-23 | 中国商用飞机有限责任公司 | Milling stability analysis method based on whole discrete strategies |
| CN110162733B (en) * | 2018-02-13 | 2023-06-02 | 中国商用飞机有限责任公司 | Milling stability analysis method based on integral discrete strategy |
| CN110750891A (en) * | 2019-10-12 | 2020-02-04 | 北京理工大学 | Parallel synchronous orthogonal turning and milling flutter stability lobe graph prediction method |
| CN110750891B (en) * | 2019-10-12 | 2021-09-21 | 北京理工大学 | Parallel synchronous orthogonal turning and milling flutter stability lobe graph prediction method |
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| CN104680000B (en) | 2018-03-09 |
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