Summary: Put forward to use be based on grey number (the gray of GM) model forecasts a method to forecast cutting tool durability, the gray that introduced GM model establishs modular method, the cutting tool durability that cuts steel easily through cutting forecasts example to analyse test and verify the feasibility of this method and forecast precision. 1 foreword the cutting function of the machinability for stuff of accurate assessment metal or cutting tool material, normally need undertakes cutting tool wears away experiment, choose different cutting speed to undertake cutting experiments inside limits of speed of commonly used cutting namely, get a group of cutting tool wear away V-T curve, and the basis is given grind blunt standard scale to give curve of cutting tool durability. Cost of this kind of assessment technique of cutting tool durability that is based on cutting experiment is higher (especially to valuable data) , take time is arduous, and come true hard below some circumstances. For this, numerous researcher tries proved recipe way quickly to all sorts of cutting tool durability (if end panel turning is trial law, conic turning trial law, ceaseless quicken law, radio isotope law to wait) undertook experimenting in great quantities study, but these experiment methods all are put in different limitation. In recent years also investigator uses linear to return to a law to forecast cutting tool durability, but should obtain idealer forecast a result, wear away normally in cutting tool level needs to choose 8 ～ at least 10 measure a place and ask this phase has better linear. Be based on ash to count a model (GM model) forecast a method from 20 centuries theory of 80 time earlier gray is founded since, already received success application in a lot of domains. In the process building a model of gray model, gray theory can be developed adequately and use a few data to show information of information and concealed mediumly, according to behavior characteristic data finds out element itself or the maths between the element concerns, extraction builds modular place to need variable, through building the model of differential equation trends of discrete data, know dynamic behavior of the system and development climate. This method has the following characteristic: ① place wants information content less (normally only should 4 above data can build modular) ; ② does not require the transcendental trait that foregone and primitive data distributings, through finite second generate, can distributing random (or obedient random distributings) the original series translate into of aleatoric and smooth disperse is orderly alignment. ③ builds modular precision taller, can retain primary system trait, system of can better report is effective circumstance. The article introduces GM model, in the cutting function experiment that cutting steel material easily, undertook forecasting to giving the cutting tool durability that grinds blunt standard to fall, obtained satisfactory result. The gray that durability of 2 cutting tool forecasts builds the GM(1 that modular method forms by equation of odd variable and first-order differential, 1) gray model is gray theory in more commonly used forecast a model. The cutting tool durability that is based on this model forecasts the method that build a model to be as follows: Press above all normally cutting tool wears away experiment method undertakes cutting experiments, every cutting period of time namely observation face of the knife after cutting tool wears away value VB, enter when cutting tool wear away normally after level, can mix according to record data beforehand the cutting tool of set grinds blunt standard to undertake forecasting to cutting tool durability. The original series that sets observation is X(0)={X(0)(VBi) | I=1, 2, ... , in N}(1) type, x(VBi) is cutting time, face of the knife after VBi is as corresponding as this hour cutting tool wears away value. Be based on gray to be forecasted theoretically ask primitive data is to wait normally spatio-temporal be apart from, but the issue is involved in the article in, face of the knife after cutting time and cutting tool wears away the concern data of the value is satisfying this one requirement hard usually. Because this still needs will hind knife face wears away the commutation that be worth VB is from 1 begin, with 1 increase by degrees, the series that contains an integer, commutation formula is Pi=VBi-VB1c+1(i=1, 2, ... , in N)VB2-VB1(2) type, c is adjust coefficient, can take a cost according to actual condition, limits extraction a cost is 0<c<2. Use interpolation computation to go out be less than Pi and be close to Pi most the cutting time that integral dot handles. Set Ip to be less than Pi and be close to Pi most integral, of time of cutting of point of the point that beg Ip inside the random sex that inserts formula to be X(0)(ip)=X(0)(Pi-1)+iP-Pi-1[X(0)(Pi)-X(0)(Pi-1)]Pi-Pi-1(3) to convert original series to lose, will random alignment turns into have compasses series, need to undertake cumulative data processing to its series normally, the expression that makes cumulative processing to X(0)(ip) is X(1)(ip)=ip ∑ : K=1X(0)(k)(ip=1, 2, ... , after the expression that N)(4) makes cumulative processing to X(0)(Pi) is handled through above for X(1)(Pi)=X(1)(ip)+(Pii-ip)X(0)(Pi)(5) , can make series of coarse primitive data disperse turns into commonly smooth disperse series. After condition of contented and smooth sex, can build forecast model GM(1 basically, 1) , its expression is ^X(1)(t+1)=[X(0)(P1)-u]e-at+uaa(6) type in, a, U knows parameter to wait for argue, can get through matrix operation according to the least square method (expression summary) . To built GM(1, 1) forecasts a model to undertake precision examines and evaluate, if model precision does not accord with a requirement, can use incomplete to differ alignment to build GM(1, 1) model undertakes correction to former model, in order to raise its precision. GM(1, when 1) model satisfies precision to ask, its are reductive data and forecast estimation to calculate formula to be ^X(0)(t+1)=[u-aX(0)(P1)]e-at(t=2, 3, ... , when N)(7) casts cutting time, take T=n+1, to it corresponding Pn+1=Pn+1, by type (7) can seek a Pn+1 cost. By type (2) begs the VBn+1 value that give to be namely with what forecast correspondence of cutting time ^ X(0)(t+1) hind knife face wear extent. Need explains, forecast precision to rise further, the article used ash waiting for dimension to count GM(1 of fill vacancies in the proper order, 1) model, use foregone series to build a GM(1 above all namely, 1) model, the method is narrated to beg before pressing give to calculate a value, in forecasting this the value to fill to be listed into datum next, to make the dimension such as alignment, need at the same time purify a the oldest data; Build GM(1 again on this foundation next, 1) model, beg a the next to calculate a value, fill its in series, at the same time purify a the oldest data... , with this analogize, through forecasting the metabolism of grey number, one by one is forecasted, ordinal fill vacancies in the proper order, till calculate a value,achieve given grind blunt standard till. 3 forecast example and effect analysis to use afore-mentioned gray to forecast a method to cutting two kinds of brands (the cutting tool durability that Y15, Y15b) cuts steel easily undertakes forecasting. It is facilitating contrast, through quadruplet cutting the experiment obtained the cutting tool wear curve of cutting whole journey. Cutting experiment condition is: The first group: Do cutting Y15, ap=1mm, f=0. 2mm/r, v=70m/min; The 2nd group: Do cutting Y15b, ap=1mm, f=0.
2mm/r, v=70m/min; The 3rd group: Do cutting Y15, ap=1mm, f=0.
2mm/r, v=50m/min; Do cutting Y15b, ap=1mm, f=0.
2mm/r, v=50m/min. Bit material all is W18Cr4V, hardness HRC64.
5 ～ 65.
3, grind blunt standard VB=0.
3mm. Watch test data is primitive the first group of VB are worth alignment 0.
0640.
080.
0970.
T4 of 12 cutting time.
35.
77.
610.
The 2nd group of 7 VB are worth 0.
040.
0650.
0870.
T48 of 1 cutting time.
512.
The 3rd group of 516 VB are worth 0.
1050.
120.
1420.
quadruplet VB is worth T24345063 of 153 cutting time 0.
0420.
0620.
0860.
T14 of 11 cutting time.
62642.
360 enter in cutting tool wear away normally after level, take 4 data to undertake forecasting as original series only, till achieve given grind blunt standard till. The original series of test data sees right table. Cutting tool wears away actual measurement curve and cutting tool wear away forecast a curve. Comparative curve is knowable: The first, the 3rd group of experiments (cutting Y15 cuts steel easily) cutting tool durability forecasts a result more accurate. The cutting tool durability of the first group of experiments forecasts error E=2.
42min(absolute value, similarly hereinafter) , relative to error E=8.
5% ; The cutting tool durability of the 3rd group of experiments forecasts error E=5.
27min, relative to error E=2.
9% . The 2nd, quadruplet experiments (cutting Y15b cuts steel easily) cutting tool durability forecasts an error bigger. The cutting tool durability of the 2nd group of experiments forecasts error E=14.
98min, relative to error E=29.
25% ; The cutting tool durability that quadruplet experiments forecasts error E=36min, relative to error E=18% . Look from actual measurement curve, the 2nd, quadruplet experiment is given grind blunt standard to slant apparently big, before cutting tool wear extent reachs this value, already entered wear away quickly level. To this kind of change of wear curve, forecast very hard apparently according to finite foregone data only. When undertaking forecasting to a system, as spatio-temporal elapse, a few inaccuracy decide disturb to the element will enter a system and produce an effect to the system, because this forecasts a model,be in good position impossibly all the time. Although use ash waiting for dimension to count the method such as fill vacancies in the proper order to be able to rise further,forecast precision, but the prospective hour that forecast is further, forecast the interval that be worth ash bigger (when curvilinear happening changes quickly especially such) . By the graph 1 visible, grind cutting tool blunt level set to be in VB=0.
26mm left and right sides is relatively reasonable, right now the cutting tool durability of the 2nd group of experiments forecasts error E=9.
2min, relative to error E=19.
87% ; The cutting tool durability that quadruplet experiments forecasts error E=2.
23min, relative to error E=1.
22% . Although the 2nd group calculates a value the error is largish, but still be inside acceptable range, according to forecast precision level at present to differentiate, still had belonged to forecast. Forecast example result by above knowable, in permit error range inside, use gray model to forecast result of cutting tool durability pretty good. Forecast methodological photograph to compare with what be based on cutting experiment completely, can save cutting time 1 ～ 2 times, can save data of many cutting test at the same time, because this has actual application value on the project. 4 conclusion forecast precision with what gray model forecasts cutting tool durability taller, but relatively cutting tool of bona fide report wears away actual condition and development trend, forecast methodological photograph to compare with convention, can shorten data of experiment time, managing test, improve experiment efficiency, economic value is had on the project. When using gray model to forecast cutting tool durability, it is normal to need to be entered in cutting tool only wear away 4 ～ are extracted when level 5 data that measure a place can, primitive data demand is little, build a model handy easy travel. Wear curve is in the cutting tool that should forecast to wear away normally inside level limits when, forecast a result more accurate, calculate a value opposite error (absolute value) the biggest do not exceed 20% , had belonged to forecast. CNC Milling