Plan to combine aspect of tooth of radial shave tine with the least square method

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Summary: The tooth flank that puts forward a kind to plan to add up to the disperse on cutting tool tooth flank to nod with the least square method plans to add up to a method, gave out the simple expression of tooth flank. Should plan to add up to curved surface to be able to be used at process of tine of fictitious and radial shave, generate thereby by shave gear tooth flank, through be being compared with involute tooth flank, can examine the validity that cutting tool designs. Keyword: Curved surface of radial shave tine plans to add up to a least square method fictitious examine Fitting Of Points On Profile Of Radial ShavingCutters By Least Square MethodLiu Qiong Et AlAbstract: The Method Of Fitting Points On The Profile Of Radial Shaving Cutters By Using The Least Square Method Is Yielded, and A Simple Expression To Build The Gear Profile Is Also Presented. It Can Simulate The Process Of Radial Shaving Cutting, and Then Deduce The Gear Profile To Be Shaved. By Comparing With The Involute Profile, the Correction Of The Cutter Design Method Can Be Tested. Keywords: 　 of 　 of Radial Shaving Cutting Surface Fitting Least Square Method Virtual Testing 1. Preface spends principle of gear clench the teeth freely according to line contact sheet [1] the tooth flank of knife of radial shave tine that computation gives already was not involute helicoid, however the dimensional curved surface that tooth form and teeth directional pass long body, design and make what this kind of complex curved surface gives radial shave tine knife brought certain difficulty. At present the commonnest treatment method is a radial shave tine tooth face and involute helicoid undertake comparative, calculate a long body volume, use next try grind means to undertake blade is ground. This kind of method can create the waste of manpower, material resources, in making a process also hard to avoid produces an error. For the validity of system of CAD of knife of tine of shave of radial of test and verify and production treatment process, the data of face of shave tine tooth that can age according to radial shave knife CAD system is outputted or the actual measurement data that already processed face of tooth of radial shave tine, plan to combine the dimensional curved surface of cutting tool tooth flank, age according to radial shave the clench the teeth of the gear in the process and knife of radial shave tine concerns, draft this to add up to curved surface to be used at fictitious shave tine treatment, turn over solution to go out thereby by the data of fictitious tooth flank of shave gear, and with what want treatment by shave gear primitive parameter undertakes comparative, satisfy the involute tooth flank of the requirement through examining to be machined to age whether face of teeth of a cogwheel is, will judge radial shave with this system of tine knife CAD or production system is accurate, examine thereby cutting tool design, production validity, in order to avoid the waste of manpower, material resources. Be restricted as a result of conditional place at present, return the data of actual tooth flank after cannot getting radial shave ages the knife is made, but the point of discrete data of tooth flank of shave tine knife that can pass pair of radial shave to age computation of knife CAD system goes out undertakes dimensional curved surface plans to close, build tooth flank model thereby. This kind of method still can be used at examining the production validity of knife of radial shave tine, the actual tooth flank of the knife of radial shave tine after wanting to measure a treatment only, place of usable article narrates a method to build corresponding tooth flank model, turn over solution to go out by shave gear, the gear that is like fictitious treatment to go out satisfies equation of former involute helicoid, explain cutting tool makes a system accord with a requirement; Show production system or issue of design system existence otherwise. 2. The tooth flank of knife of radial shave tine plans to close to age for shave of radial of test and verify knife CAD system [2] validity, discrete data of tine knife tooth flank nods the radial shave that need ages radial shave knife CAD fastens statistical cipher out (if express,1 is shown) plan to synthesize simpler curved surface equation. To examine radial shave ages the production of the knife is correct, can measure with 3 coordinate machine the tooth flank of knife of radial shave tine that measures system of quantities to had been built, through coordinate commutation translate into expresses the form of 1 can. Express data of tooth flank of system of CAD of tine of 1 radial shave (unit: Mm) - 4. 000000 - 3. 000000 - 2. 000000 - 1. 000000 - 124. 000000 - 5. 950840 - 10. 065058 - 14. 195260 - 18. 340619 - 123. 000000 - 3. 130734 - 7. 285553 - 11. 456149 - 15. 641677 - 122. 000000 - 0. 358891 - 4. 554893 - 8. 766440 - 12. 992668 - 121. 0000002. 361937 - 1. 875851 - 6. 128923 - 10. 396397 - 120. 0000005. 0287780. 748581 - 3. 546610 - 7. 855890 - 119. 0000007. 6384033. 315147 - 1. 022775 - 5. 374442 　 set M of × of the N inside foregone and rectangular region site (Xi, yj) (I ＝ 0, 1... , n - 1; J ＝ 0, 1, ... , m - 1) the function that attend is worth Zij, its least square plans to add up to multinomial to be [3] 　 secures Y above all, tectonic to X M least square plans to add up to multinomial [4] among them each ω K (X) (K ＝ 0, 1, ... , p - 1) the multinomial that is mutual orthogonality, construct by the following recurrence formula ＝ of the K in type 1, 2, ... , p - 1. If make have a ground,least square principle can be gotten next the least square of reframe Y plans to combine polynomial form in each ψ 1 (Y) (L ＝ 0, 1, ... , q - 1) the multinomial that also is mutual orthogonality, ＝ of the L in making type by the following recurrence formula 1, 2, ... , q - 1. If make, what least square principle can get the basis to be able to get binary function finally plan to add up to multinomial to will go up again type changeover is type (1) standard multinomial form. In actual computation process, spill over to prevent operation, xi and Yj use Xi* ＝ Xi respectively (I ＝ 0, 1, ... , n - 1) with Yj* ＝ Yj (J ＝ 0, 1, ... , m - 1) replace, among them right now, n duality plans to join the multinomial rank number to add up to multinomial about drafting (P, q) , can have used rank the way that plans to close will decide, differ through using namely (P, q) undertake planning to close, find out to plan to combine precision from which taller (P, q) combination. Accordingly, the basis is given a certain number of data on face of tooth of radial shave tine are nodded (X2, y2, z2) , can beg those who get its tooth flank plan to add up to multinomial. Plan to add up to polynomial coefficient rank to express according to what computation gets 2. Express 2 much to type coefficient A12341 - 5. 322984e ＋ 00 - 4. 232273e ＋ 00 - 7. 490875e ＋ 031. 454388e - 0422. 658371e ＋ 00 - 4. 152488e - 021. 349375e - 042. 938021e - 063 - 2. 667969e - 02 - 2. 920113e - 047. 500000e - 062. 534722e - 074 - 4. 995104e - 04 - 3. 501736e - 062. 500000e - 072. 569444e - 075 - 1. 007292e - 05 - 1. 215278e - 08 - 6. 203839e - 17 - 3. 472222e - 086 - 3. 541667e - 075. 069444e - 089. 972622e - 18 - 3. 611111e - 08 　 　 3. The tooth flank of knife of radial shave tine that conclusion passes pair of radial shave to age knife CAD system is outputted is calculated result (express 1 data) undertake planning to close, analyse all sorts of differring (P, q) the discretion that plans to combine precision when combination, reach through checking computations P ＝ 6, q ＝ 4 when plan to combine precision taller, the coefficient Aij that plans to add up to multinomial right now sees a table 2, right now plan to add up to precision to be: Dt (0) 2. 233370e - 28, dt (1) ＝ 6. 011858e - 14, dt (2) ＝ 7. 105427e - 15. Among them Dt (0) the sum of squares that expresses to plan to add up to multinomial and several foothold error; Dt (1) the absolute value the sum that expresses to plan to add up to multinomial and several foothold error; Dt (2) the maximum that expresses to plan to add up to multinomial and the absolute value that figure foothold error. When P ＞ 6, q ＞ 4 when, the others each plan to add up to coefficient to all be 0, and plan to add up to precision and P ＝ 6, q ＝ 4 when identical; When P ＜ 6 or Q ＜ 4 when plan to add up to requirement of precision short of. The radial shave that builds accordingly ages tooth face equation is will bit more foregone X2, y2 coordinate value takes the place of to go up form calculation is knowable: Plan to close to passing a point, of Z2 plan to combine precision very tall, with the watch 1 in corresponding Z2 agrees almost completely, this are OK from Dt (2) the value sees; Plan to close to passing a point, of Z2 plan to add up to precision to also can be achieved decimally hind the 6th. Accordingly, use this maths model to be able to undertake parameter of feature of face of tooth of radial shave tine (wait like tooth flank Yao Fashi) computation, for fictitious make provided academic basis. This model already was used at fictitious examine in planning to close to seek operation instead, and accordingly fictitious make the involute helix tooth flank that went out to be machined with this cutting tool [4] . Beg instead go out be gotten on by shave gear tooth flank a series of dot (X1, y1, z1) , era enters type (1) , according to Z1 the value can beg a θ 1, recombine X1 value can beg a μ 1, according to type (1) can beg a Y ′ 1. Compare Y1 and Y ′ 1, from computation the result can see, its error is in 10 - 4mm less than, because face of tooth of this this radial shave tine drafts,adding up to a method is correct and feasible. This method still can apply at examining further the production validity of knife of radial shave tine, the tooth flank of knife of radial shave tine after wanting to measure a treatment actually only, can plan to add up to corresponding cutting tool tooth flank. CNC Milling CNC Machining