Weihai , China. Keywords: Propeller symmetrical machining; Parallel-serial mechanism; Motion control algorithm; Forward and inverse solution

Transcription

1 Appled Mechancs and Materals ubmtted: IN: , Vols , pp 1-10 Accepted: do:10.408/ Onlne: rans ech Publcatons, wtzerland Knematcs analyss of double-cutter & double-surface symmetrcal machnng tool for large-scale marne propeller Wang Ru 1, a, Ren Yue-yng 1, b and Huang Xn-shua 1, c 1 chool of Naval Archtecture and Ocean Engneerng, Harbn Insttute of echnology at Weha, Weha 6409, Chna a wrht@16.com, b htrenyueyng@16.com, c xns_h@16.com Keywords: Propeller symmetrcal machnng; Parallel-seral mechansm; Moton control algorthm; Forward and nverse soluton Abstract. he tradtonal large-scale marne propeller processng equpments have such dsadvantages as sngle-surface machnng, propeller blade vbraton, poor stffness, repeated clampng and long processng tme. he method of double-cutter & double-surface symmetrcal machnng s proposed for large-scale marne propeller based on the parallel-seral mechansm. he processng method can overcome proposed dsadvantages, and mprove machnng effcency and accuracy. he structural desgn of the processng equpment s ntroduced and the couplng knematcs parameters of the parallel spndle platform are analyzed. he nverse knematcs soluton and mproved forward knematcs soluton are dscussed n detal. And the forward knematcs soluton can solve a troublesome problem about the selecton of good ntal parameters. A smplfed prototype s bult to verfy the soluton method whch could be used n the future actual applcaton as a foundaton. 1 Introducton Large-scale marne propeller, shown n Fg.1, whch dameter s up to 10 meters, s the key component for warshp, workng at hgh workng speed under severe envronment. If the propeller machnng accuracy s poor, t wll cause blade cavtaton, vbraton, nose, wear, etc. hese problems wll decrease the propulsve effcency and ncrease the consumpton, and decrease the level of stealth and safety of submarnes. Fg.1 A fnshed large-scale propeller In order to make the well-desgned propeller performs optmally, 5-axs CNC machnng equpments should be used n the process [1-4]. Mult-axs machnng centers [5-7], seral manpulators [8, 9] and parallel machne tools [10-1] are used n such manufacturng area, domestc and overseas. But these equpments have several shortcomngs n ther processng: 1) For the seral processng equpments, the error transmsson chan s long, and the performance of stffness s poor. Hgh level equpments are requred wth enough scale and weght to manufacture such hgh accuracy processng large propellers. ) hough the stffness performance and the tool orentaton of the orgnal parallel processng equpments are satsfactory but ther tool s workng space s small, these parallel machne tools cannot process large-scale propellers. ) he orgnal sngle-surface machnng method would lead to blade flutter whch would have a great mpact on surface qualty. o some auxlary supports must be used to ensure that the propeller s fxed along axal and crcumferental drectons. 4) Wth All rghts reserved. No part of contents of ths paper may be reproduced or transmtted n any form or by any means wthout the wrtten permsson of rans ech Publcatons, (ID: , Pennsylvana tate Unversty, Unversty Park, UA-08/04/16,:41:5)

2 Appled Mechancs and Materals Vols the sngle-surface processng, the propeller has to be flpped and clamped twce, whch wll ncrease the processng tme, and would cause some avodable processng errors. In ths paper, a new marne propeller processng machne s ntroduced based on the parallel-seral mechansm whch can process the pressure and sucton sde surfaces of propellers smultaneously wth symmetrcal double-tools. hs new numercal control machne tool could elmnate the blade cantlever effect, weaken blade tremor, and mprove processng effcency and accuracy. hese hgh accuracy propellers can reduce shp power consumpton, and ensure the submarne s stealth and securty. Mechansm desgn of machne tool In order to mplement double-cutter & double-surface symmetrcal machnng, the mechansm s shown n Fg.. wo sets of same parallel-seral mechansms are symmetrcally arranged along the pressure sde and sucton sde surfaces of the propeller. he parallel-seral mechansm conssts of -Prsm seral mechansm and -RP parallel mechansm [1, 14]. he base of -RP parallel mechansm s assembled n the X-axs platform of the seral mechansm. he cutter has two angular rotatons (α and β ) n the X-axs and Y-axs whle changng three drvng lengths of parallel mechansm, and the cutter poston can be controlled wth the help of the conventonal three-degree-of-freedom seral mechansm. he X-axs movable platform controls the cutter movement along the propeller blade radus drecton, and the Y-axs and Z-axs movable platforms control the cutter complex movement on the propeller blade secton. he cutter 1 and cutter can process the blade pressure and sucton sde surfaces smultaneously based on the post-processng algorthm. And ths processng method can reduce blade vbraton, mprove support stffness of the propeller wthn a certan degree, shorten the processng tme effectvely and rase the processng effcency. he forward and nverse knematcs solutons of the parallel-seral mechansm are two key technologes for symmetrcal machnng, and wll be analyzed n detal n ths paper. Parallel mechansm Cutter X-axs movable platform Cutter1 Parallel mechansm X-axs movable platform Y-axs movable platform Y-axs movable platform Z-axs movable platform XB Propeller Z-axs movable platform YB ZB Machne base Machne base Movng drecton Mathematcal model of machne tool Fg. tructure of the propeller machnng tool he mathematcal model of parallel-seral mechansm s constructed and shown n Fg.. In order to analyze the knematcs of the mechansm, four relatve coordnate frames are assgned, as shown n Fg. and Fg.4. hey are mechansm base coordnate frame{ B} O B X B Y B Z B, workpece coordnate frame { W} O W X W Y W Z W, tool-holder coordnate frame { M} O M X M Y M Z M, and parallel mechansm reference coordnate frame{ } O X Y Z. In Fg.4, he spndle s assembled n the platform, whch s connected to three struts of varable length by means of ball jonts that are equally

3 14 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development spaced at a nomnal angle of10. he other end of each strut s connected to a rotatonal jont that s assembled n the X-axs movable platform. he coordnate frame { M } s fxed at the movable platform of parallel mechansm wth the X-axs pontng towards 1, the Y-axs pontng along the lne, and the Z-axs pontng to the vertcal drecton. he coordnate frame { } s fxed at the X-axs movable platform wth the same axes drectons as the coordnate frame{ B }. r and R are the rad of crcles passng through jonts and R ( = 1,,) referenced to the coordnate frame { } can be expressed by [ R ], respectvely. he postons of R 1 = 0 0 R = R R 0 R = R R 0 mlarly, the postons of r referenced to the coordnate frame { M } can be expressed by M [ r ] M r 1 = 0 0 r M = r r 0 M r = r r 0 R l1 X Z P l O Y l dh dx Xw dy XB Ow ZB Zw Yw dz Fg. chematc dagram of machnng tool structure Fg.4 chematc dagram of parallel mechansm 4 Inverse knematcs (IK) analyss of machne tool he nverse knematcs problem nvolves the computaton of the postons of the drvng unts f the cutter s poston and orentatons are known [15-18]. he tool-holder coordnate frame { M } wth respect to the parallel mechansm reference coordnate frame{ }, can be descrbed by the transformaton matrx M M k m n Cα Cβ Cγ α γ Cα Cβ γ α Cγ Cα β k m n α Cβ Cγ Cα γ α Cβ γ Cα Cγ α β k m n β Cγ β γ Cβ = = + + Where the orentaton unt vectors k = [ k k k ], m = [ m m m ], and n = [ n n n ] are the drectonal cosnes of the axes XM, YM and ZM wth respect to the coordnate frame { }. α, Cα, and cos( γ ) OB β, Cβ, γ and, respectvely. α, β and Cγ ndcate sn ( ) the spndle platform wth respect to the coordnate frame { } [11, 1]. α, cos( α ), sn ( β ), cos( β ), sn ( γ ) γ are the rotatonal angles (also called the Euler angles) of Let [ x, y, z ] be the poston vector of the orgn of the frame { M} n the frame{ }, we can YB get the Cartesan poston of the ball jonts wth respect to the frame { } M [ x, y, z ] = M + [ xo, y, ] M O z M r =1,, (1)

4 Appled Mechancs and Materals Vols As each strut s constraned by the correspondng rotatonal jont and ball jont, strut 1 can only move n the plane : 0 Ω 1 y 1 = ; strut n the plane Ω : y = x ; and strut n the plane Ω : y = x, as shown n Fg.4. Accordng to Eq. (1), three constrant equatons can thus be obtaned as γ = α () x = r(cos α )(cos β 1) / () y = r(sn α )(1 cos β ) / (4) Eq.(), Eq.() and Eq.(4) show that parameters of z, x, y and γ are all functons of ndependent knematcs α and β n ths mechansm. In other words, three degrees of freedom of the tool-holder platform are defned as one lnear moton motons, α and β. z n the Z drecton and two angular B B B B B If the cutter tp poston and cutter orentaton parameter vector [ x, y, z, α, β ] s known n the coordnate frame { B } wth the help of the CNC preprocessng system. Obvously, α β B = β. hus, the drvng postons X fame { B } can be calculated by B d X x x P O sn β cosα M O B M P d = Y yp yo sn β snα M B = α, d, d Y and d Z of X-, Y- and Z- axs movable platforms n the B d = z + O P cos β + d + d (6) Z P M c H Where P descrbes the cuter length between the tool tp P and the orgn (5) O M of frame { M}. d H s the cantlever length. In order to obtan better stffness propertes, the dstance d c along the Z-axs drecton between O M and O s an optmzed const value, as shown n Fg.4. In the parallel mechansm, the orentaton of the frame { M} s same to the cutter orentaton, and the poston and orentaton of the frame { M } can be descrbed as ( d, α, β ) wth respect to the frame { }. hree postons of ball jonts can be calculated accordng to Eqs. (1), (), (), (4). hus, three drvng strut lengths can be obtaned: l = R x, y, z (7) Accordng to the nverse knematcs analyss descrbed n the above sesson, three drvng strut lengths ( l1, l, l ) of parallel mechansm and three drvng postons ( d X, dy, d Z ) of seral mechansm B B B B B can be calculated f the cutter s poston and orentatons [ xp, yp, zp, α, β ] gven. hese drvng parameters wll be the nput of the machne tool. 5 Forward knematcs (FK) analyss of machne tool If drvng parameters ( l1, l, l ) and ( d X, dy, d Z ) of the processng equpment are known, we wsh B B B B B B to get the precse cutter poston ( P, P, P ) and cutter orentaton (, j, k ), and X Y Z B B B ( ) + ( j ) + ( k ) = 1. hs process s called forward knematcs analyss, and s the nverse process of the nverse knematcs analyss [19-1]. he forward knematcs analyss s mportant to the constructon parameters calbraton and moton control of the processng equpment. c

5 16 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development Accordng to the nverse knematcs analyss of the -RP parallel mechansm, we can get three complcated equatons. l = f ( z, α, β ) ( = 1,,) (8) Eq.(8) shows that the th drvng strut length l s a functon of the poston and orentaton parameters z, α and β of the coordnate frame { M} wth respect to the frame { }. Where l ndcates the th drvng strut length of the parallel mechansm; z s the orgn poston of the frame { M } along the Z -axs of the frame { } ; r and R are the rad of crcles passng through jonts and R ( = 1,,), respectvely; α and β are two Euler angles, and descrbed the orentaton of the frame { M } wth respect to the frame { }, respectvely. Dfferentatng Eq.(8) wth respect to tme yelds l l 1 1 α l1 β l1 z O α α M l = l α l β l zo β = J β M (9) l l z α l β l z O O z M M O M Where J s a Jacoban matrx. he mcro angular dsplacements and lnear dsplacements can be calculated by 1 [ ] α, β, z O = J l1, l, l M (10) Based on Eq.(10), a forward knematcs soluton can be developed by means of tradtonal numercal methods for -RP parallel mechansm, as descrbed n Fg.5. But ths soluton has a troublesome problems: If the nput prelmnary poston and orentaton parameters are unsutable (e.g. the nput ntal parameters s far away the destnaton parameters), the teraton tmes wll be very large, and we even fal to get the correct destnaton parameters [1,]. In ths secton, an mproved numercal method s ntroduced, and t can output the desrable destnaton parameters effcently. he mproved numercal method for -RP parallel mechansm s descrbed as shown n Fg Frstly, an prelmnary const poston and orentaton parameters ( α, β, z ) are constructed, and the correspondng three drvng strut lengths ( l, l, l ) can be calculated, then, the frst strut length 0 changed from l 1 to l 1 wth the step sze l 1. Accordng to the tradtonal forwards knematcs soluton descrbed n Fg.5, When the frst strut length reach the destnaton value l 1, we can get the ntermedate poston and orentaton parameters ( α, β, z ), then let ( α, β, z ) be new ntal 0 parameters, the second strut length changed from l to l wth the step sze l,and yelds ( α j, j, j β z ). mlarly, when the thrd strut length changed from l 0 to l wth the step sze l, and yelds the tool poston and orentaton parameters ( α k, k, k β z ). Because the ntal parameters are closer to the desrable parameters per teraton, we can get the desrable poston and orentaton parameters based on the mproved forwards knematcs soluton. When the poston and orentaton parameters ( α, β, z O ) are calculated, the X M -, Y - and Z - axs poston values of tool tp P n the coordnate frame { } can be obtaned usng Eqs. () and (4).

6 Appled Mechancs and Materals Vols Input ( l1, l, l ) Input ntal ( α, β, z ) = 0 l = IK get Cal ( l1, l, l) 1 J ( l1, l, l ) ( l1, l, l ) + 1 α = α + α β β β z z z = = + l ε Yes Output ( α, β, z ) No α α 1 β = J β z z Fg.5. radtonal flowchart of forwards knematcs soluton for -RP parallel mechansm Fg.6. Improved flowchart of forwards knematcs soluton for -RP parallel mechansm P r cos( α )(cos β 1) / P cosα sn β X PY = r sn( α )(1 cos β ) / P snα sn β P Z z P cos β Because the drvng poston parameters ( d X, dy, d Z ) of the seral X-axs, Y-axs and Z-axs movable platform are known, we can get the cutter tp poston parameters n the coordnate frame { B } by (16)

7 18 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development P P d P P d P P d d B X X X B Y = Y + Y B Z Z Z H where dh s the cantlever length, as shown n Fg.. If the workpece locatng system s gven, we can get the further nformaton about the cutter tp poston and orentaton parameters wth respect to the workpece frame{ W }. Because of lmtatons of space, we can t dscuss ths problem n detal. (17) 6 IK and FK analyss cases In order to verfy the knematcs analyss of double-cutter & double-surface symmetrcal machnng tool. A set of smplfed machne tool s constructed, as shown n Fg.7. he structural parameters of ths machnng tool are lsted: r = 140mm ; R = 80mm ; d = 60mm ; d = 409mm ; 0mm < < 650mm ; 00mm < < 00mm ; 400mm < < 650mm. d X d Y d Z H c Fg.7 mplfed propeller machnng tool able 1 shows the nverse and forwards knematcs analyss cases. Obvously, nputtng the cutter B B B B B tp poston and orentaton parameter [ x, y, z, α, β ] wth the respect of the base frame{ B }, the nverse knematcs soluton can calculated the correspondng drvng parameters X Y Z 1 Conversely, let the drvng parameters be the nput of forwards knematcs soluton, we can get the B B B B B correspondng tool tp poston and orentaton parameters [ x, y, z, α, β ]. he error between [ d, d, d, l, l, l ]. B B B B B B B B B B [ x, y, z, α, β ] and [ x, y, z, α, β ] s very small, and ndcates that the forwards knematcs soluton s avalable. he moton control experment s also done based on the smplfed machnng tool, and verfed the nverse knematcs soluton successfully

8 Appled Mechancs and Materals Vols able 1 IK and FK analyss cases of the machnng equpment toot tp poston and orentaton parameters n frame { B } B B B [ x, y, z, B B α, β ] Inverse knematcs analyss output parameters [ d, d, d, X Y Z l, l, l ] 1 Forwards knematcs analyss output parameters B B B [ x, y, z, B B α, β ] [0.11mm,18.045mm,01.01mm, 0.000,0.000 ] [0.11mm,18.045mm, mm, 41.84mm,41.84mm,41.84mm] [0.109mm, mm,01.015mm, , ] [10.06mm,9.06mm,-1.01mm, 0.045,0.46 ] [ mm,-9.01mm,0.145mm, 0.0,0.005 ] [ mm,6.585mm,54.749mm, mm,40.475mm,95.1mm] [ mm,-107.7mm,1145.0mm, 91.77mm,407.55mm, mm] [10.061mm,9.057mm,-1.008mm, 0.045,0.461 ] [ mm, mm,0.1455mm, 0.0, ] 7 Conclusons 1) Double-cutter & double-surface propeller symmetrcal machnng method s ntroduced, and a mathematcal model s adopted based on parallel-seral mechansm. he nverse knematcs soluton and mproved forwards knematcs soluton are dscussed n detal. ) he spndle frame orentaton s descrbed wth parameter α and β, whle γ s the negatve rato of α. Wth ths method, we can derve that these parameters z, α and β are ndependent n -RP parallel mechansm, and they can descrbe the poston and orentaton of the spndle platform easly. ) he mproved forwards knematcs soluton for -RP parallel mechansm can overcome the dsadvantage of selectng sutable ntal parameters, and can be referenced to other smlar parallel mechansms. 4) From mathematcal analyss cases and practcal test, t shows that the nverse and forwards knematcs solutons are feasble. hese knematcs analyss wll play key role n NC programmng and post-processng system. 5) Our future work wll manly focus on the practcal applcaton based on these algorthms, and develop the double-cutter and double-surface symmetrcal machnng control system, and enable the propeller to obtan the best effcency wth mnmum vbraton and nose. Acknowledgements hs work was fnancally supported by the Natonal Natural cence Foundaton of Chna ( ), the research and development project of handong provnce of Chna (011GGX106), the Fundamental Research Funds for the Central Unverstes (HI. NRIF. 0114). References [1] J.W. Youn, Y. Jun. A dedcated CAD/CAM system for 5-axs machnng of marne propeller // Proceedngs of the AME Desgn Engneerng echncal Conference. Pttsburgh, PA, Unted states: Amercan ocety of Mechancal Engneers, (001), p [].P. Yan,.L. L. Research on the technology of numercal control machnng to propellers. Mechancal and Electrcal Equpment, (005), p (n Chnese) [] L.X. Cao, J. Lu. An ntegrated surface modelng and machnng approach for a marne propeller. Internatonal Journal of Advanced Manufacturng echnology, 5(11) (008), p

9 10 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development [4] H.C. Kuo, W.Y. Dzan. he analyss of NC machnng effcency for marne propellers. Journal of Materals Processng echnology, 14() (00), p [5] J.H. Yoon. ool tp gougng avodance and optmal tool postonng for 5-axs sculptured surface machnng. Internatonal Journal of Producton Research, 41(10) (00), p [6] J.W. Youn. Interference-free tool path generaton n fve-axs machnng of a marne propeller. IN.J.PROD.RE., 41(18) (00), p [7] X.M. Zou. Research of 5-axs machnng technology for large-scale marne propeller. Wuhan: Huazhong Unversty of cence and echnology (007). (n Chnese) [8] he poneerng ntroducton of robot n propeller producton by Yuan Hang. Guangdong shpbuldng, 8 (010), p. 19. (n Chnese) [9] Z. Huang, Y. Huang. he desgn and applcaton of heavy-duty CNC abrasve belt grndng machne wth hgh precson controllable ptch propeller. Journal of Chongqng Unversty, 4(9) (011), p (n Chnese) [10] H. Chen, Z.X. Wang. Computer aded manufacturng of marne propellers by parallel knematcs machne// Proceedngs of the AME Manufacturng Engneerng Dvson 00. Washngton, DC., Unted states: Amercan ocety of Mechancal Engneers, (00), p [11] R. Wang. Research on the key technology of 6-P parallel platform CNC mllng machne. Harbn: Harbn Insttute of echnology, (007). (n Chnese) [1] R. Wang. G. Dng,.. Zhong. Knematcs analyss of a new 5 DOF parallel-seral machne tool based on the vector method. Key Engneerng Materals, 4 (010), p [1] M.. sa, W.H. Yuan. Dynamc modelng and analyss of PR parallel mechansm usng constraned robotc approach// AME th Bennal Conference on Engneerng ystems Desgn and Analyss, EDA010. Istanbul, urkey: Amercan ocety of Mechancal Engneers, (010), p [14]. Parasuraman, P. Chan. Development of RP parallel manpulators// nd Internatonal Conference on Computer and Network echnology, ICCN 010. Bangkok, haland: IEEE Computer ocety, (010), p [15] H.C. Jung, J.D. Hwang, K.B. Park. Development of practcal postprocessor for 5-axs machne tool wth non-orthogonal rotary axes. Journal of Central outh Unversty of echnology, 18(1) (011), p [16] M.A. Nasser, M. Eder, D. Eberts. Knematcs and dynamcs analyss of a hybrd parallel-seral mcromanpulator desgned for bomedcal applcatons// 01 IEEE/AME Internatonal Conference on Advanced Intellgent Mechatroncs: Mechatroncs for Human Wellbeng, AIM 01. Wollongong, NW, Australa: IEEE Computer ocety, (01), p [17] J.D. Hwang, H.C. Jung, K.B. Park. A study on the development of a practcal postprocessor for 5-axs machnng// AME th Bennal Conference on Engneerng ystems Desgn and Analyss, EDA010. Istanbul, urkey: Amercan ocety of Mechancal Engneers, (010), p [18] X.D. Lang, X.F. L, Y. ang. he nverse knematcs analyss of -RP DOF parallel mechansm. Appled Mechancs and Materals, (01), p [19] H.B. Cho, A. Konno, M. Uchyama. Closed-form forward knematcs solutons of a 4-DOF parallel robot. Internatonal Journal of Control, Automaton and ystems, 7(5) (009), p [0] Z.L. Wang, J.J. HE, H. hang. Forward knematcs analyss of a sx-dof stewart platform usng PCA and NM algorthm. Industral Robot, 6(5) (009), p [1] M. Dehghan, M. Eghtesad, A.A. afav. Neural network solutons for forward knematcs problem of HEXA parallel robot// Proceedngs of the Amercan Control Conference. eattle, WA, Unted states: Insttute of Electrcal and Electroncs Engneers Inc., (008), p [] A. Ghasem, Z. ohel, I. Msagh. Forward knematcs analyss of a -PR parallel manpulator. World Academy of cence, Engneerng and echnology, 61 (010), p. 9-5.

10 Advanced Manufacturng and Informaton Engneerng, Intellgent Instrumentaton and Industry Development / Knematcs Analyss of Double-Cutter & Double-urface ymmetrcal Machnng ool for Large- cale Marne Propeller /