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A new design concept for milling tools of spherical surfaces obtained by kinematic generation

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A new design concept for milling tools of spherical surfaces obtained by kinematic generation
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  A new design concept for milling tools of spherical surfacesobtained by kinematic generation C. Neagu a , M. Lupeanu a,b, ⇑ , A. Rennie b a Department of Manufacturing Technology, POLITEHNICA University of Bucharest, 313 Splaiul Independentei, Bucharest 77206, Romania b Lancaster Product Development Unit, Engineering Department, Lancaster University, Lancashire LA1 4YR, United Kingdom a r t i c l e i n f o  Article history: Received 14 September 2011Receivedinrevisedform12November2012Accepted 21 December 2012Available online 4 January 2013 Keywords: Spherical millingProcessing kinematicsTool geometryDesign methodology optimization a b s t r a c t In this paper, the authors show an srcinal methodology for optimization of tool geometryused in the milling of spherical surfaces. This methodology is based on thorough theoret-icalresearch, whichhighlightsthekinematiccharacteristicsofsphericalsurfacegenerationbymilling. Thesecharacteristics leadtotheconclusionthattheconstructiveangles of toolsused in spherical surface milling must have the same values on both cutting edges.The paper also presents a practical design methodology of these tools, both for exteriorspherical milling, and for interior spherical milling.The theoretical research was validated by manufacturing physical tools used in process-ing spherical joints from 330MW turbines, as well as in processing of sphere taps used inthe petrochemical industry.   2012 Elsevier Inc. All rights reserved. 1. Introduction Nowadays, parts withspherical surfaceshaveawide rangeof useinmanufacturing, installationsandappliances. Inaddi-tiontosectorsthatarenowconsideredwellestablished,suchasautomotivemanufacture,specialistagriculturalvehiclesandmachinery, mining, petrochemical equipment, lifting and transporting machines, spherical surface parts are becomingincreasingly more pertinent in current and emerging industry-leading technologies and applications. Amongst these are,for example, large steam-powered turbines, high precision machining tools, machines and robots for aerospace engineering,human hip implants and knee prostheses.In the aforementioned sectors and applications, spherical surface parts are used mainly in bearing construction [1–4],being preferred instead of classical parts due to their compactness, and to their technical-functional advantages, amongstwhich the most important are:   allows angular movement in any direction, which ensures taking over both radial and axial loads, under the bestconditions;   in the case of an optimal design, it shows the same rigidity in any direction in space;   less sensitive to eccentricity or angular displacements, due to fitting errors or elastic and thermal deformations fromoperation; the effect on the filmof the liquid lubricant, due to these reasons, is much lower than for cylindrical bearings,whichmadethemimposeinthemanufactureofhighprecisionmachiningtools,ofturbines,ofgyroscopesforspaceshut-tles etc.;   friction losses are very small; for spherical bearings lubricated with gases, the friction coefficient is under 5  10  6 , anadvantage exploited successfully in aerospace engineering applications. 0307-904X/$ - see front matter    2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2012.12.017 ⇑ Corresponding author at: Department of Manufacturing Technology, POLITEHNICA University of Bucharest, 313 Splaiul Independentei, Bucharest77206, Romania.Applied Mathematical Modelling 37 (2013) 6119–6134 Contents lists available at SciVerse ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm  The functional – constructive particularities of spherical bearings are classified depending on their sectoral utilisation.In the constructions of cars, tractors and agricultural machinery, lifting and transportation machines, spherical bearingswith dry friction are used [3]. These are executed with total interchangeability, the preferred fit being H7/h6. The materials used for their execution are improvement or hardening steels, alloy or non-alloy, superficially hardened, to withstand thewear and contact pressure. The core must be tenacious, to give the spherical bearing fatigue resistance.In turbine construction, spherical bearings with semi-fluid friction are used [5]. The two elements, axle-bearing, are exe- cuted in pairs which do not ensure interchangeability.Thisrequirementisdeterminedbythefactthatconjugateslipsurfaces,mustensureazerominimumclearanceandbear-ing surfaces up to (90 . . . 95%) of the total contact surface. For this purpose, spherical surfaces are executed with very high(0.5 . . . 1)  l mgeometricalshapeanddimensionalprecisionandwithroughnessof(0.1 . . . 0.2)  l m,evaluatedbyRaparameter[1].In the construction of machine tools and equipment for transportation of heavy loads, hydrostatic spherical bearings areused. The materials used for the construction of these bearings, are generally, improved alloy steels, heat treated oruntreated.Gaslubricatedspherical bearingsareusedintheconstructionof highprecisionmachinetools, ofgyroscopes[5] forspaceshuttles etc. Spherical gliding surfaces are accomplished in these cases with extremely high precision.Human knee and hip prostheses are executed from special alloys, with particular consideration given in the case of tita-nium; dimensional and shape precision of the spherical surfaces is extremely accurate, and the smoothness of the upperlayer highly advanced [6–9].Henceforth, spherical gliding surface work pieces have rigorous precision and roughness conditions, and for their man-ufacture, advanced knowledge of technology processing is required. For each individual application of the spherical surfacepart, different characteristics are necessary. The processes used to obtain the surfaces are chosen in consideration of thesecharacteristics particularized for individual applications [1,5,7,10,11].Currently, spherical surfaces are obtained by:   High speed machining:- Turning (roughing);- Milling (roughing, semi-finishing);- Grinding (semi-finishing);- Lapping (finishing);- Cold Plastic Deformation (finishing);   Electro Discharge Machining (EDM);   Electro Chemical Machining (ECM);   Additive Manufacturing Technologies (AM).Frequently utilised procedures for roughing and finishing processing of spherical surfaces are turning and milling [1,10–13]. In order to obtain high precision and superior smoothing of spherical surfaces, processes like grinding and lapping arealsoapplied. ECMis usedforindividualor forlowvolumebatchproductionof thesesurfaces. Surfacefinishingis quitegood, Nomenclature M   generator point, from the active cutting edge of the milling tool O  centre of the reference system C  1  generatrix curve C  o  directrix curve x 1 , x 0  angular velocities r  1  radius of the generating circle C 1 R  =  OA  radius of the spherical surface, the distance from the centre of the reference system to the generatrix curve C 1 K OO 1  from the centre of the generatrix curve plane to the centre of the spherical surface OXYZ   director reference system O 1  X  1 Y  1  Z  1  complementary reference system u  instantaneous rotation angle of the tool h  instantaneous rotation angle of the part b  functional angle, which remains constant V   cutting speed V  1  main cutting speed V  0  feed rate a , c , k , v  functional main angles 6120  C. Neagu et al./Applied Mathematical Modelling 37 (2013) 6119–6134  buttothedetrimentofproductivity.Largesphericalsurfacesaredifficulttoobtainwiththistechnology.EDMisalsousedformanufacturing of small surfaces in low volume batches.Inthe caseof parts withlargespherical surfaces(whichcontainthepieces rotationaxis), processingbyturningis accom-plished with difficulty. The cutting tool travels an infinite number of omothetic directrix, the cutting velocity is variable,leaning towards zero when the tip of the tool reaches the axis of the part [1]. In that respect, for achieving large sphericalsurfaces, milling is preferred. Nevertheless, in recent years advances in micro-milling have started to reveal the benefitsof processing very small spherical surfaces with milling tools [14,15].Milling of spherical surfaces is currently done with profiled tools or with ball nose milling heads [16–19]. The main dis- advantage of these tools are the uneven wear of the cutting edge due to variable cutting speed, resulting in poor quality of the spherical surface and the tools can be used only for small or medium surfaces [20,21].The methodology presented in detail in this paper introduces for the first time, the possibility of designing milling toolsfor spherical surfaces using kinematic generation.Information regarding the design methodology of tools used for milling of spherical surfaces is still commercially confi-dential by thoseutilisingthis method, andthus, in specialisedliterature, this information is quasi-inexistent. Inthis paper, ascientific and systematic methodology is presented, which ensures optimal design of tools for spherical surface milling.The newmethodologyis developed infour mainstages. The first stage implies definingthe kinematics of a spherical sur-face processed through milling. Cutting speed  V   is defined within the identified specific parameters. This allows stating theformulafor theparametricequationsofthetrajectory. Thesecondstageinvolvesdrawingthemainconclusionfromthecut-tingmovementtrajectoryanalysis.Thisleadstothedefinitionofthestepofthetrajectory,consideringthecoordinatesofthegenerator point after a complete rotation of the milling tool. The tool functional geometry is defined in the third stage andconsistsof setting:thereferencesystemsandtransformational relations;theequationof theclearancesurface;theequationof the locatingsurface; the equationof the cuttingedge; the equationof the functional baseplane; the equationof the func-tional work plane; the equation of the functional plane of the cutting edge; and expressions of functional angles. Graphicalrepresentation and interpretations are undertaken to showthe most significant findings. A correctly designed tool must en-sure the same workingconditions, meaningthat the angles onboth edges must be equal. A computer simulationexample isgivenforbettercomprehensionofthewaytheinstantaneousrotationangleofthetoolinfluencesthemodificationofthetoolfunctional geometry. The final stage is proposing the optimal designof the tools usedin millingof spherical surfaces, detail-ing the main designsteps of the newmethodology. Pictures of the physical manufacturedmillingheads are providedto val-idate the methodology.The proposedmethodologyallows manufacturing of verylarge spherical surfaces with(5 . . . 10)  l mprecisionand rough-ness of Ra= (0,4 . . . 1,6)  l m, and with contact areas of 90 –95% of the total spherical surface [1]. Using the newly developedtools, the milling process of spherical surfaces (including large) can be classified in the semi-finishing/ finishing category. 2. Kinematics of spherical surface processing through milling  Generic theory for kinematics in face and cylindrical milling is set in [10–13,22–26]. Based upon this previously estab-lished research, the authors have defined the kinematics of spherical milling.Inthe context of processing spherical surfacesthroughmilling, the  M   generator point, fromthe active cuttingedgeof thetool (Fig. 1), describes the generatrix curve  C  1 , with  x 1  angular velocity, whilst the plane of the generatrix curve has a rev-olutionmotionaround O  centre, of thesurface,with x 0  angularvelocity.Initsrevolutionmotion,theplaneof thegeneratrixcurve  C  1  intersects the plane of the directrix curve  C  o  under the angle  b , which maintains unchanged during the entire pro-cessing. The generationgeometrymust ensurethe invariable distance  K   = OO 1 , fromthe centre of the generatrix curve planetothecentreofthesphericalsurface(Fig.1).Theselectionofdistance K   conditionstheradius r  1  ofthegeneratingcircle C  1  onwhich the tool transits: r  1  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 2   K  2 p   ;  R  ¼  OA ;  r  1  ¼  O 1 M  :  ð 1 Þ Thepositionofpoint M   intheplaneofthegeneratrixcurvecanbeknowninanymomentthroughtheinstantaneousrotationangle  u  of the tool and radius  r  1 , and the position in the directrix curve plane through the instantaneous rotation angle  h  of the part and the variable radius r v  , of the directrix circumference ( r  v  = O 2 M  ).The motion of the generator point will be reported amongst two Cartesian reference systems – director reference system OXYZ   and the complementary reference system  O 1  X  1 Y  1  Z  1 . The complementary reference system, being a mobile referencesystem, needs thestabilityof the directorcosinesof its axis inaccordancewiththedirector referencesystem. Thesedirectorcosines are given by the matrix (2).  X  1  Y  1  Z  1  X cos b cos h   sin h   sin b cos h Y cos b sin h   cos h  cos b sin h  Z sin b  0  cos b ð 2 Þ Cutting speed  V   (Fig. 1) is the sum of two components: C. Neagu et al./Applied Mathematical Modelling 37 (2013) 6119–6134  6121  V   ¼  V  1  þ  V  0 :  ð 3 Þ These represent the main cutting speed ð V  1 Þ  and the feed rate ð V  0 Þ .The feed rate is a composed speed and is expressed by the following vector equation: V  0  ¼  V  01  þ x o    r  1 ;  ð 4 Þ in which  V  01  represents the speed of the complementary reference system srcin, in accordance with the director referencesystem. This velocity is expressed by the following equation: V  01  ¼  x o Ksin b sin h  i  þ x o Ksin b cos   j :  ð 5 Þ Developing the cross product  x o  r  1  the following expression is obtained: x o    r  1  ¼  x o r  1 ð cos b sin h cos u þ  cos h sin u Þ  i  þ x o r  1 ð cos b cos h cos u þ  sin h sin u Þ  j :  ð 6 Þ Taking into consideration the projections of the feed rate  V  ox ,  V  oy  and  V  oz   on the three axes of the director reference system,they can be expressed based on the above, demonstrated with the corresponding connections: V  ox  ¼  x o Ksin b sin h   x o r  1 ð cos b sin h  cos u þ  cos h sin u Þ ; V  oy  ¼  x o Ksin b cos h  þ x o r  1 ð cos b cos h  cos u   sin h sin u Þ ; V  oz   ¼  0 : ð 7 Þ The main cutting speed  V  1  is expressed through the following equation: V  1  ¼  r  1 x 1 ð cos b cos h sin u þ  sin h cos u Þ  i    r  1 x 1 ð cos b sin h sin u þ  cos h cos u Þ   j    r  1 x 1 sin b sin u :  ð 8 Þ From which the projections on the three axes can be identified: V  1  x  ¼  r  1 x 1 ð cos b cos h sin u þ  sin h cos u Þ ; V  1  y  ¼  r  1 x 1 ð cos b sin h sin u   cos h cos u Þ ; V  1  z   ¼  r  1 x 1 sin b  sin u : ð 9 Þ TothevectorialEq.(3)correspondsthethreescalarequationsoftheprojectionsontheaxesofthedirectorreferencesystem: V   x  ¼  V  1  x  þ  V  ox ;  V   y  ¼  V  1  y  þ  V  oy ;  V   z   ¼  V  1  z   þ  Voz  :  ð 10 Þ Developing Eq. (10) and using Eqs. (7) and (9) we obtain: V   x  ¼  r  1 x 1 ð cos b cos h sin u þ  sin h cos u Þ  x o ð r  1 cos b  sin h cos u þ  r  1 cos h sin u þ  Ksin b  sin h Þ ; V   y  ¼  r  1 x 1 ð cos b  sin h sin u þ  cos h cos u Þ  x o ð r  1 cos b cos h cos u   r  1 sin h sin u þ  Ksin b cos h Þ ;  ð 11 Þ V   z   ¼  r  1 x 1 sin b  sin u : Fig. 1.  Spherical surface milling kinematics: (a) generatrix and diretrix curve with reference systems; (b) cutting speed and components.6122  C. Neagu et al./Applied Mathematical Modelling 37 (2013) 6119–6134  Vector  V   is obtained as a vectorial sum of its own projections: V   ¼  V   x  i  þ  V   y   j  þ  V   z   k  ð 12 Þ and its module is given by relation (13): V   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V  2  x  þ  V  2  y  þ  V  2  z  q   :  ð 13 Þ Developing relation (13) we establish: V   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r  21 x 21  þ  r  21 sin 2 u þ ð r  1 cos b cos u þ  Ksin b Þ 2 x 2 o  þ  . . .. . .  þ 2 r  1 x 1 x o ð cos b  þ  Ksin b cos u Þ s   :  ð 14 Þ The analysis of relation (14), in accordance with Fig. 1, allows the following observations:   The multiplication  r  21 x 21  represents the square of the displacement speed of generator point  M   on the generatrix trajec-tory, meaning  V  21 ;   Theexpression r  21 sin 2 u þð r  1 cos b cos u þ Ksin b Þ 2 representsthesquareofthevariableradiusofinstantaneousdirectrixtra- jectory,meaning r  2 v  ;sothemultiplicationbetweenthisexpressionand x 2 o  representsthesquareofdisplacementspeedof generator point  M   on the instantaneous director trajectory, meaning  V  2 o ;.These being said, Eq. (12) can be written as: V   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V  21  þ  V  2 o  þ 2 r  1 x 1 x o ð r  1 cos b  þ  Ksin b cos u Þ q   :  ð 15 Þ This relation can be yet transformed, if it is taken into consideration that the angularity between the two speeds  V  1  and  V  0 results from the expression: cos d  ¼  r  1 cos b  þ  Ksin b cos u r  v  :  ð 16 Þ Thus the narrower equation that represents the cutting speed is: V   ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V  21  þ  V  2 o  þ 2 V  1 V  o cos d q   :  ð 17 Þ This equation can be obtained directly on the construction fromFig. 1, which confirms the validity of the above demonstra-tions. The path of these demonstrations was chosen, firstly because the Eq. (11) and other intermediary results will servenext, in studying the functional geometry of the tool, and secondly, due to the mathematical strictness of the given method.Starting from the relations (9), through the integration of the Eqs. (11), the parametric equations of the cutting displace- ment trajectory are obtained:  X  M   ¼  r  1 cos b cos h cos u   r  1 sin h sin u þ  Ksin b cos h ; Y  M   ¼  r  1 cos b sin h cos u þ  r  1 cos h sin u þ  Ksin b sin h ;  Z  M   ¼  r  1 sin b cos u   Kcos b : ð 18 Þ The kinematics of processing ensures the functional dependency: h  ¼  x o x 1 u :  ð 19 Þ Thus, the parametric equations of the trajectory can be expressed only by one parameter – the rotation instantaneous angleof the tool:  X  M   ¼  r  1 cos b cos u cos x o x 1 u   r  1 sin u sin x o x 1 u þ  Ksin b cos x o x 1 u ; Y  M   ¼  r  1 cos b cos u sin x o x 1 u þ  r  1 sin u cos x o x 1 u þ  Ksin b sin x o x 1 u ;  Z  M   ¼  r  1 sin b cos u   Kcos b : ð 20 Þ 3. Conclusions drawn from the cutting movement trajectory analysis Processing of spherical surfaces by milling is justified for  x 1  > x o .In Fig. 2, the curve of the cutting movement trajectory is graphically represented, projected on the XOZ plane, for certainvalues of the working parameters. These values are: R=60mm; r 1  =40mm;  b  =52 0 and  x o x 1 ¼  25750 . C. Neagu et al./Applied Mathematical Modelling 37 (2013) 6119–6134  6123
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