Cutting Temperature Measurement during Drilling of Ti6Al4V, Comparison between Modeling and Experimental Results from Thrust and Torque Point of View

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1 Cutting Tmpratur Masurmnt during Drilling of Ti6Al4V, Comparison btwn Modling and Exprimntal Rsults from Thrust and Torqu Point of Viw M. Marinscu*, M. Jrad, A. Dvillz, D. Dudzinski Mtz Univrsity, LPMM Laboratory, UV, 4, ru August Frsnl 577 Mtz, Franc * mmarinscu@univ-mtz.fr Abstract Th titanium alloys ar widly usd in th arospac industry for thir high strngth to wight ratio and for thir xcllnt corrosion but thy ar considrd as a difficult to cut matrials du to th high tmpratur attaind whil machining. In this papr, in a first part, two tmpratur masurmnt tchniqus ar brifly prsntd. On mthod givs th tmpratur distribution in th workpic along th cutting dg. For th scond tchniqu th tmpratur is masurd on th cornr of th drill. For all th xprimntal tsts th forc and torqu ar masurd. In a scond part, an analytical modl basd on a CAD dfinition of th tool is introducd. Th drill cutting dg is dividd into lmntary cutting dgs and a thrmo-mchanical modl of obliqu cutting is applid for ach of thm. Finally th modling rsults ar compard with th xprimntal ons to validat th proposd approach. ITRODUCTIO Lightwight matrials such as titanium alloys ar usd in modrn arospac structurs du to thir vry good combination of mtallurgical and physical proprtis. Each class of lightwight matrial has thir advantags and disadvantags. Titanium s advantags ar a high strngth-to-wight ratio, a low dnsity, an xcllnt corrosion and rosion rsistanc, and a low modulus of lasticity [1, 2]. Howvr titanium has bn classifid as difficult to machin du to its physical proprtis. Th tmpratur attaind whil drilling is vry important and plays an important rol on tool liv. ormally th titanium is drilld using coolants but with th chang of th nvironmntal lgislation this will b prohibitd or unrcommndd. As th titanium is igniting at cutting spds of mor that 2-25 m/min in dry conditions, th solution is to drill with minimal quantity of lubricant (MQL) intrnally through th drill. Th tmpratur masurmnt whil drilling is a srious problm for th cutting industry bcaus of th difficulty to mak in situ masurmnts. On solution is using a drill-foil thrmocoupl [3]. Th us of this mthod allows obtaining th tmpratur distribution along th cutting dg of th drill. Thrmocoupls hav bn also usd to masur th tmpratur. In som cass thrmocoupls hav bn insrtd in th workpic [4] and in othr works thy hav bn insrtd in th drill [5]. From th analytical point of viw: th drill gomtry lads to a variation in th cutting angls and th cutting vlocity along th cutting dgs. Fw workrs hav studid this procss. Som of thm wr intrstd on th drill gomtry; thy proposd gomtrical modls basd on th drills grinding paramtrs as Tsai [6] and Hsih [7]. Othrs proposd modls to prdict th torqu and thrust basd on mchanistic and xprimntal approachs as alloway [8] and Oxford [9] or analytical approachs lik Armargo [1] and Watson [11]. Howvr, in ths works, th gomtrical dscription is limitd to classical twist drill or bvl ground drill. Morovr, th machining of titanium alloys nds a nw gnration of drills with curvd cutting lips and a thinnd chisl dg, Figur 1. An analytical dscription basd on a mathmatical dfinition of tools facs and cutting angls of ths kinds of drills is hr prsntd. This dfinition allows th dtrmination of th inclination angl and th normal rak angl at any point of th cutting lips. In ordr to calculat th thrust and torqu th drill cutting lips is dividd into a sris of lmntal obliqu cutting dgs assumd to b linar and for which th cutting angls ar assumd to b constant. Th thrmomchanical cutting modl of Moufki and al. [12] is finally applid on ach lmnt and by summation th torqu and thrust ar calculatd.

TEMPERATURE AD EFFORTS MEASUREMET TECHIQUES 3.1 Drills prsntation Th xprimnts hav bn conductd on HURO KX 1 CC vrtical machining cntr with a 24 rpm 24kW spindl.

Th tools ar standard Diagr Industri (Frnch carbid and PCD tools manufacturr) drills rcommndd for th machining of titanium alloys. Th twist drills (fig.

With th fifth thrmocoupls th tmpratur on th wall of th hol, at a dpth of 2 tims th diamtr, was also obtaind.

2 TEMPERATURE AD EFFORTS MEASUREMET TECHIQUES 3.1 Drills prsntation Th xprimnts hav bn conductd on HURO KX 1 CC vrtical machining cntr with a 24 rpm 24kW spindl. A Kistlr dynamomtr has bn usd for forc and torqu masurmnts. Two diffrnt drills wr usd for th masurmnts; ach drill has a 15 hlix angl. Th tools ar standard Diagr Industri (Frnch carbid and PCD tools manufacturr) drills rcommndd for th machining of titanium alloys. Th twist drills (fig. 1 a, ar normally usd with coolant fluids and with MQL. th cutting tmpratur along th cutting dg was obtaind. With th fifth thrmocoupls th tmpratur on th wall of th hol, at a dpth of 2 tims th diamtr, was also obtaind. Th hols, in th workpic for th installation of thrmocoupls, wr prformd by lctrorosion with a.5 mm diamtr tool. Th diffrnc btwn th hol and th thrmocoupl diamtrs involvs only a 3% positioning rror rportd at th radius of th drill. Drill bit ~.1 mm T5 Workpic T4 T3 T2 T1 thrmocoupl positioning twist drill twist drill Figur 1: Th two drill bits usd for th Ti6Al4V machining All drill gomtris hav drilld 2 diamtr dpth blind hols. Th dpth of cut, for th twist drills, has bn chosn using th knowldg of th drill manufacturr. For ach drill a study about th bst cutting conditions has bn prformd in conformity with th F standard F E Th obtaind conditions ar prsntd in Tabl I. For th masurmnt of tmpratur two diffrnt tchniqus wr mployd. Drill bit Cutting spd[mm/min] Fd [mm/rot] Tabl I: Optimum drilling conditions for th thr drill bits 3.2 Thrmocoupls mountd in th workpic For th first tchniqu, fiv thrmocoupls hav bn insrtd in th workpic as shown in Figur 3a. Th distanc btwn th had of th thrmocoupls and th drill cutting dgs was.1 mm, at th nd of drilling procss. In Figur 3b a pictur of th drilling montag is prsntd. Th thrmocoupls usd hav a.25 mm diamtr in ordr to hav th lowst thrmal inrtia possibl. Finally, a distribution of workpic with mountd thrmocoupls Figur 2: Th montag for th drill insrtd thrmocoupls 3.3 Thrmocoupls mountd in th drill bit Th scond masurmnt tchniqu is prsntd, Figur 4. A.25 mm diamtr thrmocoupl has bn insrtd in th drill on on of th coolant supplying hols. And othr hol has bn drilld by lctro-rosion in th drill so that it crosss th coolant supplying hol and th cutting fac as prsntd in th Figur 3. Th hol on th cutting fac was positiond at last at.5 mm from th drill two xtrmitis. Th xact position varis from on drill bit to anothr on bcaus of th difficultis ncountrd whil drilling th hols. Figur 3 prsnts on drill with th thrmocoupl insrtd in th tool.

Figur 3: CAD prsntation of th thrmocoupl insrtion hol for 21731 drill ( CAD and ( ral pictur Th drill was moving and th workpic was fixd on th Kistlr dynamomtr, th signal transmission was assurd by a

As on hol of th tool is blockd by th thrmocoupl and as th tool holdr is a pin vic clamping systm, sam problms with th lubrication hav bn ncountrd.

[11] stablishd rlationships for th various angls on th cutting lips and th chisl dg of a bvl ground drill and h studid th ffcts of fd on th various drill cutting angls in function of th nominal

Th scond is basd on th grinding paramtrs of drills. Tsai [6] providd mathmatical modls for drill flank and drill fluts gomtris.

3 Figur 3: CAD prsntation of th thrmocoupl insrtion hol for drill ( CAD and ( ral pictur Th drill was moving and th workpic was fixd on th Kistlr dynamomtr, th signal transmission was assurd by a wir-lss connction btwn a spcial tool holdr and th DAQ systm. In Figur 4, th spcial tool holdr with th tool mountd in and th workpic mountd on th Kistlr dynamomtr is prsntd. As on hol of th tool is blockd by th thrmocoupl and as th tool holdr is a pin vic clamping systm, sam problms with th lubrication hav bn ncountrd. In ths conditions an xtrnal MQL alimntation has bn addd. [11] stablishd rlationships for th various angls on th cutting lips and th chisl dg of a bvl ground drill and h studid th ffcts of fd on th various drill cutting angls in function of th nominal paramtrs. Armargo [8] studid th ffct of th fd on th cutting and inclination angls along th cutting lips. Thy showd that th ffct of th fd can bn nglctd along th cutting lips, but not along th chisl dg. Th scond is basd on th grinding paramtrs of drills. Tsai [6] providd mathmatical modls for drill flank and drill fluts gomtris. Th xplicit quations for th drill flank contour and th drill angls wr drivd and calculatd from th grinding paramtrs. Hsih [7] usd a similar analyss, h proposd a mathmatical modl of th hlicoids grinding surfac. This modl allowd th flut and flank surfacs to b obtaind. Thn th cutting dgs, th cutting and th inclination angls could b numrically calculatd. Figur 4: Th montag for th tmpratur masurmnt THERMOMECHAICAL FORCE MODEL 3.1 Drill gomtry modl Th drill gomtry is composd of two major surfacs: th flut and th flank surfacs. Ths two surfacs dfin th drill strngth, rsistanc and classs. Two kinds of approachs hav bn usd in th dfinition of drill gomtry. Th first is basd on th nominal gomtry paramtr of drills: diamtr, point angl, hlix angl, tc. Oxford [9] obsrvd th drill gomtry; h drivd a trigonomtric rlationship to calculat th inclination angl and th cutting angls along th cutting dgs in function of drill nominal paramtrs. Watson Figur 5: CAD dfinitions for th 2 drills, ( and ( Most of th proposd modls studid th convntional twist drill; this study prsnts a gomtrical and a mchanical approach of twist drills with complx cutting dgs, in particular with curvd cutting dgs. Figur 5 givs a CAD dfinition of this kind of drills. Th gomtrical mthod proposd hr is a combination of th two approachs dscribd prviously. Th quations of flut and flank surfacs ar ob-

4 taind in function of th nominal drill paramtrs. A mathmatical analysis allowd th cutting dg and th cutting angls to b numrically dtrmind. This mthod may b applid to th majority of drills points, particularly to drills with thinnd chisl dg usd in our works. This kind of drills is widly usd in th high spd machining applications. Th obsrvation of th fluts surfacs of this drill shows that thy may b dfind by th flut contour in th initial cross-sction, Figur 6, rotating and translating with rspct to th drill axis. This movmnt is don in such mannr that whn th flut contour rotats of 2π it translats of a distanc qual to th pitch. Figur 6: Hlical rotation of th flut contour Th drill fluts ar groovs prformd in th body of th drill to allow th chip vacuation and facilitat th coolant supply. Th drill fluts ar gnrally hlical; a drill may hav two, thr or four fluts. Th hlix angl and th numbr of fluts hav a significant influnc on th drill rigidity, th chip vacuation procss and th dynamic bhavior of th drill. Th drill studid hr is a two fluts drill with a constant pitch. Th fluts surfacs of th drill, Figur 5, may b dfind by th hlical motion of a cross-sction. Thn, th hlix fluts ar formd by rotating and translating of a chosn cross-sction with rspct to th drill axis Z =Z. Th hlix motion is don in such mannr that whn th flut contour rotats of 2π it translats of a distanc qual to th pitch p. So th flut surfac can b obtaind in th rfrnc fram by th following quation: 2π z 2π z ( x, y, z) = x sin y cos + p p n i 2π z 2π z b x cos + y sin = i i = p p (1) Whr b i ar constants dtrmind by fitting from th coordinats of a numbr of points of drill flut obtaind from th CAD dfinition or dirctly from a drill by tridimnsional masuring. Th drill chosn for this study is a bvl ground drill with a thinnd chisl dg. For this typ of drill th flank fac is a plan, and, thr points ar sufficint to dfin th flank fac quation. Th coordinats of ths points may b obtaind from th CAD dfinition of th drill or vntually from a ral drill with tridimnsional masuring tchniqus. Thn, th quation of th flank surfac can b rprsntd by th quation: F x, y, z = a x + b y + c z + d = (2) f f f f In th sam way, th wb-thinnd zon of th chosn drill, Figur 5, is dfind by an additional plan rprsntd by th quation: H x, y, z = a x + b y + c z + d = (3) h h h h Th wb-thinnd zon of drill prsnt also a cylindrical rgion rprsntd by th quation: C( x, y, z) = x + y + z ( ax + by + cz) a + b + c r = (4) Cutting angls dtrmination To dtrmin th cutting angls at a currnt point P of th cutting lip, Figur 6a, w us th ISO systm of fundamntal plans. At point P, th rfrnc plan PR is normal to th cutting vlocity V dirction, it contains th drill axis. Th tool cutting dg plan PS contains th vctor tangnt to th cutting lip and th plan P is locally normal to th cutting lip. Bfor dfining ths fundamntal plans, it is ncssary to obtain th cutting lip quation. Th cutting lip corrsponds to th intrsction btwn th hlical groov of th drill, dfind by (x,y,z)=, and th flank fac, dfind by F(x,y,z)=. Thn, th cutting lip quation may b obtaind by solving simultanously ths two quations: F( x, y,z) = and ( x, y, z) = (5) It is wll-known that F (or ) is a vctor normal to th surfac F(x,y,z)= (or (x,y,z)=), thn, th unit vctors rspctivly normal to th flank fac and to th groov may b xprssd by :

5 F F + F + F F ' x x ' y y ' z z n = = (6) F F' x + F' y + F' z and: + + = = ' x x ' y y ' z z n (7) ' x + ' y + ' z At th currnt point P of th cutting lip, which is th intrsction btwn th flank fac F and th hlical groov, th unit tangnt vctor is normal to th vctors n F and n calculatd at this point, it is givn by: t = n n n n F F (8) In addition, it must b notd that th unit vctors n F and n, at point P, ar in th plan P normal to th cutting lip, it is th rason why a subscript was addd in th notation. At point P, th cutting dg plan P S contains th unit vctors t tangnt to th cutting lip and th cutting vlocity dirction which is th circumfrntial dirction dnotd θ, Figur 6a. For th currnt point P, (x,y,z) and (r,θ,z) ar rspctivly th cartsian and cylindrical coordinats. At point P, th unit vctor is along th circumfrntial dirction, and th unit vctor is along th radial dirction at point P. At this point, th ffctiv cutting vlocity is dfind by th vlocity of th workpic rlativly to th tool (hr th drill): V( workpic drill) = V V = V 2 2 r f r θ f z V V = 2π r 6; V = f 6 m s r V = V = V + V with (9) f Whr Vr and Vf ar rspctivly th rotational vlocity and th fd vlocity of th drill, is th rotational vlocity xprssd in rv/min. corrsponds to th dynamic or ffctiv cutting dirction unit vctor, it is dtrmind in th Figur 6b by th rlation: V + V r θ f z = = cos β + sin β θ V V z with Vf f tan β = = V 2π r r (1) Figur 6: Fundamntal plans ( and cutting angls ( at point P on th cutting lip In this kinmatical dscription θ corrsponds to th static cutting dirction unit vctor. Th dynamic cutting angls may b now dfind. Th unit vctor n is locally normal to th local P S plan, and thus to th cutting dg. At point P, a local orthonormal basis ( V, κ, n ) may b dfind, th unit vctor κ r is in th P S plan and in th rfrnc P R plan, and is associatd to th ffctiv local cutting dg anglκ, Figur 6b. In th normal plan P, w r introduc th unit vctor t locally tangnt to th groov and locally normal to th cutting dg. It is now possibl to dtrmin, at any point of th cutting lip, th normal rak angl dfind in th normal plan P and givn by th rlation: α 1 n t n V = cos ; n n t α if t n V r α < if t > (11) > <

6 τ Locally, th cutting procss is obliqu with an inclination angl dfind in th plan PS and obtaind by: λ S = t λ S < if t V > (12) κ t λ r S > if t V < 1 κr cos ; 3.2 Th thrmomchanical modl Th drill cutting dgs ar dcomposd into sris of linar lmntary cutting dgs corrsponding to th radial incrmnt dr, Figur 6b. For th currnt point P at radial position r, th local unit bas vctors ( V, κ, ) n r ar mployd. For an lmntary cutting dg, th chip lmnt is obtaind undr obliqu cutting conditions. Thn, th thrmomchanical modl of obliqu cutting [12] was applid for ach lmntary cutting dg. In this modl th chip formation is supposd to occur by sharing within a thin band. In this shar band, th thrmomchanical rspons of th workd matrial is dscribd by a Johnson-Cook law: n ν 1 γ ɺ γ T T = A + B + m 3 3 r 1 ln 1 ɺ γ T T m r (13) Whrτ, γ, γɺ and T rprsnt rspctivly th shar strss, th shar strain, th shar strain rat and th absolut tmpratur. Th charactristics of matrial bhavior ar th strain hardning xponnt n, th strain rat snsitivity cofficint m, th thrmal softning xponnt ν and constants A and B, γɺ, T r (rfrnc tmpratur) and T m (mlting tmpratur). At th tool-chip intrfac, th friction is supposd dscribd by a man friction cofficint µ. Th lmntary cutting forcs df, df, df κ V r n ar calculatd with th obliqu cutting modl associatd to th local cutting conditions in trms of cutting angls, cutting spd, width of cut and undformd chip thicknss. Thy ar projctd in th Tool Coordinat Systm dfind by th basis ( V, κ, ) n. Thir valus ar valuatd from th lmntary chip quilibrium; th intractions btwn th lmntary chips ar nglctd: r ( / ) = R ( / ) dr chip tool d workpic chip with dr workpic chip = df x + d z s s s s dr chip tool = df df + df n V V κr κr n (14) Whr df s and d s rprsnt rspctivly, th shar and th normal forcs applid on th lmntary chip at th xit of th shar band. Ths two componnts ar obtaind from: t dw df = τ s h cos λ sinϕ d s n (15) tan ( ϕ α ) + tan λ cosη n n f c = cosη df 1 tan λ cosη tan ( ϕ α ) s sh s f c n n Whr τ h is th shar strss at th xit of th band and t is th undformd chip thicknss masurd prpndicularly to th cutting vlocity. Th lmntary cutting rsultant dr ( chip / tool) is projctd in th Tool Coordinat Systm dfind by th basis(,, x y z ) : = + + dr chip tool df df df (16) x x y y z z Th associatd lmntary torqu and thrust ar: dc = dc = df r = y df + x df df = df z z z z θ z x y z (17) Finally, th total torqu and thrust ar obtaind by summation of all th lmntal thrust and torqu ovr th cutting lips. EXPERIMETAL AD AALYTICAL RESULTS MQL drilling tsts wr conductd on Ti6Al4V, two hlical drills wr usd. For ach tool two tmpratur masurmnt tchniqus wr mployd, for ach tchniqu th thrust and torqu dvlopd whil drilling wr rcordd. Figur 7 prsnts th masurmnts obtaind with th Kistlr dynamomtr. Th drill prsnts smallr valus for th thrust and th forc compard with th drill. In Figur 8 th tmpraturs masurd with th two tchniqus ar prsntd, and th drill has a bttr bhavior that th drill for both tchniqus.

7 () (*m) 2 FZ () Mz (*m) DRILL 6 7 Tim (s) () FZ () DRILL tim (s) Mz (.m) (*m) Figur 7: Thrust and toqu masurd whil drilling with th ( and ( drills For th first tchniqu, with th thrmocoupls insrtd in th workpic, th tmpratur is obsrvd all along th cutting dg. Th drill has a highr tmpratur, in avrag, but th drill has th most important tmpratur on th cornr of th drill, whr th drill war is th most important. For th scond tchniqu, th thrmocoupl is insrtd in th cornr of th tool. Th scond masur is in concordanc with th first on, from th bginning of th drilling th tmpratur is highr for th drill that for th drill. Evn if th drilling fforts ar mor important for th tool, h has th smallr, rcordd tmpratur. Th thrmomchanical analytical modl was usd to idntify th forc and torqu whil drilling th Ti6Al4V alloy with two diffrnt drills. Also fiv st of valus, from th litratur, wr usd for th Johnson-Cook law, Tabl 2a. Comparing diffrnt laws had allowd us to idntify th most suitabl st of valus to us with th thrmomchanical modl for Ti6Al4V drilling. Tabl 2b prsnt th masurd and analytically idntifid valus for th thrust and torqu for th two drills. Figur 9 shows th variation of th forc and torqu from on st of valus for th J-C law to anothr, rlativ to th masurd ons. Th numbr 1 to 6 on th X axis ar corrsponding to th numbrs in Tabl 3b T ( C) Drilling dpth T( C) drill drill -,1,1,3,5,7,9 1,1 Radius [r/r] Figur 8: Tmpratur masurd with ( th thrmocoupl mountd in th drill tchniqu and ( with th thrmocoupl mountd in th workpic tchniqu 1,2 1,8,6,4,2 1,2 1,8,6,4,2 1,,93 1,8 1, 1,9,93,93,92 Forc,85 Torqu,84 1,1 1, ,,97 1,13 1,14,97,88 1,,99,89,89 Forc Torqu 1,6 1, Figur 9: Rlativ variation of forc and torqu for th diffrnt sts of valus for th J-C law compard with th masurd ons 1, ( drill and ( drill

8 J-C law paramtrs A B m n ν L and Lin 1998 [13] 782,7 498,4,28,28 1 Wang t al ,5 236,6,29,355,42 So t al ,9 653,1,45,198,7 Macdougall and Harding ,3,512,15,8242 Khan t al. 24 [14] ,6349,139, Drill Drill Johnson-Cook law Forc Torqu Forc Torqu Masur (1) 85 6,9 72 5,8 L and Lin (2) 793 6,4 71 5,73 Wang and Rahman (3) 916 7,5 81 6,6 So t al (4) 79 6, ,18 Macdougall and Harding (5) 719 5, ,18 Khan t al (6) 862 6, ,18 Tabl 3: Th fiv st of valus ( for th J-C law and ( th analytical rsults COCLUSIOS In this papr th gomtry of two drills with complx cutting lips wr modld. Th cutting angls on a discrtizd modl of th drill wr calculatd from th CAD dfinition of th tools. For th analytical modl fiv diffrnt Johnson- Cook laws wr usd and th analytical rsults for ach st of valus wr compard with th masurd forc and torqu valus. Finally th L and Lin [13] and Khan and al. [14] J-C laws gav th bst approximations of th masurd valus. For th validation of th analytical approach forc, torqu and tmpratur masurmnts whil drilling with th two tools wr mad. On intrsting aspct is th fact that th tool with th highr torqu and forc rachd a maximum tmpratur smallr than th othr tool. In th analytical modl, quations hav bn introducd [12] prmitting th tmpratur idntification along th cutting dg of th drill, but th rsults obtaind with this quations ar unsatisfying, th global tmpratur is much biggr that xprimntal on and th tmpratur distribution is not cohrnt. Finally w obsrv a good rlation btwn th masurd and th prdictd cutting forcs. REFERECES [1] Ezugwu, E.O.; Wang, Z.M J. Matr. Procss. Tchnol., pp ; 1997 [2] Machado, A.R.; Wallbank, J.; Proc. Inst. Mch. Eng., pp. 53-6; 199 [3] Bono M.; i, J. Journal for manufacturing scinc and nginring;, pp ; ASME 22 [4] Li, R.; Shih, J. Machin tools & manufactur, pp ; 27 [5] Zilmann, R.P.; Wingartnr, W.L. Matrials procssing tchnology, pp [6] W. D. Tsai and S. M. Wu Int. J. Mach. Tool. Ds. Rs., Vol. 19, 1979, pp [7] Jung-Fa Hsih Intrnational Journal of Machin Tools & Manufactur, Vol. 45, 25, pp [8] D. F. alloway Trans. ASME, 1957, pp [9] Oxford Trans. ASME, 1955, pp [1] E. J. A. Armargo Int. J. Mach Tool. Ds. Rs, Vol. 12, 1972, pp [11] A. R. Watson Int. J. Mach. Tool Ds. Rs, 1985, pp [12] Moufki t al. Int. J. Mach. Tools. Manuf, 44, 24, [13] L, W. S., Lin, C. F., 1998, J Matr Procss Tchnol. 75(1-3): [14] Khan t al. Int. J. of Plasticity 2 (24)

CHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle

CHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt