Machining of Curved Geometries Knee joint prothesis Intake

PARAMETRIC SURFACES Surfaces of Known Form • Plane surface • Cylindrical surface • Conical

PARAMETRIC CURVES Parametric Representation of Curve x = f ( u ); y =

OFFEST CURVE Parametric Representation of Curve x = f ( u ); y =

TOOL SELECTION Parametric Representation of Curve x = f ( u ); y =

PARAMETRIC SURFACES Flat End Mill Ball End Mill

FREE-FORM SURFACES Parametric Surface x = f (u, w); y = g (u, w);

RULED SURFACE p(u, 1) p(u, 0) Input Two curves p(u, 0), p(u, 1) p(u,

BILINEAR SURFACE p(0, 1) p(1, 0) p(1, 1) p(0, 0)

BILINEAR SURFACE (HYPERBOLIC PARABOLOID) Four Corner Points p(0, 0), p(1, 0), p(0, 1), p(1,

BILINEAR SURFACE (HYPERBOLIC PARABOLOID)

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Machining of Curved Geometries • • Knee joint prothesis • Intake manifold • Impeller Forging die for Blade • Pelton cup • Pinion

PARAMETRIC SURFACES Surfaces of Known Form • Plane surface • Cylindrical surface • Conical surface • Spherical Surface • Toroidal Surface

PARAMETRIC CURVES Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0 u 1. 0 u=1 u=0

CNC Programming

OFFEST CURVE Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0 u 1. 0 pu = dx dy du 0 du dx du 0 pn = -

OFFEST CURVE Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0 u 1. 0 pn n = pn = nx ny 0 Parametric Representation of Offset Curve X = f ( u ) + nx r Y = g( u ) + ny r Z = 0

Machining of Curved Geometries

CNC Programming

CNC Programming Gouging

CNC Programming 1 kmax= rmin

TOOL SELECTION Parametric Representation of Curve x = f ( u ); y = g( u ); z = 0; 0 u 1. 0 pu = dx du dy du 0 (pu x puu ). (pu x puu) k 2 = puu = d 2 x du 2 d 2 y du 2 0 (pu. pu)3

PARAMETRIC SURFACES Surfaces of Known Form • Plane surface • Cylindrical surface • Conical surface • Spherical Surface • Toroidal Surface

w u

PARAMETRIC SURFACES Flat End Mill Ball End Mill

FREE-FORM SURFACES Parametric Surface x = f (u, w); y = g (u, w); z = (u, w); 0 u 1. 0; 0 w 1. 0 δx pu = δy δz δx δu δu δu pw = δy δz δw δw (pu x pw ) n = (pu x pw) δw Offset Surface X = f (u, w) + nx. r; Y = g(u, w) + ny. r; Z = (u, w) + nz. r; 0 u 1. 0; 0 w 1. 0

FREE-FORM SURFACES Parametric Surface x = f (u, w); y = g (u, w); z = (u, w); 0 u 1. 0; 0 w 1. 0 E = (pu. pu ) 2 ) (LN – M u w F = (p. p ) = k 1 k 2 K = (EG – F 2 ) G = (pw. pw ) L = (puu. n ) (EN + GL -2 FM) = 0. 5(k 1 + k 2) M = (puw. n ) H = 2 ) 2 (EG – F ww N = (p. n ) (pu x pw ) n = (pu x pw)

RULED SURFACE p(u, 1) p(u, 0) Input Two curves p(u, 0), p(u, 1) p(u, w) = (1 -w) p(u, 0) + w p(u, 1)

BILINEAR SURFACE p(0, 1) p(1, 0) p(1, 1) p(0, 0)

BILINEAR SURFACE (HYPERBOLIC PARABOLOID) Four Corner Points p(0, 0), p(1, 0), p(0, 1), p(1, 1) p(u, w) = (1 -u) (1 -w) p(0, 0) + u (1 -w) p(1, 0) + (1 -u) w p(0, 1) + u w p(1, 1)

BILINEAR SURFACE (HYPERBOLIC PARABOLOID)