Projection of Solids: Projection of Solids Hareesha N G Dept of Aeronautical Engg Dayananda Sagar College of Engg Bangalore-78 3/27/2012 1 Hareesha N G, DSCE, Blore
: SOLIDS To understand and remember various solids in this subject properly, those are classified & arranged in to two major groups. Group A Solids having top and base of same shape Cylinder Prisms Triangular Square Pentagonal Hexagonal Cube Triangular Square Pentagonal Hexagonal Cone Tetrahedron Pyramids ( A solid having six square faces) ( A solid having Four triangular faces) Group B Solids having base of some shape and just a point as a top, called apex . 3/27/2012 2 Hareesha N G, DSCE, Blore
PowerPoint Presentation: SOLIDS Dimensional parameters of different solids. Top Rectangular Face Longer Edge Base Edge of Base Corner of base Corner of base Triangular Face Slant Edge Base Apex Square Prism Square Pyramid Cylinder Cone Edge of Base Base Apex Base Generators Imaginary lines generating curved surface of cylinder & cone. Sections of solids( top & base not parallel) Frustum of cone & pyramids. ( top & base parallel to each other) 3/27/2012 3 Hareesha N G, DSCE, Blore
PowerPoint Presentation: X Y STANDING ON H.P On it’s base. RESTING ON H.P On one point of base circle. LYING ON H.P On one generator. (Axis perpendicular to Hp And // to Vp.) (Axis inclined to Hp And // to Vp) (Axis inclined to Hp And // to Vp) While observing Fv, x-y line represents Horizontal Plane. (Hp) Axis perpendicular to Vp And // to Hp Axis inclined to Vp And // to Hp Axis inclined to Vp And // to Hp X Y F.V. F.V. F.V. T.V. T.V. T.V. While observing Tv, x-y line represents Vertical Plane. (Vp) STANDING ON V.P On it’s base. RESTING ON V.P On one point of base circle. LYING ON V.P On one generator. 3/27/2012 4 Hareesha N G, DSCE, Blore
PowerPoint Presentation: STEPS TO SOLVE PROBLEMS IN SOLIDS Problem is solved in three steps: STEP 1 : ASSUME SOLID STANDING ON THE PLANE WITH WHICH IT IS MAKING INCLINATION. ( IF IT IS INCLINED TO HP, ASSUME IT STANDING ON HP) ( IF IT IS INCLINED TO VP, ASSUME IT STANDING ON VP) IF STANDING ON HP - IT’S TV WILL BE TRUE SHAPE OF IT’S BASE OR TOP: IF STANDING ON VP - IT’S FV WILL BE TRUE SHAPE OF IT’S BASE OR TOP. BEGIN WITH THIS VIEW: IT’S OTHER VIEW WILL BE A RECTANGLE ( IF SOLID IS CYLINDER OR ONE OF THE PRISMS) : IT’S OTHER VIEW WILL BE A TRIANGLE ( IF SOLID IS CONE OR ONE OF THE PYRAMIDS): DRAW FV & TV OF THAT SOLID IN STANDING POSITION: STEP 2 : CONSIDERING SOLID’S INCLINATION ( AXIS POSITION ) DRAW IT’S FV & TV. STEP 3 : IN LAST STEP, CONSIDERING REMAINING INCLINATION, DRAW IT’S FINAL FV & TV. AXIS VERTICAL AXIS INCLINED HP AXIS INCLINED VP AXIS VERTICAL AXIS INCLINED HP AXIS INCLINED VP AXIS TO VP er AXIS INCLINED VP AXIS INCLINED HP AXIS TO VP er AXIS INCLINED VP AXIS INCLINED HP GENERAL PATTERN ( THREE STEPS ) OF SOLUTION: GROUP B SOLID. CONE GROUP A SOLID. CYLINDER GROUP B SOLID. CONE GROUP A SOLID. CYLINDER Three steps If solid is inclined to Hp Three steps If solid is inclined to Hp Three steps If solid is inclined to Vp Study Next Twelve Problems and Practice them separately !! Three steps If solid is inclined to Vp 3/27/2012 5 Hareesha N G, DSCE, Blore
PowerPoint Presentation: PROBLEM NO.1, 2, 3, 4 GENERAL CASES OF SOLIDS INCLINED TO HP & VP PROBLEM NO. 5 & 6 CASES OF CUBE & TETRAHEDRON PROBLEM NO. 7 CASE OF FREELY SUSPENDED SOLID WITH SIDE VIEW. PROBLEM NO. 8 CASE OF CUBE ( WITH SIDE VIEW) PROBLEM NO. 9 CASE OF TRUE LENGTH INCLINATION WITH HP & VP. PROBLEM NO. 10 & 11 CASES OF COMPOSITE SOLIDS. (AUXILIARY PLANE) PROBLEM NO. 12 CASE OF A FRUSTUM (AUXILIARY PLANE) CATEGORIES OF ILLUSTRATED PROBLEMS! 3/27/2012 6 Hareesha N G, DSCE, Blore
PowerPoint Presentation: X Y a b c d o o’ d’ c’ b’ a’ o’ d’ c’ b’ a’ o 1 d 1 b 1 c 1 a 1 a’ 1 d’ 1 c’ 1 b’ 1 o’ 1 o 1 d 1 b 1 c 1 a 1 o 1 d 1 b 1 c 1 a 1 (APEX NEARER TO V.P) . (APEX AWAY FROM V.P.) Problem 1. A square pyramid, 40 mm base sides and axis 60 mm long, has a triangular face on the ground and the vertical plane containing the axis makes an angle of 45 0 with the VP. Draw its projections. Take apex nearer to VP Solution Steps : Triangular face on Hp , means it is lying on Hp: 1.Assume it standing on Hp. 2.It’s Tv will show True Shape of base( square) 3.Draw square of 40mm sides with one side vertical Tv & taking 50 mm axis project Fv. ( a triangle) 4.Name all points as shown in illustration. 5.Draw 2 nd Fv in lying position I.e.o’c’d’ face on xy. And project it’s Tv. 6.Make visible lines dark and hidden dotted, as per the procedure. 7.Then construct remaining inclination with Vp ( Vp containing axis ic the center line of 2 nd Tv.Make it 45 0 to xy as shown take apex near to xy, as it is nearer to Vp) & project final Fv. For dark and dotted lines 1.Draw proper outline of new view DARK. 2. Decide direction of an observer. 3. Select nearest point to observer and draw all lines starting from it-dark. 4. Select farthest point to observer and draw all lines (remaining)from it- dotted. 3/27/2012 7 Hareesha N G, DSCE, Blore
PowerPoint Presentation: Problem 2: A cone 40 mm diameter and 50 mm axis is resting on one generator on Hp which makes 30 0 inclination with Vp Draw it’s projections. h a b c d e g f X Y a’ b’ d’ e’ c’ g’ f’ h’ o’ a’ h’b’ e’ c’g’ d’f’ o’ a 1 h 1 g 1 f 1 e 1 d 1 c 1 b 1 a 1 c 1 b 1 d 1 e 1 f 1 g 1 h 1 o 1 a’ 1 b’ 1 c’ 1 d’ 1 e’ 1 f’ 1 g’ 1 h’ 1 o 1 o 1 30 Solution Steps: Resting on Hp on one generator, means lying on Hp: 1.Assume it standing on Hp. 2.It’s Tv will show True Shape of base( circle ) 3.Draw 40mm dia. Circle as Tv & taking 50 mm axis project Fv. ( a triangle) 4.Name all points as shown in illustration. 5.Draw 2 nd Fv in lying position I.e.o’e’ on xy. And project it’s Tv below xy. 6.Make visible lines dark and hidden dotted, as per the procedure. 7.Then construct remaining inclination with Vp ( generator o 1 e 1 30 0 to xy as shown) & project final Fv. For dark and dotted lines 1.Draw proper outline of new vie DARK. 2. Decide direction of an observer. 3. Select nearest point to observer and draw all lines starting from it-dark. 4. Select farthest point to observer and draw all lines (remaining) from it- dotted. 3/27/2012 8 Hareesha N G, DSCE, Blore
PowerPoint Presentation: a b d c 1 2 4 3 X Y a b d c 1 2 4 3 a’ b’ c’ d’ 1’ 2’ 3’ 4’ 45 0 4’ 3’ 2’ 1’ d’ c’ b’ a’ 4’ 3’ 2’ 1’ d’ c’ b’ a’ 35 0 a 1 b 1 c 1 d 1 1 2 3 4 Problem 3: A cylinder 40 mm diameter and 50 mm axis is resting on one point of a base circle on Vp while it’s axis makes 45 0 with Vp and Fv of the axis 35 0 with Hp. Draw projections.. Solution Steps: Resting on Vp on one point of base, means inclined to Vp: 1.Assume it standing on Vp 2.It’s Fv will show True Shape of base & top( circle ) 3.Draw 40mm dia. Circle as Fv & taking 50 mm axis project Tv. ( a Rectangle) 4.Name all points as shown in illustration. 5.Draw 2 nd Tv making axis 45 0 to xy And project it’s Fv above xy. 6.Make visible lines dark and hidden dotted, as per the procedure. 7.Then construct remaining inclination with Hp ( Fv of axis I.e. center line of view to xy as shown) & project final Tv. 3/27/2012 9 Hareesha N G, DSCE, Blore
PowerPoint Presentation: Problem 5: A cube of 50 mm long edges is so placed on Hp on one corner that a body diagonal is parallel to Hp and perpendicular to Vp Draw it’s projections. X Y b c d a a’ d’ c’ b’ a’ d’ c’ b’ a 1 b 1 d 1 c 1 a 1 b 1 d 1 c 1 1’ p’ p’ a’ 1 d’ 1 c’ 1 d’ 1 Solution Steps: 1.Assuming standing on Hp, begin with Tv,a square with all sides equally inclined to xy.Project Fv and name all points of FV & TV. 2.Draw a body-diagonal joining c’ with 3’( This can become // to xy) 3.From 1’ drop a perpendicular on this and name it p’ 4.Draw 2 nd Fv in which 1’-p’ line is vertical means c’-3’ diagonal must be horizontal. .Now as usual project Tv.. 6.In final Tv draw same diagonal is perpendicular to Vp as said in problem. Then as usual project final FV. 1’ 3’ 1’ 3’ 3/27/2012 10 Hareesha N G, DSCE, Blore
PowerPoint Presentation: Y Problem 6: A tetrahedron of 50 mm long edges is resting on one edge on Hp while one triangular face containing this edge is vertical and 45 0 inclined to Vp. Draw projections. X T L a o b c b’ a’ c’ o’ a’ a 1 c 1 o 1 b 1 a 1 o 1 b 1 90 0 45 0 c 1 c’ 1 b’ c’ o’ a’ 1 o’ 1 b’ 1 IMPORTANT: Tetrahedron is a special type of triangular pyramid in which base sides & slant edges are equal in length. Solid of four faces. Like cube it is also described by One dimension only.. Axis length generally not given. Solution Steps As it is resting assume it standing on Hp. Begin with Tv , an equilateral triangle as side case as shown: First project base points of Fv on xy, name those & axis line. From a’ with TL of edge, 50 mm, cut on axis line & mark o’ (as axis is not known, o’ is finalized by slant edge length) Then complete Fv. In 2 nd Fv make face o’b’c’ vertical as said in problem. And like all previous problems solve completely. 3/27/2012 11 Hareesha N G, DSCE, Blore
PowerPoint Presentation: FREELY SUSPENDED SOLIDS: Positions of CG, on axis, from base, for different solids are shown below. H H/2 H/4 GROUP A SOLIDS ( Cylinder & Prisms) GROUP B SOLIDS ( Cone & Pyramids) CG CG 3/27/2012 12 Hareesha N G, DSCE, Blore
PowerPoint Presentation: X Y a’ d’ e’ c’ b’ o’ a b c d e o g’ H/4 H LINE d’g’ VERTICAL a’ b’ c’ d’ o” e’ g’ a 1 b 1 o 1 e 1 d 1 c 1 a” e” d” c” b” FOR SIDE VIEW Problem 7: A pentagonal pyramid 30 mm base sides & 60 mm long axis, is freely suspended from one corner of base so that a plane containing it’s axis remains parallel to Vp. Draw it’s three views. IMPORTANT: When a solid is freely suspended from a corner, then line joining point of contact & C.G. remains vertical. ( Here axis shows inclination with Hp.) So in all such cases, assume solid standing on Hp initially.) Solution Steps: In all suspended cases axis shows inclination with Hp. 1.Hence assuming it standing on Hp, drew Tv - a regular pentagon,corner case. 2.Project Fv & locate CG position on axis – ( ¼ H from base.) and name g’ and Join it with corner d’ 3.As 2 nd Fv, redraw first keeping line g’d’ vertical. 4.As usual project corresponding Tv and then Side View looking from. 3/27/2012 13 Hareesha N G, DSCE, Blore
PowerPoint Presentation: a’ d’ c’ b’ b c d a a’ d’ c’ b’ a 1 b 1 d 1 c 1 d’’ c’’ a’’ b’’ X Y 1’ 1’ 1’ Problem 8: A cube of 50 mm long edges is so placed on Hp on one corner that a body diagonal through this corner is perpendicular to Hp and parallel to Vp Draw it’s three views. Solution Steps: 1.Assuming it standing on Hp begin with Tv, a square of corner case. 2.Project corresponding Fv.& name all points as usual in both views. 3.Join a’1’ as body diagonal and draw 2 nd Fv making it vertical (I’ on xy) 4.Project it’s Tv drawing dark and dotted lines as per the procedure. 5.With standard method construct Left-hand side view. ( Draw a 45 0 inclined Line in Tv region ( below xy). Project horizontally all points of Tv on this line and reflect vertically upward, above xy.After this, draw horizontal lines, from all points of Fv, to meet these lines. Name points of intersections and join properly. For dark & dotted lines locate observer on left side of Fv as shown.) 3/27/2012 14 Hareesha N G, DSCE, Blore
PowerPoint Presentation: a 1 h 1 f 1 e 1 d 1 c 1 b 1 g 1 1 o 1 40 0 Axis Tv Length Axis Tv Length Axis True Length Locus of Center 1 c’ 1 a’ 1 b’ 1 e’ 1 d’ 1 h’ 1 f’ 1 g’ 1 o’ 1 h a b c d e g f y X a’ b’ d’ e’ c’ g’ f’ h’ o’ a’ h’b’ e’ c’g’ d’f’ o’ 45 0 a 1 h 1 f 1 e 1 d 1 c 1 b 1 g 1 o 1 1 Problem 9: A right circular cone, 40 mm base diameter and 60 mm long axis is resting on Hp on one point of base circle such that it’s axis makes 45 0 inclination with Hp and 40 0 inclination with Vp. Draw it’s projections. This case resembles to problem no.7 & 9 from projections of planes topic. In previous all cases 2 nd inclination was done by a parameter not showing TL.Like Tv of axis is inclined to Vp etc. But here it is clearly said that the axis is 40 0 inclined to Vp. Means here TL inclination is expected. So the same construction done in those Problems is done here also. See carefully the final Tv and inclination taken there. So assuming it standing on HP begin as usual. 3/27/2012 15 Hareesha N G, DSCE, Blore