chapter – 2
DESCRIPTION
- PowerPoint PPT PresentationTRANSCRIPT
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Engineering drawing is a language of all persons involved in engineering activity. Engineering ideas are recorded by preparing drawings and execution of work is also carried out on the basis of drawings. Communication in engineering field is done by drawings. It is called as a “Language of Engineers”.
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CHAPTER – 2
ENGINEERING CURVES
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Useful by their nature & characteristics. Laws of nature represented on graph.
Useful in engineering in understanding laws, manufacturing of various items, designing mechanisms analysis of forces, construction of bridges, dams, water tanks etc.
USES OF ENGINEERING CURVES
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1. CONICS2. CYCLOIDAL CURVES3. INVOLUTE
4. SPIRAL
5. HELIX
6. SINE & COSINE
CLASSIFICATION OF ENGG. CURVES
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It is a surface generated by moving a Straight line keeping one of its end fixed & other end makes a closed curve.
What is Cone ?
If the base/closed curve is a polygon, we get a pyramid.
If the base/closed curve is a circle, we get a cone.
The closed curve is known as base.
The fixed point is known as vertex or apex. Vertex/Apex
90º
Base
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If axis of cone is not perpendicular to base, it is called as oblique cone.
The line joins vertex/ apex to the circumference of a cone is known as generator.
If axes is perpendicular to base, it is called as right circular cone.
Generator
Cone Axis
The line joins apex to the center of base is called axis.
90º
Base
Vertex/Apex
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Definition :- The section obtained by the intersection of a right circular cone by a cutting plane in different position relative to the axis of the cone are called CONICS.
CONICS
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B - CIRCLE
A - TRIANGLE
CONICS
C - ELLIPSE
D – PARABOLA
E - HYPERBOLA
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When the cutting plane contains the apex, we get a triangle as the section.
TRIANGLE
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When the cutting plane is perpendicular to the axis or parallel to the base in a right cone we get circle the section.
CIRCLE
Sec Plane
Circle
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Definition :-When the cutting plane is inclined to the axis but not parallel to generator or the inclination of the cutting plane(α) is greater than the semi cone angle(θ), we get an ellipse as the section.
ELLIPSE
α
θ
α > θ
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When the cutting plane is inclined to the axis and parallel to one of the generators of the cone or the inclination of the plane(α) is equal to semi cone angle(θ), we get a parabola as the section.
PARABOLA
θ
α
α = θ
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When the cutting plane is parallel to the axis or the inclination of the plane with cone axis(α) is less than semi cone angle(θ), we get a hyperbola as the section.
HYPERBOLADefinition :-
α < θα = 0
θθ
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CONICSDefinition :- The locus of point moves in a plane such a way that the ratio of its distance from fixed point (focus) to a fixed Straight line (Directrix) is always constant.
Fixed point is called as focus.Fixed straight line is called as directrix.
M
C FV
P
Focus
Conic CurveDirectrix
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The line passing through focus & perpendicular to directrix is called as axis.The intersection of conic curve with axis is called as vertex.
AxisM
C FV
P
Focus
Conic CurveDirectrix
Vertex
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N Q
Ratio =Distance of a point from focusDistance of a point from directrix= Eccentricity
= PF/PM = QF/QN = VF/VC = e
M P
F
Axis
C VFocus
Conic CurveDirectrix
Vertex
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Vertex
Ellipse is the locus of a point which moves in a plane so that the ratio of its distance from a fixed point (focus) and a fixed straight line (Directrix) is a constant and less than one.
ELLIPSE
M
N Q
P
C FV
Axis
Focus
EllipseDirectri
x
Eccentricity=PF/PM = QF/QN
< 1.
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Ellipse is the locus of a point, which moves in a plane so that the sum of its distance from two fixed points, called focal points or foci, is a constant. The sum of distances is equal to the major axis of the ellipse.
ELLIPSE
F1A B
P
F2
O
Q
C
D
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F1A B
C
D
P
F2
O
PF1 + PF2 = QF1 + QF2 = CF1 +CF2 = constant
= Major Axis
Q
= F1A + F1B = F2A + F2BBut F1A = F2B
F1A + F1B = F2B + F1B = AB
CF1 +CF2 = ABbut CF1 = CF2
hence, CF1=1/2AB
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F1 F2
OA B
C
D
Major Axis = 100 mm
Minor Axis = 60 mm
CF1 = ½ AB = AO
F1 F2
OA B
C
D
Major Axis = 100 mm
F1F2 = 60 mm
CF1 = ½ AB = AO
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Uses :-
Shape of a man-hole.
Flanges of pipes, glands and stuffing boxes.
Shape of tank in a tanker.
Shape used in bridges and arches.Monuments.
Path of earth around the sun.Shape of trays etc.
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Ratio (known as eccentricity) of its distances from focus to that of directrix is constant and equal to one (1).
PARABOLAThe parabola is the locus of a point,
which moves in a plane so that its distance from a fixed point (focus) and a fixed straight line (directrix) are always equal.
Definition :-
Directrix AxisVertex
M
C
N Q
FV
P
Focus
Parabola
Eccentricity = PF/PM = QF/QN = 1.
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Motor car head lamp reflector.Sound reflector and detector.
Shape of cooling towers.
Path of particle thrown at any angle with earth, etc.
Uses :-
Bridges and arches construction
Home
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It is the locus of a point which moves in a plane so that the ratio of its distances from a fixed point (focus) and a fixed straight line (directrix) is constant and grater than one.
Eccentricity = PF/PM
AxisDirectrixHyperbol
aM
C
N Q
FV
P
Focus
Vertex
HYPERBOLA
= QF/QN > 1.
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Nature of graph of Boyle’s lawShape of overhead water tanks
Uses :-
Shape of cooling towers etc.
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METHODS FOR DRAWING ELLIPSE
2. Concentric Circle Method3. Loop Method
4. Oblong Method
5. Ellipse in Parallelogram
6. Trammel Method
7. Parallel Ellipse
8. Directrix Focus Method
1. Arc of Circle’s Method
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Norm
al
P2’
R =A
1
Tangent
1 2 3 4A B
C
D
P1
P3
P2
P4 P4 P3
P2
P1
P1’
F2
P3’ P4’ P4’P3’
P2’
P1’
90°
F1
Rad =B1
R=B2
`R=A
2
O
ARC OF CIRCLE’S METHOD
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Axi
sM
inor
A BMajor Axis 7
8
91011
9
8
7
6
543
2
1
12
11
P6
P5P4
P3
P2`
P1
P12
P11
P10P9
P8
P7
6
543
2
1
12 C10
O
CONCENTRIC CIRCLE
METHOD
F2F1
D
CF1=CF2=1/2 AB
T
N
Q
e = AF1/AQ
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Normal
0
1
2
3
4
1 2 3 4 1’0’
2’3’4’
1’
2’
3’
4’
A B
C
D
Major Axis
Min
or A
xis
F1 F2
Dir
ectr
ix
E
F
S
P
P1
P2
P3
P4
Tang
ent
P1’
P2’
P3’P4’
ØØ
R=AB/2
P0
P1’’
P2’’
P3’’P4’’P4
P3
P2
P1
OBLONG METHOD
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BA
P4
P0
D
C
60°
65432
10
5 4 3 2 1 0 1 2 3 4 5 65
3210P1P2P
3
Q1Q2Q3Q4
Q5P6 Q6O
4
ELLIPSE IN PARALLELOGRAM
R4
R3 R2R1S1
S2
S3
S4
P5
G
H
I
K
JM
inor Axis
Major Axis
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P6
Nor
mal
P5’ P7’P6’
P1
TangentP1’
N
N
T
T
V1
P5
P4’
P4
P3’P2’
F1
D1
D1
R1
bac
de f
g
Q
P7P3P2
Dir
ectr
ix
R=6f`
90°
1 2 3 4 5 6 7
Eccentricity = 2/3
3R1V1
QV1 = R1V1
V1F1 = 2
Ellipse
ELLIPSE – DIRECTRIX FOCUS METHOD
R=1a
Dist. Between directrix & focus = 50 mm
1 part = 50/(2+3)=10 mm V1F1 = 2 part = 20 mmV1R1 = 3 part = 30 mm
< 45º
S
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PROBLEM :-The distance between two coplanar fixed points is 100 mm. Trace the complete path of a point G moving in the same plane in such a way that the sum of the distance from the fixed points is always 140 mm.
Name the curve & find its eccentricity.
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ARC OF CIRCLE’S METHOD
Norm
alG2’
R =A
1
Tangent
1 2 3 4A B
G
G’
G1
G3
G2
G4 G4 G3
G2
G1
G1’
G3’ G4’ G4’G3’
G2’
G1’
F2F1
R=B1
R=B2
`R=A
2O
90°
90°
dire
ctrix
100
140
GF1 + GF2 = MAJOR AXIS = 140
E
e AF1AE
e =
R=70 R=70
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PROBLEM :-3Two points A & B are 100 mm apart. A point C is 75 mm from A and 45 mm from B. Draw an ellipse passing through points A, B, and C so that AB is a major axis.
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OA B
C
7545
1
D
100
1
2 2
3
3
4
45
5
66
7
7P1
P2
P3
P4
P5
P6
P7
P8
E
8
8
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PROBLEM :-5ABCD is a rectangle of 100mm x 60mm. Draw an ellipse passing through all the four corners A, B, C and D of the rectangle considering mid – points of the smaller sides as focal points.
Use “Concentric circles” method and find its eccentricity.
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I3
CD
F1 F2P Q
R
S
50I1 I4
A BI2
O
1
1
2
2
4
4
3
3100
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PROBLEM :-1Three points A, B & P while lying along a horizontal line in order have AB = 60 mm and AP = 80 mm, while A & B are fixed points and P starts moving such a way that AP + BP remains always constant and when they form isosceles triangle, AP = BP = 50 mm. Draw the path traced out by the point P from the commencement of its motion back to its initial position and name the path of P.
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A B P
R =
50
M
N
O
1
2
1 2 6080
Q
1
2
12
P1
P2 Q2
Q1
R1
R2 S2
S1
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PROBLEM :-2Draw an ellipse passing through 60º corner Q of a 30º - 60º set square having smallest side PQ vertical & 40 mm long while the foci of the ellipse coincide with corners P & R of the set square.
Use “OBLONG METHOD”. Find its eccentricity.
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ELLIPSEP
Q
R80mm
40m
m
89mm
A
C
B
D
MAJOR AXIS = PQ+QR = 129mm
R=AB
/2
1
2
3
1’ 2’ 3’ 1’’2’’3’’
O3 O3’
O1
O2 O2’
O1’
TANGENTNORMAL
60º
30º
2
3
1
dire
ctrix
F1 F2MAJOR AXIS
MIN
OR
AXI
S
S
ECCENTRICITY = AP / AS
? ?
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PROBLEM :-4Two points A & B are 100 mm apart. A point C is 75 mm from A and 45 mm from B. Draw an ellipse passing through points A, B, and C so that AB is not a major axis.
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D
C
665
43
21
0
5 4 3 2 1 0 1 2 3 4 5 66
5
32
10P2
P3
P4
P5
Q1
Q2Q3Q4
Q5
BA O
4
ELLIPSE
100
4575
P0
P1
P6Q6
G
H
I
K
J
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PROBLEM :-Draw an ellipse passing through A & B of an equilateral triangle of ABC of 50 mm edges with side AB as vertical and the corner C coincides with the focus of an ellipse. Assume eccentricity of the curve as 2/3. Draw tangent & normal at point A.
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PROBLEM :-Draw an ellipse passing through all the four corners A, B, C & D of a rhombus having diagonals AC=110mm and BD=70mm.
Use “Arcs of circles” Method and find its eccentricity.
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METHODS FOR DRAWING PARABOLA
1. Rectangle Method
2. Parabola in Parallelogram
3. Tangent Method
4. Directrix Focus Method
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2345
0
1
2
3
4
5
6 1 1 5432 6
0
1
2
3
4
5
0
VD C
A B
P4 P4
P5 P5
P3 P3
P2 P2
P6 P6
P1 P1
PARABOLA –RECTANGLE METHOD
PARABOLA
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B
0
2’
0
2
6
C
6’
V
5
P’5
30°A X
D
1’ 2’4’
5’
3’
1
34
5’
4’3’
1’0
54
3
21
P1
P2
P3
P4
P5
P’4
P’3
P’2P’
1
P’6
PARABOLA – IN PARALLELOGRAM
P6
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BA O
V
1
8
3
4
5
2
6
7
9
10
0
1
2
3
4
5
6
7
8
9
10
0
F
PARABOLA
TANGENT METHOD
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D
D
DIR
ECTR
IX
90° 2 3 4T
TN
N
S
V 1
P1
P2
PF
P3
P4
P1’
P2’P3’
P4’
PF’
AXISR
F R2
R1
R3R4
90°R F
PARABOLA DIRECTRIX FOCUS METHOD
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PROBLEM:-A stone is thrown from a building 6 m high. It just crosses the top of a palm tree 12 m high. Trace the path of the projectile if the horizontal distance between the building and the palm tree is 3 m. Also find the distance of the point from the building where the stone falls on the ground.
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6m
ROOT OF TREE
BUILDING
REQD.DISTANCE
TOP OF TREE
3m
6m
F
A
STONE FALLS HERE
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3m
6m
ROOT OF TREE
BUILDING
REQD.DISTANCE
GROUND
TOP OF TREE
3m
6m
1
2
3
1
2
3
321 4 5 6
5
6
EF
A B
CD
P3
P4
P2
P1
PP1
P2
P3
P4
P5
P6
3 2 10
STONE FALLS HERE
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PROBLEM:-
In a rectangle of sides 150 mm and 90 mm, inscribe two parabola such that their axis bisect each other. Find out their focus points & positions of directrix.
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150 mmA
B C
D1 2 3 4 5
1
2
3
4
5
O
P1
P2
P3
P4
P5
M1’ 2’ 3’ 4’ 5’
1’
2’3’
4’5’
P1’
P2’
P3’ P
4’
P5’
90 m
m
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EXAMPLEA shot is discharge from the ground level at an angle 60 to the horizontal at a point 80m away from the point of discharge. Draw the path trace by the shot. Use a scale 1:100
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ground level BA
60ºgun shot
80 M
parabola
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ground level BA O
V
1
8
3
4
5
2
6
7
9
10
0
1
2
3
4
5
6
78
9
10
0
F
60ºgun shot
D D
VF
VE= e = 1
E
![Page 59: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/59.jpg)
Connect two given points A and B by a Parabolic curve, when:-
1.OA=OB=60mm and angle AOB=90°
2.OA=60mm,OB=80mm and angle AOB=110°3.OA=OB=60mm and angle AOB=60°
![Page 60: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/60.jpg)
60
6 0
1 2 3 4 5
Parabola
5
4
3
2
1
A
B90 °O
1.OA=OB=60mm and angle AOB=90°
![Page 61: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/61.jpg)
A
B80
60
5
43
21
1 2 3 4 5110
°
Parabola
O
2.OA=60mm,OB=80mm and angle AOB=110°
![Page 62: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/62.jpg)
54321
5
4
3
2
1
A
BO
Parabola
60
60
60 °
3.OA=OB=60mm and angle AOB=60°
![Page 63: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/63.jpg)
example
Draw a parabola passing through three different points A, B and C such that AB = 100mm, BC=50mm and CA=80mm respectively.
![Page 64: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/64.jpg)
BA
C
100
50
80
![Page 65: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/65.jpg)
1’0
0
2’
0
26
6’5
P’5
1’ 2’ 4’ 5’3’134
5’
4’3’
54
32
1
P1P
2P
3
P4
P5
P’4
P’3
P’2
P’1
P’6
P6
A B
C
![Page 66: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/66.jpg)
METHODS FOR DRAWING HYPERBOLA
1. Rectangle Method
2. Oblique Method
3. Directrix Focus Method
![Page 67: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/67.jpg)
D
F
1 2 3 4 5
5’4’3’2’
P1
P2
P3P4
P5
0
P6
P0
AO EX
B
C
Y Given Point P0
90°
6
6’
Hyperbola
RECTANGULAR HYPERBOLA
AXIS
AXIS
When the asymptotes are at right angles to each other, the hyperbola is called rectangular or equilateral hyperbola
ASYMPTOTES X and Y
![Page 68: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/68.jpg)
Problem:-
Two fixed straight lines OA and OB are at right angle to each other. A point “P” is at a distance of 20 mm from OA and 50 mm from OB. Draw a rectangular hyperbola passing through point “P”.
![Page 69: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/69.jpg)
D
F
1 2 3 4 5
5’4’3’2’
P1
P2
P3P4
P5
0
P6
P0
AO EX=20
B
CY
= 50
Given Point P0
90°
6
6’
Hyperbola
RECTANGULAR HYPERBOLA
![Page 70: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/70.jpg)
PROBLEM:-
Two straight lines OA and OB are at 75° to each other. A point P is at a distance of 20 mm from OA and 30 mm from OB. Draw a hyperbola passing through the point “P”.
![Page 71: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/71.jpg)
75 0
P4
E
6’
2’1’ P1
1 2 3 4 5 6 D
P6P5
P3P2
P0
7’P7
7C
B F
O
Y =
30
X = 20
Given Point P0
A
![Page 72: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/72.jpg)
AXIS
NORMAL
C V F1
DIR
ECTR
IXD
D
1 2 3 4
4’
3’
2’
1’P1
P2
P3
P4
P1’
P2’
P3’
P4’
T1
T2
N2
N1
TANG
ENT
s
Directrix and focus method
![Page 73: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/73.jpg)
CYCLOIDAL GROUP OF CURVESWhen one curve rolls over another curve without slipping or sliding, the path Of any point of the rolling curve is called as ROULETTE.When rolling curve is a circle and the curve on which it rolls is a straight line Or a circle, we get CYCLOIDAL GROUP OF CURVES.
SuperiorHypotrochoid
Cycloidal Curves
Cycloid Epy Cycloid Hypo Cycloid
SuperiorTrochoid
InferiorTrochoid
SuperiorEpytrochoid
InferiorEpytrochoid
InferiorHypotrochoid
![Page 74: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/74.jpg)
Rolling Circle or Generator
CYCLOID:-Cycloid is a locus of a point on the circumference of a rolling circle(generator), which rolls without slipping or sliding along a fixed straight line or a directing line or a director.
C
P P
P
R
C
Directing Line or Director
![Page 75: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/75.jpg)
EPICYCLOID:-Epicycloid is a locus of a point(P) on the circumference of a rolling circle(generator), which rolls without slipping or sliding OUTSIDE another circle called Directing Circle.
2πrØ = 360º x r/Rd
Circumference of Generating Circle
R d
Rolling Circle
r
O
Ø/2
Ø/2
P0 P0
Arc P0P0 =
Rd x Ø =
P0
![Page 76: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/76.jpg)
HYPOCYCLOID:-Hypocycloid is a locus of a point(P) on the circumference of a rolling circle(generator), which rolls without slipping or sliding INSIDE another circle called Directing Circle.`
DirectingCircle(R)
P
Ø /2 Ø /2
Ø = 360 x rRR
T
Rolling CircleRadius (r)
O
Vertical
Hypocycloid
P P
![Page 77: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/77.jpg)
If the point is inside the circumference of the circle, it is called inferior trochoid.If the point is outside the circumference of the circle, it is called superior trochoid.
What is TROCHOID ? DEFINITION :- It is a locus of a point inside/outside the circumference of a rolling circle, which rolls without slipping or sliding along a fixed straight line or a fixed circle.
![Page 78: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/78.jpg)
P0
2R or D 5
T
T
1
2
1 2 3 4 6 7 8 9 10 110 120
3
4567
8
9
1011 12
P1
P2
P3
P4
P5 P7
P8
P9
P11
P12
C0 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11
Directing Line
C12
N
N
S
S1
R
P6
R
P10R
: Given Data :
Draw cycloid for one revolution of a rolling circle having diameter as 60mm.
Rolling Circle
D
![Page 79: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/79.jpg)
C0
P0
78
P64
P1 1
2
3
C2 C3P2 C4
Problem 1:A circle of diameter D rolls without
slip on a horizontal surface (floor) by Half revolution and then it rolls up a vertical surface (wall) by another half revolution. Initially the point P is at the Bottom of circle touching the floor. Draw the path of the point P.
5
6 C1
P3
P4
P5
P7
P8
70
C 5
C 6
C 7
C 8
1 2 3 4
56
D/2
πD
/2
πD/2 D/2Floor
Wal
l
CYCLOID
56
7
8
Take diameter of circle = 40mmInitially distance of centre of
circle from the wall 83mm (Hale circumference + D/2)
![Page 80: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/80.jpg)
Problem : 2 A circle of 25 mm radius rolls on the circumference of another circle of 150 mm diameter and outside it. Draw the locus of the point P on the circumference of the rolling circle for one complete revolution of it. Name the curve & draw tangent and normal to the curve at a point 115 mm from the centre of the bigger circle.
![Page 81: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/81.jpg)
First Step : Find out the included angle by using the equation
360º x r / R = 360 x 25/75 = 120º.
Second step: Draw a vertical line & draw two lines at 60º on either sides.
Third step : at a distance of 75 mm from O, draw a part of the circle taking radius = 75 mm.
Fourth step : From the circle, mark point C outside the circle at distance of 25 mm & draw a circle taking the centre as point C.
![Page 82: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/82.jpg)
P6
P4
rP2
C1
C0
C2
C3C4 C5
C6
C7
C8
1
0
234
5
6 7
O
R d
Ø/2 Ø/2
P1 P0
P3 P5
P7P8
r rRolling Circle
r
Rd X Ø = 2πrØ = 360º x r/Rd
Arc P0P8 = Circumference of Generating Circle
EPICYCLOIDGIVEN:Rad. Of Gen. Circle (r)& Rad. Of dir. Circle (Rd) S
ºU
N
Ø = 360º x 25/75 = 120°
![Page 83: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/83.jpg)
Problem :3 A circle of 80 mm diameter rolls on the circumference of another circle of 120 mm radius and inside it. Draw the locus of the point P on the circumference of the rolling circle for one complete revolution of it. Name the curve & draw tangent and normal to the curve at a point 100 mm from the centre of the bigger circle.
![Page 84: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/84.jpg)
P0 P1
Tangent
P11
r
C0
C1
C2
C3
C4C5
C6 C7 C8 C9C10
C11
C12
P10P8
01 2
3
4
5
67
89
10
11
12P2 P3
P4P5 P6
P9P7
P12
/2
/2
=360 x 412
= 360 x rR
= 120°
R
T
T
N
S
N
Norm
alr
r
Rolling Circle
Radias (r)
DirectingCircle
O
Vertical
Hypocycloid
![Page 85: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/85.jpg)
Problem :Show by means of drawing that when the diameter of rolling circle is half the diameter of directing circle, the hypocycloid is a straight line
![Page 86: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/86.jpg)
CC1
C2
C3
C4C5
C6 C7
C9
C8
C10
C11
C12P8O
10
5
7
89
11
12
12 3
4
6P1
P2
P3 P4P5 P6 P7 P9
P10
P11
P12
Directing Circle
Rolling Circle
HYPOCYCLOID
![Page 87: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/87.jpg)
INVOLUTE DEFINITION :- If a straight line is rolled round a circle or a polygon without slipping or sliding, points on line will trace out INVOLUTES.
OR
Uses :- Gears profile
Involute of a circle is a curve traced out by a point on a tights string unwound or wound from or on the surface of the circle.
![Page 88: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/88.jpg)
PROB:
A string is unwound from a circle of 20 mm diameter. Draw the locus of string P for unwounding the string’s one turn. String is kept tight during unwound. Draw tangent & normal to the curve at any point.
![Page 89: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/89.jpg)
P12
P2
002 12
6
P11 20 9 103 4 6 8 115 7 12
DP3
P4
P5
P6
P7
P8
P9
P10
P11
123
4 5 78910
11
03
04
05
06
07
08
09
010`
011
Tangent
N
N
Normal
T
T.
![Page 90: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/90.jpg)
PROBLEM:-Trace the path of end point of a thread when it is wound round a circle, the length of which is less than the circumference of the circle.
Say Radius of a circle = 21 mm & Length of the thread = 100 mm
Circumference of the circle = 2 π r = 2 x π x 21 = 132 mm
So, the length of the string is less than circumference of the circle.
![Page 91: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/91.jpg)
P
R=7toPR=6toP
R21 0
0 1 2 3 4 5 6 7 8 P11 0
12
3
456
78
9
10
P1
P2
P3
P4
P5
P6
P7P8
L= 100 mm
R=1toP
R=2t
oP
R=3
toPR=4toPR=5toP
INVOLUTE
9
ø
11 mm = 30°Then 5 mm =
Ø = 30° x 5 /11 = 13.64 °
S = 2 x π x r /12
![Page 92: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/92.jpg)
PROBLEM:-Trace the path of end point of a thread when it is wound round a circle, the length of which is more than the circumference of the circle.
Say Radius of a circle = 21 mm & Length of the thread = 160 mm
Circumference of the circle = 2 π r = 2 x π x 21 = 132 mm
So, the length of the string is more than circumference of the circle.
![Page 93: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/93.jpg)
P13
P11
313
1415
P0P12
O
7
10 123
45689
1112 1 2
P1
P2P3P4
P5
P6
P7
P8 P9 P10
P14P
L=160
mm
R=21mm
64 5 7 8 9 101112131415
ø
![Page 94: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/94.jpg)
PROBLEM:-Draw an involute of a pantagon having side as 20 mm.
![Page 95: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/95.jpg)
P5
R=01
R=201
P0
P1
P2
P3
P4
R=3
01
R=401
R=501
234 5
1
T
TN
NS
INVOLUTEOF A POLYGON
Given : Side of a polygon 0
![Page 96: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/96.jpg)
PROBLEM:-
Draw an involute of a square
having side as 20 mm.
![Page 97: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/97.jpg)
P2
1
2 3
04
P0
P1
P3
P4
N
NS R=3
01
R=401
R=201
R=01
INVOLUTE OF A SQUARE
![Page 98: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/98.jpg)
PROBLEM:-
Draw an involute of a string unwound from the given figure from point C in anticlockwise direction.
60°
A
B
C
R2130°
![Page 99: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/99.jpg)
R21
60°
A
B
C
30°X
X+A1
XX+
A2
X+A 3
X+A5
X+A4
X+AB
R =X+AB
X+66
+B
C1
23
45
C0
C1
C2C3
C4
C5
C6
C7
C8
![Page 100: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/100.jpg)
A stick of length equal to the circumference of a semicircle, is initially tangent to the semicircle on the right of it. This stick now rolls over the circumference of a semicircle without sliding till it becomes tangent on the left side of the semicircle. Draw the loci of two end point of this stick. Name the curve. Take R= 42mm.
PROBLEM:-
![Page 101: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/101.jpg)
A6
B6
5
A
B
C
B1
A1
B2
A2
B3A3
B4
A4
B5
A5
12 3
4
5
O
1
2
3
4
6
INVOLUTE
![Page 102: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/102.jpg)
SPIRALSIf a line rotates in a plane about one of its ends and if at the same time, a point moves along the line continuously in one direction, the curves traced out by the moving point is called a SPIRAL.
The point about which the line rotates is called a POLE.
The line joining any point on the curve with the pole is called the RADIUS VECTOR.
![Page 103: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/103.jpg)
The angle between the radius vector and the line in its initial position is called the VECTORIAL ANGLE.Each complete revolution of the curve is termed as CONVOLUTION.
Spiral
Arche Median Spiral for Clock
Semicircle Quarter Circle Logarithmic
![Page 104: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/104.jpg)
ARCHEMEDIAN SPIRALIt is a curve traced out by a point moving in such a way that its movement towards or away from the pole is uniform with the increase of vectorial angle from the starting line.USES :-Teeth profile of Helical gears.Profiles of cams etc.
![Page 105: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/105.jpg)
To construct an Archemedian Spiral of one convolutions, given the radial movement of the point P during one convolution as 60 mm and the initial position of P is the farthest point on the line or free end of the line.
Greatest radius = 60 mm &Shortest radius = 00 mm ( at centre or at pole)
PROBLEM:
![Page 106: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/106.jpg)
P10
1
23
4
5
6
7
89
10
11
120
8 7 012345691112
P1
P2P3
P4
P5
P6
P7 P8P9
P11P12
o
![Page 107: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/107.jpg)
To construct an Archemedian Spiral of one convolutions, given the greatest & shortest(least) radii.
Say Greatest radius = 100 mm & Shortest radius = 60 mm
To construct an Archemedian Spiral of one convolutions, given the largest radius vector & smallest radius vector.
OR
![Page 108: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/108.jpg)
3 126
58 479
1011
2
1
34
5
6
7
89
10
11
12
P1
P2P3
P4
P5
P6
P7
P8 P9
P10
P11
P12
O
N
N
T
T
SR
min
R max
Diff. in length of any two radius vectors
Angle between them in radiansConstant of the curve =
=OP – OP3
Π/2
100 – 90=
Π/2
= 6.37 mm
![Page 109: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/109.jpg)
PROBLEM:-
A slotted link, shown in fig rotates in the horizontal plane about a fixed point O, while a block is free to slide in the slot. If the center point P, of the block moves from A to B during one revolution of the link, draw the locus of point P. OAB
40 25
![Page 110: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/110.jpg)
B A O1234567891011
P1
P2P3 P4
P5
P6
P7
P8P9
P10
P11
P12
11
21
3141
51
61
71
8191
101
111
40 25
![Page 111: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/111.jpg)
PROBLEM:-
A link OA, 100 mm long rotates about O in clockwise direction. A point P on the link, initially at A, moves and reaches the other end O, while the link has rotated thorough 2/3 rd of the revolution. Assuming the movement of the link and the point to be uniform, trace the path of the point P.
![Page 112: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/112.jpg)
A Initial Position of point PPO
P1
P2
P3
P4P5
P6
P7
P8
2134567O
1
2
3
4
56
7
8
2/3 X 360° = 240°
120º
![Page 113: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/113.jpg)
A0
Linear Travel of point P on AB = 96 =16x (6 div.)
EXAMPLE: A link AB, 96mm long initially is vertically upward w.r.t. its pinned end B, swings in clockwise direction for 180° and returns back in anticlockwise direction for 90°, during which a point P, slides from pole B to end A. Draw the locus of point P and name it. Draw tangent and normal at any point on the path of P.
P1’
A
B
A1
A2
A3
A4
A5
A6P0
P1
P2
P3
P4
P5
P6
P2’
P3’
P4’ P5’
P6’
9 6
Link AB = 96
C
Tangent
Angular Swing of link AB = 180° + 90°
= 270 ° =45 °X 6 div.
ARCHIMEDIAN SPIRAL
D
NO
RM
AL
M
N
![Page 114: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/114.jpg)
Arch.Spiral Curve Constant BC
= Linear Travel ÷Angular Swing in Radians
= 96 ÷ (270º×π /180º)
=20.363636 mm / radian
![Page 115: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/115.jpg)
PROBLEM :A monkey at 20 m slides down from a rope. It swings 30° either sides of rope initially at vertical position. The monkey initially at top reaches at bottom, when the rope swings about two complete oscillations. Draw the path of the monkey sliding down assuming motion of the monkey and the rope as uniform.
![Page 116: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/116.jpg)
θ
o
0123
4 5 67
8
9
1011
121314
15
16 17 18 1920
21
222324
23
13
2224
12
34
56
78
910
1112
1415
16
1819
2021
17
P3
P9
P15
![Page 117: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/117.jpg)
Problem : 2 Draw a cycloid for a rolling circle, 60 mm diameter rolling along a straight line without slipping for 540° revolution. Take initial position of the tracing point at the highest point on the rolling circle. Draw tangent & normal to the curve at a point 35 mm above the directing line.
![Page 118: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/118.jpg)
First Step : Draw a circle having diameter of 60 mm.
Second step: Draw a straight line tangential to the circle from bottom horizontally equal to
(540 x ) x 60 mm= 282.6 mm i.e. 1.5 x x 60 mm 360
Third step : take the point P at the top of the circle.
![Page 119: CHAPTER – 2](https://reader035.vdocument.in/reader035/viewer/2022081420/56815cab550346895dcaabff/html5/thumbnails/119.jpg)
Rolling circle
P1
P2
P3
P4
P0
P6
P7
P8
P5
P9
P10
1
2
34
5
6
7 89
10
1 2 3 4 5 6 7 8 9 100
C0 C1 C2 C3 C4
Directing line
Length of directing line = 3D/2
540 = 360 + 180
540 = D + D/2 Total length for 540 rotation = 3D/2
C5 C6 C7 C8 C9 C10S
norm
al