Lecture01 Engineering Curves

8/12/2019 Lecture01 Engineering Curves

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ME119:

Engineering Drawing & Graphics

1. Engineering Curves

Department of Mechanical Engineering

Indian Institute of Technology Bombay


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Engineering Curves

Chapters 6 & 7 cover the details on Engineering Curves.

Roughly work out all the problems given to you before you come to

the drawing hall.


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Types of Engineering Curves

Conics Circle: Ellipse:Arches, bridges, dams, monuments, manholes,

glands, stuffing boxes,

Parabola:Arches, bridges, sound & light reflectors,

Hyperbola: Cooling towers, water channels, moda, horseseat,

Line:

Cycloids & Trochoids Trajectories, Gear profiles,

Involutes & Evolutes Gear profiles,

Spirals Springs, helical gear profile, cam profile

Helices 3D curve. DNA, helicopter, springs, screws, tools, gears

Freeform curves Templates (Eg.: French curves), Mathematical: Ext. features

Cams Mechanisms

Loci Mechanisms


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Conic Sections

Intersection of a right circular cone with a plane in various

orientations.

Circle

Ellipse Parabola

Hyperbola

Line (Cone is a ruled surface)


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Conic Sections


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Conic SectionsAlgebraic definition

You are familiar with conics as 2nd degree algebraicequations.

Generic definition in polar form:


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Conic SectionsAlgebraic definition


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Conic SectionsAlgebraic definition - Ellipse


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Conic SectionsAlgebraic definition - Parabola


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Conic SectionsAlgebraic definition - Hyperbola


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Conic SectionsGeometric/ graphical definition

A conic section (or just conic)

is the intersection curve

between a conical surface

(more precisely, a right

circular conical surface) with a

plane.

Straight line is a conic. It

is a hyperbola when theplane contains the cone

axis. Alternately, it is a

circle of infinite radius.
http://localhost/var/www/apps/conversion/tmp/scratch_9//upload.wikimedia.org/wikipedia/commons/1/11/Conic_Sections.svg


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Conic SectionsGeometric/ graphical definition

A conicis the locus of a point whose distance to some point,called a focus, and some line, called a directrix, are in a

fixed ratio, called the eccentricity.

Circle (e=0) Ellipse (e1)

Line (e>1)

focus

directrix

axisvertex


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Conic Sections: EllipseBasic method (from focus f& eccentricity e)

Construct an ellipse when the distance of the focus from thedirectrix is equal to 50mm and its eccentricity is 2/3.


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Conic Sections: EllipseBasic method (from focus f& eccentricity e)


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Conic Sections: EllipseBasic methods (from focus f& eccentricity e)

F, V and RH directrix are

not required. They require

additional calculations.


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Conic Sections: EllipseArcs of circles method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using arcs of circles method.

C S


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C i S ti Elli


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C i S ti Elli


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Conic Sections: EllipseConcentric circles method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using concentric of circles method.

C i S ti Elli


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Conic Sections: EllipseConcentric circles method (from major & minor axes)

C i S ti Elli


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Conic Sections: EllipseOblong method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using oblong method.

C i S ti Elli


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Conic Sections: Ellipse


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Conic Sections: EllipseTriangle method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using oblong method.

Obtain rectangle or

parallelogram from the

triangle. Then, it is same

and oblong method.Strictly speaking, this will not be

an ellipse but a nice loop

created from the triangle.



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Conic Sections: EllipseLoop of the thread method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using loop of the thread method.



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Conic Sections: EllipseLoop of the thread method (from major & minor axes)



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Conic Sections: EllipseTrammel method (from major & minor axes)

Construct an ellipse whose semi-major axis is 60mm andsemi-minor axis 40mm using trammel method.



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Conic Sections: EllipseTrammel method (from major & minor axes)



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Conic Sections: EllipseApproximate method for a superscribing rhombus

Conic Sections: Parabola


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Conic Sections: ParabolaMethods of drawing

1. Basic method: From (i) distance of focus from the directrix (p) and(ii) eccentricity (e)

2. 2. From (i) base and (ii) axis

a. Rectangle method

b. Tangent (triangle) method

Note: The above can be used for non-rectangular parabolas too.



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Conic Sections: ParabolaBasic method (from focus f)

Construct a parabola when the distance of the focus fromthe directrix is equal to 50mm.



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Conic Sections: ParabolaRectangle method (from base & axis)

Construct a parabola inscribed in a rectangle of 120mm x90mm using the Rectangle method.



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Conic Sections: ParabolaRectangle method (from base & axis)



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Conic Sections: ParabolaTangent method (from base & axis)

Construct a parabola inscribed in a rectangle of 120mm x90mm using the tangent method.



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Conic Sections: ParabolaTangent method (from base & axis)

Conic Sections: Hyperbola


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Conic Sections: HyperbolaMethods of drawing

1. Basic method: From (i) distance of focus from the directrix (p) and(ii) eccentricity (e)

2. From (i) foci and (ii) vertices

3. From (a) asymptotes and (b) one point

Note: The above can be used for non-rectangular hyperbolas too.



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ypBasic method (from focus f& eccentricity e)

Construct a hyperbola when the distance of the focus fromthe directrix is 65mm and eccentricity is 3/2.



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ypBasic method (from focus f& eccentricity e)



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ypFrom (a) foci and (b) vertices

Construct a hyperbola from its foci and vertices.



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ypFrom (a) foci and (b) vertices



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ypFrom (a) asymptotes and (b) one point

Construct a hyperbola from its asymptotes and one point.



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From (a) asymptotes and (b) one point

Cycloids and Trochoids


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Rolling Path

(Directrix)

Position of Trace Point on the Circle (Generatrix)

On Inside Outside

Straight Line Cycloid InferiorTrochoid

Superior

Trochoid

Outside Circle Epi-Cycloid

Epi-

Inferior-

Trochoid

Epi-

Superior

Trochoid

Inside Circle Hypo-Cycloid

Hypo-

Inferior-Trochoid

Hypo-

SuperiorTrochoid

The curve traced by a point on/inside/outside of a circulardisc when it rolls on a line or outside/inside an arc.



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Construction of a cycloid

Construct a cycloid from diameter of the generating circle.



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Construction of a cycloid



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Construction of the (inferior & superior) trocoid

Construct the inferior & superior trachoid from diameter ofthe generating circle.

Same method as trachoid.

-Length of PA is 2R

-The circle to be divided is of radius R1.



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Cycloids and TrochoidsC t ti f th (i f i & i ) t id


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Alternate method:Mark the points of cycloid. Contract or extend CiPito get the

desired point.

Cycloids and TrochoidsC t ti f i /h l id


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Construction of epi-/hypo-cycloid

Construct an epi-cycloid from diameters of the generatingand directing circles.

Cycloids and TrochoidsC t ti f i /h l id


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Cycloids and TrochoidsC t ti f E i l id


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Construction of Epi-cycloid

Cycloids and TrochoidsConstr ction of epi /h po l id


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Cycloids and TrochoidsConstruction of Hypo cycloid


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Construction of Hypo-cycloid

Involutes & Evolutes


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Involute: Curve traced by the end of a piece of threadunwound from a circle or polygon while keeping the thread

tight. Alternately, it may be visualized as the curve traced by

a point in a straight line when it is rolled without slip around

a circle or polygon.

Evolute: Locus of the centre of curvature of a curve.

Involutes & EvolutesConstruction of an involute of a circle


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Construction of an involute of a circle

Construct an involute of a circle of diameterd

.

Involutes & EvolutesConstruction of an involute of a circle


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Construction of an involute of a circle

Involutes & EvolutesConstruction of an involute of a triangle


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Construction of an involute of a triangle

Involutes & EvolutesConstruction of an involute of a square


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Construction of an involute of a square

Involutes & EvolutesConstruction of an involute of a hexagon


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Construction of an involute of a hexagon

Involutes & EvolutesLocus of the tip of a line rolling over a circle


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Locus of the tip of a line rolling over a circle

Spiral


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Curve traced by a point going round a point simultaneouslymoving towards it (spiral in) or away from it (spiral out).

There could be a variety of spirals depending on the nature

of relationship between the radial and peripheral velocities.

However, the following two are the most popular:

1. Archemedian spiral: Radial displacement per revolution is

constant.

1. Logarithmic or equi-angular spiral: Ratio of the lengths ofconsecutive radius vectors enclosing equal angles is always

constant.

Archemedian Spiral


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Construct an archemedian spiral of 1.5 convolution for the

greatest and shortest radii.

Archemedian Spiral


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Archemedian Spiral


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A link of 225mm long swings on a pivot O from its vertical

position of rest to the right through an angle of 75 and

returns to its initial position at uniform velocity. During this

period, a sleeve approximated as a point P initially at a

distance of 20mm from the pivot O moving at uniform speed

along the link reaches its end. Draw the locus of P.

Archemedian Spiral


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Archemedian Spiral


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A straight link PQ of 60 mm length revolves one complete

revolution with uniform motion in CCW direction about hinge

P. During this period, an insect moves along the link from P

to Q and back to P with uniform linear speed. Draw the path

of the insect for a stationary viewer. Identify this curve.

Archemedian Spiral


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Logarithmic or Equi-angular spiral


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Construct the logarithmic spiral of one convolution, given the

shortest radius r(=10mm) and the ratio of the lengths of

radius vectors enclosing an angle (=30) is k=10/9.



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HelixOrdinary helix


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A helix has a pitch of 50mm and diameter of 75mm. Draw

its front view.

HelixOrdinary helix


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HelixConical helix


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The diameter of a cone is 75mm at the

base and its height is 100mm. The pitch

of a helix on it is 75mm. Draw its front

view.

Cams


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Disc cam is a profile whose radial distance from its centre

of rotation varies to meet the required pattern. When a

follower, which may be a sharp wedge or a roller ribs on it,

moves (translates or oscillated) in the desired manner. 3D

cam are also available which are used in various

applications such as indexing.


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Cams3D cam


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CamsPlate cam - Exercise


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Draw a cam to give the following uniform motions to a roller

of 25mm diameter:

- A rise of 40mm over 90 of revolution

- Dwell in the upper position for the next 75 of revolution

- Fall by the same distance during the next 120 ofrevolution

- In lower most position for the remaining part of revolution

Note: Least metal is 20mm and diameter of the shaft is50mm.

CamsPlate camExercise


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Locus of any Point in a Mechanism


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Mechanisms are used to convert one motion into another. It

is important to know the locus of any point of a mechanism

which may be its hinge (joint) or a point on any link.

Locus of any Point in a MechanismExercise


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Two equal cranks AB and CD connected by the link BD as

shown in Fig. 6, rotate in opposite directions. Point P is on

BD and point Q is along its extension beyond B. Draw the

loci of P and Q for one revolution of AB. AB = CD =

450mm; AC = BD = 1500mm; PD = 300mm; BQ = 300mm.

Locus of any Point in a MechanismExercise


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Conclusions


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Quickly skim through Chapters 1-5 of N.D. Bhatsbook that cover

the basic things such as

- Use of drawing instruments

- Sheet layout and Freehand sketching

- Lines, Lettering and Dimensioning

- Scales

- Geometrical constructions

Roughly work out all the problems given to you. Only if you come

prepared, you will be able to complete all problems of the sheet in

the drawing session.

Thank You!


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K.P. Karunakaran

Professor

Department of Mechanical EngineeringIndian Institute of Technology Bombay

Powai, Mumbai-400076, INDIA

Tel.: 022-25767530/ [email protected]

www.me.iitb.ac.in/~karuna

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Lecture01 Engineering Curves