LOCI OF POINTS AND GENERATED CURVES

Loci of PointsThe path traced out by a point when it moves in space, under given conditions or in accordance with a definite law, is known as locus of that point (loci is the plural of locus) The path of a point which moves according to mathematically defined conditions is known as its Locus.

LOCUSFor example, a point P moving in a plane, so that it is always at a constant distance from another fixed point O traces out a circle as its locus. Many important geometrical curves (ellipse, parabola, hyperbola, cycloidal curves) may be considered as Loci e.g, conic curves, helices, and screw threads, involutes and spiral curves.

Locus as a circle

The locus of a point P is a circle when it moves in a plane in such a way that its distance from a fixed point O, always remain constant. The fixed point O is called the centre and the constant distance OP is called the radius.P O

Locus as a straight line

The locus of a point P is a straight line when it moves in a plane in such a way that its distance from a fixed line AB is always a constant. If the fixed line AB is an arc of a circle, then the locus will be another arc drawn through the point P and having the same center as of arc AB.

Locus of P

A

B

Locus of point on a mechanism

The locus of a point in a mechanism is the path which is traced by the point when the mechanism moves through a complete cycle of operation. The method of drawing the locus of a particular point in a mechanism is to construct the mechanism in several positions. The point is plotted for each position and its locus is obtained by drawing a smooth curve through these plotted points. The mechanism in successive positions may be drawn with drawing instruments geometrically or with a paper trammel. The use of computer aided drafting renders the procedure very handy and fast.

Plane and space curves

Curve or curved line

A line which is generated by a point that moves in a constantly changing direction is called curve or curved line. The exact nature of each curve or curved line is determined by the motion of its generated point. The following are the two general classes of curves or curved lines ::

Plane curves or single-curved lines singleA line which is generated by a point that moves in a constantly changing direction in the same plane is called plane curve or single curved line.

Space curves or Double curved lines a line which is generated by a point that moves in a constantly changing direction in the space is called space curve or double curved line.

Types of plane and space curves 1.

The following are the important types of plane and space curves used in engineering practice ::Plane curves or single curved lines:(a)

Roulettes or cycloidal curves(i) (ii) (iii)

Cycloid Trochoids (superior and inferior) Epicycloid Epitrochoids (superior and inferior) Hypocycloid Hypotrochoid (superior and inferior)

(b) (c)

Involutes Spirals(i) (ii)

Archemedian spiral Logarithmic spiral

2.

Space curves or double curved lines:(a)

Helix (i) Cylindrical Helix (ii) Conical helix

1. PLANE CURVES OR SINGLE CURVED LINES

(a) CYCLOIDAL AND SPIRAL CURVESROULETTES or CYCLOIDAL CURVES Those curves which are generated by a fixed point on a rolling curve that rolls without slipping along fixed base curve. The rolling curve is called generating curve and the fixed curve is called the directing curve.

APPLICATION OF ROULETTES

CYCLOID

The curve is the locus of a point on the circumference of a circle which rolls, without slipping, along a fixed straight line.

ENGINEERING APPLICATION OF CYCLOID

The cycloid curve was formerly used more extensively in the design of gear tooth profile, but modern production methods tend to limit its applications to small gears used in instruments, for which epicycloids or hypocycloid curves are generally used.

PROBLEM

Draw a cycloid, given the diameter of a generating circle as 50 mm. also draw a tangent and normal at any given point T on the curve.

Solution - Cycloid

Solution - Steps

With center Co draw the rolling circle of 50 mm. draw a straight line, the path along which it is to roll, tangent to the circle. Fix the initial position of point which is to trace the required locus while the rolling circle make some revolution along the base line. Let it be Po. Mark a length Po Po equal to the circumference of the rolling circle, along the base line, and divide it into a number of equal parts, 12 here. Divide the circumference of the rolling circle also into the same number of equal parts.

Solution - Steps

Through division points on the rolling circle, draw lines parallel to fixed line and at the points on the fixed line erect perpendiculars to cut the horizontal center line of the rolling circle at points C1, C2, C3 etc. As the circle rolls through 1/12th of a complete revolution, the center Co will move to the position C1 and the point P will move from initial position Po to P1 and so on. Therefore, the points Po, P2, P3 etc. are plotted by the intersection of lines drawn division points 1, 2, 3 etc on the circle and the corresponding circle arcs drawn with centers C1, C2 etc, as illustrated for P4 and P5. A smooth curve joining all the 12 points plotted thus, gives the required cycloid.

Solution - StepsTangent and normal at a point on the cycloid Draw the rolling circle in such a position that It passes through T, by chain line. The normal is given by the line TN, where N is the point of contact between the rolling circle, and the fixed line. The tangent T1, T2 is perpendicular to TN at T.

Trochoids

The curve generated by a point within or outside the circle which rolls along a straight line is called trochoid. When a circle rolls, without slipping along a fixed straight line, the locus of the fixed point P not lying on the rolling circle is a trochoid. When the point P which traces the locus is outside the rolling circle, the locus produced is superior trochoid. trochoid. When the point P is inside the rolling circle the locus is inferior trochoid. trochoid. The construction of both trochoids is very similar to that used for cycloid.

Problem

Draw trochoids, given the diameter of the rolling circle as 40 mm and the fixed point P, tracing the locus, is 8 mm away from the rolling circle.

Solution - Superior Trochoid

Solution - Inferior Trochoid

Solution

The construction of both trochoids is very similar to that used for cycloid. It should be noted however, that in each case the circumference of the rolling circle is laid out along the fixed line and divided into 12 equal parts, and the circle through the given point P is divided into 12 equal parts, not the reverse.

Epicycloid

The curve generated by a point on the circumference of a rolling circle which rolls outside the directing circle is called epicycloid. When a circle rolls, without slipping, around the outside of a fixed circle, the locus of a point on the circumference of the rolling circle is called the epicycloid. The rolling circle is called generating circle and the fixed circle is called the directing circle. circle.

Problem

Draw an epicycloid, given the radii of rolling and directing circles as r = 30 mm and R = 120 mm, respectively. Also draw a normal and a tangent at any point Q on the curve.

Solution - Epicycloid

1. PLANE CURVES OR SINGLE CURVED LINES

(b) InvoluteA curve traced out by an end of a piece of string when unwound from a circle or a polygon is called involute. When a straight line rolls, without slipping, on a curve, the locus of any point on the straight line is an involute to the curve. The involute to a circle is the locus of the end of a taut string as it is unwound from the surface of a cylinder or base circle.

Engineering application of involute

Involute of a circle is used as the profile of gear teeth. Cams are often designed to the involute shape because it ensures rolling contact between the roller and the follower at constant speed. The involute of a circle can be drawn by drawing tangents at various points on the circumference of the circle and making the various points at corresponding distances along their Tangents. While the involute of any polygon can be drawn by extending its sides, keeping the corners of polygon as successive sides of the polygon thereby terminating on the extended sides

Problem

Draw an involute to a circle of 50 mm. Also draw a tangent and normal to it, at any given point on it.

Solution Involute to a circle

Problem

Draw the involute of a circular arc which subtends an angle (90 degrees here) at the center of the circle of 120 mm.

Involute to a circular arc

Problem

Draw an involute to an equilateral triangle of 20 mm side.

Involute of a triangle

1. PLANE CURVES OR SINGLE CURVED LINES

(c) Spirals

A curve generated by a point moving continuously in one direction along a rotating line is called spiral. The point or the end about which the line rotates is called pole. The line joining any point on the spiral curve with the pole is called the radius vector and the angle between this and the line in its initial position is called the vector angle. When line completes one revolution, the moving point is said to have traced out one revolution. A spiral may take any number of revolutions before reaching the pole, but there will be as many convolutions as the number of revolutions.

Archimedean Spiral

The curve traced out by a point moving with uniform velocity along a line which is also rotating with uniform velocity is called Archimedean spiral. It is the locus of a point P which moves at a steady rate along a line, while the line rotates at uniform speed about center, O , such that for each angular displacement of the line, the linear displacement of the point is constant.

Engineering applications of Archimedean Spirals

They are used in the construction of cams, threads of scroll chucks and in some other simple devices.

Problem

Construct an Archimedean spiral of two convolutions, given the greatest and the shortest radii as 84 mm and 12 mm, respectively.

Archimedean Spiral ( Two Convolutions )

Problem

Construct an Archimedean spiral of one convolution , given the radial movement of the point P during one convolution as 60mm and the initial position of P as pole O.