# Loci of Points – Engineering Drawing / Technical Drawing Notes

### Loci of Points on a Moving Link in Simple Mechanisms

**Mechanism** is the combination of various links so paired that the motion is completely constrained. Mechanism are used to transform the motion, e.g. rotary to reciprocating, reciprocating to rotary, rotary to oscillating, etc. Thus, mechanism are used to convert available motion into useful motion.

A **locus **is the path of a point which moves in space. Here, we are interested to find the loci of points on following simple mechanisms.

- Slider crank mechanism
- Four bar mechanism
- Crank-Connecting rod-trunnion mechanism
- Combination of above mechanisms

The locus of a point P moving in a plane about another point C in such a way that its distance from it is constant, is a circle of radius equal to CP. Refer the figure given below.

The Locus of a point P moving in a plane in such a way that its distance from a fixed line AB is constant, is a line through P, parallel to the fixed straight line. Refer the figure given below.

The locus of a point P moving in a plane in such a way that its distance from a fixed arc AB of a circle is constant, is the another arc drawn through P with the same centre. Refer the figure given below.

Some typical locus problems such as construction of ellipse, parabola, hyperbola, cycloid, spirals, etc. are already discussed in the chapter.

In simple mechanisms, it is often necessary to know the paths of points on their moving parts. These are determined by assuming a number of different positions of the moving parts and then locating the corresponding positions of the points.

### Examples: Loci of Points

**Example 1 – **Loci of Points

**In a slider-crank mechanism, the connecting rod BC is 100 mm long and the crank AB is 20 mm long. The slider C is sliding on a straight path passing through the point A. Draw the locus of the midpoint P of the connecting rod BC for one complete revolution of the crank OB.**

### Procedure:

- Draw a circle of radius AB = 20 mm with centre A. The circle is the locus of the point B.
- Divide the circle into 12 equal parts and mark them as B1, B2, B3, … B12 as shown in the figure in the direction of rotation. AB1, AB2, ….., AB12 represent the positions of the crank during the rotation.
- Draw a horizontal line through A on which the slider C is to travel.
- With centre B1 and radius BC (BC = 100 mm), cut the path of C at a point C1. Draw a line joining B1 and C1. Again, with centre B1 and radius BC/2 = 100/2 = 50 mm, cut the line B1C1 at a point P1.
- Similarly, with centres B2, B3, ….B12 and radius BC, cut the path of C at points C2, C3, ….C12. Draw lines joining B2 and C2, B3 and C3, ……, B12C12 at points P2, P3, ….P12 respectively.
- Draw a smooth curve through P1, P2, ….. ,P12 to get the locus of the point P.

**Note:- The distance C1C7 is the travel of the slider and is equal to twice the length of the crank.**

**Example 2 – Loci of Points**

**In the off-set crank mechanism, shown in the figure, the slider end C moves in guides along the line DE, 15 mm below the axis A of the crank-shaft. Crank AB is 20 mm long and connecting rod BC is 100 mm long. Draw he locus of a point P, 35 mm from B along CB produced. Determine also the travel of the slider.**

### Procedure:-

- Draw a circle of radius AB = 20 mm with centre A. The circle is the locus of the point B.
- Divide this line into 12 equal parts and mark them as B1,B2,. ..B12 as shown in the figure in the direction of rotation. AB1,AB2, ….AB12 represent the positions of the crank during the rotation.
- Draw a horizontal line through A.Draw another line DE parallel to this line and at a offset of 15 mm as shown. The slider C travels along this line PE and hence, it represents path of C.
- With centre B1 and radius BC, cut the path of C at a point C1. Draw a line joining B1 and C1. Extend this line C1B1 and mark on it point P1 such that B 1P1 = 35 mm.
- Similarly, with centres B2, B3, …., B 12 and radius BC, cut the path of C at points C2,C3, …C12. Draw lines joining B2 and C2, B3 and C3, …B 12 and C12.Next,with B2,B3,… .,B12 as centres and radius BP = 35 mm, cut the line produced C2B2,C3B3, ….,C12B12 at points P1, P2, …,P12 respectively.
- Draw a smooth curve through P1, P2, …, P12 to get the locus of the point P.
- The distance C1C7 is the travel of the slider.

**Example 3 – Loci of Points**

**In the mechanism shown in the figure, the connecting rod is constrained to pass through the trunnion (guide) at D. Trace the locus of the end C and a point P on BC for one complete revolution of the crank.**

### Procedure:-

- Draw a circle of radius AB = 25 mm with centre A. The circle is the locus of the point B.
- Divide this circle into 12 equal parts and mark them as B1,B2, …B12 as shown in the figure in the direction of rotation. AB1,AB2, ….,AB12 represent the positions of the crank during the rotation.
- Draw the location of trunnion D as shown. From B1, B2, ….B12 draw lines passing through D and of 140 m length to get points C1, C2 , ….C12 respectively.
- On lines B1C1, B2C2, …,B 12C12 mark points P1,P2, ….P 12 respectively at a distance of 25 mm from B towards C on the rod.
- Join the points P1, P2 , …P12 and C1 , C2, …C 12 by means of a smooth curve to get the loci of the points P and C respectively.

**Example 4 – Loci of Points**

**The crank AB rotates about A and the connecting rod BC slides in the same plane on the curved surface of a shaft (40 mm diameter and centre D) as shown in the Figure 3.7(A). Draw the locus of the end C and the point P beyond CB and 25 mm from B for one complete revolution of the crank AB.**

### Procedure:

- Draw a circle of radius AB = 35 mm with centre A. The circle is the locus of the point B.
- Divide this circle into 12 equal parts and mark them as B1, B2, ….., B 12 as shown in the figure in the direction of rotation. AB1, AB2, …,AB12 represent the positions of the crank during the rotation.
- Draw, another circle of radius 20 mm and with centre D, by taking distance AD = 75 mm as shown.
- Note draw lines tangent to the circle with centre D, on the upper side from B1, B2, …..B12 and mark them C1, C2 ,. .., C12 respectively by taking BC = 130 mm.
- Extend C1B1, C2B2 , ..,C12B12 upto P1, P2, .. .P12 respectively by taking BP = 25 mm or CP = 155 mm.
- Join points C1, C2, ….C12 and P 1, P2, …., P12 to get the loci of the points C and P respectively.

**Example 5 – Loci of Points**

**ABCD is a four bar mechanism with the link AC as the fixed link. Driving crank AB is 40 mm long. Driven crank DC is also40 mm long. Connecting link BC is 130 mm long. Distance between A and D is 130 mm. Two cranks rotate in the opposite directions. Draw the loci of points P and Q for one complete revolution of the driving crank. The point P is the midpoint of the connecting link BC and point Q is 30 mm from C on BC extended.**

### Procedure:-

- Draw a line AD of length t 30 mm. Draw a circle of radius AB = 40 mm with centre A. The circle is the locus of point B.
- Draw another circle of radius DC = 40 mm with centre D. The circle is the locus of the point C.
- Divide circle with centre A into 12 equal parts and mark them as B1, B2, …., B12 as shown in the figure in the clockwise direction of rotation AB1, AB2, ……, AB12 represent the positions of the crank during the rotation.
- With centre BJ and radius BC, cut the path of Cat a point CJ .Draw a line joining B1 and C1• Extend this line B1C1 and mark on it point Q1 such that C1Q1 = 30 mm. Mark also point P1 such that it is the midpoint of BC.
- Similarly, with centres B2, B3, …, B12 and radius BC, cut the path of C at points C2, C3, ….,C12 by considering that C rotates in the anticlockwise direction (i.e. opposite to the direction of rotation of B). Draw lines joining B2 and C2, B3 and C3, …., B12 and C12 • Next, with C2 , C3, . . ., C12 as centres and radius CQ = 30 mm, cut the line produced B2 C2, B3C3, ….., B12 C12 at points Q2, Q3, ….Q12 respectively. Also mark points P2, P3, ….P12 as midpoints of B2 C2, B3C3, ….. B12C12
- Join Points P1, P2, …P12 and Q1,Q2, ….Q12 by means of a smooth curve to get the loci of the points P and Q respectively.

**Example 6 – Loci of Points**

**In the figure shown below crank OA revolves around ‘O’ while rod AB is constrained to pass through ‘s’. For one complete revolution of rank; draw the locus of point ‘B’ and point ‘P’ on the extension of the rod AB.**

### Procedure:-

- Draw a circle of radius OA = 15 mm with centre O. The circle is the locus of the point A.
- Divide this circle into 12 equal parts and then as A1, A2, …, A12 as shown in the figure in the direction of rotation. OA 1, OA2, ….., OA12 represent the positions of the crank along the rotation.
- Draw the location of the trunnion S as shown from A 1, A2,….,A12 draw lines passing through Sand AB of 35 mm length to get points Bp.B2,…., B12 respectively.
- On lines A1B1, A2 B2…..mark points P1, P2, ……, P12 respectively at a distance of 10 mm from A and on extension of BA
- Join the poin0ts P1, P2, ….., P12 and B1 , B2 , …B12 by means of a smooth curve to get the locus of the points P and B respectively.