kinematic pair

KINEMATIC PAIR

• A kinematic pair is a joint of two links that permits relative motion.

• When any two links are connected in such a way that their relative motion is completely or successfully constrained, they form a kinematic pair.

• Practical examples: ■ In a reciprocating steam engine (Fig.1.1), the kinematic pairs existing are:

(a) Crank and connecting rod,

(b) Connecting rod and piston rod, and

(c) Piston and engine cylinder.

■ The crank and connecting rod of the steam engine are said to form a kinematic pair, because (i) they are in contact, and (ii) they have relative motion between them.

1. Classification of Kinematic Pairs

Kinematic pairs may be classified into many types based on the three considerations, as shown in Fig.1.10.

I. Kinematic Pairs Depending upon Relative Motion between Links

1. Sliding (or Prismatic) Pair (P)

• When two links have a sliding motion relative to each other, it is known as sliding or prismatic pair.

• It is designated by the letter P.

• As shown in Fig.1.11, the bar 1 is constrained to have a sliding motion relative to bearing 2, forms, a sliding pair.

• Since the relative motion between links 1 and 2 can be expressed by a single co-ordinate '3', therefore the sliding pair has single degree of freedom.

• Practical examples:

■ Piston and cylinder;

■ Cross-head and guides in a steam engine;

■ Ram and its guides in a shaper;

■ Tail stock on the lathe bed, etc.

2. Turning (or Revolute) Pair (R)

• When two elements are connected such that one element revolves around the other, it forms a turning pair.

• It is designated by the letter R.

• The turning pair is also known as a revolute, a hinge or a pin-jointed pair.

• The shaft 1 with two collars rotates in a bearing 2, as shown in Fig.1.12, forms a turning pair.

• Since the relative motion between links 1 and 2 can be expressed by a single co-ordinate 'θ', therefore the turning pair has a single degree of freedom.

• Practical examples:

■ Lathe spindle supported in the head stock;

■ Crank shaft in a journal bearing in an engine;

■ Cycle wheels turning over their axles;

■ Arbor supported between the arbor support and column of a milling machine; etc.

3. Screw (or Helical) Pair (S)

• In a screw pair, one link is constrained to have a combination of turning and sliding motion relative to the other link.

• It is designated by the letter S.

• In Fig.1.13, the links 1 and 2 form a screw pair.

• Practical examples:

■ Nut and bolt

■ Lead screw of a lathe with nut.

■ Threaded spindle and movable jaw in a bench vice; etc.

• Though the screw pair allows rotation as well as translation, it has one degree of freedom because the relative movement between 1 and 2 can be expressed by a single co-ordinate 'θ' or 's'. These two co-ordinates are related by the relation:  where L is lead of the screw.

4. Cylindrical Pair (C)

• In a cylindrical pair, one link is constrained to have a combination of translation and rotation motion relative to the other link.

• It is designated by the letter C.

• As shown in Fig.1.14, the links 1 and 2 form a cylindrical pair, as it allows both translation and rotation motion parallel to the axis of rotation.

• Unlike screw pair, the cylindrical pair has two degrees of freedom. Because two independent co-ordinates 's' and 'θ' are required to specify its relative motion.

5. Spherical (or Globular) Pair (G)

• In a spherical pair, one link is constrained to swivel in or about the other fixed point.

• It is designated by the letter G.

• A ball and socket joint as shown in Fig.1.15, represents the spherical pair.

• Since the complete description of motion of spherical pair requires three independent variables (two angles α and ϕ) to define the direction Ou and the angle θ of rotation about Ou. Therefore the spherical pair has three degrees of freedom.

• Practical examples:

■ Attachment of a car mirror;

■ Ball and socket joint;

■ Pen supporters joint in a pen stand; etc.

6. Rolling Pair

• When one link is free to roll over the other, it forms a rolling pair.

• A ball and roller bearing as shown in Fig. 1.16 forms a rolling pair. As shown in Fig.1.16, (i) the ball and the bearing forms a rolling pair; and (ii) the ball and the shaft also forms another rolling pair.

• Practical examples:

■ Ball and roller bearings;

■ A lawn mover rolling over a lawn;

■ A road roller rolling over the ground; etc.

II. Kinematic Pairs Depending upon Nature of Contact between Links

1. Lower Pair

• If a kinematic pair in motion has a surface or area contact between the two links, it is called a lower pair.

• It may noted that all sliding pairs, turning pairs, screw pairs, cylindrical pairs and spherical pairs form lower pairs.

• Pairs shown in Fig.1.11, 1.12, 1.13, 1.14 and 1.15 are examples of lower pair.

• Practical examples: Nut and bolt, bolt and socket joint, shaft rotating in bearing, piston reciprocating in a cylinder, etc.

2. Higher Pair

• If a kinematic pair in motion has a line or point contact between the two links, it is called a higher pair.

• The cam and follower arrangement shown in Fig.1.17(a) is an example of higher pair, as there is a line contact between them.

• Practical examples:

■ Cam and flat faced follower, roller bearings, and most gears (Fig.1.17(b)) have line contact.

■ Cam and knife edge follower, ball bearings, and teeth of skew helical gears have point contact.

III. Kinematic Pairs Depending upon Mechanical Arrangements for Constraining Motion

1. Closed (or Self-clòsed) Pair

• When two links of a pair are held together mechanically, they constitute a closed pair.

• Practical examples: All lower pairs are self-closed pairs.

2. Unclosed (or Open or Force-closed) Pair

• When two links of a pair are not held together mechanically, they constitute an unclosed pair. In this case, the contact between the two links is maintained by the forces exerted by spring and gravity.

• Practical examples: The cam and follower mechanism (Fig.1.17(a)) forms an unclosed pair.