Kinematics of Machines. Brown Hills College of Engineering & Technology

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Gertrude Lang
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Introduction: mechanism and machines, kinematic links, kinematic pairs, kinematic chains, plane and space mechanism, kinematic inversion, equivalent linkages, four link planar mechanisms, mobility

1 Introduction: mechanism and machines, kinematic links, kinematic pairs, kinematic chains, plane and space mechanism, kinematic inversion, equivalent linkages, four link planar mechanisms, mobility and range of movement, straight line mechanisms, steering mechanisms, pantograph, problems. Introduction: The subject Theory of Machines may be defined as that branch of Engineering-science, which deals with the study of relative motion between the various parts of a machine, and forces which act on them. The knowledge of this subject is very essential for an engineer in designing the various parts of a machine. Classification of Theory of Machines as following four branches: 1. Kinematics: It deals with the relative motion between the various parts of the machines. 2. Dynamics: It deals with the forces and their effects, while acting upon the machine parts in motion. 3. Kinetics: It deals with the inertia forces which arise from the combined effect of the mass and motion of the machine parts. 4. Statics: It deals with the forces and their effects while the machine parts are at rest. The mass of the parts is assumed to be negligible. Mechanisms is a combination of rigid or restraining parts or bodies from which the machine is assembled, this is done by making one of the parts as fixed, and the relative motion of other parts is determined with respect to the fixed part. Example, Slider-crank mechanism used in internal combustion engine or reciprocating air compressor, where the rotary movement of the crank is converted through the connecting rod into the reciprocating motion of the slider, or vice-versa. Fig shows; Link-1 is fixed, Link-2 is Crank, Link-3 is Connecting rod and Link-4 is piston which slides in a cylinder. However, the term linkage has been widely used as a synonym for the word mechanism. Machine is a combination of the mechanisms which receives energy and transforms it into some useful work from which we reduce the human efforts. A machine consists of a number of parts or bodies. Kinematic Link Each part of a machine, which moves relative to some other part, is known as a kinematic link. A link may consist of several parts, which are rigidly fastened together, so that they do not move relative with another part. For example, in a reciprocating steam engine, piston, piston rod and crosshead constitute one link; connecting rod with big and small end bearings constitute a second link; crank, crank shaft and flywheel a third link and the cylinder, engine frame and main bearings a fourth link. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 1

Types of Links 1. Rigid link: A rigid link is one which does not undergo any deformation while transmitting motion. 2.

For example, belts, ropes, chains and wires are flexible links and transmit tensile forces only. 3.

jacks and brakes. Kinematic Pair The two links of a machine, when in contact with each other, are said to form a pair. If the relative motion between them is completely or successfully constrained (i.

2 Types of Links 1. Rigid link: A rigid link is one which does not undergo any deformation while transmitting motion. 2. Flexible link: A flexible link is one which is partly deformed in a manner not to affect the transmission of motion. For example, belts, ropes, chains and wires are flexible links and transmit tensile forces only. 3. Fluid link: A fluid link is one which is formed by having a fluid in a container and the motion is transmitted through the fluid by pressure or compression only, as in the case of hydraulic presses, jacks and brakes. Kinematic Pair The two links of a machine, when in contact with each other, are said to form a pair. If the relative motion between them is completely or successfully constrained (i.e. in a definite direction), the pair is known as kinematic pair. Classification of Kinematic Pairs A. According to Nature of Relative Motion 1. Turning Pair / Revolute Pair: When the two elements of a pair are connected in such a way that one can only turn or revolve about a fixed axis of another link, the pair is known as turning pair. Turning pair has a single degree of freedom. 2. Sliding Pair / Prismatic Pair: When the two elements of a pair are connected in such a way that one can only slide relative to the other, the pair is known as a sliding pair. Sliding pair has a single degree of freedom. 3. Screw Pair: When the two elements of a pair are connected in such a way that one element can turn about the other by screw threads, the pair is known as screw pair. The lead screw of a lathe with nut, and bolt with a nut are examples of a screw pair. Screw pair has a single degree of freedom. 4. Cylindrical Pair: When the two elements of a pair are connected in such a way that one element in rotation or translation, parallel to the axis of rotation to the other element, the pair is known as cylindrical pair. Cylindrical Pair has a two degree of freedom. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 2

5. Rolling pair: When the two elements of a pair are connected in such a way that one roll over another fixed link, the pair is known as rolling pair.

Spherical pair: When the two elements of a pair are connected in such a way that one element (with spherical shape) turns or pivots about the other fixed element, the pair formed is called a

Two coordinates x and y describe the relative translation in the xyplane and the third describe the relative rotation about the z-axis. B. According to the type of contact between the links: 1.

3 5. Rolling pair: When the two elements of a pair are connected in such a way that one roll over another fixed link, the pair is known as rolling pair. Ball and roller bearings are examples of rolling pair. 6. Spherical pair: When the two elements of a pair are connected in such a way that one element (with spherical shape) turns or pivots about the other fixed element, the pair formed is called a spherical pair. The ball and socket joint, attachment of a car mirror, pen stand etc., are the examples of a spherical pair. 7. Planar pair: it has a three degree of freedom. Two coordinates x and y describe the relative translation in the xyplane and the third describe the relative rotation about the z-axis. B. According to the type of contact between the links: 1. Lower pair: When the two elements of a pair have a surface or area contact when relative motion takes place and the surface of one element slides over the surface of the other, the pair formed is known as lower pair. It will be seen that sliding pairs, turning pairs, cylindrical pairs, spherical pairs, planar pairs and screw pairs form lower pairs. 2. Higher pair: When the two elements of a pair have a line or point contact when relative motion takes place and the motion between the two elements is partly turning and partly sliding, then the pair is known as higher pair. A pair of friction discs, toothed gearing, belt and rope drives, ball and roller bearings and cam and follower are the examples of higher pairs. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 3

Kinematic Chain A kinematic chain is an assembly of links in which the relative motions of the links are

In case, the motion of a link results in indefinite motions of other links, it is a non-kinematic chain

However, some authors prefer to call all chains having relative motions of the links as kinematic chains.

A kinematic chain is a series of links connected by kinematic pairs.

otherwise it is termed an open chain shown in fig 2.

4 Kinematic Chain A kinematic chain is an assembly of links in which the relative motions of the links are possible and the motion of each relative to the other is definite [Figs (a), (b), and (c)]. In case, the motion of a link results in indefinite motions of other links, it is a non-kinematic chain [Fig.d)]. However, some authors prefer to call all chains having relative motions of the links as kinematic chains. A redundant chain does not allow any motion of a link relative to the other [Fig. (e)]. A kinematic chain is a series of links connected by kinematic pairs. The chain is said to be closed if every link is connected to at least two other links shown in fig 1, otherwise it is termed an open chain shown in fig 2. fig 1 fig 2 A link which is connected to only one other link is known as a singular link. If it is connected to two other links, it is called a binary link. Similarly, if a link is connected to three other links, it is referred to as a ternary link, and so on. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 4

Plane Mechanisms If all points of a mechanism move in parallel planes, then it is defined as a plane mechanism. A simple plane mechanism is shown in Fig.

A very common example of a space mechanism, known as Hooke's joint, is shown in Fig.

In this process, the relative motions of the links of the mechanisms produced remain unchanged. A slider-crank chain mechanism has the following kinematic inversions: 1.

5 Plane Mechanisms If all points of a mechanism move in parallel planes, then it is defined as a plane mechanism. A simple plane mechanism is shown in Fig. where all points move in parallel planes. Space Mechanisms A space mechanism is one in which all points of the mechanism do not move in parallel planes. A very common example of a space mechanism, known as Hooke's joint, is shown in Fig. Kinematic Inversion This process of fixing different links of the same kinematic chain to produce distinct mechanisms is called kinematic inversion. In this process, the relative motions of the links of the mechanisms produced remain unchanged. A slider-crank chain mechanism has the following kinematic inversions: 1. First Inversion This inversion is obtained when link 1 is fixed and links 2, 3 and 4 are made the crank, connecting rod and the slider shown in Fig (a) Applications: 1. Reciprocating engine, 2. Reciprocating compressor as shown in Fig. (b), if it is a reciprocating engine, 4 (piston) is the driver and if it is a compressor, 2 (crank) is the driver. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 5

2. Second Inversion: This inversion is obtained when link 2 is fixed and links 3, 4 and 1 are made the crank, slider and connecting rod shown in Fig (a) fig (a) fig (b) Applications: 1.

The forward stroke takes a little longer and cuts the metal whereas the return stroke is idle and takes a shorter period. Working: Slider 4 rotates in a circle about A and slides on link 1 fig. (b).

6 2. Second Inversion: This inversion is obtained when link 2 is fixed and links 3, 4 and 1 are made the crank, slider and connecting rod shown in Fig (a) fig (a) fig (b) Applications: 1. Whitworth quick-return mechanism shown in Fig (b), 2. Rotary engine Whitworth Quick-Return Mechanism: It is a mechanism used in workshops to cut metals. The forward stroke takes a little longer and cuts the metal whereas the return stroke is idle and takes a shorter period. Working: Slider 4 rotates in a circle about A and slides on link 1 fig. (b). C is a point on link 1 extended backwards where link 5 is pivoted. The other end of link 5 is pivoted to the tool, the forward stroke of which cuts the metal. The axis of motion of slider 6 (tool) passes through O and is perpendicular to OA, the fixed link. The crank 3 rotates in the counter-clockwise direction. Initially, let the slider 4 be at B' so that C be at C'. Cutting tool 6 will be in the extreme left position. With the movement of the crank, the slider traverses the path B'BB" whereas point C moves through C'CC". Cutting tool 6 will have the forward stroke. Finally, slider B assumes the position B" and cutting tool 6 is in the extreme right position. The time taken for the forward stroke of slider 6 is proportional to the obtuse angle B" AB' at A. Similarly, slider 4 completes the rest of the circle through path B"B'" B' and C pass through C"C"'C'. There is backward stroke of tool 6. The time taken in this is proportional to the acute angle B"AB' at A. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 6

3. Third Inversion This inversion is obtained when link 3 is fixed and links 2, 4 and 1 are made the crank, oscillates and connecting

Crank and Slotted-Lever Mechanism Oscillating Cylinder Engine: As shown in fig.

The piston reciprocates inside the cylinder pivoted to the fixed link 3.

7 3. Third Inversion This inversion is obtained when link 3 is fixed and links 2, 4 and 1 are made the crank, oscillates and connecting rod shown in Fig (a) Applications: 1. Oscillating Cylinder Engine, 2. Crank and Slotted-Lever Mechanism Oscillating Cylinder Engine: As shown in fig. (b), link 4 is made in the form of a cylinder and a piston is fixed to the end of link 1. The piston reciprocates inside the cylinder pivoted to the fixed link 3. The arrangement is known as oscillating cylinder engine, in which as the piston reciprocates in the oscillating cylinder, the crank rotates. 4. Fourth Inversion This inversion is obtained when link 4 is fixed and links 3, 2 and 1 are made the oscillates about the fixed pivot B on link 4, oscillates about B and end 0 and link 1reciprocate along the axis of the fixed link 4shown in Fig (a) fig (a) or fig (b) Application: Hand Pump Fig. (b) Shows a hand-pump. Link 4 is made in the form of a cylinder and a plunger fixed to the link 1 reciprocates in it. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 7

8 Equivalent Mechanisms It is possible to replace turning pairs of plane mechanisms by other type of pairs having one or two degrees of freedom, such as sliding pairs or cam pairs. This can be done according to some set rules so that the new mechanisms also have the same degrees of freedom and are kinematically similar. This process is called equivalent mechanism. Sachin Chaturvedi Lecturer in Department of Mechanical Engineering 8

September 20, Chapter 5. Simple Mechanisms. Mohammad Suliman Abuhaiba, Ph.D., PE

September 20, Chapter 5. Simple Mechanisms. Mohammad Suliman Abuhaiba, Ph.D., PE Chapter 5 Simple Mechanisms 1 Mohammad Suliman Abuhaiba, Ph.D., PE 2 Assignment #1 All questions at the end of chapter 1 st Exam: Saturday 29/9/2018 3 Kinematic Link or Element kinematic link (link) or