Theory of Machines and Mechanism Lecture 1

Lecture 1

Łukasz Jedliński, Ph.D., Eng. Basic terms and definitions

Theory of Machines and Mechanism may be defined as that branch of engineering science, which deals with the study of relative motion between the various parts of 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. Basic terms and definitions

Machines and mechanisms have been devised by people since the dawn of history. The ancient Egyptians devised primitive machines to accomplish the building of the pyramids and other monuments. Though the wheel and pulley (on an axle) were not known to the Old Kingdom Egyptians, they made use of the lever, the inclined plane (or wedge), and probably the log roller. The origin of the wheel and axle is not definitively known. Its first appearance seems to have been in Mesopotamia about 3000 to 4000 B.C.

A Mesopotamian wheel

Pulley systems Basic terms and definitions

Theory of Machines and Mechanism may be subdivided into the following four branches:

Kinematics - is that branch of theory of machines which is responsible to study the motion of bodies without reference to the forces which are cause this motion, i.e it’s relate the motion variables (displacement, velocity, acceleration) with the time.

Dynamics - is that branch of theory of machines which deals with the forces and their effects, while acting upon the machine parts in motion.

Kinetics – it deals with inertia forces which occur due to combined effect of mass and motion of machine elements.

Statics - is that branch of theory of machines which deals with the forces and their effects, while the machine parts are rest. Basic terms and definitions

The role of kinematics is to ensure the functionality of the mechanism, while the role of dynamics is to verify the acceptability of induced forces in parts. The functionality and induced forces are subject to various constraints (specifications) imposed on the design.

The objective of kinematics is to develop various means of transforming motion to achieve a specific kind needed in applications. For example, an object is to be moved from point A to point B along some path. The first question in solving this problem is usually: What kind of a mechanism (if any) can be used to perform this function? And the second question is: How does one design such a mechanism? The objective of dynamics is analysis of the behavior of a given machine or mechanism when subjected to dynamic forces. For the above example, when the mechanism is already known, then external forces are applied and its motion is studied. The determination of forces induced in machine components by the motion is part of this analysis. Basic terms and definitions

A mechanism is a group of links connected to each other by joints, to form a kinematic chain with one link fixed, to transmit force and motion. It is design to do a specific motion.

A kinematic chain is defined as an assemblage of links and joints. Kinematic chains or mechanisms may be either open or closed.

Openmechanism Closedmechanism

A machine is an assemblage of links that transmits and/or transforms forces, motion and energy in a predetermined manner, to do work. The term machine is usually applied to a complete product. Basic terms and definitions

The similarity between machines and mechanisms is that they are both combinations of rigid bodies and the relative motion among the rigid bodies are definite. The difference between machine and mechanism is that machines transform energy to do work, while mechanisms so not necessarily perform this function.

Machine - car Mechanism - gearbox Basic terms and definitions

Structure – is an assemblage of a number of rigid bodies having no relative motion between them and meant for carrying load having straining action.

Vehicle Frame

Bridge Basic terms and definitions

Kinematic link (or simply link , element ), definitions: 1. A rigid body that provides connections to other rigid bodies by at least two joints. Not always true. (Theoretically, a true rigid body does not change shape during motion. Although a true rigid body does not exist, mechanism links are designed to minimally deform and are considered rigid.) Elastic parts, such as springs, are not rigid and, therefore, are not considered links. They have no effect on the kinematics of a mechanism and are usually ignored during kinematic analysis. They do supply forces and must be included during the dynamic force portion of analysis. Beter definition: 2. It is the individual part or component of the mechanism, most often rigid, indivisible in terms of the function which meets in the mechanism.

4 – Flexible link

3 – Fluid link Basic terms and definitions

A link may consist of several parts, which are rigidly fastened together, so that they do not move relative to one another e.g. piston rod.

Piston rod Piston rod – kinematic representation A A

Part 4 B Part 1 B

Part 3

Part 2 Basic terms and definitions

Types of links: Binary link - one with two nodes. Ternary link - one with three nodes. Quaternary link - one with four nodes.

Different classification :

Simple link Complex link Basic terms and definitions

Types of links: Crank – a link that makes a complete revolution and is pivoted to ground. Rocker – a link that has oscillatory (back and forth) rotation and is pivoted to ground. Coupler (or connecting rod) as a link that has complex motion and is not pivoted to ground. Ground (base , frame ) is defined as any link or links that are fixed (non moving) with respect to the reference frame. Note that the reference frame may in fact it self be in motion. Basic terms and definitions

Joint (kinematic pair, pair) - connection between links (two or more), which permits relative motion between them and physically adds some constraint(s) to this relative motion.

Classification of joints according to the : 1. Type of relative motion between the elements. 2. Type of contact between the elements. 3. Type of closure between the elements. 4. Number of links joined. Basic terms and definitions

1. According to the type of relative motion between the elements a) Revolute (Turning ) 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. b) Sliding (Prismatic, Prism ) 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. c) Cylindrical pair. Permits angular rotation and an independent sliding motion. Thus, the cylindrical pair has two degrees of freedom. d) Rolling pair. When the two elements of a pair are connected in such a way that one rolls over another fixed link, the pair is known as rolling pair. Ball and roller bearings are examples of rolling pair. e) Screw (helical ) 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. f) Spherical pair. When the two elements of a pair are connected in such a way that one element (with spherical shape) turns or swivels 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. g) Planar pair (flat ). It is seldom found in mechanism in its undisguised form, except at a support point. It has three degrees of freedom, that in, two translations and a rotation. Basic terms and definitions a) Revolute (Turning, Pin ) pair.

b) Sliding (Prismatic, Prism ) pair.

c) Cylindrical pair Basic terms and definitions d) Rolling pair.

e) Screw (helical ) pair

f) Spherical pair.

g) Planar pair (flat) Basic terms and definitions

2. The kinematic pairs according to the type of contact between the elements may be classified as discussed below (Reuleaux ): a) Lower pair. When the two elements of a pair have a surface 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: revolute, sliding, screw, cylindrical, sphere, planar pairs form lower pairs. b) 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.

Cam and follower Spur gear Belt drive Basic terms and definitions

3. The kinematic pairs according to the type of closure between the elements may be classified as discussed below : a) Self closed pair. When the two elements of a pair are connected together mechanically in such a way that only required kind of relative motion occurs, it is then known as self closed pair. The lower pairs are self closed pair. b) Force – closed pair. When the two elements of a pair are not connected mechanically but are kept in contact by the action of external forces, the pair is said to be a force-closed pair. The cam and follower in an example of force closed pair, as it is kept in contact by the force exerted by spring and gravity.

Cam and follower - forced closed Spherical pair – self closed Basic terms and definitions

4. Classification of joints according to the number of links joined. The order of a joint is defined as the number of links joined minus one. The combination of two links has order one and it is a single joint.

one-pin joint

Joint of order one

two-pin joints

Joint of order two (multiple joints) Basic terms and definitions

Degree of freedom (DOF) or mobility (M) The number of inputs that need to be provided in order to create a predictable output; also: Number of independent coordinates needed to define the position of the element/mechanism.

Rigid body in 2D space (plane) – 3 DOF Rigid body in 3D space – 6 DOF Basic terms and definitions

Planar Mechanisms

Kutzbach’s modification of Gruebler’s equation

M = 3( L– 1) – 2 J1 –J2

M = degree of freedom or mobility L = number of links (with a frame)

J1 = number of 1 DOF joints J2 = number of 2 DOF joints

Spatial Mechanisms

M = 6( L– 1) – 5 J1 – 4 J2 – 3 J3 – 2 J4 –J5 Basic terms and definitions

Number of degrees of freedom for plane mechanisms

The number of degrees of freedom or mobility (M) for some simple mechanisms having no higher pairs (i.e. p4 = 0) as shown in figure are determined as follows: a) The mechanism has three links and three binary joints i.e. n = 3 and p5 = 3 M = 3 (3 – 1) – 2 x 3 = 0 b) The mechanism has four links and four binary joints i.e. n = 4 and p5 = 4 M = 3 (4 – 1) – 2 x 4 = 1 c) The mechanism has five links and five binary joints i.e. n = 5 and p5 = 5 M = 3 (5 – 1) – 2 x 5 = 2 d) The mechanism has five links and four binary joints (because there are two joints at B and D, and

two ternary joints at A and C) i.e. n = 5 and p5 = 6 M = 3 (5 – 1) – 2 x 6 = 0 d) The mechanism has six links and eight binary joints (because there are four ternary joints at A, B, C

and D) i.e. n = 6 and p5 = 8 M = 3 (6 – 1) – 2 x 8 = -1 Basic terms and definitions

Number of degrees of freedom for plane mechanisms

It may be noted that:

. When M = 0, then the mechanism forms a structure and no relative motion between the links is possible Fig. a and d;

. When M = 1, then the mechanism can be driven by a single input motion, as shown Fig. b;

. When M = 2, then two separate input motions are necessary to produce constrained motion for the mechanism, as shown in Fig. c;

. When M = -1 or less, then there are redundant constrains in the chain and it forms a statically indeterminate structure, as shown in Fig. e. Basic terms and definitions

M ≤ 0 Mechanism with zero, or negative, degrees of freedom are termed locked mechanisms. These mechanisms are unable to move and form a structure.

M > 1 Mechanism with multiple degrees of freedom need more than one driver to precisely operate them. In general, multi-degree of freedom mechanism offer greater ability to precisely position a link. There is no requirement that a mechanism have only one DOF, although that is often desirable for simplicity. Basic terms and definitions

Kutzbach’s–Gruebler’s formula for mechanism mobility does not take into account the specific geometry (size or shape) of the links, only the connectivity of links and the type of connections (constraints). Grübler equation not always works.

Structure Mechanism DOF real = 0 DOF real= 1 DOF real = 1 DOF real = 1 DOF from equation = DOF from equation = 3 x 3 – 2 x 3 – 1 = 2 DOF from equation = 3 x 6 – 2 x 8 – 1 = 1 3 x 6 – 2 x 8 – 1 = 1 Basic terms and definitions

DOF real = 1 DOF real = 1 DOF from equation = 3 x 3 – 2 x 3 – 2 = 1 DOF from equation = 3 x 6 – 2 x 6 – 8 = -2 Basic terms and definitions

DOF real = 1 DOF real = 1 DOF from equation = 3 x 4 – 2 x 6 = 0 DOF from equation = 3 x 1 – 2 x 2 = -1 Basic terms and definitions

Grashof’s law

As is clear, the motion of links in a system must satisfy the constraints imposed by their connections. However, even for the same chain, and thus the same constraints, different motion transformations can be obtained. This is demonstrated in figure on the next slide, where the motions in the inversions of the four-bar linkage are shown. In this figure s identifies the smallest link, l is the longest link, and p, q are two other links. Basic terms and definitions

Crank -rocker mechanism

Double-crank mechanism Double-rocker mechanism Basic terms and definitions

Grashof’s law

From a practical point of view, it is of interest to know if for a given chain at least one of the links will be able to make a complete revolution. In this case, a motor can drive such a link. The answer to this question is given by Grashof’s law, which states that for a four-bar linkage, if the sum of the shortest and longest links is not greater than the sum of the remaining two links, at least one of the links will be revolving. Grashof’s law (condition) is expressed in the form: s = length of shortest link s + l ≤ p + q l = length of longest link p = length of one remaining link q = length of other remaining link

Reuleaux approach the problem somewhat differently but, of course, obtains the same results. The conditions are: s + l +p ≥ q s + l - p ≤ q s + q +p ≥ l s + q - p ≤ l Basic terms and definitions

These four condition are illustrated below by demonstrating what happens if the conditions are not met.

s + l + p < q s + l - p > q

s + q +p < l s + q - p < l Basic terms and definitions

An ideal system of classification of mechanisms does not exist. Many attempts have been made, none has been very successful in devising a completely satisfactory method.

Example mechanism

Beam engine Basic terms and definitions

Example mechanism

Coupling rod of a locomotive Basic terms and definitions

Example mechanism

Slider-crank mechanism Basic terms and definitions

Example mechanism

Scotch yoke mechanism Basic terms and definitions

Example mechanism

Geneva mechanism