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Theory of Machines and Mechanism

Lecture 1

Łukasz Jedliński, Ph.D., Eng.

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

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.


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.


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.

Open mechanism Closed mechanism

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.


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

Vehicle Frame


Structure – is an assemblage of a number of rigid bodies having no relative motion

between them and meant for carrying load having straining action.

Bridge

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

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

A

B

A

B

Piston rod – kinematic representation

Part 1

Part 2

Part 3

Part 4


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


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.

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.


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.


a) Revolute (Turning, Pin) pair.

b) Sliding (Prismatic, Prism) pair.

c) Cylindrical pair


d) Rolling pair.

e) Screw (helical) pair

f) Spherical pair.

g) Planar pair (flat)


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

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

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.


two-pin joints

one-pin joint

Joint of order one

Joint of order two (multiple joints)

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

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

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


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.


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.



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.

DOF real = 1

DOF from equation = 3 x 3 – 2 x 3 – 1 = 2 DOF from equation =

3 x 6 – 2 x 8 – 1 = 1

DOF real = 1

DOF from equation =

3 x 6 – 2 x 8 – 1 = 1

Structure

DOF real = 0Mechanism

DOF real= 1


DOF real = 1

DOF from equation = 3 x 3 – 2 x 3 – 2 = 1

DOF real = 1

DOF from equation = 3 x 6 – 2 x 6 – 8 = -2


DOF real = 1

DOF from equation = 3 x 4 – 2 x 6 = 0

DOF real = 1

DOF from equation = 3 x 1 – 2 x 2 = -1

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.



Crank -rocker mechanism

Double-crank mechanism Double-rocker mechanism

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 + l ≤ p + q

s = length of shortest link

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


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

s + q - p < ls + q +p < l

These four condition are illustrated below by demonstrating what happens

if the conditions are not met.

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

Example mechanism


Coupling rod of a locomotive

Example mechanism


Slider-crank mechanism

Example mechanism


Scotch yoke mechanism

Example mechanism


Geneva mechanism

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