Introduction(lec1), Lecture notes of Machine Design

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Introduction

ME 561P Machine Design I

 Machine Design is the creation of new and better

machines and improving the existing ones.

 A new or better machine is one which is more

economical in the overall cost of production and

operation.

 From the study of existing ideas, a new idea has to be

conceived.

 The measurement of physical quantities is one of the most important operations in engineering.

 Every quantity is measured in terms of some arbitrary, but internationally accepted units, called fundamental units.

 Some units are expressed in terms of other units, which are derived from fundamental units, are known as derived units e.g. the unit of area, velocity, acceleration, pressure, etc.

Review of Fundamental and Derived Units

 It is an important factor in the field of Engineering science, which may be defined as an agent, which produces or tends to produce, destroy or tends to destroy motion.

 According to Newton’s Second Law of Motion, the applied force or impressed force is directly proportional to the rate of change of momentum.

Force

 

 

Force F ma F kma

a t

v v

ma t

mv v t

mv mv

mv mv

t v v

a

v

v

m

Momentum mv

f i

f i f i

f i

i f

f

i

  







 

 

 













,

Rateof changeof momentum

Changeof momentum

timereq'dtochangefrom to

constantacceleration

finalvelocityof thebody

initialvelocityof thebody

Let massof thebody

F ma

k

toa bodyof unit mass.

such thatit producesa unitacceleration

For thesakeof convenience,the unitof forceadoptedis

constant of proportionality

In SI system of units, the unit of force is called newton

(briefly written as N). A newton may be defined as the force,

while acting upon a mass of one kg, produces an acceleration

of 1 m /s^2 in the direction in which it acts. Thus

1  1  1 2 ^12

s

kg m s

N  kg m  

 It has been established since long that a rigid body is composed of small particles.

 If the mass of every particle of a body is multiplied by the square of its perpendicular distance from a fixed line, then the sum of these quantities (for the whole body) is known as mass moment of inertia of the body. It is denoted by I.

 Consider a body of total mass m. Let it be composed of small particles of masses m 1, m 2, m 3, m 4, etc. If k 1, k 2, k 3, k 4, etc., are the distances from a fixed line, as shown, then the mass moment of inertia of the whole body is given by

Mass Moment of Inertia

I  m 1 k 12  m 2 k 22  m 3 k 32  m 4 k 42 ...

 If the total mass of a body may be assumed to concentrate at one point (known as centre of mass or centre of gravity), at a distance k from the given axis, such that

 The distance k is called the radius of gyration.

 It may be defined as the distance, from a given reference, where the whole mass of body is assumed to be concentrated to give the same value of I.

 The unit of mass moment of inertia in SI system is kg-m^2.

2

2 4 4

2 3 3

2 2 2

2 1 1

2 ...

I mk

mk m k m k m k m k



    

 It may be defined as the product of force and the perpendicular distance of its line of action from the given point or axis.

 A little consideration will show that the torque is equivalent to a couple acting upon a body.

 The Newton’s second law of motion when

applied to rotating bodies states, the

torque is directly proportional to the rate

of change of angular momentum.

Mathematically,

Torque

 

T I

dt

d

dt

d

T I

I

dt

d I

Torque T

: angularaccelerati on

Since isconstant,

 Whenever a force acts on a body and the body undergoes a displacement in the direction of the force, then work is said to be done. For example, if a force F acting on a body causes a displacement x of the body in the direction of the force, then

 If the force varies linearly from zero to a maximum value of F, then

 When a couple or torque ( T ) acting on a body causes the angular displacement (θ) about an axis perpendicular to the plane of the couple, then

Work

Workdone Fx

x

F

x

F

Work done 

abody through 1.

Itistheworkdonebyaforceof 1 whenitdisplaces

such that 1 1

InSIsystem,theunitof workis or joule, ;

Workdone

m

N

N-m J

N-m J

T

 It may be defined as the capacity to do work.

 The energy exists in many forms e.g. mechanical, electrical, chemical, etc.; but we are mainly concerned with mechanical energy.

 The mechanical energy is equal to the work done on a body in altering either its position or its velocity.

 The following three types of mechanical energies are important from the subject point of view :

1. Potential energy. It is the energy possessed by a body, for doing work, by virtue of its position.

Energy

PE  Wh  mgh , N  m

2. Strain energy. It is the potential energy stored by an elastic body when deformed.

 A compressed spring possesses this type of energy, because it can do some work in recovering its original shape.

 Thus, if a compressed spring of stiffness ( s ) N per unit deformation ( i.e. extension or compression) is deformed through a distance x by a weight W, then

 In case of a torsional spring of

stiffness ( q ) N-m per unit angular

deformation when twisted

through an angle θ radians, then

W sx

trainenergy workdone Wx sx N m

S 2

2

S train energy  workdone  q 

Notes : 1. When a body of mass moment of inertia I (about a given axis) is rotated about that axis, with an angular velocity ω, then it possesses some kinetic energy. In this case,

2. When a body has both linear and angular motions, e.g. wheels of a moving car, then the total kinetic energy of the body is equal to the sum of linear and angular kinetic energies. 3. The energy can neither be created nor destroyed, though it can be transformed from one form into any of the forms, in which energy can exist. This statement is known as ‘ Law of Conservation of Energy ’. 4. The loss of energy in any one form is always accompanied by an equivalent increase in another form.

 When work is done on a rigid body, the work is converted into kinetic or potential energy or is used in overcoming friction.

 If the body is elastic, some of the work will also be stored as strain energy.

2

KE of rotation  I 

2 2

Total KE  mv  I 