12101dep-notice_04242021.pdf

DYNAMICS OF MACHINES (BME-28)
B.Tech (Fifth Sem.)
Dr. Sanjay Mishra
Associate Professor, MMMUT Gorakhpur

Syllabus
Dynamics of Machines (BME 28)
UNIT-I
STATIC & DYNAMIC FORCE ANALYSIS Static equilibrium of two/three force members, Static
equilibrium of member with two forces and torque, Static force analysis of linkages, D’Alembert’s
principle, Equivalent offset inertia force, Dynamic force analysis of four link mechanism and slider crank
mechanism, Dynamically equivalent system
TURNING MOMENT & FLYWHEEL Engine force analysis-Piston and crank effort, Turning moment
on crankshaft, Turning moment diagrams-single cylinder double acting steam engine, four stroke IC
engine and multi-cylinder steam engine, Fluctuation of energy, Flywheel and its design
UNIT-II
Governors Terminology, Centrifugal governors-Watt governor, Dead weight governors-Porter & Proell
governor, Spring controlled governor-Hartnell governor, Sensitivity, Stability, Hunting, Isochronism,
Effort and Power of governor
Gyroscopic Motion Principles, Gyroscopic torque, Effect of gyroscopic couple on the stability of aero
planes, ships& automobiles

UNIT-III
BALANCING OF MACHINES Static and dynamic balancing, Balancing of several masses rotating in the
same plane and different planes, Balancing of primary and secondary forces in reciprocating engine, Partial
balancing of two-cylinder locomotives, Variation of tractive force, swaying couple, hammer blow,
Balancing of two cylinder in-line engines
MECHANICAL VIBRATIONS Introduction, Single degree free & damped vibrations of spring-mass
system, Logarithmic decrement, Torsional vibration, Forced vibration of single degree system under
harmonic excitation, Critical speeds of shaft
UNIT-IV
Friction Introduction Friction in journal bearing-friction circle, Pivots and collar friction-Flat and conical
pivot bearing Flat collar bearing, Belt drives-types, material, power transmitted, ratio of driving tensions for
flat belt, centrifugal tension, initial tension, rope drive-types Laws of friction, Efficiency on inclined plane,
Screw friction, Screw jack, Efficiency, Friction in journal bearing-friction circle, Pivots and collar friction-
Flat and conical pivot bearing, Flat collar bearing
Clutches, Bakes & Dynamometers Single and multiple disc friction clutches, Cone clutch, Brakes-types,
Single and double shoe brake, Simple and differential Band brake, Band and Block brake, Absorption and
transmission dynamometers, Prony brake and rope brake dynamometers

1. Ability to carry out static and dynamic force analysis of four bars mechanism and slider crank
mechanism, and design of flywheels.
2. To understand types of centrifugal governors, the effects of characteristic parameters and controlling
force diagrams and principles of gyroscopic effect and its engineering applications.
3. To Understand the balancing of rotating and reciprocating masses and ability to analyze single degree
freedom systems subjected to free, damped and forced vibrations as well as calculation of critical
speeds of shaft.
4. To Understand the applications of friction in pivot and collar bearings, belt drives, clutches, brakes
and dynamometers.
Course Outcome

The objective of kinematics is to develop various means of transforming motion to
achieve a specific kind of applications.
The objective of dynamics is analysis of the behavior of a given machine or
mechanism when subjected to dynamic forces.
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.
Kinematics and Dynamics: Difference

The term machine is usually applied to a complete product. A car is a machine.
Similarly, a tractor, a combine, an earthmoving machine, etc are also machine. At
the same time, each of these machines may have some devices performing specific
functions, like a windshield wiper in a car, which are called mechanisms.
The distinction between the machine/mechanism and the structure is more
fundamental. The former must have moving parts, since it transforms motion,
produces work, or transforms energy. The latter does not have moving parts; its
function is purely structural, i.e., to maintain its form and shape under given
external loads, like a bridge, a building, or an antenna mast.
Kinematics and Dynamics: Difference

Dynamics of Machine: Analyses the forces and couples on the members of the machine due to external forces
(static force analysis), also analyses the forces and couples due to accelerations of machine members ( Dynamic
force analysis)
Rigid Body: Deflections of the machine members are neglected in general by treating machine members as rigid
bodies (also called rigid body dynamics).
 The link must be properly designed to withstand the forces without undue deformation to facilitate proper
functioning of the system.
 In order to design the parts of a machine or mechanism for strength, it is necessary to determine the forces
and torques acting on individual links. Each component however small, should be carefully analysed for its
role in transmitting force.
 The forces associated with the principal function of the machine are usually known or assumed.

Forces acting on machine elements
 Joint forces (or Reaction forces): the action and reaction between the bodies involved will be through the
contacting kinematic elements of the links that form a joint. The joint forces are along the direction for
which the degree-of-freedom is restricted.
 Physical forces
 Friction or resisting force
 Inertial forces

Reactions for different types of support

Principle of Transmissibility: The point of application of a force can be transmitted
anywhere along its line of action but within the body
Principle of Superposition: The effect of a force
on a body remains unaltered if we add or subtract
any system which is in equilibrium. It is very
useful in application of parallel transfer of force.

Equivalent systems of forces: Two system are said to be equivalent if they can be
reduced to the same force-couple System at a given point.
Two force system act on the same rigid body are equivalent if the sums of the forces
(resultant) and sums of the moments about a point are equal.

Apart from static forces, mechanism also experiences inertia forces when subjected to acceleration, called dynamic
forces.
Static forces are predominant at lower speeds and
Dynamic forces are predominant at higher speeds.
 Force analysis helps to determine the forces transmitted from one point to another, essentially from input to
output.
 It is the starting point for strength design of a component/ system, basically to decide the dimensions of the
components
 Force analysis is essential to avoid either overestimation or under estimation of forces on machine member.
Overestimation: machine component would have more strength than required. Over design leads to heavier
machines, costlier and becomes not competitive
Underestimation: leads to design of insufficient strength and to early failure.
IMPORTANCE OF FORCE ANALYSIS

 A machine is a device that performs work by transferring energy by means of mechanical forces from a
power source to a driven load. It is necessary in the design of a mechanism to know the manner in which
forces are transmitted from the input to output so that the components of mechanism can be properly sized
to withstand the stresses induced.
 All links have mass, and if links are accelerating, there will be inertia forces associated with this
motion. If the magnitudes of these inertia forces are small relative to the externally applied loads, then
they can be neglected in the force analysis. Such an analysis is referred to as a STATIC FORCE
ANALYSIS.

Static Equilibrium of two force members
 There are many types of structural elements. The support condition has a significant
influence on the behavior of the specific element. It is advantageous to identify certain
types of structural elements which have distinct characteristics.
 If an element has pins or hinge supports at both ends and carries no load in-between, it
is called a two-force member. These elements can only have two forces acting upon
them at their hinges.
 If only two forces act on a body that is in equilibrium, then they must be equal in
magnitude, co-linear and opposite in sense. This is known as the two-force
principle.

 The two-force principle applies to ANY member or structure that has only two forces acting on it. This is
easily determined by simply counting the number of places where forces act on that member.
(REMEMBER: reactions are considered to be forces!) If they act in two places, it is a two-force member.
 One of the unique aspects of these members is the fact that the line of action of the resultants of the
forces acting on the two ends of the member MUST pass along the center line of the structural element.
If they did not, the element would not be in equilibrium.
 Most, but not all, two-force members are straight. Straight elements are usually subjected to either
tension or compression. Those members of other geometries will have bending across (or inside) their
section in addition to tension or compression, but the two-force principle still applies.
 Some common examples of two-force members are columns, struts, hangers, braces, pinned truss
elements, chains, and cable-stayed suspension systems.

Equilibrium of a Two-Force Body
4 - 18
• Consider a plate subjected to two forces F1 and F2
• For static equilibrium, the sum of moments about
A must be zero. The moment of F2 must be zero. It
follows that the line of action of F2 must pass
through A.
• Similarly, the line of action of F1 must pass through
B for the sum of moments about B to be zero.
• Requiring that the sum of forces in any direction be
zero leads to the conclusion that F1 and F2 must
have equal magnitude but opposite sense.

A member under the action of two forces will be in
equilibrium if
– the forces are of the same magnitude,
– the forces act along the same line, and
– the forces are in opposite directions.

 If three non-parallel forces act on a body in equilibrium, it is known as a three-force
member.
 The three forces interact with the structural element in a very specific manner in order
to maintain equilibrium.
 If a three-force member is in equilibrium and the forces are not parallel, they must
be concurrent. Therefore, the lines of action of all three forces acting on such a member
must intersect at a common point; any single force is therefore the equilibrant of the
other two forces.
 A three-force member is often an element which has a single load and two reactions.
These members usually have forces which cause bending and sometimes additional
tension and compression.
 The most common example of a three-force member is a simple beam.
Static Equilibrium of three force members

Equilibrium of a Three-Force Body
4 - 21
• Consider a rigid body subjected to forces acting at
only 3 points.
• Assuming that their lines of action intersect, the
moment of F1 and F2 about the point of intersection
represented by D is zero.
• Since the rigid body is in equilibrium, the sum of the
moments of F1, F2, and F3 about any axis must be
zero. It follows that the moment of F3 about D must
be zero as well and that the line of action of F3 must
pass through D.
• The lines of action of the three forces must be
concurrent or parallel.

THREE FORCE MEMBER
A member under the action of three forces will be in
equilibrium if –
the resultant of the forces is zero, and –
the lines of action of the forces intersect at a point
(known as point of concurrency).

D’Alembert’s Principle
Inertia force:
 Inertia is a property of matter by virtue of which a body resists any change in velocity.
 It is an imaginary force which acts on a rigid body and brings it in equilibrium.
 It is mathematically equal to the accelerating force in magnitude but opposite in direction
Inertia force (Fi)= -(accelerating force)=- m x a ( a = linear acceleration of the CG of the body)
Inertia torque:
 Inertia torque resists any change in the angular velocity of the body.
 Inertia torque brings the body in equilibrium when applied on it.
 Inertia torque is equal to accelerating couple in magnitude but opposite in direction
Inertia Torque (Ti)= - ( I x α ) where (I =mass moment of inertia of the body about an axis passing through the CG
of the body and perpendicular to the plane of the rotation of the body and α = angular acceleration
I=mk2 (m= mass of body and k is radius of gyration)

D’Alembert’s Principle states that the resultant force acting on a body together with the inertia force
are in equilibrium. It is used to convert the dynamic problem into equivalent static problem.
Fr +F i=0 where Fr is the resultant external force act on the body and Fi is inertial force.
According to Newton’s second law of motion
Resultant force = m x a
Fr = ma
Fr - ma=0
But Fr + Fi=0
Therefore Fi= -(m x a)

Equivalent offset inertia force:
 In plane motion involving acceleration the inertia force acts on a body through the its centre of mass.
 If the body acted upon by forces such that resultant do not pass through the centre of mass, a couple will also act on the body.
 It is necessary to replace the inertial force and inertia couple by an equivalent offset inertia force which can account for both.
Resultant force (F) due to large number of forces acting on the body does
not pass through the CG and it is at the distance h from CG
Consider two equal opposite forces F at G so that the rigid body is now
acted upon by
i) a couple of magnitude F x h in anti clockwise
ii) a force of magnitude F passing through G
The couple F x h causes angular acceleration of the rigid body
Force F causes linear acceleration of CG of the body.
Hence we will have two equations

Corresponding to couple:
Couple = I x α
F x h= m k2 x α
Corresponding to force:
Force = m x a
You can find the a and α for the above two equation. ( if F,h,m and k are known)
But if only a and α are known then you have to find F and h
F= mx a
h= (I x α)/F =(mk2 x α)/F
h is the distance of the resultant force from CG

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