MODULE-1 THEORY OF

1. MODULE-1
INTRODUCTION TO FINITE ELEMENT METHOD
www.cambridge.edu.in
Department of
Mechanical Engineering
FINITE ELEMENT METHOD
18ME61
MANJUNATHA T V
Assistant Professor

2. Course Outline
Module1: Introduction , Velocity and Acceleration analysis of planar
mechanisms
Module 2: Static force analysis, Dynamic force analysis
Module 3: Spur Gears, Gear Trains
Module 4: Balancing of Rotating Masses, Reciprocating Masses, Governors.
Module 5: Free vibrations, Forced vibrations.
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3. Module-1 Introduction
Introduction: Mechanisms and machines, Kinematic pairs-types, degree of freedom,
Kinematic chains and their classification, Kinematic inversions,
Velocity and Acceleration analysis of planar mechanisms Graphical method: Velocity and
Acceleration Analysis of Mechanisms Velocity and acceleration analysis of four bar
mechanism, slider crank mechanism. Mechanism illustrating Corioli’s component of
acceleration. Angular velocity and angular acceleration of links, velocity of rubbing.
Velocity and Acceleration Analysis of Mechanisms (Analytical Method):Velocity and
acceleration analysis of four bar mechanism, slider crank mechanism using complex
algebra method.
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4. Module-1 Introduction
Mechanics: It is that branch of scientific analysis which deals with motion, time and
force.
Mechanism: A mechanism is a constrained kinematic chain. This means that the motion
of any one link in the kinematic chain will give a definite and predictable motion relative
to each of the others. Usually one of the links of the kinematic chain is fixed in a
mechanism.
A mechanism with four links is known as
simple mechanism.
the mechanism with more than four links is known as
compound mechanism.
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5. Module-1 Introduction
 Structure: It is an assemblage of a number of resistant bodies (known as
members) having no relative motion between them and meant for carrying
loads having straining action. A railway bridge, a roof truss, machine frames
etc., are the examples of a structure.
Machine: A machine is a mechanism or collection of mechanisms, which
transmit force from the source of power to do some particular type of
work. “Though all machines are mechanisms, all mechanisms are not
machines”.
Machine is an assemblage of parts that transmit forces, motion and energy
in a predetermined manner.
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6. Basic Definitions
 Joint: A connection between two links that allows motion between the links. The motion allowed may be
rotational (revolute joint), translational (sliding or prismatic joint), or a combination of the two (roll-slide
joint).
Kinematic Link or Element Each part of a machine, which moves relative to some other part, is known as a
kinematic link (or simply link) or element.
1. Rigid link. A rigid link is one which does not undergo any deformation while transmitting motion. Strictly
speaking, rigid links do not exist. However, as the deformation of a connecting rod, crank etc. of a
reciprocating steam engine is not appreciable, they can be considered as rigid links.
 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 receptacle and the motion is transmitted
through the fluid by pressure or compression only, as in the case of hydraulic presses, jacks and brakes.
Binary link: Link which is connected to other links at two points. =1
Ternary link: Link which is connected to other links at three points.=2
Quaternary link: Link which is connected to other links at four points.=3
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7. Module-1 Introduction
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kinematic pair : When two kinematic links are connected in such a way that their motion is either
completely or successfully constrained, these two links are said to form a kinematic pair. (i.e. in a definite direction),
the pair is known as kinematic pair.
Kinematic pairs can be classified according to
i) Nature of contact: 1.Lower pair(surface or area contact)
2.Higher pair(point or line contact)
ii) Nature of mechanical constraint or relative motion between the elements
Sliding pair. Turning pair (revolute pair). Cylindrical pair.
Rolling pair. Spherical pair. Helical pair or screw pair.

8. Module-1 Introduction
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iii) Nature of relative motion or type of closure: Closed pair, Unclosed or force closed pair

9. Module-1: Types of constrained motion
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Constrained Motion: In a kinematic pair, if one element has got only one definite motion relative to the other, then the motion is
called constrained motion.
(a) Completely constrained motion. If the constrained motion is achieved by the pairing elements themselves, irrespective of
direction of force applied then it is called completely constrained motion. EX: The piston and cylinder (in a steam engine)
form a pair and the motion of the piston is limited to a definite direction
(b) Successfully constrained motion. If constrained motion is not achieved by the pairing elements themselves, but by some
other means, then, it is called successfully constrained motion. Eg. Foot step bearing, where shaft is constrained from
moving upwards, by its self weight.
(c) Incompletely constrained motion. When relative motion between pairing elements takes place in more than one direction, it
is called incompletely constrained motion. Eg. Shaft in a circular hole.

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Kinematic Chain : When the kinematic pairs are coupled in such a way that the last link is joined to the first link to transmit
definite motion (i.e. completely or successfully constrained motion), it is called a kinematic chain. or
In other words, a kinematic chain may be defined as a combination of kinematic pairs, joined in such a way that each link
forms a part of two pairs and the relative motion between the links or elements is completely or successfully
For a kinematic chain N = 2 P – 4 = 2 (j + 2) / 3
Where N = no. of links, P = no. of Pairs and j = no. of joints
LHS > RHS, then the chain is locked
LHS = RHS, then the chain is constrained
LHS < RHS, then the chain is unconstrained
The most important kinematic chains are those which consist of
four lower pairs, each pair being a sliding pair or a turning pair.
The following three types of kinematic chains with four lower
pairs are important from the subject point of view :
1. Four bar chain or quadric cyclic chain,
2. Single slider crank chain, and
3. Double slider crank chain.

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 Four bar chain (mechanism) is the simplest and the basic kinematic chain
and consists of four rigid links which are connected by pin joints.
 Each of them forms a turning pair at A, B, C and D.
The link that makes a complete revolution is called a Crank.
The shortest link, will make a complete revolution relative to the other three
links crank or driver. In Fig., AB (link 2 ) is a crank.
 link BC (link 4) which makes a partial rotation or oscillates is known as
Lever Or Rocker Or Follower
 link CD (link 3) which connects the crank and lever is called Connecting
Rod Or Coupler.
 The fixed link AD (link 1) is known as Frame Of The Mechanism.
 In this type of chain all four pairs will be turning pairs,The four links may
be of different lengths.
 Grashof ’s law : According to Grashof ’s law for a four bar mechanism “the
sum of the shortest and longest link lengths should not be greater than the
sum of the remaining two link lengths” if there is to be continuous relative
motion between the two links.
 S + L ≤ P + Q L=length of the longest link S=length of the
shortest link
1. Four bar chain (mechanism)=4T

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Inversions of mechanism:
A mechanism is one in which one of the links of a kinematic chain is fixed. Different mechanisms can be
obtained by fixing different links of the same kinematic chain. These are called as inversions of the mechanism
By fixing each link at a time we get as many mechanisms as the number of links, then each mechanism is called
‘Inversion’ of the original Kinematic Chain.
 Inversions of four bar chain mechanism:
Inversions of four bar chain mechanism: There are three inversions:
1) Beam Engine or Crank and lever mechanism.
2) Coupling rod of locomotive or double crank mechanism.
3) Watt’s straight line mechanism or double lever mechanism.

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Beam Engine or Cranks and Lever Mechanism

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Beam Engine or Cranks and Lever Mechanism
A part of the mechanism of a beam engine (also known as cranks and lever mechanism) which consists of
four links is shown in Fig. In this mechanism, when the crank rotates about the fixed centre A, the lever
oscillates about a fixed centre D. The end E of the lever CDE is connected to a piston rod which reciprocates
due to the rotation of the crank.

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Coupled wheel of locomotive
. Coupling rod of locomotive: In this mechanism the length of link AD = length of link BC. Also length of link AB = length
of link CD. When AB rotates about A, the crank DC rotates about D. this mechanism is used for coupling locomotive wheels.
Since links AB and CD work as cranks, this mechanism is also known as Double Crank Mechanism.
the links AD and BC (having equal length) act as cranks and are connected to the respective wheels. The link CD acts as a
coupling rod and the link AB is fixed in order to maintain a constant centre to Centre distance between them. This
mechanism is meant for transmitting rotary motion from one wheel to the other wheel.

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Watt's indicator mechanism or double lever mechanism
A Watt‟s indicator mechanism (also known as Watt's straight line
mechanism or double lever mechanism) which consists of four links is
shown in Fig.
The four links are: fixed link at A, link AC, link CE and link BFD. It
may be noted that BF and FD form one link because these two parts have
no relative motion between them. The links CE and BFD act as levers.
The displacement of the link BFD is directly proportional to the
pressure of gas or steam which acts on the indicator plunger. On any
small displacement of the mechanism, the tracing point E at the end of
the link CE traces out approximately a straight line.

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2. The Single slider-crank Mechanism=3T+1S
When one of the turning pairs of a four-bar chain is replaced by a sliding pair, it becomes a single slider-crank chain or
simply a slider-crank chain. the purpose of this mechanism is to convert rotary motion to reciprocating motion and vice
versa
1. Reciprocating IC engine mechanism

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2. Oscillating cylinder engine
In this mechanism Link 3 forming the turning pair is fixed and it corresponds to the connecting rod of a reciprocating steam
engine mechanism. When the crank (link 2) rotates, the piston attached to piston rod (link 1) reciprocates and the cylinder
(link 4) oscillates about a pin pivoted to the fixed link at A.
Slider-crank chain

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Slider-crank chain:. Crank and slotted lever mechanism
In this mechanism, the link AC (i.e. link 3) forming the turning pair is fixed, as shown in Fig. The link 3 corresponds to the
connecting rod of a reciprocating steam engine. The driving crank CB revolves with uniform angular speed about the fixed
center C. A sliding block attached to the crank pin at B slides along the slotted bar AP and thus causes AP to oscillate about
the pivoted point A. A short link PR transmits the motion from AP to the ram which carries the tool and reciprocates along
the line of stroke R1R2. The line of stroke of the ram (i.e. R1R2) is perpendicular to AC produced
In the extreme positions, AP1 and AP2 are tangential to the circle and the cutting tool is at the end of the stroke. The forward
or cutting stroke occurs when the crank rotates from the position CB1 to CB2 (or through an angle β) in the clockwise
direction. The return stroke occurs when the crank rotates from the position CB2 to CB1 (or through angle α) in the
clockwise direction. Since the crank has uniform angular speed, therefore, Therefore, when the crank rotates uniformly, we
get
Time to cutting = α = α
Time of return β = 360 – α
This mechanism is used in shaping machines, slotting machines and in rotary engines, Length of stroke = 2AP*(CB/AC).

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Slider-crank chain:. Crank and slotted lever mechanism

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Slider-crank chain:. Whitworth Quick Return Mechanism
This mechanism used in shaping and slotting machines.
In this mechanism the link CD (link 2) forming the turning pair is fixed; the driving crank CA (link 3) rotates at a
uniform angular speed. The slider (link 4) attached to the crank pin at a slides along the slotted bar PA (link 1) which
oscillates at D. The connecting rod PR carries the ram at R to which a cutting tool is fixed. The motion of the tool is
constrained along the line RD produced.

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Slider-crank chain: GNOME ENGINE
https://youtu.be/SYYVxnLNzBI?si=3idH_FiMVybWNPJy
It consists of seven cylinders in one plane and all revolves about fixed center D, as shown in Fig. 5.25, while the crank
(link 2) is fixed. In this mechanism, when the connecting rod (link 4) rotates, the piston (link 3) reciprocates inside the
cylinders forming link 1.

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Double Slider Crank Chain::2T+2S
A four bar chain having two turning and two sliding pairs such that two pairs of the same kind are adjacent is known as
double slider crank chain
Elliptical Trammel : This inversion is obtained by fixing the slotted plate (link 4). • fixed plate or link 4 has two straight
grooves cut in it, at right angles to each other. • link 1 and link 3, are known as sliders and form sliding pairs with link 4. •
link AB (link 2) is a bar which forms turning pair with links 1 and 3. • When the links 1 and 3 slide along their respective
grooves, any point on the link 2 such as P traces out an ellipse on the surface of link 4

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Double Slider Crank Chain::2T+2S
Scotch yoke mechanism:

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Double Slider Crank Chain::2T+2S

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Intermittent motion mechanisms
Intermittent motion mechanisms An intermittent-motion mechanism is a linkage which converts continuous motion into
intermittent motion. These mechanisms are commonly used for indexing in machine tools.
Geneva wheel mechanism: link A is driver
and it contains a pin which engages with the slots in
the driven link B. The slots are positioned in such a
manner, that the pin enters and leaves them
tangentially avoiding impact loading during
transmission of motion. In the mechanism shown,
the driven member makes one-fourth of a revolution
for each revolution of the driver. The locking plate,
which is mounted on the driver, prevents the driven
member from rotating except during the indexing
period.

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Intermittent motion mechanisms
Ratchets are used to transform motion of rotation or translation into intermittent rotation or translation. In the fig., A is the
ratchet wheel and C is the pawl. As lever B is made to oscillate, the ratchet wheel will rotate anticlockwise with an
intermittent motion. A holding pawl D is provided to prevent the reverse motion of ratchet wheel

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A pantograph is an instrument used to reproduce to an enlarged or a reduced scale and as exactly as possible the path
described by a given point. It consists of a jointed parallelogram ABCD as shown in Fig. It is made up of bars connected by
turning pairs. A pantograph is mostly used for the reproduction of plane areas and figures such as maps, plans etc., on
enlarged or reduced scales.
Pantograph

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Straight line motion mechanisms are mechanisms, having a point that moves along astraight line, or nearly along a straight
line, without being guided by a plane surface.
Condition for exact straight line motion:
If point B moves on the circumference of a circle with center O and radius OA, then, point C, which is an extension of AB
traces a straight line perpendicular to AO, provided product of AB and AC
Straight Line Motion Mechanisms

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Peaucellier exact straight line motion mechanism
.
Here, AE is the input link and point E moves along a circular path of radius AE = AB. Also, EC = ED = PC =
PD and BC = BD. Point P of the mechanism moves along exact straight line, perpendicular to BA extended To
prove B, E and P lie on same straight line:
Triangles BCD, ECD and PCD are all isosceles triangles having common base CD and apex points being B, E
and P. Therefore points B, E and P always lie on the perpendicular bisector of CD. Hence these three points
always lie on the same straight line.
To prove product of BE and BP is constant.

32. Module-1 Degrees of freedom/mobility of a mechanism:
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Degrees of freedom/mobility of a mechanism: It is the number of inputs (number of independent coordinates) required to
describe the configuration or position of all the links of the mechanism, with respect to the fixed link at any given instant.
If we apply the Kutzbach criterion to planer mechanism, then equation of Kutzbach criterion will be modified and that
modified equation is known as Grubler‟s Criterion for planer mechanism. Therefore in planer mechanism if we consider the
links having 1 to 3 DOF, the total number of degree of freedom of the mechanism considering all restraints will becomes
Grubler’s equation: Number of degrees of freedom of a mechanism is given by
F = 3(n-1)-2l-h.
Where, F = Degrees of freedom
n = Number of links = n2 + n3 +……+nj, where, n2 = number of binary links, n3 = number of ternary links…etc.
l = Number of lower pairs, which is obtained by counting the number of joints. If more than two links are joined together at
any point, then, one additional lower pair is to be considered for every additional link.
h = Number of higher pairs

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