Introduction

Hamming Number Technique For Epicyclic Gear Train

Generation of epicyclic gear trains with up to eight-links has been reported by many investigators . All the methods need tests for isomorphism in order to isolate distinct gear kinematic chains. Unlike linkages, the graphs of gear kinematic chains need be tested for displacement and rotational isomorphism. This has encouraged some of the investigators to define new graphs. Status of the present work is to generate distinct graphs and then go for reverse transformation. All this will become more significant only when the designer is able to compare the numerous distinct gear trains with the same number of links and degree-of-freedom and know the relative merits or demerits without having to actually design, fabricate and test them for the specified performance.

John Holland with his colleagues and students had developed genetic algorithms which consider the process of competition, reproduction and the struggle for survival. David Goldberg has made major contributions in this area and in an introductory chapter genetic algorithms with a group of four 5-bit binary strings were illustrated.

However, with the increasing applications of neutral networks in engineering and realizing its potentials and lack of its applications in this area of kinematics an attempt was made by the author to develop an algorithm to study topological characteristics of kinematic chains (linkages). The present work extends the above algorithm to epicyclic gear trains taking care of the fact that the graphs representing gear trains contain edges of different types unlike in linkages. In addition the concept of parallelism is introduced since characteristics like speed ratios, transmission efficiency etc., are related to parallelism in graphs (gear trains). Numerical strings are proposed which will test both displacement and rotational isomorphism uniquely with least effort. Also, the fitness-a concept that emerges from the genetic algorithm is related to parallelism so that without any extra computational effort, the distinct gear trains can be compared for their anticipated behavior (speed ratios and transmission efficiencies).

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Literature Review Generation of epicyclic gear trains with up to eight-links has been reported recently. Almost all the work reported so far deals with the generation of distinct graphs and then go for reverse transformation. It is desirable to know the characteristics inherent to the structure so that best chain can be selected from the numerous distinct chains (gear trains) with the same number of links and degree-of-freedom. In order to accomplish this, quantitative measures are developed in a very simple way, using some of the principles of the genetic process. These measures are used to test isomorphism and to know relatively the characteristics like speed ratios and transmission efficiency.

Problem Identification

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The gear trains especially epicyclic gear trains are having wide application in robotic manipulators so as to obtain the necessary motions of the end effector to perform various task assigned to the robots. But we have infinite arrangement of Epicyclic planetary gear trains with same number of elements and Degree of freedom. It is very difficult to select suitable gear train for specific purpose

To overcome this problem we are having Hamming number Technique (detecting for Isomorphism) Proposed by A.C.Rao , a kinematician.

This method is proposed for isomorphic gear trains which shall bevery few out of the infinite gear trains.

An isomorphism is a kind of mapping between objects, which shows a relationship between two operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, isomorphic structures are structurally identical.

Functionally similar structures are called isomorphic.

Methodology

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1.3 GENERAL TERMINOLOGY

1.3.1 Kinematics – the study of constrained motion without regard to forces that cause

motion.

1.3.2 Kinetics - the study of the forces that result from, or lead to, the motion of rigid

bodies. A kinetic analysis will let us know the internal and external forces that are

developed in a mechanism as well as tell us how much power we must provide to operate a

mechanism to perform a particular function. This information is essential in order to conduct the

detail design of the individual components of the mechanism.

1.3.3 STATICS – Science of bodies at rest or forces in equilibrium.

1.3.4 DYNAMICS – Branch of mechanics that deals with motion, and the motion of bodies or

matter under the influence of forces.

1.3.5 Link - Assumed rigid body with two or more nodes (attachment points). A link can be

classified by the number of nodes it contains. A binary link has two nodes, ternary three nodes,

quaternary - four nodes, and so on.

1.3.6.a Binary link – a link with two joint connections

Fig-1.1

1.3.6.b Ternary link - a link with three joint connections

1.3.6.c Quaternary Link- a link with four

connections

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Fig-1.3

1.3.7 Kinematic Joint - A connection between two links which allows some relative motion. A

joint can be classified by the number of DOF it permits. A pin joint between two links has one DOF

(rotation). Another single DOF joint would be a square pin in a slot. The DOF would be the position of

the pin within the slot with respect to one end of the slot. An example of a joint with two DOF would

be a round pin in a slot. The pin can translate along the slot as well as rotate about its own axis.

1.3.8 Types of Joints

Pin Joints - joints that allow only one degree of freedom and the only motion allowed is pure

rotation.

Slider Joints - joints that allow only one degree of freedom and the only motion allowed is

translational rotation.

Half Joints - joints that allow two degrees of freedom and the motion is complex motion.

1.3.9 Dyad (kinematic pair) – A pair of rigid bodies connected by a kinematic joint i.e. two links

connected by a joint.

Fig-1.4

1.3.10 Pairs, Higher Pairs, Lower Pairs and Linkages

A pair is a joint between the surfaces of two rigid bodies that keeps them in contact and relatively

movable. For example, in Fig.-1.5, door jointed to the frame with hinges makes revolute joint (pin

joint), allowing the door to be turned around its axis. Fig1.5 shows skeletons of a revolute joint. Fig-

1.5 is used when both links joined by the pair can turn. Fig-1.5 is used when one of the link jointed by

the pair is the frame.

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Fig.-1.5 Revolute pair

In Fig-1.6 sash window can be translated relative to the sash. This kind of relative motion is called a

prismatic pair. Its skeleton outlines are shown in b, c and d. c and d are used when one of the links is

the frame.

Fig-1.6 Prismatic pair

Generally, there are two kinds of pairs in mechanisms, lower pairs and higher pairs. What

differentiates them is the type of contact between the two bodies of the pair. Surface-contact pairs

are called lower pairs. In planar (2D) mechanisms, there are two subcategories of lower pairs --

revolute pairs and prismatic pairs, as shown in Fig-1.6 and Fig-1.7 respectively. Point-, line-, or curve-

contact pairs are called higher pairs. Fig-1.7 shows some examples of higher pairs Mechanisms

composed of rigid bodies and lower pairs are called linkages.

a. Cam and Fllower b.Gear Teeth

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Fig-1.7 Higher pairs

1.3.11 kinematic chain - A collection of links and joints connected in a manner to provide a

controlled output in response to a supplied input.

Fig-1.8

1.3.11.a Open kinematic chain (loop) – One that does not connect back onto itself

1.3.11.b Closed kinematic chain (loop) – One that does connect back onto itself

1.3.12 Degrees of Freedom (DOF)

The number of input motions that must be provided in order to provide the desired

output, OR

The

Equation:

F=3(N-1)- 2j

N = Total number of links (including a fixed or single ground link)

J = Total number number of independent coordinates required to define the position and

orientation of an object

For a planar mechanism, the degree of freedom (mobility) is given by

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Gruebler’s of joints (some joints count as ½, 1, 2, or 3)

• Example: Slider-crank N=4, J=4, F=1

• Example: 4-Bar linkage N=4, J=4, F=1

fig-1.9

Fig-1.10

Now that we have defined degrees of freedom, We can look at the illustration above and determine

the degrees of freedom of each.

1.3.13 Joints: Single Degree-of-Freedom

Lower Pairs (first order joints)

• All are full-joints (counts as 1 in Kutzbach’s Equation)

Revolute (R)

• Also called a pin joint or a pivot, take care to ensure that the axle member is firmly anchored in one

link, and bearing clearance is present in the other link

• Washers make great thrust bearings

• Snap rings keep it all together

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Fig -1.11

Prismatic (P)

• Also called a slider or sliding joint, beware Saint-Venant!

Helical (H)

• Also called a screw, beware of thread strength, friction and efficiency

Fig-1.13

1.3.14 Joints: Multiple Degree-of-Freedom

1.3.14.a Lower Pairs

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Cylindrical (C) 2 DOF

A multiple-joint (counts as 2 in Gruebler’s Equation)

Fig-1.14

Spherical (S) 3 DOF

A multiple-joint not used in planar mechanisms

Fig-1.15

Planar (F) 3 DOF

A multiple-joint (counts as 3 in Gruebler’s Equation)

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Fig-1.16

1.3.14.b Higher Pairs

Link against a plane

• A force is required to keep the joint closed (force closed)

• A half-joint (counts as 0.5 in Gruebler’s Equation)

Fig-1.17

Pin-in-slot

• Geometry keeps the joint closed (form closed)

• A multiple-joint (counts as 2 in Gruebler’s Equation)

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Fig-1.18

Second order pin joint, 3 links joined, 2-DOF

A multiple-joint (counts as 2 in Gruebler’s Equation)

Fig-1.19

1.3.15 Mechanism - A kinematic chain that has had one link rigidly attached to the reference

frame (ground).

Fig-1.20

1.3.16 Machine

A collection of mechanisms arranged to transmit forces and do work.Now that we understand what

a mechanism is, we can define the degrees of freedom of a mechanism as the number of inputs

needed to define the output state. In the designing a mechanism, you have some idea of your

desired output. You want to achieve the output with a minimum number of inputs. Each input (i.e., a

motor or actuator) requires a controller, adds weight and cost to the system, requires maintenance

and is a potential source of breakdown. Therefore, it is desireable to design mechanisms that require

as few inputs as possible, ideally a mechanism has just one DOF to achieve the desired output. Of

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course this is not always possible, particularly for mechanisms that must perform highly complex

functions, but many useful mechanisms have a single DOF.

1.3.17 MACHINES AND MECHANISMS

A mechanism is a kinematic chain in which at least on link has been "grounded" or attached to the

frame of reference.

A machine is a combination of resistant bodies arranged to compel the mechanical forces of nature

to do work accompanied by determinate motions.

Machines are devices used to alter, transmit, and direct forces to accomplish a specific objective. A

chain saw is a familiar machine that directs forces to the chain with the objective of cutting wood. A

mechanism is the mechanical portion of machine that has the function of transferring motion and

forces from a power source to an output. It is the heart of the machine. For the chain saw, the

mechanism takes power from a small engine and delivers it to the cutting edge of the chain.

Fig 1.21 illustrates a track hoe that is driven by hydraulic cylinders and other linkages used for its

operation. Although the entire device is called a machine, the parts that take the power from the

cylinders and drive the raising and lowering of the arm are mechanisms.

A mechanism can be considered rigid parts that are arranged and connected so that they produce

the desired motion of the machine. The purpose of the mechanism in fig 1.21 is to dig a portion

of earth and placed it at some other desired place. Mechanism analysis ensures that the device

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will exhibit motion that will accomplish the desired purpose of a machine.

Fig 1.21 Track hoe mechanism

1.3.18 Mechanisms and Structures

fig-1.22

A mechanism is defined by the number of positive degrees of freedom. If the assembly has zero or

negative degrees of freedom it is a structure.

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A structure is an assembly that has zero degrees of freedom. An assembly with negative degrees of

freedom is a structure with residual stresses.

Fig-1.23

A Stephenson's six-bar linkage has the two ternary links joined at all three points to binary links.

A Watt's six-bar linkage also has two ternary links and four binary links but the two ternaries are

actually pinned together at one point.

1.3.19 Step by Step machine fabrication

1. Machine parts are known as “elements”

2. Two elements in relative motion and in contact are known as a “pair”

3. The element joining pairs together is known as a “link”.

4. A group of links and elements that are joined together is a “kinematic chain”.

5. Fix one link of the kinematic chain and the chain becomes a “mechanism”

6. Apply force with the mechanism and it becomes a ”machine”

1.3.20 Types of Motion

1.3.20.a Pure Translation

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All points on the body describe parallel paths. A reference line drawn on the body changes its linear

position but does not change it angular orientation.

1.3.20.b Pure Rotation

The body possesses one point which has no motion with respect to the stationary frame of

reference. All other points on the body describe arcs about that center. A reference line drawn on

the body through the center changes only its angular orientation.

1.3.20.c Complex motion

A simultaneous combination of rotation and translation. Any reference line drawn on the body will

change both its linear position and its angular orientation. Points on the body will travel in non-linear

paths, and there will be, at every instant, a center of rotation which will constantly change location.

1.3.21 Skeleton diagram – A succinct schematic drawing of a kinematic chain

1.3.21.a Skeleton Outline

Fig-1.24 Skeleton outline

For the purpose of kinematic analysis, a mechanism may be represented in an abbreviated, or

skeleton, form called the skeleton outline of the mechanism. The skeleton outline gives all the

geometrical information necessary for determining the relative motions of the links. In Fig-1.24, the

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skeleton outline has been drawn for the slider-crank mechanism This skeleton contains all necessary

information to determine the relative motions of the main links, namely, the length AB of the crank;

the length BC of the connecting rod; A the location of the axis of the main bearing; and the path AC

of point C, which represents the wrist-pin axis.

1.3.22 KINEMATIC INVERSION

The first step in drawing a kinematic diagram is selecting a member to serve as a fixed link or frame.

In some cases, the selection of a frame is arbitrary. As different links are chosen as a frame, the

relative motion of the links is not altered, but the absolute motion can be drastically different,

depending on the link selected as the frame. However the relative motion is often the desired result.

Since the relative motion of the links does not change with the selection of a frame, the choice of a

frame link is often not important. Utilizing alternate links to serve as the fixed link is termed as

“kinematic inversion”.

The mechanisms in Figures 3(b-e) are kinematic inversions of each other. This means that they are all

derived from the same kinematic chain and they di_er from one another only in which of the links in

the kinematic chain is _xed to ground, or made the ground link (a.k.a. frame). Likewise, the

mechanisms in Figures 4(b-d) are kinematic inversions of each other. Kinematic inversions exist for

any kinematic chain, not just for the four bar chain. Clearly, the number of kinematic inversions for

any particular kinematic chain is equal to the number of links in the chain. The relative motion of the

links in a set of kinematic inversions is the same, however, the absolute motion of the links varies,

since they are referenced to a different fixed body in each case.

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1.3.23.The Grashoff Condition

Without knowing how the links of a four-bar mechanism are connected, we can tell something about

how it will behave just by knowing the lengths of the individual links and investigating the Grashof

condition. Let the length of the shortest and longest links be denoted by S and L, respectively. The

intermediate links will be labeled P and Q. If we compare the quantity S+L with P+Q, we can get a tell

if any of the links will be able to rotate or not. The Grashof condition states that if S+L<P+Q then the

mechanism is a Grashof mechansim and at least one link will be capable of a full revolution.

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Fig-1.30

If S+L>P+Q then the mechanism is non-Grashof and all combinations of the links will be double

rockers and none of the links will be capable of a full rotation.

Fig-1.31

If S+L=P+Q

then we have a special case Grashof and the mechanism will have changeovers where it can switch

configurations and the output can be indeterminate.

1.3.25 Kinematic Analysis and Synthesis

In kinematic analysis, a particular given mechanism is investigated based on the mechanism

geometry plus other known characteristics (such as input angular velocity, angular acceleration, etc.).

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Kinematic synthesis, on the other hand, is the process of designing a mechanism to accomplish a

desired task. Here, both choosing the types as well as the dimensions of the new mechanism can be

part of kinematic synthesis.

1.3.24 The scope of synthesis may be classified as :

1.3.24.a. Type synthesis : - The beginning phase of mechanism design is to select the type of

mechanism. This phase usually involves design factors such as manufacturing process, material,

safely, reliability, space and economy etc. The study of kinematic is usually slightly involved in type

synthesis.

1.3.24.b Number synthesis : - The second step is number synthesis which deals with the

number of links and joints required to obtain a certain mobility/degree of freedom.

1.3.24.c Dimensional synthesis : - The third step is dimensional synthesis is which

size/length of the links, location and orientation of links and pivots are determined.

1.3.24.d Static machine design : - The fourth step in the determination of the cross-section

of the links on the basis of the static-machine design criteria based on static force and then carried

out analysis to check the mechanism developed and change the parameters listed in sr. no. 1 to 4, if

necessary.

1.3.24.e Dynamic design : - The model of machine is developed and inertia forces, jerks,

acceleration, and frequency response are taken into consideration in dynamic design.

1.3.24.f. Other design parameters : - The other design parameters are design for

manufacture, design for maintenance, design for assemble, design for cost, light/heavy weight,

quality, reliability, safety, human-machine environment, accessibility etc.

MODELLING APPROACH OF KINEMETIC CHAIN & MECHANISM

4.0 INTRODUCTION

Structural analysis and synthesis of the kinematic chain and mechanism has been the subject of a

number of studies in recent years. Structural synthesis of kinematic chains and mechanism is needed

for:

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1. Determining all possible structurally distinct kinematic chains with a given number of links and

given degree of freedom.

2. Determining all possible structurally distinct mechanism that can be derived from a given

kinematic chain.

The most important problem in structural synthesis is detection of distinct kinematic chain. For this

purpose a number of methods have been proposed. Most of this methods are based on adjacency

matrix or distance matrix.

4.1 Degree of links d(li)

The degree of links actually represents the type of links like binary, ternary, quaternary links etc. and

let us designated as d (l)

d(Ii) = 2, for binary link

d(Ii) =3, for ternary link

d(Ii) =4, for quaternary link

d(Ii) =n ,for n-nary link

4.2 Degree Vector

Degree vector represents the degree of individual link of kinematics chain and defined as

v = { d(l1), d(l2), d(l3 ), ……………d(ln) }

Where, n = number of links in a kinematics chain

vi = degree of ith link in the associated kinematics chain.

For example the degree vector of the chain shown in fig. 4.6(b), written as

V = { 3 3 3 3 2 2 2 2 }

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4.3 SHORTEST PATH DISTANCE

The path between two links vi and vj is an alternating sequence of links and joints starting from link i

and terminating at link j. The sum of the joints in a path is called the path length. There may be many

paths of different lengths to go from vi to vj. The path of smallest length is called shortest path. The

shortest distance between link i and j is the sum of the joints in the shortest path from link I to link j.

For example in Fig.4.1 the shortest path between link 1 & 2 is 1-5-2 having shortest path distance

equal to the sum of joints two while shortest path between links 1 & 4 is 1-6-4 having shortest path

distance equal to two.

Fig.4.1

4.4 Shortest path distance matrix

This matrix is used to represent the kinematic chain. We may use a kinematic chain

(shown above) to represent shortest path distance matrix.

D= { dij} = Shortest path between ith link & jth link..

dii = 0

D= { dij}m×n

Where dii = 0

dij = shortest path distance between link i and link j .

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4.5 Interactive effect or relative weight of degree of links

Critical study of the kinematic structure has revealed that there is a strong correlation between the

kinematic structure and the machine performance like wear , reliability , susceptibility to

manufacturing error because of link tolerances and bearing clearances. . On the other hand degree of

link also represents the number of simple joints which are actually the sum of links.

For example, in this fig.4.2 the links 1, 3, 4 and 2 have degree equal to 3

Fig.4.2

and all these links represents ternary links. And other links represent binary link. The distance matrix

does not take the influence of joints formed by different types of links.

4.6 RELATIVE WEIGHT OF THE DEGREE OF LINKS

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Critical study of the structure of kinematic chains it is revealed that there is a strong correlation

between the structure of kinematic chain and its performances like wear, reliability, susceptibility to

manufacturing error because of link tolerances and joint/bearing clearances, compactness, degree of

similarity and dissimilarity [ 11,12,13 ]. Therefore, the information about the types of the links

directly connected to each other with a joint is also introduced in the modified adjacency matrix in

the form of relative importance of the degree of the i th link to jth link and vice versa. The relative

weight of the degree of the links w ij is defined as the ratio between degree of i th link and degree of jth

link and given as :

wij = d (Ij) / d(Ij)

wji = d(Ij) / d(Ii)

4.7 MUTUAL INTERACTIVE EFFECT OF RELATIVE WEIGHTS (wij)

This is the average of the relative weights of two links and is given as:

Wij = (wij + wji)/2

Wij = ½ [d (Ii) / d (Ij) + d (lj)/d (li)

For example, the link 1 and link 2 directly connected to each other through a simple joint in the

kinematic chain shown in figure 1(a) have the degree of the link 3 and 2 respectively. Therefore the

average of relative weights of link 1 and link 2 will be

Wij = ½ [2/3 + 3/2] = 13/12

If the two directly connecting links have the same degree, the average of the relative weights will be

one.

4.8 Weighted Distance Matrix [WD]

In a mechanism design problem, systematic steps are: type synthesis, structural/number synthesis

and dimensional synthesis. Structural synthesis of the kinematic chain- and mechanism has been the

subject of a number of studies in recent years. One important aspect of structural synthesis is to

develop the all-possible arrangements of kinematic chains and also their derived mechanisms for a

given number of links, joints and degree of freedom so that the designer has the liberty to select the

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best or optimum mechanisms according 10 his requirements. In the course of development of

kinematic chains and mechanisms duplication may be possible For this reason, lots of methods have

been proposed by many researchers to check for duplication or, in other words, to detect the

isomorphism of kinematic chains Most of these methods are based on the adjacency matrix [1] and

the distance matrix [2] for determining the structurally distinct mechanisms of a kinematic chain, the

link disposition method (3], the How matrix method [4] and the row sum of extended distance matrix

methods [5] are used. Minimum code [6], characteristic polynomial of matrix [7], identification code

[8], link path code [9] summation polynomial [10] etc are used to characterize the kinematic chains.

With regard to these methods, there is either a lack of uniqueness or they take too much

time .Determination of all distinct mechanisms from an n-link kinematic chain using flow matrix [4] is

a very lengthy, process because n-f1ow, matrices are required. Row sum of extended adjacency

matrix method [5] identifies 10 distinct mechanisms derived from the family of 6-link, 1-F, kinematic

chains but it distinguish 71 distinct mechanisms derived from the family of 8-link. I-F, kinematic

chains instead of the 71, reported by other researchers, Hence there is a need to develop a

computationally efficient method for determining the distinct mechanisms of a kinematic chain.

In the present work, a new method is proposed to determine the distinct mechanisms of a kinematic

chain. Critical study of kinematic chain and mechanism structure has revealed that the performance

of the joints, is affected by the degree of the links (types of links) .For this purpose, the interaction

effect of the degree of connecting links on the joints has been taken into consideration, while

defining the extended adjacency matrix [WD] along with the structural information of the links and

joints From the [WD] matrix the two structural invariants WD∑ and WDmax are derived based on the

eigen spectrum of the [WD] matrix using the software, MATLAB. These structural invariants are the

same for identical or structural]y equivalent mechanisms and different for distinct mechanisms.

[WD] = {gig} n×n Where; gig = = { dij }× (WIJ )

To determine the elements gig of [WD] matrix, it is observes that when

vi = vj

WiJ = 1 , hence gig = dij

Hence following deduction are made

gii = dii = 0 when i=j

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gig = dij when vi = vj

gij = dij * Wij when vi ≠ vj

Using the facilities existed in software MATLAB the all elements of [WD] matrix are determined

easily.

The basic weighted matrix is given below.

0 g12 g13 g14 g15 g1n

g21 0 g23 - - g2n

WD = g31 - 0 - - g3n

g41 - - - - g4n

gn1 gn2 gn3 - - 0 n×n

4.9 STRUCTURAL INVARIANTS ( WD∑ and WD max )

Eigen values are the characteristics invariants of the chain. Eigen spectra have been determined

from (0,1) adjacency matrices. The proposed [WD] matrix provides entirely different set of eigen

value of kinematic chain. . To make this [WD] matrix eigen spectrum as a powerful single number

characteristics index , new composite invariants are proposed. These indices are sum of all all

absolute eigen values ∑ WD and max absolute eigen value WDmax of WD matrix. The eigen value of

[WD ] matrix are obtained by using MATLAB software . It is hoped that these invariants are capable

of characterizing all kinematic chains uniquely.

4.9 CHARECTERISTIC EQUATION OF [WD] MATRIX

The characteristic polynomial [1] is generally derived from (0,1) adjacency matrix. The roots of n th

order characteristic polynomial are the set of n-eigen values called as eigen spectrum. Many

researchers have reported co-spectral graphs (the non isomorphic graphs having same eigen

spectrum) derived from (0,1) adjacency matrix. Proposed [WD] matrix have additional information

about the types of links existing in a kinematic chain in the form of mutual interactive effect,

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therefore it is expected that the characteristic polynomial and its roots will be unique to clearly

identity the kinematic graphs & kinematic chains and even co-spectral graphs.

The characteristic equation of [WD] matrix is given as :

│WD –λ 1 │ = 0

= λn + a1 λn-1+ a2λn-2+……………………+ an-1 λ+an

= ∏ (λ - λj)nj

j = 1

The two important properties of eigen spectrum are

1. The sum of the eigen values is the trace of [WD] matrix.

2. The product of eigen values is the determinant of [WD]matrix.

4.10 Identification of distinct mechanisms

The basis of the kinematic chain. By kinematic inversion, each chain leads to as many mechanisms as

there are links in the chain. Some of the mechanism / inversion are equivalent and therefore should

be counted as once. For this reason, no appropriate mathematical equation has yet been found

which tells how many different types of mechanism, each having the same number of links and same

number of joints, can be derived from a given kinematic chain. Therefore it is essential to developed

an efficient analytical method by which the number of distinct mechanisms for a given family of a

kinematic chains. In this chapter an effort is made to determine the number of distinct mechanism

form kinematic chain.

Observing the structural invariants for the above eight mechanisms, it is found that the structural

invariants or link -1 and link - 4 are the same, hence they are treated as structurally equivalent links

and form only two distinct mechanisms .

e (Number of equivalent links)

d (Number of distinct links)

T= e+d

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T= Total distinct mechanisms

Result & Discussions We have verified the HAMMING NUMBER TECHNIQUE FOR EPICYCLIC GEAR

TRAINS upto seven elements mechanisms. Our aim is to develop this technique upto 10

links.This technique has a number of applications in detecting the isomorphism among

various mechanisms containing same number of elements. As if the two or more mechanisms

having same number of elements, are isomorphic, then their function will be same so there is

no need to use two mechanisms of same function, but to use only one of them. To reduce this

unnecessary and time taking exercise, our HAMMING NUMBER TECHNIQUE is very

useful. By use of this technique we can easily identify the non isomorphic mechanisms and

can save time and money also.

This technique has following applications:-

1.The designer is able to compare the numerous distinct gear trains with the same number of links and degree-of-freedom (d.o.f) and know the relative merits or demerits without having to actually design, fabricate and test them for the specified performance.

2.To develop an algorithm to study topological characteristics of kinematic chains (linkages).

3. Numerical strings are proposed which will test both displacement and rotational isomorphism uniquely with least effort.

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4. The fitness-a concept that emerges from the genetic algorithm is related to parallelism so that without any extra computational effort, the distinct gear trains can be compared for their anticipated behavior.

5. Speed ratios and transmission efficiency can be easily determined with the help of this technique.

Conclusion & Scope of further work It is proved that the hamming number test for isomorphism among epicyclic gear trains is definitive. It is applicable to chains with high degree of freedom can also be dealt with.

Isomorphism among planetary gear kinematic chain graphs with different types of edges can be tested easily with certainly by the Hamming number technique. Inversion or distinct gear trains are revealed without any additional effort. Secondary Hamming number test is not necessary in case of trains with less number of elements.

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