CHAPTER 1

INTRODUCTION

Gears are most commonly used for power transmission in all the modern devices. These

toothed wheels are used to change the speed or power between input and output. They have

gained wide range of acceptance in all kinds of applications and have been used extensively in

the high-speed marine engines.

In the present era of sophisticated technology, gear design has evolved to a high

degree of perfection. The design and manufacture of precision cut gears, made from

materials of high strength, have made it possible to produce gears which are capable of

transmitting extremely large loads at extremely high circumferential speeds with very

little noise, vibration and other undesirable aspects of gear drives.

A gear is a toothed wheel having a special tooth space of profile enabling it to mesh

smoothly with other gears and power transmission takes place from one shaft to other by means

of successive engagement of teeth.

Gears operate in pairs, the smallest of the pair being called “pinion” and the larger one

“gear”. Usually the pinion drives the gear and the system acts as a speed reducer and torque

converter.

1

1.1 ADVANTAGES OF GEAR DRIVES:

The following are the advantages of the gear drives compared to other drives.

Gear drives are more compact in construction due to relatively small centre distance.

Gears are used where the constant velocity ratio is desired.

Gears can be operated at higher speeds.

It has higher efficiency.

Reliability in service.

It has wide transmitted power range due to gear shifting facility.

Gear offers lighter loads on the shafts and bearings.

Gear can change the direction of axis of rotation.

1.2 DISADVANTAGES OF GEAR DRIVES:

Not suitable for the shafts which are at longer center distance.

Manufacturing is complex. It needs special tools and equipment.

Require perfect alignment of shafts.

Requires more attention to lubrication.

The error in cutting teeth may cause vibration and noise during operation.

1.3 GENERAL CLASSIFICATION OF GEARS:

Although the types and the modalities of gear design vary widely for mechanical power

transmission, gears are generally categorized into the following types.

1.3.1 ACCORDING TO THE POSITION OF AXES OF THE SHAFTS:

The axes of the two shafts between which the motion is to be transmitted, may be

( a) Parallel, (b) Intersecting, and (c) Non-Intersecting and Non-parallel

1.3.2 ACCORDING TO THE PERIPHERAL VELOCITY OF THE GEARS:

The gears, according to the peripheral velocity of the gears, may be classified as: (a) Low

Velocity, (b) Medium Velocity and (c) High Velocity.

2

1.3.3 ACCORDING TO TYPE OF GEARING:

The gears, according to the type of gearing may be classified as (a) External Gearing, (b)

Internal Gearing, (c) Rack and Pinion.

1.3.4 ACCORDING TO THE POSITION OF THE TEETH ON THE GEAR SURFACE:

The teeth on the gear surface may be (a) Straight (b) Inclined (c) Curved.

1.3.5 ACCORDING TO THE POSITION OF AXES OF THE SHAFTS:

Spur Gears: Spur gears are the simplest and most common type of gear. Their general form is

a cylinder or disk. The teeth project radially, and with these "straight-cut gears", the leading

edges of the teeth are aligned parallel to the axis of rotation. These gears can only mesh

correctly if they are fitted to parallel axles. Spur gears on non-parallel shafts can mesh, but only

point contact will be achieved, not line contact across the full width of the tooth; also the length

of the path of contact may be too short. This causes impact stress and noise. Spur gears make a

characteristic whine at high speeds and can not take as much torque as helical gears because

their teeth are receiving impact blows.

Fig: 1.1 Figure showing different lines of contact

3

Helical Gears: Helical gears offer a refinement over spur gears. The leading edges of the

teeth are not parallel to the axis of rotation, but are set at an angle. Since the gear is curved, this

angling causes the tooth shape to be a segment of a helix. The angled teeth engage more

gradually than do spur gear teeth. This causes helical gears to run more smoothly and quietly

than spur gears. Helical gears also offer the possibility of using non-parallel shafts.

With parallel helical gears, each pair of teeth first makes contact at a single point at one side

of the gear wheel; a moving curve of contact then grows gradually across the tooth face. It may

span the entire width of the tooth for a time. Finally, it recedes until the teeth break contact at a

single point on the opposite side of the wheel. Thus force is taken up and released gradually.

.

Fig: 1.2 Pitch point

Double Helical Gears:

Double helical gears, also known as herringbone gears, overcome the problem of axial thrust

presented by 'single' helical gears by having teeth that set in a 'V' shape. Each gear in a double

helical gear can be thought of as two standards, but mirror image, helical gears stacked. This

cancels out the thrust since each half of the gear thrusts in the opposite direction. They can be

directly interchanged with spur gears without any need for different bearings.

Bevel Gears:

Bevel gears are essentially conically shaped, although the actual gear does not extend all the

way to the vertex (tip) of the cone that bound it. With two bevel gears in mesh, the vertices of

their two cones lie on a single point, and the shaft axes also intersect at that point. The angle

4

between the shafts can be anything except zero or 180 degrees. Bevel gears with equal numbers

of teeth and shaft axes at 90 degrees are called miter gears.

Fig: 1.3 Bevel Gears

Worm Gears:

Worm gear is a special case of a spiral gear in which the larger wheel usually has hollowed or

concave shape such that a portion of the pitch diameter of the other gear is enveloped on it. The

smaller of the two wheels called the worm which also has a large spiral angle. The shafts may

have any angle between them, but normally it is 90°.The system can be non-throated, single

throated, double throated types.

Hypoid Gears:

These are similar to spiral bevel gears, but have non-intersecting axes i.e. the axes of pinion

is offset relative to the gear axis. However the plane containing the two axes is usually at right

angles to each other.

5

CHAPTER 2

HELICAL GEARS

Helical gears offer a refinement over spur gears. The leading edges of the teeth are not parallel

to the axis of rotation, but are set at an angle. Since the gear is curved, this angling causes the

tooth shape to be a segment of a helix. Helical gears can be meshed in a parallel or crossed

orientations. The former refers to when the shafts are parallel to each other; this is the most

common orientation. In the latter, the shafts are non-parallel.

The angled teeth engage more gradually than do spur gear teeth causing them to run more

smoothly and quietly. With parallel helical gears, each pair of teeth first make contact at a single

point at one side of the gear wheel; a moving curve of contact then grows gradually across the

tooth face to a maximum then recedes until the teeth break contact at a single point on the

opposite side. In spur gears teeth suddenly meet at a line contact across their entire width

causing stress and noise. Spur gears make a characteristic whine at high speeds and can not take

as much torque as helical gears. Whereas spur gears are used for low speed applications and

those situations where noise control is not a problem, the use of helical gears is indicated when

the application involves high speeds, large power transmission, or where noise abatement is

important. The speed is considered to be high when the pitch line velocity exceeds 25 m/s

Quite commonly helical gears are used with the helix angle of one having the negative of the

helix angle of the other; such a pair might also be referred to as having a right-handed helix and

a left-handed helix of equal angles. The two equal but opposite angles add to zero: the angle

between shafts is zero – that is, the shafts are parallel. Where the sum or the difference (as

described in the equations above) is not zero the shafts are crossed. For shafts crossed at right

angles the helix angles are of the same hand because they must add to 90 degrees.

6

2.1 Helical gear nomenclature

Helix angle, ψ 

Angle between a tangent to the helix and the gear axis. Is zero in the limiting case of a

spur gear.

Normal circular pitch, pn 

Circular pitch in the plane normal to the teeth.

Transverse circular pitch, p 

Circular pitch in the plane of rotation of the gear. Sometimes just called "circular pitch".

pn = pcos(ψ)

2.2 Helical Gear geometrical proportions p = Circular pitch = d g. p / z g = d p. p / z p

p n = Normal circular pitch = p .cosβ

P n =Normal diametrical pitch = P /cosβ

p x = Axial pitch = p c /tanβ

m n =Normal module = m / cosβ

α n = Normal pressure angle = tan -1 ( tanα.cos β )

β =Helix angle

d g = Pitch diameter gear = z g. m

d p = Pitch diameter pinion = z p. m

7

a =Center distance = ( z p + z g )* m n /2 cos β

a a = Addendum = m

a f =Dedendum = 1.25*m

CHAPTER 3

MARINE ENGINES

3.1 RECIPROCATING DIESEL ENGINES:

The engine used in 99% of modern ships is diesel reciprocating engines. The rotating

crankshaft can power the propeller directly for slow speed engines, via a gearbox for medium

and high speed engines, or via an alternator and electric motor in diesel-electric vessels.

The reciprocating marine diesel engine first came into use in 1903 when the diesel electric

river tanker Vandal was put in service by Branobel. Diesel engines soon offered greater

efficiency than the steam turbine, but for many years had an inferior power-to-space ratio.

Diesel engines today are broadly classified according to:

(i) Their operating cycle (a) two-stroke (b) four-stroke

(ii) Their construction (a) Crosshead (b) trunk (c) opposed piston

(iii) Their speed

o Slow speed: any engine with a maximum operating speed up to 300 revs/minute,

although most large 2-stroke slow speed diesel engines operate below 120

revs/minute. Some very long stroke engines have a maximum speed of around 80

revs/minute. The largest, most powerful engines in the world are slow speed, two

stroke, and crosshead diesels.

o Medium speed: any engine with a maximum operating speed in the range 300-

900 revs/minute. Many modern 4-stroke medium speed diesel engines have a

maximum operating speed of around 500 rpm.

o High speed: any engine with a maximum operating speed above 900 revs/minute.

8

Most modern larger merchant ships use either slow speed, two stroke, crosshead engines, or

medium speed, four stroke, trunk engines. Some smaller vessels may use high speed diesel

engines.

The size of the different types of engines is an important factor in selecting what will be

installed in a new ship. Slow speed two-stroke engines are much taller, but the area needed,

length and width, is smaller than that needed for four-stroke medium speed diesel engines. As

space higher up in passenger ships and ferries is at a premium, these ships tend to use multiple

medium speed engines resulting in a longer, lower engine room than that needed for two-stroke

diesel engines. Multiple engine installations also give redundancy in the event of mechanical

failure of one or more engines and greater efficiency over a wider range of operating conditions.

As modern ships' propellers are at their most efficient at the operating speed of most slow

speed diesel engines, ships with these engines do not generally need gearboxes. Usually such

propulsion systems consist of either one or two propeller shafts each with its own direct drive

engine. Ships propelled by medium or high speed diesel engines may have one or two

(sometimes more) propellers, commonly with one or more engines driving each propeller shaft

through a gearbox.

9

CHAPTER 4

FINITE ELEMENT METHODS

The finite element method is numerical analysis technique for obtaining approximate

solutions to a wide variety of engineering problems. Because of its diversity and flexibility as an

analysis tool, it is receiving much attention in engineering schools and industries. In more and

more engineering situations today, we find that it is necessary to obtain approximate solutions to

problems rather than exact closed form solution.

It is not possible to obtain analytical mathematical solutions for many engineering problems.

An analytical solutions is a mathematical expression that gives the values of the desired

unknown quantity at any location in the body, as consequence it is valid for infinite number of

location in the body. For problems involving complex material properties and boundary

conditions, the engineer resorts to numerical methods that provides approximate, but acceptable

solutions.

The fundamental areas that have to be learned for working capability of finite element

method include:

MATRIX ALGEBRA.

SOLID MECHANICS.

VARIATION METHODS.

COMPUTER SKILLS.

Matrix techniques are definitely most efficient and systematic way to handle algebra of finite

element method. Basically matrix algebra provides a scheme by which a large number of

equations can be stored and manipulated. Since vast majority of literature on the finite element

method treats problems in structural and continuum mechanics, including soil and rock

mechanics, the knowledge of these fields became necessary. It is useful to consider the finite

element procedure basically as a Variation approach. This conception has contributed

significantly to the convenience of formulating the method and to its generality.

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4.1 TERMS COMMONLY USED IN FINITE ELEMENT METHOD:

DESCRITIZATION: The process of selecting only a certain number of discrete points in

the body can be termed as Descritization.

CONTINUUM: The continuum is the physical body, structure or solid being analyzed.

NODE: The finite elements, which are interconnected at joints, are called nodes or nodal

points.

ELEMENT: Small geometrical regular figures are called elements.

DISPLACE MODELS: The nodal displacements, rotations and strains necessary to specify

completely deformation of finite element.

DEGREE OF FREEDOM: The nodal displacements, rotations and strains necessary to

specify completely deformation of finite element.

LOCAL COORDINATE SYSTEM: Local coordinate system is one that is defined for a

particular element and not necessary for the entire body or structure.

GLOBAL SYSTEM: The coordinate system for the entire body is called the Global

coordinate system.

NATURAL COORDINATE SYSTEM: Natural coordinate system is a local system, which

permits the specification of point within the element by a set of dimensionless numbers,

whose magnitudes never exceeds unity.

INTERPOLATION FUNCTION: It is a function, which has unit value at one nodal point and

a zero value at all other nodal points.

ASPECT RATIO: The aspect ratio describes the shapes of the element in the assemblage

for two dimensional elements; this parameter is defined as the ratio of largest dimension

of the element to the smallest dimension.

FIELD VARIABLES: The principal unknowns of a problem are called the field variables.

11

PROCESS OF FINITE ELEMENT ANALYSIS: Fig: 3.1

12

Physical Problems

Mathematical Model

Governed by

Different Equations,

Assumptions In

Geometry

Kinematics

Material Law

Loading

Boundary

Conditions

etc

Change of Physical Problems

Finite Element Solution Choice of

Finite Element Mesh Density Solution

Parametric Representation

Loading Boundary

Interpolation of Results

Assessment of accuracy of Finite Element Solution of Mathematical Model

Design Improvement

Improve Mathematical Model

Define Mesh

Define Analysis

4.2 GENERAL DESCRIPTION OF FEM:

In the finite element method, the actual continuum of body of matter like solid, liquid or gas

is represented as an assemblage of sub divisions called Finite elements. These elements are

considered to be inter connected at specified points known as nodes or nodal points. These

nodes usually lie on the element boundaries where an adjacent element is considered to be

connected. Since the actual variation of the field variables (like Displacement, stress,

temperature, pressure and velocity) inside the continuum are is not know, we assume that the

variation of the field variable inside a finite element can be approximated by a simple function.

These approximating functions (also called interpolation models) are defined in terms of the

values at the nodes. When the field equations (like equilibrium equations) for the whole

continuum are written, the new unknown will be the nodal values of the field variable. By

solving the field equations, which are generally in the form of the matrix equations, the nodal

values of the field variables will be known. Once these are known, the approximating function

defines the field variable throughout the assemblage of elements.

The solution of a general continuum by the finite element method always follows as orderly

step-by-step process. The step-by-step procedure for static structural problem can be stated as

follows:

Step 1:- DISCRITIZATION OF STRUCTURE (DOMAIN).

The first step in the finite element method is to divide the structure of solution region in to sub

divisions or elements.

Step 2:- SELECTION OF PROPER INTERPOLATION MODEL.

Since the displacement (field variable) solution of a complex structure under any specified

load conditions cannot be predicted exactly, we assume some suitable solution, within an

element to approximate the unknown solution. The assumed solution must be simple and it

should satisfy certain convergence requirements.

13

Step 3:- DERIVATION OF ELEMENT STIFFNESS MATRICES (CHARACTERISTIC MATRICES) AND

LOAD VECTORS.

From the assumed displacement model the stiffness matrix [K(e)] and the load vector P(e) of

element ‘e’ are to be derived by using either equilibrium conditions or a suitable Variation

principle.

Step 4:- ASSEMBLAGE OF ELEMENT EQUATIONS TO OBTAIN THE EQUILIBRIUM EQUATIONS.

Since the structure is composed of several finite elements, the individual element stiffness

matrices and load vectors are to be assembled in a suitable manner and the overall equilibrium

equation has to be formulated as

[K]φ = P

Where [K] is called assembled stiffness matrix,

Φ is called the vector of nodal displacement and

P is the vector or nodal force for the complete structure.

Step 5:- SOLUTION OF SYSTEM EQUATION TO FIND NODAL VALUES OF DISPLACEMENT (FIELD

VARIABLE)

The overall equilibrium equations have to be modified to account for the boundary conditions

of the problem. After the incorporation of the boundary conditions, the equilibrium equations

can be expressed as,

[K]φ = P

For linear problems, the vector ‘φ’ can be solved very easily. But for non-linear problems,

the solution has to be obtained in a sequence of steps, each step involving the modification of

the stiffness matrix [K] and ‘φ’ or the load vector P.

14

Step 6:- COMPUTATION OF ELEMENT STRAINS AND STRESSES.

From the known nodal displacements, if required, the element strains and stresses can be

computed by using the necessary equations of solid or structural mechanics.

In the above steps, the words indicated in brackets implement the general FEM step-by-step

procedure.

4.3 ADVANTAGES OF FEM:

The FEM is based on the concept of discretization. Nevertheless as either a variational or

residual approach, the technique recognizes the multi dimensional continuity of the body not

only does the idealizations portray the body as continuous but it also requires no separate

interpolation process to extend the approximate solution to every point within the continuum.

Despite the fact that the solution is obtained at a finite number of discrete node points, the

formation of field variable models inherently provides a solution at all other locations in the

body. In contrast to other variational and residual approaches, the FEM does not require trail

solutions, which must all, apply to the entire multi dimensional continuum. The use of separate

sub-regions or the finite elements for the separate trial solutions thus permits a greater flexibility

in considering continuum of the shape.

Some of the most important advantages of the FEM derive from the techniques of

introducing boundary conditions. This is another area in which the method differs from other

variational or residual approaches. Rather than requiring every trial solution to satisfy the

boundary conditions, one prescribes the conditions after obtaining the algebraic equations for

assemblage.

No special techniques or artificial devices are necessary, such as the non-cantered difference

equations or factious external points often employed in the finite difference method.

The FEM not only accommodates complex geometry and boundary conditions, but it also has

proved successful in representing various types of complicated material properties that are

difficult to incorporate in to other numerical methods. For example, formulations in solid

mechanics have been devised for anisotropic, nonlinear, hysteretic, time dependant or

temperature dependant material behavior.

15

One of the most difficult problems encountered in applying numerical procedures of

engineering analysis is the representation of non-homogeneous continua. Nevertheless the FEM

readily accounts for non-homogeneity by the simple tactic of assigning different properties

within an element according to the pre selected polynomial pattern. For instance it is possible to

accommodate continuous or discontinuous variations of the constitutive parameters or of the

thickness of a two-dimensional body.

The systematic generality of the finite element procedure makes it a powerful and versatile

tool for a wide range of problems. As a result, flexible general-purpose computer programs can

be constructed. Primary examples of these programs are several structural analysis packages

which include a variety of element configurations and which can be applied to several categories

of structural problems. Among these packages are ASKA, STRUDL, SAP and NASTRAN &

SAFE. Another indicator of the generality of the method is that programs developed for one

field of engineering have been applied successfully to problems in different field with a little or

no modification.

Finally an engineer may develop a concept of the FEM at different levels. It is possible to

interpret the method in physical terms. On the other hand the method may be explained entirely

in mathematical terms. The physical or intuitive nature of the procedure is particularly useful to

the engineering student and practicing engineer.

4.4 LIMITATION:

One limitation of finite element method is that a few complex phenomenons are not

accommodated adequately by the method as its current state of development. Some examples of

such phenomenon form the realm of solid mechanics are cracking and fracture behavior, contact

problems, bond failures of composite materials, and non-linear material behavior with work

softening. Another example is transient, unconfined seepage problems. The numerical solution

of propagation or transient problem is not satisfactory in all respects. Many of these

phenomenon’s are presently under research and refinements of the methods to accommodate

these problems better can be expected.

The finite element method has reached a high level of development as a solution technique.

However the method yields realistic results only if the coefficients or material parameters, which

describe the basic phenomenon, are available. Material non-linearity in solid mechanics is a

16

notable example of a field in which our understanding of material behavior has lagged behind

the development of the analytical tool. In order to exploit fully the power of the finite element

method, significant effort must be direct towards the development of constitutive laws and the

evaluation of realistic coefficients and material parameters.

Even the most efficient finite element computer code requires a relatively large amount of

computer memory and time. Hence use of this method is limited to those who have access to

relatively large, high-speed computers. The advent of time-sharing, remote batch processing

and computer service bureaus or utilities ahs alleviated this restriction to some degree. In

addition the method can be applied indirectly to common engineering problems by utilizing

tables, graphs and other analysis aids that have been generated by finite element codes.

The most tedious aspects of the use of finite element methods are the basic processes of sub-

dividing the continuum and of generating error free input data for the computer. Although these

processes may be automated to a degree they have been totally accomplished by computer

because some engineer judgement may be employed in the discretization. Error in the input data

may go undetected and erroneous result obtained there form may appear acceptable.

Consequently it is essential that the engineer/programmer provide checks to detect such errors.

In addition to check internal code, an auxiliary routine that reads the input data and generates a

computer plot of the discretized continuum is desirable. This plot permits a rapid visual check of

the input data.

Finally as for any approximate numerical method the results of the finite element analysis

must be interpreted with care. We must be aware of the assumptions employed in the

formulation, the possibility of the numerical difficulties and the limitations in the material

characterizations used. A large volume of solution information is generated by a finite element

routine but this data is worthwhile only when its generation and interpretation are tempered by

proper engineering judgment.

17

4.5 BASIC APPROACH TO FEA SOFTWARE:

Basic approach for any finite element analysis (FEA) can be divided in to three parts

Pre – processors

Solver

Post – Processor

4.5.1 PRE – PROCESSOR:

Pre – processor mainly contains building of model, meshing, assigning material properties

etc.

4.5.1.1 BUILDING OF MODEL:

Geometry is usually difficult to describe, as it has to be as close to as real. Since it has to

take real world loads and boundary conditions, it is equal to prototype simulated in computer.

4.5.1.2 CREATION OF FINITE ELEMENT MODEL FOR MESHING:

After assigning material properties and structural properties to the model, meshing is done.

Meshing is the process of dividing the model into finite number of finite sized elements.

FEA uses complex system of points called nodes, which make grid called MESH. This mesh

is programmed to contain material and structural properties, which define how the structure will

react to certain loading conditions. Nodes are assigned at a certain density throughout the

material depending on the anticipated stress levels of a particular area. Regions that will receive

large amounts of stress usually have a higher node density than those, which experience little or

no stress.

Points of interest may consist or fracture point of previously tested material, fillets, corners,

complex details and high stress areas.

The mesh acts like a spider web in that form each node, there extends a mesh element to each

of the adjacent nodes. The web of vectors is what carries the material properties to the object,

creating many elements.

18

Each FEA programs may came with an element library or one is constructed over time.

Elements are thought to fill the whole experiment; hence choice of elements very critical.

While meshing it has to be decided whether we have to go for finer mesh or coarser one. The

finer the mesh, the better is the result; but longer is the analysis time. Therefore a compromise

between accuracy and solution speed is usually made.

The mesh may be created manually or automatically. In the manually created mesh, we will

notice that the elements are similar at joints. This is known as mesh refinement and it enables

the stress to be captured at the geometric discontinuities (the junction).

Manual meshing is a tedious process for models with any degree of complications but inbuilt

mesh engine creates mesh automatically thus reducing the time. These automatic meshing has

imitations as to regards mesh quality and solution accuracy.

4.5.2 SOLVER: Solvers are geometric task oriented. These are developed for specific

applications. Solvers are designed based on continuum approach where in construction of mass,

momentum and energy equation of state, thermodynamic equations as and when required for

each of the elements and the solution is obtained by interpreting these solutions. The solution to

these equations essentially depends on two methods

1. Implicit

2. Explicit

Choice of the method is based on the nature of problems.

The main goal of a finite element analysis is to examine a structure or a component response

to certain loading conditions. Therefore specifying the proper loading conditions is a key step in

the analysis. These loading conditions may be static, dynamic or transient whose nature may be

linear or non-linear.

The word loads in FEA includes the applied forcing functions externally or internally.

For example loads in structural are displacements, forces, pressures, temperatures and

gravity. In thermal analysis the applied loads are temperatures, heat flow rates convections and

heat generation.

Solver reads the model created by the pre-processor and formulates the mathematical

representation of the model. All the parameters defined in the pre-processing stage are used to

19

do this. If the model is correct the solver proceeds to form the element stiffness matrix for the

problem. After calculating the element stiffness matrix it calculates the results like

displacements, pressure, stresses, strains etc at each node within the component or continuum.

All these results are sent to a results file, which may be read by the Post-Processor.

4.5.3. POST – PROCESSOR:

Here the results of the analysis are read and interpreted. They can be presented in the form of

a table, a contour plot, deformed shape of the component to the mode shapes and natural

frequencies if frequencies are involved. Other results are available for fluids, thermal and

electrical analysis types.

Contour plots are usually more effective way of viewing for structural type problems. Slices

can be made through 3-D models to facilitate the viewing of internal stress patterns.

Post – Processor includes the calculations of stresses and strains I any of the x, y or z

directions or induced in a direction at an angle to the coordinate axis. The principal stresses and

strains may also be plotted or if required the yield stress and strains according to the main

theories if failure (von mises, st-venani, tresea etc) can be plotted.

Other information such as the strain energy, plastic strain and creep strain may be obtained

for certain types of analyses.

20

CHAPTER 5

FEA SOFTWARE – ANSYS

5.1 INTRODUCTION TO ANSYS:

ANSYS Stands for Analysis System Product.

Dr. John Swanson founded ANSYS. Inc in 1970 with a vision to commercialize the concept

of computer simulated engineering, establishing himself as one of the pioneers of Finite Element

Analysis (FEA). ANSYS inc. supports the ongoing development of innovative technology and

delivers flexible, enterprise wide engineering systems that enable companies to solve the full

range of analysis problem, maximizing their existing investments in software and hardware.

ANSYS Inc. continues its role as a technical innovator. It also supports a process-centric

approach to design and manufacturing, allowing the users to avoid expensive and time-

consuming “built and break” cycles. ANSYS analysis and simulation tools give customers ease-

of-use, data compatibility, multi platform support and coupled field multi-physics capabilities.

5.2 EVOLUTION OF ANSYS:

ANSYS has evolved into multipurpose design analysis software program, recognized around

the world for its many capabilities. Today the program is extremely powerful and easy to use.

Each release hosts new and enhanced capabilities that make the program more flexible, more

usable and faster. In this way ANSYS helps engineers meet the pressures and demands modern

product development environment.

5.3 OVERVIEW OF THE PROGRAM

The ANSYS program is flexible, robust design analysis and optimization package. The

software operates on major computers and operating systems, from PCs to workstations and to

super computers. ANSYS features file compatibility throughout the family of products and

across all platforms. ANSYS design data access enables user to import computer aided design

models in to ANSYS, eliminating repeated work. This ensures enterprise wide, flexible

engineering solution for all ANSYS user.

21

User Interface:

Although the ANSYS program has extensive and complex capabilities, its organization and

user-friendly graphical user interface makes it easy to learn and use.

There are four graphical methods to instruct the ANSYS program:

1. Menus.

2. Dialog Boxes

3. Tool bar.

4. Direct input of commands.

Menus:

Menus are groupings of related functions or operating the analysis program located in

individual windows. These include:

Utility menu

Main menu

Input window

Graphics window

Tool bar

Dialog boxes

Dialog boxes:

Windows present the users with choices for completing operations or specifying settings.

These boxes prompt the user to input data or make decisions for a particular function.

Tool bar:

The tool bar represents a very efficient means for executing commands for the ANSYS

program because of its wide range of configurability. Regardless of how they are specified,

commands are ultimately used to supply all the data and control all program functions.

Output window:

It records the ANSYS response to commands and functions.

22

Graphics window:

Represents the area for graphic displays such as model or graphically represented results of

an analysis. The user can adjust the size of the graphics window, reducing or enlarging it to fit to

personal preferences.

Input window:

Provides an input area for typing ANSYS commands and displays program prompt messages.

Main menu:

Comprise the primary ANSYS functions, which are organized in pop-up side menus, based

on the progression of the program.

Utility menu:

It contains ANSYS utility functions that are mapped here for access at any time during an

ANSYS session. These functions are executed through smooth, cascading pull down menus that

lead directly to an action or dialog box.

Processors:

ANSYS functions are organized into two groups called processors. The ANSYS program has

one pre-processor, one solution processor; two post processors and several auxiliary processors

such as the design optimizer. The ANSYS pre-processor allows the user to create a finite

element model to specify options needed for a subsequent solution. The solution processor is

used to apply the loads and the boundary conditions and then determine the response of the

model to them. With the ANSYS post processors, the user retrieves and examines the solutions

results to evaluate how the model responded and to perform additional calculations of interest.

Database:

The ANSYS program uses a single, centralized database for all model data and solution

results. Model data (including solid model and finite element model geometry, materials etc) are

written to the database using the processor. Loads and solution results data are written using the

23

solutions processor. Post processing results data are written using the post processors. Data

written to the database while using one processor are therefore available as necessary in the

other processors.

File format:

Files are used, when necessary, to pass the data from part of the program to another, to store

the program to the database, and to store the program output. These files include database files,

the results file, and the graphics file and so on.

5.4 REDUCING THE DESIGN AND MANUFACTURING COSTS USING ANSYS

(FEA):

The ANSYS program allows engineers to construct computer models or transfer CAD

models of structures, products, components, or systems, apply loads or other design performance

conditions and study physical responses such as stress levels, temperature distribution or the

impact of lector magnetic fields.

In some environments, prototype testing is undesirable or impossible. The ANSYS program

has been used in several cases of this type including biomechanical applications such as high

replacement intraocular lenses. Other representative applications range from heavy equipment

components, to an integrated circuit chip, to the bit-holding system of a continuous coal-mining

machine.

ANSYS design optimization enables the engineers to reduce the number of costly prototypes,

tailor rigidity and flexibility to meet objectives and find the proper balancing geometric

modifications.

Competitive companies look for ways to produce the highest quality product at the lowest

cost. ANSYS (FEA) can help significantly by reducing the design and manufacturing costs and

by giving engineers added confidence in the products they design. FEA is most effective when

used at the conceptual design stage. It is also useful when used later in manufacturing process to

verify the final design before prototyping.

24

Program availability:

The ANSYS program operates on Pentium based PCs running on Wndows95 or Windows

NT and workstations and super computers primarily running on UNIX operating system.

ANSYS Inc. continually works with new hardware platforms and operating systems.

Analysis types available:

1. STRUCTURAL STATIC ANALYSIS.

2. STRUCTURAL DYNAMIC ANALYSIS.

3. STRUCTURAL BUCKLING ANALYSIS.

LINEAR BUCKLING

NON LINEAR BUCKLING

4. STRUCTURAL NON LINEARITIES

5. STATIC AND DYNAMIC KINEMATICS ANALYSIS.

6. THERMAL ANALYSIS.

7. ELECTROMAGNETIC FIELD ANALYSIS.

8. ELECTRIC FIELD ANALYSIS

9. FLUID FLOW ANALYSIS

COMPUTATIONAL FLUID DYNAMICS

PIPE FLOW

10. COUPLED-FIELD ANALYSIS

11. PIEZOELECTRIC ANALYSIS.

5.5 TYPES OF STRUCTURAL ANALYSIS:

Structural analysis is the most common application of the finite element method. The term

structural (or structure) implies civil engineering structures such as bridges and buildings, but

also naval, aeronautical and mechanical structures such as ship hulls, aircraft bodies and

machines housings as well as mechanical components such as pistons, machine parts and tools.

There are seven types of structural analyses available in ANSYS. One can perform the

following types of structural analyses. Each of these analysis types are discussed in detail as

follows.

25

1. Static analysis

2. Modal analysis

3. Harmonic analysis

4. Transient dynamic analysis

5. Spectrum analysis

6. Buckling analysis

7. Explicit dynamic analysis

5.5.1 STRUCTURAL STATIC ANALYSIS:

A static analysis calculates the effects of steady loading condition on a structure, while

ignoring inertia and damping effects such as those caused by time varying loads. A static

analysis can, however include steady inertia loads (such as gravity and rotational velocity), and

time varying loads that can be approximated as static equivalent loads (such as the static

equivalent wind and seismic loads commonly defined in many building codes.)

5.6 PROCEDURE FOR ANSYS ANALYSIS:

Static analysis is used to determine the displacements, stresses, strains and forces in

structures or components due to loads that do not induce significant inertia and damping effects.

Steady loading in response conditions are assumed. The kinds of loading that can be applied in

a static analysis include externally applied forces and pressures, steady state inertial forces such

as gravity or rotational velocity imposed (non-zero) displacements, temperatures (for thermal

strain).

A static analysis can be either linear or non linear. In our present work we consider linear

static analysis.

The procedure for static analysis consists of these main steps:

1. Building the model.

2. Obtaining the solution.

3. Reviewing the results.

26

5.6.1 BUILD THE MODEL:

In this step we specify the job name and analysis title use PREP7 to define the element types,

element real constants, material properties and model geometry element types both linear and

non-linear structural elements are allowed. The ANSYS element library contains over 80

different element types. A unique number and prefix identify each element type.

E.g. BEAM 94, PLANE 71, SOLID 96 and PIPE 16

MATERIAL PROPERTIES:

Young’s modulus(EX) must be defined for a static analysis . If we plan to apply inertia

loads(such as gravity) we define mass properties such as density(DENS). Similarly if we plan to

apply thermal loads(temperatures) we define coefficient of thermal expansion(ALPX).

5.6.2 OBTAIN THE SOLUTION:

In this step we define the analysis type and options, apply loads and initiate the

finite element solution. This involves three phases:

Pre – processor phase

Solution phase

Post-processor phase

27

The following table shows the brief description of steps followed in each phase:

PREPROCESSOR

PHASE

SOLUTION PHASE POST-PROCESSOR

PHASE

GEOMETRY

DEFINITIONS

ELEMENTMATRIX

FORMULATION

POST SOLUTION

OPERATIONS

MESH GENERATION OVERALL MATRIX

TRIANGULARIZATION

POST DATA PRINT

OUTS(FOR REPORTS)

MATERIAL (WAVE FRONT) POST DATA

DEFINITIONS SCANNING POST DATA

DISPLAYS

CONSTRAINT

DEFINITIONS

DISPLACEMENT.

STRESS,ETC

LOAD DEFINITION CALCULATION

MODEL DISPLAYS

Table 4.1

5.6.2.1 PRE – PROCESSOR:

Pre processor has been developed so that the same program is available on micro, mini,

super-mini and mainframe computer system. This slows easy transfer of models one system to

other.

Pre processor is an interactive model builder to prepare the FE (finite element) model and

input data. The solution phase utilizes the input data developed by the pre processor, and

prepares the solution according to the problem definition. It creates input files to the temperature

etc., on the screen in the form of contours.

28

GEOMETRICAL DEFINITIONS:

There are four different geometric entities in pre processor namely key points, lines, areas

and volumes. These entities can be used to obtain the geometric representation of the structure.

All the entities are independent of other and have unique identification labels.

MODEL GENERATIONS:

Two different methods are used to generate a model:

Direct generation.

Solid modeling

With solid modeling we can describe we can describe the geometric boundaries of the model,

establish controls over the size and desired shape of the elements and then instruct ANSYS

program to generate all the nodes and elements automatically. By contrast, with the direct

generation method, we determine the location of every node and size, shape and connectivity of

every element prior to defining these entities in the ANSYS model. Although, some automatic

data generation is possible (by using commands such as FILL, NGEN, EGEN etc) the direct

generation method essentially a hands on numerical method that requires us to keep track of all

the node numbers as we develop the finite element mesh. This detailed book keeping can

become difficult for large models, giving scope for modeling errors. Solid modeling is usually

more powerful and versatile than direct generation and is commonly preferred method of

generating a model.

MESH GENERATION:

In the finite element analysis the basic concept is to analyze the structure, which is an

assemblage of discrete pieces called elements, which are connected, together at a finite number

of points called Nodes. Loading boundary conditions are then applied to these elements and

nodes. A network of these elements is known as Mesh.

29

FINITE ELEMENT GENERATION:

The maximum amount of time in a finite element analysis is spent on generating

elements and nodal data. Pre processor allows the user to generate nodes and elements

automatically at the same time allowing control over size and number of elements. There

are various types of elements that can be mapped or generated on various geometric

entities.

The elements developed by various automatic element generation capabilities of pre

processor can be checked element characteristics that may need to be verified before the finite

element analysis for connectivity, distortion-index, etc.

Generally, automatic mesh generating capabilities of pre processor are used rather

than defining the nodes individually. If required, nodes can be defined easily by defining

the allocations or by translating the existing nodes. Also one can plot, delete, or search

nodes.

BOUNDARY CONDITIONS AND LOADING:

After completion of the finite element model it has to constrain and load has to be

applied to the model. User can define constraints and loads in various ways. All constraints

and loads are assigned set 1D. This helps the user to keep track of load cases.

MODEL DISPLAY:

During the construction and verification stages of the model it may be necessary to

view it from different angles. It is useful to rotate the model with respect to the global

system and view it from different angles. Pre processor offers this capability. By windowing

feature pre processor allows the user to enlarge a specific area of the model for clarity

and details. Pre processor also provides features like smoothness, scaling, regions, active

set, etc for efficient model viewing and editing.

30

MATERIAL DEFINITIONS:

All elements are defined by nodes, which have only their location defined. In the

case of plate and shell elements there is no indication of thickness. This thickness can be

given as element property. Property tables for a particular property set 1-D have to be

input. Different types of elements have different properties for e.g.

Beams : Cross sectional area, moment of inertia etc

Shells : Thickness

Springs : Stiffness

Solids : None

The user also needs to define material properties of the elements. For linear static

analysis, modules of elasticity and Poisson’s ratio need to be provided. For heat transfer,

coefficient of thermal expansion, densities etc are required. They can be given to the

elements by the material property set to 1-D.

5.6.2.2 SOLUTION:

The solution phase deals with the solution of the problem according to the problem

definitions. All the tedious work of formulating and assembling of matrices are done by the

computer and finally displacements are stress values are given as output. Some of the

capabilities of the ANSYS are linear static analysis, non-linear static analysis, transient dynamic

analysis, etc.

5.6.2.3 POST – PROCESSOR:

It is a powerful user-friendly post-processing program using interactive colour graphics. It

has extensive plotting features for displaying the results obtained from the finite element

analysis. One picture of the analysis results (i.e. the results in a visual form) can often reveal in

seconds what would take an engineer hour to asses from a numerical output, say in tabular form.

The engineer may also see the important aspects of the results that could be easily missed in a

stack of numerical data.

31

Employing state of art image enhancement techniques, facilities viewing of:

Contours of stresses, displacements, temperatures, etc.

Deform geometric plots

Animated deformed shapes

Time-history plots

Solid sectioning

Hidden line plot

Light source shaded plot

Boundary line plot etc.

32

CHAPTER 6

THEORETICAL DESIGN CALCULATIONS

6.1 CALCULATIONS FOR STEEL [40 Ni2 Cr1 Mo28 STEEL]:

Input parameters:

Power = P=9000 KW

Speed of Pinion: N= 3500 r.p.m

Gear Ratio, i=7

Helix Angle, β=25°

Material Selection:

The material for pinion & Gear is 40 Ni2 Cr1 Mo28 steel.

Its design compressive stress & bending stresses are

[σc=11000kgf/cm2 ], [σb=4000kgf/cm2] [Table 7,PSG 8.5]

Properties for 40 Ni2 Cr1 Mo28 Steel:

Density of 40 Ni2 Cr1 Mo28 Steel (ρ) = 7860 kg / m3

Young’s Modulus = 215*10 5 N /m m2

Poisson’s ratio (ν) = 0.30

i= 7

ψ=0.3 [Table 9,10 PSG 8.14]

[MT]= MTkd k

MT=97420 KW/N

kd k=1.3

[MT]= MT kD k

=(97420 x 9000 x 1.8)/3500

33

= 325661.14 kgf-cm

Now, minimum centre distance based on the surface compressive strength is given by

a≥(i+1) 3√[0.7/σc]2 x E[Mt]/(iψ)

a≥88.41 cm

Minimum module based on beam strength:

mn ≥1.15cosβx3√[ MT/(YV σb ψmZ1 )] [Table 8,PSG 8.13 A]

Let Z1 =18, ψm =10 [Table 11,PSG 8.14]

Virtual number of teeth ZV =Z1/cos3β =18/0.744=25

Lewis form factor YV =0.154 -0.912/ ZV

=0.4205

mn ≥1.15cos25x3√[ 325661.14/(0.4205*4000*10*18 )]

≥1.0676 cm

mn ≥10.67 mm

But for mn =11-16 mm, σc and σb are ≥ [σc ]&[σb ] also FS <FD which makes design unsafe.

So mn =18mm=1.8 cm

No of teeth on pinion= Z1=2acosβ/ mn (i+1)=12.

But in order to avoid interference, Z1 is taken as 18.

No of teeth on gear, Z2 =i Z1 =126

Diameter of the pinion= (mn* Z1 )/ cos β =1.8*18/cos25°

= 35.74 cm

Diameter of the gear= i*d1 =7*35.74=250.24 cm

Centre distance, a = (d1+ d2)/2 =142.99 cm

Face width, b= ψa= 0.3*142.99= 43 cm

34

(or)

b= ψm mn =10*1.8=18

Taking maximum value of both b is 43 cm.

Checking Calculations:

σc =0.7*[( i+1)/a ] √(i+1)*E*[MT] / (ib) ≤ [σc ]

σb = 0.7(i+1) [MT] / (ab mn YV ) ≤ [σb]

Based on the compressive stress :

σc = [0.7*8*√(8*2.15*10 6 *325661.14) / (7*43)]=180.90 N/mm2

Based on the bending stress :

σb = [(0.7*8*325661.14) / (88.4*43*0.4205)]=228.35 N/mm2

From the calculations, σc & σb values are less than the [σc ] & [σb] values of given material, i.e 40

Ni2 Cr1 Mo28. Therefore our design is safe.

Addendum, mn =18 mm,

Dedendum=1.25* mn=22.5 mm,

Tip circle diameter of the pinion= d1+(2*addendum) =357.4+3.6 = 393 mm

Tip circle diameter of the gear= d2+(2*addendum) =2502+3.6 = 2538 mm

Root circle diameter of pinion= d1-(2*addendum) =357.4-3.6 = 312.4 mm

Root circle diameter of gear= d2-(2*addendum) =2502-3.6 = 2457.4 mm

When the gear transmits power P, the tangential force produced due to the power is given by,

FT=(P*KS)/V

V=(π DP NP )/(60*1000) = (357.4*3500* π)/(60*1000)=65.42 m/s

FT=(9000*1000*2)/65.42=275145.21 N

35

Lewis derived the equation for beam strength assuming the load to be static. When the gears

are running at high speeds, the gears may be subjected to dynamic effect. To account for the

dynamic effect, a factor known as CV known as velocity factor or dynamic factor is considered.

The design tangential force along with dynamic effect is given by

FD= FT * CV = (P* KS * CV )/V

The velocity factor CV is developed by Barth. It depends on the pitch line velocity and the

workmanship of the manufacturer and it is given by

CV = (5.5+√V)/5.5 for V>20m/s.

FD= FT * CV = (P* KS * CV )/V , where FT=275145.21 N, KS=2, V=65.42 m/s

CV=(5.5+√65.42)/5.5 = 2.47

FD=275,145*2.47 = 679771.90 N

According to Lewis equation, the beam strength of helical gear tooth is given by

FS= [σe]*b* π* mn* YV = (1.75*341)*430* π*18*0.4205=6101677.663N

( or)

FS= [σb]*b* π* mn* YV = 4000*43* π*1.8*0.4205=4089938.94N

Since, FS > FD , Our design is safe.

When the power is transmitted through gears, apart from static loads produced by the power,

some dynamic loads are also applied on the gear tooth due to reasons like inaccuracies of tooth

spacing, irregularities of tooth profiles and deflections of tooth under load.

Considering the above conditions, Buckingham derived an equation to find out the maximum

load acting on the gear tooth which is given by

FD= FT + FI Where FD = Maximum Dynamic Load

FT =Static load produced by power

FI=Incremental load due to dynamic action

36

Incremental load depends on the pitch line velocity, face width of a gear tooth, gear

materials, accuracy of cut and the tangential load and it is given by

FI= [0.164Vm (cbcos2β+ FT) cosβ] / [0.164Vm+1.485√ cbcos2β+ FT]

Where Vm is the pitch line velocity in m/s

b is the face width of the gear tooth in mm

c is the dynamic factor or deformation factor in N/mm.

Deformation factor c can also be selected from table 41, 42 PSG 8.50

Here c = 11860*e, c= 11860*0.026=308.36 N/mm.

FT=137572.60 N

Vm=65.42 m/s=65.42*103 mm/s

b=38.16 cm=381.6mm

β=25°

FD = FT + FI = FT + [0.164Vm (cbcos2β+ FT) cosβ] / [0.164Vm+1.485√ cbcos2β+ FT]

=137572.60+ [0.164*65.42(308.36*430*cos 2 25+137572.60) cos25]

[0.164*65.42+(1.485√ 308.36*430*cos225+137572.60)]

=137572.60+[2396542.95/747.98]

=140776.62 N

Since FS > FD , our design is safe.

One of the most important gear failures is the failure of gear tooth due to pitting. This pitting

failure occurs when the contact stresses between the two meshing teeth exceed the surface

endurance strength of the material. In order to avoid this type of failure, the proportions of the

gear tooth and the surface properties such as surface hardness, should be selected in such a way

that the wear strength of the gear is more than the effective load between the meshing teeth.

Based on Hertz theory of contact stresses, Buckingham derived an equation for wear strength

of gear tooth which is given by FW= (D1*b*Q*KW)/cos2β

37

Where, FW is the max or limiting load for wear in N.

D1 is the pitch circle diameter of the pinion (mm)

b is the face width of the pinion(mm)

Q is the ratio factor.

Q=Ratio factor= 2i / (i+1) =1.75

KW=2.553N/mm2 [Table 25.37 JDB]

D1=357.4 mm

b = 430 mm

FW = (D1*b*Q*KW)/cos3β

= (357.4*430*1.75*2.553)/cos225

=900086.75N

FD=140776.62N Since FW > FD our design is safe.

6.1.1DESIGN OF HOLE:

Design is based on the torque transmitted by the shaft

Torque transmitted T= (60*P)/2πN= (60*9000*103)/(2π*3500)=24555.3N-mm

We know that torque transmitted by the shaft T= (πd3*fs)/16=d=125 mm

Key dimensions:

Width of key (w) =d/4=32 mm

Thickness of key (t) =d/6=18 mm

38

6.2 CALCULATIONS FOR CERAMICS [98%Al2O3 , 0.4-0.7% Mn, 0.4-0.7%Mg ]:

Input parameters:

Power = P=9000 KW

Speed of Pinion: N= 3500 r.p.m

Gear Ratio, i=7

Helix Angle, β=25°

Material Selection:

Let the material for pinion & Gear is Aluminum Alloy.

Its design compressive stress & bending stresses are

[ [σc=25000kgf/cm2 ], [σb=3500kgf/cm2] [Table 7,PSG 8.5]

Properties for Aluminum Alloy:

Density of Aluminum Alloy (ρ) = 3900 kg / m3

Young’s Modulus = 340*10 3 N /m m2

Poisson’s ratio (ν) = 0.220

i= 7

ψ=0.3 [Table 9,10 PSG 8.14]

[MT]= MTkd k

MT=97420 KW/N

kd k=1.3

[MT]= MT kD k

=(97420 x 9000 x 1.8)/3500

= 325661.14 kgf-cm

39

Now, minimum centre distance based on the surface compressive strength is given by

a≥(i+1) 3√[0.7/σc]2 x E[Mt]/(iψ)

a≥59.59 cm

Minimum module based on beam strength:

mn ≥1.15cosβx3√[ MT/(YV σb ψmZ1 )] [Table 8,PSG 8.13 A]

Let Z1 =18, ψm =10 [Table 11,PSG 8.14]

Virtual number of teeth ZV =Z1/cos3β =18/0.744=25

Lewis form factor YV =0.154 -0.912/ ZV

=0.4205

mn ≥1.15cos25x3√[ 325661.14/(0.4205*3500*10*18 )]

≥1.11 cm

mn ≥11.16 mm

But for mn =11-16 mm, FS <FD which makes design unsafe.

So mn =18mm=1.8 cm

No of teeth on pinion= Z1=2acosβ/ mn (i+1)=12.

But in order to avoid interference, Z1 is taken as 18.

No of teeth on gear, Z2 =i Z1 =126

Diameter of the pinion= (mn* Z1 )/ cos β =1.8*18/cos25°

= 35.74 cm

Diameter of the gear= i*d1 =7*35.74=250.24 cm

Centre distance, a = (d1+ d2)/2 =142.99 cm

Face width, b= ψa= 0.3*142.99= 43 cm

(or)

40

b= ψm mn =10*1.8=18

Taking maximum value of both b is 43 cm.

Checking Calculations:

σc =0.7*[( i+1)/a ] √(i+1)*E*[MT] / (ib) ≤ [σc ]

σb = 0.7(i+1) [MT] / (ab mn YV ) ≤ [σb]

Based on the compressive stress :

σc = [((0.7*8)/60)*√(8*340*10 4 *325661.14) / (7*43)]=150.65 N/mm2

Based on the bending stress :

σb = [(0.7*8*325661.14) / (60*43*0.4205)]=218.94 N/mm2

From the calculations, σc & σb values are less than the [σc ] & [σb] values of given material, i.e 40

Ni2 Cr1 Mo28. Therefore our design is safe.

Addendum, mn =18 mm,

Dedendum =1.25* mn=22.5 mm,

Tip circle diameter of the pinion= d1+(2*addendum) =357.4+3.6 = 393 mm

Tip circle diameter of the gear= d2+(2*addendum) =2502+3.6 = 2538 mm

Root circle diameter of pinion= d1-(2*addendum) =357.4-3.6 = 312.4 mm

Root circle diameter of gear= d2-(2*addendum) =2502-3.6 = 2457.4 mm

When the gear transmits power P, the tangential force produced due to the power is given by,

FT=(P*KS)/V

V=(π DP NP )/(60*1000) = (357.4*3500* π)/(60*1000)=65.42 m/s

FT=(9000*1000*2)/65.42=275145.21 N

Lewis derived the equation for beam strength assuming the load to be static. When the gears

are running at high speeds, the gears may be subjected to dynamic effect. To account for the

dynamic effect, a factor known as CV known as velocity factor or dynamic factor is considered.

41

The design tangential force along with dynamic effect is given by

FD= FT * CV = (P* KS * CV )/V

The velocity factor CV is developed by Barth. It depends on the pitch line velocity and the

workmanship of the manufacturer and it is given by

CV = (5.5+√V)/5.5 for V>20m/s.

FD= FT * CV = (P* KS * CV )/V , where FT=275145.21 N, KS=2, V=65.42 m/s

CV=(5.5+√65.42)/5.5 = 2.47

FD=275,145*2.47 = 679771.90 N

According to Lewis equation, the beam strength of helical gear tooth is given by

FS= [σe]*b* π* mn* YV = (1.75*100)*430* π*18*0.4205=1789348.28N

( or)

FS= [σb]*b* π* mn* YV = 3500*43* π*1.8*0.4205=3578690.65N

Since, FS > FD , Our design is safe.

When the power is transmitted through gears, apart from static loads produced by the power,

some dynamic loads are also applied on the gear tooth due to reasons like inaccuracies of tooth

spacing, irregularities of tooth profiles and deflections of tooth under load.

Considering the above conditions, Buckingham derived an equation to find out the maximum

load acting on the gear tooth which is given by

FD= FT + FI Where FD = Maximum Dynamic Load

FT =Static load produced by power

FI=Incremental load due to dynamic action

Incremental load depends on the pitch line velocity, face width of a gear tooth, gear

materials, accuracy of cut and the tangential load and it is given by

42

FI= [0.164Vm (cbcos2β+ FT) cosβ] / [0.164Vm+1.485√ cbcos2β+ FT]

Where Vm is the pitch line velocity in m/s

b is the face width of the gear tooth in mm

c is the dynamic factor or deformation factor in N/mm.

Deformation factor c can also be selected from table 41, 42 PSG 8.50

Here c = 11860*e, c= 11860*0.026=308.36 N/mm.

FT=137572.60 N

Vm=65.42 m/s=65.42*103 mm/s

b=38.16 cm=381.6mm

β=25°

FD = FT + FI = FT + [0.164Vm (cbcos2β+ FT) cosβ] / [0.164Vm+1.485√ cbcos2β+ FT]

=137572.60+ [0.164*65.42(308.36*430*cos 2 25+137572.60) cos25]

[0.164*65.42+(1.485√ 308.36*430*cos225+137572.60)]

=137572.60+[2396542.95/747.98]

=388329.19 N

Since FS > FD , our design is safe.

One of the most important gear failures is the failure of gear tooth due to pitting. This pitting

failure occurs when the contact stresses between the two meshing teeth exceed the surface

endurance strength of the material. In order to avoid this type of failure, the proportions of the

gear tooth and the surface properties such as surface hardness, should be selected in such a way

that the wear strength of the gear is more than the effective load between the meshing teeth.

43

Based on Hertz theory of contact stresses, Buckingham derived an equation for wear strength of

gear tooth which is given by FW= (D1*b*Q*KW)/cos2β

Where, FW is the max or limiting load for wear in N.

D1 is the pitch circle diameter of the pinion (mm)

b is the face width of the pinion(mm)

Q is the ratio factor.

Q=Ratio factor= 2i / (i+1) =1.75

KW=2.553N/mm2 [Table 25.37 JDB]; D1=357.4 mm; b = 430 mm

FW = (D1*b*Q*KW)/cos3β = (357.4*430*1.75*2.553)/cos225 =900086.75N

FD=140776.62N Since FW > FD our design is safe.

6.2.1DESIGN OF HOLE:

Design is based on the torque transmitted by the shaft

Torque transmitted T= (60*P)/2πN= (60*9000*103)/(2π*3500)=24555.3N-mm

We know that torque transmitted by the shaft T= (πd3*fs)/16=d=125 mm

Key dimensions:

Width of key (w) =d/4=32 mm

Thickness of key (t) =d/6=18 mm

44

CHAPTER 7

DEFINITION OF PROBLEM

Marine engines are among heavy-duty machineries, which need to be taken care of in the best

way during prototype development stages. These engines are operated at very high speeds which

induce large stresses and deflections in the gears as well as in other rotating components. For the

safe functioning of the engine, these stresses and deflections have to be minimized.

In this project, we have performed static-structural analysis on a high speed helical gear used

in marine engines, using different materials. The results obtained in the static analysis are

compared with those obtained in theoretical and a conclusion has been drawn on the material to

be used for the gear.

7.1 ELEMENT CONSIDERED FOR STATIC ANALYSIS:

According to the given specifications the element type chosen is SOLID 45. SOLID45 is

used for the 3-D modeling of solid structures. The element is defined by eight nodes having

three degrees of freedom at each node: translations in the nodal x, y, and z directions. The

element has plasticity, creep, swelling, stress stiffening, large deflection, and large strain

capabilities.

The element is defined by eight nodes and the orthotropic material properties. Orthotropic

material directions correspond to the element coordinate directions. The element coordinate

system orientation is as described in Coordinate Systems.

The following Figure shows the schematic diagram of the 8-noded static solid element.

Fig: 7.1 SOLID-45, 8-NODE STATIC ELEMENT

45

7.2 ASSUMPTIONS AND RESTRICTIONS:

Zero volume elements are not allowed. Elements may be numbered or may have the planes

IJKL and MNOP interchanged. Also, the element may not be twisted such that the element has

two separate volumes. This occurs most frequently when the elements are not numbered

properly.

All elements must have eight nodes. A prism-shaped element may be formed by defining

duplicate K and L and duplicate O and P node numbers. A tetrahedron shape is also available.

The extra shapes are automatically deleted for tetrahedron elements.

7.3 MATERIALS USED:

Table 6.1

MATERIALS YOUNGSMODULES

(N/mm2)

DENSITY

(Kg/m

t3)

POISSIONSRATIO

Steel 215*105 7850 0.30

Aluminum alloy 340*105 3900 0.220

7.4 MESH GENERATION:

Before building the model, it is important to think about whether a free mesh or a mapped

mesh is appropriate for the analysis. A free mesh has no restrictions in terms of element shapes

and has no specified pattern applied to it.

Compared to the free mesh, a mapped mesh is restricted in terms of the element shape it

contains and pattern of the mesh. A mapped mesh contains either only quadrilateral or only

triangular element, while a mapped volume mesh contains only hexahedral elements. In

addition, a mapped mesh typically has a regular pattern, with obvious rows of elements.

For mapped mesh, we must build the geometry as a series of fairly regular volumes and/or

areas that can accept a mapped mesh. The type of mesh generation considered here is a free

mesh since the 3D figure is not a regular shape. Solid 45 is used to model in ANSYS.

46

7.5 BOUNDARY CONDITIONS:

A). Geometry boundary conditions:

The shaft is fixed at the centre along with its key.

B) Loads applied:

FT=137572.60 N

FR =55248.70 N

Finally the boundary conditions are verified before going for a solution.

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CHAPTER 8

ANSYS RESULTS

8.1 RESULTS FOR STEEL:

Fig: 8.1 meshed gear model

Fig: 8.2 boundary conditions & Forces

48

Fig: 8.3 deformed model of helical gear with steel as a material

Fig: 8.4 stresses in X-direction of model of helical gear with steel as a material

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Fig: 8.5 stresses in Y-direction of model of helical gear with steel as a material

Fig: 8.6 stresses in Z-direction of model of helical gear with steel as a material

50

Fig: 8.7 Von Misses stresses of model of helical gear with steel as a material

51

8.2 RESULTS FOR CERAMICS:

Fig: 8.8 the stresses in X-direction of model of helical gear with ceramics as a material

Fig: 8.9 stresses in Y-direction of model of helical gear with ceramics as a material

52

Fig: 8.10 stresses in Z-direction of model of helical gear with ceramics as a material

Fig: 8.11 Von Misses stresses of model of helical gear with ceramics as a material

53

CHAPTER 9

DISCUSSION OF RESULTS

9.1 RESULTS FOR STEEL AS MATERIAL: The following table shows the

comparison between theoretical results & experimental results.

TABLE 9.1

PARAMETER DESIGN STRESSES INDUCED STRESSES

BENDING STRESS 400 N/mm2 228.35 N/mm2

COMPRESSIVE STRESS 1100 N/mm2 178.59 N/mm2

TABLE 9.2

PARAMETER THEORETICAL

RESULTS

ANSYS RESULTS

DEFLECTION 0.091958 mm

BENDING STRESS 228.35 N/mm2 225.271 N/mm2

COMPRESSIVE STRESS 178.59 N/mm2 165.359 N/mm2

VON MISES STRESSES σ yield=972 N/mm2 245.63N/mm2

From the table 9.1 we observe that the bending & compressive stresses

obtained practically from the ANSYS are much lower than those of the

results obtained theoretically. Thus the design is safe from the structural point

of view.

From the table 9.2 we observe that the induced bending & compressive

stresses are much lower than the design stresses. Thus the design is safe from

the structural point of view.

The maximum deflection is found to be 0.091958mm which is well with in

the permissible limits. Thus the design is safe based on rigidity point of view.

The induced von mises stresses with magnitude of 245.63N/mm2 are much

lower than the yield stress i.e. 972 N/mm2 according to the manufacturer’s

specifications.

54

9.2 RESULTS FOR CERAMICS [98% Al2O3] AS MATERIAL:

The following table shows the comparison between theoretical results & experimental results.

TABLE 9.3

PARAMETER DESIGN STRESSES INDUCED STRESSES

BENDING STRESS 350 N/mm2 220.35 N/mm2

COMPRESSIVE STRESS 2500 N/mm2 150.30 N/mm2

TABLE 9.4

PARAMETER THEORETICAL

RESULTS

ANSYS RESULTS

DEFLECTION 0.058752 mm

BENDING STRESS 228.35 N/mm2 225.271 N/mm2

COMPRESSIVE STRESS 178.59 N/mm2 165.359 N/mm2

VON MISES STRESSES σ yield=972 N/mm2 245.63N/mm2

From the table 9.3 we observe that the bending & compressive stresses

obtained practically from the ANSYS are much lower than those of the

results obtained theoretically. Thus the design is safe from the structural point

of view.

From the table 9.4 we observe that the induced bending & compressive

stresses obtained are much lower than those of the design stresses. Thus the

design is safe from the structural point of view.

The maximum deflection is found to be 0.058752mm which is well with in

the permissible limits. Thus the design is safe based on rigidity point of view.

The induced von mises stresses with magnitude of 260.92/mm2 is far below

the yield stress i.e. 2000 N/mm2 according to the manufacturer’s

specifications.

Thus the helical gear parameters that constitute the design are in turn safe

based on the strength and rigidity.

55

COMPARISION OF RESULTS

Table 9.5 Comparison of Results

PARAMETERS STEEL CERAMICS

DEFLECTIONS 0.091958 mm 0.058752 mm

BENDING STRESS 225.271 N/mm2 205.576 N/mm2

COMPRESSIVE STRESS 165.359 N/mm2 139.076 N/mm2

The results obtained above are less than material properties as mentioned before. Hence

the design is safe and optimum.

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CHAPTER 10

MANUFACTURING OF HELICAL GEAR

GEAR HOBBING:

Hobbing is a machining process for making gears, splines, and sprockets on a hobbing

machine, which is a special type of milling machine. The teeth or splines are

progressively cut into the workpiece by a series of cuts made by a cutting tool called a

hob. Compared to other gear forming processes it is relatively inexpensive but still quite

accurate, thus it is used for a broad range of parts and quantities.

Hobbing process uses a hobbing machine with two non-parallel spindles, one mounted

with a blank workpiece and the other with the hob. The angle between the hob's spindle

and the workpiece's spindle varies, depending on the type of product being produced. For

example, if a spur gear is being produced, then the hob is angled equal to the helix angle

of the hob; if a helical gear is being produced then the angle must be increase by the

same amount as the helix angle of the helical gear. The two shafts are rotated at a

proportional ratio, which determines the number of teeth on the blank; for example, if the

gear ratio is 40:1 the hob rotates 40 times to each turn of the blank, which produces 40

teeth in the blank. Note that the previous example only holds true for a single threaded

hob; if the hob has multiple threads then the speed ratio must be multiplied by the

number of threads on the hob. The hob is then fed up into workpiece until the correct

tooth depth is obtained. Finally the hob is fed into the workpiece parallel to the blank's

axis of rotation.

Up to five teeth can be cut onto the work piece at the same time. Often times multiple gears are cut at the same time. For larger gears the blank is usually gashed to the rough shape to make hobbing easier.

10.1ADVANTAGES OF GEAR HOBBING PROCESS:

High productivity rate Economical and efficient operation

Accuracy

Close tolerances

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Versatility of operations

Smooth finishes

10.2 HOBBING MACHINE:

Modern hobbing machines, also known as hobbers, are fully automated machines that

come in many sizes, because they need to be able to produce anything from tiny

instrument gears up to 10 ft (3.0 m) diameter marine gears. Each gear hobbing machine

typically consists of a chuck and tailstock, to hold the workpiece or a spindle, a spindle

on which the hob is mounted, and a drive motor.

For a tooth profile which is a theoretical involute, the fundamental rack is straight-sided,

with sides inclined at the pressure angle of the tooth form, with flat top and bottom. The

necessary addendum correction to allow the use of small-numbered pinions can either be

obtained by suitable modification of this rack to a cycloidal form at the tips, or by

hobbing at other than the theoretical pitch circle diameter. Since the gear ratio between

hob and blank is fixed, the resulting gear will have the correct pitch on the pitch circle,

but the tooth thickness will not be equal to the space width.

Hobbing machines are characterised by the largest module or pitch diameter it can

generate. For example, a 10 in (250 mm) capacity machine can generate gears with a 10

in pitch diameter and usually a maximum of a 10 in face width. Most hobbing machines

are vertical hobbers, which means the blank is mounted vertically. Horizontal hobbing

machines are usually used for cutting longer workpieces; i.e. cutting splines on the end

of a shaft.

58

Fig 10.1: 3D model of gear hobbing process

10.3 Hob

The hob is the cutter used to cut the teeth into the workpiece. It is cylindrical in shape

with helical cutting teeth. These teeth have grooves that run the length of the hob, which

aid in cutting and chip removal. There are also special hobs designed for special gears

such as the spline and sprocket gears.

The cross-sectional shape of the hob teeth are almost the same shape as teeth of a rack

gear that would be used with the finished product. There are slight changes to the shape

for generating purposes, such as extending the hob's tooth length to create a clearance in

the gear's roots. Each hob tooth is relieved on the back side to reduce friction.

Most hobs are single-thread hobs, but double-, and triple-thread hobs increase production

rates. The downside is that they are not as accurate as single-thread hobs.

59

Fig 10.2 A gear hob in a hobbing machine with a finished gear

10.4 Types of hobs:

Roller chain sprocket hobs

Worm wheel hobs

Spline hobs

Chamfer hobs

Spur and helical gear hobs

Straight side spline hobs

Involute spline hobs

Serration hobs

Semitopping gear hobs

10.5 MACHINING PROCESS:

Hobbing uses a hobbing machine with two non-parallel spindles, one mounted with a

blank workpiece and the other with the hob. The angle between the hob's spindle and the

workpiece's spindle is set same as that of helix angle of the helical gear in this case it is

( β=25°).The material selected for the manufacturing of prototype is mild steel.

The two shafts are rotated at a proportional ratio, which determines the number

of teeth on the blank.. Here the ratio is 18:1, which means for every rone rotation of hob

18 teeth are cut onto the workpiece. The hob is then fed up into workpiece until the

correct tooth depth is obtained. Finally the hob is fed into the workpiece parallel to the

blank's axis of rotation.

60

Fig 10.3: WMW vertical Gear hobbing Machine

10.6 MACHINE SPECIFICATIONS:

Type of machine: Gear Hobbing Machine - Vertical

Make: WMW (KARL-MARX-STADT)

Type: ZFWZ 500 x 5

Year of manufacture: 1959/2005

Type of control: conventional

Technical details

Max wheel diameter: 210 mm 

61

Gear width: 195 mm 

Max module: 6 (6.25; 6.5; 6.75)  

Min module: 1 (1.25; 1.5; 1.75;)  

Table size round: 460mm   

Total power requirement: 3.5 kW 

Weight of the machine: 2 ton 

Dimensions of the machine: L: 2.5 x B: 1.6 x H: 1.8 m 

Details of motor:

Electric drive with the motor capacity of 2.2 kW

rotation: 1500 rpm

rapid traverse: 1.0 kW

CHAPTER 11

CONCLUSION

Bending and compressive stresses were obtained theoretically & by using

Ansys software for both ceramic& steel.

From the table 8.5, it is observed that bending and compressive stresses of

ceramics are less than that of the steel.

Weight reduction is a very important criterion in the marine gears.

Hence ceramic material is preferred.

62

CHAPTER 12

FUTURE SCOPE OF THE WORK

In the present investigation of static analysis, a high speed helical gear, is analyzed

by FEM package ANSYS.

As a future work, modal analysis and harmonic analysis of the gear can be performed

to find out the mode shapes and natural frequencies of the gear.

63

REFERENCES

1. THIRUPATHI R. CHANDRUPATLA & ASHOK D.BELEGUNDU .,

INTRODUCTION TO FINITE ELEEMENT IN ENGINEERING, Pearson ,2003

2. JOSEPH SHIGLEY, CHARLES MISCHIKE ., MECHANICAL

ENGINEERING DESIGN, TMH,2003

3. MAITHRA ., HANDBOOK OF GEAR DESIGN,2000

4. V.B.BHANDARI., DESIGN OF MACHINE ELEMENTS,TMH,2003

5. R.S.KHURMI ., MACHINE DESIGN, SCHAND,2005

6. DARLE W DUDLEY.,HAND BOOK OF PRACTICAL GEAR DESIGN,1954

64

7. ALEC STROKES., HIGH PERFORMANCE GEAR DESIGN,1970

8. www.matweb.com

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