MODELLING AND FINITE ELEMENT

ANALYSIS OF

REAR AXLE HOUSING FOR CHEVY CARS

PROJECT REPORT

Submitted by

S.SURESH

Register No: AC09MED012in partial fulfillment for the award of the degree

of

MASTER OF ENGINEERINGIn

ENGINEERING DESIGN

ADHIYAMAAN COLLEGE OF ENGINEERING

(Autonomous)

(Accredited by NBA, National Board of Accreditation)

(An ISO 9001:2000 Certified Institution)

HOSUR.

APRIL-2011

ANNA UNIVERSITY OF TECHNOLOGY, COIMBATORE.

APRIL-2011

BONAFIDE CERTIFICATE

This is to certify that project report titled “MODELLING AND FINITE

ELEMENT ANALYSIS OF REAR AXLE HOUSING FOR CHEVY CARS” is the

bonafide work of Mr.S.SURESH who carried out the project work under my supervision.

Certified further, that to the best of my knowledge the work reported herein does not form

part of any other project report or dissertation on the basis of which a degree or award was

conferred on an earlier occasion of this or any other candidate.

SIGNATURE OF GUIDE SIGNATURE OF HOD

Prof.K.MYLSAMY M.E.,[Ph.D.,] Prof.CHANNANKAIAH ME.[Ph.D.,]

Assistant Pofessor, Head of the Department,

Department of mechanical Engg, Department of mechanical Engg,

Adhiyamaan college of Engineering, Adhiyamaan college of Engineering,

Hosur. Hosur.

ACKNOWLEDGEMENT

We wish to express our heartfelt gratitude to the Department of Mechanical

Engineering, Adhiyamaan College of Engineering for their continued support, in technical

expertise we aspire to excel this techno savvy world.

We would like to thank our honourable heartfelt support from our beloved and

respected principal Dr.G.RANGANATH, M.E., Ph.D., M.I.S.T.E, E.I.E, C.Engg

(INDIA)., for setting up an excellent atmosphere in this institution.

We wish to express the deepest gratitude to our head of the department

Prof.CHANNANKAIAH M.E., [Ph.D] who has been inspirational and supportive

throughout our project being there with us whenever we needed his expertise and helping

us to complete the project successfully.

And also we specially thank our department staffs, lab instructors and attenders. They

helped us externally throughout the project. Finally I thank my fellow colleagues who

helped me whenever I was struck with some problems and doubts.

CONTENTS

Chapter Description Page

No

Abstract

CHAPTER 1 INTRODUCTION

1.1 Axle Housing 1

1.2 Leak Testing 3

1.3 Dunk Testing 4

1.4 Objective of Project 5

1.5 Organization of Thesis 5

CHAPTER 2 LITERATURE REVIEW 7

2.1 Introduction 7

2.2 Scope of thé Pressent Work 8

2.3 Methodology 8

CHAPTER 3 STRUCTURAL VIBRATION 9

3.1 Introduction 9

3.2 Vibration, Resonance and Mode shapes 9 3.3

Formal Approach 10

3.4 Application of Frequency Analysis 11

CHAPTER 4 FINITE ELEMENT METHOD 13

4.1 Introduction 13

4.2 Basic Concept 13

4.3 Need for Finite Element Method 15

4.4 FEM in Structural Analysis 16

4.5 Applications of FEM in Engineering

4.5.1 Available Commercial FEM Software Packages 16

ABSTRACT

Axle housing assemblies are well known structures which are in common use in

most vehicles. Such axle assemblies include a number of components which are adapted to

transmit rotational power from an engine of the vehicle to the wheels thereof. Before final

assembly of axle housing it has to be analysed to make sure that it withstand impact and

heavy load for safety.Engineering stress estimation is very essential to find safety of the

structure. Stress analysis gives prior idea of the structure for optimum results. Many

methods like Analytical, Experimental and Numerical methods are available to estimate

stress and strain estimates on the problem. But Analytical methods are suitable for simple

problems, and Experimental methods are difficult to apply and will not complete

information of the problem. Due to this numerical methods are dominated in the stress

analysis field. Implementation of Finite Element Methods for structural analysis is

possible due to the emergence of fast computing technology. The accuracy of the

numerical methods directly depends on the quality of the mesh. Generally quad or

Hexahedra elements gives much better accuracy compared to tri or Tetra Hedra elements.

So Hypermesh, meshing software along with Nastran is considered for analysis. In the

present work, analysis is carried out on Rear Axle Housing for Stress capability. The Rear

Axle Housing is having a uniform thickness of 4.5 mm. But a groove which is spread on

the bottom of the member throughout the length, thickness is around 2.5 mm. The

member needs to be tested for stress condition. Also Modal analysis is carried out to check

for any possible resonance. The member will be shell meshed using hyper mesh and is

imported to Nastran for analysis. Possible stress concentration regions, load carrying

capacity and zones of improvements will be suggested from the analysis.

CHAPTER – 1

INTRODUCTION

CHAPTER – 1

INTRODUCTION

1.1. AXLE HOUSING

Axle assemblies are well known structures which are in common use in most

vehicles. Such axle assemblies include a number of components which are adapted to

transmit rotational power from an engine of the vehicle to the wheels thereof. Typically,

an axle assembly includes a differential which is supported within a non-rotating carrier.

The differential is connected between an input drive shaft extending from the vehicle

engine and a pair of output axle shafts extending to the vehicle wheels. The axle shafts are

contained in respective non-rotating tubes which are secured to the carrier. Thus, rotation

of the differential by the drive shaft causes corresponding rotation of the axle shafts. The

carrier and the tubes form housing for these drive train components of the axle assembly,

in asmuch as the differential and the axle shafts are supported for rotation.

Axle housings are generally classified into two basic types. The first axle housing

type is a unitized carrier construction, commonly referred to as Spicer type axle assembly.

In this structure, the carrier (which houses the differential) is directly connected to the two

tubes (which house the axle shafts). An opening is provided at the rear of the carrier to

permit assembly of the differential therein. This opening is closed by a cover during use.

The second axle housing type is a separable carrier construction. In this structure, the

axle tubes are connected together by a central member which is formed separate and apart

from the carrier. This central member is generally hollow and cylindrical in shape, having

a large generally circular opening formed there through. During assembly, the differential

is first assembled within the carrier, and then the carrier is secured to the central member.

The overall shape of this type of axle housing (i.e., the generally round shape of the central

member and the elongated tubes extending there from) generally resembles the shape of a

banjo musical instrument. Hence, this type of axle housing is commonly referred to as

banjo-type axle housing. Banjo-type axle housings are advantageous because the carrier

and differential can be removed from the axle assembly for service without disturbing the

other components.

One known structure for banjo-type axle housing is formed by splitting one end of

each of two tubes, spreading the two split ends apart, and securing the spread ends

together to form a hollow cylindrical central member. The central member includes

rearward and forwardly facing openings. A rear mounting plate and cover are secured over

the rearward facing opening and a forward mounting plate is secured over the forwardly

facing opening. The forward mounting plate includes a generally oval shaped opening

which receives a differential and carrier assembly. Typically, a pair of baffle plates is

secured within the axle housing central member to cover the interior ends of the axle

tubes. The baffle plates have apertures formed there through which the axle shafts extend.

The baffle plates function to prevent the splashing of differential lubricant out of the

central member into the axletubes.

The above-described banjo-type axle housing has been in common use for years.

However, it has been found that under typical vertical loading conditions, the axle housing

develops tensile stresses within the curved portions of the spread apart tube ends which

form the central member. These tensile stresses can cause the central member of the axle

housing to fracture at or near the curved portions. Thus, it would be desirable to provide

an improved structure for a banjo-type axle structure which is resistant to these tensile

stresses and, therefore, has a longer useful life. Also, it would be desirable to provide an

improved structure for a banjo-type axle housing which is simple and inexpensive in

construction

GENERAL MODEL OF REAR AXLE HOUSING:

PARTS OF AXLE HOUSING

Figure 1.1 shows the pictorial view of the rear axle housing model considered for the

problem. A groove (notch) is there at the center through out the length of axle housing for

welding both halves and for lubrication purpose.

Before final assembly of axle housing it has to be tested to make sure that no leakage

is there, during leak testing a huge pressure will be exerted, therefore we need to analyze

the housing for safety .

Figure No. 1.1 Parts of Axle housing

The Rear Axle housing of a special car is to be analyzed for both static and modal analysis

to avoid failure conditions.. The main objective of the project is to model the rear axle

housing and test it under distributed pressure load to find structural safety due to a groove

provided in the member. Also Modal analysis is carried out to avoid resonance conditions

of the problem.

1.2. LEAK TESTING FOR REAR AXLE HOUSING

Leak Testing is the branch of nondestructive testing that is concerned with the

escape of liquids, vacuum or gases from sealed components or systems. This article will

cover the reasons for leak testing and some of the technology behind the science.

Like other forms of nondestructive testing, leak testing has a great impact on the

safety or performance of a product. Reliable leak testing saves costs by reducing the

number of reworked products, warranty repairs and liability claims. The time and money

invested in leak testing often produces immediate profit.

The three most common reasons for performing a leak test are

Material Loss - With the high cost of energy, material loss is increasingly important. By

leak testing, energy is saved not only directly, through the conservation of fuels such as

gasoline and LNG but also indirectly, through the saving of expensive chemicals and even

compressed air.

Contamination - With stricter environmental regulations, this reason for testing is

growing rapidly. Leakage of dangerous gases or liquids pollutes and creates serious

Personnel hazards.

Reliability - Component reliability has long been a major reason for leakage testing. Leak

tests operate directly to assure serviceability of critical parts from pacemakers to

refrigeration units.

The present work of Leak testing is to find the leakage in the members (Flaws in the

structure) which will hamper the functioning of the machine or structure. Here component

to be leak tested will put in to a tank filled with water. The object will be immersed in the

water tank. Water bubbles indicate the type of leakage and extent of leakage. From the

special equipment attached to the Leak Testing Machine flaws or leakage points will be

identified.

Leak testing equipment is a type of nondestructive testing equipment used to

measure the escape of liquids, vacuum or gases from sealed components or systems. Some

configurations require a separate leak detector or sensor as an input. They are often

equipped with various other components such as pumps, calibrators, gages and cases. A

leak is a hole or porosity in an enclosure capable of passing a fluid from the higher

pressure side to the lower pressure side.

There are many basic leak test methods and a few variations. The most familiar are

dunk testing, pressure decay, mass flow, mass spectrometer and ultrasonic. Dunk testing is

still the most popular method, with pressure decay and mass flow rapidly gaining in use.

Mass flow is the de facto test method of choice in automotive applications. An exception

is pressure decay testing on a brazing fixture.

1.2.1. Dunk Testing

Figure No. 1.2 Dunk Tester

A dunk tester (Figure 1.2) inspects an automotive radiator at a radiator repair shop. If

it leaks, the bubbles show where and it can be repaired. Source: Stewart Ergonomics Inc.

Dunk testing, sometimes called bubble testing, is used for applications that do not

require high sensitivity. With dunk testing, the part under test is pressurized; submerged in

a liquid—typically water—while the operator looks for bubbles. Bubbles form at the

source of the leak as a result of air pressure, and the amount of bubbles per minute can

signify the size of the leak. Automotive radiators often are checked for leaks this way. If a

leak is present, the bubbles indicate where and the leak can be repaired. Leak testing

works best when speed is not a factor. On a production line where test time is critical, leak

testing is not the best choice.

This is an example of a familiar process for low-volume applications and repairs but

an inappropriate use in high-volume applications. High-speed leak testing in a production

line situation hampers the operator’s ability to accurately identify bubbles. However, dunk

testing can be used on fuel tank filler assemblies and fuel tanks themselves & axle

housings.

One advantage of water dunking is temperature stability. The large volume does not

change temperature, which affects most of the more sophisticated testers.

1.3 . Objective of the Project

The objective of the project is to

To develop the FE model

Find out static stress level in the housing

Find out the deformation

Find out the natural frequencies

1.4 . Organization of the Thesis

This chapter presents general overview of the axle housing and leak testing. Types of

axle hosing and their construction are discussed. Also brief introduction to leak testing is

discussed.

Chapter two discussions is done on works reported in the literature and state of

art, regarding the axle housing analysis, also scope and methodology of present work is

given.

The chapter three contains a brief introduction regarding vibration resonance and mode

shapes.

In chapter four discussions regarding Finite element Method and Finite element solver

Msc Nastran is carried out.

The chapter five contains details regarding geometry of axle hosing, type and number of

elements used for analysis.

Chapter six static analysis is carried out in which assumptions, material properties and

boundary conditions are described.

The chapter seven contains results and discussion for static analysis for different cases.

In chapter eight modal analysis is carried out to compute the natural frequencies and

mode shapes of a structure also validation of result is carried out.

The chapter nine contains the conclusion of the thesis and directions to future work

A list of reference is provided at the end of the thesis, and this consists of list of

published papers in journals and conferences, and list of books.

Now, the next chapter discusses the review of the literature on axle housing analysis.

Summary

In this chapter study about types of axle housing is carried out. There are mainly

two types one is unitized type and other is split (banjo) type. For the thesis banjo type axle

hosing is considered. Before final assembly axle housing has to be checked for porosities,

therefore it is subjected to leak testing a non-destructive testing method, during which a

pressure is imposed on surface of axle housing, we need to study the effect this pressure

on axle housing. This will be studied in coming chapters. This chapter also explains

organization of thesis.

CHAPTER – 2

LITERATURE REVIEW

CHAPTER – 2

LITERATURE REVIEW2.1. Introduction

Systematic review of the literature concerned to static and modal analysis of rear

axle housing has been presented in this chapter. The nomenclature used by various authors

in their original work has been retained in this chapter as such.

Stress analysis gives prior idea of the structure for optimum results and modal

analysis is important in machines where there is likely to be cyclic out of balance forces,

such as in rotating machinery. Modal analysis is carried out to find the natural frequency

of the system to avoid resonance conditions in the operations. The works reported in

literature are discussed below

(Hong Su, Ph.D. 2000) Have presented “Automotive CAE Durability Analysis Using

Random Vibration Approach”, and concluded that the frequency domain method can

improve our understanding of system dynamic behaviors, in terms of frequency

characteristics of both structures and loads, and their couplings.

(Ji-xin Wang, Guo-qiang Wang, Shi-kui Luo, Dec-heng Zhou 2002) Have presented

“Static and Dynamic Strength Analysis on Rear Axle of Small Payload Off-highway

vehicles” and they concluded that FEA helps to avoid expensive and time-consuming

development loops and also allow the number of high-cost test carriers to be substantially

reduced,

(Badiola, Virginia, Pintor, Jesús María, Gainza, Gorka 2004) Have presented “Axle

Housing And Unitize Bearing Pack Set Modal Characterization” and concluded that error

obtained between FEA and experimental modal analysis is acceptable.

(Yuejun E. Lee, Sree Sreedhar, D. Marla and C. Pawlicki Visteon Corporation 2006) have

presented “Automotive Axle Simulation and Correlation” and good correlation results

have been achieved.

2.2. Scope of the Present Work

In the literature, it has been observed that, the static strength and dynamic

characteristics of rear axle are analyzed typical load cases. According to the analytical

results, the weak locations of rear axle are obtained and the modified design has been

determined. Expensive and time-consuming development loops can be avoided using CAE

package and the design period is shortened. In the present work, rear axle housing is

statically tested for two cases with uniform housing thickness of 4.5mm and housing with

a grove at center of 2.5mm thickness through out length.

2.3. Methodology

The model is built using Pro/ENGINEER modeling software.

The model is imported to Hypermesh for meshing.

Meshing of the structure using quad elements for proper quality control.

Exporting the model into Patran (pre processor)/ Nastran for further analysis

Structure is analyzed for full thickness of 4.5 mm

The model is analyzed for 2.5 mm thickness along the small groove region.

The model is tested for Natural frequencies to avoid resonant conditions.

Summary In this chapter literature survey has been carried out, which concludes that

FEA helps to avoid expensive and time consuming loops and frequency domain method

improve our understanding of system dynamic behavior. Also scope and methodology of

work is presented.

CHAPTER – 3

STRUCTURAL VIBRATION

CHAPTER – 3

STRUCTURAL VIBRATION

3.1. Introduction

This chapter introduces Vibration, Resonance, and mode Shapes. Also introduces

the application of frequency analysis.

3.2. Vibration, Resonance and Mode Shapes

Structural vibration problems present a major hazard and design limitation for a very

wide range of engineering products. On the other hand, in a number of structures,

structural integrity is of paramount concern, and a thorough and precise knowledge of the

dynamic characteristics is essential. There is also a wider set of components or assemblies

for which vibration is directly related to performance, either by virtue of causing

temporary malfunction or by creating disturbance, discomfort or noise. Therefore, it is

important that the vibration levels encountered in service or operation be anticipated and

brought under satisfactory control. A comprehensive study of the vibration phenomena

includes determining the nature and extent of vibration response levels and verifying

theoretical models and predictions. A significant amount of applied technology pertaining

to vehicle dynamics has emerged over the last 20 years or so. The advent of finite element

analysis as a tool to study vehicle’s vibration and dynamic aspect has further accelerated

growth in this field.

Vibration is the study of the repetitive motion of objects relative to a stationary

frame of reference or nominal position (usually equilibrium). Vibration is evident

everywhere and in many cases greatly affects the nature of engineering designs. The

vibration properties of engineering devices are often limiting factors in their performance.

Vibration can be harmful and should be avoided, or it can be extremely useful and desired.

In either case, knowledge about vibration- how to analyze, measure and control is useful.

A comprehensive understanding of structural dynamics is essential to the design and

development of new structures, and solving noise and vibration problems on existing

structures. Modal analysis is an efficient tool for describing, under-standing, and modeling

structural behavior. The study of modal analysis is an excellent means of attaining a solid

understanding of structural dynamics.

Vibration usually becomes a concern when its amplitude grows large enough to

cause either excessive stress, or if it disturbs the people in, on or near the vibrating

object(s). As far as most structures are concerned, vibration will disturb the people around

the structure long before stress becomes an issue. There are many items of equipment

(balances, microscopes, cameras, transmission equipment etc.) that are very sensitive to

vibration.

Modal analysis is important in machines where there is likely to be cyclic out of

balance forces, such as in rotating machinery (engines electric & pneumatic motors,

generators, industrial equipment, etc.) and fluid flow applications (due to alternating

vortex shedding). The chief aim of any vibration analysis is to ensure that the system is not

subjected to dangerous resonant condition during the range of operation. A point to note is

that although the response of the system is time dependent, any excitation will be

harmonic and the solution may be obtained using the eigenvalue approach. It is important

to note that many applications fall in a category beyond this range, and full dynamic

analysis are required.

3.3. Formal Approach

If the system is given some initial disturbance, then it will vibrate at some frequency

known as its natural frequency. The natural frequency of a system is defined as the

frequency at which the system oscillates if the forcing function is identically zero. If

harmonic loading is applied, the solution becomes transient in nature, but modal analysis

can still be carried out for systems.

It may be recalled that the square of the natural frequency is referred to as an

eigenvalue. For a single mass-spring system, there is one eigenvalue, for distributed mass

systems (all practical applications), an infinite number of eigenvalues exist. The lowest

natural frequency, usually referred to as the fundamental frequency, has the lowest

potential or strain energy, and hence the reason why it is often regarded as the ‘lazy

mode’.

The fundamental frequency is usually the one of most interest to design engineers, as

most systems are designed to operate below it. Oftentimes, an operating frequency is

higher than the fundamental, hence as the equipment speeds up or slows down; it

experiences a momentary ‘shudder’ period as it passes through the resonance zone. There

is a corresponding mode shape which describes the displacement of the system due to the

vibration.

Eigenvalues are otherwise known as latent roots and characteristic values, the square

root of the eigenvalue is known as a natural frequency or resonant frequency. There are

also a number of terms used to describe mode shapes, they are also known as eigenvectors,

normal modes, characteristic vectors or latent vectors. The first five modes of vibration for

an aerofoil are given below.

Figure No. 3.1 Different Mode Shapes

3.4. Application of Frequency Analysis

In many practical problems the natural frequencies and mode shapes are required.

Designers use modal analysis to determine if there are any natural frequencies within the

range of operation. Alternatively, measured mode shapes and natural frequencies of a

structure can be compared with those predicted by FEA in a condition monitoring program

to verify structural integrity.

There are also situations where the response of the structure to a particular forcing

excitation is required. This is usually found using a technique known as modal

superposition. The overall response is described in terms of a sum of modal responses,

with the contribution of a particular mode given by the proximity of the forcing frequency

to the natural frequency and the amount of damping present in the system. The response is

dominated by modes close to the excitation frequency and therefore the modal series is

often truncated to reduce computation. Modal superposition methods can only be applied

in application with a harmonic excitation; otherwise the response becomes non-linear and

cannot be solved using the eigenvalue extraction approach.

The results from a forced harmonic analysis can be used to determine whether the

displacement of a particular structure is within acceptable limits. By calculating the stress

induced by the vibration it is also possible to predict the fatigue life of a particular

component.

Frequency based analysis perform eigenvalue extraction to calculate the natural

frequencies and corresponding mode shapes of a ‘free system’ (i.e. with no time dependent

loads applied).

Modal-dynamic analysis is transient in nature. They give the response for the model

as a function of time where a cyclic (sinusoidal) load is applied to the structure. Modal-

dynamic analysis is also referred to as forced harmonic response analysis. Complex

displacements and phase angles are evaluated and deflection & stresses may be calculated

at specific times. This analysis type is formulated on the principle of modal superposition,

and so a natural frequency analysis must be carried out first. The modal amplitudes are

integrated through time & the response is subsequently evaluated. This analysis solution

must be linear in nature (in time domain), as superposition & eigenvalue extraction

techniques cannot be applied to non-linear time domain applications.

Summary In chapter discussion about vibration, resonance and mode shapes is done. After

discussion it concludes that vibration becomes a concern when its amplitude grows large

to cause stress.

CHAPTER – 4

FINITE ELEMENT METHOD

CHAPTER – 4

FINITE ELEMENT METHOD This chapter gives brief introduction of Finite Element method and Msc Nastran a FEM software package

4.1. Introduction

The digital computer has exerted a most profound impact on the engineering and

scientific communities. The finite element method implemented on a computer in the form

of general-purpose program provides a broad foundation for engineering analysis. To

heighten the understanding of the behavior of the structure or a machine component, the

analyst has at his disposal three standard tools,

1. Analytical methods.

2. Numerical methods.

3. Experimental techniques.

Analytical methods provide quick and close form of solutions, but they treat only

simple geometries and capture only the idealized structural theory. Using the experimental

techniques, representative or full-scale models can be tested. Experimentation is costly,

however both in terms of the test facilities the model instrumentation and the actual test.

Relative to analytical methods numerical methods require very few restrictive

assumptions and can treat complex geometries. They are far cost effective than

experimental techniques. The current interest in the engineering community for

development and application of computational tools based on numerical methods is

thereby justified. The most versatile numerical method in the hands of engineers is Finite

element method (FEM).

4.2. Basic Concept

It is appropriate to give here an outline of the widely used finite element method

for the analysis of solids and structures. The structure or machine component under

consideration is discretised as assemblage of finite elements of different types (one, two,

three dimensional, beam, plate and shell elements) of different shapes (triangular,

quadrilateral, tetrahedron, pentahedral etc) and orders (linear, quadratic, cubic etc). These

elements are interconnected at their nodes. The finite element model is analyzed using the

basic rules of solid and structural mechanics, namely equilibrium of forces and

compatibility of displacements. The displacement field within each element is

approximated in terms of the nodal displacements using interpolation functions known as

shape functions. Using the assumed displacement function the strain field, the stress field

and finally the strain energy stored in the elements can all be expressed in terms of nodal

displacements. The total potential energy of the body is calculated as the sum of the strain

energy of the elements plus work potential of the externally applied loads. Minimization

of the potential energy function with respect to the nodal displacement results in a system

of algebraic equations. Solutions to these equations provide the nodal displacement. The

strains and stresses at any point within each element are then calculated using the now

known displacements. The above finite element procedure of artificially subdividing the

given continuum into convenient sub domains and assuming separate displacement

functions for each can be termed piecewise Rayleigh-Ritz procedure.

Consider an elastic rod of uniform cross sectional area A and length L, connecting

grid points one and two as shown in Figure. 4.1 The rod is subjected to an axial load and is

in static equilibrium.

Figure.4.1 Extensional elastic rod

X=0

1 L 2

F1 F2

U1 U2

Axial translations U1 and U2 are the only permitted displacements at grid points 1

and 2. Thus this element is said to have two degree of freedom. The goal is to find an

equation relating force to displacement for each degree of freedom. For static equilibrium,

summing forces in x- direction requires

Or

Assume that the rod changes length by an amount dL due to axial load, strain in the rod

can be related to displacement by the definition of simple strain.

Assume that the material of the rod is homogeneous, isotropic and linear. For such a

material axial strain is related to axial stress by

= E

By definition, axial (normal) stress is given by axial force divided by area. Thus,

(GRID1) =

(GRID2) =

From above equations following relation is obtained.

[F] = [K] [U]

Where,

[K] = Element stiffness matrix

[F] = Vector of forces

[U] = Vector of unknown displacements resulting from [F]

Each type of element has its own elemental stiffness matrix. Stiffness matrices for more

complex elements (general beams, plates and solids) are determined using procedures

based on energy principles.

4.3. Need for Finite Element Method

Design and analysis

Computer simulations

FEM/FEA is the most widely applied computer simulation method in engineering

Closely integrated with CAD/CAM applications

4.4. FEM in Structural Analysis

Procedures

Divide structure into pieces (elements with nodes)

Describe the behavior of the physical quantities on each element

Connect (assemble) the elements at the nodes to form an approximate system of

equations for the whole structure

Solve the system of equations involving unknown quantities at the nodes (e.g.,

displacements)

Calculate desired quantities (e.g., strains and stresses) at selected elements

Computer Implementations

Preprocessing (build FE model, loads and constraints)

FEA solver (assemble and solve the system of equations)

Post processing (sort and display the results)

4.5. Applications of FEM in Engineering

Mechanical/Aerospace/Civil/Automobile Engineering

Structure analysis (static/dynamic, linear/nonlinear)

Thermal/fluid flows

Electromagnetic

Geomechanics

Biomechanics

4.5.1. Available Commercial FEM Software Packages

ANSYS (General purpose, PC and workstations)

SDRC/I-DEAS (Complete CAD/CAM/CAE package)

NASTRAN (General purpose FEA on mainframes)

ABAQUS (Nonlinear and dynamic analyses)

COSMOS (General purpose FEA)

ALGOR (PC and workstations)

PATRAN (Pre/Post Processor)

HyperMesh (Pre/Post Processor)

Dyna-3D (Crash/impact analysis)

4.6. Finite Element Solver MSC/Nastran

4.6.1 Introduction

MSC/NASTRAN is the industry’s leading general-purpose finite element computer

program. MSC/NASTRAN has proven its accuracy and effectiveness over and over. It has

remained the leading FEA program by constantly evolving to take advantage of the latest

analytical capabilities and algorithms for structural analysis.

MSC/NASTRAN offers a wide variety of analysis types, including linear static,

normal modes, buckling, heat transfer, dynamics, frequency response, transient response,

random response, response spectrum analysis, and aero-elasticity. Virtually any material

type can be modeled, including composites and hyper elastic materials.

MSC/NASTRAN is written primarily in FORTRAN and has over a million lines of

code. MSC/NASTRAN is composed of a large number of building blocks called modules.

A module is a collection of FORTRAN subroutines designed to perform a specific task

such as, processing model geometry, assembling matrices, applying constraints, solving

matrices, calculating output quantities, conversing with the database, printing the solution,

and so on. An internal language called the Direct Matrix Abstraction Program (DMAP)

controls the modules.

4.7. Finite Element Model using NASTRAN

An overview of the various categories of information needed to create a finite

element model is as follows

:

4.7.1. Coordinate System

MSC/NASTRAN has a built-in rectangular Cartesian system called the basic

coordinate system, also called the default coordinate system

4.7.2. Model Geometry

Model geometry is defined in MSC/NASTRAN with grid points. A grid point is a

point on or in the structural continuum to which finite elements are attached. A simple

model may have only a handful of grid points; a complex model may have many tens or

thousands. The structure's grid points displace with the loaded structure. Each grid point

of the structural model has six possible components of displacement: three translations (in

the x-, y-, or z-directions) and three rotations (about the x-, y-, or z-axes). These

components of displacement are called degrees of freedom (DOFs).

4.7.3. Finite Elements

Once the geometry of the structural model has been established, the grid points are

connected by finite elements. A thorough understanding of the nature of the structure is

required to properly choose the type and quantity of elements-no finite element program

can independently decide the above factors.

MSC/NASTRAN has an extensive library of finite elements covering a wide range of

behavior. Overview of some of the elements is given below:

Line elements

Beam element (CBEAM): The beam element is defined with a CBEAM entry and

its properties are defined with a PBEAM or PBCOMP entry. The beam element

includes extension, torsion, bending in two perpendicular planes and the associated

shears.

Bar element (CBAR): The bar element is defined with a CBAR entry and its

properties are defined with a PBAR entry. The bar element is one-dimensional

bending element which is prismatic, and for which the elastic axis, gravity axis and

shear center all coincide.

Bend element (CBEND): The bend element is defined with a CBEND entry and its

properties are defined with a PBEND entry. The bend element is a one-

dimensional bending element with a constant radius of curvature. The bend

element may be used to analyze either curved beams or pipe elbows. The bend

element includes extension, torsion, bending in two perpendicular planes, and the

associated transverse shear.

Rod element (CROD, CONROD, and CTUBE): The rod element is defined with a

CROD entry and its properties with a PROD entry. The rod element includes

extensional and torsional

Properties. The CONROD entry is an alternate form that includes both the

connection and property information on a single entry. The tube element is defined

with CTUBE entry, and its properties with a PTUBE entry.

Surface elements

Shear panel element (CSHEAR): The shear panel element is defined with a

CSHEAR entry and its properties with a PSHEAR entry. A shear panel is a two

dimensional structural element that resists the action of tangential forces applied to

its edges, and the action of normal forces if effectiveness factors are used on the

alternate form of the PSHEAR bulk data entry.

Shell element (CTRIA, CQUAD): MSC/NASTRAN includes two different shapes

of isoparametric shell elements (triangular and quadrilateral) and two different

stress systems (membrane and bending). There are in all a total six different forms

of shell elements that are defined by connection entries as follows:

CTRIA3: isotropic triangular element with optional coupling of bending and

membrane stiffness.

CTRIA6: isotropic triangular element with optional coupling of bending and

membrane stiffness and optional mid-side nodes.

CTRIAR: isotropic triangular element with no coupling of bending and membrane

stiffness the membrane stiffness formulation includes rotation about the normal to

the plane of the element.

CQUAD4: isotropic quadrilateral element with optional coupling of bending and

membrane stiffness.

CQUAD8: isotropic quadrilateral element with optional coupling of bending and

membrane stiffness and optional mid-side nodes.

CQUADR: isotropic quadrilateral element with no coupling of bending and

membrane stiffness the membrane stiffness formulation includes rotation about the

normal to the plane of the element.

The properties for the above elements are defined on the PSHELL or PCOMP

entry. Anisotropic material may be specified for all shell elements. Transverse shear

flexibility may be included for all bending elements on an optional basis.

Conical shell element (RINGAX): The properties of the conical shell elements are

assumed to be symmetrical with respect to the axis of the shell. The conical shell

element cannot be combined with other types of elements. The geometry of a

problem using the conical shell element is described with RINGAX entries instead

of GRID entries.

Solid elements

MSC/NASTRAN includes three different solid polyhedron elements,

which are defined on the following bulk data entries.

CTETRA: Four sided solid element with 4 to 10 grid points.

CPENTA: Five sided solid element with 6 to 15 grid points.

CHEXA: Six sided solid element with 8 to 20 grid points.

4.7.4. Loads

MSC/NASTRAN is capable of simulating a variety of loads; some of them being:

Concentrated forces and moments.

Distributed loads on bars and beams.

Pressure loads on plate and solid surfaces.

Gravity loads-for example, the response of a structure to its own weight.

Loads due to acceleration.

Enforced displacements.

4.7.5. Boundary Conditions

Structures respond to loads by developing reactions at their point or points of

constraint. In most cases, boundary conditions are modeled in MSC/NASTRAN by

constraining appropriate degrees of freedom to zero displacement.

4.7.6 Material Properties

MSC/NASTRAN can model a wide range of material properties. The material

property definition entries are used to define the properties for each of the materials used

in the structural model.

The MAT1 entry is used to define the properties for isotropic materials and may be

referenced by any of the structural elements.

The MAT2 entry is used to define the properties of anisotropic materials for

triangular and quadrilateral membrane and bending elements. The MAT2 entry

specifies the relationship between the inplane stresses and strains. It may also be

used for anisotropic transverse shear.

The MAT3 entry is used to define the properties for orthotropic materials used in

the modeling of axisymmetric shells. This entry may only be referenced by

CTRIAX6 entries.

The MAT8 entry is used to define the properties of orthotropic materials used in

the modeling of quadrilateral and triangular shell elements for composite

structures.

The MAT9 entry is used to define the properties of anisotropic materials for the

CHEXA, CPENTA, and CTETRA elements.

4.8 MSC/NASTRAN input file

The MSC/NASTRAN input file may be created by using either a finite

element pre-processor or by inputting manually the required data. The input file consists

of five distinct sections; namely

NASTRAN Statement …… Optional

File Management Statements …… Optional

Executive Control Statements …… Required Section

CEND …… Required Delimiter

Case Control Commands …… Required Section

BEGIN BULK …… Required Delimiter

Bulk Data Entries …… Required Section

ENDDATA …… Required Delimiter

4.8.1 NASTRAN Statement

The NASTRAN statement is optional and is used to modify certain operational

parameters (also called system cells). Examples include aspects of working memory,

data-block size, data-block parameters, machine specific issues, numerical methods, etc.

The NASTRAN statement is not needed in most runs.

4.8.2. File Management Section

The File Management Section (FMS) is also optional, it is used primarily to attach

or initialize MSC/NASTRAN databases and FORTRAN files.

4.8.3. Executive Control Section

The primary function of this section is to specify the type of analysis to be

performed.

4.8.4 Case Control Section

Entries in the Case Control Section are called commands. The Case Control

Section is used to specify and control the type of analysis output required.

Eg. DISPLACEMENT = ALL, SPCFORCE = n, STRESS = NONE

Case Control commands also manage sets of Bulk Data input, define analysis sub cases

and select loads and boundary conditions.

4.8.5 Bulk Data Section

Bulk Data entries contain everything required to describe the finite element model-

geometry, coordinate systems, finite elements, element properties, loads, boundary

conditions, and material properties. The last entry in this section is the ENDDATA

command.

4.9. MSC/NASTRAN Output Files

The various files created by MSC/NASTRAN on the submission of an input file

(*.dat) upon successful execution of the job are:

*. BALL Contains permanent data for database runs.

*. F04 Contains database file information and a module execution summary.

*. F06 Contains the MSC/NASTRAN analysis results.

*. LOG Contains system information and system error messages.

MASTER and DBALL files can be automatically deleted (scratched) upon completion of

the run by adding the statement SCR=YES to the execution command

Conclusion In this chapter a brief introduction to Finite Element Model is presented. Relative to

analytical methods numerical methods require few assumptions can treat complex

geometries. They are cost effective than experimental techniques. The most versatile

numerical method is Finite element method. Implementation of Finite Element Method for

structural analysis is possible due to emergence of fast computing technologies. MSC

Nastran is one of the available commercial Finite Element software and is accurate and

effective software. It offers wide variety of analysis types, including linear static, normal

modes, buckling, heat transfer, dynamics, frequency response, etc.

CHAPTER – 5

FINITE ELMENT MODEL5.1. Introduction

In chapter-4 we have studied about Finite Element Method and Solver Finite

Element MSC/Nastran. This chapter contains detailed model geometry, type of element

and number of nodes and elements used for analysis.

PRO/ENGINEER

Pro/ENGINEER (Pro/E for short) is a commercial CAD/CAM package that is

widely used in industry for CAD/CAM applications. It is one of the new generation

of systems that not only over a full 3-D solid modeller, in contrast to purely 2-D

and surface modellers, but also parametric functionality and full associativity. This

means that explicit relationships can be established between design variables and

changes can be made at any point in the modelling process and the whole model is

updated.

The method of constructing a model of an object is very similar to that followed in

the production of a physical component. For example the manufacture of the shaped

block in Figure 1 would start with the choice of construction environment, the

selection of a piece of stock material followed by a series of manufacturing processes,

e.g. milling, drilling, welding/sticking. Pro/E has direct analogues for most of these

operations as various types of FEATURES which can be combined to generate a

complete representation of a PART, Pro/E's terminology for a single component.

Features fall into three main categories, Construction, Sketched and Pick/Placed.

FEATURES

CONSTRUCTION FEATURES

These features are purely used as an aid to the construction of the part, a number

of various forms are available the most commonly used are the:

Csys Coordinate systems which aid in the orientation of additional features and the

assembly of the part in to subsequent assemblies. CSYS feature is normally

the _rst feature in a part de_nition and is used as the basis for the placement

of all subsequent features.

Datums These are an extension of the idea of construction lines as used on a

traditional drawing. The most used type is a DATUM PLANE which allows a

2-D reference plane to be de_ned in space. Additional forms include DATUM

AXES, DATUM POINTS and DATUM CURVES. It is normal to add three

DEFAULT datum planes, immediately after the initial coordinate system, to

e_ectively generate default x-y, x-z and y-z planes.

SKETCHED FEATURES

These features are so named because they all involve the use of the SKETCHER

mode within ProE, (see below for more details on its use). The main features that

use this functionality are:

Protrusion Using this feature material can be added to/removed from a part by sketching

a cross-section and then extruding/revolving/sweeping the section to produce

a 3-D solid/cut. A solid protrusion is normally the _rst non-constructional

feature in a part, and is used to produce the base solid entity of the part. In

the material removal mode the action is similar to a turning, saw or milling

cut.

Rib This allows the user to produce a thin rib or web. This is a limited version of

the protrusion function.

PICK & PLACE FEATURES

Pick and place features tend to refer to simple or standard operations, e.g. the

production of HOLES, ROUNDS and CHAMFERS. The action to produce the

required e_ect has been preprogrammed into ProE, thus requiring the user to

indicate the position of the operation on the existing model.

MODIFICATION OF FEATURESThe parametric nature of ProE means that the modi_cation of features is relatively

easy, individual features can be selected and the associated parameters/dimensions

6

changed. However, it should be noted that ProE produces a HISTORY based model

in which features can be dependant on one or more previous features for their

de_nition, e.g. a chamfer on an edge generated by a cut or protrusion. These

PARENT-CHILD dependencies mean that when a parent feature is modi_ed its

children are automatically revised to reect the changes. N.B. Care should be taken

not to remove references used by child features.

5.2. Model Geometry

The rear axle housing is modeled using PRO/ENGINEER modeling software.

All the major components are built using curves, surfaces and volumes. The model is

exported to hypermesh for meshing. The meshed model is taken to Patran for application

of boundary conditions and material properties. The structure has been analyzed for two

conditions of slot thickness and results are presented as follows.

Figure 5.1 shows built up model of PRO/ENGINEER software imported to

Patran. The member is having a total length of 1.2 m with cross sections. Most of the

structure is made of 4.5 mm thick sheet. In the dome region thickness is around 2mm. The

thickness is changed to around 6.5 at the axle ends. A Thin groove is provided in the

structure for alignment. But this region is the more stress concentration region.

Figure 5.2 shows Sectional view of the member. Differential with rear axle will be

housed in the above structure.

5.3. Finite Element Model

Figure 5.3 shows fine meshed with quad elements rear axle housing in Hypermesh.

A total of 21089 nodes and 20964 elements are used. The structure is divided into

components and meshed. The above colors shows components created for meshing of the

object. Shell elements are used for meshing of the object.

FIGURE NO.5.1 FRONT VIEW OF REAR AXLE HOUSING

Figure No. 5.2 Sectional view of Rear Axle Housing

.

Figure No. 5.3 Finite Element model of Rear Axle Housing

Conclusion This chapter defines the model geometry, and both front view and c/s view of model

with dimensions are presented. Also meshed model is shown with all information.

CHAPTER – 6

STATIC ANALYSIS

6.1. Introduction

In chapter-4 discussion regarding Finite Element Method and Finite Element

Solver MSC/Nastran is carried out and in chapter-5 details regarding Finite Element

model is presented. This chapter presents details regarding assumptions made, material

properties and boundary conditions applied for analysis

6.2. Assumptions

The load is distributed as uniformly distributed load

Torsion load effect on rear axle housing is neglected

Frictional effects in transferring the loads are neglected.

6.3. Material Properties

Properties of material used for the problem are given in table 6.1. The material used is

steel (c-45).

Table 6.1. Material Properties

Material Properties for C - 45

Modulus of Elasticity [N/mm2] 206*103

Poisson's Ratio 0.3

Density(kg/mm3) 7.8E-6

Yield Stress 353 Mpa

Permissible Stress 140 Mpa

6.4. Boundary conditions

Figure No.6.1 Rear Axle Housing with boundary conations

Figure 6.1 shows boundary conditions of the structure. The leak test load of 10000

N is converted to distributed load (pressure) and applied on he top surface bottom is

supported. The surface area of load applied is 1865 mm2 the red color shows applied

pressure boundary conditions.

Conclusion In this chapter information regarding assumptions made, material properties and

boundary conditions required for analysis is presented.

CHAPTER – 7

RESULTS AND DISCUSSION7.1. Introduction

The geometry of rear axle housing is modeled in chapter 5 and analyzed in chapter

6. The static and dynamic analysis is carried out with the boundary conditions of a load of

10000N is applied at the top surface and the bottom surface is constrained, for the rear

axle housing, static analysis is done for following cases.

1. Von-mises Stress analysis of rear axle housing with uniform thickness of 4.5mm.

2. Von-mises Stress analysis of rear axle housing with a groove of thickness 2.5mm

at the center through out length.

3. Deformation of rear axle housing with uniform thickness of 4.5mm.

4. Deformation of rear axle housing with a groove of thickness 2.5mm at the center

through out length.

5. Von-mises Stress analysis of dome.

6. Von-mises Stress analysis of axle ends.

7. Von-mises Stress analysis of loading region.

8. Von-mises Stress analysis of groove region with 4.5mm thickness.

9. Von-mises Stress analysis of groove region with 2.5mm thickness.

In chapter 8 discussions regarding dynamic analysis (modal analysis) is done to find

the natural frequency of the system to avoid resonance conditions in the operations.

7.2 Results and Discussion

Figure No. 7.1 Von-mises Stress distribution of axle housing with uniform thickness of

4.5 mm

Case 1. Von-mises Stress analysis of rear axle housing with uniform thickness of

4.5mm.

The von-mises stress is plotted for the rear axle housing subjected to load of 10000N

applied at top surface. The load (pressure) is distributed uniformly at the top surface with

area of 1865 mm2. The bottom surface is constrained.

The figure 7.1 shows the distribution of von-mises stress. From the figure 7.1 it is

clear that the von-mises stress is observed to be maximum near to the dome and top

surface. As we move to away from the dome the magnitude of the stress decreases to

wards the axle end and bottom surface.

The maximum stress of 27.8 Mpa is developed near dome region and a minimum

stress of -3.10 Mpa is developed at the axle ends as it is away from loading region. At the

dome region stress varies between 1.85 Mpa to -3.10 Mpa. The stress level around groove

region varies between 7.42 Mpa t0 5.56 Mpa.

Figure No. 7.2 Von-mises Stress distribution with a groove of thickness 2.5 mm

Case 2. Von-mises Stress analysis of rear axle housing with a groove of thickness

2.5mm at the center through out length.

The figure 7.2 shows the distribution of von-mises stress with a groove thickness of

2.5mm. The groove is made at the center of housing throughout the length for alignment

purpose. The analysis is made to find the effect of this groove on the structure.

The figure 7.2 shows the distribution of von-mises stress. From the figure 7.2 it is

clear that the von-mises stress is observed to be maximum near to the dome and top

surface. As we move to away from the dome the magnitude of the stress decreases to

wards the axle end and bottom surface.

The maximum stress of 28.1 Mpa is developed near dome region and a minimum

stress of -3.10 Mpa is developed at the axle ends.

On comparison of case 1 and case 2 a slight increase of stress from 27.8 Mpa to 28.1

Mpa is observed near the loading region. So effect of groove is almost negligible.

Figure No. 7.3 Deformation of rear axle housing with uniform thickness of 4.5 mm

Case 3. Deformation of rear axle housing with uniform thickness of 4.5 mm

The deformation result is plotted for the rear axle housing subjected to load of

10000N applied at top surface. The load (pressure) is distributed uniformly at the top

surface with area of 1865 mm2. The bottom surface is constrained.

The figure 7.3 shows the deformation results. From the figure 7.3 it is clear that the

deformation is observed to be maximum near to the dome and top surface. As we move to

away from the dome the magnitude of the deformation decreases to wards the axle ends.

Fig 7.3 shows maximum displacement of 0.0239 mm due to the applied leak test

load, and minimum displacement of -5.12*10-9 mm is observed near the axle ends. At the

axle end the deformation value is 0.00159 mm and at the dome region deformation varies

from 0.00637 mm to 0.00478 mm.

Figure No. 7.4 Deformation of rear axle housing with groove of thickness of 2.5 mm

Case 4. Deformation of rear axle housing with a groove of thickness 2.5 mm at the

center through out the length.

The figure 7.4 shows the deformation results with a groove thickness of 2.5mm.

The groove is made at the center of housing throughout the length for alignment purpose.

The analysis is made to find the effect of this groove on the structure.

From the figure 7.4 it is clear that the deformation is observed to be maximum near

to the dome and top surface. As we move to away from the dome the magnitude of the

deformation decreases to wards the axle ends.

Maximum deformation is 0.0243mm near the dome region, and minimum

deformation is -3.96*10-9 mm

On comparison of case 3 and case 4 a slight increase of deformation value from

0.0239 to 0.0243 is observed near the loading region. So effect of groove is almost

negligible.

Figure No. 7.5 Von-mises Stress distribution in the dome region

Figure No. 7.6 Von-mises Stress Distribution In The Axle Ends

Figure No. 7. 7 Von-mises Stress Distribution in the Load Region

Case 5, 6 and 7. Von-Mises Stress Distribution In Different Parts.

In this section von-mises stress distribution in different parts of housing are

discussed.

Figure 7.5 shows von-mises stress in the dome region dome provides housing for

differential unit. Maximum stress is around 7.62 Mpa which is observed between the

junctions to rear axle housing tapered region.

Figure 7.6 shows von-mises stress results for the end region. Here stress is very

small ( around 0.161 Mpa) since it is away from the loading region and also thickness is

more on this region.

Figure 7.7 Shows von-mises stress results in the loading region (top surface).

Maximum stress can be observed in the loading region which is around 27.8 Mpa.

Minimum stress can be observed at the end region toward axle end.

Figure No. 7.8 Vonmises Stress In The Groove Region With 4.5 mm Thickness

Figure No. 7.9 Von-mises Stress in the Notch Region with 2.5 mm Thickness

Case 8. Von-mises Stress analysis of groove region with 4.5mm thickness.

The von-mises stress is plotted for the groove region of rear axle housing with

thickness of 4.5 mm. The axle housing is subjected to load of 10000N applied at top

surface. The load (pressure) is distributed uniformly at the top surface with area of 1865

mm2. The bottom surface is constrained.

The analysis is carried out to study the effect of this region thickness on stress

generation will be considered. The result shows the structure is very safe as the stress is

with in the working range.

The stress in the groove region is shown in Fig 9.8. Since thickness is considered

equal to 4.5 mm not much stress is observed in this region. The max stress is 7.14 Mpa

and minimum stress is 0.0682 Mpa.

Case 9. Von-mises Stress analysis of groove region with 2.5mm thickness

The von-mises stress is plotted for the groove region of rear axle housing with

thickness of 2.5 mm, situated at the center and is spread over the length of housing as

shown in Figure 1.1.

The stress in the slot is region is shown in fig 9.9. Since thickness is small (2.5 mm), the

stresses are appreciable in this region. The problem has been carried out to find the effect

of this slot on the leak test condition of the structure. The maximum stress is 18 Mpa and

minimum stress is 0.0681 Mpa.

But the result shows the structure is very safe as the stress is with in the working range.

Comparing the results of case 8 and case 9 maximum stress increases from 7.14 Mpa to

18 Mpa, that means if the groove thickness changes from 4.5 mm to 2.5 mm the stress

increases by 60%.

The above analysis results shows structure is very safe even under lesser slot thickness,

as the developed stresses are well with in working range of stress. So problem is safe.

Summary

In this chapter results for different cases are plotted and discussion on each case is

made. Here axle housing is analyzed for two conditions first with uniform housing

thickness of 4.5 mm and next with a groove at the center of thickness 2.5 mm which

spread throughout length of housing. From analysis it is concluded that variation of stress

is negligible and within working range when groove is considered. If stress distribution in

groove region is considered stress level increases by 60% compared with uniform

thickness of housing but the stress is within working range.

CHAPTER – 8

MODAL ANALYSIS8.1. Introduction

In the previous chapter discussion has been made on static analysis, eligibility of the

static strength of rear axle cannot prove that it will never break. In reality, the rear axle

housing is loaded with kinds of stimulations, which result in breakages such as resonance,

fatigue etc. It is very significant to analyze dynamic characteristic for the chosen design of

rear axle housing. Therefore in this chapter Modal analysis of rear axle housing is

performed.

Normal modes analysis computes the natural frequencies and mode shapes of a structure.

The natural frequencies are the frequencies at which a structure will tend to vibrate if

subjected to a disturbance.

Modal analysis is carried out to find the natural frequency of the system to avoid

resonance conditions in the operations. Maximum frequency of operation in leak testing

(Ch 1.2) using the machine is 8 Hz. So the system should have above this value to avoid

resonance conditions. Normal mode analysis is carried out using Patran-Nastran software

after applying proper boundary conditions.

Msc-Nastran has been used as the solver to run FEA modal analysis. Geometrical models

are developed through PRO/ENGINEER and are exported as IGES files to

Hypermesh and then to Nastran. Since in modal analysis the mass of the model plays an

important role, it is necessary to work with the whole model. Due to this, several

simplifications have been made in the geometrical model in order to minimize the

computational cost, as for example eliminate fillet radius, draft angles, etc.

The frequencies for first five mode shapes are given in table 8.1

Table 8.1 – Natural frequencies of the system

Mode Natural Frequencies Hz

1 12.1

2 20.56

3 28.663

4 35.184

5 38.112

Table 8.1 shows natural frequency of the system is above the maximum applied

frequencies. So System will work properly with out any resonance nature.

8.2. Mode Shapes

Figure No. 8.1 Mode shape 1

Figure No. 8.2 Mode Shape 2

Figure No. 8.3 Mode Shape 3

Figure No. 8.4 Mode Shape 4

Figure No. 8.5 Mode Shape 5

Fig 8.1 shows 1st mode shape for the natural frequency. Mode shape is the

deformation of the structure under the particular natural frequency. The above mode

shows vertical natural of vibration with reference to its axis. The natural frequency for this

mode is 12.1Hz and maximum deformation is 0.614mm near to dome region.

Mode shape for the 2nd natural frequency is in Fig 8.2. This shows flexural mode of

vibration at this frequency. The natural frequency for this mode is 20.56Hz and maximum

deformation is 0.506mm at the edge of top surface.

Fig 8.3 mode shows twisted nature of the vibration at this frequency. Since here no

damping conditions are assumed for natural frequency estimation, the obtained

deformation values are useful to give nature of vibration at that frequency rather then

vibration amplitude. The natural frequency for this mode is 28.6Hz and maximum

deformation is 0.664mm.

Fig 8.4 shows torsional mode of the system at this frequency value of 35.184 Hz.

The nature of animation helps us to provide some kind of arrest to prevent the member to

move in that direction. The natural frequency for this mode is 35.184Hz and maximum

deformation is 1.09mm.

Fig 8.5 shows maximum amplitude of vibration at the dome section. All these mode

are tentative values rather then actual values. The natural frequency for this mode is

38.112 Hz and maximum deformation is 2.55 mm.

8.3. Validation of Results Modal analysis capacity of Nastran is demonstrated by a known example due to

complexity involved in estimating natural frequencies of live systems which are made of

many elements. Generally complex live systems are validated by experimental methods

using Tachometers (Single Reed and Multiple Reed). Here a problem of 5meter height

and 1X1 m cross section member with density equals to 7800 Kg/m3 and Young’s

Modulus=200Gpa. Natural frequency due to self weight can be estimated as

fn =( 1/2 ) sqrt(3K/M) [1]

Here K= AE/L for axial vibration

K=1x200x109/5 = 40x10 9 N/m

M= density*volume=7800x5x1x1=39000 Kg

fn = ( 1/2 )sqrt(3x40x109/39000) =279.19 Hz

Nastran Result

Figure No. 8.6 Natural Frequency And Mode Shape

Fig 8.6 shows Patran – Nastran results for the above said model data. Generally 3

dimensional results depend on number of elements. Here solution is 293.28 Hz against the

theoretical value of 279.19 Hz.

Error is around 5%.

((293.28-279.19)/279.19).=0.05

Summary

In this chapter different mode shapes are presented and discussed. From discussion

axle housing is safe during leak testing. Also result validation is carried out.

CHAPTER – 9

CONCLUSIONS

The Rear Axle Housing has been built using PRO/ENGINEER modeling software. The

model has been imported to Hypermesh in IGES format. The meshed model has been

exported to Patran for application of boundary conditions and execution. The problem has

been solved for two conditions of slot thickness. Initially problem is solved for 4.5 mm

uniform thickness to verify stress condition. Next the problem is solved with groove

region of thickness equal to 2.5. A Total leak testing load of 10000 N is applied as

distributed Pressure on the top surface of area 1865 mm2. The bottom of the structure is

supported for this load. For the two conditions of thickness of strip, the results are

presented for Von-mises, and Deformation. The results show marginal reduction in stress

and deformation results by varying the thickness along the strip. But the structure is safe in

both the cases from Von-mises. The modal is further tested for natural frequency

conditions to avoid the resonance conditions and results are presented. All the relevant

pictures are presented. A theoretical validation also carried out to demonstrate the ability

of Patran-Nastran software in solving the engineering problem with very near accuracy.

SCOPE FOR FURTHER WORK

The Structure can be further tested for dynamic conditions of loads

The strip thickness can be optimized for better results

The member can be tested for actual vertical loads from transmission

The usage composite members can be checked for better results and light weight

The support conditions can be varied and checked for better stress and deformation

results.

REFERENCES

1. Hong Su, (2000), “Automotive CAE Durability Analysis Using Random Vibration

Approach”, CAE Tools and Methods Group Advanced Technology Office, Visteon

Corporation.

2. Badiola, Virginia, Pintor, Jesús María, Gainza, Gorka “Axle Housing And Unitize

Bearing Pack Set Modal Characterisation” Dana Equipamientos S.A., España, Universidad

Pública de Navarra, Dpto. Ingeniería Mecánica, Energética y de Materiales, España,

Centro de Innovación Tecnológica de Automoción de Navarra (CITEAN), España-

F2004F461

3. Brian J. Schwarz & Mark H. Richardson” EXPERIMENTAL MODAL ANALYSIS”

Vibrant Technology, Inc. Jamestown, California 95327

4. Badiola, Virginia*, 2Pintor, Jesús María, 3Gainza, Gorka” AXLE HOUSING AND

UNITIZE BEARING PACK SET MODAL CHARACTERISATION”

1Dana Equipamientos S.A., España, 2Universidad Pública de Navarra, Dpto. Ingeniería

Mecánica, Energética y de Materiales, España, 3Centro de Innovación Tecnológica de

Automoción de Navarra (CITEAN), España

5. Yuejun E. Lee, Sree Sreedhar, D. Marla and C. Pawlicki Visteon Corporation

“Automotive Axle Simulation and Correlation”

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Science and Engineering, Jilin University, Nanling Campus, No. 142 Renmin Street,

Changchun 130025, PR China

7. Qiang Zhang, Jian-min Xue “Static and Dynamic Strength Analysis on Rear Axle of

Small Payload Off-highway Dump Trucks” Inner Mongolia North Hauler Joint Stock Co.

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