11667

Kingdom of Saudi Arabia

Ministry of Higher Education

Umm Al-Qura University

Faculty of Science

Department of Mathematics

Numerical Comparison of Methods for Solving the BVP of

Rotating Variable Thickness Disk

A Thesis Submitted in Partial Fulfillment of the Requirements for

the Degree of Master of Science in Applied Mathematics

Prepared by

Suzan Abdullah Eid Al-Ahmadi

Supervised by

Prof. Dr. Ashraf Mobarez Zenkour

Prof. of Applied Mathematics

Department of Mathematics

Faculty of Science

King AbdulAziz University

1432 A.H. - 2011 A.D.

i

CHAPTER ZERO: Introduction ……………………………….. 1

CHAPTER ONE: Basic Concepts …………………………….. 17

1.1 Cylindrical coordinates……………….. 17

1.2 Strain-displacement relations…………. 19

1.3 Equilibrium equations………………… 21

1.4 Hooke's law (stress-strain relations)….. 23

1.5 Navier-Stokes equations………………. 28

CHAPTER TWO: Boundary-Value Problems and Some Numerical

Solutions…………………………………… 30

2.1 Differential equations…………………. 30

2.2 Boundary-value problems…………….. 32

2.3 Numerical solutions............................... 34

2.3.1 Finite difference method……... 34

2.3.2 Modified Runge-Kutta's method 39

2.4 Least square method………………….. 43

2.5 Richardson's extrapolation method…... 50

CHAPTER THREE: Two Solutions for the BVP of a Rotating Variable-

Thickness Solid Disk…………………….. 55

3.1 Basic equations………………………. 55

3.2 Analytical solution…………………… 57

3.3 Finite difference method…………….. 62

3.4 Numerical examples and discussion…. 64

3.5 Conclusion ……………………………. 74

ii

CHAPTER FOUR: Elastic Stresses in Rotating Variable-Thickness

Annular Disks using Different Methods….. 75

4.1 Basic equations………………………... 75

4.2 Analytical solution…………………….. 84

4.3 Numerical solutions…………………… 85

4.3.1 Finite difference method………. 85

4.3.2 Modified Runge-Kutta's method 88

4.4 Numerical examples and discussion…... 90

4.4.1 Finite difference method………... 90

4.4.2 Modifed Runge-Kutta's method... 101

4.5 Conclusions…………………………… 118

General Conclusions……………………………………………….. 120

References………………………………………………………….. 122

Arabic Summary...…………………………………………………. د

1

The theoretical and experimental investigations on the rotating solid

and annular disks have been widespread attention due to the great practical

importance in mechanical engineering. Rotating disks have received a

great deal of attention because of their widely used in many mechanical

and electronic devices. They have extensive practical engineering

application such as in steam turbine (see Figure 0.1) which is a mechanical

device that extracts thermal energy from pressurized steam, and converts it

into rotary motion. About 80% of all electricity generation in the world is

by use of steam turbine.

Figure 0.1: A rotor of a modern steam turbine used in a power plant.

2

Gas turbine, Figure 0.2, also called a combustion turbine, is a rotary

engine that extracts energy from a flow of combustion gas. It is used to

power aircraft, trains, ships, generators, and even tanks.

Figure 0.2: A typical axial-flow gas turbine turbojet.

Turbo generator (see Figure 0.3) is a turbine directly connected to an

electric generator for the generation of electric power. Large steam

powered turbo generators (steam turbine generators) provide the majority

of the world's electricity and they are used by steam powered turbo-electric

ships.

Figure 0.3: A turbo generator.

3

Flywheel (see Figure 0.4) is a mechanical device with a significant

moment of inertia used as a storage device for rotational energy. Recently,

flywheels have become the subject of extensive research as power storage

devices for uses in vehicles.

Figure 0.4: Flywheel stationary engine.

Figure 0.5: An automobile engine.

4

Internal combustion engine, Figure 0.5, is an engine in which the

combustion of a fuel occurs with an oxidizer in a combustion chamber. It

dominates as a power supply for cars, aircraft, and boats, from the smallest

to the biggest.

Turbojets, as given in Figure 0.6, are the oldest kind of general purpose

of jet engines. They are very common in medium rang cruise missiles, due

to their high exhaust speed, low frontal area and relative simplicity.

Figure 0.6: A cross-section of a turbojet engine.

Figure 0.7: Internal combustion piston engine.

Reciprocating engine (see Figure 0.7), also often known as a piston

engine, is a heat engine that uses one or more reciprocating pistons to

5

convert pressure into a rotating motion. It is used in some application such

as to drive many modern torpedoes or as pollution free motive power.

Centrifugal compressors, as given in Figure 0.8, sometimes referred to

radial compressors, are a special class of radial-flow work-absorbing turbo

machinery that include pumps, fans, blowers and compressors.

Figure 0.8: Jet engine cutaway showing the centrifugal compressor.

Figure 0.9: A disk brake on a car.

6

Disk brake, Figure 0.9, is a device for slowing or stopping rotation of a

wheel. Disk brakes were most popular on sports cars when they were first

introduced, since these vehicles are more demanding about brake

performance. Now, disks become the more common in most passenger

vehicles.

Gears (see Figure 0.10) are rotating machine part having cut teeth, or

cogs, which mesh with another toothed part in order to transmit torque.

Two or more gears working in tandem are called a transmission and can

produce a mechanical advantage through a gear ratio and thus may be

considered as a simple machine.

Figure 0.10: Helical gears.

Pumps, as given in Figure 0.11, are devices used to move fluids, such as

liquids, gases or slurries. Ship propeller (see Figure 0.12) is a type of fan

that transmits power by converting rotational motion into thrust.

The problems of rotating solid and annular disks have been performed

under various interesting assumptions and the topic can be easily found in

most of the standard elasticity books. The problems of rotating variable-

7

thickness annular and solid disks are rare in the literature. Most of the

research works are concentrated on the analytical solutions of rotating

isotropic disks with simple cross-section geometries of uniform thickness

and especially variable thickness. The material density of these rotating

disks is taken to be either constant or specifically variable. The analytical

elasticity solutions of such rotating disks are available in many books of

elasticity.

Figure 0.11: A large, electrically driven pump.

Figure 0.12: Ship propeller.

8

Recent studies indicate that the stresses in variable-thickness rotating

solid and annular disks are much lower than those in constant thickness

disks at the same angular velocity. Hence, for a better utilization of the

material, it is appropriate to allow variation in the thickness of the disk.

Investigations pertaining to the behavior of rotating disks within elastic

zone can be traced back to Thompson [1], wherein he provided a numerical

approach to the turbine disk. Manson [2] has presented a finite difference

solution of the equilibrium and compatibility equations for elastic stresses

in a symmetrical disk, and subsequently Manson [3] has reported a

simplified method for determining the disk profile under the combination

of centrifugal and thermal loading. Leopold [4] has solved similar

problems for disks with variable thickness using semi-graphical method.

Theoretical studies pertaining to elastic and elastic-plastic stress analysis of

rotating disk made of isotropic material were also available in the literature

(Timoshenko and Goodier [5]; Reid [6]; Reddy and Srinath [7]; Rees [8];

Ugural and Fenster [9]; Srinath [10]). Sherbourne and Murthy [11] have

applied dynamic relaxation technique effectively to analyze non-linear

material behavior coupled with variable geometry in rotating disks.

Laszlo [12] has first reported theoretical analysis of rotating disks in

plastic regime or in region of permanent deformation. Millenson and

Manson [13] have analyzed the stress distribution in rotating disk under

conditions of plastic flow and creep. Lee [14] has presented an exact

solution based on deformation theory of plasticity with axial symmetry in

strain hardening range and subsequently reported a partly linearized

solution of plastic deformation of rotating disk considering finite strain.

Manson [15] has also presented solutions for disks of work hardening

materials based on von-Mises theory and deformation theory of plasticity.

Koiter [16] has obtained a closed form solution for a compressible elastic-

9

plastic tube of non-hardening materials made the first original application

of incremental strain theory of plasticity. In a later work, the generalization

of the theory to a singular yield surface was achieved by allowing the yield

condition to be specified by several yield functions (see Koiter [17]). Study

of non-linear behavior found resurgence in 1980s when Gamer [18] has

reported that the stress distribution in a rotating solid disk obtained by

several researchers and also given in many textbooks on plasticity, is not

meaningful since the corresponding displacement field is incompatible

with the necessary continuity requirements at the elastic-plastic interface.

Later, considering the fact that the plastic core of the disk consists of

two parts with different forms of yield condition. Gamer [18] has obtained

a consistent analytical solution for the elastic-plastic response of a rotating

uniform thickness solid disk using Tresca's yield condition and its

associated flow rule. Gamer [19, 20, 21] has also studied the analytical

solutions of such disks with a linear strain-hardening material behavior

using same yield condition. Güven [22, 23] has extended this work to

annular and solid disks of variable-thickness and variable-density, obtained

their analytical solutions using the same material behavior, yield condition,

and to fully plastic variable-thickness solid disks with constant thickness in

the central portion (see also Güven [24]). You et al. [25] and You and

Zhang [26] have applied a polynomial stress-plastic strain relation to

obtain the approximate analytical solution for rotating solid disks of

uniform thickness with nonlinear strain hardening materials. Güven [27]

has considered an annular disk profile in exponential form and studied the

effect of application of external pressure analytically. Güven [28] has also

investigated the deformation of constant thickness rotating annular disks

with rigid inclusion in the fully plastic state. He has obtained an analytical

solution by using Tresca's yield condition and assuming linear strain

hardening.

11

Rees [29] has studied elastic-plastic deformation of rotating solid and

annular uniform-thickness disks made of elastic-perfectly plastic material.

Eraslan [30] has extended the work of Rees [29] to variable-thickness solid

disks made of elastic linearly hardening materials and studied inelastic

stress state of solid disks with exponential thickness variation using both

Tresca's and von-Mises criterion. Eraslan [31] has studied inelastic

deformations of constant and variable-thickness rotating annular disks with

rigid inclusion using Mises-yield criterion. Eraslan [32] has obtained exact

solutions to thermally induces axe-symmetric purely elastic stress

distributions in non-uniform heat-generating composite tubes. Eraslan [33]

has presented the analytical solution of elastic-plastic rotating annular

disks with variable-thickness in a parabolic form. Eraslan and Argeso [34]

have calculated the elastic and plastic limit angular speed for rotating disks

of variable-thickness in power function form.

Eraslan and Orcan [35] have studied the elastic-plastic deformation of

variable-thickness solid disks having concave profiles. They have

presented an analytical solution for elastic and plastic deformation of

linearly hardening rotating solid disk of variable thickness in an

exponential form. Eraslan and Orcan [36] have also obtained an analytical

solution for elastic-plastic deformation of a linearly hardening rotating

solid disk of variable thickness in a power function form. In another paper,

Eraslan [37] has presented an analytical solution for rotating disks with

elliptical thickness variation. Apatay and Eraslan [38] have presented

analytical solutions for elastic deformation of rotating solid and annular

disk with parabolically varying-thickness with free, radially constrained

and pressurized boundary conditions. Bhowmick et al. [39] have employed

an approximate solution to the rotating disk behavior under externally

loaded condition. Eraslan et al. [40] have studied elasto-plastic

deformation of variable-thickness annular disks subjected to external

11

pressure based on von-Mises yield criterion, deformation theory of

plasticity and Swift's hardening law.

Farshad [41] has investigated the influence of bi-modulus material

behavior on tress field of a rotating solid disk. Parmaksizoğlu and Güven

[42] have considered a rotating annular disk having non-homogeneous bi-

modulus material behavior. Güven and Parmaksizoğlu [43] have

considered a rotating anisotropic annular disk of variable-thickness and

variable-density. Tutuncu [44] has investigated the influence of anisotropy

on stresses in rotating disks and used a laminated plate theory. Güven et al.

[45] have investigated an elastic-plastic rotating annular disk problem.

Zenkour [46] has obtained the thermo-elastic solution for variable-

thickness annular disks. Zenkour [47] has presented accurate elastic

solutions for the rotating variable-thickness and/or uniform-thickness

orthotropic circular cylinders containing a uniform-thickness solid core of

rigid or homogeneously isotropic material. In Zenkour [48], the closed

form solutions for the rotating exponentially graded annular disks

subjected to various boundary conditions are obtained. Zenkour and Allam

[49] have developed analytical solution for the analysis of deformation and

stresses in elastic rotating visco-elastic solid and annular disks with

arbitrary cross-sections of continuously variable-thickness. Allam et al [50]

have studied stresses and deformation of a rotating circular disk carrying a

steady current and coated with a coaxial thick visco-elastic material under

the influence of a steady current.

Reddy et al. [51] have studied axisymmetric bending and stretching of

functionally graded (FG) solid and annular circular plates. Bayat et al. [52]

have developed a new set of equilibrium equations with small and large

deflections in FG rotating disk with axisymmetric bending and steady-state

thermal loading.

12

Lee et al. [53] have obtained an elastic solution for pure bending

problem of simply supported transversely isotropic circular plates with

elastic compliance coefficients being arbitrary functions of the thickness

coordinate. Chen et al. [54] have obtained a three-dimensional analytical

solution for transversely isotropic FG disk rotating at a constant angular

velocity. Chen and Chen [55] have derived three-dimensional analytical

solution of the elastic equations for transversely isotropic FG rotating plate

by means of direct displacement method. Li et al. [56] have used stress

function method and presented a set of elasticity solutions for the

axisymmetric problem of transversely isotropic simply supported and

clamped edge FG circular plates subjected to a transverse load.

Ruhi et al. [57] have presented a semi-analytical solution for thick

walled finitely-long cylinders made of functionally graded material (FGM)

under thermo-mechanical load. Fukui and Yamanaka [58] have studied the

effects of the gradation of components on the strength and deformation of

thick walled FG tubes under mechanical load such as internal pressure with

plane strain conditions. Fukui et al. [59] have extended their previous work

by considering a thick-walled FG tube under uniform thermal loading.

They investigated the effect of graded components on residual stresses.

Durodola and Attia [60, 61] have presented a finite element analysis for

FG rotating disks using commercial software package. Kordkheili and

Naghdabadi [62] have presented a semi-analytical thermo elastic solution

for hollow and solid rotating axisymmetric disks made of FGMs under

plane stress condition. The results were compared with those of Durodola

and Attia [60, 61] under the central loading.

Jahed and Sherkatti [63] have applied the variable material properties

(VMP) method and obtained stresses for an inhomogeneous rotating disk

with variable-thickness under steady temperature field assuming the

13

material properties as field variables. Jahed and Shirazi [64] have

evaluated the temperature in a rotating disk during heating and cooling

using VMP method. Farshi et al. [65] have also used VMP method and

obtained optimal profile for an inhomogeneous non-uniform rotating disk.

Jahed et al. [66] have analyzed an inhomogeneous disk to model to achieve

minimum weight of the disk with variable-thickness. They have obtained

stresses in the rotating disk under a steady temperature field using the

VMP method.

Durodola and Adlington [67] have carried out a predicative assessment

of the effect of various forms of gradation of material properties on

deformation and stresses in rotating axisymmetric disks and rotors. Horgan

and Chan [68, 69] have investigated the pressured FGM hollow cylinder

and disk problems and the stress response of FGM isotropic liner elastic

rotating disk. Ha et al. [70] have calculated the stress and strength ratio

distributions of the rotating composite flywheel rotor of varying material

properties in the radial direction. Güven and Çelik [71] have investigated

the effects of material inhomogeneity on the transverse vibrations of the

rotating solid disk of constants-thickness.

Eraslan and Akis [72] have obtained closed-form solutions for FG

rotating solid shafts and disks by assuming that Young's modulus E is

either an exponential or a parabolic function of the radius r. You et al. [73]

have derived a closed-form solution for FG rotating disks subjected to a

uniform temperature. The temperature is change by taking Young's

modulus, the thermal expansion coefficient, and the mass density to vary

according to power-law functions of the radius. Vivio and Vullo [74, 75]

have studied stresses and strains in variable-thickness annular and solid

rotating elastic disks subjected to thermal loads and having a variable-

density along the radius. Zenkour [76, 77] has investigated the stress

14

distribution in rotating three-layer sandwich solid disks with face sheets

made of different isotropic materials and a FG core.

Zenkour [78] has obtained a thermoelastic solution for FG rotating

annular disks with uniform angular velocity and subjects to a steady-state

thermal load. In Bayat et al. [79] the theoretical formation for bending

analysis of FG rotating disks based on first order shear deformation theory

is presented. Bayat et al. [80] have presented a theoretical solution for

thermoelastic analysis of FG rotating disk with variable-thickness based on

first order shear deformation theory. Bayat et al. [81] have presented an

analysis of FG rotating disk with variable-thickness subjected to

centrifugal body and thermal loading. In Bayat et al. [82], an elastic

solution for axisymmetric rotating disks made of FGMs with variable-

thickness are presented.

As many rotating components in use have complex cross-sectional

geometries, they cannot be dealt with using the existing analytical

methods. Numerical methods, such as finite element method (Zienkiewicz

[83]), the boundary element method (Banerjee and Butterfield [84]) and

Runge-Kutta's algorithm (You et al. [85]; Hojjati and Hassani [86]; Hojjati

and Jafari [87]) can be have applied to cope with these rotating

components. However, as Sterner et al. [88] have pointed out these

numerical analyses usually require extensive computer resources, are

tedious to perform due to extensive meshing requirements and are

expensive, making them unsuitable for preliminary design type analysis.

Therefore, Sterner et al. [88] have developed a unified numerical method

for elastic analysis of rotating disks with general, arbitrary configuration

based on the repeated application of truncated Taylor's expansion. Hojjati

and Hassani [86] have used both finite element method and Runge-Kutta's

algorithm for stress-strain analysis of rotating discs with non-uniform

15

thickness and density. Hojjati and Jafari [89] have solved the rotating

annular disks with uniform and variable- thickness and densities using

homotopy perturbation method and Adomian's decomposition method.

Hojjati and Jafari [87] have presented the analytical solutions for the

elastic-plastic rotating annular disks of variable-thickness and density

using the homotopy perturbation method and Runge-Kutta's algorithm.

You et al. [85] have developed a unified numerical method for the analysis

of deformation and stresses in elastic-plastic rotating disks with arbitrary

cross-sections of continuously variable-thickness and arbitrarily variable-

density made of nonlinear strain-hardening materials. In a recent paper,

Zenkour and Mashat [90] have presented both analytical and numerical

solutions for the analysis of deformation and stresses in elastic rotating

disks with arbitrary cross-sections of continuously variable-thickness.

The aim of this thesis is to study and compare various analytical and

numerical solutions for the boundary-value problems (BVP) of rotating

solid and annular disks with variable-thickness.

The thesis consists of an introduction, four chapters, general

conclusion, references, English, and Arabic summaries.

In Chapter 1, we state the cylindrical coordinates and the basic field

equations such as the strain-displacement relations (Cauchy's relations),

equilibrium equations, the stress-strain relations (Hooke's law) and Navier-

Stokes equations.

In Chapter 2, we state some general forms of differential equations

and different kinds of the boundary-value problems (BVPs). Some

numerical solutions are also stated. The finite difference method and the

modified Runge-Kutta's method are both discussed. In addition, the least

square and Richardson’s extrapolation method are also presented.

16

In Chapter 3, we derived a unified governing equation from the basic

equations of the rotating variable-thickness solid disk and the proposed

stress-strain relationship. The analytical solution for rotating solid disk

with arbitrary cross-section of continuously variable-thickness is

presented. Next, finite difference method (FDM) is introduced to solve the

governing equation. A comparison between both analytical and numerical

solutions has been made.

In Chapter 4, two different annular disks for the radially varying

thickness annular disks are given. The numerical solutions such as finite

difference method (FDM) and modified Runge-Kutta's method (R-K)

solutions as well as the analytical solution are available for the first annular

disk while the analytical solution is not available for the second annular

disk. Both analytical and numerical results for radial displacement, stresses

and strains are obtained for the first annular disk of variable-thickness. The

accuracy of the present numerical solutions is discussed and their ability of

use for the second rotating variable-thickness annular disk is investigated.

Finally, the distributions of the radial displacement, stresses and strains are

presented and the appropriate comparisons and discussions are made at the

same angular velocity.

71

The solution for many problems in elasticity requires to use a

curvilinear cylindrical coordinates. It is therefore necessary to

have the field equations expressed in terms of such coordinate

system. The purpose of this chapter is to state the cylindrical

coordinates and the basic field equations in cylindrical

coordinates.

§

Many applications in elasticity theory involve domains that have curved

boundary surfaces, commonly including circular, cylindrical and spherical

surfaces. To formulate and develop solutions for such problems, it is

necessary to use curvilinear coordinate systems (see Martin [91]). We will

review one of the most common curvilinear systems the Cylindrical

coordinate. The Cylindrical coordinate system uses ),,( zr as shown in

Figure 1.1.

A cylindrical coordinate system is a three-dimensional coordinate

system that specifies point positions by the distance from a chosen

reference axis, the direction from the axis relative to a chosen reference

direction, and the distance from a chosen reference plane perpendicular to

the axis. The latter distance is given as a positive or negative number

depending on which side of the reference plane faces the point.

71

Figure 1.1: Cylindrical coordinate system.

The origin of the system is the point where all three coordinates can be

given as zero. This is the intersection between the reference plane and the

axis. The axis is variously called the cylindrical or longitudinal axis, to

differentiate it from the polar axis, which is the ray that lies in the reference

plane, starting at the origin and pointing in the reference direction. The

distance from the axis may be called the radial distance or radius, while the

angular coordinate is sometimes referred to as the angular position or as

the azimuth. The radius and the azimuth are together called the polar

coordinates, as they correspond to a two-dimensional polar coordinate

system in the plane through the point, parallel to the reference plane. The

third coordinate may be called the height or altitude (if the reference plane

is considered horizontal), longitudinal position, or axial position.

The three coordinates ),,( zr of a point P are defined as:

The radial distance r is the Euclidean distance from the z-axis to the

point P.

71

The azimuth is the angle between the reference direction on the

chosen plane and the line from the origin to the projection of P on

the plane.

The height z is the signed distance from the chosen plane to the point

P.

Relations between the Cartesian and Cylindrical systems are given by:

From Cylindrical to Cartesian

.

,sin

,cos

zz

ry

rx

(1.1.1)

From Cartesian to Cylindrical

.

,tan

,

1

22

zz

x

y

yxr

(1.1.2)

Cylindrical coordinates are used mostly for volume and surface area

analysis of revolved solids. Examples of resolved solids are cone sections,

hyperboloids of one sheet, sphere sections, elliptical and circular cylinders.

§

We now pursue the development of the strain-displacement relations

(Cauchy's relations) in cylindrical coordinates. Starting with the form

,)(2

1ε Tuu (1.2.1)

02

where ε is the strain matrix and u is the displacement gradient matrix

and Tu )( is its transpose. The displacement vector and strain tensor can

be expressed by

,zzrr eueueuu ,ε

zzzr

zr

rzrr

(1.2.2)

where ie are the unit basis vectors in the curvilinear system.

The derivative operation in cylindrical coordinates can be expressed by

,1

zr ez

ue

u

re

r

uu

zrz

rrrr ee

r

uee

r

uee

r

u

zz

rrr ee

u

ree

uu

reeu

u

r

111

.zzz

zrzr ee

z

uee

z

uee

z

u

(1.2.3)

Placing this result into the strain-displacement from, Eq. (1.2.3), gives the

desired relations in cylindrical coordinates. The individual scalar equations

are given by

,r

urr

,

1

uu

rr ,

z

uzz

,1

2

1

r

u

r

uu

rr

r

,1

2

1

zz

u

rz

u

.2

1

r

u

z

u zrzr (1.2.4)

07

§

As mentioned in the introduction, in order to solve many elasticity

problems, formulation must be done in curvilinear coordinates typically

using cylindrical or spherical systems. Thus by following similar methods

as used with strain-displacement relations, we now wish to develop

expressions for the equilibrium equations in curvilinear cylindrical

coordinates. By using a direct vector/matrix notation, the equilibrium

equations can be expressed as

.0σ F (1.3.1)

where jiij eeσ is the stress matrix or dyadic, and F is the body force

vector. The desired curvilinear expression can be obtained from Eq. (1.3.1)

by using the appropriate form for .σ

Figure 1.2: Stress components in cylindrical coordinates.

00

Cylindrical coordinates were originally presented in Figure 1.1. For

such a system, the stress components are defined on the differential

element shown in Figure 1.2, and thus the stress matrix is given by

.σ

zzzr

zr

rzrr

(1.3.2)

Now, the stress can be expressed in terms of traction components as

,σ zzrr eTeTeT (1.3.3)

where

.

,

,

zzzrzrz

zzrr

zrzrrrr

eeeT

eeeT

eeeT

(1.3.4)

The divergence operation in the equilibrium equations can be written as

,11

σz

TT

rT

rr

T zr

r

)(1

zrzrrrzrzr

rr eee

re

re

re

r

z

zrrr

r eeeeer

1

.zzz

rrz e

ze

ze

z

(1.3.5)

Combining this result into Eq. (1.3.1) gives the vector equilibrium equation

in cylindrical coordinates. The three scalar equations expressing

equilibrium in each coordinate direction then became

,0)(11

rr

rzrr Frzrr

02

,021

F

rzrrr

zr

.011

zrz

zzrz Frzrr

(1.3.6)

It is interesting to note that the equilibrium equations in cylindrical

coordinates contain additional terms not involving derivatives of stress

components. The appearance of these terms can be explained

mathematically because of the curvature of the space. However, a more

physical interpretation can be found by redeveloping these equations

through a simple force balance analysis on the appropriate differential

element. In general, Eqs. (1.3.6) look much more complicated when

compared to the Cartesian form. However, under particular conditions, the

curvilinear forms lead to an analytical solution that could not be reached by

using Cartesian coordinates.

§ ( )

In order to construct a general three dimensional constitutive law for

linear elastic materials, we assume that each stress component is linearly

related to each strain component

,222 161514131211 zrzrzrr CCCCCC

,222 262524232221 zrzrzr CCCCCC

,222 363534333231 zrzrzrz CCCCCC

,222 464544434241 zrzrzrr CCCCCC

,222 565554535251 zrzrzrz CCCCCC

,222 666564636261 zrzrzrzr CCCCCC (1.4.1)

02

where the coefficients ijC are material parameters and the factors of 2 arise

because of the symmetry of the strain. Note that this relation could be

expressed by writing the strains as a linear function of the stress

components. These relations can be cast into a matrix format as

.

2

2

2

............

..................

..................

..................

...............

.........

6661

21

161211

zr

z

r

z

r

zr

z

r

z

r

CC

C

CCC

(1.4.2)

The above relations can also be expressed in standard tensor notation by

writing

,klijklij C (1.4.3)

where ijklC is a fourth-order elasticity tensor whose components include

all the material parameters necessary to characterize the material. Based on

the symmetry of the stress and strain tensors, the elasticity tensor must

have the following properties:

.jiklijkl CC (1.4.4)

In general, the fourth-order tensor ijklC has 81 components. However, Eq.

(1.4.4) reduces the number of independent components to 36, and this

provides the required match with form of Eqs. (1.4.1) or (1.4.2). The

components of ijklC or equivalently ijC are called elastic moduli and have

units of stress (force/area). In order to continue further, we must address

the issues of material homogeneity and isotropy.

If the material is homogenous, the elastic behavior dose not vary

spatially, and thus all elastic moduli are constant. For this case, the

02

elasticity formulation is straightforward, leading to the development of

many analytical solutions to problems of engineering interest. A

homogenous assumption is an appropriate model for most structural

applications, and thus we primarily choose this particular case for

subsequent formulation and problem solution.

Similar to homogeneity, another fundamental material is isotropy. This

property has to do with differences in material moduli with respect to

orientation. For example, many materials including crystalline minerals,

wood, and fiber-reinforced composites have different elastic moduli in

different directions. Materials such as these are said to be anisotropic. Note

that for most real anisotropic materials there exist particular directions

where the properties are the same. These directions indicate material

symmetries. However, for many engineering materials (most structural

metals and many plastics), the orientation of crystalline and grain

microstructure is distributed randomly so that macroscopic elastic

properties are found to be essentially the same in all directions. Such

materials with complete symmetry are called isotropic. As expected, an

anisotropic model complicates the formulation and solution of problems.

The tensorial form of Eq. (1.4.3) provides a convenient way to establish

the desired isotropic stress-strain relations. If we assume isotropic

behavior, the elasticity tensor must be the same under rotating of the

coordinate system. The most general form that satisfies this isotropy

condition is given by

,jkiljlikklijijklC (1.4.5)

where , and are arbitrary constants.

Using the general form of Eq. (1.4.5) in stress-strain relation, Eq.

(1.4.3), gives

02

,2 ijijkkij (1.4.6)

where we have relabeled particular constants using and . The elastic

constant is called lamé's constant and is referred to as the shear

modulus or modulus of rigidity.

Eq. (1.4.6) can be written out in individual scalar equations as

,2)( rzrr

,2)( zr

,2)( zzrz

,2 rr

,2 zz

.2 zrzr (1.4.7)

Relations given in Eqs. (1.4.6) or (1.4.7) are called the generalized

Hooke's law of linear isotropic elastic solids. They are named after Robert

Hooke who in 1678 first proposed that the deformation of an elastic

structure is proportional to the applied force. Notice the significant

simplicity of the isotropic form when compared to the general stress-strain

low originally given by Eq. (1.4.1). It should be noted that only two

independent elastic constants are needed to describe the behavior or

isotropic materials.

Stress-strain relations of Eqs. (1.4.6) or (1.4.7) may be inverted to

express the strain in terms of the stress. In order to do this, it is convenient

to use the index notation from Eq. (1.4.6) and set the two free indices the

same to get

.)23( kkkk (1.4.8)

01

This relation can be solved for kk and substituted back into Eq. (1.4.6) to

get

,222

1

ijkkijij

(1.4.9)

which is more commonly written as

,1

ijkkijijEE

(1.4.10)

where

,)23(

E (1.4.11)

is called the modulus of elasticity or Young's modulus, and

,)(2

(1.4.12)

is referred to as Poisson's ratio. The index notation relation of Eq. (1.4.10)

may be written out in component (scalar) form giving the six equations

,)(1

zrrE

,)(1

rzE

,)(1

rzzE

,2

11

rrr

E

,2

11zzz

E

.2

11zrzrzr

E

(1.4.13)

01

Constitutive form of Eqs. (1.4.10) or (1.4.13) again illustrates that only two

elastic constants are needed to formulate Hooke's law for isotropic

materials. By using any of the isotropic forms of Hooke's law, it can be

shown that the principal axes of stress coincide with the principal axes of

strain. This result also holds for same but not all anisotropic materials.

§

We now wish to develop the reduced set of field equations solely in

terms of the displacements. This system is referred to as the displacement

formulation and is most useful when combined with displacement-only

boundary conditions. This is easily accomplished by using the strain-

displacement relations in Hooke's law to give

),( ,,, ijjiijkkij uuu (1.5.1)

which can be expressed as six scalar equations

,21

)(1

r

u

z

uu

rru

rrrz

rr

,1

21

)(1

u

urz

uu

rru

rrr

zr

,21

)(1

z

u

z

uu

rru

rrzz

rz

,1

r

u

r

uu

rr

r

,1

z

zu

rz

u

.

r

u

z

u zrzr (1.5.2)

01

Using these relations in the equilibrium equations gives the result

,0)( ,, ikikkki Fuu (1.5.3)

which are the equilibrium equations in terms of the displacements and are

referred to as Navier's or Lamé's equations. This system can be expressed

in vector form as

,0)()(2 Fuu (1.5.4)

or written out in terms of the three scalar equations

z

uu

rru

rrr

u

rr

uu z

rr

r

1)(

1)(

222

2

,0 rF (1.5.5)

z

uu

rru

rrr

u

rr

uu z

rr

1

)(11

)(2

22

2

,0 F (1.5.6)

,01

)(1

)(2

z

zrz F

z

uu

rru

rrzu

(1.5.7)

where the Laplacian is given by

.11

2

2

2

2

22

zrrr

rr

(1.5.8)

Navier's equations are the desired formulation for the displacement

problem, and the system represents three equations for the three unknown

displacement components. This system is still difficult to solve, and

additional mathematical techniques have been developed to further

simplify these equations for problem solution. Common methods normally

employ the use of displacement potential functions.

03

The purpose of this chapter is to state some general forms of

differential equations and some kinds of boundary-value

problems (BVPs). In addition, this chapter states some numerical

solutions. The FDM as well as the modified R-K method are

presented. In addition, the least square and Richardson

extrapolation are also presented.

§

To obtain accurate numerical solutions to differential equations

governing physical systems has always been an important problem with

scientists and engineers. These differential equations basically fall into two

classes, ordinary and partial, depending on the number of independent

variables present the differential equations: one for ordinary and more than

one for partial (see Jain [92]).

The general form of the ordinary differential equation can be written as

,][ gL (2.1.1)

where L is a differential operator and g is a given function of the

independent variable r. The order of the differential equation is the order of

its highest derivative and its degree is the degree of the derivative of the

highest order after the equation has been rationalized. If no product of the

03

dependent variable )(r with itself or any of its derivatives occur, the

equation is said to be linear, otherwise it is nonlinear. A linear differential

equation of order m can be expressed in the form

),()()(][

0

)( rgrrfLm

p

pp

(2.1.2)

in which )(rf p are known functions. The general nonlinear differential

equation of order m can be written as

,0,,...,,, )()1( mmrF (2.1.3)

or

,,...,,,)( )1()( mm rfr (2.1.4)

which is called a canonical representation of differential equation given in

Eq. (2.1.3). In such a form, the highest order derivative is expressed in

terms of the lower order derivatives and the independent variable. The

general solution of the mth order ordinary differential equation contains m

independent arbitrary constants. In order to determine the arbitrary

constants in the general solution if the m conditions are prescribed at one

point, these are called initial conditions. The differential equation together

with the initial conditions is called the initial-value problem. Thus, the mth

order can be expressed as

.1,...,2,1,0,)(

,,...,,,)(

)(00

)(

)1()(

mpr

rfr

pp

mm

(2.1.5)

If the m conditions are presented at more than one point, these are

called boundary conditions. The differential equation together with the

boundary conditions is known as the boundary-value problem.

03

§

A general boundary-value problem can be represented symbolically as

,,,...2,1,][

,][

mgU

gL

(2.2.1)

where L is an mth order differential operator, g is a given function and U

are the boundary conditions. We shall use r as an independent variable for

the boundary-value problem.

If L represents an mth order linear differential operator and ][U

represent two point boundary conditions, then Eq. (2.2.1) can be expressed

in the form

)(

0

)(][ vm

vv rfL

],,[),()(...)()( )(10 barrgrfrfrf m

m (2.2.2)

.,...,2,1,)()(][1

0

)(,

)(, mbbaaU

m

k

kk

kk

(2.2.3)

For m=2q, the k boundary conditions which are linearly independent

and contain only derivatives up to )1( q th order are called the essential

boundary conditions, and the remaining )2( kq boundary conditions are

termed the suppressible boundary conditions.

The simplest boundary-value problem is given by second order

differential equation

],,[),()()()( 012 barrgrfrfrf (2.2.4)

with one of the three boundary conditions given below.

The boundary conditions of the first kind are:

00

(i) 1)( a and .)( 2 b

The boundary conditions of the second kind are:

(ii) 1)( a and .)( 2 b

The boundary conditions of the third kind, sometimes called Sturm's

boundary conditions, are:

(iii) 110 )()( aaaa and .)()( 210 bbbb

where 100 ,, aba and 1b are all positive constants.

In Eq. (2.2.1) if ,0)( rg the differential equation is called

homogeneous; otherwise it is inhomogeneous. Similarly, the boundary

conditions are called homogeneous when are zero; otherwise

inhomogeneous. The boundary-value problem is called homogeneous if the

differential equation and the boundary conditions are homogeneous. A

homogeneous boundary-value problem )0,0)(( rg possesses only a

trivial solution .0)( r We, therefore, consider those boundary-value

problem in which a parameter occurs either in differential equation or in

the boundary conditions, and we determine values of , called eigenvalues,

for which the boundary-value problem has a nontrivial solution. Such a

solution is called eigen-function and the entire problem is called an

eigenvalues or characteristic value problem.

In the boundary-value problems, the arbitrary constants in the solution

are determined from the conditions given at more than one point.

Therefore, it is possible for more than one solution to exist or no solution

may exist.

In general, a boundary-value problem does not always have a unique

solution. However, the existence and uniqueness of the solution for a

03

special class of boundary-value problems, called class M, can be

established.

A boundary-value problem will be called of class M if it is of the form

,)(,)(),,( 21 barf (2.2.5)

and,

(i) the initial-value problem

,)(,)(),,( 1 Aaarf (2.2.6)

with A arbitrary, has a unique solution, and ),( rf is such that

(ii) ),( rf is continuous

0),( rf for ),(],,[ bar . (2.2.7)

In what follow we will discuss some boundary-value problems and we

will investigate their analytical and numerical solutions.

§

The finite difference method (FDM) was first developed by A. Thom in

the 1920s under the title “the method of square” to solve nonlinear

hydrodynamic equations (Thom and Apelt [93]). The finite difference

techniques are based upon the approximations that permit replacing

differential equations by finite difference equations. These finite difference

approximations are algebraic in form, and the solutions are related to grid

points (see Linz and Wang [94]).

Thus, a finite difference solution basically involves three steps:

1. Dividing the solution into grids of nodes.

03

2. Approximating the given differential equation by finite difference

equivalence that relates the solutions to grid points.

3. Solving the difference equations subject to the prescribed boundary

conditions and/or initial conditions.

Figure 2.1: Common two-dimensional grid patterns

FDM scheme:

Differential equations

estimating derivatives numerically

finite difference equations.

The error in a method's solution is defined as the difference between its

approximation and the exact analytical solution. The two sources of error

in finite difference methods are round-off error, the loss of precision due to

computer rounding of decimal quantities, and truncation error or

discretization error, the difference between the exact solution of the finite

03

difference equation and the exact quantity assuming perfect arithmetic

(that is, assuming no round-off).

Figure 2.2: The finite difference method

To use a finite difference method to attempt to solve (or, more

generally, approximate the solution to) a problem, one must first discretize

the problem's domain. This is usually done by dividing the domain into a

uniform grid (see Figure 2.2). Note that this means that finite-difference

methods produce sets of discrete numerical approximations to the

derivative, often in a “time-stepping” manner.

An expression of general interest is the local truncation error of a

method. Typically expressed using Big-O notation, local truncation error

refers to the error from a single application of a method. That is, it is the

quantity ii fRf )( if )( iRf refers to the exact value and if to the

numerical approximation. The remainder term of a Taylor polynomial

...)(!2

)(

!1

)()()( 200

00

RRf

RRf

RfRRf

),()(!

)( 0)(

RRRn

Rfn

nn

(2.3.1)

03

is convenient for analyzing the local truncation error. Using the Lagrange

form of the remainder nR from the Taylor polynomial for )( 0 RRf

which is

.,

)!1()( 00

1)1(

0 RRRRn

fRRR n

n

n

(2.3.2)

The dominant term of the local truncation error can be discovered. For

example, again using the forward-difference formula for the first

derivative, knowing that ),()( 0 RiRfRf i

,)(!2

)()()()( 2

000 Rif

RiRfRfRiRf

(2.3.3)

and with some algebraic manipulation, this leads to

,!2

)()(

)()(0

00 Rif

RfRi

RfRiRf

(2.3.4)

and further noting that the quantity on the left is the approximation from

the finite difference method and that the quantity on the right is the exact

quantity of interest plus a remainder, clearly that remainder is the local

truncation error. A final expression of this example and its order is:

).()()()(

000 RORf

Ri

RfRiRf

(2.3.5)

This means that, in this case, the local truncation error is proportional to

the step size R . Ignoring the )( RO term, we get

.)()(

)( 000

Ri

RfRiRfRf

(2.3.6)

The expression on the right side of Eq. (2.3.6) is the forward difference

approximation to f at 0R . It is computed using only values of f and

converges to the derivative with order one. In an almost identical way we

get the backward difference

03

.)()(

)( 000

Ri

RiRfRfRf

(2.3.7)

The forward and backward difference rules have relatively low accuracy,

but it is not hard to get better approximations. Again, from Taylor's

Theorem we know that

),)(()(2

)())(()()( 320

000 RORiRf

RiRfRfRiRf

(2.3.8)

),)(()(2

)())(()()( 320

000 RORiRf

RiRfRfRiRf

(2.3.9)

so that, neglecting the higher order terms,

.)(2

)()()( 00

0Ri

RiRfRiRfRf

(2.3.10)

This is the centered difference approximation to the first derivative. It

has second order convergence and normally gives a much more accurate

answer than either Eq. (2.3.6) or Eq. (2.3.7).

Many other formulas can be developed along these lines. For example,

the expression

.)(2

)(3)(4)2()( 000

0Ri

RfRiRfRiRfRf

(2.3.11)

is another forward difference approximation for the first derivative.

We can also derive approximations for higher derivatives this way.

Taking more terms in the Taylor expansions

3020000 )(

6

)()(

2

)())(()()( Ri

RfRi

RfRiRfRfRiRf

),)(( 4RO (2.3.12)

03

3020000 )(

6

)()(

2

)())(()()( Ri

RfRi

RfRiRfRfRiRf

),)(( 4RO (2.3.13)

we find that

).)(()(

)()(2)()( 2

2000

0 RORi

RiRfRfRiRfRf

(2.3.14)

Thus,

.)(

)()(2)()(

2000

0Ri

RiRfRfRiRfRf

(2.3.15)

This is a very useful centered difference approximation to the second

derivative that has second order accuracy.

Consider a general second order equation (see Jain [92])

,,, UURfU .,0 bRR (2.3.16)

with the initial conditions

,)( 00 URU .00)( URU (2.3.17)

We define

,,,!2

)( 2

1 iii UURfR

K

,,,!2

)(1

2112122

2

2

K

R

bUKaURaURaRf

RK iiii

,22111 KWKWURUU iii

).(1

22111 KWKWR

UU ii

(2.3.18)

33

where Wbaa ,,, 21212 and W are arbitrary constants to be determined.

The Taylor series expansion gives

...,!4

)(

!3

)(

!2

)( 432

1

mv

iiiiii UR

UR

UR

URUU

.....!3

)(

!2

)( 32

1

mviiiii U

RU

RURUU (2.3.19)

where

)),(),(,( iiii RURURfU

,)( iUURi fffUfU

URUUURUUUURRmvi ffffUfUfffUfU 222[ 22

.])( iUUURU fffffUff (2.3.20)

We may write Eq. (2.3.20) as

,ii fU

,ii DfU

,2UiiUi

mvi ffDfffDU

where

.U

fU

UR

D ii

Equation (2.3.19) becomes

iiiiii fDR

DfR

fR

URUU 2432

1!4

)(

!3

)(

!2

)(

..., UiiU ffDff

33

...!3

)(

!2

)( 232

1

UiiUiiiii ffDfffDR

DfR

RfUU

(2.3.21)

Simplifying 2K , we get

UiUiRi ffbfUafaRfK

R212222 2

1

)(

2

URiUUiUUiRR fUaffbfUafa

R 222214

1222

22

2

212

!2

)(

),)(( 321212212 ROffafUfbaffba UiUUiiURi

or

Uiiii ffafDaR

DfaR

fR

K 2122

2

4

2

32

24

)(

2

)(

2

)(

),)(( 5RO (2.3.22)

where we have used

.2

1212 ba (2.3.23)

The substitution of 1K and 2K in Eq. (2.3.18) yields

iiiii DfWaR

fWWR

URUU 22

3

21

2

12

)()(

2

)(

),)(()(4

)( 5212

2222

4

ROffaWfDaWR

Uii

iiii DfWaR

fWWRUU 22

2

2112

)()(

2

1

).)(()(4

)( 4212

2222

3

ROffaWfDaWR

Uii

(2.3.24)

33

On comparing Eq. (2.3.24) with Eq. (2.3.21), we obtain

,121 WW ,221 WW

,3

122 Wa .122 Wa (2.3.25)

The coefficients of 4)( R in 1iU and of 3)( R in 1iU of equation

(2.3.24) will not match with the corresponding coefficients in Eq. (2.3.21)

for any choice of 2a , 21a , 2W and 2W .Thus the local truncation error is

))(( 4RO in U and ))(( 3RO in U . A simple solution of Eq. (2.3.23)

and Eq. (2.3.25) may be written as

,2

121 WW

,3

2212 aa ,

3

421 b

,2

11 W

.2

32 W

Thus the Runge-Kutta method Eq. (2.3.18) becomes

,,,!2

)( 2

1 iii UURfR

K

,3

4,

3

2

3

2,

3

2

!2

)(11

2

2

K

RUKURURRf

RK iiii

.2

1211 KKURUU iii

.32

1211 KK

RUU ii

(2.3.26)

30

The Runge-Kutta method using four K's is given by

,,,2

)( 2

1 iii UURfR

K

,1

,4

1

2

1,

22

)(11

2

2

K

RUKURU

RRf

RK iiii

,1

,4

1

2

1,

22

)(21

2

3

K

RUKURU

RRf

RK iiii

,2

,,2

)(33

2

4

K

RUKURURRf

RK iiii

,3

13211 KKKURUU iii

.223

143211 KKKK

RUU ii

(2.3.27)

A mathematical procedure for finding the best-fitting curve to a given

set of points by minimizing the sum of the squares of the offsets (the

residuals) of the points from the curve. The sum of the squares of the

offsets is used instead of the offset absolute values because this allows the

residuals to be treated as a continuous differentiable quantity. However,

because squares of the offsets are used, outlying points can have a

disproportionate effect on the fit, a property that may or may not be

desirable depending on the problem at hand.

33

Figure 2.3: Least squares fitting

Figure 2.4: Least squares offsets

In practice, the vertical offsets from a line (polynomial, surface,

hyperplane, etc.) are almost always minimized instead of the perpendicular

offsets. This provides a fitting function for the independent variable that

estimates for a given (most often what an experimenter wants),

allows uncertainties of the data points along the - and -axes to be

incorporated simply, and also provides a much simpler analytic form for

the fitting parameters than would be obtained using a fit based on

33

perpendicular offsets. In addition, the fitting technique can be easily

generalized from a best-fit line to a best-fit polynomial when sums of

vertical distances are used. In any case, for a reasonable number of noisy

data points, the difference between vertical and perpendicular fits is quite

small.

The least square method is a very popular technique used to compute

estimations of parameters and to fit data. It is one of the oldest techniques

of modern statistics as it was first published in 1805 by the French

mathematician Legendre in a now classic memoir. Nowadays, the least

square method is widely used to find or estimate the numerical values of

the parameters to fit a function to a set of data and to characterize the

statistical properties of estimates. It exists with several variations: Its

simpler version is called ordinary least squares (OLS), a more sophisticated

version is called weighted least squares (WLS), which often performs

better than OLS because it can modulate the importance of each

observation in the final solution. Recent variations of the least square

method are alternating least squares (ALS) and partial least squares (PLS).

The linear least squares fitting technique is the simplest and most

commonly applied form of linear regression and provides a solution to the

problem of finding the best fitting straight line through a set of points. In

fact, if the functional relationship between the two quantities being graphed

is known to within additive or multiplicative constants, it is common

practice to transform the data in such a way that the resulting line is a

straight line, say by plotting T vs instead of T vs in the case of

analyzing the period T of a pendulum as a function of its length . For this

reason, standard forms for exponential, logarithmic, and power laws are

often explicitly computed. The formulas for linear least squares fitting

were independently derived by Gauss and Legendre. Vertical least squares

33

fitting proceeds by finding the sum of the squares of the vertical deviations

E of a set of n data points

,),....,,,(2

21 Nii aaafE (2.4.1)

from a function f . Note that this procedure does not minimize the actual

deviations from the line (which would be measured perpendicular to the

given function). In addition, although the unsquared sum of distances

might seem a more appropriate quantity to minimize, use of the absolute

value results in discontinuous derivatives which cannot be treated

analytically. The square deviations from each point are therefore summed,

and the resulting residual is then minimized to find the best fit line. This

procedure results in outlying points being given disproportionately large

weighting.

The condition for E to be a minimum is that

,0

ia

E ,0

ib

E (2.4.2)

for Ni ...,,2,1 . For a linear fit,

,),( babaf (2.4.3)

so

,)(),(

1

2

N

iii babaE (2.4.4)

,)1()(2

1

N

iii ba

a

E (2.4.5)

.)()(2

1

N

iiii ba

b

E (2.4.6)

33

Setting 0

b

E

a

E, and dividing by 2 yields

,

11

N

ii

N

iibaN (2.4.7)

.

11

2

1

N

iii

N

ii

N

ii ba (2.4.8)

In matrix form,

,

1

1

1

2

1

1

N

iii

N

ii

N

ii

N

ii

N

ii

b

aN

(2.4.9)

or

.

1

1

1

1

2

1

1

N

iii

N

ii

N

ii

N

ii

N

iiN

b

a

(2.4.10)

The 22 matrix inverse is

,1

111

111

2

12

11

2

N

ii

N

ii

N

iii

N

iii

N

ii

N

ii

N

ii

N

ii

N

ii

NN

b

a

(2.4.11)

we define the mean (or the expected values)of and by writing a line

above and : thus

,

1

N

ii (2.4.12)

33

.

1

N

ii (2.4.13)

The mean is the average value of the data. Now using Eqs. (2.4.12) and

(2.4.13) in Eq. (2.4.11) we obtain

,2

1

2

11

2

2

11

2

111

2

1

NN

aN

ii

N

iii

N

ii

N

ii

N

ii

N

iii

N

ii

N

ii

N

ii

(2.4.14)

.2

1

2

12

11

2

111

N

N

N

N

bN

ii

N

iii

N

ii

N

ii

N

ii

N

ii

N

iii

(2.4.15)

These can be rewritten in a simpler form by defining the sums of squares

(see Kenney and Keeping [95])

,)( 2

1

2

1

2 NssN

ii

N

ii

(2.4.16)

,)( 2

1

2

1

2

Nss

N

ii

N

ii (2.4.17)

,))((

11

Nss

N

iii

N

iii (2.4.18)

which are also written as

,N

ssV

(2.4.19)

,N

ssV (2.4.20)

33

.),cov(N

ss

(2.4.21)

Here, ),cov( is the covariance and V and V are variances. Note

that the quantities

N

iii

1

and

N

ii

1

2 can also be interpreted as the dot

products

,

1

2

N

ii (2.4.22)

N

iii

1

. (2.4.23)

In terms of the sums of squares, the regression coefficient b is given by

.),cov(

ss

ss

Vb

(2.4.24)

and a is given in terms of b using Eq. (2.4.7) as

.ba (2.4.25)

The overall quality of the fit is then parameterized in terms of a quantity

known as the correlation coefficient, defined by

,

22

ssss

sscc

(2.4.26)

which gives the proportion of ss which is accounted for by the

regression.

Let be i the vertical coordinate of the best-fit line with -coordinate i ,

so

,ˆii ba (2.4.27)

33

then the error between the actual vertical point i and the fitted point is

given by

.iiie (2.4.28)

Now define 2s as an estimator for the variance in ie ,

.2

1

22

N

i

i

N

es (2.4.29)

Then s can be given by

.22

2

N

ss

ssss

N

bsssss

(2.4.30)

(see Acton [96]; Gonick and Smith [97]).

The standard errors for a and b are

,1

)(2

ssNsaSE (2.4.31)

.)(ss

sbSE (2.4.32)

Richardson's extrapolation is used to generate high-accuracy results

while using low-order formulas. Although the name attached to the method

refers to a paper written by Richardson and Gaunt in 1927, the idea behind

the technique is much older (see Burden and Faires [98]).

Extrapolation can be applied whenever it is known that an

approximation technique has an error term with a predictable form, one

33

that depends on parameter, usually the step size . Suppose that for each

number 0 we have formula )(N that approximates an unknown

value M and that the truncation error involved with the approximation has

the form

,...)( 33

221 KKKNM (2.5.1)

for some collection of unknown constants ...,,, 321 KKK .

Since the truncation error is )(O , we would expect, for example, that

,01.0)01.0(,1.0)1.0( 11 KNMKNM (2.5.2)

and, in general, ,)( 1 KNM unless there was a large variation in

magnitude among the constants ...,,, 321 KKK .

The object of extrapolation is to find an easy way to combine the rather

inaccurate )(O approximations in an appropriate way to produce

formulas with a higher-order truncation error. Suppose, for example, we

can combine the )(N formulas to produce an )( 2O approximation

formula, ),(ˆ N for M with

,...ˆˆ)(ˆ 33

22 KKNM (2.5.3)

for some, again unknown, collection of constants ...,ˆ,ˆ21 KK . Then we

have

,ˆ0001.0)01.0(ˆ,ˆ01.0)1.0(ˆ22 KNMKNM (2.5.4)

and so on.

If the constants 1K and 2K are of roughly the same magnitude, then

the )(ˆ N approximations are much better than the corresponding )(N

approximations. The extrapolation continues by combining the )(ˆ N

33

approximations in a manner that produces formulas with )( 3O truncation

error, and so on.

To see specifically how we can generate these higher-order formulas,

let us consider the formula for approximating M of the form

....)( 33

221 KKKNM (2.5.5)

Since the formula is assumed to hold for all positive , consider the result

when we replace the parameter by half its value. Then we have the

formula

....8422

3

3

2

21

KKKNM (2.5.6)

Subtracting Eq. (2.5.3) form twice this equation eliminates the term

involving 1K and gives

....42

)(2

2 33

32

2

2

KKNNM (2.5.7)

To facilitate the discussion, we define )()(1 NN and

.)(22

)(2

2)( 111112

NNNNNN (2.5.8)

Then we have the )( 2O approximation formula for M:

....4

3

2)( 3322

2 KK

NM (2.5.9)

If we now replace by 2

in this formula, we have

....32

3

82

33222

KKNM (2.5.10)

30

This can be combined with Eq. (2.5.9) to eliminate the 2 term.

Specifically, subtracting Eq. (2.5.9) from 4 times Eq. (2.5.10) gives

,...8

3)(

243 33

22

KNNM (2.5.11)

and dividing by 3 gives an )( 3O formula for approximating M:

....83

)(2

2

3322

2

KNN

NM (2.5.12)

By defining

.3

)(2

2)(

22

23

NN

NN

(2.5.13)

We have the )( 3O formula

....8

)( 333

KNM . (2.5.14)

The process is continued by constructing an )( 4O approximation

,7

)(2

2)(

33

34

NN

NN

(2.5.15)

and an )( 5O approximation

,15

)(2

2)(

44

45

NN

NN

(2.5.16)

and so on. In general, if M can be written in the form

33

),()(1

1

mm

j

jj OKNM

(2.5.17)

then for each mj ...,,3,2 , we have an )( jO approximation of the form

.12

)(2

2)(

1

11

1

j

jj

jj

NN

NN

(2.5.18)

55 * The results of this chapter is already appeared in Ref. [99].

In this chapter, a unified governing equation is firstly derived

from the basic equations of the rotating variable-thickness solid

disk and the proposed stress-strain relationship. The outer edge of

the solid disk is considered to have free boundary conditions. The

analytical solution for rotating solid disk with arbitrary cross-

section of continuously variable-thickness is presented. Next, the

finite difference method (FDM) is introduced to solve the

governing equation. A comparison between both analytical and

numerical solutions is made. Finally, some application examples

are given to demonstrate the validity of the proposed method.

§

As the effect of thickness variation of rotating solid disks can be taken

into account in their equation of motion, the theory of the variable-

thickness solid disks can give good results as that of uniform-thickness

disks as long as they meet the assumption of plane stress. The present solid

disk is considered as a single layer of variable thickness. After considering

this effect, the equation of motion of rotating disks with variable thickness

can be written as

,0)()()(d

d 22 rrhrhrrhr

r (3.1.1)

65

where r and are the radial and circumferential stresses, h(r) is the

variable-thickness of the disk, r is the radial coordinate, is the material

density of the rotating solid disk and is the constant angular velocity.

The relations between the radial displacement u and the strain

components are irrespective of the thickness of the rotating solid disk.

They can be written as

,,d

d

r

u

r

ur (3.1.2)

where r and are the radial and circumferential strains, respectively.

The above geometric relations lead to the following condition of

deformation harmony:

.0)(d

d rr

r (3.1.3)

For the elastic deformation, the constitutive equations for the variable-

thickness solid disk can be described with Hooke's law

,,EE

rrr

(3.1.4)

where E is Young's modulus and is Poisson's ratio. Introducing the stress

function and assuming that the following relations hold between the

stresses and the stress function

.d

d

)(

1,

)(

22rrrhrrh

r

(3.1.5)

Substituting Eq. (3.1.5) into Eq. (3.1.4), one obtains

65

.)(d

d

)(

11

,d

d

)()(

11

22

22

rrrhrrhE

rrrhrrhE

r

(3.1.6)

§

The substitution of Eq. (3.1.6) into Eq. (3.1.3) produces the following

confluent hypergeometric differential equation for the stress function

:)(r

)(d

d

)(

)1()(

d

d

)()(

)(d

d

2r

rrEhr

rrrhrh

rhr

E

r

)()(

)1()(

)()(

)(d

d

22r

rrEhr

rrhrrh

rhr

E

r

.02)(

)(d

d22

22

2

22

E

rr

rh

rr

E

r

E

r

(3.2.1)

The above equation may be simplified to be

2

22

2

)(

)(d

d)(

d

d)(

)(

)(2)(d

d

rEh

rr

rhr

rrh

rEh

rrhrr

r

.0)1(

)(

)()()(d

d22

2

E

r

rrEh

rrhrrhr

(3.2.2)

65

Then, multiplying by )(rrEh , one obtains

.0)3(d

d1

d

d

d

d1

d

d 322

22

rh

r

h

h

r

rr

h

h

rr

rr

(3.2.3)

The boundary conditions for the rotating solid disk are

.at0

,0at

br

r

r

r

(3.2.4)

The thickness of the solid disk is assumed to vary nonlinearly through

the radial direction. It is assumed to be in terms of a simple exponential

power law distribution according to the following case:

,e)( 0

k

b

rn

hrh

(3.2.5)

Figure 3.1 (a): Variable-thickness solid disk profiles for k = 0.7 and n = 2.

65

Figure 3.1 (b): Variable-thickness solid disk profiles for k = 1.5 and n = 2.

Figure 3.1 (c): Variable-thickness solid disk profiles for k = 2.5 and n = 0.5.

56

where 0h is the thickness at the middle of the disk, n and k are geometric

parameters and b is the outer radius of the disk (see Figure 3.1). The value

of n equal to zero represents a uniform-thickness solid disk while the value

of k equal to unity represents a linearly decreasing variable-thickness solid

disk. For small k and large n (k = 0.7 or 1.5 and n = 2) the profile of the

solid disk is concave while it is convex for large k and small n (k = 2.5 and

n = 0.5). It is to be noted that the parameter n determines the thickness at

the outer edge of the solid disk relative to 0h while the parameter k

determine the shape of the profile.

Introducing the following dimensionless forms:

brR / ,

,)3( b

),(1

)(2

0

rbh

R

),,(),(221 r

E

).,(1

),(221 r

(3.2.6)

Then, Eq. (3.2.3) may be written in the following simple form

.0e)1(d

d)1(

d

d 32

22

RRknR

RknRR

RknRkk (3.2.7)

The general solution of the above equation can be written as

Rjiji

RWFCRMRR

knk

d)()()(e)( ,1,22

,d)()()( ,2,

Rjiji MFCRW (3.2.8)

56

where 1C and 2C are arbitrary constants, is a dummy parameter, jiM ,

and jiW , are Whittaker's functions

),,,()(),,,()( ,,k

jik

ji nRjiWRWnRjiMRM (3.2.9)

in which

.0,1

,2

1 k

kj

ki

(3.2.10)

In addition, the function )(RF is given in terms of Whittaker's functions

by

.)()()()()1(

e)(

,1,,1,

222

RWRkMRMRW

RRF

jijijiji

Rknk

(3.2.11)

The substitution of Eq. (3.2.8) into Eq. (3.1.5) with the aid of the

dimensionless forms given in Eq. (3.2.6) gives the radial and

circumferential stresses in the following forms:

R

jijiR

WFCRMRRknk

d)()()(e)( ,1,1

122

,d)()()( ,2,

Rjiji MFCRW (3.2.12)

.3d

de)(

2

2

R

RR

knR (3.2.13)

Here, the first derivative of the stress function )(R with respect to R may

be given easily by using Eq. (3.2.8). Note that the first derivatives of

Whittaker's functions jiM , and jiW , can be represented by

56

.)()()(d

d

,)()()(d

d

,1,21

,

,121

,21

,

RWRWinRR

kRW

R

RMjiRMinRR

kRM

R

jijik

ji

jijik

ji

(3.2.14)

Finally, the dimensionless strains and the corresponding radial

displacement may be obtained easily using Eq. (3.2.12) and Eq. (3.2.13) as

well as the dimensionless forms given in Eq. (3.2.6). Therefore, all of

stress function, stresses, strains, and radial displacement may be

determined completely after applied the dimensionless of the boundary

conditions given in Eq. (3.2.4).

§

The finite difference method presented in Chapter 2 is used here to

solve the present problem numerically. The resolution of the elastic

problem of rotating solid disk with variable thickness is to solve a second-

order differential equation (3.2.7) under the given boundary conditions

0)1()0( such that )0()0( 21 . Equation (3.2.7) can be written

in the following general form:

),()()( RsRqRp (3.3.1)

where the prime )( denotes differentiation with respect to R and

.e)(,1

)(,1

)(2

RRsR

RknRq

R

knRRp

knRkk

(3.3.2)

It is clear that the above problem has a unique solution because

),(),( RqRp and )(Rs are continuous on [0, 1] and 0)( Rq on [0, 1]. The

linear second-order boundary value problem given in Eq. (3.3.1) requires

that difference-quotient approximations be used for approximating and

. First we select an integer 0N and divided the interval [0, 1] into

56

)1( N equal subintervals, whose end points are the mesh points

,RiRi for ,1,...,1,0 Ni where )1/(1 NR . At the interior mesh

points, ,iR ,,...,2,1 Ni the differential equation to the approximated is

).()()()()()( iiiiii RsRRqRRpR (3.3.3)

If we apply the centered difference approximations of )( iR and )( iR

to Eq. (3.3.3), we arrive at the system:

12

1 )(2

1)()(2)(2

1

iiiiii Rp

RRqRRp

R

),()( 2iRsR (3.3.4)

for each ....,,2,1 Ni The N equations, together with the boundary

conditions

,0

,0

1

0

N

(3.3.5)

are sufficient to determine the unknowns ,i 1...,,2,1,0 Ni . The

resulting system of Eq. (3.3.4) is expresses in the tri-diagonal NN -

matrix form:

,BA (3.3.6)

where

....,,2,1),()(

,2,...,,4,3,,2...,,2,1,0

,...,,3,2),(2

1

,1...,,2,1),(2

1

,...,,2,1),()(2

2

,,

1,

1,

2,

NiRsRB

ijNjNiAA

NiRpR

A

NiRpR

A

NiRqRA

ii

ijji

iii

iii

iii

(3.3.7)

56

The solution of the finite difference discretization of the two-point linear

boundary value problem can therefore be found easily even for very small

mesh sizes.

§

Some numerical examples for the rotating variable-thickness solid disks

will be given according the analytical and numerical solutions )3.0( .

According to Eq. (3.2.6), the stress function, radial displacement, strains

and stresses of the rotating variable-thickness solid disk are determined as

per the analytical solution are compared with those obtained by the

numerical FDM solution.

The results of the present FDM investigations for the stress function

for the rotating variable-thickness solid disk with k = 2.5 and n = 0.5 are

reported in Table 3.1. For this example, N = 9, 19, 39 and 79, so R has

the corresponding values 0.1, 0.05, 0.025 and 0.0125, respectively. The

FDM gives results compared well with the exact solution, especially for

small values of R . The relative error between the exact method and the

FDM with 0125.0R at 6.0iR for example, may be less than

51062.2 .

56

Table 3.1: Dimensionless stress function of a rotating variable-

thickness solid disk (k = 2.5, n = 0.5).

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0

0.001318247

0.002635012

0.003948803

0.005258119

0.006561453

0.007857295

0.009144132

0.010420449

---

---

---

---

---

---

---

0.010333909

---

---

---

0.005341761

---

---

---

0.010398523

---

0.002632275

---

0.005253982

---

0.007852303

---

0.010414946

0.001317819

0.002634312

0.003947918

0.005270818

0.006560297

0.007856045

0.009142869

0.010419072

0.0125

0.0250

0.0375

0.0500

0.0625

0.0750

0.0875

0.1000

0.011684731

0.012935466

0.014171142

0.015390252

0.016591293

0.017772771

0.018933196

0.020071091

---

---

---

---

---

---

---

0.019984044

---

---

---

0.015367662

---

---

---

0.020049282

---

0.012929711

---

0.015384444

---

0.017767073

---

0.020065635

0.011683316

0.012934026

0.014169696

0.015388799

0.016589850

0.017771346

0.018931799

0.020069727

0.1125

0.1250

0.1375

0.1500

0.1625

0.1750

0.1875

0.2000

0.021184986

0.022273424

0.023334964

0.024368176

0.025371649

0.026343989

0.027283820

0.028189791

---

---

---

---

---

---

---

0.028132394

---

---

---

0.024349525

---

---

---

0.028175489

---

0.022268320

---

0.024363514

---

0.026339842

---

0.028186218

0.021183663

0.022272149

0.023333741

0.024367011

0.025370546

0.026342953

0.027282855

0.028188899

0.2125

0.2250

0.2375

0.2500

0.2625

0.2750

0.2875

0.3000

0.029060568

0.029894846

0.030691343

0.031448803

0.032166002

0.032841742

0.033474861

0.034064227

---

---

---

---

---

---

---

0.034048816

---

---

---

0.031439588

---

---

---

0.034060453

---

0.029891894

---

0.031446504

---

0.032840118

---

0.034063288

0.029059753

0.029894109

0.030690687

0.031448230

0.032165512

0.032841338

0.033474542

0.034063994

0.3125

0.3250

0.3375

0.3500

0.3625

0.3750

0.3875

0.4000

0.034608743

0.035107349

0.035559023

0.035962780

0.036317677

0.036622811

0.036877324

0.037080400

---

---

---

---

---

---

---

0.037107585

---

---

---

0.035964469

---

---

---

0.037087271

---

0.035107098

---

0.035963207

---

0.036623899

---

0.037082122

0.034608596

0.035107288

0.035559047

0.035962888

0.036317868

0.036623085

0.036877677

0.037080832

0.4125

0.4250

0.4375

0.4500

0.4625

0.4750

0.4875

0.5000

55

Table 3.1: Continued

Analytical FDM

iR

0125.0R 025.0R 05.0R 1.0R

0.037231269

0.037329209

0.037373544

0.037363648

0.037298942

0.037178902

0.037003051

0.036770967

0.037231777

0.037329791

0.037374198

0.037364369

0.037299729

0.037179570

0.037003958

0.036771929

---

0.037331532

---

0.037366529

---

0.037182291

---

0.036774810

---

---

---

0.037375156

---

---

---

0.036786324

---

---

---

---

---

---

---

0.036832186

0.5125

0.5250

0.5375

0.5500

0.5625

0.5750

0.5875

0.6000

0.036482282

0.036136679

0.035733896

0.035273729

0.034756025

0.034180690

0.033547685

0.032857027

0.036483294

0.036137738

0.035734998

0.035274868

0.034757198

0.034181892

0.033548911

0.032858271

---

0.036140911

---

0.035278284

---

0.034185494

---

0.032862001

---

---

---

0.035291940

---

---

---

0.032876919

---

---

---

---

---

---

---

0.032936503

0.6125

0.6250

0.6375

0.6500

0.6625

0.6750

0.6875

0.7000

0.032108789

0.031303102

0.030440152

0.029520183

0.028543493

0.027510437

0.026421427

0.025276927

0.032110047

0.031304368

0.030441421

0.029521449

0.028544751

0.027511682

0.026422653

0.025278128

---

0.031308165

---

0.029525248

---

0.027515416

---

0.025281728

---

---

---

0.029540441

---

---

---

0.025296129

---

---

---

---

---

---

---

0.025353741

0.7125

0.7250

0.7375

0.7500

0.7625

0.7750

0.7875

0.8000

0.024077458

0.022823594

0.021515961

0.020155240

0.018742161

0.017277507

0.015762106

0.014196841

0.024078628

0.022824727

0.021517051

0.020156281

0.018743148

0.017278432

0.015762966

0.014197627

---

0.022828123

---

0.020159404

---

0.017281209

---

0.014199988

---

---

---

0.020171897

---

---

---

0.014209433

---

---

---

---

---

---

---

0.014247568

0.8125

0.8250

0.8375

0.8500

0.8625

0.8750

0.8875

0.9000

0.012582635

0.010920462

0.009211340

0.007456327

0.005656526

0.003813078

0.001927164

0

0.012583344

0.010921087

0.009211874

0.007456766

0.005656864

0.003813309

0.001927282

0

---

0.010922960

---

0.007458083

---

0.003814000

---

0

---

---

---

0.007463350

---

---

---

0

---

---

---

---

---

---

---

0

0.9125

0.9250

0.9375

0.9500

0.9625

0.9750

0.9875

1.0000

55

Richardson extrapolation method is applied here with ,05.0,1.0R

025.0 and 0.0125 the obtained results are listed in Table 3.2. These

extrapolations are given, respectively by

,3

)1.0()05.0(4Ext1

RR iii (3.4.1a)

,3

)05.0()025.0(4Ext2

RR iii (3.4.1b)

,3

)025.0()0125.0(4Ext3

RR iii (3.4.1c)

.15

ExtExt16Ext 12

4ii

i

(3.4.1d)

Table 3.2 shows that all extrapolations results are correct to the decimal

places listed. In fact, if sufficient digits are maintained, the approximation

of i4Ext gives results those agree with the exact solution with maximum

difference error of 9100.1 at some of the mesh points.

Table 3.2: Dimensionless stress function of a rotating variable-

thickness solid disk using Richardson's extrapolation method with

different values of ΔR (k = 2.5, n = 0.5).

iR

i1Ext i2Ext i3Ext i4Ext

Analytical

0.0 0 0 0 0 0

0.1 0.010420061 0.010420421 0.010420447 0.010420444 0.010420449

0.2 0.020071028 0.020071086 0.020071091 0.020071090 0.020071091

0.3 0.028189855 0.028189794 0.028189792 0.028189790 0.028189791

0.4 0.034064332 0.034064233 0.034064229 0.034064227 0.034064227

0.5 0.037080500 0.037080406 0.037080401 0.037080400 0.037080400

0.6 0.036771037 0.036770972 0.036770969 0.036770967 0.036770967

0.7 0.032857058 0.032857029 0.032857028 0.032857028 0.032857027

0.8 0.025276926 0.025276927 0.025276928 0.025276927 0.025276927

0.9 0.014196825 0.014196839 0.014196840 0.014196840 0.014196841

1.0 0 0 0 0 0

55

Table 3.3: Dimensionless stress function of a rotating variable-

thickness solid disk (k = 0.7, n = 2).

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.0

0.003891568 0.003891627 0.003891774 0.003892229 0.003893631 0.1

0.006897877 0.006898133 0.006898890 0.006901872 0.006913732 0.2

0.008971625 0.008972022 0.008973209 0.008977938 0.008996897 0.3

0.010102762 0.010103227 0.010104619 0.010110178 0.010132490 0.4

0.010321685 0.010322155 0.010323563 0.010329193 0.010351793 0.5

0.009683519 0.009683947 0.009685228 0.009690352 0.009710919 0.6

0.008256930 0.008257280 0.008258329 0.008262527 0.008279371 0.7

0.006116796 0.006117044 0.006117787 0.006120761 0.006132695 0.8

0.003339509 0.003339638 0.003340026 0.003341577 0.003347797 0.9

0 0 0 0 0 1.0

Table 3.4: Dimensionless stress function of a rotating variable-

thickness solid disk (k = 1.5, n = 2).

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.0

0.005416705 0.005416605 0.005416315 0.005415269 0.005412300 0.1

0.009932747 0.009933554 0.009935982 0.009945759 0.009985706 0.2

0.013033300 0.013034871 0.013039590 0.013058534 0.013135277 0.3

0.014512487 0.014514484 0.014520482 0.014544543 0.014641868 0.4

0.014415043 0.014417117 0.014423346 0.014448330 0.014549332 0.5

0.012959495 0.012961364 0.012966973 0.012989467 0.013080378 0.6

0.010460010 0.010461482 0.010465900 0.010483617 0.010555201 0.7

0.007259771 0.007260750 0.007263688 0.007275470 0.007323059 0.8

0.003682061 0.003682530 0.003683938 0.003689581 0.003712369 0.9

0 0 0 0 0 1.0

55

Tables 3.3 and 3.4 present the results of investigations for the stress

function for the rotating variable-thickness solid disks with k = 0.7, n =

2 and k = 1.5, n = 2, respectively.

Now the least square method and curve fitting are used for the discrete

results of the stress function . So, one can get easily the radial

displacement, strains and stresses since we have as a continuous

function of R. The distributions of the stress function, radial and

circumferential stresses are presented in Figure 3.2. The numerical FDM

solution is compared with the exact analytical solution for the rotating

variable-thickness solid disk with k = 2.5 and n = 0.5. It can be seen that

the FDM can describe the stress function, radial displacement, strains and

stresses through the thickness of the rotating solid disk very well enough.

For the sake of completeness and accuracy, additional results for the

stress function, radial stress and circumferential stress are presented in

Figures 3.3-3.5 for different values of the geometric parameters k and n.

Figure 3.3 shows the stress function through the radial direction of the

rotating solid disk with k = 2.5, n = 0.5; k = 0.7, n = 2 and k = 1.5, n = 2.

Similar results for the radial 1 and the circumferential 2 stresses are

plotted in Figures 3.4 and 3.5. Figure 3.3 shows that the stress function

increases as k increases and this irrespective of the value of n. Figures 3.4

and 3.5 show that k = 2.5, n = 0.5 gives the largest stresses. The

intersection of the two cases k = 0.7, n = 2 and k = 1.5, n = 2 may be

occurred at R = 0.1 for the radial stress and at R = 0.15 for the

circumferential stress.

56

Figure 3.2: Stress function , radial stress 1 and circumferential stress

2 for the variable-thickness solid disk.

Figure 3.3: Stress function of the variable-thickness solid disk for

different values of k and n.

56

Figure 3.4: Radial stress 1 in the variable-thickness solid disk for

different values of k and n.

Figure 3.5: Circumferential stress 2 in the variable-thickness solid disk

for different values of k and n.

56

Figure 3.6: Radial displacement U of the variable-thickness solid disk for

different values of k and n.

Figure 3.7: Radial strain 1 in the variable-thickness solid disk for

different values of k and n.

56

Figure 3.8: Circumferential strain 2 in the variable-thickness solid disk

for different values of k and n.

The radial displacement, radial strain as well as circumferential strain are

also presented in Figures 3.6-3.8 for different values of the geometric

parameters k and n. Figure 3.6 shows the radial displacement U through the

radial direction of the rotating solid disk for different values of k and n.

Similar results for the radial strain 1 and the circumferential strain 2 are

plotted in Figures 3.7 and 3.8. We notice as show in Figure 3.6 that, the

radial displacement U increases as k increases and this irrespective of the

value of n. Figures 3.7 and 3.8 show that k = 2.5, n = 0.5 gives the largest

strains. The intersection of the two cases k = 0.7, n = 2 and k = 1.5, n = 2

may be occurred at R = 0.08 for the radial strain and at R = 0.18 for the

circumferential strain.

56

It is clear that, the FDM gives stress function, radial strain and stresses

consequently, with excellent accuracy with the exact analytical solution. In

fact, FDM may be failed to get accurate radial displacement and

circumferential strain, especially, at the outer edges of the solid disk (see

Figures 3.6 and 3.8). However, in some cases of rotating variable-thickness

solid disks, the analytical solutions are not available. So, one can trustily

use the present FDM solutions.

§

The rotating solid disk with variable thickness is treated herein. By

introducing a suitable stress function, the governing equation is derived

from the equation of motion of rotating disk, compatibility equation and

the proposed stress-strain relationship. Both the analytical and numerical

solutions are presented. The calculation of the rotating solid disk is turned

into finding the solution of a second-order differential equation under the

given conditions at the center and the outer edge of the disk. The numerical

solution is based upon the finite difference method. The governing

equation is solved analytically with the help of Whittaker's functions and a

number of numerical examples are studied. The results of the two solutions

at different disk configurations are compared. The proposed FDM

approach gives very agreeable results to the analytical solution and so it

may be used for different problems that analytical solutions are not

available.

75 * The results of this chapter is already sent for possible publication in a suitable journal.

In this chapter, two different annular disks for the radially varying

thickness are given. The numerical finite difference method

(FDM) and modified Runge-Kutta's method (R-K) solutions as

well as the analytical solution are available for the first disk while

the analytical solution is not available for the second annular disk.

Both analytical and numerical results for radial displacement,

stresses and strains are obtained for the first annular disk of

variable thickness. The accuracy of the present numerical

solutions is discussed and their ability of use for the second

rotating variable-thickness annular disk is investigated. Finally,

the distributions of radial displacement, stresses and strains are

presented and the appropriate comparisons and discussions are

made at the same angular velocity.

§

As the effect of thickness variation of rotating annular disks can be

taken into account in their equation of motion, the theory of the variable-

thickness annular disks can give good results as that of the uniform-

thickness annular disks as long as they meet the assumption of plane stress.

67

After considering this effect, the equation of motion of rotating disks with

variable thickness can be written as

,0)()()(d

d 22 rrhrhrrhr

r (4.1.1)

where r and are the radial and circumferential stresses, r is the radial

coordinate, is the density of the rotating disk, is the constant angular

velocity, and h(r) is the thickness which is function of the radial coordinate

r.

The relations between the radial displacement u and the strains are

irrespective of the thickness of the rotating disk. They can be written as

,,d

d

r

u

r

ur (4.1.2)

where r and are the radial and circumferential strains, respectively.

For the elastic deformation, the constitutive equations for the rotating

disk can be described with Hooke's law

,,EE

rrr

(4.1.3)

where E is Young's modulus and is Poisson's ratio.

The substitution of Eq. (4.1.2) into Eq. (4.1.3) produces the constitutive

equations for r and as:

.1

,1

2

2

dr

du

r

uE

r

u

dr

duEr

(4.1.4)

The thickness of the annular disk is given using various distributions

through the radial direction as follows:

66

Figure 4.1 (a): Variable-thickness annular disk 1 profile for k = 0.4.

Figure 4.1 (b): Variable-thickness annular disk 1 profile for k = 4.0 .

67

Figure 4.1 (c): Variable-thickness annular disk 1 profile for k = 0.2.

Figure 4.1 (d): Variable-thickness annular disk 1 profile for k = 2.0 .

68

Disk 1: Nonlinearly distribution

.)(

2

0

k

b

rhrh

(4.1.5)

Disk 2: Exponentially distribution

.e2)( 0

kk

b

a

b

rn

hrh (4.1.6)

For both disks, 0h is the thickness at the inner edge of the disk, n and k are

geometric parameters, a is the inner radius of the disk and b is the outer

radius of the disk. The value of n equal to zero represents a uniform-

thickness annular disk. It is to be noted that the parameter n determines the

thickness at the outer edge of the annular disk relative to 0h while the

parameter k determine the shape of the profile. For disk 1, the geometric

parameter k is given according to (k = 2.0 ) and (k = 4.0 ) For positive

values of k, the profile is concave whereas it may be convex for negative

values of k (see Figure 4.1). For disk 2, the geometric parameters k and n

are given according to three different sets. For small k and large n (k = 1.4,

n = 2, and k = 0.8, n = 1.2) the profile of the annular disk is convex while it

is concave for large k and small n (k = 2, n = 0.6) (see Figure 4.2).

To simplify the solving process, we introduce the following

dimensionless variables:

).,(1

),(,/,/

),,(),(),(,)1(

2

2

21

2212

2

r

r

baAbrR

Eru

b

EUb

(4.1.7)

78

Figure 4.2 (a): Variable-thickness annular disk 2 profile for k = 1.4

and n = 2.

Figure 4.2 (b): Variable-thickness annular disk 2 profile for k = 0.8

and n = 1.2.

78

Figure 4.2 (c): Variable-thickness annular disk 2 profile for k = 2

and n = 0.6.

The substitution of Eq. (4.1.5) into Eq. (4.1.1) with the aid of r and

given in Eq. (4.1.4) produces the following confluent hypergeometric

differential equation for the radial displacement )(ru according to the

annular disk 1

)(1

21)(1 2

2

0

2

2

2

2

0

rudr

dE

b

rh

krudr

drE

b

rh

kk

.0)()1(

12 222

02

2

0

rb

rhru

r

Eb

rh

k

k

k

(4.1.8)

78

Then multiplying (4.1.8) by

Eb

rh

r

k2

0

2)1( , we obtain

)()12()()12()(2

22 rukru

dr

drkru

dr

dr

.0)1( 223

E

r (4.1.9)

Using the dimensionless variables given in (4.1.7) we obtain

E

Ukb

E

UdR

dRkb

E

UdR

dbR

)12()12( 2

22

222

.023

E

bR (4.1.10)

Multiplying (4.1.10) by2b

E gives

.0)12(d

d)12(

d

d 3

2

22 RUk

R

URk

R

UR (4.1.11)

In a similar way, we can obtain the following differential equation for the

annular disk 2. By the substitution of Eq. (4.1.6) into Eq. (4.1.1) with the

aid of r and given in Eq. (4.1.4), we get

)(

1)(2)(

200 rudr

dErfhrfk

b

rnh

k

)()1(

)(2)(200 ru

r

Erfhrfk

b

rnh

k

78

,01

)()(2

)(22

2

2

022

0

rudr

drErfh

rrfh (4.1.12)

where

.)(

kk

b

a

b

rn

erf (4.1.13)

Then multiplying (4.1.12) by

Erfh

r

)(2

)1(

0

2

we obtain the following simple form:

)(

2)(

)(2)(

)(2

22 ru

dr

dr

rf

rfrfb

rnk

rudr

dr

k

.0)1(

)(2)(

)(2)(223

E

rru

rf

rfrfb

rnk

k

(4.1.14)

Using the dimensionless variables given in (4.1.7) we obtain

U

eE

ekenRb

dR

dUe

kenR

eE

Rb

dR

Ud

E

kk

kkkk

kk

kk

kk

ARn

ARnARnk

ARn

ARnk

ARn

bR

)(

)()(2

)(

)(

)(

2

2

2

2

2

2

2

22

.023

E

R (4.1.15)

78

Multiplying (4.1.15) by2b

E gives the final differential equation for disk 2

as follows

R

UR

ee

eeeknR

R

UR

kk

kkk

nAnR

nAnRnAk

d

d

2

2

d

d

2

22

.0

2

2 3

RU

ee

eeeRknkk

kkk

nAnR

nAnRnAk (4.1.16)

§

The analytical solution for Eq. (4.1.11) may be available while it is not

available for Eq. (4.1.16). The modified numerical solutions may be

available for both cases. Firstly, we will get both the analytical and

numerical solutions for Eq. (4.1.11). If the numerical solutions are

compared will with the analytical, we can use them to solve Eq. (4.1.16)

for the second case.

The analytical solution for Eq. (4.1.11) concerning disk 1 can be written as

),()()()( 2211 RPRFcRFcRU (4.2.1)

where 1c and 2c are arbitrary constants and )(1 RF and )(2 RF are given

by:

,)( 22211

2 kkkkRRF

,22212

2 kkkkRRF

.

262)(

3

k

RRP

(4.2.2)

78

The substitution of Eq. (4.2.2) into Eq. (4.1.4) with the aid of the

dimensionless forms given in Eq. (4.1.7) gives the radial and

circumferential stresses in the following forms:

.)()()()()()(

)(

,)()()()()()(

)(

222

1112

222

1111

R

RP

dR

RdP

R

RF

dR

RdFc

R

RF

dR

RdFcR

R

RP

dR

RdP

R

RF

dR

RdFc

R

RF

dR

RdFcR

(4.2.3)

Finally, the dimensionless strains may be obtained easily using Eq.

(4.2.3) as well as the dimensionless forms given in Eq. (4.1.7). So, all of

radial displacement, stresses and strains can be completely determined

under the traction conditions on the inner and outer surfaces of the annular

disk. They can be expressed as

).1(at)0(0

),2.0(at)0(0

RbrUu

ARarUu (4.2.4)

The finite difference approach presented in Chapter 2 is used here to

solve the present problem numerically.

The resolution of the elastic problems of rotating annular disks with

variable thickness are to solve both second-order differential equations

(4.1.11) and (4.1.16) with the aid of the boundary conditions given in Eq.

(4.1.7). Equations (4.1.11) or (4.1.16) can be written in the following

general form:

),()()( )()()( RsURqURpU ll

ll

ll .2,1l (4.3.1)

77

where the prime )( denotes differentiation with respect to R and l denotes

the disk number. The functions )(,)(,)( RsRqRp lll may be given by

For annular disk 1:

,)(

)(,)21(

)(,)21(

)(21211 kR

RRs

R

kRq

R

kRp

(4.3.2)

For annular disk 2:

,

2

21)(2

kk

kkk

nAnR

nAnRnAk

ee

eeeknR

RRp

,

2

21)(

22

kk

kkk

nAnR

nAnRnAk

ee

eeeRkn

RRq

.)(2 RRs (4.3.3)

It is clear that the above problems have unique solutions because

),(),( RqRp ll and )(Rsl are continuous on [0.2, 1] and 0)( Rql on [0.2,

1]. The linear second-order boundary value problems given in (4.3.1)

requires that difference-quotient approximations be used for approximating

)(lU and )(lU . First we select an integer 0N and divided the interval

[0.2, 1] into )1( N equal subintervals, whose end points are the mesh

points ,RiRi for ,1...,,1,0 Ni where )1/(1 NR . At the interior

mesh points, ,iR ,...,,2,1 Ni the differential equation to the approximated

is

),()()()()()( )()()(ili

lili

lili

l RsRURqRURpRU (4.3.4)

If we apply the centered difference approximations of )()(i

l RU and

)()(i

l RU to Eq. (4.3.4), we arrive at the system:

76

)(1

)(2)(1

)(2

1)()(2)(2

1l

iill

iill

iil URpR

URqRURpR

),()( 2il RsR (4.3.5)

for each ....,,2,1 Ni The N equations, together with the boundary

conditions

.0

,0

1

0

NU

U (4.3.6)

Are sufficient to determine the unknowns ,iU 1...,,2,1,0 Ni . The

resulting system of Eq. (4.3.5) is expresses in the tri-diagonal NN -

matrix form:

,BUA (4.3.7)

where

.2,1,...,,2,1),()(

,2,1,2,...,,4,3,,2...,,2,1,0

,2,1,...,,3,2),(2

1

,2,1,1...,,2,1),(2

1

,2,1,...,,2,1),()(2

2

,,

1,

1,

2,

lNiRsRB

lijNjNiAA

lNiRpR

A

lNiRpR

A

lNiRqRA

ili

ijji

ilii

ilii

ilii

(4.3.8)

The solutions of the finite difference discretization of the two-point linear

boundary value problems can therefore be found easily even for very small

mesh sizes.

77

The modified Runge-Kutta's method presented in Chapter 2 may be

used here to solve both differential equations (4.1.11) and (4.1.16) with the

aid of the boundary conditions given in Eq. (4.1.7). Equations (4.1.11) or

(4.1.16) can be written in the following general form:

.2,1,,, )()()( lUURfU lll

l (4.3.9)

where the prime )( denotes differentiation with respect to R and l denotes

the disk number. The functions lf may be given by

,)(

)21()21(

2)1(

2)1(

1 kR

RU

R

kU

R

kf

(4.3.10)

)2(2

2

21U

ee

eeeknR

Rf

kk

kkk

nAnR

nAnRnAk

.

2

21 )2(

2RU

ee

eeeRkn

Rkk

kkk

nAnR

nAnRnAk

(4.3.11)

Runge-Kutta's iterative formulae for the second-order differential

equations are

,3

1 )(3

)(2

)(1

)()()(1

lllli

li

li

KKKURUU

.223

1 )(4

)(3

)(2

)(1

)()(1

llllli

li

KKKKR

UU

(4.3.12)

where R is the increment of the distance along the radial direction of the

rotating annular disks, and 4,3,2,1,)(

jKlj can be determined with the

following relations

78

,,,2

)( )()(2

)(1

li

liil

lUURf

RK

,1

,4

1

2

1,

22

)( )(1

)()(1

)()(2

)(2

lli

lli

liil

lK

RUKURU

RRf

RK

,1

,4

1

2

1,

22

)( )(2

)()(1

)()(2

)(3

lli

lli

liil

lK

RUKURU

RRf

RK

.2

,,2

)( )(3

)()(3

)()(2

)(4

lli

lli

liil

lK

RUKURURRf

RK

(4.3.13)

The numerical simulation starts from the inner boundary, where a trial

value of the first derivative of the radial displacement is assumed. With a

small distance R , the radial displacement and its first derivative at the

new position can be obtained using Eq. (4.3.12) and the radial displacement

is calculated. According to the difference between the computed radial

displacement and the known radial displacement at the outer boundary, the

initial trial value of the first derivative of the radial displacement at the

inner boundary is modified and the next iteration is carried out in the same

way. This iterative process is performed until both the boundary conditions

are simultaneously satisfied. Once the radial displacement is obtained, the

stresses and the strains in the rotating annular disks can be obtained using

Eq. (4.1.4) and Eq. (4.1.3) with the aid of the dimensionless given in Eq.

(4.1.7). This process may be easily done after getting the continuous form

of the radial displacement using the curve fitting and least square method.

So, all other quantities can be easily determined.

88

§

Some numerical examples for the rotating variable-thickness annular

disks will be given according the analytical and numerical solutions

)3.0( . According to Eq. (4.1.7), the radial displacement, stresses and

strains of the rotating variable-thickness annular disk are determined as per

the analytical solutions are compared with those obtained by the numerical

FDM solutions for disk 1 in Figures 4.3–4.9. The inner and outer radii of

the disk are taken to be a = 0.2 b (R = A = 0.2) and b (R = 1), and the

results are given in terms of the rotating angular velocity.

The results of the present FDM investigations of the radial displacement

,U for the annular disk 1 with different cases of the geometric parameters

k are reported in Tables 4.1, 4.2, 4.3, 4.4. For these examples, N = 8, 16, 32

and 64, so R has the corresponding values 0.1, 0.05, 0.025 and 0.0125,

respectively. The FDM gives results compared well with the exact

solution, especially for small values of R . The relative error between the

exact solution and the FDM one with 0125.0R at 6.0R may be

61061.6 for k = 4.0 (Table 4.1); 41032.2 for k = 4.0 (Table 4.2);

51013.5 for k = 2.0 (Table 4.3); and finally may be 41075.1 for k =

2.0 (Table 4.4).

88

Table 4.1: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 4.0 )

using finite difference method with different values of R .

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2000

0.001470751 0.001470384 --- --- --- 0.2125

0.002931746 0.002931115 0.002929253 --- --- 0.2250

0.004384915 0.004384095 --- --- --- 0.2375

0.005831236 0.005830283 0.005827474 0.005816901 --- 0.2500

0.007270953 0.007269912 --- --- --- 0.2625

0.008703742 0.008702645 0.008699407 --- --- 0.2750

0.010128823 0.010127695 --- --- --- 0.2875

0.011545055 0.011543916 0.011540554 0.011527877 0.011486712 0.3000

0.012951008 0.012949872 --- --- --- 0.3125

0.014345014 0.014343893 0.014340587 --- --- 0.3250

0.015725212 0.015724114 --- --- --- 0.3375

0.017089579 0.017088513 0.017085368 0.017073536 --- 0.3500

0.018435963 0.018434932 --- --- --- 0.3625

0.019762100 0.019761109 0.019758187 --- --- 0.3750

0.021065633 0.021064683 --- --- --- 0.3875

0.022344126 0.022343221 0.022340556 0.022330583 0.022299350 0.4000

0.023595080 0.023594219 --- --- --- 0.4125

0.024815936 0.024815121 0.024812728 --- --- 0.4250

0.026004089 0.026003321 --- --- --- 0.4375

0.027156893 0.027156171 0.027154052 0.027146194 --- 0.4500

0.028271668 0.028270991 --- --- --- 0.4625

0.029345700 0.029345068 0.029343218 --- --- 0.4750

0.030376252 0.030375665 --- --- --- 0.4875

0.031360564 0.031360020 0.031358429 0.031352622 0.031336430 0.5000

88

Table 4.1: Continued

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0.032295855 0.032295353 --- --- --- 0.5125

0.033179328 0.033178867 0.033177523 --- --- 0.5250

0.034008172 0.034007750 --- --- --- 0.5375

0.034779561 0.034779177 0.034778064 0.034774101 --- 0.5500

0.035490659 0.035490312 --- --- --- 0.5625

0.036138621 0.036138309 0.036137409 --- --- 0.5750

0.036720592 0.036720314 --- --- --- 0.5875

0.037233710 0.037233464 0.037232760 0.037230376 0.037226336 0.6000

0.037675107 0.037674892 --- --- --- 0.6125

0.038041909 0.038041724 0.038041197 --- --- 0.6250

0.038331238 0.038331080 --- --- --- 0.6375

0.038540210 0.038540078 0.038539711 0.038538618 --- 0.6500

0.038665938 0.038665831 --- --- --- 0.6625

0.038705534 0.038705449 0.038705223 --- --- 0.6750

0.038656102 0.038656040 --- --- --- 0.6875

0.038514750 0.038514707 0.038514603 0.038514506 0.038518179 0.7000

0.038278579 0.038278554 --- --- --- 0.7125

0.037944690 0.037944683 0.037944682 --- --- 0.7250

0.037510184 0.037510192 --- --- --- 0.7375

0.036972158 0.036972180 0.036972266 0.036972873 --- 0.7500

0.036327711 0.036327745 --- --- --- 0.7625

0.035573938 0.035573983 0.035574137 --- --- 0.7750

0.034707936 0.034707991 --- --- --- 0.7875

0.033726800 0.033726863 0.033727066 0.033728093 0.033734890 0.8000

0.032627626 0.032627695 --- --- --- 0.8125

0.031407507 0.031407581 0.031407818 --- --- 0.8250

0.030063540 0.030063617 --- --- --- 0.8375

0.028592817 0.028592898 0.028593149 0.028594317 --- 0.8500

88

Table 4.1: Continued

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0.026992436 0.026992430 --- --- --- 0.8625

0.025259489 0.025259483 0.025259820 --- --- 0.8750

0.023391072 0.023391067 --- --- --- 0.8875

0.021384280 0.021384276 0.021384588 0.021385627 0.021391127 0.9000

0.019236210 0.019236206 --- --- --- 0.9125

0.016943956 0.016943952 0.016944219 --- --- 0.9250

0.014504614 0.014504612 --- --- --- 0.9375

0.011915283 0.011915281 0.011915479 0.011916126 --- 0.9500

0.009173058 0.009173056 --- --- --- 0.9625

0.006275037 0.006275036 0.006275145 --- --- 0.9750

0.003218318 0.003218318 --- --- --- 0.9875

0 0 0 0 0 0.1000

Table 4.2: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 4.0 )

using finite difference method with different values of R .

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.006318406 0.006319823 0.006324098 0.006341554 0.006416461 0.3

0.014582647 0.014586212 0.014596940 0.014640340 0.014821081 0.4

0.023478567 0.023484234 0.023501276 0.023569983 0.023852901 0.5

0.031163058 0.031170274 0.031191961 0.031279254 0.031636653 0.6

0.035385152 0.035392942 0.035416350 0.035510473 0.035894534 0.7

0.033557488 0.033564501 0.033585569 0.033670231 0.034014924 0.8

0.022804891 0.022809421 0.022823026 0.022877677 0.023099839 0.9

0 0 0 0 0 1.0

88

Table 4.3: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 2.0 )

using finite difference method with different values of R .

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.010041299 0.010041411 0.010041782 0.010043771 0.010057939 0.3

0.020343564 0.020344437 0.020347087 0.020358162 0.020408370 0.4

0.029553817 0.029555317 0.029559843 0.029578343 0.029657343 0.5

0.036056881 0.036058731 0.036064301 0.036086902 0.036181342 0.6

0.038135919 0.038137804 0.038143474 0.038166400 0.038261175 0.7

0.034022491 0.034024083 0.034028870 0.034048183 0.034127528 0.8

0.021917214 0.021918179 0.021921079 0.021932767 0.021980592 0.9

0 0 0 0 0 1.0

Table 4.4: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk (k = 2.0 )

using finite difference method with different values of R .

Analytical FDM

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.007405693 0.007407059 0.007411182 0.007428019 0.007500009 0.3

0.016396849 0.016400030 0.016409601 0.016448292 0.016608721 0.4

0.025527401 0.025532162 0.025546471 0.025604118 0.025840498 0.5

0.032944472 0.032950243 0.032967580 0.033037311 0.033321674 0.6

0.036519308 0.036525287 0.036543241 0.036615388 0.036908666 0.7

0.033914044 0.033919238 0.033934831 0.033997454 0.034251489 0.8

0.022622846 0.022626097 0.022635855 0.022675029 0.022833712 0.9

0 0 0 0 0 1.0

88

Table 4.5: Dimensionless radial displacement U of a rotating variable-

thickness annular disk 1 (k = 4.0 ) using Richardson extrapolation method

with different values of R .

Analytical i4Ext i3Ext i2Ext i1Ext iR

0 0 0 0 0 0.2

0.011545055 0.011544991 0.011545036 0.011544779 0.011541599 0.3

0.022344126 0.022344073 0.022344109 0.022343880 0.022340995 0.4

0.031360564 0.031360521 0.031360550 0.031360365 0.031358019 0.5

0.037233710 0.037233677 0.037233699 0.037233555 0.037231723 0.6

0.038514750 0.038514725 0.038514742 0.038514635 0.038513281 0.7

0.033726800 0.033726784 0.033726795 0.033726724 0.033725827 0.8

0.021384280 0.021384272 0.021384278 0.021384242 0.021383793 0.9

0 0 0 0 0 1.0

Richardson extrapolation method with ,1.0R ,05.0 ,025.0 and

0.0125 and the obtained results are listed in Table 4.5. These

extrapolations are given, respectively by

,3

)1.0()05.0(4Ext1

RURU iii (4.4.1a)

,3

)05.0()025.0(4Ext 2

RURU iii (4.4.1b)

,3

)025.0()0125.0(4Ext3

RURU iii (4.4.1c)

.15

ExtExt16Ext 12

4ii

i

(4.4.1d)

Table 4.5 shows that all extrapolations results are correct to the decimal

places listed. In fact, if sufficient digits are maintained, the approximation

87

of i4Ext gives results those agree with the exact solution with maximum

difference error of 61054.5 at some of the mesh points.

Now the least square method and curve fitting are used for the discrete

results of the radial displacement U . So, one can get easily the stresses and

strains since we have U as a continuous function of R. The distributions of

the radial displacement, radial and circumferential stresses are presented in

Figure 4.3 for k = 4.0 and in Figure 4.4 for k = 4.0 . The numerical FDM

solution is compared with the exact analytical solution for the rotating

variable-thickness annular disk 1. It can be seen that the FDM can describe

the radial displacement, stresses and strains through the thickness of the

rotating annular disk very well enough.

86

Figure 4.3: Radial displacement U , radial stress 1 and circumferential stress

2 for disk 1, k = 0.4 (Analytical and FDM solutions).

Figure 4.4: Radial displacement U , radial stress 1 and circumferential stress

2 for disk 1, k = 4.0 (Analytical and FDM solutions).

87

Figure 4.5: Radial displacement U of the variable-thickness annular disk 1 for

two values of k (Analytical and FDM solutions).

Figure 4.6: Radial stress 1 in the variable-thickness annular disk 1 for two

values of k (Analytical and FDM solutions).

88

Figure 4.7: Circumferential stress 2 in the variable-thickness annular disk 1

for two values of k (Analytical and FDM solutions).

Figure 4.8: Radial strain 1 in the variable-thickness annular disk 1 for two

values of k (Analytical and FDM solutions).

888

Figure 4.9: Circumferential strain 2 in the variable-thickness annular disk 1 for

two values of k (Analytical and FDM solutions).

For the sake of completeness and accuracy, additional results for the

radial displacement, radial stress, circumferential stress, radial strain and

circumferential strain are presented in Figures 4.5–4.9 for two values of the

geometric parameters k. Figure 4.5 shows the radial displacement U

through the radial direction of the rotating annular disk 1 with k = 2.0 .

Similar results for the radial 1 and the circumferential 2 stresses are

plotted in Figures 4.6 and 4.7. In addition, similar results for the radial 1

and the circumferential 2 strains are plotted in Figures 4.8 and 4.9. Figure

4.5 shows that the maximum values for the radial displacement U occur at

7.0R for both k = 2.0 and k = 2.0 . Figure 4.6 shows that the radial

stress 1 for k = 2.0 is greater than the corresponding one for k = 2.0

when 48.02.0 R and 19.0 R . However, Figure 4.7 gives

888

circumferential stress 2 for k = 2.0 greater than that of k = 2.0 when

64.02.0 R and 191.0 R .

Once again, the radial strain 1 for k = 2.0 is greater than the

corresponding one for k = 2.0 during 9.046.0 R as shown in Figure

4.8. In addition, Figure 4.9 shows that the maximum value of the

circumferential strains 2 occur at 55.0R for k = 2.0 and at 6.0R for

k = 2.0 .

It is clear that, the FDM gives radial displacement, stresses and strains

consequently, with very good accuracy with the exact analytical solution.

So, it will trustily used to find the solution for Eq. (4.1.16) of the rotating

variable-thickness annular disk 2. As a result, the radial displacement,

stresses and strains in the rotating variable-thickness disk 2 are plotted in

Figures 4.17–4.22 according to all cases.

Some numerical examples for the rotating variable-thickness annular

disk will be given according the analytical and numerical solutions

)3.0( . Results determined as per the analytical solutions are compared

with those obtained by the numerical modified Runge-Kutta's method(R-K)

solutions in Figures 4.10–4.16 for disk 1. The inner and outer radii of the

disk are taken to be a = 0.2 b (R = A = 0.2) and b (R = 1), and the results

are given in terms of the rotating angular velocity

The results of the present investigations for the radial displacement U of

the annular disk 1 are reported in Tables 4.6–4.9 (k = 4.0 and k = 2.0 ).

For these examples, N = 8, 16, 32 and 64, so R has the corresponding

values 0.1, 0.05, 0.025 and 0.0125, respectively. The modified R-K method

gives results compared well with the exact solutions, especially for small

888

values of R . The relative error between the exact method and the

modified R-K method with 0125.0R at 6.0R may be 71083.4

for k = 4.0 (Table 4.6); 71036.3 for k = 4.0 (Table 4.7); 71082.5

for k = 2.0 (Table 4.8); and finally may be 71064.3 for k = 2.0

(Table 4.9).

888

Table 4.6: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 4.0 )

using modified Runge Kutta's method with different values of R .

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2000

0.001470751 0.001470744 --- --- --- 0.2125

0.002931746 0.002931725 0.002931725 --- --- 0.2250

0.004384915 0.004384885 --- --- --- 0.2375

0.005831236 0.005831204 0.005831204 0.005831204 --- 0.2500

0.007270953 0.007270924 --- --- --- 0.2625

0.008703742 0.008703716 0.008703716 --- --- 0.2750

0.010128823 0.010128799 --- --- --- 0.2875

0.011545055 0.011545032 0.011545032 0.011545032 0.011545032 0.3000

0.012951008 0.012950985 --- --- --- 0.3125

0.014345014 0.014344989 0.014344989 --- --- 0.3250

0.015725212 0.015725185 --- --- --- 0.3375

0.017089579 0.017089552 0.017089552 0.017089552 --- 0.3500

0.018435963 0.018435936 --- --- --- 0.3625

0.019762100 0.019762074 0.019762074 --- --- 0.3750

0.021065633 0.021065607 --- --- --- 0.3875

0.022344126 0.022344101 0.022344101 0.022344101 0.022344101 0.4000

0.023595080 0.023595055 --- --- --- 0.4125

0.024815936 0.024815911 0.024815911 --- --- 0.4250

0.026004089 0.026004065 --- --- --- 0.4375

0.027156893 0.027156870 0.027156870 0.027156870 --- 0.4500

0.028271668 0.028271644 --- --- --- 0.4625

0.029345700 0.029345677 0.029345677 --- --- 0.4750

0.030376252 0.030376230 --- --- --- 0.4875

0.031360564 0.031360542 0.031360542 0.031360542 0.031360542 0.5000

888

Table 4.6: Continued

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0.032295855 0.032295834 --- --- --- 0.5125

0.033179328 0.033179308 0.033179308 --- --- 0.5250

0.034008172 0.034008152 --- --- --- 0.5375

0.034779561 0.034779541 0.034779541 0.034779541 --- 0.5500

0.035490659 0.035490640 --- --- --- 0.5625

0.036138621 0.036138602 0.036138602 --- --- 0.5750

0.036720592 0.036720574 --- --- --- 0.5875

0.037233710 0.037233692 0.037233692 0.037233692 0.037233692 0.6000

0.037675107 0.037675090 --- --- --- 0.6125

0.038041909 0.038041893 0.038041893 --- --- 0.6250

0.038331238 0.038331222 --- --- --- 0.6375

0.038540210 0.038540195 0.038540195 0.038540195 --- 0.6500

0.038665938 0.038665924 --- --- --- 0.6625

0.038705534 0.038705519 0.038705519 --- --- 0.6750

0.038656102 0.038656089 --- --- --- 0.6875

0.038514750 0.038514737 0.038514737 0.038514737 0.038514737 0.7000

0.038278579 0.038278566 --- --- --- 0.7125

0.037944690 0.037944678 0.037944678 --- --- 0.7250

0.037510184 0.037510173 --- --- --- 0.7375

0.036972158 0.036972148 0.036972148 0.036972148 --- 0.7500

0.036327711 0.036327701 --- --- --- 0.7625

0.035573938 0.035573928 0.035573928 --- --- 0.7750

0.034707936 0.034707927 --- --- --- 0.7875

0.033726800 0.033726792 0.033726792 0.033726792 0.033726792 0.8000

0.032627626 0.032627618 --- --- --- 0.8125

0.031407507 0.031407500 0.031407500 --- --- 0.8250

0.030063540 0.030063532 --- --- --- 0.8375

0.028592817 0.028592811 0.028592811 0.028592811 --- 0.8500

888

Table 4.6: Continued

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0.026992436 0.026992430 --- --- --- 0.8625

0.025259489 0.025259483 0.025259483 --- --- 0.8750

0.023391072 0.023391067 --- --- --- 0.8875

0.021384280 0.021384276 0.021384276 0.021384276 0.021384276 0.9000

0.019236210 0.019236206 --- --- --- 0.9125

0.016943956 0.016943952 0.016943952 --- --- 0.9250

0.014504614 0.014504612 --- --- --- 0.9375

0.011915283 0.011915281 0.011915281 0.011915281 --- 0.9500

0.009173058 0.009173056 --- --- --- 0.9625

0.006275037 0.006275036 0.006275036 --- --- 0.9750

0.003218318 0.003218318 --- --- --- 0.9875

0 0 0 0 0 0.1000

Table 4.7: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 4.0 )

using modified Runge Kutta's method with different values of R .

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.009551535 0.009551520 0.009551520 0.009551520 0.009551520 0.3

0.018077905 0.018077889 0.018077889 0.018077889 0.018077889 0.4

0.025116278 0.025116265 0.025116265 0.025116265 0.025116265 0.5

0.029787275 0.029787265 0.029787265 0.029787265 0.029787265 0.6

0.030980716 0.030980708 0.030980708 0.030980708 0.030980708 0.7

0.027406526 0.027406521 0.027406521 0.027406521 0.027406521 0.8

0.017613812 0.017613809 0.017613809 0.017613809 0.017613809 0.9

0 0 0 0 0 1.0

887

Table 4.8: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 2.0 )

using modified Runge Kutta's method with different values of R .

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.010041299 0.010041270 0.010041270 0.010041270 0.010041270 0.3

0.020343564 0.020343533 0.020343533 0.020343533 0.020343533 0.4

0.029553817 0.029553791 0.029553791 0.029553791 0.029553791 0.5

0.036056881 0.036056860 0.036056860 0.036056860 0.036056860 0.6

0.038135919 0.038135903 0.038135903 0.038135903 0.038135903 0.7

0.034022491 0.034022480 0.034022480 0.034022480 0.034022480 0.8

0.021917214 0.021917208 0.021917208 0.021917208 0.021917208 0.9

0 0 0 0 0 1.0

Table 4.9: Numerical and analytical results for the dimensionless radial

displacement U of a rotating variable-thickness annular disk 1 (k = 2.0 )

using modified Runge Kutta's method with different values of R .

Analytical R-K

iR 0125.0R 025.0R 05.0R 1.0R

0 0 0 0 0 0.2

0.007405693 0.007405678 0.007405678 0.007405678 0.007405678 0.3

0.016396849 0.016396833 0.016396833 0.016396833 0.016396833 0.4

0.025527401 0.025527387 0.025527387 0.025527387 0.025527387 0.5

0.032944472 0.032944460 0.032944460 0.032944460 0.032944460 0.6

0.036519308 0.036519298 0.036519298 0.036519298 0.036519298 0.7

0.033914044 0.033914037 0.033914037 0.033914037 0.033914037 0.8

0.022622846 0.022622843 0.022622843 0.022622843 0.022622843 0.9

0 0 0 0 0 1.0

886

The numerical applications will be carried out for the radial

displacement, stresses and strains

The distribution of the radial displacement and stresses are presented in

Figure 4.10 for k = 2.0 and in Figure 4.11 for k = 2.0 . The modified R-K

numerical solution is compared with the exact analytical solution for the

rotating variable-thickness annular disk 1. It is clear that, the modified R-K

method gives all results with excellent accuracy when compared with the

exact analytical solution.

887

Figure 4.10: Radial displacement U , radial stress 1 and circumferential stress

2 for disk 1, k = 2.0 (Analytical and R-K solutions).

Figure 4.11: Radial displacementU , radial stress 1 and circumferential stress

2 for disk 1, k = 2.0 (Analytical and R-K solutions).

888

Figure 4.12: Radial displacement U of the variable-thickness annular disk 1 for

two values of k (Analytical and R-K solutions).

Figure 4.13: Radial stress 1 in the variable-thickness annular disk 1 for two

values of k (Analytical and R-K solutions).

888

Figure 4.14: Circumferential stress 2 in the variable-thickness annular disk 1

for different values of k (Analytical and R-K solutions).

Figure 4.15: Radial strain 1 in the variable-thickness annular disk 1 for two

values of k (Analytical and R-K solutions).

888

Figure 4.16: Circumferential strain 2 in the variable-thickness annular disk 1

for two values of k (Analytical and R-K solutions).

For the sake of completeness and accuracy, additional results for the

radial displacement, radial stress, circumferential stress, radial strain and

circumferential strain are presented in Figures 4.12–4.16 for two values of

the geometric parameters k. Figure 4.12 shows the radial displacement U

through the radial direction of the rotating annular disk 1 with k = 4.0 .

Similar results for the radial 1 and the circumferential 2 stresses are

plotted in Figures 4.13 and 4.14. In addition, similar results for the radial

1 and the circumferential 2 strains are plotted in Figures 4.15 and 4.16.

Figure 4.12 shows that the maximum values for the radial displacement U

occur at 66.0R for k = 4.0 and at 72.0R for k = 4.0 .

Figure 4.13 shows that the radial stress 1 for k = 4.0 is greater than

the corresponding one for k = 4.0 when 92.047.0 R . Also, Figure 4.14

shows that the circumferential stresses 2 for k = 4.0 are greater than

those for k = 4.0 when 63.02.0 R and 192.0 R .

888

However, Figures 4.15 shows that the radial strain 1 for k = 4.0 is

greater than that for k = 4.0 when 91.045.0 R . Figure 4.16 shows that

the circumferential strain 2 has two maximum values at 55.0R and

65.0R for k = 4.0 and k = 4.0 , respectively.

It can be seen from Figures 4.10–4.16 that the modified R-K method

can describe all results through the thickness of the rotating annular disk 1

very well enough. This puts into evidence the great role played by the

modified R-K method in the modeling of rotating variable-thickness

annular disks. So, it will trustily used to find the solution for Eq. (4.1.16) of

the rotating variable-thickness annular disk 2. As a result, the radial

displacement, stresses and strains in the rotating variable-thickness disk 2

are plotted in Figures 4.17–4.22 according to all cases.

The radial displacement U of the present FDM and modified R-K

method for the rotating variable-thickness annular disk 1 with k = 3.0 is

reported in Table 4.10. Additional results for the rotating variable-

thickness annular disk 2 with different cases of the geometric parameters k

and n are reported in Tables 4.11–4.13. For these examples, R have the

corresponding values 0.05, 0.025 and 0.0125. The absolute difference error

between the FDM method and the modified R-K method with 0125.0R

at 6.0R may be 71003.8 for the rotating variable-thickness annular

disk 1 (Table 4.10).

For the rotating variable-thickness annular disk 2, the absolute

difference errors between the FDM and the modified R-K method with

0125.0R at 6.0R are 61084.3 , 61083.4 and 61088.4 for k

= 2, n = 0.6 (Table 4.11), k = 1.4, n =2 (Table 4.12) and k = 0.8, n = 1.2

(Table 4.13), respectively.

888

Table 4.10: Numerical results for the dimensionless radial displacement

U of a rotating variable-thickness annular disk 1 (k = 3.0 ) using finite

difference method and modified Runge-Kutta's method with different

values of R .

R-K FDM

iR 0125.0R

025.0R 05.0R 0125.0R

025.0R 05.0R

0 0 0 0 0 0 0.2

0.010781538 0.010781538 0.010781538 0.010781089 0.010779729 0.010774912 0.3

0.021350021 0.021350021 0.021350021 0.021350058 0.021350149 0.021351079 0.4

0.030483888 0.030483888 0.030483888 0.030484393 0.030485886 0.030492326 0.5

0.036684477 0.036684477 0.036684477 0.036685280 0.036687668 0.036697594 0.6

0.038365453 0.038365453 0.038365453 0.038366354 0.038369044 0.038380082 0.7

0.033905194 0.033905194 0.033905194 0.033905993 0.033908380 0.033918114 0.8

0.021665590 0.021665590 0.021665590 0.021666087 0.021667574 0.021673616 0.9

0 0 0 0 0 0 1.0

Table 4.11: Numerical results for the dimensionless radial displacement

U of a rotating variable-thickness annular disk 2 (k = 2, n = 0.6) using finite

difference method and modified Runge-Kutta's method with different

values of R .

R-K FDM

iR 0125.0R 025.0R 05.0R 0125.0R

025.0R 05.0R

0 0 0 0 0 0 0.2

0.019725494 0.019725494 0.019725494 0.019720284 0.019704787 0.019644494 0.3

0.032371119 0.032371119 0.032371119 0.032365790 0.032349896 0.032287646 0.4

0.039920734 0.039920734 0.039920734 0.039916071 0.039902138 0.039847362 0.5

0.042650255 0.042650255 0.042650255 0.042646417 0.042634941 0.042589705 0.6

0.040385019 0.040385019 0.040385019 0.040382058 0.040373197 0.040338203 0.7

0.032796136 0.032796136 0.032796136 0.032794095 0.032787987 0.032763822 0.8

0.019483847 0.019483847 0.019483847 0.019482787 0.019479614 0.019467043 0.9

0 0 0 0 0 0 1.0

888

Table 4.12: Numerical results for the dimensionless radial displacement

U of a rotating variable-thickness annular disk 2 (k = 1.4, n = 2) using finite

difference method and modified Runge-Kutta's method with different

values of R .

R-K FDM

iR 0125.0R

025.0R 05.0R 0125.0R

025.0R 05.0R

0 0 0 0 0 0 0.2

0.022014588 0.022014588 0.022014588 0.022009059 0.021992604 0.021928343 0.3

0.034902930 0.034902930 0.034902930 0.034896925 0.034879002 0.034808470 0.4

0.042048345 0.042048345 0.042048345 0.042042757 0.042026057 0.041960064 0.5

0.044272218 0.044272218 0.044272218 0.044267385 0.044252931 0.044195665 0.6

0.041592580 0.041592580 0.041592580 0.041588719 0.041577165 0.041531314 0.7

0.033680710 0.033680710 0.033680710 0.033677994 0.033669864 0.033637563 0.8

0.020024225 0.020024225 0.020024225 0.020022803 0.020018544 0.020001610 0.9

0 0 0 0 0 0 1.0

Table 4.13: Numerical results for the dimensionless radial displacement

U of a rotating variable-thickness annular disk 2 (k = 0.8, n = 1.2) using

finite difference method and modified Runge-Kutta's method with different

values of R .

R-K FDM

iR 0125.0R 025.0R 05.0R 0125.0R

025.0R 05.0R

0 0 0 0 0 0 0.2

0.021154179 0.021154179 0.021154179 0.021148100 0.021129991 0.021059214 0.3

0.033897407 0.033897407 0.033897407 0.033890965 0.033871729 0.033796064 0.4

0.041231100 0.041231100 0.041231100 0.041225289 0.041207919 0.041139349 0.5

0.043745679 0.043745679 0.043745679 0.043740799 0.043726202 0.043668452 0.6

0.041327629 0.041327629 0.041327629 0.041323827 0.041312452 0.041267374 0.7

0.033588323 0.033588323 0.033588323 0.033585704 0.033577864 0.033546761 0.8

0.020008980 0.020008980 0.020008980 0.020007631 0.020003591 0.019987551 0.9

0 0 0 0 0 0 1.0

888

Figure 4.17: Radial displacement U , radial stress 1 and circumferential stress

2 for disk 2 (FDM and R-K solutions).

Figure 4.18: Radial displacement U of the variable-thickness annular disk 2 for

different values of k and n (FDM and R-K solutions).

887

Figure 4.19: Radial stress 1 in the variable-thickness annular disk 2 for

different values of k and n (FDM and R-K solutions).

Figure 4.20: Circumferential stress 2 in the variable-thickness annular disk 2

for different values of k and n (FDM and R-K solutions).

886

Figure 4.21: Radial strain 1 in the variable-thickness annular disk 2 for

different values of k and n (FDM and R-K solutions).

Figure 4.22: Circumferential strain 2 in the variable-thickness annular disk 2

for different values of k and n (FDM and R-K solutions).

887

Furthermore, the plots of different results for the variable-thickness

annular disk 2 are given in Figures 4.17–4.22 for various values of k and n.

Figure 4.17 shows that both the modified R-K method and FDM give

accurate results. It is obvious in Figures 4.17–4.22 that the absolute

difference error between the two methods may be neglected. Figure 4.18

shows that the case of k = 1.4, n = 2 gives the largest displacements while

the case of k = 2, n = 0.6 gives the smallest ones. The difference between

cases may be increases at 6.0R .

The same discussion may be available for the remainder Figures. The

difference between cases may be increase at 2.0R (the inner edge) as

show in Figures 4.19 and 4.21. Also it is increasing at 3.0R as given in

Figure 4.20. Finally, this difference error is increasing at 4.0R as shown

in Figure 4.22.

§

This chapter presents analytical and numerical solutions for BVP of

rotating variable-thickness annular disks with a general, arbitrary

configuration. The governing equation is derived from the equilibrium

equation and the stress-strain relationship. The calculation of the rotating

annular disk is turned into finding the solution of a second-order

differential equation under the given conditions at two boundary fixed

points. Finite difference method and modified Runge-Kutta's method are

introduced to solve the governing equation for two types of annular disks,

in which the analytical solution of one of them only is available. A number

of numerical examples is studied. The results from the analytical and

numerical FDM or modified R-K method solutions are compared. The

proposed FDM and modified R-K method approaches give very agreeable

results to the analytical solution. The FDM and modified R-K method

888

presented here may be trustily used for the boundary-value problems that

their analytical solutions are not available.

021

The object of this thesis is to study the rotating of variable-thickness

solid and annular disks. The rotating variable-thickness disks are subjected

to different boundary conditions at the inner and outer edges. The

boundary values problems presented herein are solved for solid and

annular disks at the same angular velocity.

The governing equations of the presented disks are solved analytically

and numerically. The analytical solution is presented with the help of

Whittaker's functions. However the numerical solutions are based upon

both the finite difference method (FDM) as well as the modified Runge-

Kutta's (R-K) method.

Firstly, we discussed the rotating of variable thickness solid disk. The

outer edge is free of prescribed forces and so the radial stress should be

vanished. The results for stress function and the corresponding stresses are

presented in Chapter 3. Additional results for radial displacement and

strains are also presented. The results presented in this chapter are given by

using the analytical and numerical FDM solutions. The FDM solution

gives results compared well with those given by the analytical one except

for minor cases at the outer edge of the solid disk.

Secondly, the rotating variable-thickness annular disk is presented in

Chapter 4. The inner and outer edges of the annular disk are clamped and

so the radial displacement should be vanished. Two annular disks with

variable configuration are discussed. The analytical solution is available for

disk 1. Both the modified R-K method and FDM solutions are also used.

020

They give very accurate results comparing with the analytical solution. The

results are presented in some tables and plotted in some figures.

The analytical solution is not available for dealing with disk 2. So, the

two numerical solutions are used only. In addition, some numerical results

are tabulated and shown graphically using all solution for disk 1. However,

only the numerical solutions are presented graphically for disk 2.

It is to be noted that the relative error between any of the numerical

solutions and the analytical solution is very small. Also the absolute

difference error between the two numerical solutions is very small and may

be neglected. The modified R-K method still gives results more accurate

than the FDM comparing with the analytical solution.

Moreover, the effects of different parameter are also investigated

throughout this thesis. For example, the effect of the geometric parameters

n and k for solid and annular disks is discussed and three cases for different

values of these parameters are studied. These cases show the concavity and

convexity of the solid and annular variable-thickness disks.

211

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No. 2, pp. 145–153 (2011).

المملكة العربية السعودية

وزارة التعليم العالي

جامعة أم القرى

كلية العلوم التطبيقية

قسن العلوم الرياضية

مكار متغير الس و مقارنة عددية لطرق حل مسألة القيمة الحدية لقرص د

بحث جكويلي هقدم ليل درجة الواجسحير

في

(جطبيقية رياضيات)الرياضيات

هقدم هي

عبد هللا عيد األحودي بث سوزاى

إشرافجحث

ورقهبارز ز بي أشرف/ األسحاذ الدكحور

الرياضيات الحطبيقية أسحاذ

قسن الرياضيات

كلية العلوم

جاهعة الولك عبد العسيس

م 3122 -هـ 1432

بسم اهلل الرمحن الرحيم

شهد اهلل أنه ال إله إال هى واملالئكة ( )وأولى العلم قائما بالقسط

81آية رقم ) آل عمران )

ب

إن انز عه انعطاء تال حذد

يع نعح انتعة انجذ اقار أي داعثا جاح تعذ نإ

انحاج تانسجد انذعاء ي تتثعا تأعى قهتى طشق ف نإ

إن أت أي أخات أخ

ذأ

تانجم عتشافإأل تاكست انعهح تقهث حث شكش

ـج

فانحذ . ايمء يات األسضيمء ف يمء انسااخ يثاسكا طثا ا كثش انحذ هلل حذا

سأن أتاو ز انشسانح انعهح إتفضم ي عه عى تأشكش ن تعان عه يا ان آخشا هلل أل

.سثحا أ فع تا تقثها خانصح نج انكشى

،أششف يثاسص صقس / يتا انتقذش نسعادج األستار انذكتستجضم انشكش اإل أتجكا

تج إسشادف انز نى ذخش جذا األستار تقسى انشاضاخ تجايعح انهك عثذ انعضض،

أفاد ي يع عه انغضش فأسأل هللا عض جم أ جض ،حض انجد إنز انشسانح إلخشاج

. جعم كم يا أفاد ت ف يضا حساتأع كم خش

انشكش انصل نز انجايعح انشايخح جايعح أو انقش يثهح ف أساتزتا األفاضم

.انعهو انشاضحعه سأسى سئس سئسح قسى

تقسى األستار، عثذ انعط يحذ عثذ هللا / انذكتساألستار انشكش نسعادج خانص كا أتقذو ت

نتحه يشقح انسفش يشاجعح انشسانح حضس نهتحكى تكشي انشاضاخ تجايعح انطائف،

.تاقشح انشسانح

تقسى األستار انساعذ ،يشانخانذ عثذ انعظى /نسعادج انذكتس انشكشخانص كا أتقذو ت

.اتياقشقثل تشاجعح انشسانح نتكشي تجايعح أو انقش انعهو انشاضح

تفاا أخات أخ انز شجع نانذ انذتانعشفا تخانص انشكش كا أتقذو

.ف يساعذت

د

سائضت اىظخت اىحيقت اسة اىذ األقشاص حه اىظشت اىؼيت اىبحدأطبحج

حذ أ .اىنانتاىذست اىؼيت فألخا ظشا خشاس اال اسة حاصث اخا األقشاص اىذ

ضداد ح ،اىارس اىذست ف خخيف األصضة اىنانت اىنشبائت إسخخذاا حؼذداسغ بسبب

.لأت ز األقشاص ػذا حن خغشة اىس

:اىخالطت

أ اىحيه اىؼذدت اىسخخذت حؼط دقت ػاىت خائش شبت خطابقت غ أت اىذساست

اىحو اىخحيي فإ ن االػخاد ػيا ف دساست بؼغ اىسائو اىخاطت باألقشاص اىذسات

.اىخ ظؼب اىحظه ػي حو ححيي ىال اىظخت اىحيقت خغشة اىس

مزىل اىؼذدت ىسائو اىق اسخخاس بؼغ اىحيه اىخحييت اىشساىت ز ذف اى

.اىنتب طشق اىحو اث قاساىل ػو مافت اس ظج أ حيق خغش اىس اىحذت ىقشص د

اصغ اىسخخذت يخض ػا زيت باىش فظه أسبؼححخ اىشساىت ػي قذت ػات

:ي ام ؼشػا سبؼاأل فظهاى. اىؼشبتخ االضيضت باىيغ

:األه ىفظوا

سطات اىؼادالث األساست حشو ػالقاث حؼشف اإلحذاراث اإل فظوحاه زا اى

( كقا )اإلصاد -فؼاه، ؼادالث االحضا، ػالقاث اإل(ػالقاث مش) اإلصاحت-فؼاهاإل

".سمسخ شاف"ؼادالث

:زااى ىفظوا

ح ػشع بؼغ طس اىؼادالث اىخفاػيت بؼغ أاع سائو اىق فظوف زا اى

قذ حج اقشت طشقت . اىحذت اىخخيفت باإلػافت إى رىل ح ػشع بؼغ طشق اىحو اىؼذدت

ـ

اىشبغح ػشع طشقت باإلػافت إى رىل . اىؼذىت" محا"" سش"اىفشق اىحذد طشقت

.حج قاست اىخائش ػذدا ححييا ".سخشاسد س"األطغش طشقت االسخقشاء ىـ

:ىذاىزا فظواى

ح اسخخاس طسة حذة ىؼادىت حشمت دسا قشص ظج خغش ىفظوف زا ا

قذ ح "(. ك"قا )فؼاه اإلصاد ل باسخخذا اىؼادالث األساست اىؼالقاث ب اإلاىس

باإلػافت إى . ل ر قطغ ػشع اخخاسحقذ اىحو اىخحيي ىذسا قشص ظج خغش اىس

Finite Difference Methodرىل ح حقذ اىحو اىؼذد باسخخذا طشقت اىفشق اىحذد

(FDM) .اس حج قاست اىحيه اىخحييت ا فظوف ات زا اى ىؼذدت ىيقشص اىظج اىذ

.لخغش اىس

قذ ىحظ أ اىحو اىؼذد ؼط خائش حناد حن خطابقت غ اىحو اىخحيي ػذ ششط

اس جظحذت ؼت ا ػذا بؼغ اىحاالث ػذ اىحافت اىخاسصت ىيقشص اى ح قاست اىخائش .اىذ

.ػذدا ححييا ػشػا باا

:ابغشاى ىفظوا

حذ ح األخز . لاىشابغ بذساست دسا بؼغ األقشاص اىحيقت خغشة اىس فظوخخض اى

اسإلف ا قذ ح اىحظه ػي اىخائش اىخحييت ىيحاىت . ػخباس حاىخ خخيفخ ىيقشص اىحيق اىذ

اس مزىل ح اىحظه ػي اىحيه اىؼذدت باسخخذا :األى ىيقشص اىحيق اىذ

اىفشق اىحذد طشقت.Finite Difference Method (FDM)

اىؼذىت " محا"" سش"طشقتModified Runge-Kutta's Method (R-K).

قذ حج دساست دقت اىحيه اىؼذدت بقاسخا باىحو اىخحيي اىاحش ف ز اىحاىت دساست

أخشا ح ػشع . د اىحو اىخحييإنات اسخخذا اىحيه اىؼذدت ىيحاىت األخش حذ خؼزس ص

اس خغش صاداث اإلبؼغ حصؼاث اإلصاحت اإل فؼاالث خاله اىبؼذ اىقطش ىيقشص اىحيق اىذ

.قاست اىخائش ػذدا ححييا ػشػا باا .اىضات تل ػذ فس اىسشػاىس