Two solutions for the BVP of a rotating variable-thickness solid disk

doi:10.4236/ns.2011.32021

Vol.3, No.2, 145-153 (2011) Natural Science

http://dx.doi.org/10.4236/ns.2011.32021

Two solutions for the BVP of a rotating

variable-thickness solid disk

Ashraf M. Zenkour1,2, Suzan A. Al-Ahmadi1

1Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia; Coresponding Author:

zenkour@kau.edu.sa

2Department of Mathematics, Faculty of Science, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt; zenkour@sci.kfs.edu.eg

Received 23 November 2010; revised 25 December 2010; accepted 28 December 2010.

ABSTRACT

This paper presents the analytical and numeri-

cal solutions for a rotating variable-thickness

solid disk. The outer edge of the solid disk is

considered to have free boundary conditions.

The governing equation is derived from the ba-

sic equations of the rotating solid disk and it is

solved analytically or numerically using finite

difference algorithm. Both analytical and nu-

merical results for the distributions of stress

function and stresses of variable-thickness solid

disks are obtained. Finally, the distributions of

stress function and stresses are presented and

the appropriate comparisons and discussions

are made at the same angular velocity.

Keywords: Rotating, Solid Disk, Variable

Thickness, Analytical Method, Finite Difference

Method

1. INTRODUCTION

The theoretical and experimental investigations on the

rotating solid disks have been widespread attention due

to the great practical importance in mechanical engi-

neering. Rotating disks have received a great deal of

attention because of their widely used in many me-

chanical and electronic devices. They have extensive

practical engineering application such as in steam and

gas turbines, turbo generators, flywheel of internal com-

bustion engines, turbojet engines, reciprocating engines,

centrifugal compressors and brake disks. The problems

of rotating solid disks have been performed under vari-

ous interesting assumptions and the topic can be easily

found in most of the standard elasticity books [1,2]. For

a better utilization of the material, it is necessary to al-

low variation of the effective material or thickness prop-

erties in one direction of the solid disk.

The problems of rotating variable-thickness solid disks

are rare in the literature. Most of the research works are

concentrated on the analytical solutions of rotating iso-

tropic disks with simple cross-section geometries of

uniform thickness and specifically variable thickness.

The solution of a rotating solid disk with constant thick-

ness is obtained by Gamer [3,4] taking into account the

linear strain hardening material behavior. The inelastic

and viscoelastic deformations of rotating variable-

thickness solid disks have been presented in the litera-

ture [5-8]. Eraslan [5], and Eraslan and Orcan [6] have

analytically studied rotating disks of exponentially

varying thickness and of linearly strain hardening mate-

rial. Eraslan [7] has presented the stress distributions in

elastic-plastic rotating disks with elliptical thickness pro-

files using Tresca and von Mises criteria. Zenkour and

Allam [8] have developed analytical solution for the

analysis of deformation and stresses in elastic rotating

viscoelastic solid and annular disks with arbitrary cross-

sections of continuously variable thickness.

As many rotating components in use have complex

cross-sectional geometries, they cannot be dealt with

using the existing analytical methods. Numerical meth-

ods, such as the finite element method [9], the boundary

element method [10] and Runge-Kutta’s algorithm [11],

can be applied to cope with these rotating components.

You et al. [11] have numerically studied rotating solid

disks of uniform thickness and constant density as well

as annular disks of variable thickness and variable den-

sity. In a recent paper, Zenkour and Mashat [12] have

presented both analytical and numerical solutions for the

analysis of deformation and stresses in elastic rotating

disks with arbitrary cross-sections of continuously vari-

able thickness.

In this article, a unified governing equation is firstly

derived from the basic equations of the rotating variable-

thickness solid disk and the proposed stress-strain rela-

tionship. The analytical solution for rotating solid disk

with arbitrary cross-section of continuously variable

thickness is presented. Next, the finite difference method

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

146

(FDM) is also introduced to solve the governing equa-

tion. A comparison between both analytical and numeri-

cal solutions is made. Finally, a number of application

examples are given to demonstrate the validity of the

proposed method.

2. BASIC EQUATIONS

As the effect of thickness variation of rotating solid

disks can be taken into account in their equation of mo-

tion, 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 vari-

able thickness. After considering this effect, the equation

of motion of rotating disks with variable thickness can

be written as



22

d0,

dr

hrhh r

r





 (1)

where r



and





are the radial and circumferential

stresses, h is the variable thickness of the disk, r is the

radial coordinate,



is the material density of the ro-

tating 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

uu

rr





 (2)

where r



and





are the radial and circumferential

strains, respectively. The above geometric relations lead

to the following condition of deformation harmony:



d0.

dr

r







 (3)

For the elastic deformation, the constitutive equations

for the variable-thickness solid disk can be described

with Hooke’s law

,,

rr

rEE







 







(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

22

1d

,.

d

rr

hrh r





 

 (5)

Substituting Eq.5 into Eq.4 , one obtains

22

11 d,

d

11d .

d

rr

Ehrhr

r

Ehrhr















 























(6)

3. FORMULATION AND ANALYTIC

ELASTIC SOLUTION

The substitution of Eq.6 into Eq.3 produces the fol-

lowing confluent hypergeometric differential equation

for the stress function ():r





2

23

ddd

1dd

d

13 0.

d

rh

rr

hr r

r

rh hr

hr

















 





(7)

The boundary conditions for the rotating solid disk are

at 0,

0at .

r

rb









 (8)

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 distribu-

tion according to the following case:



0e,

k

r

nb

hrh







 (9)

where 0

h 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 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 vari-

able-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

0

h while the parameter k determine the shape of the

profile.

Introducing the following dimensionless forms:



 



2

0

12 2

/

3,

1,

,,,

1

,,.

r

Rrb

b

Rr

bh

E



 



 

 



 







(10)

Then, Eq.7 may be written in the following simple

form



2

2 3

2

dd

11e0.

d

k

kknR

RknRRknRR

R











(11)

The general solution of the above equation can be

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

147

(a)

(b)

(c)

Figure 1. Variable-thickness solid disk profiles for (a) k = 0.7

and n = 2, (b) k = 1.5 and n = 2 and (c) k = 2.5 and n = 0.5.

written as

  



 



22

,1 ,

,2 ,

ed

d,

k

kn R

R

ij ij

R

ij ij

RRMRC FW

WRCF M













 



















(12)

where 1

C and 2

C are arbitrary constants,



is a

dummy parameter, ,ij

M

and ,ij

W are Whittaker’s

functions











,,

,, ,,, ,

kk

ij ij

MRMijnRWRWijnR

(13)

in which

11

,,0.

2

ijk

kk





  (14)

In addition, the function



F

R is given in terms of

Whittaker’s functions by

  

22

2

,1,,1,

e.

1

k

kn

R

ijijij ij

R

FR WRMRkM RWR









(15)

The substitution of Eq.12 into Eq.5 with the aid of the

dimensionless forms given in Eq.10 gives the radial and

circumferential stresses in the following forms:

  



 



22

1

1,1,

,2 ,

ed

d,

k

kn R

R

ijij

R

ijij

RRMRC FW

WRCFM





 

















(16)



2

d

e.

d3

k

nR R

RR









 (17)

Here, the first derivative of the stress function





R

with respect to R may be given easily by using Eq.12.

Note that the first derivatives of Whittaker’s functions

,ij

M

and ,ij

W can be represented by



 

1

,,

2

1

1,

2

1

,,1,

2

d

,

d.

d

k

ij ij

ij

k

ijiji j

k

MR nRiMR

RR

ijM R

k

WRnRiWRWR

RR











 











(18)

Finally, the stress function





R and consequently

the stresses





1R



and





2R



may be determined

completely after applied the dimensionless of the

boundary conditions given in Eq.8.

4. FINITE DIFFERENCE ALGORITHM

The resolution of the elastic problem of rotating solid

disk with variable thickness is to solve a second-order

differential equation, Eq.11, under the given boundary

conditions









010



 such that





10







20



.

Eq.11 can be written in the following general form:





,pRqR sR

 

  (19)

where the prime ()



denotes differentiation with re-

spect to R and

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

148



2

1,

e.

k

nR

knR

pR R

kn R

qR R

sR R















(20)

It is clear that the above problem has a unique solution

because

 

,,pRqR and





s

R are continuous on

[0,1] and



0qR on [0,1]. The linear second-order

boundary value problem given in Eq.19 requires that

difference-quotient approximations be used for ap-

proximating 

 and 

. First we select an integer

0N and divided the interval [0,1] into





1N



equal subintervals, whose end points are the mesh points

,

i

RiR for 0,1,..., 1,iN where





1/ 1RN .

At the interior mesh points, ,

i

R 1, 2,...,,iN the dif-

ferential equation to the approximated is

 





.

 

  

iiiiii

RpRRqR RsR (21)

If we apply the centered difference approximations of



i

R



 and



i

R



 to Eq.21, we arrive at the system

(see Eq.22):

for each 1, 2,...,.iN The N equations, together with

the boundary conditions

0

1

0,

N



(23)

are sufficient to determine the unknowns ,

i



0,1, 2,...,1iN

. The resulting system of Eq.22 is ex-

presses in the tri-diagonal NN



-matrix form:

,

A

B (24)

where

2

,

,1

,,

2

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

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

2

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

2

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

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

ii i

ij ji

ii

ARqRiN

R

ApRiN

R

ApRiN

AAiN jNji

BRsRiN









 



 

  

 

(25)

The solution of the finite difference discretization of

the two-point linear boundary value problem can there-

fore be found easily even for very small mesh sizes.

5. NUMERICAL EXAMPLES AND

DISCUSSION

Some numerical examples for the rotating variable-

thickness solid disks will be given according the ana-

lytical and numerical solutions





0.3



. According to

Eq.10, the stress function , the radial stress 1



and

the circumferential stress 2



determined as per the

analytical solution are compared with those obtained by

the numerical FDM solution.

The results of the present investigations for the stress

function



are reported in Table 1 for rotating vari-

able-thickness solid disk with k = 2.5 and n = 0.5. For

this example, N = 9, 19, 39 and 79, so R has the cor-

responding values 0.1, 0.05, 0.025 and 0.0125, respec-

tively. 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 0.0125,R



 may be less than 4

1.3 10



.

Richardson’s extrapolation method is applied here

with 0.1,0.05,0.025,R



 and 0.0125 and the obtained

results are listed in Table 2. These extrapolations are

given, respectively by



 

1

4 0.050.1

Ext ,

3

ii

i

RR 

 (26a)



 

2

4 0.0250.05

Ext ,

3

ii

i

RR 

 (26b)



 

3

4 0.01250.025

Ext ,

3

ii

i

RR 

 (26c)

21

4

16 ExtExt

Ext ,

15

ii

i



 (26d)

Table 2 shows that all extrapolations results are cor-

rect to the decimal places listed. In fact, if sufficient dig-

its are maintained, the approximation of 4

Ext i gives

results those agree with the exact solution with maxi-

mum difference error of 9

1.0 10

 at some of the mesh

points. Additional results for the stress function



are

reported for rotating variable-thickness solid disk with k

= 0.7 and n = 2 in Table 3 and with k = 1.5 and n = 2 in

Table 4. Once again, the FDM gives results compared

well with the exact analytical solution, especially for

small values of R



.

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 and circumferential stresses



 



22

11

12 1,

22

iiii iii

RR

pRR qRpRR sR





 



 

 



 (22)

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

149

Table 1. Dimensionless stress function



of a rotating variable-thickness solid disk (k = 2.5, n = 0.5).

FDM

i

R

0.1R 0.05R 0.025R



 0.0125R





Analytical

0 0 0 0 0 0

0.0125

0.0250

0.0375

0.0500

0.0625

0.0750

0.0875

0.1000

---

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

0.001318247

0.002635012

0.003948803

0.005258119

0.006561453

0.007857295

0.009144132

0.010420449

0.1125

0.1250

0.1375

0.1500

0.1625

0.1750

0.1875

0.2000

---

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

0.011684731

0.012935466

0.014171142

0.015390252

0.016591293

0.017772771

0.018933196

0.020071091

0.2125

0.2250

0.2375

0.2500

0.2625

0.2750

0.2875

0.3000

---

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

0.021184986

0.022273424

0.023334964

0.024368176

0.025371649

0.026343989

0.027283820

0.028189791

0.3125

0.3250

0.3375

0.3500

0.3625

0.3750

0.3875

0.4000

---

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

0.029060568

0.029894846

0.030691343

0.031448803

0.032166002

0.032841742

0.033474861

0.034064227

0.4125

0.4250

0.4375

0.4500

0.4625

0.4750

0.4875

0.5000

---

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

0.034608743

0.035107349

0.035559023

0.035962780

0.036317677

0.036622811

0.036877324

0.037080400

0.5125

0.5250

0.5375

0.5500

0.5625

0.5750

0.5875

---

0.037375156

---

0.037331532

---

0.037366529

---

0.037182291

---

0.037231777

0.037329791

0.037374198

0.037364369

0.037299729

0.037179570

0.037003958

0.037231269

0.037329209

0.037373544

0.037363648

0.037298942

0.037178902

0.037003051

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

150

0.6000 0.036832186 0.036786324 0.036774810 0.036771929 0.036770967

0.6125

0.6250

0.6375

0.6500

0.6625

0.6750

0.6875

0.7000

---

0.032936503

---

0.035291940

---

0.032876919

---

0.036140911

---

0.035278284

---

0.034185494

---

0.032862001

0.036483294

0.036137738

0.035734998

0.035274868

0.034757198

0.034181892

0.033548911

0.032858271

0.036482282

0.036136679

0.035733896

0.035273729

0.034756025

0.034180690

0.033547685

0.032857027

0.7125

0.7250

0.7375

0.7500

0.7625

0.7750

0.7875

0.8000

---

0.025353741

---

0.029540441

---

0.025296129

---

0.031308165

---

0.029525248

---

0.027515416

---

0.025281728

0.032110047

0.031304368

0.030441421

0.029521449

0.028544751

0.027511682

0.026422653

0.025278128

0.032108789

0.031303102

0.030440152

0.029520183

0.028543493

0.027510437

0.026421427

0.025276927

0.8125

0.8250

0.8375

0.8500

0.8625

0.8750

0.8875

0.9000

---

0.014247568

---

0.020171897

---

0.014209433

---

0.022828123

---

0.020159404

---

0.017281209

---

0.014199988

0.024078628

0.022824727

0.021517051

0.020156281

0.018743148

0.017278432

0.015762966

0.014197627

0.024077458

0.022823594

0.021515961

0.020155240

0.018742161

0.017277507

0.015762106

0.014196841

0.9125

0.9250

0.9375

0.9500

0.9625

0.9750

0.9875

1.0000

---

0

---

0.007463350

---

0

---

0.010922960

---

0.007458083

---

0.003814000

---

0

0.012583344

0.010921087

0.009211874

0.007456766

0.005656864

0.003813309

0.001927282

0

0.012582635

0.010920462

0.009211340

0.007456327

0.005656526

0.003813078

0.001927164

0

Table 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).

i

R

1

Ext i 2

Ext i 3

Ext i 4

Ext i 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

A. M. Zenkour et al. / Natural Science 3 (2011) 145-153

151

Table 3. Dimensionless stress function



of a rotating variable-thickness solid disk (k = 0.7, n = 2).

FDM

Analytical

0.0125R





0.025R





0.05R

0.1R

i

R

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 4. Dimensionless stress function



of a rotating variable-thickness solid disk (k = 1.5, n = 2).

FDM

Analytical

0.0125R



 0.025R



 0.05R 0.1R

i

R

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

since we have  as a continuous function of R. The

distributions of the stress function, radial and circum-

ferential stresses are presented in Figure 2. The numeri-

cal 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 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 and stresses are presented

in Figures 3-5 for different values of the geometric pa-

rameters k and n. Figure 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.

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