### Volume 13, Issue 1, Winter and Spring 2005, Page 1-69

##### VIBRATIONAL CHARACTERISTICS OF A ROTATING SHAFT CONTAINING A TRANSVERSE CRACK

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 1-16

DOI:
10.33899/rengj.2005.45557

ABSTRACT:

The influence of a transverse crack upon the dynamic behavior of a rotating

shaft is studied. Introduction of such a crack results in lower transverse natural

frequencies due to the added local flexibility. The strain energy release function is

related to the compliance of the cracked shaft that is to the local flexibility due to

introduction of crack. This function is related to the stress intensity factor, which for

transverse of a shaft with a crack has a known expression. As a result, the local

flexibility of the shaft due to the presence of the crack has been computed. This result,

can be further utilized to yield the dynamic response of a shaft with complex geometry.

Starting from the equation of motion for the shaft under bending to derive the

expression of calculating the natural frequency of the shaft.

Two cases of fixing the shaft are suggested in this study to investigate and

analyze the vibration characteristics of the shaft with and without cracks. The

fundamental natural frequency showed strong dependence on the crack depth, This

dependence is smaller as the order of the frequency increase. Experimental results are in

close agreement with those practical from the theoretical analysis. Finally, the results

showed that the change in dynamic response due to the crack is high enough to allow

the detection of the crack and estimation of its magnitud

##### Microprocessor-Based Controller for the Uninterruptible Power Supply Voltage Regulation

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 1-17

DOI:
10.33899/rengj.2005.45576

Abstract

Good regulation and stability are important factors to be considered in

designing of the uninterruptible power supplies systems, which depend on the load

requirements. On the other side, the cost factor roles the proper design selection of

the uninterruptible power supply, especially for the commercial applications. A pulse

width modulation uninterruptible power supplies are considered to have good

features over the rival one.

This paper presents a suggested method for the controlling of the

uninterruptible power supplies to regulate the output voltage, by using an easy

practicable, low cost, and one-sensor, microprocessor-based regulator. This

regulator circuit depends on minimizing the hardware complicity with efficient

software. The practical results show that a good and reliable regulation performance

in the applications when the fluctuation in both input DC voltage and load occurred,

such as the applications using the solar cells or batteries as the input voltage source

supplying variable load conditions.

2004/12/ 2004

##### THE MEMORIZATION BEHAVIORS OF DIFFERENT MIOS STRUCTURES

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 17-24

DOI:
10.33899/rengj.2005.45562

ABSTRACT: In this chapter the various kinds of charge storage cells are discussed

as a result of examining many samples with different structures. The C-V, I-V and R-V

measurements of the structures confirm the memorization capability of MIOS devices.

The examined structures reveal three kinds of memory actions. The first one is the

charge storage capability which can be shown through (C-V) curve shifting as the device

was exposed to certain stress for a certain time. The second is the electronic switching

that is demonstrated by the fact that the switching between ON and OFF states and

back to original state can only be obtained by inverting the polarity of the applied bias

voltage. The third kind of memorization action is that the device can be switched into a

variety of stable intermediate resistance states. The new resistance state is determined

by the height of the programming pulse applied to the device. This memory action is

noticed from R-V characteristic and known as a nonvolatile analogue memory behavio

##### The Effect of the Rail Materials and the Superconducting Coils on the Lifting Force of the Magnetically Leviated Trains

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 18-32

DOI:
10.33899/rengj.2005.45578

been studied the effect of the rail materials and the

superconducting coils on the lifting force of the magnetically leviated trains.

Concentration on the Super-conducting coils for creation of the lifting force was

done also, in order to minimize the current without decreasing the lifting force a

new frame coil was designed such that we take benefit of all the allowed area of

the base.

The rail is a material tape in which lifting force is formed when a magnet

is moved over it. Detailed study of the rail materials which can be used as rails

was done, lifting force of the Aluminum and Copper was studied. The lifting force

##### Bifurcation and Voltage C ollapse in the Electrical Power Systems

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 25-41

DOI:
10.33899/rengj.2005.45565

13 1 2005

Regional Referred Scientific Journal

Published

by the College of Engineering

Mosul University

Volume 13 Number2 1 2005

Abbas Fadheel Dawood

For Enquires write to Editor In Chief.

AL-RAFIDAIN ENGINEERING

College of Engineering / Mosul University/ Mosul-IRAQ

http://rafidain.5u.com

E-mail: rafiengg2004@yahoo.com

The views expressed in the issue are those of the authors

and do not reflect the views of the Editorial Board or the

Policy of the College of Engineering.

Secretary:

Professor Dr. Sabah Mohammed Jamel

Dr. Sami Abdul Mawjoud

Dr. Abdullah Y. Tayib

Dr. Ahmed Khorsheed Al-Sulaifanie

Al-Rafidain Engineering Vol. 13 No. 1 2005

ENGLISH SECTION

CONTENTS

No. Title Page No.

1. Vibrational characteristics of Rotating Shaft

Containing a Transverse Crack

By

Mohammed Jamel, S. , Al – Sarraf, Z.S. and ALRawi,

M.N . . . . . . . . . . . 1

2. The Memorization Behaviors of different MIOS

Structures

By

Mohamad , W.F. and Ali, L.S. . . . . . . . . . . . . . 17

3. Bifurcation and Voltage Collapse in the Electrical

power Systems

By

Al- Sammak, A.N.B. . . . . . . . 25

4. Performance of Outdoor MIMO System and Effect

of Antenna Separation

By

Abosh, A.M. ……..

42

5. Characteristics of Flow Over Normal and Oblique

Weirs with semicircular Crests

By

Noori, B.M.A and Chilmeran, T.A.H. ………

49

6. Coefficient of Discharge of Chimney Weir Under

Free and Submerged Flow Conditions

By

Hayawi, H.A.M., Yahya, A.A.g and Hayawi,

G.A.M. 62

Al_Rafidain engineering Vol.13 No.1 2005

1

VIBRATIONAL CHARACTERISTICS OF A ROTATING

SHAFT CONTAINING A TRANSVERSE CRACK

Dr. SABAH MOHAMMED JAMEL

PROFESSOR

ZIAD SHAKEEB AL-SARRAF MOHAMMED NAJEEB AL-RAWI

ASSISTANT LEACTURE ASSISTANT LEACTURE

Mechanical Engineering Department, College of Engineering, Mosul University

ABSTRACT:

The influence of a transverse crack upon the dynamic behavior of a rotating

shaft is studied. Introduction of such a crack results in lower transverse natural

frequencies due to the added local flexibility. The strain energy release function is

related to the compliance of the cracked shaft that is to the local flexibility due to

introduction of crack. This function is related to the stress intensity factor, which for

transverse of a shaft with a crack has a known expression. As a result, the local

flexibility of the shaft due to the presence of the crack has been computed. This result,

can be further utilized to yield the dynamic response of a shaft with complex geometry.

Starting from the equation of motion for the shaft under bending to derive the

expression of calculating the natural frequency of the shaft.

Two cases of fixing the shaft are suggested in this study to investigate and

analyze the vibration characteristics of the shaft with and without cracks. The

fundamental natural frequency showed strong dependence on the crack depth, This

dependence is smaller as the order of the frequency increase. Experimental results are in

close agreement with those practical from the theoretical analysis. Finally, the results

showed that the change in dynamic response due to the crack is high enough to allow

the detection of the crack and estimation of its magnitude.

الصفات الاهتزازية للعمود الدوار المتضمن شقًا مستعرضًا

أ.د. صباح محمد جميل ملا علي

السيد زياد شكيب عبد الباقي الصراف السيد محمد نجيب عبد الله الراوي

مدرس مساعد مدرس مساعد

في هذا البحث تم دراسة تأثير الشق العرضي على السلوك الداينميكي للعمود الدوار. ابتداءًا من معرفة

ان وجود الشق في العمود يقلل من قيمة الترددات الطبيعية العرضية من خلال وجود المرونة الموقعية المتكونة

بمنطقة الشق . كما ان دالة طاقة الانفعال المتحررة والمتعلقة بمنطقة الشق من خلال مطاوعة الشق للعمود

الدوار خلال وجود المرونة الموقعية ، فإن هذه الدالة مرتبطة بمعامل شدة الاجهاد كتعبير لوجود الشق العرضي

. لذا فقد تم حساب المرونة الموقعية للعمود خلال منطقة الشق . حيث ان النتائج التي تم الحصول عليها يمكن

الاستفادة منها في تحديد الاستجابة الداينميكية للعمود ذات التركيب المعقد .

ابتداءًا من معادلة الحركة للعمود المتعرض للانحناء فقد تم تحليل وحساب معادلة التردد الطبيعي

لظروف تثبيت العمود . وقد اعتمدت الدراسة من خلال اخذ حالتين لتثبيت العمود ومن ثم دراسة وتحليل

B & K ) الصفات الاهتزازية للعمود في حالة وجود وعدم وجود شق وذلك باستخدام جهاز محلل الاهتزازات

لتحليل الموجات الاهتزازية حيث بينت النتائج أن التردد الطبيعي الاساسي يعتمد بشكل كبير (Type 2515

على عمق الشق وهذا الاعتماد يقل كلما زادت قيمة الأس للتردد نفسه .

بينت النتائج أخيرًا أن التغير في الاستجابة الداينميكية خلال وجود الشق يعد كافيًا في الكشف وتقدير

قيمة الشق .

Submitted 20 th Feb. 2004 Accepted 4th Jan 2005

Al_Rafidain engineering Vol.13 No.1 2005

2

NOMENCLATURE:

A Cross-section area of the shaft (m2)

a Crack depth (mm)

C Local flexibility (Compliance) (m/N)

C1, C2, C3, C4 Constant

Cw Wave velocity (m/sec)

D Diameter of the shaft (mm)

E Young modulus of elasticity (N/m2)

I Second moment of area (m4)

KIII Stress intensity factor

L Length of the shaft (m)

M Bending moment (N.m)

P(x) Uniform load (N)

R Radius of the shaft (mm)

r Distance to the crack

T Torque (N.m)

V Shear force (N)

ν Poisson’s ratio

X Displacement (mm)

Y Deflection (mm)

Greek

n β Frequency factor

ρ Density (kg/m3)

ω Frequency with crack (HZ)

o ω Frequency without crack (HZ)

n ω Natural Frequency (HZ)

1. INTRODUCTION

Since the mid-seventies the dynamic behavior of cracked shaft has been

investigated increasingly because damages in turbines, generators, pumps, and other

machines occurred quite often. This caused costly shutdowns of entire plants and was

sometimes followed by the total loss of the machine. Fracture of a shaft which means

crack are originated at points of stress concentration either inherent in design or

introduced during fabricate on or operation.

Cracks defined as micro or macro interrupt the continuums are in principle

unavoidable. Also the initiation occurs during the vibration especially when the shaft is

unbalance or mis-align [1].

Singularity in elastic structure can introduce their dynamic behavior. Jones and

O’Donnnell [2] showed that axisymmetric solids have considerable local flexibility at

their junctures. Cracks are associated with local flexibilities that can introduce

considerable local flexibilities, which influence considerably the dynamic response of

Al_Rafidain engineering Vol.13 No.1 2005

3

the structure. Such analyses have been reported for turbine vanes [3], welded plates [4]

and for framed structures [5]. It was shown experimentally that changes in natural

frequencies due to cracks can be safely detected in certain machines and structures and

their magnitude can be estimated. Cracks often appear in a variety of machinery.

2. LOCAL FLEXIBILITY

Sih and Loeber [6,7] studied the somewhat similar problem of transverse wave

scattering about a penny-shaped crack. They studied the scattering of a given wave due

to the penny-shaped crack by way of the field equation solved by a finite Hankel

transform. Although the same procedure could be used for problem at hand, and the

energy method was utilized, based on the wealth of data existing for the strain energy

release function. Also by using the vibration analyzer devise to calculate the frequency

of the shaft then to make a relation between the change in frequency with the

compliance.

A transverse crack of depth (a) is considered on a shaft of radius (R). The shaft

has local flexibility due to crack; it’s depending on the direction of the applied forces.

We considered just only bending deformation, and the axial force which give coupling

with transverse motion of the cracked shaft will not considered here, also shear stress

are not considered. Therefor the shaft is bent by a pure bending moment.

The strain energy in the shaft due to a torque T is

2

T 2C U = (1)

where C is the local flexibility (compliance) of the shaft due to crack.

2( )

1

2

1 2

a R a

T C

A

G U

∂ −

∂

=

∂

∂

=

π (2)

Hertzberg [8] suggested that by measuring the flexibility of a test specimen or a

component model, with various crack depth (a), the value of the gradient

a

C

∂

∂ as function

of (a) could be determined, leading to the determination of the strain energy release

function. Miller [9] demonstrated that the energy release rate G could be related to the

stress intensity factor K as

2μ

2

III G = K (3)

Where μ is the shear modulus and the mode III stress intensity factor III K is defined by

the relation

Al_Rafidain engineering Vol.13 No.1 2005

4

0

2

sin

2

( ) terms of order r

r

KIII + + ⎥⎦

⎤

⎢⎣

= ⎡

θ

π

τ (4)

Giving the shear stresses in the vicinity of the crack at distance (R) from its tip.

Equation (2) and (3) yield

2( ) 2

2

R a

T

K

da

dC = III π −

μ (5)

Integrating

= ∫ −

a

III R a da

T

C K

0

2

2

2(π )

μ (6)

An expression is needed for the stress intensity factor III K for the problem at

hand. For a shaft with a crack Bueckner [10] has outlined a method for the

determination of K as a function of the crack depth. Benthem and Koiter [11] have

approximated the stress intensity factor K for a solid cylinder with a crack through the

following expression: -

⎥⎦

⎤

⎢⎣

= ⎡ + 2 + 2 + 3 + 4 + 20.208 5

2

1

128

35

16

5

8

3

2

1 1

8

K 3 λ λ λ λ λ λ (7)

where λ = (R - a) /R, Fig. 1.

Fig 1. Geometry of a shaft with a transverse crack

The dimensionless stress intensity factor K is defined by the relation

R r

a R a

R a

K T

2

( ) 1

( )

2 1/ 2

3 ⎥⎦

⎤

⎢⎣

⎡ −

−

=

π

τ (8)

Therefore, comparison with equation (4) yields

2R

a

Al_Rafidain engineering Vol.13 No.1 2005

5

K

R

a R a

R a

K T III

1/ 2

3

( )

( )

2

⎥⎦

⎤

⎢⎣

⎡ −

−

=

π (9)

The flexibility ( C ) in dimensionless form becomes

∫ −

−

=

a

K a da

R a

R

R

R a

R

a

R

R C

0

2

3

3 5

( )

( )

1 2( )

4

π μ π

(10)

The integral has the value

3. EQUATION OF MOTION

In order to find the special function of the natural frequency of the shaft it

should be calculate the equation of motion that representing the analysis of the shaft

under the shear and bending effects, then to find the deflection of the shaft. A horizontal

shaft is used with a length (L), and a uniform distributed load P (x) on the whole shaft.

This accomplished by take a segment from the shaft in order to study the force effect.

Fig. 2.

Fig 2. Effect of Forces on the rotating shaft

We assume that the bending moment of the shaft is M (x,t) and the shear force is

V (x,t), then the segment have a displacement (x) from the left end and have a length

(dx). By taking the summation of the forces in the vertical direction equal to zero.

V + Pdx − (V + dV ) = 0 (11)

0.0017(1 / ) 0.008(1 / ) 0.092

0.0086(1 / ) 0.0044(1 / ) 0.0025(1 / )

0.035(1 / ) 0.01(1 / ) 0.029(1 / )

7 9

3 4 6

4 2

+ − + − −

+ − + − + −

= − − + − + − +

a R a R

a R a R a R

C a R a R a R

M(x,t) M(x,t) +dM(x,t)/dx

V(x,t) V(x,t) +dV(x,t)/dx

P(x,t)

dx

x

y(x,t)

Al_Rafidain engineering Vol.13 No.1 2005

6

And the summation of the moments applied on the shaft equal to zero.

M − (M + dM) +Vdx = 0 (12)

From the equation (11) and (12), we get

2

2 ( , ) ( , )

x

P x t M x t

∂

∂

= (13)

From the strength of materials [12], the following relation ship is derived

between the elastic curve (curvature) and bending moment, and also bending stiffness

(EI) of the shaft as below.

EI

M

R

1 =

(14)

After applying the equation above we get the function of fourth order as,

4

4 ( , ) ( , )

x

P x t EI y x t

∂

∂

= (15)

Depending on the Newton‘s second law the equation (15) written as

2

2 ( , ) ( , ) ( )

t

P x t A x dx y x t

∂

∂

= ρ (16)

where ρ = density of the shaft metal

Applying Equation (16) and (15) in equation (13), get

2

2

4

4 ( , ) ( ) ( , ) ( , ) ( )

t

P x t A x dx y x t

x

EI x y x t

∂

∂

= −

∂

∂

ρ (17)

For the case of the free vibration, P (x,t)=0. Then equation (17) may be written as

( , ) ( , ) 0

2

2

4

4

2 =

∂

∂

+

∂

∂

t

y x t

x

C y x t w (18)

A

C EI w ρ

= (19)

where w C =Wave velocity

Al_Rafidain engineering Vol.13 No.1 2005

7

Initial conditions

Since, equation (18) involves a second order derivative to time and a fourth order

derivative with respect to (x), two initial conditions and four boundary conditions are

needed for finding a solution. This is accomplished by using the separation of variables

technique.

Let a function y (x,t) be the product of two separate functions, one with respect

to (x) and the another with respect to (t).

y(x,t) (x).g (t) n n =φ (20)

now,

( ) . ( )

2

2

2

2

x

t

g t

t

y

n

n φ

∂

∂

=

∂

∂

(21)

( ) . ( )

4

4

4

4

g t

x

x

x

y

n

n

∂

∂

=

∂

∂ φ

(22)

Substitution in equation (18), to get

. ( ) 0

( )

. ( )

( )

2

2

4

4

2 =

∂

∂

+

∂

∂

x

t

g t g t

x

C x n

n

n

n

w φ

φ

(23)

2

2

2

4

2 4 ( )

( )

( ) 1

( ) n

n

n

n

n

w

t

g t

x g t

x

x

C ω

φ

φ

= −

∂

∂

=

∂

∂

(24)

( ) 0

( ) 4

4

4

− =

∂

∂

x

x

x

n n

n β φ

φ

(25)

where , 2

2

4

w

n

n C

ω

β =

then the general solution become

y(x,t) C cosβx C sinβx C coshβx C sinhβx 1 2 3 4 = + + + (26)

where C1, C2, C3, C4 constants, and the natural frequency is given by:-

Al_Rafidain engineering Vol.13 No.1 2005

8

4

2

AL

EI

n nL ρ

ω = β (27)

4. THEORETICAL AND EXPERIMENTAL INVESTIGATION

The research contributes a study on a model of two cases of fixing end condition

of rotating shaft to analysis the vibration behavior and effect on the dynamic

characteristics with and without a crack, and also to find the change of local flexibility

(compliance).

The First Case:

(Fixed simply supported shaft)

The shaft is fixed to the left end and the other end is simply supported Fig. 3.

The shaft used in this study, had a diameter (D=8 mm), and a length is (L=0.6m),

density ( ρ =7800 Kg/m3), the Young modulus (E=207*109 N/m2), second moment of

area (I=π*D4/64 m4), and Frequency factor ( nL1 β =3.926602)[3].

Fig 3. Model of the first state of fixing the shaft

First the value of the natural frequency of the shaft from equation (27) was

calculated, In case of no crack. Then the calculation repeated experimentally to find the

natural frequency of the rotating shaft, by means of vibration analyzer (B & K Type

2515) to investigate the vibration characteristics of shaft.

Second, it is used equation (10) to find the local flexibility of the shaft and

change this property by increasing the depth of crack, and effect this on the vibration

response of the shaft. Then in order to find the values of frequency due to crack

theoretically, we can put the values get from equation (10) in the relation between the

local flexibility (C) and change in natural frequency with and without a crack [13].

L R2 R1

M W(N/m)

x

Al_Rafidain engineering Vol.13 No.1 2005

9

⎥ ⎥ ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣

⎡

−

⎟ ⎟⎠

⎞

⎜ ⎜⎝

⎛

= 1 1

2

o

D

C L

ω

ω

Experimental model Fig. 4 is accomplished where a D.C Motor through

coupling connects the shaft to the left is made for this purpose, and pinned to the right

end by use bracket. The speed of shaft must be controlled by using voltage variable

transformer (shown below) how give the range between (0-250) volts. The calculation

of frequency was taken using a portable vibration analyzer (B & K Type 2515) to

investigate the vibration spectrum of rotating shaft with and without crack. The

vibration signal was received from the accelerometer that put contact to the near point

of the rotating shaft (the accelerometer was fixed on the bearing of shaft). Then

transform to the vibration analyzer to analysis by using (F.F.T.) relation, and the output

shown by the digital monitor screen of the vibration device.

Fig 4.Experimental model of the shaft

The experimental result is plotted in Fig. 5.Vs the theoretical function. In the

experiment, shafts were firstly examined by calculating the natural frequency and

investigate the vibration dynamic behavior. After that the crack was made by means of

saw-cut which is supported transversely to the center of the shaft, also the depth of

crack in this study is taken between (a/D = 0 - 0.75).

Al_Rafidain engineering Vol.13 No.1 2005

10

Fig 5.Frequency drop vs crack depth ratio

Due to figure above the measured value of the frequency change ω /ω o against

the relative crack depth is done, here ω is the transverse natural frequency with the

crack and o ω the same frequency without the crack. There will be some deference

between the theoretical and experimental results of change of natural frequency with

crack depth ratio.

So this is normally because due to experimental part there is some parameter

effect to the frequency like the effect of rotation speed of the shaft, the accuracy of

crack depth. Also in case of low lubrication of the shaft and bearings then the friction

occurred and causes change in frequency values, in spite of the fixing of accelerometer

to the near point of the rotating shaft. All these points will causes some difference

between the theoretical and experimental results, but on the other side the experimental

results are in close agreement with the theoretical.

The Second Case:

(Fixed simply supported shaft with a concentrated load at the center)

The shaft is fixed to the left end and the other end is simply supported to the

right but with concentrated load (rotor disk) at the center Fig. 6. The shaft was used in

this study, with diameter (D=8 mm), and a length is (L=0.6,0.5,0.4m), density ( ρ =7800

Kg/m3), the Young modulus (E=207*109 N/m2). The disk have a mass (m=0.139 Kg)

and the density of disk ( ρ =2770 Kg/m3), the second moment of area (I=π*D4/64 m4),

Frequency factor ( nL1 β =3.926602).

0.0 0.2 0.4 0.6 0.8 1.0

(a/ D)

0.90

0.92

0.94

0.96

0.98

1.00

1.02

(ω/ωο)

D=8 mm

L=600 mm

Present Work (Theoritical)

Present Work (Experimental)

(Theoretical

Al_Rafidain engineering Vol.13 No.1 2005

11

Fig 6. Model of the second state of fixing the shaft

As in the first case, we calculate the value of the natural frequency of the shaft in

case of no crack and no concentrated load; this is accomplished by using equations

mentioned before. Then we repeat it after fixing the disk in the middle of shaft length,

then make a crack near the disk position and calculate the change of frequency due to

variable crack depth Fig. 7. Using three lengths of shaft (0.6,0.5,0.4m) does this and the

mass of disk is (0.139 Kg).

Second we calculate experimentally the value of the natural frequency of the

shaft and change this value in case of no crack, no load (disk) and with disk and crack

for three length of shaft as in Fig. 8.The theoretical and experimental results of figures

(7,8) below show a good close in values, and there will be some difference because due

to experimental part of the effect of rotation speed of the shaft, and the accuracy of

crack depth. Also the fixed of the accelerometer to the near point of the rotating shaft.

All will exhibit some change in values in comparison with theoretical results.

Fig 7.Effect the crack and the load (disk)

On the change of frequency (Theoretical)

L/2 R R 2 1

M

W(N/m)

x

L/2

W1

350 400 450 500 550 600 650

Length (mm)

40

60

80

100

120

140

160

180

Frequency (Hz)

Theoritical

m1=0.139 Kg

Perfect Shaft

With a Disk

With a Disk & Crack

Theoretical

Al_Rafidain engineering Vol.13 No.1 2005

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Fig 8. Effect the crack and the load (disk)

On the change of frequency (Experimental)

5. THE COMPLIANCE (LOCAL FLEXIBILITY)

To find the local flexibility of the rotating shaft theoretically, we used equation

(10) to find the local flexibility of the shaft and note this change of property by

increasing the depth of crack and effect this on the vibration response of the shaft.

In this work it suggest the range of crack depth between (0-0.75 a/D) then put

these values in equation (10) was derived before, in order to find the change of

flexibility due to increasing crack. In experiment we used the relation between the local

flexibility (C) and change in natural frequency with and without a crack [13]. Fig. 9.

Fig 9. Theoretical against Experimental results for

The dimensionless crack depth against local flexibility

350 400 450 500 550 600 650

Length (mm)

40

60

80

100

120

140

160

180

Frequency (Hz)

Experimental

m1=0.139 Kg

Perfect Shaft

With a Disk

With a Disk & Crack

1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3

Local Flexibility (C)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(a / D)

Theoritical Work

Experimental Work

Theoretical

W k

Al_Rafidain engineering Vol.13 No.1 2005

13

From the results of change of frequency and equation (10) the cracked shaft

local flexibility (compliance) was computed and entered in figure above, as a function

of the crack depth. At small crack depths (a/D) there is a considerable discrepancy

between theoretical and experimental results which was to be expected due to the

difficulty in accurate measurement of small frequency differences which appear for

cracks with (a/D) in the range (0-0.4).

The dimensionless Local flexibility functions, equation (10) are plotted In Fig.

10. As in figure below,

Fig 10. Dimensionless Flexibility of the Cracked Shaft

In Bending, and In Tension

They observed the difference between the values of flexibility that calculated

from the equation (10) theoretically and compare it by the values contributed by

Dimarogonas [13] who study the compliance of the stationary cracked shaft with open

crack.

The results showed the variable in points were gets from analytical solution

because in this study the shaft was under bending only and other property was neglected

.So these assumptions will leads to some difference between the two studies, but they

have the same behavior in changing the local flexibility due to the crack depth ratio.

6. DISCUSSION

The natural frequency of a rotating shaft found to be considerably influenced by

the presence of a transverse crack. The quantitative evaluation of this effect based on

the derivation of an equation of motion to derive the formula of calculating the natural

frequency of the rotating shaft. Also it is depending on the strain energy function to get

the integral relation between the local flexibility and the stress intensity factor.

By the present method, it is notice from the curves mentioned above that the

natural frequency of the rotating shaft will be decreased by increasing the depth of crack

1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3

Local Flexibility (C)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(a / D)

Theoritical Work Bending)

Dimarogonas (Tension)

Theoretical Work (Bending)

Al_Rafidain engineering Vol.13 No.1 2005

14

refer to the changing of the vibration spectrum of the shaft Fig (11). Also increasing the

crack depth rapidly decreases the values of frequencies. This is done by using a portable

vibration analyzer (B & K Type 2515) with magnetic acceleration which support to the

near point of rotating shaft

It is noticed that the vibration characteristic of the rotating shaft changes due to

the crack depth, which causes reduction in natural frequencies. Also this lower of

frequency will increase by increasing the crack depth so it’s used the ratio of the crack

to the diameter of the shaft in the range (0-0.75).

The effect of adding the mass on the shaft (rotor disk) causes to decrease the

values of natural frequencies, And by increasing the crack depth on the shaft which lead

to lowered in natural frequencies due to supporting the disk. As mentioned above.

The crack on the rotating shaft will change in some property like the local

flexibility. So the local flexibility of a shaft in bending due to the crack is evaluated

from the theoretical and experimental results relating to the derivation of the strain

energy release function to the crack depth, contributed by some authors.

These methods can have many practical applications because there is a wealth of

analytical results for strain energy release function. For present work it’s noticed that

local flexibility increased by increasing the crack depth and this observed by calculating

the theoretical values of local flexibility from the equation derived above. And for the

experimental results the calculate the values of natural frequencies of the rotating shaft

with and without crack then used the expression which depend on finding the flexibility

from the change of frequencies to get the local flexibility.

It is noticed that through the crack detectability. Cracks of smaller than 0.2

relative crack depth can be identified only in a quite environment by a skilled observer.

For such depths above 0.2 the identification is very easy.

For industrial application this level of crack detectability is rather adequate for

most application. Moreover, careful measurement and good knowledge of the uncracked

shaft behavior might render the method applicable even for relative crack depths of the

order of 0.1.

Finally, this work will represent a technique for non-destructive testing methods

depending on, use the vibration analysis and the spectrum of vibration and monitoring it

on a screen. So it can used also for identification of the location and the magnitude of

the crack on a rotating shaft, without direct inspection, even at running conditions. It

allows also for continuous monitoring in shaft in service, especially for machine which

has welded rotors and frequent inspections are impractical.

Al_Rafidain engineering Vol.13 No.1 2005

15

7.REFERENCES

[1] Donald, J. and Wulpi, “Failures of Shafts”, Metallurgical Consultant, 2000.

[2] Jones, D. P. and O’Donnel, W.J. “Local flexibility for axisymmetric junctures”.

Trans ASME J. Engng Ind. 1-5 (1971).

[3] Rao, S.S., “Mechanical Vibration”, 3rd edition, 1995.

[4] Chondros, T. G. and Dimarogonas, A. D. “Identification of cracks in welded joints

of complex structures”, J. Sound and Vibration 69, pp. 531-538. 1980.

[5] Chondros, T. G. and Dimarogonas, A. D. “Identification of cracks in circular plates

welded at the contour”, ASME J. paper No.79-DET-106. Design Engng Tech,

Conf., St. Louis, and U.S.A. (Sept. 1980).

[6] Sih, G. C. and Loeber, J. E. “Vibration of an Elastic Solid Containing a Penny-

Shaped Crack”. J. Acoust. Soc. Am.44, pp. 1237-1245 (1968).

[7] Loeber, J. F. and Sih, G. C. “wave Scattering about a Penny-Shaped Crack on a

Bimaterial Interface, in Dynamic Crack Propagation”. (Ed. G. Sih), pp. 513-528,

Nordhoff, Leyden (1973).

[8] Richard. W. Hertzberg. “Deformation and Fracture Mechanics of Engineering

Materials”. 4th edition. John Wiley & Sons, Inc. (1996).

[9] Miller, K. J., “An Introduction to Fracture Mechanics”, Mechanical and Thermal

behavior of Metallic Materials, pp. 97-131, (1982).

[10] Bueckner, H. F. “Field Singularities and related integral representations”. In

Methods of Analysis and Solution of Crack Problems, (Ed. G. Sih), pp. 239.

Noordhoff, Leyden. (1973).

[11] Benthem, J. P. and Koiter, W. T. “Asymptotic approximations to crack problems”.

ibid, pp. 174-198. (1981).

[12] Morrow, H. W., “Static and Strength of Materials”, 3rd edition, Prentice Hall,

(1998).

[13] Dimarogonas, A., and Massouros, G., “Torsional Vibration of a Shaft with a

Circumferential Crack”, Engineering Fracture Mechanics, Vol. 15, No.34, pp.

439-444, (1981).

Al_Rafidain engineering Vol.13 No.1 2005

16

200

250

300

350

400

450

500

mplitude (nm/s)

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s) (

x/L)=0.5

(2a/D)=0.19

Freq.=87.4 Hz

(x/L)=0.5

(2a/D)=0.476

Freq.=69.6 Hz

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

Without Crack

Freq.=91.2 Hz

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

(x/L)=0.5

(2a/D)=0.19

Freq.=87.4 Hz

(x/L)=0.5

(2a/D)=0.476

Freq.=69.6 Hz

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

(x/L)=0.1

(2a/D)=0.19

Freq.=90 Hz

(x/L)=0.1

(2a/D)=0.476

Freq.=88.2 Hz

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

0 20 40 60 80 100 120 140 160 180 200

Frequency (Hz)

0

50

100

150

200

250

300

350

400

450

500

Amplitude (nm/s)

(x/L)=0.3

(2a/D)=0.19

Freq.=90 Hz

(x/L)=0.3

2a/D)=0.476

Freq.=82.4 Hz

Without

Crack

(a/D)=0

With Crack

(a/D)=0.1

90.2

With Crack

(a/D)=0.3

Freq.=86.73

With Crack

(a/D)=0.5

Freq.=84.6

With Crack

(a/D)=0.7

Freq.=82.32

With Crack

(a/D)=0.75

Freq.=72.6

Fig 11. Experimental measurement of spectrum

vibration

for several crack depth of shaft

Al_Rafidain engineering Vol.13 No.1 2005

17

THE MEMORIZATION BEHAVIORS OF DIFFERENT

MIOS STRUCTURES

W. F. MOHAMAD L. S. ALI

ELECTRICAL ENGINEERING DEPARTMENT

COLLEGE OF ENGINEERING

UNIVERSITY OF MOSUL

ABSTRACT: In this chapter the various kinds of charge storage cells are discussed

as a result of examining many samples with different structures. The C-V, I-V and R-V

measurements of the structures confirm the memorization capability of MIOS devices.

The examined structures reveal three kinds of memory actions. The first one is the

charge storage capability which can be shown through (C-V) curve shifting as the device

was exposed to certain stress for a certain time. The second is the electronic switching

that is demonstrated by the fact that the switching between ON and OFF states and

back to original state can only be obtained by inverting the polarity of the applied bias

voltage. The third kind of memorization action is that the device can be switched into a

variety of stable intermediate resistance states. The new resistance state is determined

by the height of the programming pulse applied to the device. This memory action is

noticed from R-V characteristic and known as a nonvolatile analogue memory behavior.

مختلفة MIOS سلوكيات الخزن والذاكرة في تراكيب

د. وكاع فرمان محمد د. لقمان سفر علي

في هذا البحث تمت دراسة مختلف أنواع خلايا الخزن وذلك بعد فحص نماذج ذات تراكيب مختلفة. نتائج

أظهرت التراكيب المفحوصة .(MIOS) تدعم امكانية الخزن في نبائط ال (R-V) وال (C-V) وال (I-V) قياسات ال

ثلاثة أنواع من عمليات الخزن والذاكرة؛ النوع الأول هو امكانية خزن الشحنات في التركيبة والتي يمكن ملاحظتها

بعد تعريض النبيطة الى اجهاد كهربائي لزمن معين. والنوع الثاني هو اظهاره عمل (C-V) من خلال زحف منحني ال

ومن ثم الرجوع الى (ON) وال (OFF) مفتاح الكتروني والذي يمكن ملاحظته من خلال تحول المفتاح بين حالتي ال

الحالة الأصلية وذلك بعد قلب القطبية للفولتية المسلطة. والنوع الثالث للخزن والذاكرة هو امكانية استخدام النبيطة

حيث يمكن (OFF) و (ON) كمفتاح الكتروني يمكن تحويله بين حالات مقاومية مختلفة ومستقرة تتوسط حالتي ال

تحديد المقاومة للحالة الجديدة من ارتفاع نبضة البرمجة المسلطة. لقد لوحظت عملية الخزن هذه من خلال دراسة

وهذه تعرف بسلوكية الذاكرة التناظرية الغير متطايرة. .(R-V) خصائص ال

Submitted 23 rd March. 2004 Accepted 2nd Dec 2004

Al_Rafidain engineering Vol.13 No.1 2005

18

1- INTRODUCTION

Essentially the memory devices are structures whose resistance and capacitance vary

with magnitude and polarity of applied voltages [1]. The storage devices may be volatile or

nonvolatile. They can be used as an analogue or digital memories. The MOS structure is an

important type of the memory devices. Recently the shunt capacitance and shunt conductance

of such structures have been studied and investigated thoroughly [2,3]. The retention and

endurance of charges in the non-volatile memories depend on the oxide layer of the device.

The oxide layer is the most important part in the MOS structure. This layer limits the type of

the storage device. It is known that the leakage current is responsible for enhanced charge loss

in flash EEPROM memory. The leakage current is a tunneling process via neutral traps. The

leakage current induced by Fowler-Nordheim (FN) stress in MOS capacitors increases

drastically when the oxide thickness decreases [2,3]. The MOS device is essential structure in

flash EEPROM memory. It is more important to study the factors and parameters which

influence switching and retention of memorization in MIOS structures.

2- MIOS DEVICE FABRICATION

The MIOS devices used in the present investigation were fabricated as follows:

After the wet chemical treatment of the silicon wafers have been carried out, thermal

oxides were grown thermally at 800 oC in dry oxygen for time intervals 15 mins, 25 mins and

35 mins that yield silicon dioxide of thicknesses 7.75 nm, 15.5 nm and 21.7 nm respectively.

The oxide thickness tox was calculated from C-V measurement realized at 100 KHz. We are

aware that this method gives a rough estimation of the oxide thickness, but for this work we

do not need a precise measurement of oxide thickness.The wet chemical treatment was

repeated for cleaning only the back sides of all silicon wafers after thermal silicon dioxide

(SiO2)th growth. Then aluminum was thermally vacuum evaporated on the back side of all

wafers as a back contact with thickness of 200 nm. Post-metallization annealing was carried

out under vacuum for 60 mins at 400 oC, for making a good ohmic contact between silicon

and aluminum as a back contact.Then thermal vacuum evaporated (SiO)d film of 100 nm

thickness was deposited with a rate of 0.2 nm/sec on a part of the thermal grown silicon

dioxide (SiO2)th using a suitable mask to form (SiO)d 100 nm second insulator layer.For other

samples the second insulator layer was fabricated by thermal vacuum evaporated (SiO2)d

films of 100 nm thickness with deposition rate of 0.2 nm/sec on the thermal grown silicon

dioxide (SiO2)th to form (SiO2)d 100 nm.For each kind of the MIOS devices, two types of gate

contacts were fabricated. For some devices a strip of NiCr of 40 nm was deposited with a rate

of 0.2 nm/sec on the second insulator layer using a suitable metallic mask with an aperture of

2 mm width and 20 mm length.In the last step, for all devices, aluminum gate contacts of 200

Al_Rafidain engineering Vol.13 No.1 2005

19

nm thick were thermally vacuum deposited through the metallic mask with ( 1 and 2 mm)

diameter holes.

3- MIOS CHARGE STORAGE CAPABILITY

For the MIOS (Al/(SiO2)d100 nm/(SiO2)th7.75 nm/p-Si) structure the high frequency (1

MHz) capacitance voltage (C-V) curves were measured before and after stress voltage to

evaluate the effect of the stress on the capacitors as shown in Figs.(1) and (2). From the high

frequency C-V curves, the characteristics of the flat-band voltage shifts were obtained. The

distribution of the generate dinterface-statesdensities were calculated. Before stressing, oxide

charges are found to be 1.63 × 1011 charge / cm2.After the stress of – 10 V for 1000 sec, the CV

curve indicates the presence of the positive charge in the dioxide. The change in oxide

charges are calculated after the stress and are found to be equal to ΔVFB × Cacc, i.e. (2.6 × 1011

charge/cm2).That occurred because of tunneling of holes from p-type silicon substrate into the

gate structure [4]. Comparing the two C-V characteristics for strip gate and dot gate samples,

it is clear that the shift window in the dot gate sample approaches 2.5 V while in the strip gate

sample is about 2 V.This is attributed to the more recombination of electron injected from

metal gate with stored positive charges, and the more tunneling back of holes near Si/SiO2

interface into Si substrate in the strip gate sample after removing a stress voltage because of

larger area and larger defects. Hence, the density of remained store charges will be less [5].

4- MIOS DIGITAL PROGRAMMABLE RESISTOR MEMORY

Switching action of the two kinds of devices has been studied after exposing them to a

stress voltage of 10 V for 1000 sec. The experimental I-V curves for each device in “OFF”

and “ON” states are illustrated in Figs.(3) and (4). It is clear from both figures that these

devices exhibit memory switching [6]. Both the ON-state and the OFF-state characteristics

extrapolate through the I-V origin. The on-state is thus retained once the bias is removed,

giving a non-volatile, memory switching. By applying a negative bias the device can be

switched from conducting ON-state back to the OFF-state. From the two characteristics

shown, the behavior of each device differs from the other. The switching voltage from the

OFF-state (line AB) to ON-state (line CAD) for the device of SiO deposited insulator is

between (5-6) V, while that for SiO2 deposited insulator is between (7-8) V. In the reverse

direction the switching voltage from the ON-state (line CAD) to the OFF-state (line EA) for

the device of SiO deposited insulator is between – 3V and – 4V, while that for SiO2

deposited insulator is between – 6V and – 7 V.

The two devices are of the same thermal tunnel silicon dioxide of

7.75 nm thickness. The difference in the switching voltages is attributed to the second

deposited insulator difference, because both deposited insulators (SiO and SiO2) have the

same thickness (100 nm). The forming effect in SiO deposited layer happens at a voltage less

than that of SiO2 deposited layer, i.e. the insulation reliability of SiO is less than that of SiO2

Al_Rafidain engineering Vol.13 No.1 2005

20

[7]. Although the programming mechanism of this memory device is not yet understood fully

[1], it is thought that the current in a formed device is carried by a filament which is less than

1 μm in diameter. Formation of a filament may be associated with a diffusion of the top metal

into the insulator layer, resulting in a dispersion of metallic atoms in the insulating (SiO and

SiO2) matrix [8].

-7 -6 -5 -4 -3 -2 -1 0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gate voltage (v)

C/Cacc

Before stress

After stress of

-10 v 1000 sec

Cacc=31 nF/cm2

Area=3.14× 10-2 cm2

Deposited SiO2

TH=100 nm

Fig.(5-5) MIOS C-V characteristics for thermal

SiO2 TH=7.75 nm with dot gate

-7 -6 -5 -4 -3 -2 -1 0 1

0.4

0.5

0.6

0.7

0.8

0.9

1

Gate voltage (v)

C/Cacc

Before stress

After stress of

-10 v 1000 sec

Cacc=31 nF/cm2

Area=0.4 cm2

Deposited SiO2 TH=100 nm

Fig.(5-4) MIOS C-V characteristics for thermal

SiO2 TH=7.75 nm with strip gate

(1)

(2)

Al_Rafidain engineering Vol.13 No.1 2005

21

-10 -5 0 5 10

-6000

-4000

-2000

0

2000

4000

6000

8000

Gate voltage (v)

Gate current (μA)

ON state

OFF state

After stress of 10 v 1000 sec

Deposited SiO2 TH=100 nm

Fig.(5-6) MIOS I-V characteristics for thermal

SiO2 TH=7.75 nm with dot gate

-4 -2 0 2 4 6

-4000

-3000

-2000

-1000

0

1000

2000

Gate voltage (v)

Gate current (μA)

Deposited SiO TH=100 nm

After stress of 10 v 1000 sec

ON state

OFF state

Fig.(5-7) MIOS I-V characteristics for thermal

SiO2 TH=7.75 nm with dot gate

A B

C

E

D

A B

C

D

E

(3)

(4)

Al_Rafidain engineering Vol.13 No.1 2005

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5- MIOS ANALOGUE PROGRAMABLE RESISTOR MEMORY

Non-volatile memory switching has been observed in

Al/(SiO2)d 100 nm/(SiO2)th 21.7 nm/(p-Si) (MIOS) structure. Evidence for filamentary

conduction is found for devices that are in their low impedance state. The switching

phenomenon requires the existence of two impedance states which are stable at zero applied

bias. The device tested showed memory switching and their initial state was one of high

resistance. Fig.(5) shows analogue switching characteristic of Al/(SiO2)d 100 nm/(SiO2)th 21.7

nm/(n-Si) (MIOS) device.

After the device was exposed to stress voltage of 40 V for 1000 sec., the device

displayed a non-volatile, analogue memory behavior. The resistance state is determined by the

height of the programming pulse applied to the device. The range of programming voltages

that can be applied is referred to as the programming window. The operation of the device

involves the following processes [1]:

1. Forming: This is an only one time process in which a stress of 40 V for 1000 sec is

applied across the device electrodes. This creates a vertical deep conducting channel of

submicron width, which can be programmed to a value in the range 500 Ω to 600 KΩ.

2. Writing: To decrease the device resistance, positive “write” pulses are applied.

3. Erasing: To increase the device resistance, negative “erase” pulses are applied.

4. The device resistance can be “read” using a voltage of less than 0.2 V without causing

reprogramming.

0.5 1 1.5 2 2.5 3 3.5

0

100

200

300

400

500

600

700

Pulse height (v)

Bulk resistance of the structure (KΩ )

+ve writing pulses

-ve erasing pulses

Deposited SiO2 TH=100 nm

After stress voltage +40 v

1000 sec

Fig.(5-22) MIOS bulk resistance versus applied pulse

height for thermal SiO2 TH=21.7 nm with dot gate

(5)

Al_Rafidain engineering Vol.13 No.1 2005

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The programming pulses (write or erase), which range between 1 V and 3 V, are typically 500

nsec width. In Fig.(5) the device resistance is seen to increase from 500 Ω toward 600 KΩ

depending on the height

of the erase negative pulse. The magnitude of write positive pulse is used to set the final

resistance of the device. The programming window is 2 V.

It is thought [9] that the current in a formed device is carried by a filament, which is

less than 1 μm in diameter. Formation of a filament may be associated with a diffusion of the

top metal into the amorphous SiO2 layers, resulting in a dispersion of metallic atoms in the

insulating

SiO2 matrix [10]. At Si-SiO2 interface, when the device is in the high resistance state, it is

characterized by a large device voltage and low device current. In this state the semiconductor

under the tunnel oxide is deep depleted since any minority charge at Si-SiO2 interface is

effectively drained away by the tunnel-oxide. At switching point the device becomes unstable

due to the initiation of a regenerative feedback mechanism [3], which collapses the width of

the deep-depletion region to its strong-inversion value.

6- CONCLUSIONS

The examined devices manifest three kinds of memorization phenomena. The first one is the

charge storage capability which can be noticed through C-V curve displacement when stressing

the device. The second is the digital memory switching which is demonstrated by the fact that the

switching between ON and OFF states and back can only be obtained by inverting the polarity of

applied bias voltage. The third kind of memorization noticed in this work is that a device can be

switched into a variety of stable intermediate resistance states. The new resistance states could be

determined by the height of the programming applied pulses. This phenomenon is known as the

analogue memorization.

7- REFERENCES

[1] A. F. Murray and L. W. Buchan, “A users guide to non-volatile on-chip analogue memory”,

Electronics & Communication Engineering Journal PP. 53-63, April, 1998.

[2] P. L. Swart and C. K. Cmpbell, “Effect of losses and parasitic on

a voltage-controlled tunable distributed RC notch filter” IEEE J. Solid-State Circuits,

Vol. SC-8, No. 1, PP. 35-36, 1973.

[3] J. G. Simmons, L. Faraone, U. K. Mishra, and F. L. Hsueh, “Determination of the

switching criterion for metal/tunnel oxide/n/ p+ silicon switching devices”, IEEE

Electron Device Letters, Vol. EDL-2, No. 5, PP. 109-112, 1981.

Al_Rafidain engineering Vol.13 No.1 2005

24

[4] A. Meinertzhgen, C. Petit, M. Jourdain, and F. Mondon, “Anode hole injection and stress

induced leakage current decay in metal-oxide-semiconductor capacitors”, Solid-State

Electronics Vol. 44, PP. 623-630, 2000.

[5] T. Y. Huang and W. W. grannemann, “Non-volatile memory properties of metal / SrTiO3 /

SiO2 / Si structures”, Thin Solid Films, 87, PP. 159-165, 1982.

[6] J. M. Shannon and S. P. Lau, “Memory switching in amorphous silicon-rich silicon

carbide”, Electronics Letters Vol. 35, No. 22, PP. 1976-1977, 1999.

[7] H. F. Wolf “Semiconductors” Copy right 1971, by John Wiley & Sons, Inc.

[8] G. Dearnaley, D. V. Morgan, and A. M. Stoneham, “A model for filament growth and

switching in amorphous oxide films”, J. Non-Crystalline Solids 4, PP. 593-612, 1970.

[9] H. Kroger and H. A. Ricahrd Wegener, “Memory switching in

polycrystalline silicon films”, Thin Solid Films, 66, PP. 171-176,

1980.

[10] D. V. Morgan, A. E. Guile and Y. Bektore, “Switching times and arc cathode emitting

site life-times for aluminum oxide films”, Thin Solid Films, 66, PP. L 35-L 38, 1980.

Al_Rafidain engineering Vol.13 No.1 2005

25

Bifurcation and Voltage C ollapse in the

Electrical Power Systems

Mr. Ahmed N. B. Alsammak, M.Sc.

Electrical Engineering Department

University of Mosul

Mosul – Iraq

Abstract:

Voltage stability is indeed a dynamic problem. Dynamic analysis is

important for a better understanding of voltage instability process. In this work

an analysis of voltage stability from bifurcation and voltage collapse point of

view based on a center manifold voltage collapse model. A static and dynamic

load models were used to explain voltage collapse. The basic equations of a

simple power system and load used to demonstrate voltage collapse dynamics

and bifurcation theory. These equations are also developed in a manner, which is

suitable for the Matlab-Simulink application. As a result detection of voltage

collapse before it reach the critical collapse point was obtained as original point.

Keywords: Power System Stability, Voltage Stability, Voltage Collapse,

Bifurcation, Reactive Power Compensation and Matlab-Simulink.

التشعيب وانهيار الفولتية لأنظمة القدرة الكهربائية

أحمد نصر بهجت السماك

قسم الهندسة الكهربائية

جامعة الموصل

الملخص:

استقرارية الفولتية هي بالتأكيد مسألة ديناميكية لذا فالتحليل الديناميكي مهم جدًا لفهم

عمليات عدم استقرارية الفولتية في هذا البحث تم تحليل استقرارية الفولتية بالتركيز على

نقطة انهيار الفولتية والتشعيب وكذلك على نوع أو سبب هذا الانهيار ووضح كذلك تأثير

استخدام الأحمال الثابتة والمتحركة والمتمثل بالمحركات الحثية. المعادلات الأساسية لنموذج

نظام القدرة والأحمال (المستخدمة لتوضح أو شرح انهيار الفولتية والتشعيب) تمت معاملتها

بطريقة بحيث تكون مناسبة في تحليلات برنامج الماتلاب. كشف انهيار الفولتية قبل حدوثها

هي نقطة اصيلة في هذا البحث.

Submitted 23 rd Feb. 2004 Accepted 2nd Dec 2004

Al_Rafidain engineering Vol.13 No.1 2005

26

List of symbols:

V = Amplitude terminal load voltage (p.u.).

δ = Internal terminal load voltage angle in degree.

Em = Amplitude of generator internal voltage (p.u.).

δm = Internal generator voltage angle in degree.

Eo = infinity bus or slack bus voltage (p.u.).

C = compensated load capacitor in p.u.

Yo = Amplitude of equivalent impedance for the transformer

and transmission line in p.u.

Ym = Amplitude of equivalent impedance for the generator, transformer

and transmission line in p.u.

M =Generator moment of inertia p.u.

dm = damping coefficient

Pm = Mechanical power.

P&Q = Real and reactive power load demand respectively.

Kpw, Kpv,Kqw,Kqv and Kqv2 = Constant parameters for the real and reactive load power.

ω = Speed and equal toδ& .

1. Introduction:

The continuing interconnections of bulk power systems, brought about by

economic and environmental pressures, have led to an increasingly complex

system that must operate ever closer to limits of stability. This operating

environment has contributed to the growing importance of the problems

associated with the dynamic stability assessment of power systems. To a large

extent, this is also due to the fact that most of the major power system

breakdowns are caused by problems relating to the system dynamic responses. It

is believed that new types of instability emerge as the system approaches the

limits of stability.

One type of system instability, which occurs when the system is heavily

loaded, is voltage collapse. This event is characterized by a slow variation in the

system operating point, due to increase in loads, in such a way that voltage

magnitudes gradually decrease until a sharp, accelerated change occurs.

Voltage collapse in electric power systems has recently received

significant attention in the literature (see, e.g., [1] for a synopsis), this has been

attributed to increases in demand which result in operation of an electric power

system near its stability limits. A number of physical mechanisms have been

identified as possibly leading to voltage collapse. From a mathematical

perspective, voltage collapse has been viewed as arising from a bifurcation of the

power system governing equations as a parameter is varied through some critical

value. In several papers [9-15], voltage collapse is viewed as an instability

which coincides with the disappearance of the steady state operating point as a

Al_Rafidain engineering Vol.13 No.1 2005

27

system parameter, such as a reactive power demand is quasistatically varied. In

the language of bifurcation theory, these papers link voltage collapse to a fold or

saddle node bifurcation of the nominal equilibrium point.

Dobson and Chiang [2] presented a mechanism for voltage collapse, which

postulates that this phenomenon occurs at a saddle node bifurcation of

equilibrium points. They employed the Center Manifold Theorem to understand

the ensuing dynamics, In the same paper., they introduced a simplex example

power system containing a generator, an infinite bus and a nonlinear load (as

shown in Fig.(1)). The saddle node bifurcation mechanism for voltage collapse

postulated in Ref.[2] was investigated for this example in [3] and in [4].

All essential distinction exists between the mathematical formulation of

voltage collapse problems and transient stability problems. In studying transient

stability [5,6], one often interested in whether or not a given power system can

maintain synchronism (stability) after being subjected to a physical disturbance

of moderate or large proportions. The faulted power system in such a case has

been perturbed in a severe way from steady-state, and one studies the possibility

of the post-fault system returning to steady-state (regaining synchronism). In the

voltage collapse scenario, however, the disturbance may be viewed as a slow

change in a system parameter, such as a power demand. Thus, the disturbance

does not necessarily perturb the system away from steady-state. The steady-state

varies continuously with the changing system parameter until it disappears at a

saddle node bifurcation point. It is therefore not surprising that saddle node

bifurcation is being studied as a possible route to voltage collapse [7].

In this paper a suitable model is set up to analyze the power system in [2].

This model is then used with the some cases such as change in load and in the

reactive load power as well as using constant and dynamic load, as induction

motor.

The basic equations of the power system and load are also developed in a

manner, which is suitable for the Matlab-Simulink application [8] and not

depended on ready programs (compact program package) such as Auto [16]. The

computer results show that voltage collapse may be studied before bifurcation

with a static model and after bifurcation with a dynamic model so the goal of this

work is to show the richness of the qualitative behaviors, which may occur near

voltage collapse, and to illustrate their effect on system trajectories.

2. Saddle-Node Bifurcations &Voltage Collapse

A saddle-node bifurcation is the disappearance of a system equilibrium as

parameters change slowly. The saddle-node bifurcation of mot interest to power

system engineers occurs when a stable equilibrium at which the power system

operates disappears [1]. The consequence of this loss of the operating

equilibrium is that the system state changes dynamically. In particular, the

dynamics can be such that the system voltages fall in a voltage collapse. Since a

saddle-node bifurcation can cause a voltage collapse there for it is useful to study

saddle-node bifurcations of power system models in order to avoid these

collapses, such as using PID controller to control saddle-node bifurcations [17].

Al_Rafidain engineering Vol.13 No.1 2005

28

3. Reactive Power and Voltage Collapse:

Voltage collapse typically occurs in power systems which are heavily

loaded, faulted and/or have reactive power shortages. Voltage collapse is system

instability that it involves many power system components and their variables at

once. Indeed, voltage collapse often involves an entire power system, although it

usually has a relatively larger involvement in one particular area of the power

system [1].

Although many other variables are typically involved, some physical

insight into the nature of voltage collapse may be gained by examining the

production, transmission and consumption of reactive power. Voltage collapse is

associated with the reactive power demands of loads not being met because of

limitations in the production and transmission of reactive power. Limitations are

the productions of reactive power include generator and SVC reactive power

limits and the reduced reactive power produced by capacitors at low voltages.

The primary limitations on the transmission of reactive power are the high

reactive power loss on heavily loaded lines and line outages. Reactive power

demands of loads increases with the increasing of load, motor stalling, or

changes in load composition such as an increased proportion of compressor load.

4. The Model

This section summarizes an example from [4] to illustrate how voltage

collapse model applies to the power system model shown in Fig.(1). One

generator is a slack bus and the other generator has constant voltage magnitude

E, and angle dynamics given by the swing equation:

M⋅δ&&m=−dm⋅ω +Pm+Em⋅V⋅Ym⋅sinδ( −δm−θm)+Em2 ⋅Ym⋅sinθ(m) …..(4.1)

where M, dm, and Pm, are the generator moment of inertia, damping coefficient

and mechanical power respectively.

The load model includes a dynamic induction motor based on a model of

Walve [13] with a constant PQ load in parallel. The induction motor model

specifies the real and reactive power demands P and Q of the motor in terms of

load voltage V and frequency δ& . The combined model for the motor and the PQ

load [2] is:

P Po P1 K K (V T V) pw pV

= + + ⋅δ& + + ⋅ & …..(4.2)

2

2 Q Qo Q1 K K V K V qw qV qV = + + ⋅δ&+ ⋅ + ⋅ …..(4.3)

where Po, Qo are the constant real and reactive powers of the motor and P1, Q1

represent the PQ load.

From eq.(4.3):

Kqw

− Kqv ⋅V − Kqv2⋅V 2 +Q −Qo −Q1

δ& = …..(4.4)

Substituted eq. (4.4) in eq.(4.2) we get:

T KqwKpv

V KpwKqv V KpwKqv KqwKpv V Kpw Qo Q Q Kqw Po P P

⋅ ⋅

⋅ ⋅ + ⋅ − ⋅ ⋅ + ⋅ + − − ⋅ + −

= 2 ( ) ( 1 ) ( 1 ) 2

& ..(4.5)

Al_Rafidain engineering Vol.13 No.1 2005

29

thus,

δ&m =ωm …..(4.6)

From eq.(4.1)&(4.6) we get:

M

m dm ωm Pm Em V Ym sin(δ δm θm) Em2 Ym sin(θm)

ω

− ⋅ + + ⋅ ⋅ ⋅ − − + ⋅ ⋅

& = …..(4.7)

In eq.(4.3) Q1 is chosen as the system parameter so that increasing Q1

corresponds to increasing the load reactive power demand. The load also

includes a capacitor C as part of its constant impedance representation in order to

maintain the voltage magnitude at a normal and reasonable value. It is

convenient to derive the Thevenin equivalent circuit with the capacitor. The

adjusted values are:

1 2 cos( )

'

2

2

o

Yo

C

Yo

C

Eo Eo

⋅ θ

⋅

+ −

= …..(4.8)

' 1 2 cos( ) 2

2

o

Yo

C

Yo

Yo Yo C ⋅ θ

⋅

= ⋅ + − …..(4.9)

⎟ ⎟ ⎟ ⎟

⎠

⎞

⎜ ⎜ ⎜ ⎜

⎝

⎛

− ⋅

⋅

= + −

1 cos( )

sin( )

' tan 1

o

Yo

C

o

Yo

C

o o

θ

θ

θ θ …..(4.10)

The real and reactive powers supplied to the load by the network are:

P(δ ,V) = −Eo'⋅Yo'⋅V⋅sin(δ +θo')−Em⋅V⋅Ym⋅sinδ( −δm+θm)

+V2 ⋅(Yo'⋅sinθ( o')+Ym⋅sinθ(m)) …..(4.11)

Q(δ ,V) = Eo'⋅Yo'⋅V⋅cosδ( +θo')+Em⋅V⋅Ym⋅cosδ( −δm+θm)

−V2 ⋅(Yo'⋅cosθ( o')+Ym⋅cosθ(m)) …..(4.12)

In order to compute bifurcation value Q1 and the associated bifurcation

equilibrium point, the following approximate formulas [15] are used as shown in

appendix (A) equation (A3). The bifurcation value is:

Qo + Q1 − (− Kqv + Eo'⋅Yo'+Em⋅Ym)⋅V + (Kqv2 + Yo'+Ym)⋅V 2 = 0…..(4.13)

and the voltage magnitude at the bifurcation equilibrium point is:

( )

( ) Qo

Kqv Yo Ym

Q Kqv Eo Yo Em Ym −

⋅ + +

− + ⋅ + ⋅

=

4 2 '

' ' 2

*

1 …..(4.14)

Formulas (4.13) and (4.14) are derived from the approximate static model given

in Ref.[15]:

( )

(Kqv Yo Ym)

V Kqv Eo Yo Em Ym

⋅ + +

− + ⋅ + ⋅

=

4 2 '

* ' ' …..(4.15)

The last three equations show the relationship between the bifurcation point and

certain load, transmission network and generator parameters.

Al_Rafidain engineering Vol.13 No.1 2005

30

5. Bifurcations

Consider the modified power system model described by Ref.[2] which is

given by (4.1), (4.2)&(4.3) in the general form:

x& = F(x,λ ) …..(5.1)

where x is the state vector and λ is a time-varying parameter vector. Specifically,

in the power system model described in section (4), x = (δ,ω,V) and λ denotes

the parameter vector that includes real and reactive power demands at each load

bus. The parameters in (5.1) are subject to variation and, as a result, changes may

occur in the qualitative structure of the solutions of the static equation associated

with (5.1), i.e., solutions of F(x,λ)=0 for certain values of λ. For example, a

change in the number of solutions for x may occur as the parameters vary. As a

result, the dynamic behavior of (5.1) may be altered.

Bifurcation theory [1] is concerned with branching of the static solutions of

(5.1) and, in particular, it is interested in how solutions x(λ) branch as λ varies.

These changes, when they occur, are called Bifurcations and the parameter

values at which a bifurcation happens are called bifurcation values.

It is important in our following analysis of voltage collapse to distinguish

two different periods: the period before bifurcation and the period after

bifurcation. Power systems are normally operated near a stable equilibrium

point. As system parameters change slowly, the stable equilibrium point changes

position but remains a stable equilibrium point. This situation may be modeled

with the static model F(x, λ)=0 by regarding F(x, λ)=0 as specifying the position

of the stable equilibrium point x as a function of λ. (Here it would be more

precise to call F(x, λ)=0 a quasistatic model since λ varies and causes

corresponding variations in (x). This model may also be called parametric load

flow model. Exceptionally, variation in λ will cause the stable equilibrium point

to bifurcate. The stable equilibrium point of (5.1) may then disappear or become

unstable depending on the way in which the parameter is varied and the specific

structure of the system.

After the bifurcation, the system state will evolve according to the

dynamics of (5.1). (Some types of bifurcation result in the persistence of the

stable equilibrium point even after the bifurcation and the static model apply just

as before the bifurcation. However, we do not expect this sort of bifurcation to

be typical in power systems.) To summarize, analysis of a typical bifurcation of a

Eo∠0 C Em∠δm

V∠δ

Fig.(1) Power System Model.

⎟⎠

⎞

⎜⎝

∠⎛ − −

2

π

θo Yo ⎟⎠

⎞

⎜⎝

∠⎛ − −

2

π

Ym θm

~ Load ~

Al_Rafidain engineering Vol.13 No.1 2005

31

stable equilibrium point in a power system with slowly moving parameters has

two parts:

(1) Before the bifurcation when the (quasi) static model applies.

(2) After the bifurcation when the dynamical model (5.1) applies.

The current research on voltage collapse uses the static model and only considers

the system before the bifurcation.

6. Simulation Procedure

In this work, the voltage stability procedure used to perform the simulation

by the proposed model would be presented by a simple block diagram as shown

in Fig.(2). The simulation have been made with the use of the step-by-step

solution with using ode15s, ordinary differential equations which used to solve

stiff problem with good accuracy. The used program is Matlab 6.0 [8] to which

fast and accuracy results could be obtained. The differential equations from 4.1

to 4.7 are arrangement in such away by using the following figure to the results.

Fig.(2) Simulink model for the sample system.

Al_Rafidain engineering Vol.13 No.1 2005

32

7. Results and Discussion

A model of the sample system shown in Fig.(1) and foregoing equations

are used to illustrate the process of voltage collapse. For the bifurcations

analysis, Fig.(3) shows the bifurcation diagram, which is appears the relates

between voltage magnitude V and reactive power demand Q1. This figure

investigates a generic mechanism leading to disappearance of stable equilibrium

points and the consequent system dynamics for one-parameter dynamical

systems. To simplify the discussion, note first that in Fig.(3) which shows the

relation between six bifurcation’s depicted. For Q1<10.95,a stable equilibrium

point exists. (Upper left in Fig.(3)). As Q1 is increased, an unstable (“subcritical”)

Hopf bifurcation is encountered at point Q1=10.95. As Q1 is increased

further the stationary point regains stability at Q1=Q1*=11.42 through a stable

(“supercritical”) Hopf bifurcation.

Fig.(4) shows an example of a typical voltage collapse, for the fourth order

models, phenomenon after a saddle node bifurcation. The initial conditions used

to generate the simulations are: δm=0.35,ω =0.001,δ =0.14,andV =0.9. Note the

oscillatory nature of solution due to varying in Q1, where Q1=11.25+0.005t The

previous example, Fig.(4), demonstrated the center manifold model for the

Stable equilibrium

Unstable equilibrium

Saddle-node bifurcation.

Saddle-node for periodic orbits bifurcation.

Period-doubling bifurcation.

Hopf bifurcation.

Stable periodic orbit.

Unstable periodic orbit.

Fig.(3) Bifurcation diagram.

Al_Rafidain engineering Vol.13 No.1 2005

33

dynamics of voltage collapse after a sad

##### Modeling Deficit Irrigation Water Requirement For Maize in Mosul Region

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 33-42

DOI:
10.33899/rengj.2005.45581

Abstract

A computerized model with (Microsoft QuickBasic version 1.1) was proposed

for simulating the effect of deficit irrigation for maize crop during spring and autumn

seasons in Mosul region. The simulation is based on 16 years of climatological data

for the period ( 1985-2000 ) for Mosul meteorological station , which includes daily

maximum and minimum temperatures , maximum and minimum relative humidity ,

wind speed at 2m height , and sunshine hours , which is used to calculate daily

reference evapotranspiration with Penman-Monteith equation .The model predicts

yield reduction by changing irrigation depth for three different irrigation methods

(sprinkler , furrow and drip) .The rainfall is divided into three classes which represent

three regions (wet ,semi- arid and arid )

##### Characteristics of Flow Over Normal and Oblique Weirs With Semicircular Crests

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 49-61

DOI:
10.33899/rengj.2005.45568

characteristics of free flow over normal and oblique

weirs with semicircular crests are studied experimentally. For this purpose, forty eight

weir models were constructed and tested. The first twelve models were normal weirs in

which the crest radius was varied three times; 5cm, 7.5cm and 10cm. For each crest

radius, the weir height was varied four times; 35cm, 30cm, 25cm and 20cm. The

remaining models were oblique weirs. The oblique angle was varied three times; 60°, 45°

and 30°. In weirs of the same oblique angle, the crest radius and weir height were varied

similarly to those of normal weirs.The experimental results showed that for normal weirs,

the discharge coefficient (Cdw) increases with the increase of head to crest height ratio

(h/P) for the same height of weir. In case of oblique weirs, it was found that (Cdw)

decreases with the increase of (h/P) values and weirs of small oblique angle (α ) give high

values of (Cdw ).For normal weirs, the discharge magnification factor (QNC /QNS) and

performance increase as values of (h/P) increase. Normal weirs of semicircular crests

perform better than those of sharp crested weirs for all values of weir height and crest

radius tested in this study. While, for oblique weirs the discharge magnification factor

(QOB/QNS) and performance increased with the decrease of (h/P) values. As (h/P) value

approaches zero, the discharge magnification factor approaches the length magnification

of the weir. Weirs of small oblique angles give high discharge magnification factor and

high performance.A simple procedure was applied for the hydraulic design of oblique

weirs. The design method yields the final dimensions of a weir and predicts its headdischarge

curve for the whole range of operation.

##### COEFFICIENT OF DISCHARGE OF CHIMNEY WEIR UNDER FREE AND SUBMERGED FLOW CONDITIONS

*AL-Rafdain Engineering Journal (AREJ)*,
Volume 13, Issue 1, Pages 62-69

DOI:
10.33899/rengj.2005.45573

Abstract:

The main objective of this investigation is to study experimentally the water surface

profiles and to obtain convenient expressions for the estimation of discharge

coefficients (Cd) for free flow over chimney weir and the discharge factor (q/q1) for

submerged flow. Four chimney weir models with different vertex angles were

constructed and tested, the surface water profiles, for all models were smooth upstream

and fall suddenly downstream the model and at a high discharge it become concave

while at law discharge the water surface profile become convex. The coefficient of

discharge for free flow increase with the decrease of the upstream head and with the

decrease of half vertex angle (θ ). While the discharge factor for submerged flow

increase with the decrease of the submergence ratio (h2/h1). Two general expressions

were optioned, one, for the estimation of Cd with respect of (h/p),(w/p) and θ for free

flow conditions and the other for estimation of the coefficient factor (q/q1) with respect

to (h2/h1) and (h1/p).

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