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**ISSN**: 18130526

**Publisher**: Mosul University

**Faculty**: Engineering

**Language**: Arabic and English

**This journal is Open Access
**

Al Rafidain Engineering Journal

Abstract

The first issue of Al Rafidain Engineering Journal published in 1993 by the college of engineering – University of Mosul. The journal is publishing at a rate of six issues in the year (Bi-Monthly).

The journal publishes the referred original and valuable engineering research papers.

Al Rafidain engineering journal includes the following titles:

• Architectural Engineering

• Civil Engineering

• Computer Engineering

• Electrical Engineering

• Environmental Engineering

• Mechanical Engineering

• Megatronic Engineering

• Water Resources Engineering

The aim of publishing the journal is to develop the knowledge in the fields of applied engineering science.

• irrigation and drainage engineering

• Computer Engineering

Target domain and

Rivers Engineering magazine aims to develop knowledge in the field of engineering and science related to it. Should contribute to the article submitted for publication in the development of engineering sciences in various fields will be considered in the dissemination of innovative and distinctive articles in these areas. Articles which are referred to the magazine for the purpose of evaluating the arbitrators with the reputation and extensive experience in the field of jurisdiction has been accepted for publication or apologize. And will be re-papers not accepted for publication to their owners.

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ealrafidain@yahoo.com

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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

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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

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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

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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

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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 12 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 22 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 23 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

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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). معامل