Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods

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1 ngineering doi:0.436/eng Published Online April 0 ( Stress Function of a otating Variable-Thicness Annular Dis Using xact and Numerical Methods Abstract Ashraf M. Zenour Daoud S. Mashat Department of Mathematics Faculty of Science King AbdulAziz University Jeddah Saudi Arabia Department of Mathematics Faculty of Science Kafrelsheih University Kafr l-sheih gypt -mail: zenour@gmail.com eceived January 3 0; revised March 8 0; accepted March 3 0 In this paper the exact analytical and numerical solutions for rotating variable-thicness annular dis are presented. The inner and outer edges of the rotating variable-thicness annular dis are considered to have free boundary conditions. Two different annular diss for the radially varying thicness are given. The numerical unge-kutta solution as well as the exact analytical solution is available for the first dis while the exact analytical solution is not available for the second annular dis. Both exact and numerical results for stress function stresses strains and radial displacement will be investigated for the first annular dis of variable thicness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thicness annular dis is investigated. Finally the distributions of stress function displacement strains and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity. Keywords: otating Annular Dis Variable Thicness unge-kutta Method. Introduction The annular diss have received a great deal of attention because of their widely used in many mechanical and electronic devices such as computer dis memory units circular saws and turbine rotors. The problems of rotating annular diss can be easily found in most of the standard elasticity boos []. Most of the research wors are concentrated on the analytical solutions of rotating diss with simple cross-section geometries of uniform thicness and especially variable thicness [3-8]. In [6] the rotating annular diss are analytically studied and closed form solutions for the rotating diss subected to various boundary conditions are obtained. In [7] the rotating viscoelastic solid and annular diss of exponentially varying thicness are studied by analytical means. The problem of an annular dis subected to purely radial temperature variation is treated in [8]. The analytical solution for the analysis of thermal deformations and stresses in elastic annular diss with arbitrary cross-sections of continuously variable thicness is presented. As many rotating components in use have complex cross-sectional geometries they cannot be dealt with using the existing analytical methods. Numerical methods such as the finite element method [9] the boundary element method [0] and unge-kutta s algorithm [-4] can be applied to cope with these rotating components. Both finite element method and unge-kutta's algorithm are used in [4]. Some additional numerical methods are used in the literature. In [5] the rotating annular diss with uniform and variable thicnesses and densities are solved using homotopy perturbation method and Adomian s decomposition method. This wor is a predictive assessment of the stresses in and deformation of a rotating annular dis with variable thicness variation. Two methods of analysis namely the analytical method and unge-kutta numerical method of governing differential equations were used. This study concentrated first on the analytical solution for the rotating annular dis with arbitrary cross-section of continuously variable thicness. A unified governing equation will be first derived from the basic equations of the rotating diss and the proposed stress-strain relationship. Next unge-kutta method is introduced to solve the Copyright 0 Scies.

2 A. M. ZNKOU T AL. 43 governing equation. A comparison between both analytical and numerical solutions is made. The accuracy of the numerical solution is used to find the stress function stresses strains and displacement of rotating variable-thicness annular dis whose analytical solution is not available. Finally a number of numerical examples are given to demonstrate the validity of the proposed method.. Basic quations As the effect of thicness variation of rotating annular diss can be taen into account in their equation of motion the theory of the variable-thicness annular diss can give good results as that of the uniform-thicness annular diss as long as they meet the assumption of plane stress. After considering this effect the equation of motion of rotating diss with variable thicness can be written as d hrr h h r 0 () d r where r and are the radial and circumferential stresses r is the radial coordinate is the density of the rotating dis is the constant angular velocity and h is the thicness which is function of the radial coordinate r. The relations between the radial displacement u and the strains are irrespective of the thicness of the rotating dis. They can be written as d u u r () d r r where r and are the radial and circumferential strains respectively. The above geometric relations lead to the following condition of deformation harmony: d r r 0. (3) d r For the elastic deformation the constitutive equations for the rotating dis can be described with Hooe s law r r r (4) where is Young s modulus and is Poisson s ratio. Introducing the stress function and assuming that the following relations hold between the stresses and the stress function d r r. (5) hr h d r Substituting quation (5) into quation (4) one obtains d r r hr h d r d r. h d r hr The substitution of quation (6) into quation (3) produces the following confluent hypergeometric differential equation for the stress function (): r r d r dhd r d r h dr dr r d h h r h d r The thicness of the annular dis is assumed to vary nonlinearly through the radial direction. It is assumed to be in terms of a simple exponential power law distribution according to the following two types: Dis : Dis : r a n e b b h. 0 h r (6) (7) (8) r a n b b h0 e. h r For both diss h 0 is the thicness at the inner edge of the dis n and are geometric parameters a is the inner radius of the dis and b is the outer radius of the dis (see Figures and ). The value of n equal to zero represents a uniform-thicness annular dis. It is to be noted that the parameter n determines the thicness at the outer edge of the annular dis relative to h 0 while the parameter determine the shape of the profile. The thicness of the annular dis may be decreases at the outer edge (see Figure ) while the thicness of the annular dis may be increases at the outer edge (see Figure ). The value of equal to unity represents a linearly decreasing variable thicness for the annular dis while it represents a linearly increasing variable thicness for the annular dis. The geometric parameters and n are given according to three cases. In case : =.5 n = in case : =.5 n = 0.5 and in case 3: = 0.6 n =. For small and large n (case 3) the profile of the annular dis is concave while it is convex for large and small n (case ). For large and n (case ) the profile of the annular dis may be changing its shape from convexity to concavity. In addition the profile of the annular dis is convex for small and large n (case 3) while it is concave for large and small n (case ). Also for large and n (case 3) the profile of the annular dis may change its shape from concavity to convexity. (9) Copyright 0 Scies.

3 44 A. M. ZNKOU T AL. (a) (a) (b) (b) (c) Figure. Variable-thicness annular dis profiles for (a) case (b) case and (c) case 3. (c) Figure. Variable-thicness annular dis profiles for (a) case (b) case and (c) case 3. Copyright 0 Scies.

4 A. M. ZNKOU T AL. 45 To simplify the solving process we introduce the following dimensionless variables: 0 b 3 U u r r b r b A a b r r. bh (0) Then quation (7) may be written in the following simple forms according to the two cases: d d n d d () n A 3 n e 0 for annular dis and n A d n e d d n A d e () n A n e n A 3 e 0 n A e for annular dis. 3. xact and Numerical Solutions The exact general solution for quation () may be available while it is not available for quation (). The modified unge-kutta numerical solution may be avail- where able for both cases. Firstly we will get both the exact and numerical solutions for quation (). If the numerical solution is compared well with the exact one we can use it to solve quation () for the second case. 3.. xact Solution The exact general solution for quation () can be written as n na e cw c M e P i i (3) where c and c are arbitrary constants and Wi and M i are Whittaer s functions (see Abramowitz and Stegun [6]) M M i z W W i z (4) i i in which i z n. (5) It is to be noted that Whittaer s functions may be given by z M i z e z H i z (6) z Wi z e z K i z where H and K are the hypergeometric and Kummer s functions respectively with n > 0. It is to be noted that for real values of i and Whittaer s functions Wi and M i converge for. P in quation (3) can be written as The term d d (7) P W P M M P W n e P. M W M W ( ) i i i i (8) The substitution of quation (3) into quation (5) with the aid of the dimensionless forms given in quation (0) gives the radial and circumferential stresses in the following forms: i i n na e e cw c M P (9) n A d e. d 3 Note that the first differentiations of Whittaer s func- tions W and M i d Wi n i Wi Wi d d M i n i Mi i Mi d i. are given by (0) Copyright 0 Scies.

5 46 A. M. ZNKOU T AL. Finally the dimensionless strains and the corresponding radial displacement may be obtained easily using quation (9) as well as the dimensionless forms given in quation (0). So all of stress function stresses strains and radial displacement can be completely determined under the traction conditions on the inner and outer surfaces of the annular dis. They can be expressed as 0 0 at r a A () 0 0 at r b. 3.. Numerical Solution The modified unge-kutta algorithm may be used here to solve both the differential quations () and () with the aid of the boundary conditions given in quation (). quations () or () can be written in the following general form: s s s s f s. () where the prime denotes differentiation with respect to and s denotes the dis number. The functions f s may be given by s s s s i i i K f n f e (3) f n n A n A n e n A e n A n e n A e n A e. (4) unge-kutta s iterative formulae for the second-order differential equation are [-4] s s s s s s i i K K K 3 K 4 6 s s s s s s i i i K K K3 6 (5) where is the increment of the distance along the radial direction of the rotating annular diss and s K 34 can be determined with the following relations s s s s s K fsi i i i K s s s s s s K3 fsi i i K i K 4 s s s s s s K4 fsi i i K i K3. (6) The numerical simulation starts from the inner boundary where a trial value of the first derivative of the stress function is assumed. With a small distance the stress function and its first derivative at the new position can be obtained using quation (5) and the radial stress is calculated. According to the difference between the computed radial stress and the nown radial stress at the outer boundary the initial trial value of the first derivative of the stress function at the inner boundary is modified and the next iteration is carried out in the same way. This iterative process is performed until both the boundary conditions are simultaneously satisfied. Once the stress function is obtained the stresses strains and displacement in the rotating annular diss can be obtained using quations () (5) and (6) with the aid of the dimensionless given in quation (0). This process may be easily done after getting the continuous form of the stress function using the curve fitting and least square method. So all other quantities can be easily determined. 4. Numerical xamples and Discussion Some numerical examples for the rotating variablethicness annular dis will be given according the exact and numerical solutions ( 0.3). esults determined as per the exact analytical solution are compared with those obtained by the numerical unge-kutta (-K) solution in Figures 3-0 for dis. Additional results for stress function radial displacement strains and stresses of the rotating variable-thicness annular dis using the numerical -K solution are also presented in Figures -6. The inner and outer radii of the dis are taen to be a = 0. b ( = A = 0.) and b ( = ) and the results are given in terms of the rotating angular velocity. The numerical applications will be carried out for the radial displacement strains stress function and stresses Copyright 0 Scies.

6 A. M. ZNKOU T AL. 47 Figure 3. Stress function Ф radial stress σ and circumferential stress σ in the variable-thicness annular dis. Figure 6. Stress function Ф of the variable-thicness annular dis for all cases. Figure 4. Displacement U radial strain ε and circumferential strain ε in the variable-thicness annular dis. Figure 7. adial stress σ in the variable-thicness annular dis for all cases. Figure 5. Displacement U of the variable-thicness annular dis for all cases. Figure 8. Circumferential stress σ in the variable-thicness annular dis for all cases. Copyright 0 Scies.

7 48 A. M. ZNKOU T AL. Figure 9. adial strain ε in the variable-thicness annular dis for all cases. Figure 0. Circumferential strain ε in the variable-thicness annular dis for all cases. that being reported herein with help of quation (0) are in the following dimensionless forms: U U 00. (7) The distribution of the stress function and stresses are presented in Figure 3 while the distribution of the radial displacement and strains are presented in Figure 4. The results are given for rotating variable-thicness annular dis with =.5 and n = (case ). The numerical -K solution is compared with the exact analytical solution. It is clear that -K method gives all results with excellent accuracy when compared with the exact analytical solution. The distributions of the stress functions radial stress circumferential stress radial displacement radial strain as well as circumferential strain through the radial direction of the annular dis are all presented in Figures 5-0 respectively for different cases of the geometric parameters and n. Figure 5 shows that the radial displacement U has minimum value near the inner edge (at = 0.5) and maximum value near the outer edge (at = 0.9) for all cases of the parameters and n. Cases and 3 (n = ) are coincided with each other when = 0.. Case with small n (n = 0.5) gives the highest values of the displacement. Figure 6 shows that the stress function has its maximum at the mid-plane of the dis ( = 0.55) for case while its maximum occurs at ( = 0.45) and ( = 0.5) for cases and 3 respectively. Once again cases and 3 are coincided with each other when = 0.7. Case with small n gives the highest values of the stress function. The radial stress and circumferential stress in the rotating variable-thicness annular dis are plotted in Figures 7 and 8. The maximum value of does not occur at the mid-plane of the dis. It occurs at = 0.33 = 0.43 = 0.35 for cases 3 and respectively. The maximum value of occurs at the inner surface of the dis while the smallest value occurs at outer surface. Case gives the smallest stresses. The same conclusion may be applied on the plots of strains in Figures 9 and 0. It can be seen from Figures 3-0 that the -K method can describe all results through-the-thicness of the rotating annular dis very well enough. This puts into evidence the great role played by -K method in the modeling of rotating variable-thicness annular diss. So it will trustily used to find the solution for quation () of the rotating variable-thicness annular dis. As a result the stresses and stress function in the rotating variable-thicness dis are plotted in Figures -3 according to all cases. However the radial displacement and strains are plotted in Figures 4-6. Figure. Stress function Ф radial stress σ and circumferential stress σ in the variable-thicness annular dis for case. Copyright 0 Scies.

8 A. M. ZNKOU T AL. 49 Figure. Stress function Ф radial stress σ and circumferential stress σ in the variable-thicness annular dis for case. Figure 5. Displacement U radial strain ε and circumferential strain ε in the variable-thicness annular dis for case. Figure 3. Stress function Ф radial stress σ and circumferential stress σ in the variable-thicness annular dis for case 3. Figure 6. Displacement U radial strain ε and circumferential strain ε in the variable-thicness annular dis for case Conclusions Figure 4. Displacement U radial strain ε and circumferential strain ε in the variable-thicness annular dis for case. This paper presents an exact analytical solution for a rotating variable-thicness annular dis. Also it presents a unified numerical method based on unge-kutta forth-order method for the elastic calculation of different rotating annular diss with a general arbitrary configuration. The governing equation was derived from the equilibrium equation and the stress-strain relationship. The calculation of the rotating annular dis was turned into finding the solution of second-order differential equations under the given conditions at two boundary fixed points. unge-kutta method algorithm was introduced to solve the governing equation for two types of annular diss in which the exact solution of one of them only is available. A number of numerical examples were studied. The results from the exact analytical and -K solutions Copyright 0 Scies.

9 430 A. M. ZNKOU T AL. were compared. The proposed -K approach gives very agreeable results to the exact analytical solution. The modified -K method presented here may be trustily used for the boundary value problems that their exact solutions are not available. 6. eferences [] S. P. Timosheno and J. N. Goodier Theory of lasticity McGraw-Hill New Yor 970. [] S. C. Ugral and S.K. Fenster Advanced Strength and Ap- plied lasticity lsevier New Yor 987. [3] U. Güven lastic-plastic Stresses in a otating Annular Dis of Variable Thicness and Variable Density International Journal of Mechanical Sciences Vol pp [4] U. Güven On the Stress in the lastic-plastic Annular Dis of Variable Thicness under xternal Pressure International Journal of Solids and Structures Vol pp [5] U. Güven Stress Distribution in a Linear Hardening Annular Dis of Variable Thicness Subected to xternal Pressure International Journal of Mechanical Sciences Vol pp doi:0.06/s (97) [6] A. M. Zenour Analytical Solutions for otating xponentially-graded Annular Diss with Various Boundary Conditions International Journal of Structural Stability and Dynamics Vol pp doi:0.4/s [7] A. M. Zenour and M. N. M. Allam On the otating Fiber-einforced Viscoelastic Composite Solid and Annular Diss of Variable Thicness International Journal for Computational Methods in ngineering Science & Mechanics Vol pp. -3. doi:0.080/ [8] A. M. Zenour Thermoelastic Solutions for Annular Diss with Arbitrary Variable Thicness Structural ngineering and Mechanics Vol pp [9] O. C. Zieniewicz The Finite lement Method in ngineering Science McGraw-Hill London 97. [0] P. K. Baneree and. Butterfield Boundary lement Methods in ngineering Science McGraw-Hill New Yor 98. [] M. L. James G. M. Smith and J. C. Wolford Applied Numerical Methods for Digital Computation Harper & ow New Yor 985 [] L. H. You Y. Y. Tang J. J. Zhang and C.Y. Zheng Numerical Analysis of lastic-plastic otating Diss with Arbitrary Variable Thicness and Density International Journal of Solids and Structures Vol pp doi:0.06/s (99)00308-x [3] C. F. Gerald and P. O. Wheatley Applied Numerical Analysis 6th dition Addison-Wesley California 00. [4] M. H. Hoati and A. Hassani Theoretical and Numerical Analysis of otating Discs of Non-Uniform Thicness and Density International Journal of Pressure Vessels and Piping Vol pp doi:0.06/.ipvp [5] M. H. Hoati and S. Jafari Semi-xact Solution of lastic Non-Uniform Thicness and Density otating Diss by Homotopy Perturbation and Adomian s Decomposition Methods. Part I: lastic Solution International Journal of Pressure Vessels and Piping Vol pp doi:0.06/.ipvp [6] M. Abramowitz and I. Stegun Handboo of Mathematical Functions 5th Printing US Government Printing Office Washington DC 966. Copyright 0 Scies.