Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 279 New Finite Difference Derivation For Calculation of Natural Frequency of Sector Steel Plate Dr. Sadjad A. Hemzah Civil Engineering Department College of Engineering, University of Al-Qadisiyah Al-Qadisiyah, Iraq E-mail: sah246@gmail.com Received 1 March 2016 Accepted 17 April 2016 Abstract A free vibration analysis of isotropic thin circular plate with various edge conditions have been studied in the present work. This study involves the obtaining of natural frequencies by solving the mathematical model that governs the vibration behavior of the plate using finite difference method. The numerical results of natural frequencies of circular plate are presented for different cases such as aspect ratio, curvature effect, grid size and boundary conditions. A good results was obtained from finite difference procedure compared with that obtained from the finite element analysis using Abaqus Package program. Keywords: Curved Plate, Free Vibration, Thin Plate, Finite Deference Analysis , Numerical Analysis. اشتقاق جديد بطريقة الفروقات المحددة لحساب التردد الحر للصفائح الحديدية المنحنية الخالصة ملت الدراسة اشت. النحيفة الدائرية الصفائح تحليل في المحددة الفروقات طريقة حساب االهتزاز الحر باستخدام دراسة البحث هذا في تم النتائج المتحصلة من .الفروقاتطريقة على ايجاد االهتزاز الحر من خالل حل المعادالت الرياضية الخاصة للصفائح المقوسة باستخدام , درجة التقوس , حجم التقسيم العرض الى الطول نسبةالتحليل باستخدام طريقة الفرقات المحدةة لهذه الصفائح ولحاالت مختلفة مثل العناصر طريقة من المتحصلة النتائج مع جيدا توافقا المحددة الفروقات طريقة من المتحصلة النتائج أعطت . وع المساندللصفيحة و ن .الدراسة هذه ضمن المأخوذة المتغيرات أنواع ولكافة ABAQUS برنامج وباستخدام المحددة .الفروقات المحددة ,التحليل العددي طريقة، األلواح النحيفة ، التردد الحراأللواح المقوسة ، كلمات البحث: 1. Introduction mailto:sah246@gmail.com Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 280 Plates, as structural elements, are extensively used in many fields of engineering including aerospace, civil structures, hydraulic structures, containers, ships, instruments, and machine parts. When in service, they are subjected to dynamic loadings the effect of which is very critical. Much research has been conducted into plate behavior, using a wide range of methods. An excellent monograph of the early literature relating to vibration analysis of plates was published by Leissa [1] . The small thickness makes the plate susceptible to various types of effective such as buckling modes and vibration modes. In engineering application, however, plate problems often involve consideration of dynamic disturbances, produced by time-dependent external forces or displacements. Dynamic loads may be created by moving vehicles, wind gusts, unbalanced machines, etc. Most researchers [1][2][3] , have used classical thin plate theory in their formulations to study the plate response; where the flexural vibration of the thin plate is characterized by a fourth-order partial differential equation. A direct solution of such equation might be difficult and most of the reported solutions are based on numerical methods such as finite difference method [4,9] , and finite element method [4], [5] . A number of approaches were proposed by different researchers to solve the differential equation of plats. Finite Element and Finite Difference methods are the well-known approaches to be the most widely used numerical procedures to find the solution of the mentioned differential equation. Finite Element method advantageous is that it is very suitable for practical engineering problems of complex geometries. However, the computational complexity involved in this method constitutes the main disadvantage of this technique, especially in real-time application. On the other hand, the method is fast enough to analyze, relatively easy to program, and also seems to be more convenient for uniform structures such as plate system. The main serious impediment of this method is that it is not suitable for problems with complex and irregular. Moreover, since the finite difference method is difficult to vary the size of the difference in particular regions, it is not suitable for problems with rapidly changing variables such as stress concentration problems. However, because of the geometry uniformity of the thin plates, finite difference method seems to be more applicable and faster to calculate deformations, forces, stresses and strains and natural frequencies. The objective of the present study is to develop an accurate and efficient method for determining the natural frequencies of isotropic curved thin plate with different boundary conditions, aspect ratios, curvature and grid size. 2. Description of The Problem The curved plate considered in this study is an isotropic plate with constant thickness (h) and has an inner radius Rin and outer radius Rout as shown in Fig. (1). The circular plate divided by a polar mesh. The interval of mesh in the direction of the radius is (β), while (φ) is the angle of each interval of mesh in the direction of central angle, the interval between the nodes in the perpendicular direction on the radius is denoted as (γ i-1 , γ i , γ i+1). 3. Governing Equations and Finite Difference Formulation and Solution The governing equation of motion in polar coordinate for a curved isotropic plate of uniform thickness and without in plane load is [7] : Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 281 where: q= is distributed load applied on the plate surface ρ= density of the material h= thickness of the plate D= flexural rigidity These variables could be expressed as follows: therefore, the final governing equation of motion of isotropic circular plate will be as follows: The solution of equation ( 4 ) may be accomplish by finite difference method, as shown in Fig. ( 2 ) : where: λ = ω 2 . ρ.h Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 282 By applying the finite difference scheme at the interior nodes of the divided plate, the following system of simultaneous linear equations in matrices will be obtained [ 6 ] Where{w} is column matrix whose elements, ωi, represent the amplitude of the free vibration, [ K] is a square matrix obtained from the finite difference expression of the biharmonic operator , while[B] is a diagonal matrix representing the constants in the term of Eq. (7), and λ= ω2.ρ. Notice that Eq (7) is an Eigen-value problem. For a given thickness (h) and plate–aspect ratio a b (b= radius* substantial angle(rad)), the Eigen-value (ω) can be determined numerically by using any relevant technique. The smallest Eigen-value gives the most (fundamental) free vibration factor. 4. The Eigen-value Problem Solution The Eigen-value problem may be solved by various numerical or analytical techniques. Numerical methods for nonlinear problems usually depend on iterative procedure. The procedure of the adopted numerical method that is used in this study is outlined, as follows (8) : 1. Compute the stiffness matrix [K] from applying the coefficient patterns for finite difference operators Fig.(3) at the interior nodes of the plate. 2. Compute the geometry matrix [B] from applying the coefficient patterns for finite difference operators Fig.(3) at the interior nodes of the plate. 3. Compute the inverse matrix for the stiffness matrix [K] (by Gauss-Jordan or other suitable technique) 4. Make [C] matrix by multiplying the inverse matrix [K] -1 by the geometry matrix, as follows: [C]=[K] -1 [B] Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 283 [C]{w}=λ * [I] {w} (C-λ * [I]){w}=0 where λ* = = where the largest Eigen-value ( ) of the matrix [K]-1is the reciprocal of the smallest value of the Eigen-value(λ) of the matrix [K]. By applying the following steps (a-d) to Eq. (10), the minimum Eigen-value can be obtained: a- Assume initial trial vector which may be taken equal to 1.0 b- Substitute the vector at Eq.(10) c- Approximate value of ( ) is obtained by dividing the first element of the column matrix[C] by w1, ( ) = First row of [C] / w1 where w1 is the first element of the matrix d- The second approximate value of the characteristic vector is obtained by: These steps can be continued until the errors become sufficiently small where the used error criteria is the average of the sum of the absolute differences: Thus; the natural frequency of isotropic curved thin plate with constant thickness will be: 5. Boundary Conditions The situation of the boundary condition of the circular plate should be characterized to find the solution. Thus the boundary conditions of a circular plate with a radius (r) may be defined as: 1. Fixed edge (clamped edge) : Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 284 2. Simply supported edge: 6. Numerical Results The derived numerical schemes for determining the natural frequency (w) were programmed in order to analyze different cases for an isotropic curved plate. Results of the numerical analysis for the present investigation was compared with those obtained from the finite element analysis program Abaqus. Abaqus provides both triangular and quadrilateral shell elements with linear interpolation and user- choice of larger- and small-strains formulations. For most of applications large-strain shell elements (S4R, S3R, and SAX1) are appropriate with these considerations. In the following study the S4R (4 node shell element) is used in the analysis. The present study was performed for a steel plate with the following properties: - Inner radius = 1 m and outer radius = 2 m. - modulus of elasticity E = 200 GPa. - poisons ratio υ = 0.3 . - thickness t = 10 mm . Several parameters were concerned in this study. These parameters are : 1. Mesh Size 2. Aspect Ratio and boundary conditions 3. Curvature effect  Mesh size In order to investigate the effect of mesh used in the presented finite difference method, a curved plate of angle 38.197 ○ and with the previous properties was analyzed with different mesh sizes and for two types of boundary conditions, simply support and Clamped support. The results of convergence as a function of mesh size for both types of boundary conditions were shown in table(1) and table (2). Also Fig. (3) and (4) show the convergence of the calculated natural frequency with those of Abaqus F.E. Program, and Fig. (5) shows a sample of the deflection values of SSSS plate for both finite difference and finite element results. It can be noticed that the natural frequency value obtained from both finite difference and finite element methods goes to a value of approximately 19.81 and 35.84 for both simply and Clamped supports, also, the (15×15) mesh gives a difference of about 0.35% and 0.93% of final predicted values of both two types of supports. However, a mesh of 10×10 is very useful to be used in the analysis procedure because its saving time and it has a difference around 1% of predicted values of natural frequencies. Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 285  Aspect Ratio and boundary conditions The effect of both aspect ratio and boundary condition was investigated. The aspect ratio for the curved plate was taken as (a/b) where a represent the length of the plate in radial direction and b represents the average width of the plate which can be obtained by multiplying the average radius by the radian value of the substantial angle. The studied aspect ratios were starts from (0.5 to 3) and for four types of boundary conditions surrounding the plate. These boundary conditions are : 1- SSSS ( simply supported all around) 2- CCCC ( all edges are Clamped) 3- SCSC (both bottom and top edges are simple and both left and right edges are Clamped) 4- CSCS (both bottom and top edges are Clamped and both left and right edges are simple) Table (3) and Fig. (6) show the natural frequencies results of the present finite difference procedure with different types of support conditions and aspect ratios. Note that S and C symbols represent simple and Clamped support. From these results it can be seen that values of the natural frequency of a plate of boundary conditions (SCSC) are closed to a plate of Clamped supports in lower values of aspect ratios and they are closed to a simply support plate in a higher values of aspect ratios. While , a plate of boundaries (CSCS) has an adverse behavior within the used range of aspects ratios.  Curvature effect The curvature effect was studied by increasing the average radius of the plate (i.e radius at the center of the plate) from 1 to ∞ so that the aspect ratio was kept to be equal to 1 with a length of 1m. So when R goes to ∞, the plate will be like a rectangular plate of dimensions (1x1)m . Also two types of boundary conditions was taken into account in studying the curvature effect, simply and Clamped supports. The results of curvature effect on both types of boundary conditions are listed in table (4). While Fig. (7) shows these results on a logarithmic x-axis represents the radius value and natural frequencies value represented on the y-axis. From the results shown below it can be seen clearly that when R goes to ∞, the plate will behave as a square plat of dimensions 1x1m which has a natural frequency of 19.739 and 35.841for both simple and Clamped supports respectively. In the current finite difference scheme the calculated natural frequency has a difference of round 0.7% from the analytical values of square plate. 5. Conclusions The free vibration of thin curved circular isotropic plate was investigated herein. A finite difference approach was presented to analyze the plate. The validity of the adopted approach was compared with finite element approach which was done by using software package (ABAQUS). The investigation was carried out to simulate the natural frequency of thin curved plate with different types of boundary conditions, aspect ratios, curvature and different mesh size. The following are the main points concluded after studying the results obtained from the present study: 1. The results of the adopted finite difference approach of analysis showed good agreement with those obtained by finite element analysis (ABAQUS). Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 286 2. The schemes of finite difference approach which were derived in the present study can be used to analyze curved circular thin plate with a good agreement results. 3. The present study shows the validity of the derived finite difference schemes with a very small error and for any suitable mesh size if compared with finite element analysis program package results 4. Finite Difference Method is more efficient for such problems than Finite Elements Method (software package), since F.D.M gives good agreement of results with less time of calculations. 5. The finite difference method gives an under estimations value for natural frequency which it is more safe than finite element method which give an over estimation values for such structures . References [1] A.W. Leissa: "Vibration of plates". NASA SP-160 (1969). [2] Stephen P. Timoshenko, S. Woinowsky-Krieger: "Theory of plates and shells". McGraw-Hill (1981). [3] A. W. Leissa: "The Free Vibration of Rectangular plates". Journal of Sound and Vibration, 31(3), pp. 257--293 (1973). [4] Klaus-Jurgen Bathe: "Finite Element Procedures". Prentice-Hall, Inc. (1996). [5] J. N. Reddy: "Finite Element Method". McGraw-Hill, Second Edition, (1993). [6] Gorman,D.J. “An Exact Analytical Approach to the Free Vibration Analysis of Rectangular Plates with Mixed Boundary Conditions.” J. Sound and Vibration, Vol.93, No.2, 1984, pp.235-247. [7] S. Timoshenko and S. Woinowsky-Krieger "Theory of Plates and Shells", McGraw-Hill, second edition, 1987 [8] Harik, I.E., Liu, X., Balakrishnan, N., “Analytical Solution to Free Vibration of rectangular plates.” J. Sound and Vibration, Vol.153, No.1, 1992, pp.51-62. [9] Husain M.H., Alwash N.A., Amash H.K, “Free Flexural Vibration of Rectangular Thin Plate with Tapered Thickness.” Eng. Of Technology, Vol.21, No.10, 2002, pp.736-745. Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 287 Table (1): Finite difference and F.E. Results for a simply supported curved plate Mesh Size ω (F.D) (rad/sec) ω (F.E) (rad/sec) 4x4 18.693 21.087 5x5 19.067 20.655 6x6 19.277 20.416 7x7 19.408 20.271 8x8 19.497 20.178 9x9 19.560 20.113 10x10 19.607 20.066 11x11 19.644 20.031 12x12 19.674 20.003 13x13 19.699 19.981 14x14 19.721 19.964 15x15 19.739 19.950 Table (2): Finite difference and F.E. Results for a Clamped supported curved plate Mesh Size ω (F.D) (rad/sec) ω (F.E) (rad/sec) 4x4 28.949 42.198 5x5 31.019 39.822 6x6 32.341 38.684 7x7 33.230 38.035 8x8 33.853 37.628 9x9 34.306 37.354 10x10 34.644 37.162 11x11 34.903 37.021 12x12 35.106 36.915 13x13 35.268 36.833 14x14 35.399 36.768 15x15 35.507 36.716 Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 288 Table (3): Finite difference natural frequencies in terms of of curved plate with different boundary conditions and aspect ratios Support Type SSSS CCCC SCSC CSCS Aspect Ratio ω (rad/sec) ω (rad/sec) ω (rad/sec) ω (rad/sec) 0.5 47.45008 88.66515 81.68419 53.33016 0.75 27.08775 48.76832 42.69226 34.12574 1 19.60723 34.64397 27.87902 27.88447 1.25 16.08721 28.66401 20.82557 25.24501 1.5 14.16622 25.79708 17.04361 23.91319 1.75 13.00771 24.27092 14.84224 23.1532 2 12.25685 23.38445 13.47623 22.67953 2.25 11.74301 22.83161 12.58302 22.36439 2.5 11.37613 22.46648 11.97287 22.14402 2.75 11.10513 22.21385 11.54042 21.98377 3 10.89932 22.03229 11.22416 21.86353 Table (4): Finite difference natural frequencies in terms of of curved plate with different boundary conditions and different radiuses R(m) Theta (deg) CCCC SSSS ω (rad/sec) ω (rad/sec) 1 57.29578 35.06038 19.99886 1.5 38.19719 34.64397 19.73921 2.5 22.91831 34.45288 19.66049 3.5 16.37022 34.40288 19.64656 4.5 12.7324 34.38265 19.6421 5.5 10.41741 34.37249 19.6402 10.5 5.456741 34.35769 19.63795 15.5 3.696502 34.35468 19.6376 20.5 2.794916 34.35359 19.63749 30.5 1.87855 34.35279 19.63742 50.5 1.13457 34.35237 19.63738 100.5 0.570107 34.35219 19.63737 ∞ 0.057267 34.35213 19.63736 Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 289 Figure (1): Geometry of the curved plate and node distribution λ Wi + Wi = 0 Figure (2): Finite difference scheme of curved plate differential equation Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 290 Figure(3): Finite difference versus Finite Element Results for a simply supported curved plate Figure(4): Finite difference versus Finite Element Results for a Clamped supported curved plate Al-Qadisiyah Journal For Engineering Sciences, Vol. 9……No. 2 ….2016 291 F.D F.E Figure (5): Deflection values for first mode of SSSS plate for finite difference and finite element Figure (6): Finite difference natural frequencies of curved plate with different boundary conditions and aspect ratios Figure (7): Finite difference natural frequencies of curved plate with different boundary conditions and different radiuses