Kiriş ve Plak Elemanlarda Dinamik Karakteristiklerin Belirlenmesi için Kullanılan Bazı Sayısal Yaklaşımların Karşılaştırılması

Son yıllarda, temel denklemlerin ve sınır koşullarının kodlanması, hesaplama süresi ve algoritma karmaşıklığı azaltmak, çözümün doğruluğunu artırmak ve hızlı yakınsamasını sağlamak, çözümün kararlılığı artırmak vb. nedenlerden ötürü çeşitli türdeki doğrusal ve doğrusal olmayan denklemleri çözebilmek için birçok sayısal yöntem geliştirilmiştir. Bu çalışmada, literatürde sıkça kullanılan sayısal yöntemlerden; diferansiyel kareleme (DKY), diferansiyel dönüşüm (DDY) ve sonlu farklar (SFY) yöntemleri algoritmaları ile kısaca anlatılmış ve kiriş ve plakanın modal analizi için uygulanarak sonuçları birbirleriyle karşılaştırılmıştır. Seçilen yapısal elemanlarda kesme gerinmesi etkileri ihmal edilmiş, plaka elemanlar ise basit mesnetli sınır koşulu kullanılarak tek boyutlu duruma indirgenmiştir. Bu varsayımlar altında, anlatılmakta olan sayısal yöntemler uygulanarak boyutsuz doğal frekanslar hesaplanarak tablolaştırılmış ve mod şekilleri çizdirilmiş. Kullanılan yöntemlerin gücünü ve doğruluğunu anlamak için, yüksek titreşim modlarında sayısal sonuçlar irdelenmiş ve DDY'nin daha hızlı ve daha doğru çözümler verdiği, DQM'nin sonuçlarının ise seçilen düğüm noktaları dağılımına bağlı olduğu ve dolasyısıyla DDY'den daha az doğru olduğu görülmüştür. Ancak, uygulama kolaylığı ve çok boyutlu durumlar için doğru sonuçlar DKY'nin olumlu özellikleridir.

Anahtar Kelimeler:

Dinamik Karakteristikler, Diferansiyel Kareleme, Diferansiyel Dönüşüm, Euler Kiriş, Kirchhoff Plaka, Sonlu Farklar.

Comparison of Some Numerical Approaches for Determination of Dynamic Characteristics in Beam and Plate Elements

In the last few decades, many numerical methods have been developed and employed to solve for various types of linear and nonlinear equations due to challenges in the aspect of the implementation of governing equations and boundary conditions, computation time, algorithm complexity, accuracy, convergency, stability of the solution and so on. Of the numerical methods in the open literature, differential quadrature (DQM), differential transform (DTM), and finite difference (FDM) methods are expressed briefly with their algorithms and compared to each other for the modal analysis of beam and plate elements. For simplicity, shear strains effects are neglected for the chosen structural elements, and plate element is reduced to one-dimensional case up to chosen simply-supported boundary condition. Under these assumptions, computed non-dimensional natural frequencies by applying concerned methods are tabulated, and mode shapes are plotted. To understand the strength and accuracy of employed methods, numerical results in the high vibration modes are investigated, and it is seen that DTM gives faster and more accurate solutions while the results of DQM depend on chosen grid distribution and has less accurate than DTM. However, the ease of implementation and accurate results for multi-dimensional cases are pros properties of the DQM.

Keywords:

Dynamic Characteristics, Differential Quadrature, Differential Transform, Euler Beam, Finite Difference, Kirchhoff Plate.,

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Bellman, R., Kashef, B.G., Casti, J. (1972). Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics, 10, 40-52.
Civan, F., Sliepcevich, C.M. ( 1984). Differential Quadrature for Multidimensional Problems. Journal of Mathematical Analysis and Applications, 101, 423-443.
Wang, X., Bert, W. (1993). A new approach in Applying Differential Quadrature to Static and Free Vibrational Analyses of Beams and Plates. Journals of Sound and Vibration, 162, 566-572.
Wang, X., Bert, C.W., Striz, A.G. ( 1993). Differential Quadrature Analysis of Deflection, Buckling, and Free Vibration of Beams and Rectangular Plates. Computers and Structures, 48, 473-479.
Du, H., Lim, M. K., Lin, R. M. (1994). Application of Generalized Differential Quadrature Method to Structural Problems. International Journal of Numerical Methods in Engineering, 37, 1881-1896.
Du, H., Lim, M. K., Lin, R. M. (1995). Application of Differential Quadrature to Vibration Analysis. Journal of Sound and Vibration, 181, 279-293.
Malik, M., Bert, C. W. (1996). Implementing Multiple Boundary Conditions in the DQ Solution of Higher Order PDEs: Application to Free Vibration of Plates. International Journal for Numerical Methods in Engineering, 39, 1237-1258.
Bert, C.W., Malik, M. (1996). Semianalytical Differential Quadrature Solution for Free Vibration Analysis of Rectangular Plates. AIAA Journal, 34(3), 601-606.
Shu, C., Du, H. (1997). Implementation of Clamped and Simply Supported Boundary Conditions in The GDQ Free Vibration Analysis of Beams and Plates. International Journal of Solid Structures, 34(7), 819-835.
Tornabene, F., Viola, E., Inman, D. J. (, 2009). 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. Journal of Sound and Vibration, 328 (3), 259-290.
Arikoglu, A., Ozkol, I. (2012). Vibration Analysis of Composite Sandwich Plates by the Generalized Differential Quadrature Method. AIAA Journal, 50 (3), 620-630.
Tornabene, F., Fantuzzi, N., Bacciocchi, M., Dimitri, R. (2015). Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method. Thin Walled Structures, 97, 114-129.
Yavuz, M. T., Ozkol, I. (2021). Free Vibration Analysis of a Rotating Double Tapered Beam with Flexible Root via Differential Quadrature Method. Aircraft Engineering and Aerospace Technology, 93( 5), 900-914.
Zhou, J. K. (1986). Differential Transformation and Its Application for Electrical Circuits. Huazhong University Press, Wuhan, China.
Malik, M., Dang, H. H. (1998). Vibration Analysis of Continuous Systems by Differential Transformation. Applied Mathematics and Computation, 96, 17-26.
Malik, M., Allali., M. (2000). Characteristic Equations of Rectangular Plates by Differential Transformation. Journal of Sound and Vibration, 233(2), 359-366.
Chen, C. K., Ho, S. H. ( 1996). Application of Differential Transformation Method to Eigenvalue Problems. Applied Mathematics and Computation, 79, 173-188.
Chen, C. K., Ho, S. H. (1999). Solving Partial Differential Equations by Two Dimensional Differential Transform Method. Applied Mathematics and Computation, 106, 171-179.
Yeh, Y. L., Jang, M. J., Wang, C. C. (2006). Analyzing the free vibrations of a plate using finite difference and differential Transformation Method. Applied Mathematics and Computation, 178, 493-501.
Yalcin, S., Arikoglu, A., Ozkol, I. (2009).Free Vibration Analysis of Circular Plates by Differential Transformation Method. Applied Mathematics and Computation, 212, 377-386.
Jang, M. J., Chen, C. L., Liu, Y. C. (2001). Two-dimensional Differential Transform for Partial Differential Equations. Applied Mathematics and Computation, 123, 109-122.
Arikoglu, A., Ozkol, I. (2010). Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method. Composite Structures, 92 (12), 3031-2039.
Shu, C. (2000). Differential Quadrature and Its Application in Engineering. Springer.
Hatami, M., Ganji, D. D., Sheikholesmani, M. (2017). Differential Transformation Method for Mechanical Engineering Problems. Academic Press.
Blevins, R. D. (2001). Formulas for natural frequency and mode shape. Krieger Publishing.
Leissa, A.W. (1973). The Free Vibration of Rectangular Plates. Journal of Sound and Vibration, 31(3), 257-293.