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

Öz

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

Öz

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.

Anahtar Kelimeler

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

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