Değişken Kesitli Eğri Eksenli Çubukların Düzlem Dışı Titreşimlerinin Matrikant Yöntemiyle İncelenmesi

Abstract

Bu çalışmada eğri eksenli değişken kesitli düzlemsel çubukların düzlem dışı titreşim davranışları ele alınmaktadır. Çubukların statik davranışlarını ifade eden denklemlerden hareketle dinamik davranışlarını ifade eden denklemlere ulaşılmıştır. Çalışmanın amacı; eğri eksenli sürekli değişken kesitli düzlemsel çubukların düzlem dışı titreşim problemlerini, başlangıç değerleri yöntemi (matrikant) ile kayma deformasyonu ve hem eğilme hem de burulma dönme eylemsizliği etkilerini de dikkate alarak, analitik olarak çözmektir. Birinci bölümde, çubuk teorisi üzerine kısa bir giriş yapılmış ve çalışmanın amacı ve kapsamı belirtilmiştir. İkinci bölümde, eğri eksenli çubuklar ile ilgili literatürdeki çalışmalar incelenmiş ve bu çalışmalarda kullanılan yöntemler kısaca anlatılmıştır. Genelde sonlu elemanlar ve çeşitli enerji metotları kullanılarak yapılan çözümlerin birçoğunda eksenel uzama, kayma deformasyonu ve dönme eylemsizliği etkilerinin ihmal edilmekte olduğu gözlenmiştir. Üçüncü bölümde eğri eksenli çubukların genel denklemleri( yer değiştirme, denge ve bünye) ile bu denklemlere bağlı olarak düzlem içi ve düzlem dışı titreşim denklemlerine yer verilmiştir. Dördüncü bölümde değişken kesitli eğrisel çubukların düzlem dışı titreşimlerinin analitik çözümü matrikant yöntemi kullanılarak elde edilmiştir. Matrikant yöntemi hakkında bilgi verilip, basit bir sayısal örnekle açıklanmaya çalışılmış ve yöntemin titreşim problemlerine uygulanışı detaylı bir şekilde ele alınmıştır. Beşinci bölümde, literatürdeki çalışmalar incelenip, matrikant yöntemi kullanılarak bulunan sonuçlarla karşılaştırılmıştır. Diferansiyel denklemleri çözerken Matlab paket programından faydalanılmış ve çözümü yaparken kullandığımız program ile ilgili çeşitli optimizasyonlar yapılarak en hızlı ve en yakın çözüm için hangi parametrelerin kullanılması gerektiği hesaplanmıştır. Altıncı bölümde, bazı sayısal örnekler sonlu elemanlar yöntemi ile çözülmüştür. Ansys ve Abaqus analiz programları kullanılarak elde edilen sonuçlar literatürdeki ve matrikant yöntemindeki sonuçlarla karşılaştırılmıştır. Yedinci bölümde ise çalışmanın sonuçları tartışılmış ve bu konudaki çalışmaların nasıl geliştirilebileceği konusunda bazı öneriler sıralanmıştır.

The purpose of the present study is to give the approximate solution to the governing equations of out-of-plane dynamic problems of a curved beam with varying cross-sections and to exhibit the advantages of the solution. These governing equations solved by using the optimizated matricant method and compared between the other studies results. In the vibration problems of this study, the effects of axial extension, shear deformation and rotatory inertia due to torsional vibrations are taken into account. The matricant method is used in order to solve the governing differential equations. In the first chapter, general concepts of the theory of beams and the aim of the present study are given.In the second chapter, the studies in the literature on dynamic problems of curved beams with varying cross-section are reviewed. Reviewing the literature has shown that although a few papers deal with the out of plane dynamic behavior of arches with varying cross-section, many excellent papers are present with in-plane dynamic behavior of arches with non-uniform cross-section and variable curvatures. With the advancement of computer technology, arch problems are solved widely by using finite element method and many finite elements were developed for this purpose. If the behavior of the arch is non-planar, then a usual finite element model of this arch becomes very complicated with many degrees of freedom. In these cases, it is evident that a model, which, with a relatively few degrees of freedom, offers a good description of the behavior of the arch, is needed. In the third chapter, the governing differential equations of curved beam are given for the static and dynamic problems. The beam is represented by a space curve whose every point is coupled with a rigid orthonormal vector diad. The vectors are chosen to be perpendicular to the tangent vector of the space curve in the initial state and they represent the cross-section of the beam. In the deformed configuration, these directors still remain unit and perpendicular each other because of the assumption of a rigid cross-section. The matricant (initial value) method explained and applied for the vibration problems of curved beams in the fourth chapter. In this method, the curvature and the cross-section of the arch are considered as variable. The axial extension, shear deformation and rotatory inertia effects are also considered in the governing differential equations of free vibrations. In the fifth chapter, the problems in the literature which interest of dynamic behavior of curved beams with continuous varying cross-section and variable curvature are solved by using the matricant method and the comparisons between the results are given in the tables and figures. Clamped-clamped, clamped-free and free-free boundary conditions are studied for different opening angles. The effects of step ratio, location of the step, boundary condition and opening angle on frequency coefficients are studied. Mostly use clamped-clamped boundary conditions, because the axial extension, shear deformation and rotatory inertia effects are more changeable than other beam types. The mode transition phenomenon is also investigated and the mode shapes are given in figures. The transformation phenomenon is characterized by the sharp increase in frequencies of modes that occurs at certain combinations of curvature and length of the beam. This increase in mode frequencies is accompanied by a significant change in the mode shapes. There is still no comprehensive analysis of the transformation phenomenon and there are no proper explanations and methods for prediction the frequencies of an arch. This is possibly due to the fact that numerical simulations, commonly employed for the analyses, provide little analytical insight into the vibrational problem. Also the Matlab code, which we use to solve matricant equations, is optimized. All results and parameters are observed. Best parameters for this program are chosen by solution time and accuracy. This process is never done in previous literature and it is a really good example for further studies. In the sixth chapter, general information about the finite element analysis and ABAQUS, which is a commercially available finite element program, are given. The experimental studies and numerical solutions of curved beams are given. The experimental results are compared with the theoretical solution for two different curved beams at different boundary conditions. Thanks to ABAQUS the natural frequencies and mode shapes of beams are also compare with the experimental results. The results show that experimental, analytical and finite element solutions are in good agreement with each other. Also the results show that the geometry of the archs are really important to get a healthy solutions. There were some differencies between circular and elliptic archs. The study results are discussioned and some suggestions are made to improve method in advance in the last chapter. According to that with the improvements of Matlab program code it is more easy to work on the other geometries like parabola or helical.