Doğru ve eğri eksenli çubuklardan oluşan çerçevelerin titreşimleri
Vibrations of frames consisting of straight and curved axis beams
Yazar: NİHAT ÖZTEN
Danışman: PROF. DR. EKREM TÜFEKCİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Katı Cisimlerin Mekaniği Bilim Dalı
Derece: Yüksek Lisans, 2020
Konu: Makine Mühendisliği = Mechanical Engineering

Özet:
Çubuk, en basit ve temel yapı elemanı olması sebebiyle, araştırmacıların yıllardır üzerinde çalıştığı temel bir konu olmuştur. Günümüzde birçok kullanım yeri olduğu için bu konudaki çalışmalar da hala devam etmektedir. 
Çubuklar ile ilgili yapılan çalışmalara bakıldığında ana konuyu belirleyen bazı farklılıklar olduğu görülmektedir. Çubuklar, ekseninin şekline göre, kesitlerinin şeklinde göre, kesitlerinin çubuk boyunca sabit veya değişken olmasına göre farklı yapılarda olabilir. Mevcut çalışmaların çoğunda, bu gibi farklılıklardan biri veya birkaçı seçilerek, ele alınan belirli bir çubuğun statik veya dinamik problemleri incelenmiştir. 
Bu çalışmada, doğru eksenli ve eğri eksenli çubuklardan oluşan çerçeve sisteminin hem düzlem içi hem de düzlem dışı titreşim davranışları ele alınmaktadır. Uçak, gemi, otobüs, tren gövdelerinde bu şekilde çerçeve sistemler kullanılmaktadır. 
Çubuğun titreşimlerini ifade eden genel denklemler, eksen uzamasına, kayma deformasyonuna ve dönme eylemsizliğine ait etkileri göz önüne almaktadır.  Titreşim probleminin denklemleri Matlab paket programı yardımıyla çözülmüştür. 
Birinci bölümde, çubuk ve bu çalışmadaki amaç hakkında bilgiler verilerek konuya giriş yapılmıştır. 
İkinci bölümde, doğru eksenli ve eğri eksenli çubukların titreşimleri ile ilgili yapılmış olan çalışmaların özetlenmesi yapılmıştır. Yapılan çalışmalarda, eksen uzama, kayma deformasyonu, dönme eylemsizliği etkilerinden bir veya birkaçının göz ardı edilerek basitleştirme yapıldığı görülmüştür. Fakat yapılan bu basitleştirmeler, problemi, gerçek durumdan bir miktar uzaklaştırmaya sebebiyet vermektedir. Literatürdeki çalışmalar incelendiğinde, elde edilen denklemlerin çözümü için, Ritz-Galerkin metodu, enerji denklemleri ve sonlu elemanlar gibi yöntemlerin kullanıldığı görülmüştür. 
Üçüncü bölümde, eğri eksenli çubukların genel denklemleri verilmiştir. Bu denklemler, hem düzlem içi hem de düzlem dışı için çözülmüştür. Problemlerin çözümünde başlangıç değerleri yöntemi kullanılmıştır. Ulaşılan çözüm, herhangi bir sınır şartına bağlı değildir. 
Çalışmanın dördüncü bölümünde, doğru eksenli çubukların genel denklemleri ile bu denklemlerin çözümleri verilmiştir. Çubuk, elastik, izotrop ve homojen malzemeye sahip olarak değerlendirilmiştir.
Beşinci bölümde, doğru eksenli ve eğri eksenli çubuklardan oluşan çerçeve sistemi ile ilgili çeşitli örnekler çözülmüştür. Elde edilen sonuçlar, hem sonlu elemanlar analiz programı ile hem de literatürde bulunan mevcut değerler ile karşılaştırılmıştır. Bazı parametrelerin frekans değerleri üzerine etkisi de karşılaştırmalı olarak grafiklerle sunulmuştur. 
Altıncı bölümde, çalışma sırasında göz önünde bulundurulan durumlar belirtilmiş, çalışmada ulaşılan sonuçlar tartışılmış ve sonraki araştırmalar için konu ile ilgili yeni önerilerde bulunulmuştur.

Abstract:
Beam is the most basic structural elements, so researchers have been working on this topic for years. Since there are many areas of usage, the analysis of beams has a great importance and studies on static and dynamic problems of beams are still continuing.
There are some differences that identify the main subject when current studies are examined. These are shape of the beam axis, shape of the beam cross section, variability on the section of the beam. Most of the current studies focused on only one of these differences.
Equations of vibrations are very complex equation when all effects are taken into account. These effects are axial extension, shear deformation and rotatory inertia. Neglecting these effects simplifies equations, so the solution of the problem becomes easier. However, this situation takes away from the reality of the solution and causes to reach approximate solutions. 
Many studies have examined about the in plane and out of plane vibrations of beams. Considering the literature review, it can be seen that Ritz-Galerkin method, energy methods or finite element method are mostly used. Finite element method, which has become widespread with the development of computer technology, finds application in most of the studies. 
In this study, in plane and out of plane vibrations of the frame system consisting of straight axis and curved axis beams are presented. Aircraft, ship, bus, train bodies are used this frame systems. As a simple example, lamp post on the streets have this frame system. The governing differential equations are solved exactly using the initial value method for both in plane and out of plane situations. Obtained solutions does not depend on any boundary conditions. Various samples about frame system are solved for some boundary conditions and they are compared with finite element results and literature studies. The equations of the vibration problem are solved with the help of Matlab software. 
In the first chapter, general informations about the beam element and the purpose of this study are stated. Although there is a lot of study about the analysis of straight and curved beams, most of them have neglected the effects of axial extension, shear deformation and rotatory inertia. Thanks to the neglect of these effects, equations have become simple. In previous studies, various approximate solution methods have been used such as differential transformation method, differential quadrature method and finite element method. Nowadays, finite element method is widely used with the development of computer technology. Also, finite element method seen as one of the most powerful tools for solid bodies and structural mechanics. 
In the second chapter, a review of the literature about straight and curved axis beams and frame systems is given. Many studies used Euler-Bernoulli beam theory. In this theory, cross-section is perpendicular to bending line, so shear deformation and rotary inertia are neglected. On the other hand, Timoshenko beam theory takes into account shear deformation and rotatory inertia. Timoshenko beam theory gives more realistic results, especially in large cross section on beams. Even so a few studies have taken these effects into account. Ritz-Galerkin method, energy methods and finite element method are used to solve the obtained equations. 
Considering the literature review, it is better understood why this topic was chosen. In this study, straight axis beam and curved axis beam are examined together without neglecting the effects of axial extension, shear deformation and rotatory inertia. Analytical solution was obtained with the method of initial values. The cross section is constant for the exact solution of all problems in the study.  
In the third chapter, the general equations of a curved beam and solution of equations are given for both in plane and out of plane dynamic problems. Loading and any boundary conditions does not affect the solution. With the fundamental matrix and initial values vector, displacement components and internal load can be calculated. After the static equations of the beam, dynamic behavior of the beam can be reached thanks to the D'Alembert principle.
In the fourth chapter, the governing differential equations of a straight beam and solution of equations are given. Material of the beam is assumed to be elastic, homogeneous and isotropic. Also, displacements and deformations are considered too small. Equations of straight axis beam are similar to equations of curved axis beam, so this chapter has been added to ensure the integrity. The main difference is the coordinate axis. In the equations of the straight axis beam, -x, -y, -z coordinates are used instead of -n, -b, -t coordinates.
In the fifth chapter, various samples about the vibrations of the frame system consisting of straight axis and curved axis beams are solved. The matrix of coefficient is given and the boundary conditions are defined. 
There are four different problem types. First problem is the frame system that contains one straight axis beam and one curved axis beam. In the second problem, there are two different curved axis beam and one straight axis beam. Third problem has different three curved axis beams and these beams are tangent to each other at the junction points. Also, this problem is solved for different boundary conditions. Fourth problem has two symmetrical straight axis beam and one curved axis beam. This problem is compared with a study in the literature, so it has a different cross-section that the other problems. 
The coefficients matrix is written for the curved axis beam and straight axis beam. The boundary conditions used in the problems are defined. Boundary conditions and continuity conditions are defined separately for all four problems. Cross section dimensions and material information of the beam are given. Obtained analytical results were compared with the finite element analysis results and all results are compatible. Fourth problem is also compared with an existing study in literature. 
There are two different parametric studies. The effects of some parameters on frequency values are presented comparatively in graphs. In the first parametric study, opening angle of the curved axis beam and radius of the curved axis beam are changing. The length of the straight axis beam is constant. Three-dimensional surface graphics of the first three modes were drawn. Similar interpretations can be made for all three modes. In the second parametric study, the length of the straight axis beam, the length of curved axis beam, and opening angle are constant. Inner and outer diameter dimensions of curved axis beam have changed with constant wall thickness, so slenderness of the curved axis beam is changed. Since, dimensions of the straight axis beam are constant, the slenderness ratio of two beams is changed. The graph shows the effect of this ratio change on frequency values for first eight modes. 
In the sixth chapter, factors which considered during this study are indicated. It is explained that four different frame systems are solved similar to street lamps and obtained results for all problems are compatible with analytical results. Obtained results in this study are discussed, and suggestions are made for the future researches.