Kademeli ve düz kiriş yapılarının termal etki altındaki titreşim davranışının incelenmesi
Investigation of the vibrational behavior of stepped and straight beam structures under thermal effect
Yazar: FURKAN ALTINTAŞ
Danışman: PROF. DR. METİN ORHAN KAYA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı / Uçak ve Uzay Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2022
Konu: Uçak Mühendisliği = Aeronautical Engineering
Dizin: Kiriş teorisi = Beam theory ; Kirişler = Beams ; Kompozit malzemeler = Composite materials ; Titreşim = Vibration

Özet:
Kiriş yapılarının başta havacılık olmak üzere otomotiv ve inşaat sektöründe oldukça yaygın bir kullanım alanı vardır. Genel olarak, boyuna ve enine dik yükleri destekleyen kiriş yapılarının uzunluğu kesit ölçülerine göre oldukça büyüktür. Kirişler, eksenine dik olan yükleri taşır ve bu nedenle yükler uzunluğa dik doğrultudadır. Kullanım alanı oldukça geniş olan kiriş yapılarının analizinde çeşitli metotlar ve teoriler bulunmaktadır. Bu çalışma kapsamında Euler-Bernoulli ve Timoshenko kiriş teorileri incelenmiş, kirişlerin titreşim davranışının incelenmesi amacıyla titreşim denklemleri elde edilmiştir. Bir dizi diferansiyel denklemden oluşan titreşim denklemlerinin çözümü için oldukça yaygın ve pratik çözüm yöntemi olan Diferansiyel Dönüşüm Metodu (DDM) kullanılmış, sonuçlar analitik çözüm ile kıyaslanmıştır. Diferansiyel Dönüşüm Metodu için MATLAB kodu oluşturulmuş, çözümler geliştirilen kod yardımıyla elde edilmiştir. Bu teorilerin haricinde günümüzde kullanımı yaygınlaşan Sonlu Elemanlar Yöntemi(SEY) ile de sonuçlar elde edilmiş ve analitik sonuçlar ile karşılaştırma yapılmıştır. Sonlu Elemanlar Yöntemi'nde ABAQUS paket programı kullanılmış, kiriş modellemeleri bu yazılım ile yapılmıştır.Uçak yapıları yüksek mukavemete ve yorulma dayanıma sahip oldukça hafif yapılardan oluşmaktadır. Bu bağlamda tez çalışmasında incelenmek üzere kiriş yapı malzemeleri olarak çeşitli metalik ve kompozit malzemeler belirlenmiştir. Kompozit malzemelerin elastik karakteristikleri belirlenmesi için kompozit teorisi kullanılarak laminaların mikromekanik ve makromekanik analizi yapılmıştır.Sıcaklıkla ilgili fenomenlerin yapılar üzerindeki etkisi oldukça geniş bir çalışma alanına sahiptir. Sıcaklıktaki değişiklik, kirişin titreşim davranışında büyük bir değişikliğe sebep olabilir. Bu tür yapıların dinamik davranışları yapının termal genleşmesine ve malzeme özelliklerine bağlı olarak sıcaklık etkisiyle değişmektedir.Bu tez çalışmasında sıcaklık değişiminin kirişlerin davranışına etkisini incelemek amacıyla elde edilen titreşim denklemine sıcaklık terimi eklenmiş ve çözümler yinelenmiştir. Belirlenen beş farklı sıcaklık değişimi için sonuçlar karşılaştırmalı olarak verilmiştir. Kesiti, uzunluğu boyunca değişken olan kirişlere kademeli kirişler denmektedir. Uçaklarda kullanılan kanat yapılarını ele aldığımızda, uçak gövdesinden uzaklaşıldıkça kanat kesiti küçülmekte dolayısı ile kesit değişmektedir. Bu yapılar da çalışma kapsamına dahil edilmiş, düz kirişlere ek olarak kademeli kirişler de incelenmiştir. İncelenmek amacıyla uzunluğu boyunca farklı oranlarda kesiti ve malzemesi değişen çeşitli kirişler oluşturulmuş, titreşim denklemleri elde edilmiştir. Düz kirişler, ankastre mesnet-serbest uç, ankastre mesnet-ankastre mesnet, ankastre mesnet-kayar mesnet ve sabit mesnet-kayar mesnet olmak üzere dört farklı sınır şartında incelenirken kademeli kirişler ise ankastre mesnet-ankastre mesnet sınır şartında incelenmiştir. Sonuç bölümü oldukça detaylı oluşturulmuştur. Her bir malzeme ve kesit için beş farklı sıcaklık değişimi için sonuçlar Euler-Bernoulli, Timoshenko kiriş teorisi ve SEY çözümü yapılarak tablolar ile verilmiştir. Sıcaklık etkisinin kirişin titreşim davranışına etkisinin ve farklı sınır şartlarının incelenmesi ile diğer çalışmalar için temel oluşturması amaçlanmıştır.

Abstract:
Beam structures have a very common usage area in the automotive and construction sectors, especially in aviation. For this reason, free and forced vibrations of beams have been studied by many researchers and important studies have been put forward. Elements such as reinforcement and main beams that make up aircraft structures, and horizontal columns used in buildings and bridges are examples of beams. In general, the length of beam structures supporting vertical longitudinal and transverse loads is quite large relative to their cross-sectional dimensions. Beams carry loads perpendicular to their axis and therefore loads are in the direction perpendicular to the length. There are various methods and theories in the analysis of beam structures, which are widely used. Euler-Bernoulli and Timoshenko beam theories are frequently used in studies on this subject. Shear stress is not taken into account in Euler-Bernoulli beams, while shear stresses are taken into account in Timoshenko beams.Within the scope of this study, Euler-Bernoulli and Timoshenko beam theories were examined and vibration equations were obtained in order to examine the vibration behavior of beams. Differential Transformation Method (DTM), which is a very common and practical solution method, was used for the solution of vibration equations consisting of a series of differential equations, and the results were compared with the analytical solution. The differential transformation method is one of the effective tools used to solve general differential equations. The transformation technique in this method is based on the Taylor series expansion. By using the differential transformation technique, ordinary and partial differential equations are transformed into algebraic expressions using Taylor series expansion. By solving differential equations with the help of this method, the processing time is shortened most of the time and accurate results are 
achieved. MATLAB code was created for Differential Transformation Method and solutions were obtained with the help of the developed code. Apart from these theories, results were obtained with the Finite Element Method (FEM), which has become widespread today as a result of technological developments, and comparisons were made with analytical results. In Finite Element Method, ABAQUS package program was used and beam models were made with this software.The results obtained by beam theories and finite element method have been verified by comparing with literature studies. The effect of the slenderness ratio was examined in different boundary conditions with beam theories and finite element method, and a suitable slenderness ratio was selected for the study. In this study, convergence study was carried out for the differential transformation method used in the solution of Euler Bernoulli and Timoshenko beam theories, and the required number of n was decided depending on the desired mode number. Vibration is one of the important phenomena affecting engineering structures. Understanding the vibration behavior of mechanical and structural systems is essential for the safe design, construction and operation of various machines and structures. Failure of many mechanical and structural elements and systems can be attributed to vibration. For example, blade and disc failures in steam and gas turbines and structural failures in airplanes are often associated with vibration and the resulting fatigue phenomenon. In this study, general information about vibration is mentioned. Vibration problems are classified and information about vibration movements is given. Metallic and composite materials are widely used in aerospace, both because of their broad physical, mechanical and chemical characteristics, and because of the modification of their properties by structural arrangement. In this sense, the most important reason for the increase in the use of composite materials is the high strength/weight ratio. By reducing its weight, it allows more cargo, fuel and passengers to be carried. Composite and metal parts are very strong and have very high fatigue strength. In addition, the ability of composite materials to take complex and aerodynamic profile shapes easily creates another advantage. When we look at the 
materials used in aircraft construction, it is seen that aluminum has been mostly adhered to for many years. Today, the production of important parts such as wings is carried out with this material. In this context, various metallic and composite materials were determined as beam building materials to be examined in the thesis study. General information about composite materials is given, classification of composite structures is mentioned. Micromechanical and macromechanical analysis of laminae were performed using composite theory to determine the elastic characteristics of composite materials. Within the scope of this analysis, equations describing the relationship between the displacement and deformation of the layers and laminae were obtained. By making a literature search, the most used layer sequences were determined, and various laminates were created using these sequences. It was aimed to find the equivalent elasticity constants of these laminates, and in this context, the A, B and D matrix, which defines the force-moment and shape-displacement relationship, was obtained.The effect of temperature-related phenomena on structures has a wide field of study. The change in temperature can cause a large change in the vibration behavior of the 
beam. The effect of temperature is taken into account in order to effectively design the buildings, and the designs are shaped accordingly. For example, satellites are subject to thermal cycling due to periodic entries and exits into the Earth's shadow. In addition, the operating conditions of automobile and aircraft structures are in a wide temperature range. The dynamic behavior of such structures changes with the effect of temperature depending on the thermal expansion of the structure and the material properties. In this thesis, the temperature term was added to the vibration equation obtained in order to examine the effect of temperature change on the behavior of beams and the solutions were repeated. The results for the five different temperature changes determined are given comparatively.Beams whose cross section is variable along their length are called stepped beams. When we consider the wing structures used in airplanes, the wing section gets smaller as you move away from the aircraft body, so the section changes. Aircraft wings are 
important structures that affect performance both structurally and aerodynamically. For this reason, it is necessary to analyze these structures and determine their static and dynamic behavior under the influence of force. In this case, the wings can be modeled and analyzed using stepped beam structures. These structures are also included in the scope of the study, in addition to straight beams, stepped beams are also examined. In order to be examined, various beams with varying cross-section and material along their length were formed and vibration equations were obtained. While 
the straight beams are examined in four different boundary conditions: clamped-free end, clamped-clamped, clamped-roller and pinned-roller, stepped beams are examined in the clamped-clamped boundary condition.The conclusion section is very detailed. The results for five different temperature changes for each material and section are given in tables by making Euler-Bernoulli, Timoshenko beam theory and FEM solution. It is aimed to form the basis for other studies by examining the effect of temperature on the vibration behavior of the beam and different boundary conditions. In this study, it is thought that the thermal effect acting on the beams occurs due to a uniform temperature difference. In future studies, different cases of the effect of temperature difference on the beam can be examined. In addition, the mesh can be expanded by using various beam theories and differential equation solving methods.