Kesiti kademeli değişen eğri eksenli düzlemsel kompozit çubukların düzlem dışı serbest titreşimlerinin kesin çözümü
Exact solution of out-of-plane free vibrations of stepped planar composite curved beams
Yazar: ANIL ÇETİNKOL
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:
Mühendislik uygulamaları açısından kirişler, yapısal eleman olarak önemli bir yere sahiptir. Literatürde genellikle düz kompozit çubuklar ele alınmış, kademeli ve eğri eksenli düzlemsel kompozit çubuklar üzerine yapılan çalışma çok azdır? Bu çalışmada kademeli ve eğri eksenli düzlemsel kompozit çubukların titreşim davranışlarının incelenmesi amaçlanmıştır. Bu çalışmada da kiriş eleman özelinde, ortotropik malzeme özelliği gösteren kompozit çubuklar ele alınmıştır 
Birinci bölümde, çalışmanın konusu olan çubuk ve çubuk teorisi hakkında bilgi verilmiş ve devamında çalışmanın amacına ulaşmasında ne tür parametreler kullanılacağı ve hedef çıktılar belirlenmiştir.
İkinci bölümde, eğri eksenli düzlemsel kompozit çubukların dinamik davranışlarıyla ilgili yapılan çalışmalar incelenmiş ve ilgili çalışmaların yöntemi ve çıktıları hakkında bilgiler verilmiştir. Çubuğun kesiti kademeli olarak değişen çalışmalarına literatürde az olduğu gözlemlenmiştir.
Üçüncü bölümde, elastik çubukların genel teorisi noktasından başlayan teorik çalışmalar önce izotrop malzemeden yapılmış eğri eksenli çubuğun statik davranışları, sonrasında D'Alambert prensibi kullanılarak izotrop malzemeden yapılmış eğri eksenli çubuğun dinamik davranışları ve son olarak da kompozit malzemeden yapılmış eğri eksenli çubuğun statik ve dinamik genel diferansiyel denklemleri sunulmuştur. Burada düzlem içi ve düzlem dışı denklemler bir arada verilmiştir. İlgili denklemler çıkarılırken kompozit tabakaların özellikleri, kayma deformasyonu ve dönme eylemsizliği etkileri hakkında bilgiler verilmiştir.
Dördüncü bölümde, bir önceki bölümde elde edilen diferansiyel denklemlerden çubuğun düzlem dışıyla ilgili olan denklem takımları için kesin çözüm verilmiştir. Hem sabit kesitli hem de tek kademeli çubuklar için denklemlerin nasıl formüle edildiği ve çözüldüğü açıklanmıştır. Denklemlerdeki kayma deformasyonu ve dönme eylemsizliği ile ilgili olan ifadeler ve denklem takımlarındaki etkileri belirtilmiştir. Bölümün devamında çalışmadaki problemler tanıtılmış ve çalışmayla ilgili olan mesnetleme tipi, narinlik oranı, kademe konumu, kademe oranı, tabaka kodu, kirik açıklığı, elastisite modülü oranı gibi parametrelerin neler olduğu açıklanmıştır.
Beşinci bölümün ilk kısmında, sabit kesitli çember eksenli kompozit çubuğun ilk beş moduna ait boyutsuz frekansların narinlik oranı, kiriş açıklığı, tabaka kodu ve elastisite modülü oranıyla nasıl değiştiği grafikler halinde verilmiştir. Grafiklerde ayrıca farklı sınır şartları ile kayma deformasyonu ve dönme eylemsizliği etkileri gösterilmiştir. İkinci kısımda ise tek kademeli yapıda olan çubuk için değerlendirme yapabilmek adına yeterli olduğu düşünülen ilk beş mod için boyutsuz frekans değerleri ele alınmıştır. Farklı narinlik oranı, kiriş açıklığı, tabaka kodu gibi parametrelerin yanı sıra kademe oranı, kademe konumu gibi kesiti kademeli değişen çubuklar için önemli olan parametrelerin boyutsuz frekansları nasıl etkilediği grafikler halinde verilmiştir. Sabit kesitli çubukta olduğu gibi farklı sınır şartları ile kayma deformasyonu ve dönme eylemsizliği etkileri incelenmiştir. Sunulan birçok grafik, tasarım aşamasında mühendislerin ve çalışanların faydalanabileceği birer kaynak haline gelmiştir.
Altıncı bölümde ise literatürde, bu çalışmada olduğu gibi kademeli yapıda kompozit çubukla ilgili sayısal bir çalışmaya rastlanmamasından ötürü, gelecekte yapılabilecek muhtemel çalışmalara referans olması amacıyla endüstriyel uygulamalarda kullanılan farklı malzeme tipleri için sayısal çalışmalar yapılmıştır. Ayrıca daha önce izotrop malzemelerle ilgili yapılan benzer çalışmalarla karşılaştırmalar yapılıp, tabakalı kompozit malzemeden ve izotrop malzemeden yapılmış eğri eksenli çubukların frekans değerlerindeki farklılıklar açıklanmıştır.

Abstract:
In aspect of engineering applications, beams as structural element have a large and important place. Composite materials, another engineering phenomena with superior material properties, continue to expand and consolidate its position in structural areas with developing industrial applications. In this study, laminated composite arches that is orthotropic material was choosen as the main subject. While straight composite beam are widely used in the literature, stepped and curved laminated composite arches that is rarely seemed were studied and it is aimed to investigate the vibration behavior.
In engineering world the definition of orthotropic materials also includes the laminated composite materials. But also each laminae of laminated composite can be orthotropic  material in itself. In generally the material properties of a laminae has three different Elastic Modulus if it is not considered as a two-dimensional material. In this study the material was taken into account as three-dimensional due to reach free vibrations of beams out-of-plane. Otherwise there were no result could not be observed in terms of vibration behaviour. These three different Elastic Modulus values are result of orientation of reinforcement in a matrix. Throughout this study it is considered like that every laminae of laminated beam is made of same material with each other. It is another thing that makes the calculation of vibration easier.
In the first chapter, the information about rod and rod's theory was given. Then, the target outputs and the parameters that will be used to achieve aim of the study were determined. 
In the second chapter, the investigations about the dynamic behavior of composite curved beams were examined and the method and outputs of related studies were given. The studies on curved composite beams which have stepped or graded cross-sections were found as few as possible.
In the third chapter, in the theoretical studies starting from the general theory of elastic rods, differential equations were presented for firstly the static behaviors of curved beams made of isotropic material, then dynamic behaviour of curved beams made of isotropic material using D'Alambert Principle, and finally the static and dynamic behaviors of curved beams made of laminated composite material. 
There are some few theory about the beam element and one of them is selected to make comparison with other studies which used same theory. First, for the theory the static behaviour equations of beam were obtained. To obtain the static behaviour equations, fundemental equations for the beam, such as equilibrium equation, replacement-strain equations and constitutive equations, were used. Both in-plane and out-of-plane behaviour equations were given together. It was seen that isotropic and orthotropic material have same equations for in-plane and out-of-plane at the beginning of the study.
The equations obtained were used to have dynamic equations in terms of vibration behaviour. For getting dynamic equations D'Alambert Principle was selected. With this principle the equations of beam were obtained depending on the coordinates and time. But due to the vibration is considered as the behaviour of the beam depending ob the small time span, the time expressions were neglected. Thus, the equations were only depended on the coordinate expressions. Throughout the study time expressions were not used.
The assumptions made were not only for the beam which is made of isotropic material but also for the beam which is made of orthotropic or laminated material. Due to nature of laminated material the constants of material properties were not only one for each definition. The orthotropic material has three different values for elastic modulus, shear modulus, Poisson's ratio. These different values of material constants were used in proper way in each related equation of the static behaviour equation set. However, it is also possible to take these relationships into account for dynamic behaviour of a laminated composite material.
In the same way D'Alambert Principle was applied to the laminated composite material. Thus, final equation set was obtained for the laminated composite beam with linear axis. 
Due to the scope of the study is on curved beams, it had to be performed for beams which has curved axis as well. At that point the exact solution method were used. As required by the exact solution definition, the new assumptions were done. One of them is that the cross-section area of the beam must be constant through the tangential axis. The other one is that the curvature of the tangential axis must be constant. So, the beam becomes a complete circle or part of a circle. Only with these two design parameters can the exact solution be reached. 
Information about the properties of composite layers and effects of shear deformation and rotatory inertia were detailed. The researchers who contribute on the solution of the problems were cited as reference.
The expressed differential equations were obtained in cases where the laminae angles of laminated material are 0º or 90º and symmetric in both geometry and material properties about the middle surface. One of the reasons for these assumptions was that it can be similar to the studies in the literature. Another was to be able to simplify general differential equations. When these assumpstions are applied to the study it makes the shear stress between the laminates neglected. 
In the fourth chapter, the exact solution was obtained for the sets of equations which related to the out-of-plane vibrations of  curved beam from the differential equations obtained in the previous chapter. It was explained how to formulate and solve equations for both uniform cross-section and single stepped beams. For a curved beam, exact solution can only be used when its cross-section area and curvature is constant value. The governing equations for the out-of-plane vibrations of curved beams were solved exactly by using the initial value method. The required inital values are found by equaitons obtained from boundary conditions. The expressions related to shear deformation and rotatory inertia in equations and their effects on equation sets are indicated. In the following section, the problems in the study were introduced. The parameters such as support type, slenderness ratio, step position, step ratio, layer orientation, opening angle, elastic modulus ratio were explained. 
In the first part of the fifth chapter, it was given how graphs of dimensionless frequencies of the first five modes of uniform curved beams change with different slenderness ratio, opening angle, layer orientation and elastic modulus ratio. The graphs also showed  effects of shear deformation and rotatory inertia assumptions. In the second part of the fifth chapter, dimensonless frequency values of first five modes of a single stepped curved beams were discussed. In addition to effects of parameters such as different slendeness ratio, opening angle, layer orientation, effects of parameters which are important for curved beams with varying cross-section such step ratio, step position were used in graphs. The effects of shear deformation and rotational inertia with different boundary conditions were investigated. The graphics presented as outputs of this chapter became a resource that engineers and employees can use during the design phase.
In the sixth chapter, since no numerical study on composite curved beams with stepped cross-section as in this study was found, numerical studies were conducted for different material types used in industrial applications in order to be a reference for possible future studies. In addition, comparisons in frequency values were made between isotropic and laminated composite materials for similar works that is studied before.
To expansion the scope of the study some new design parameters could be taken into account in the future studies. Adding the shear stress between laminaes into equations the behaviour of the laminated composite beam, which has laminae with different fiber angles than 0º and 90º, could be performed. 
The other design to work with could be a laminated material with an anti-symmetry laminae pattern. With this type of design the in-plane vibrations of beam affect the out-of-plane vibrations of beam. This requires more comprehensive equation set that define the behaviour of material. Due to the interplanar effect both in-plane and out-of-plane equation should be solved together. Their effect on each other cannot be ignored.
Another design to work with could be beam whose cross-sectional area is constantly changing. For such a element, dividing the beam into small pieces can provide an approximate solution. Dividing into small pieces means that there will be many sections with different cross-sectional area and there should be continuity between them.