Değişken kesitli kirişlerin titreşimleri (kesin çözüme dayalı incelemeler)
Vibrations of tapered beams (parametric studies based on exact solution)
Yazar: SEVİNÇ AYCAN YETİM
Danışman: PROF. DR. ÖZGÜR TURHAN
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Makine Dinamiği, Titreşimi ve Akustiği Bilim Dalı
Derece: Yüksek Lisans, 2022
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Değişken kesit = Variable cross section ; Kirişler = Beams

Özet:
Değişken kesitli kirişler; makina, inşaat, havacılık gibi birçok mühendislik alanında en temel yapısal veya makina bileşeni olarak kullanılırlar. Mekanik sistemlerde, arızalara, performans düşüklüğüne ve gürültüye sebep olan titreşim, özellikle çalışma koşullarında meydana gelebilecek hasarların önlenebilmesi açısından incelenmesi gereken en önemli mühendislik problemlerinden biridir. Titreşim karakteristiklerinin gerçeğe yakın olarak öngörülebilmesi ve tasarımın en baştan bu bilinçle gerçekleştirilmesi, titreşim kaynaklı problemlerin bir felakete sebep olmadan önce önlenebilmesi açısından büyük öneme sahiptir. Bu sebeple, genellikle taşıyıcı yapı olarak modellenen kirişlerin titreşim problemleri önemli bir araştırma konusudur.
Bu tezde, mühendislik uygulamalarında sıklıkla karşılaşılan değişken kesitli kirişlerin titreşim problemleri üzerine çalışılmıştır. Eni sabit, yüksekliği doğrusal olarak değişen ve eni ve yüksekliği aynı oranda değişen kirişlerin eğilme titreşimleri problemi ele alınmış, çeşitli sınır koşulları altında, yalın, üzerinde bir noktasal kütle taşıyan ve lineer bir yay ile desteklenmiş kirişlerin boyutsuz doğal frekansları ve kip biçimleri kesin çözüme dayalı olarak belirlenmiştir. Kalın ucu gömülü – ince ucu serbest, ince ucu gömülü – kalın ucu serbest, mafsallı – mafsallı, gömülü – gömülü, kalın ucu gömülü – ince ucu mafsallı, ince ucu gömülü – kalın ucu mafsallı, kalın ucu mafsallı – ince ucu serbest, ince ucu mafsallı – kalın ucu serbest ve serbest – serbest olmak üzere, olası dokuz sınır koşulu için kirişlerin doğal frekansları ve kip biçimleri elde edilmiştir.
Tez; giriş, matematiksel formülasyon, parametrik incelemeler ve sonuçlar olmak üzere 4 ana bölümden oluşmaktadır. Giriş bölümünde, çalışmanın tanımı ve amacı açıklanarak, değişken kesitli kirişler üzerine yapılan çalışmalar sunulmuş, bu çalışmanın literatüre katkısı belirtilmiştir. Matematiksel formülasyon bölümünde, önce, tez kapsamında ele alınan, eni sabit, yüksekliği doğrusal olarak değişen ve eni ve yüksekliği aynı oranda değişen kirişlerin bilinen kesin çözümü özetlenmiş, sonrasında bu çözüm, üzerinde bir noktasal kütle taşıyan ve lineer bir yay ile desteklenmiş kiriş problemlerine uyarlanmıştır. Parametrik incelemeler bölümünde, sırasıyla, yalın, üzerinde bir noktasal kütle taşıyan ve lineer bir yay ile desteklenmiş kirişlerde kesit değişim ölçüsü, boyutsuz kütle büyüklüğü, boyutsuz yay katsayısı, boyutsuz kütle konumu ve boyutsuz yay konumunun, boyutsuz doğal frekanslar ve kip biçimleri üzerine etkisi, olası dokuz sınır koşulu için parametrik olarak incelenmiş, sonuçlar tablolar ve grafikler halinde sunulmuştur. Son olarak sonuçlar bölümünde, çalışmadan elde edilen sonuçlar derlenmiş, bu çalışmanın geliştirilebilmesi için ileride yapılabilecek çalışmalar için önerilerde bulunulmuştur.

Abstract:
Variable cross-section beams are used as main structural or machine components in many engineering fields such as mechanical, civil and aerospace engineering. Vibrations, which result in malfunction, decrease in operating performance and noise in mechanical systems, is one of the most important engineering problems that should be considered in order to prevent these to occur in working conditions. Accurate prediction of the vibration characteristics of common design elements like beams is, therefore, of utmost importance for the design process and constitutes a subject of continuous research interest.
In this study, the exact solution of the free flexural vibration problem of tapered beams is studied. The problem of flexural vibrations of tapered beams, in which the depth is varying linearly whereas the width remains constant along the length and in which both the depth and the width vary linearly, are considered. Non-dimensional natural frequencies and mode shapes of plain tapered beams under various boundary conditions, of those carrying a point mass and supported by a linear spring are obtained based on exact solution. 
The study has a two-fold goal. In the first place, it aims to extract as much general instruction as possible from the exact solution of the tapered beam frequency analysis. To this end, the frequency analysis results are presented for each case, in the form of graphs showing the dependence of the natural frequencies on a selected system parameter; thus facilitating general conclusions to be drawn on the effect of the considered parameter on natural frequencies. In the second place, the obtained exact results are expected to constitute a sound basis of reference for studies based on approximate methods. To this end, it includes a large amount of tabulated numerical results corresponding to various parameter combinations of the related systems.
Nine possible boundary conditions are considered for each case. These are clamped-free (both, thick and thin end clamped configurations), pinned-pinned, clamped-clamped, clamped-pinned (two configurations), pinned-free (two configurations) and free-free boundary conditions.
Problem formulations and results are all given in non-dimensional form, to facilitate generalizations and quick reference. 
The thesis consists of four chapters. Namely, "Introduction", "Mathematical Formulation", "Parametric Studies" and "Conclusions". 
In the first chapter, the description and purpose of the study are explained and researches on variable cross-section beams are presented. A literature survey has shown that although there are many studies on tapered beams, most of them are based on approximate methods such as Galerkin, Rayleigh–Ritz and Finite Elements Methods. The limited number of researches based on exact solution are reviewed and summarized.
In the second chapter, a problem formulation is given, covering two different kinds of tapered beams: Those with only the depth is linearly variable (labelled n=1 in the text) and those with both depth and width are linearly variable (labelled n=2).The exact solution of the problem as given by Koloušek is first described and then adapted to the problems of tapered beams carrying a point mass and to those supported by a linear spring. 
In the third chapter, parametric studies are carried out that demonstrate the effect of various system parameters on natural frequencies and mode shapes and results are presented in the form of tables and graphs.
In the first part of the third chapter, the effect of taper ratio on natural frequencies and mode shapes in a plain tapered beam is investigated for nine possible boundary conditions. The first seven non-dimensional natural frequencies are computed for each boundary condition combination. The results are presented on Tables 3.3-3.8 and C1-C6 for certain selected values of the taper ratio . Along with, Figures 3.1, 3.2, 3.4, 3.6, 3.8, 3.10 and 3.12 are given depicting the dependence of the natural frequencies on that parameter, making it possible to draw general conclusions on the effect of taper on the natural frequencies. As a result, it is concluded that an increase in the taper ratio has the effect of reducing all the dimensionless natural frequencies (non-dimensionalized with respect to the thickest cross section features) in both n=1 and n=2 cases, except the fundamental frequency of the fixed-free beam with thick end fixed, which increases with taper ratio. In addition, the first six mode shapes of the beam with different boundary conditions are also obtained and presented on the same graph for quick comparison for three different taper ratio values, which are α=0.5, α=1 and α=1.5. These graphs are presented on Figures 3.3, 3.5, 3.7, 3.9, 3.11 and 3.13. 
In the second part of the chapter, a beam carrying a point mass is studied for the nine possible boundary conditions listed above and the effect of taper ratio, mass and location of the point mass on vibration characteristics is investigated. Again, the first seven non-dimensional frequencies and the first six mode shapes are obtained for each case and results are presented both in tabular form and in the form of graphs.
First, the mass and the location of the point mass are fixed, and the effect of taper ratio on the natural frequencies and mode shapes is investigated. Results are presented both in tabular form and in the form of graphs depicting the dependence of the natural frequencies on the taper ratio. Tables 3.9-3.14 and D1-D6 and Figures 3.14, 3.16, 3.18, 3.20, 3.22 and 3.24 present the results related to natural frequencies while figures 3.15, 3.17, 3.19, 3.21, 3.23 and 3.25 depicting the results related to natural mode shapes.
Then, the taper ratio and the location of the point mass are fixed, and the effect of the point mass is investigated. Results are again given in both tabular form and in the form of graphs depicting the dependence of the natural frequencies on the added lumped mass. Tables 3.15-3.20 and D7-D12 and Figures 3.26, 3.28, 3.30, 3.32, 3.34 and 3.36 present the results related to natural frequencies while Figures 3.27, 3.29, 3.31, 3.33, 3.35 and 3.37 depicting those related to natural mode shapes. 
And last, the taper ratio and the point mass are fixed, and the effect of the point mass location is investigated.  Tables 3.21-3.26 and D13-D18 and Figures 3.38-3.49 present the results related to natural frequencies.
The third part of parametric studies consists of investigations on tapered beams supported by a linear spring. Four different boundary conditions (clamped-free, pinned-pinned, clamped-clamped and clamped-pinned) are considered and the effect of taper ratio, spring constant and location of the spring is considered. The first seven non-dimensional frequencies and the first six mode shapes are obtained for each case. The results of the investigation are presented on Tables 3.27-3.38 and Figures 3.50-3.66.  
In the final chapter of the thesis a general evaluation of the study and some suggestions for future studies are presented.