Eksenel fonksiyonel derecelendirilmiş rotor pallerinin titreşim analizi
Vibration analysis of axially functionally graded rotor blades
Yazar: BURAK KILIÇ
Danışman: DR. ÖĞR. ÜYESİ ÖZGE ÖZDEMİR
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Uçak ve Uzay Mühendisliği Ana Bilim Dalı / Uçak ve Uzay Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2019
Konu: Makine Mühendisliği = Mechanical Engineering ; Uçak Mühendisliği = Aircraft Engineering

Özet:
Helikopterler, üzerlerinde yer alan rotor sisteminin dönmesi sonucunda oluşan kaldırma kuvvetiyle hareket eden hava taşıtlarıdır. Helikopterler hareket esnasında çeşitli nedenlerden dolayı titreşime maruz kalırlar. Görülen titreşimin temel kaynakları; motor devri ve frekansı, aerodinamik kuvvetler ve kendi kendini tahrik eden titreşimlerdir. Helikopterlerde titreşim kaynakları incelendiğinde en önemli titreşim kaynağının rotor ve paller olduğu görülür. Pal tasarımının doğru yapılmaması durumunda uçuş esnasında meydana gelen aeroelastik kuvvet ve momentler önce rotora, rotordan da iletim mili vasıtasıyla dişli kutusu ve güç sistemine aktarılır. Bu sebeple helikopter gövdesinin büyük bir kısmında titreşim meydana gelmiş olur. Bu problemin ortadan kaldırılması için pal tasarımı ve malzeme seçiminin doğru yapılması ve pal üzerinde yüzey kontrol elemanları kullanarak aerodinamik etkileri azaltmak gerekmektedir.
Tasarım çalışmasının ilk adımı olan titreşim ve doğal frekans hesaplamalarında birçok nümerik ve yaklaşık hesaplama metodu kullanılmaktadır. Bu yöntemlerin başında Sonlu Elemanlar Metodu (SEM) gelmektedir. Sonlu Elemanlar Metodu (SEM), kompleks bir mühendislik probleminin sonlu sayıda alt bileşene bölünerek her bir bileşenin ayrı ayrı ele alınması sonucunda kabul edilebilir yakınsaklıkta yaklaşık çözümlerin elde edilmesidir.
Söz konusu tez çalışmasında bünyesinde iki farklı malzemeyi farklı kompozisyonlarda barındıran ve malzeme özelliklerinin eksen boyunca değiştiği fonksiyonel derecelendirilmiş pallerin serbest titreşimi davranışlarının incelenmesine yönelik bir hesaplama modülü geliştirilmiştir. Hesaplama modülü, Mathematica programında yapılacak kodlama çalışmaları sonucunda geliştirilmiş ve farklı malzeme özelliklerine bağlı olarak palin doğal frekans değerleri incelenmiştir. Geliştirilen modülün hesaplama yöntemi Sonlu Elemanlar Metodu olacak ve hesaplama çalışmaları sonucunda elde edilecek olan doğal frekans değerleri, literatür çalışmalarına ait sonuçlarla kıyaslanarak doğrulamalar yapılmıştır.
Mathematica programı yardımıyla geliştirilen hesaplama modülü ile; farklı kiriş teorileri, farklı malzeme özellikleri ve homojenlik durumu, farklı sınır şartları, dönme hareketi ve merkezkaç kuvveti gibi birçok farklı durum için sayısal modelleme yapılmış ve bu modellemelerle titreşim analizi yapılarak doğal frekans değerleri bulunmuştur.
Yapılan çalışmalar sonucunda elde edilen doğal frekans değerleri, literatürde yer alan ilgili değerlerle kıyaslanmış ve böylece geliştirilen sayısal modelin doğrulaması yapılmıştır. Elde edilen sonuçlar karşılaştırıldığı zaman her farklı durum ve kiriş teorisi için de değerler arasındaki fark ve hata oranlarının kabul edilebilir seviyelerde olduğu görülmüştür.

Abstract:
Helicopters are rotating aircraft systems. Helicopters are subject to vibration for various reasons during movement. The main sources of vibration; engine speed and frequency, aerodynamic forces and self-driving vibrations. When the sources of vibration in helicopters are examined, it is seen that the most important vibration source is rotor and rotor blades. If the rotor blade design is not done correctly, the aeroelastic forces and moments that occur during the flight are transferred to the rotor and to the gearbox and power system via the transmission shaft from the rotor. For this reason, vibration occurs in a large part of the helicopter body. In order to eliminate this problem, it is necessary to minimize the aerodynamic effects of rotor blade design and material selection using surface control elements on the rotor blade.
In order to perform the vibration analysis in the rotor blade design process correctly, it is necessary to determine the natural frequency values and vibration types, to determine the variable forces and moments, to investigate the solutions for damping and to design the appropriate damping system. In addition, fatigue and life calculations under a certain working condition and physical tests should be performed and the results should be compared with theoretical calculations and validation studies should be done.
Many numerical and approximate calculation methods are used in the vibration and natural frequency calculations which are the first step of the design study. Finite element method (FEM) is the beginning of these methods. The finite element method (FEM) is to obtain approximate solutions in acceptable convergence by dividing a complex engineering problem into a finite number of sub-components by considering each component separately.
In this thesis, a calculation module has been developed for examining the free vibration behavior of the functionally graded rotor blades, which contain two different materials in different compositions and whose material properties change along the axis. The calculation module was developed as a result of coding studies in the Mathematica program and the natural frequency values of the rotor blade were investigated depending on the different material properties. The calculation method of the developed module will be a finite element method (FEM) and the natural frequency values that will be obtained as a result of the calculation studies are compared with the results of the literature studies.
The thesis focuses on axially functional graded beams showing different boundary conditions and movement characteristics. Functionally graded material (FGM) technology has begun to take place in engineering applications at a near future. Functionally graded materials (FGM) resulting from the development of composite material technology is a new generation material that enables the advanced engineering applications by reflecting the mechanical, physical and chemical properties of the materials it contains. Nowadays, the material properties and material composition of functionally graded materials (FGM) whose production areas are increasing and application fields are increasing day by day with the development of additive manufacturing technology and powder metallurgy are changing throughout the structure. The material change takes place in the form of a gradient connected to a function. Due to these properties, the functional graded materials are used and applied in many different sectors such as aerospace, automotive and medical.
Elements such as bars, columns and beams are frequently used in engineering applications. Various beam theories are used in the analysis and calculation of these elements. The assumptions and calculation methods of each beam theory differ. For this reason, the most appropriate beam theory should be applied by taking into consideration the element properties and acceptances. Beam theories, such as rotor blades and airplane wings, provide a one-dimensional modeling of structures that are very large in length. Within the scope of the study, the beam model was developed by using Euler - Bernoulli and Timoshenko beam theories.
The Euler - Bernoulli Beam Theorem is a theory developed by Leonard Euler and Daniel Bernoulli, and is also known as thin bar theory. In this theory, the sections perpendicular to the plane and beam axis remain the same before and after bending. The reason for this situation is the neglect of the deformations due to shear in theory. The Timoshenko Beam Theorem is a theorem developed by Stephen Timoshenko and also known as thick beam theory. In Timoshenko Beam Theory, unlike the Euler - Bernoulli Beam Theory, shear stresses and deformations due to these stresses are considered. Thus, the sections which are perpendicular to the neutral axis rotate at a certain angle after bending. Therefore, the effect of shear stresses and deformations due to these stresses are also taken into consideration in the calculations. In addition, the shear correction factor is used for the Timoshenko beam to be a constant value.
Within the scope of the thesis, axial functional graded beams were modeled according to the Euler - Bernoulli and Timoshenko beam theories and vibration analyzes were performed. In these studies, beam conditions with different boundary conditions and material properties were investigated. For the creation of mathematical model and solution, finite element method (FEM) was used. The related formulas and equations of the finite element method (FEM) and beam theories were transferred to the software environment with the help of Mathematica program and a calculation module which can perform free vibration analysis according to the different beam characteristics was developed. Calculation results were compared with the literature. Studies conducted in this context; Free vibration analysis of homogeneous and uniform beams and free vibration analysis of axially functional graded beams.
With the calculation module developed with the help of Mathematica program; Numerical modeling has been made for different beam theories, different material properties and homogeneity status, different boundary conditions, rotational motion and centrifugal force.
Natural frequency values obtained as a result of the studies are compared with the related values in the literature and thus the numerical model developed has been verified. When the results were compared, it was found that the difference between the values and the error rates for each different state and beam theory were acceptable. Clamped-free and clamped-clamped boundary conditions were applied. In addition,the situation of turning the beam or not has been examined. Different power-law exponent values representing the distribution of the material on the beam are examined and the effect of this value on natural frequency is examined.
The calculation module developed in this sense has been a successful finite element solution which can calculate the natural frequency value according to different boundary conditions, different geometric properties (length and thickness) and different material gradients of axially functional graded beams. It is foreseen that with the specializations and improvements to be made on the module, different vibration conditions such as forced vibration and successful results can be obtained for different systems.