Dalgalanan hızla dönen kirişlerde kaotik titreşimler
Chaotic vibrations in rotating beams with fluctuating speed
- Tez No: 834766
- Danışmanlar: PROF. DR. ÖZGÜR TURHAN
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Makine Mühendisliği, Uçak Mühendisliği, Physics and Physics Engineering, Mechanical Engineering, Aeronautical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2023
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Lisansüstü Eğitim Enstitüsü
- Ana Bilim Dalı: Makine Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Makine Dinamiği, Titreşimi ve Akustiği Bilim Dalı
- Sayfa Sayısı: 173
Özet
Türbinli jet motorları, rüzgâr türbinleri, helikopter ve uçak pervaneleri, kompresörler gibi önemli mühendislik uygulamaları nedeniyle rotor-pala sistemlerinin titreşimleri önemli bir araştırma konusu oluşturur. Titreşimlerden ortaya çıkan problemlerin, çalışma koşullarında sistem üzerinde hasar oluşturabilir. Bunun sonucu olarak yüksek maliyetlere sebep ve hayatî riskler meydana geleceğinden bu problemleri önlemek amacıyla titreşimlerin ele alınması gerekir. Bundan dolayı bu sistemlerin henüz prototip aşamasında titreşimlerden kaynaklanan problemlerin gerçek bir felakete sebep olmasına engel olmak için titreşim karakteristiklerinin öngörülmesi çok önemlidir. Bu çalışmada ilk olarak, rijit bir göbeğe ankastre bağlı elastik bir kiriş şeklinde modellenen rotor-pala sistemlerinde, rotorun bağlı olduğu milin burulma titreşimlerinden dolayı rotorda meydana gelen hız dalgalanmalarının palanın eğilme titreşimleri üzerindeki etkileri irdelenmiştir. Konsol kiriş sınır koşullarına sahip palanın titreşimleri, dönen elastik bir kirişin üçüncü dereceye kadar lineer olmayan terimlerin hesaba dahil edildiği düzlem içi eğilme titreşimleri olarak modellenmiştir. Kısmî türevli bir integro-diferansiyel denklemden ibaret olan hareket denklemi boyutsuz parametreler kullanılarak boyutsuzlaştırıldıktan sonra Galerkin yöntemi kullanılarak ayrıklaştırılmıştır ve sonlu sayıda denklemden oluşan bir adi diferansiyel denklemler takımı elde edilmiştir. Böylece kirişin hem parametre tahriği hem de dış zorlamaya maruz kaldığı tespit edilmiştir. Daha sonra hareketi kontrol eden kontrol 5 tane parametre sırasıyla boyutsuz göbek yarıçapı α, boyutsuz ortalama dönme hızı β0, boyutsuz dalgalanma katsayısı δ, boyutsuz viskoz sönüm faktörü ζ ve boyutsuz dalgalanma frekansı ω̅ olarak belirlenmiştir. Böylece bu parametrelerin hareket üzerinde etkileri gözlemlenebilir duruma gelmiştir. Ardından bir, iki ve üç serbestlik dereceli modeller üzerinde durularak titreşim davranışlarının Lyapunov üsleri hesabına dayalı kaos kartları, çatallanma diyagramları ve Poincaré tasvirleri vb. kaos kriterleri açısından değerlendirilerek kaotik sonuçlar verdiğine dair önemli bulgulara rastlanmıştır. Bölüm 1'de çalışmanın amacı belirtilerek literatürde yapılmış olan çalışmalar özetlenmiştir ve ardından çalışmanın gerekliliği ortaya konmuştur. Bölüm 2'de rotor hız dalgalanması probleminin teorisine ve kaos kriterlerine yer verilmiştir. Elde edilen hız dalgalanması probleminin Bölüm 3'te bir, iki ve üç serbestlik dereceli modellerine ilişkin incelemeler yapılarak bunlara ait kaos kartları verilmiştir. Kaos kartları kullanılarak modellerin karşılaştırılması yapılmıştır. Bölüm 4'te çalışmadan elde edilen sonuçlar özetlenerek ileride yapılacak olan çalışmalara ilişkin önerilerde bulunulmuştur.
Özet (Çeviri)
Dynamics and especially vibrations of rotor-blade systems, which are used in engineering applications such as turbine jet engines, wind turbines, helicopter and aircraft propellers, compressors have long been a subject of constant research interest. Problems arising from the mechanical vibrations should be handled in order to prevent possible damages causing both high costs and catastrophes to the system under operating conditions. Considering these facts, it is very important to anticipate vibration characteristics in order to prevent a real disaster caused by vibration problems at the prototype phase. In this study, chaotic vibrations on the nonlinear model of blade connected to a rotor with angular velocity fluctuations are studied. The rotor is considered as a rigid hub, while the blade attached to the rotor is modeled as a homogeneous Euler-Bernoulli beam with uniform cross-section and Kelvin-Voigt material model with viscous damping. The chaotic behaviors in the in-plane bending vibrations of the beam caused by the harmonically varying angular velocity of the hub are considered in discrete submodels obtained from the continuous model in terms of certain control parameters of the equation of motion. The main purpose of the thesis is firstly to determine the possible chaotic vibration behaviors of single, two and three degrees-of-freedom discrete sub-models generated from the continious beam-hub model, which represents the nonlinear dynamics of rotor-blade systems; then to decide the most ideal model by determining to what the extent the chaos cards obtained from models with different degrees-of-freedom confirm each other. The thesis study is organized under four main titles:“Introduction”,“Theory”,“Angular Velocity Fluctuation Problem”,“Conclusions”The first part of the thesis, the“Introduction”section, summarizes the studies in the literature. Due to the detection of chaotic behaviors in single and two degree of freedom nonlinear models of beams, the possibility of chaotic behavior of rotating beams has emerged. The aim was to detect the chaotic behaviors of one, two, and three degree of freedom submodels of a rotating beams attacted to hub representing rotorblade systems, emphasizing the importance of nonlinear vibrations of this system, which has been overlooked in the literature. The second part of the thesis, the“Theory”section covers the methods and approaches used in theory and the chaos criteria. In Section 2.1, the bending vibrations of a rotating rotor-blade system with angular velocity fluctuations are modeled as in-plane bending vibrations of a uniform crosssection elastic Euler-Bernoulli beam rotating with a rigid hub. Geometrical and dynamical nonlinearities, up to the third-degree terms are taken into account. By using the Euler-Bernoulli beam theory and adding Kelvin-Voigt viscoelastic material damping to model, equation of motion (EOM) of continuous system is obtained. Then, the it is non-dimensionalized by using dimensionless quantities and parameters. The related dimensionless integro-differential equation of the continous model is discretized using the Galerkin method by selecting the eigenfunctions of a linear stationary cantilever beam as weight functions, and a finite set of nonlinear ordinary differential equations are obtained. Then, system parameters which governing the motion were determined as dimensionless hub radius α, average rotation speed β0, dimensionless fluctuation coefficient δ, viscous damping factor ζ, and dimensionless fluctuation frequency ω̅. In addition, single, two and three degrees-of-freedom models are derived using the discretized equation of motion. Section 2.2 includes how chaos criteria, such as position-time and velocity-time graphs, phase diagrams, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, were obtained and interpreted. In the third part of the thesis, the“Angular Velocity Fluctuation and Chaos”section the probability of chaotic vibrations in a rotating beam due to angular velocity fluctuation will be investigated. The analyzes will be carried out on the single, two and three degrees-of-freedom models obtained in Section 2.1, respectively. For this purpose, each of the chaos criteria introduced in Section 2.2 will be applied separately to ensure the reliability of the results to be obtained. In Section 3.1, the single degree of freedom model is discussed. In Section 3.1.1, Three of the control parameters governing the motion are selected as α=2, β=3, ζ=0.1 and the chaos chart based on Lyapunov exponent calculation in the other two parameters ω̅-δ plane is shown in Figure 3.1. Thus, chaotic motion is observed when either one or both of the ω and δ parameters are large. By selecting ω̅=4, the bifurcation diagram in Figure 3.2 is obtained when the bifurcation parameter δ. Thus, the parameters that cause periodic and chaotic motion are determined. In Table 3.1, sample cases and their Poincaré points, motion periods, Lyapunov exponents and behaviors are given, and some sample cases, which are given in Figures 3.4-3.8, are selected and the parameters that cause periodic and chaotic motion are shown. These are also confirmed using the chaos criteria such as positiontime, velocity-time, phase diagram, and Poincaré section. So, it has been determined that there are chaotic behaviors of the single degree-of-freedom model. In Section 3.1.2, α=1, ζ=0.1, ω̅=7 are selected and the chaos chart in δ–β0 plane given in Figure 3.9. Then, chaotic motion is observed when either one or both of the δ and β0 parameters are large, also periodic motion inter-areas are shown in the chaotic areas. Crisis are determined. In Table 3.2, sample cases are given. In Section 3.1.3, α=1, δ=1.6, ζ=0.1 are selected and the chaos chart in β0-ω̅ plane given in Figure 3.13. Chaotic motion is not observed in regions where dimensionless fluctuation frequency ω̅ and/or average rotational speed β0 is small, it is seen that chaotic motion can occur only if at least one of them is large. For a result of the system where ω̅ cannot be selected, only β0 can be changed in order to avoid from chaotic behaviors. In Table 3.3, sample cases are given. In Section 3.2, the two degrees of freedom model is discussed. In Section 3.2.1, again α=2, β=3, ζ=0.1 are selected and the chaos chart in ω̅-δ plane given in Figure 3.17. Then, chaotic motion is observed when either one or both of the ω̅ and δ parameters are large. By selecting ω̅=8, bifurcation diagram in Figure 3.19 is obtained when the bifurcation parameter δ. Period-doubling parameters and also Feigenbaum constant are obtained. In Table 3.4, sample cases are given in Figures 3.20-3.31 and six of them are discussed in terms of the other chaos criteria. Both of the Galerkin coordinates are coupled each other. In Section 3.2.2, α=1, ζ=0.1, ω̅=7 are selected and the chaos chart in δ–β0 plane given in Figure 3.32. Then, chaotic motion is observed when either one or both of the δ and β0 parameters are large, also periodic motion inter-areas are shown in the chaotic areas. Crisis are determined also. By selecting δ=3, bifurcation diagram in Figure 3.33 is obtained when the bifurcation parameter β0. Period-doubling parameters are determined. In Table 3.5, sample cases are given in Figures 3.25-3.38 and six of them are discussed in terms of the other chaos criteria. and it is shown that the chaos card and the bifurcation diagram confirm each other. Both of the Galerkin coordinates are coupled each other. In Section 3.2.3, β0=5, ζ=0.1, ω̅=1.6 are selected and the chaos chart in δ-α plane given in Figure 3.40. Then, chaotic motion is observed when both inward(α0) oriented beam also when δ=0 fluctuation does not exist and there are no chaotic behaviors. In Table 3.6, sample cases are given in Figures 3.41-3.44. When α=0 there is no hub and again there are no chaotic behaviors. Periodic behavior occurs when α is large and δ is small. In Section 3.2.4, β0=2.4, δ=1.8, ζ=0.1 are selected and the chaos chart in ω̅-α plane given in Figure 3.45. Chaotic motion is not observed in regions where fluctuation frequency ω̅ is small, dimensionless hub radius α is positive, and where ripple frequency ω̅ is large and dimensionless core radius |α| is small. There is a possibility that chaotic motion may occur in regions other than this. By selecting α=2, bifurcation diagram in Figure 3.46 is obtained when the bifurcation parameter ω̅. Some crisis regions are determined and sample cases are given in Table 3.7. In Section 3.2.5, δ=1.6, ζ=0.1, ω̅=6 are selected and the chaos chart in β0-α plane given in Figure 3.52. Chaotic motion is observed in regions where the average rotational speed β0 is large and the dimensionless hub radius α is small, and in the regions where mean rotation speed β0 is small and the dimensionless hub radius α is large, chaotic motion is only observed if both of these are large(|α|>0) can occur. Thus, since chaotic motion behavior can occur in both α>0 (outward-oriented beam) and α
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