Dalgalanan hızla dönen kirişlerde kaotik titreşimler
Chaotic vibrations in rotating beams with fluctuating speed
Yazar: UĞUR TATLIER
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, 2023
Konu: Fizik ve Fizik Mühendisliği = Physics and Physics Engineering ; Makine Mühendisliği = Mechanical Engineering ; Uçak Mühendisliği = Aeronautical Engineering
Dizin: Doğrusal olmayan dinamik = Nonlinear dynamics ; Doğrusal olmayan titreşimler = Nonlinear vibrations ; Kaotik titreşim = Chaotic vibration ; Mekanik titreşim = Mechanical vibration

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.

Abstract:
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(α<0) and
outward(α>0) 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<0 (inwardoriented beam) cases, chaos analysis should be performed for both problems.
In Section 3.2.6, α=1, δ=1.6, ζ=0.1 are selected and the chaos chart in β0-ω̅ plane given
in Figure 3.58. 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. By selecting ω̅=5, bifurcation diagram in Figure 3.23 is obtained when the
bifurcation parameter β0. Period-doubling parameters and some crisis regions are
determined. In Table 3.9, sample cases are given and it is shown that the chaos card
and the bifurcation diagram confirm each other.
In Section 3.3, the three degrees of freedom model is discussed. One more time α=2,
β=3, ζ=0.1 selected and the chaos chart in ω̅-δ plane given in Figure 3.65. Then,
chaotic motion is observed when either one or both of the ω̅ and δ parameters are large.
By selecting ω̅=9, bifurcation diagram in Figure 3.66 is obtained when the bifurcation
parameter δ. Period-doubling parameters and also Feigenbaum constant are obtained.
In Table 3.10, sample cases are given in Figures 3.68-3.85 and four of them are
discussed in terms of the other chaos criteria. All three of the Galerkin coordinates are
correlated.
In Section 3.4, The chaos cards obtained in Section 3.1, Section 3.2 and Section 3.3
are plotted on the same parametric plane and compared.
In Section 3.4.1, The chaos cards obtained in Section 3.1.1, Section 3.2.1 and Section
3.4 are plotted on the same parametric plane ω̅-δ by setting the α=2, β=3, ζ=0.1. There
is a serious jump in transition from the chaos card of the single degree of freedom
model to the two degree of freedom model. When switching from the chaos card of
the two-degree-of-freedom model to the three-degrees-of-freedom model, there is not
much jump. The time spent solving the problem of the three-degree-of-freedom model
is too much and the chaos chart gain is low. In addition, even if the time spent on
solving the problem of the single degree of freedom model is very small, it has been
decided that the most optimal model is the model with 2 degrees of freedom, since it
has a significant difference from the others.
In Section 3.4.2 δ-β0 by setting the α=1, ζ=0.1 ω̅=6. There is a serious jump in
transition from the chaos card of the single degree of freedom model to the two degree
of freedom model in Figure 3.89. Similar effects but detailed results are obtained from
two degree of freedom model.
In Section 3.4.3 δ-β0 by setting the α=1, ζ=0.1 ω̅=6. There is a serious jump in
transition from the chaos card of the single degree of freedom model to the two degree
of freedom model in Figure 3.90. Again similar effects but detailed results are obtained
from two degree of freedom model.
In the fourth and last part of the thesis, "Conclusion" part includes results obtained
from the study are summarized and suggestions are made for future studies.