Akışkan Taşıyan Ankastre Bir Borunun Dinamik Analizi Ve Bulanık Mantık Tabanlı Uyarlamalı Kontrolü

Abstract

Akışkan taşıyan boruların dinamik analizi ve titreşim kontrolü uzun süreden beri araştırmacıların üzerinde durduğu bir konu olmuştur. Akışkan taşıyan boru sistemlerinin günümüzdeki kullanım alanlarının artmasıyla yapılan bu dinamik ve titreşim analizlerinin önemi gün geçtikçe artmaktadır. Artan bu önem nedeniyle bu konu hakkındaki araştırmalar tüm hızıyla devam etmektedir. Bu çalışmada akışkan taşıyan bir ankastre borunun literatürde bulunan serbest titreşim denklemi kullanılarak sonlu elemanlar metodunun zayıf formülasyon tekniği ile titreşim biçimlerine ayrıklaştırılmıştır. Sistem parametrelerinin sistem üzerindeki etkileri incelenmiştir. Daha sonra sistemin zorlanmış titreşim denklemi elde edilerek sistem dinamiği incelenmiş ve elde edilen sistem dinamiğine göre sistemin dalgalanma kararsızlığının karakteristiği incelenmiştir. Son olarak sistemin dinamik denklemi göz önünde bulundurularak yapısal titreşimleri bastırmak için bir uyarlamalı titreşim denetleyicisi tasarlanmış ve bu denetleyicinin uygulanabilirliği üzerinde durulmuştur. İlk bölümde bu konuyla ilgili daha önce yapılmış çalışmalar özetlenmiştir. İkinci bölümde ele alınan sistem tanımlanmış ve literatürde var olan serbest titreşim denklemi alınarak bu denklemin titreşim biçimlerine ayrıştırılması üzerinde durulmuştur. Sistem serbest titreşim denklemi sonlu elemanlar metodu kullanılarak titreşim biçimlerine ayrıklaştırılmıştır. Daha sonra akışkan taşıyan boru elemanlara ayrılmış ve her eleman için kütle, jiroskopik ve rijitlik matrisleri bulunmuştur. Bulunan bu matrisler sistemin sınır koşullarına uygun bi şekilde birleştirilerek tüm sistemin global matrisleri bulunmuştur. Bu global matrisler kullanılarak sistemin kritik doğal frekanslarına ulaşılmıştır. Sistem haraket denklemi ve sınır koşulları nümerik hesaplamalarda işlem kolaylığı sağlanması açısından boyutsuz parametrelere dönüştürülmüştür. Üçüncü bölümde ise sistemin dinamik özelliklerini incelemek amacıyla kritik doğal frekansların sistem parametreleri ile değişimi gözlemlenmiştir. Akışkan akış hızı, borunun elastiklik modülü, boru malzemesinden kaynaklanan yapısal sönüm katsayısı gibi sistem parametrelerinin etkilerini incelemek amacıyla simülasyonlar yapılmıştır. Simülasyonlar sonucunda akış hızı gibi sistem parametrelerinin sistem dinamiği üzerinde büyük etkileri olduğu görülmüştür. Ayrıca yine bu ikinci bölümde sistem parametrelerinin değişiminin sistemin açık çevrim cevabını üzerindeki etkileri incelenmiştir. Yapılan simülasyonlar sonucunda akışkan hızının sistemi kararsızlığa sürükleyen en önemli etken olduğu gözlemlenmiş ve akışkan hızının artmasıyla doğal frekansların düştüğü, sistemin de kararsız bir davranış sergilediği görülmüştür. Dördüncü bölümde sistemi kararsız yapan akış hızının (kritik hız) yukarı çekilmesi ve sistemde oluşan titreşimleri bastırmak amacıyla PID ve bulanık mantık tabanlı uyarlamalı kontrol olmak üzere iki çeşit denetleyici tasarlanmış ve sistemin kritik akış hızı aşıldığında kararlılığının korunması için bu denetleyicilerin performansları incelenmiştir. PID kontrol altında sistem düşük akış hızlarında iyi sonuçlar vermesine rağmen akış hızı yükseldiğinde titreşimleri bastırmakta başarısız olmuştur. Tasarlanan bulanık mantık tabanlı uyarlamalı kontrolün ise kritik akış hızı aşılsa bile başarılı sonuçlar verdiği ve titreşimleri bastırdığı ya da makul değerlere sönümlediği gözlemlenmiştir. Beşinci bölümde ise yapılan çalışma özetlenerek, elde edilen sonuçlar verilmiş ve gelecekte yapılabilecek çalışmalar hakkında bilgiler verilmiştir.

Fluid-conveying pipes are found in many practical applications such as exhaust pipes in engines, stacks of flue gases, air-conditioning ducts, pipes carrying fluids in chemical and power plants, risers in offshore platforms, and tubes in heat exchangers and power plants. Because of the wide usage area of these systems, the researchers have worked on this subject for a long time and there are lots of works about dynamics and stability analysis of fluid conveying pipes in the literature. The flow of fluid through a pipe can impose pressures on the walls of the pipe causing it to deflect under certain flow conditions. This defection of the pipe may lead to structural instability of the pipe. The fundamental natural frequency of a pipe generally decreases with increasing velocity of fluid flow. The pipe becomes susceptible to resonance or fatigue failure if its natural frequency falls below certain limits. With large fluid velocities the pipe may become unstable. There are certain practical examples where flow velocity reaches a very high value such as feed lines of rocket motors and water turbines. The purpose of the study presented in this thesis is to investigate the dynamic behavior of fluid conveying pipe and propose suitable control strategies in order to eliminate or suppress its vibration. The system under consideration consists of a uniform, straight, vertical cantilever pipe which conveys incompressible fluid. The x-axis is vertical and extend along the pipe. y-axis is horizontaland perpendicular to the x-axis. When determinig the forces acting on pipe and fluid element some assumptions are taken into account such as the fluid is incompressible, the fluid flow in the pipe is uniform, time-dependent harmonic are added on the fluid flow speed, the pipe isn t changed its shape by effect of pressure, all movements at the system occur at x-y surface, the pipe is made up of aliminium and all movements are small. Governing equation of motion for free transverse vibration is derived by using Newtonian approach. The governing equation of motion is discretized to obtain mass, gyroscopic and stiffness matrices using weak formulation of the finite element method. The pipe is divided four elements and mass, gyroscopic and stiffness matrices of each elements are found. After that global mass, gyroscopic and stiffness matrices of the system are found via assembly process. The natural frequencies of the system are then determined for the cases where the pipe conveys fluid with zero and nonzero velocities. The approximate analytical solutions obtained for the pipe conveying fluid with zero velocity are then compared to the numerical results found in this thesis validate the developed finite element model of the system. The results show that natural frequencies obtained through analytical and numerical analysis are in good agreement. In Chapter 3, effects of the systems parameters such as speed of fluid flow, internal structural damping of the pipe’s material, external damping, pipe’s mass, fluid’s mass and length of the pipe on the natural frequencies and open loop response of the system are investigated. The results show that the flow speed is one of the most important parameters which plays very vital role for the stability of the pipe. The critical flow speed is therefore determined. At the critical speed, first natural frequency is being zero and if the flow speed exceeds the critical value, the system becomes unstable. It is observed that when mass of the pipe, mass of the fluid and length of the pipe are increased, the natural frequencies of the system decrease and the system becomes more susceptible to instability. When the coefficients of internal structural damping of the pipe s material and external damping due to friction of the pipe with surrounding stationary fluid medium are increased the natural frequencies of the system decrease and the system becomes more susceptible to instability. The open loop response of the system means the time-dependent change of the cantilever pipe s tip displacements. It is observed that when the mass of the pipe and mass of the fluid are increased, the amplitude of vibration at the tip of the pipe decrease but oscillation of the tip s vibration increase. Increment at the oscillation of the tip s vibration causes to increment at the settling time of the open loop response of the systems. When the length of the pipe is increased, the amplitude of the pipe vibration and settling time of the open loop response increase and the system becomes unstable. The effects of the amplitude and circular frequency of the harmonic on the open loop response of the system are small. In Chapter 4, the vibration control of the system is studied. The system shows unstable behavior when the flow velocity exceeds the critical value. The control is important to suppress the pipe vibration and to improve the stability boundaries. First, a reduced order PI controller is applied to the system. The PI controller s inputs are linear and angular displacement of cantilever pipe s free end, linear and angular velocity of free end and liner displacement and velocity of fourth node of the finite element model of the system. Output of the controller is control forces which are applied by the actuators. Two actuators use in the system One of them is at the free end of the pipe, other one is at the fourth node. It is shown that it is effective to suppress vibration of the system only for the flow velocities less then critical speed. When the flow velocity exceeds the critical speed the effectiveness of the controller deteriorates. In order to improve the stability margin a fuzzy logic based adaptive controller is also developed in Chapter 4. Fuzzy logic controller is choosed because of it s advanteges as compared with the other controllers. These advantages are explained such as it is very robust, it can be easily modified, it is very quick and cheaper to implement. All the state’s displacements and velocities and fluid flow velocity are considered the adaptive controller inputs. It is known that feedbacking all state s displacements and velocities is not practical way of the control pipe s vibration. Because when the all state s variables are fed back, the number of used actuators increase. Because of this the number of the feedback signal is tried to reduce. Firstly all state s displacements and velocities are fed back and observed the controller performance. Then state s feedback signals are reduced and the control performances are compared. At the end of this process the fuzzy logic based adaptive controller s inputs are fluid flow velocity, linear and angular displacement of cantilever pipe s free end, linear and angular velocity of free end and liner displacement and velocity of fourth node of the finite element model of the system. Output of the controller is control forces which are applied by the actuators. The fuzzy logic controllers involve four parts. First part is fuzzification unit. The crisp inputs are fuzzificate at this unit. Second unit is information unit. Information unit contains two parts such as rule base and data base. Membership functions of the inputs and outputs of the controller is setted in the data base. Rule base is very important for controller performance. Rule base is constituted by an expert. It is known that the expert’s knowledge is very important for this type of controller design and its performance. Third part of the fuzzy logic controller is inference unit. Mamdani type inference system is used to control the pipe s vibrations. Last part is defuzzification unit. The fuzzy value convert crisp value in this unit. Center of area method is used to defuzzificate the output. The fuzzy logic based adaptive controller is tested for different flow velocities. Adaptive controller behaves like a PID controller at low flow velocities. At higher flow velocities it successfully suppresses the vibration and holds the system response in the stable region. In Chapter 5, the results of the simulations and suggestions for future works are indicated.