Burun iniş takımı sisteminde shimmy davranışının modellenmesi, analizi, testi ve kontrolü
Modelling, analysis, test, and control of the shimmy behavior in nose landing gear system
Yazar: SENA KOÇAK
Danışman: DOÇ. DR. ALİ FUAT ERGENÇ
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Mekatronik Mühendisliği Ana Bilim Dalı / Mekatronik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2021
Konu: Havacılık Mühendisliği = Aeronautical Engineering ; Mekatronik Mühendisliği = Mechatronics Engineering
Dizin: Mekanik titreşim = Mechanical vibration ; İniş takımı = Landing gear

Özet:
Titreşim, mekanik bir sistemdeki salınım hareketlerini tanımlayan bir terimdir. Shimmy ise hava ve kara araçlarında görülebilen bir titreşim davranışı olarak bilinmektedir. Bu davranış, tekerlek ile yer etkileşiminden kaynaklanan ve kendinden uyarlamalı bir titreşim hareketidir. Shimmy sonucu araçlarda meydana gelen yüksek frekanslı ve genlikli hareket, yolcu konforunu azaltmakta ve hatta kazalara/hasara yol açabilmektedir. Literatürde, shimmy fenomenin incelenmesi üzerine çeşitli çalışmalar tasarım parameterlerini optimize etmek ve aşınma gibi problemleri öngörmek adına sürdürülmektedir. Bu çalışmada ise hava araçlarının burun iniş takımlarındaki shimmy davranışının modellenmesi, analizi, testi ve kontrolü üzerine çeşitli çalışmalar yapılmıştır.
Literatürde, shimmy dinamiğinin incelenmesi üzerine çeşitli matematiksel modeller ve yaklaşımlar kullanılmaktadır. Bu yaklaşımlar, shimmy titreşimin hangi parametre ve koşullar altında tetikleneceğini anlamaya yönelik olarak sunulmuştur. Farklı model yaklaşımlarının bulunmasının sebebi ise shimmy titreşimini incelerken kullanılan tekerlek modellerinin (point contact, straight strecthed, rigid ring gibi) değişiklik göstermesidir. Bu çalışmada, burun iniş takımında shimmy davranışın modellenmesi sürecinde gecikmeli bir model yaklaşımı kullanılmıştır. Von Schlippe'nin tekerlek modeli, literatürde tekerlek hafızası olarak bilinen ve tekerlek elastikliğinden kaynaklanan gecikme etkisi göz önünde bulundurularak yeniden düzenlenmiştir. Bu modeller, shimmy fenomenine ait dinamiğin anlaşılmasında önemli bir yere sahiptir. Oluşturulan gecikmeli model, kurgulanan çeşitli senaryolar yardımıyla test edilmiştir. Ardından, gecikmeli model ve klasik gecikmesiz model için lineerleştirme çalışmaları yapılmıştır.
Bu çalışmada, gecikmeli ve gecikmesiz shimmy matematiksel modellerine ait analizler yapılmıştır. Gecikmesiz lineer model için kararlılık haritaları ve özdeğer trendleri oluşturulmuştur. Gecikmeli nonlineer model için zaman serisi ve limit çevrimleri verilmiştir. Ayrıca gecikmeli lineer model için CTCR yardımıyla kararlılık tablosu ve QPmR yardımıyla ise spektrum analizleri sunulmuştur. Analizler sonucunda tasarım ve süreç parametrelerin shimmy üzerine direkt etkileri gözlemlenmiştir.
Bu tez çalışmasında, modelleme ve analiz çalışmalarını doğrulama amacıyla bir test düzeneği kurulmuştur. Bu test düzeneği; farklı hız, yük ve tasarım parametrelerinin denenebilmesine imkan sağlamıştır. Böylece koşulların shimmye etkisi parametrik olarak da incelenebilmiştir. Yapılan çeşitli test sonuçları ile kullanılan matematiksel model çıktılarının uyumlu olduğu görülmüştür. Bu sayede, kullanılan model doğrulanmıştır.
Gecikmeli shimmy modelini kontrol etmek amacıyla kararlaştırıcı değişken sönümleme tabanlı bir kontrol yöntemi (Stabilizing Variable Damping Based Control – SVDC) önerilmiştir. Bu yöntemde, sönümleme parametresi kontrolör olarak kullanılmıştır. Kapalı çevrim kontrol sisteminin kararlı bölgede kalmasını sağlamak adına QPmR algoritması kullanmış, parametreye ait sınırlar belirlenmiştir. Kontrolör parametresinin hesaplanmasında çok amaçlı optimizasyon kullanmış ve titreşim enerjisi ile kontrol işaretinin ISE değerleri minimize edilmiştir. QPmR ile elde edilen kararlılık sınırları kısıt olarak kullanılmıştır. Kontrolörün etkinliği; ISE değerleri, RMS değerleri, bozucu bastırma ve sönümleme performansı açısından değerlendirilmiştir. Kontrol işlemi sonucunda başarılı bir titreşim bastırma performansı elde edilmiştir.

Abstract:
The term vibration states a mechanical phenomenon whereby oscillations occur about an equilibrium point. Whereas, shimmy is known as a vibration behavior observed in air and land vehicles. Shimmy describes the self-excited oscillatory movement in aircraft landing gears, tires in motorcycles, and trailers.
Earlier researches related to shimmy phenomenon aims design optimization and wear or malfunction prediciton in vehicle components. In this thesis, shimmy behavior of the nose landing gear of aircrafts is modelled, analyzed, tested, and controlled.
Shimmy is a critical vibration problem that may be observed in landing gear systems during taxi, landing, and take-off. This behavior may be identified as an evident high frequency vibratory motion spreading from the nose of the aircraft to the tail. Shimmy vibration occurs also in the frequency range 10 and (to) 30 Hz and arises from the interaction between tire and road [1]. The source of energy of the shimmy is forward movement of the aircraft. Even though shimmy is generally observed in the nose landing gear, main landing gear may be exposed to the severe vibrations.
Sources of shimmy oscillation are various. Oscillations may be triggered with the combination of structural, tire, and process variables such as mass, damping coefficient, geometrical properties, velocity, the excitation force, also nonlinear effects like friction, tire elasticity, and freeplay. It may be experienced in different speed ranges and under different conditions, therefore it is still mentioned as a phenomenon in the literature due to its nonlinear and uncertain nature. This vibration in the aircraft originated by shimmy may reduce passenger comfort and be strong enough to prevent the pilot to steer the aircraft correctly. As a result, excessive wear on the wheel, malfunction of the landing gear and even accidents may occur.
In literature, shimmy dynamics may be investigated via several mathematical models and approaches to comprehend which condition and parameter combination may trigger the shimmy oscillation. The reason for the various model approaches is that wheel models used when examining the shimmy vibration may vary. There are variety of mathematical modelling concepts of the wheel is used in order to examine shimmy behavior. Tire approximations such as point contact, stretched string, straight tangent, and rigid ring were integrated in shimmy mathematical model by researchers frequently. Point contact model states that the tire responds with a delay when lateral force is applied to the tire. Fundamental part of point contact approximation is this delay. On the other hand, stretched string tire model treats tire as a massless elastic string with a finite length. Straight tangent tire model could be defined as a linear approximation of the stretched string approach. Unlike stretched string model, straight tangent approximation concerns only the lateral deflection of the leading point of the tire. Furthermore, rigid ring model is developed in order to describe the dynamic behavior of the tire and comprise dynamics (gyroscopic moment and resonances) of the and contact patch.
In this thesis, Von Schlippe's approach [2] (stretched string) is integrated into dynamic shimmy model. According to Von Schlippe, contact length betwen tire and roas is 2𝑎 and there is no slip. Also, it is considered that the slip angle of the tire remains in a narrow region. Besides, the study focused on the delay effect originated from the elastic structure of tire which is called tire memory. The nonsliding contact points, that are in the contact region conserving their position data with respect to the axes of inertial system, cause memory effect [3]. This delay effect integrated into mathematical model to express the lateral deflection of the rear contact point.
Numerous test scenarios are designed and the performance of the shimmy model is measured for different runway conditions, tire wear and a wide range of velocity profiles. Test cases are designed to simulate the actual situations that may occur during taxiing. Tire wear, rough runway, acceleration in rough runway, deceleration after lateral bump on tire, and acceleration in potholed road scenarios are modelled and shimmy behaviors are examined and discussed in this conditions.
The complexity and nonlinearity of the mathematical shimmy models encourage researchers to linearize the both delayed and nondelayed (models) systems. The idea of linearization is to form an understanding of the dynamics and characteristics of the system. Linearization also allows to analyze the stability of the system, and to comment on design parameters and operating points. The fundamental knowledge about shimmy behavior of the nose landing gear system is exerted from linearized models. Mathematical shimmy model is linearized via Taylor series expansion under the assumption of small change in slip angle.
Prediction and prevention of nose landing gear shimmy unstability should be considered in aircraft design process. Therefore, simulation and analysis are essential tools to examine the influence of landing gear design parameters (with process parameter such as load and velocity) on shimmy. The effects of changes in the parameters like damping coeeficient, spring coefficent, moment of inertia, load, caster length, rake angle and relaxation length and initial conditions is investigated for the delayed nonlinear shimmy model in this study. The results reveal the time histories, stability characteristic and the amplitudes of the limit cycles for nonlinear shimmy dynamics. In simulations, aircraft taxiing with varied initial steering angle and velocity are handled. The stability of this characteristic equation of linear nondelayed shimmy model is determined through the location of the eigenvalues. Positive real values of eigenvalues cause the instability. In addition, a stability map is constructed by forming a set defined as a Cartesian product of two desired parameters in specified ranges and checking their eigenvalues. The locations of the real parts of the eigenvalues with parameter changes also provide us a perspective about the stable design conditions of the nose landing gear system. Both stability analyzes are discussed with the view of landing gear design and optimization approach. Delayed modelling approach for shimmy is not common due to its cumbersome stability analysis of transcendental characteristic equations which is inherently in place when there is time delay in the system. In literature, delayed tire model is presented but their stability analysis is limited. CTCR method with Kronecker multiplication extension is suggested to address this problem in this work. CTCR method computes the stability status of a delayed system within the specified range of delay value [4]. Stability switching tables are introduced in order to determine shimmy free conditions via comparing nominal delay (2𝑎/𝑉 [5]) and critical delay values computed by the CTCR. Spektrum analysis of delayed shimmy model is also completed via QPmR algorithm. Quasi-Polynomial-mapping based Root-finder (QPmR) aims to calculate all the zeros of a quasi-polynomial located in a predefined region of the complex plane [6]. The movement of the roots of characteristic polynomial with system parameter variation presented via an implementation of QPmR. The relation between design (structure and tire) and continuation parameter change and stability is discussed.
The given methods have approaches and their outputs differ depending on the intended use. Any of these analyzing methods can be preferred by considering the needs, calculation load/time and accuracy. These methods are presented in order to contribute to the design optimization process while providing information about the stability of the system.
It is very important to verify the theoretical models and approaches to understand shimmy behavior through experiments. In this thesis, a shimmy test bench is constructed in order to verify modelling and analysis studies. The test bench allows user to vary caster length, load, and velocity. Shimmy tests are pursued with an aircraft tire "Aeroclassic 11x4-500 8 PLY" for acquiring a better understanding about shimmy of nose landing gear tire. Preliminary experiment are also conducted to obtain system and tire parameters. Smiley and Horne proposed a tire parameter calculation methodology for aircraft tires based on experiments and tire type [7]. Half contact length, relaxation length, side force derivative and moment derivative are derived via this study. Shimmy tests are carried out with various configuration in order to validate theoretical shimmy model. Tire deflection, conveyor velocity, vertical load, and steering angle are obtained via meausrement system and collected via Micro850 Progrramble Logic Controller System. Results reveal that the mathematical shimmy model is coherent with the experimental output. Comments about shimmy experiments are also introduced and "Lessons Learned" related with the model – experimental data incompability are discussed.
Shimmy may cause unstable motion results in decrease passenger comfort and pilot visibility, tire wear, malfunction in the landing gear system, and even accidents. Consequently, shimmy vibration should be controlled and damped. Implementation of passive damping strategies is a basic solution for absorbing shimmy vibration via shimmy damper and design parameters such as rake angle and caster length. Passive structures are designed for specific runway, load and velocity conditions. Therefore, damping performance of the gear system with passive dampers may worsen for some taxi scenearios. Semiactive and active control methods are implemented to the landing gear system for such cases. Active shimmy damper utilizes the sensor output form landing gear to determine adequate torque generated by an actuator to damp shimmy oscillation. Among all vibration damping methods, active damping proposes better performance even under poor runway, higher loading, and extreme taxi velocity conditions. In this thesis, a stabilizing optimal controller is designed by using damping coefficient as a controller. In this case, additional damping coefficient symbolizes the shimmy damper and offers simpilicity for controller design process. Stable range of controller parameter is computed via QPmR in order to ensure the stability of the delayed closed loop system. The controller parameter is determined by multiobjective optimization which concerns mathematical optimization problems with more than one objective function to be optimized simultaneously. Sequential Quadratic Programming (SQP) is chosen as optimization algorithm. Integral square error of relative vibration energy of the nose landing gear system and control signal are optimized as objective function. Stable range of controller parameter computed by QPmR is the constraint of optimization problem. Stability range for controller parameter 𝑘𝑐 is also calculated for various load/velocity conditions. This metodology is proposed with the name of Stabilizing Variable Damping Based Control (SVDC). The efficiency of the controller is evaluated in regards to ISE values of objective functions, RMS values of open and closed loop system output, disturbance behavior, and supression performance. Shimmy behavior of the closed loop system is improved via increased suppression and reduced energy consumption.