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Bu tezde, iniş takımlarının yer rezonansına olan etkileri incelenmiştir. Öncelikle helikopterler ve helikopterlerde kullanılan iniş takımları okuyucuya aktarılmış, ardından iniş takımlarının tasarımındaki birkaç temel hedef açıklanmıştır.Yer rezonansı modeli sekiz serbestlik dereceli olarak kurulmuştur. Analiz için gerekli olan gövde direngenlik matrislerinin tesir katsayıları yardımıyla veya dinamik indirgeme yöntemiyle elde edilebileceği ve özellikle düşük frekanslı titreşim modlarında, helikopter gövdesinin bir rijit cisim olarak davrandığı gösterilmiştir. Analiz için gerekli olan sönüm matrisi Rayleigh sönüm modeli ile modellenmiş ve aşırı gövde sönüm oranlarında kararsızlık bölgelerinin birbirleriyle birleştiği ve kararlılık için gerekli sönüm oranının çok yüksek mertebelere çıktığı gözlemlenmiştir.Kızak ile zemin arasında kullanılan sınır şartlarında, kısıtlanan serbestlik dereceleri arttıkça kızağın direngenliği artmaktadır. Kızağın, gövde direngenliği ile karşılaştırılabilir mertebelere yaklaştıran bu durum, sonlu elemanlar modeli (SEM) ile analitik modelleme için kullanılan basitleştirilmiş model (BM) arasındaki özdeğerler farklılıklarını artırmaktadır.Özellikle karmaşık mod biçimlerine sebep olan veya başka bir değişle, özvektörlerini birçok serbestlik derecesine dağıtan kızak tasarımlarının, yer rezonansı probleminde ortaya çıkan kararsızlık miktarlarını azalttığı ve buna bağlı olarak daha az sönüm gereksinimlerine ihtiyaç duyulduğu gösterilmiştir. In this thesis, the effects of landing gear on ground resonance problem is inspected. First of all, helicopter and helicopter landing gear systems are illustrated to the reader and after, the main aims of designing a landing gear is explained.Ground resonance model is created with eight degrees of freedom(DOF). It can be seen that the required stiffness matrix of helicopter fuselage for ground resonance problem, can be determined using influence coefficient or dynamic reduction technics because of rigid body behaviour of helicopter fuselage especially in low frequency dynamics.Required damping matrix for ground resonance problem is constructed using Rayleigh damping model. One of the main findings of these type of ground resonance model which includes multi DOF fuselage model, if damping ratio of fuselage is very high comparing to blade in-plane damping ratio, some unstable region can join together and this leads to increase in required damping level for fuselage.Finally, landing gear which transmits some of its kinetic energy on untriggered DOFs, reduce required damping levels. This is the main design rule for decreasing required damping level and reducing ground resonance danger 121

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2010, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2010, Otomotiv sektöründe kabin içi gürültü ve titreşimin azaltılması, ürün kalitesini arttırmak ve müşteri memnuniyetini sağlamak için temel gerekliliklerden biri haline gelmiştir. Taşıt içi akustik konfor gittikçe büyük bir önem kazanırken, kabin içi tasarımının yanı sıra, amacına uygun ve özel olarak üretilen akustik malzemelere duyulan ihtiyaç da artmaktadır. Ürün geliştirme sürecinde, malzemenin akustik performansını etkileyen malzeme özelliklerinin çok olması ve akustik performansı belirlemek için yapılan testlerin oldukça uzun sürmesi karşılaşılan sorunların başında gelmektedir. Bu yüksek lisans tez çalışmasında, Comet Trim akustik malzeme modelleme yazılımı ve makroskopik özellikler kullanılarak tek ve çok katmanlı gözenekli malzemelerin ses yutma katsayısı ve ses iletim kaybının hesaplanması amaçlanmıştır. Modelleme ile elde edilen sonuçlar, empedans tüpüyle yapılan test sonuçlarıyla karşılaştırıldığında, önerilen modelleme programıyla elde edilen değerlerin mühendislik açısından yeterli yakınsaklığı sağladığı gözlenmiştir. Bu çalışmada, malzemelerin akustik performansının belirlenmesi ve geliştirilmesi için akustik malzeme modelleme programlarının kullanılmasının zaman ve iş gücü açısından kazanç sağladığı sonucuna varılmıştır., Noise and vibration reduction in the passenger compartment has become one of the essential requirements to enhance the product quality and ensure customer satisfaction in the automotive industry. Acoustic comfort inside the vehicle has gained significant importance and in addition to cabin interior design, the need for purpose-specific and specially manufactured acoustic materials is also on the rise. The vast number of material properties affecting material acoustic performance and the time consuming tests required to determine the acoustic performance are the primary problems encountered in the product development process. In this dissertation, calculation of sound absorption coefficient and sound transmission loss of the single and multilayer porous materials using Comet Trim acoustic material modelling software and macroscopic properties was aimed for. Comparing the model-based results to the impedance tube test results, it has been observed that the recommended modelling software provides sufficient convergence. In this study, it has been concluded that the acoustic material modelling software provides advantages in terms of time and effort in the determination and development of the acoustic performance of porous materials., Yüksek Lisans, M.Sc.

Ses kalitesi, özellikle beyaz eşya sektöründe, pazar önceliği açısından önemli bir belirleyici nitelik halini almıştır. Bu çalışmanın amacı, farklı model çamaşır makinalarının ses kalitelerini karşılaştırmalı olarak belirlemek, ses kalitesi ile ilgili nesnel ve öznel parametrelerin ilişkilerini ortaya koyarak bu yönde bir matematik model oluşturmak, tasarım parametreleri ile ses kalitesi arasındaki ilişkiyi ortaya koymaktır. Bu amaçla farklı model sekiz adet çamaşır makinası seçilmiş ve ses kalitesi yazılımı yardımıyla bunların psiko-akustik metrikleri belirlenmiştir. Elde edilen bu veriler ?ana bileşenlerin analizi? yoluyla ana bileşenlerine ayrılmış, bu sayede sesi nesnel olarak azami düzeyde temsil edebilecek en az sayıda parametre belirlenmiştir. Ayrıca bu makinaların ses kaliteleri, oniki kişilik bir jüri tarafından zorlanmış seçim ve anlamsal fark yöntemleriyle öznel olarak değerlendirilmiştir. Ardından psiko-akustik nesnel metriklerle, jüri testinden elde edilen öznel veriler doğrusal regresyon yardmıyla ilişkilendirilerek bir matematik model kurulmuştur. Makinanın su alma, yıkama ve sıkma aşamalarında sesin hangi özelliklerinin insan algısı açısından beğenilir nitelikte olduğu grafik yöntemlerle irdelenmiştir. Çamaşır makinasının dinamik davranışını önemli ölçüde etkileyen sönüm faktörü ile ilişkili makina parçası olan kazan amortisörü seçiminin ses kalitesine etkilerini araştırmak amacıyla, aynı tip çamaşır makinasına farklı tip amortisörler bağlanarak sıkma aşamasında ses kayıtları alınmış, bu yeni ses örnekleri önceden elde etmiş olduğumuz modele konarak ses kalitesine ait nesnel ve öznel parametrelerin ne şekilde etkilendiği incelenmiştir. Sound quality has become an important factor in gainig market advantage especially in household appliances. The aim of this study is to determine the sound quality of different washing machines and to obtain a mathematical model by putting forward the relation between the objective and subjective aspects of sound quality and its relation to the design parameters. Eight different model washing machines are selected and psycho-acoustic metrics are determined by the help of the sound quality software. This data is then processed by principal components analysis and reduced to fewer number of variables which still can describe the properties of the sound. Meanwhile, a jury test is conducted with twelve jurors to determine the subjective ratings of the sounds. Then the objective metrics are correlated with the subjective jury tests by linear regression technique to obtain a mathematical model. Graphical methods are used to demonstrate which properties of the sounds that belong to working stages such as ?water filling?, ?washing? and ?spin extraction? are preferred by the human perception. Finally, the effects of the damping factor on the sound quality is investigated by applying dampers that have different damping characteristics on same type of washing machine, and the recorded sound samples are processed by our mathematical model to obtain the influence of the subjective and objective parameters of sound. 81

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2006, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2006, Reaktif ve sönümlü susturucuların analitik yöntemlerle akustik modellenmesi ve bulunan sonuçların deneyle kanıtlanması bu tezin konusunu oluşturmaktadır. Bu tezin amacı, bir susturucu performans kriteri olan ses iletim kaybı için gerekli araçları geliştirerek, egzos tasarımcılarının tasarımda kullanabilecekleri yararlı sonuçları ortaya koymaktır. Böylece, ülkemizdeki egzos susturucusu tasarımcıları, kullanılacağı yer ve amaçlara göre en uygun susturucunun tasarımı için hangi matematiksel yaklaşımlar ve deneysel yöntemler kullanabileceklerini toplu olarak bu tezde bulabileceklerdir. Reaktif tip susturucuların temel prensibi, susturucudan geçen gürültü azaltımı için, akustik enerjinin kaynağa geri yansıtılmasıdır. Sönümlü tip susturucuların temel prensibi ise, susturucudaki akustik enerjinin azaltımının susturucu boyunca kullanılan çeşitli yutucu malzemelerle azaltılmasıdır. Egzos sistemini oluşturan elemanların akustik olarak modellenmesinde çeşitli teorik yöntemler ve deneysel yöntemler kullanılmaktadır. İki boyutlu analitik yöntem, sonlu elemanlar yöntemi ve transfer matris yöntemi bu teorik yöntemlere örnek olarak verilebilir. Teorik incelemelerde, egsoz sistemlerinin analizinde sınır frekansına kadar olan frekans alanı içinde çok iyi sonuçlar veren transfer matris yöntemi tercih edilmiştir. Bunun sebebi, transfer matris yönteminin iki boyutlu analitik yönteme göre daha kolay uygulanabilir olmasıdır. Buna ek olarak, sonlu elemanlar yöntemine göre daha küçük boyutlu matrisler ortaya çıktığı için bilgisayarlarda daha hızlı ve daha rahat kullanılabilmesi de bir başka tercih sebebidir. Deneysel incelemelerde ise, transfer matris yöntemini esas alan “iki yük yöntemi”, “iki kaynak yöntemine” tercih edilmiştir. Bunun sebebi egzos uç borusundaki empedansı değiştirmenin kaynağı değiştirmekten daha kolay olmasıdır., The subject of this thesis is the acoustic modeling of reactive and dissipative mufflers by making use of the analytical and the experimental methods. The purpose of this thesis is to develop useful tools for the exhaust designers in the design of exhaust mufflers by making use of the sound transmission loss concept as a muffler performance criterion. Hence, the exhaust system manufacturers in our country can select the best method and the most appropriate muffler type depending on the usage area and the size of the vehicle as well as the testing method by using the information provided in this thesis. The fundamental principle of reactive mufflers is the reflection of the acoustic energy back to the source in order to attenuate the noise passing through the muffler. The fundamental principle of dissipative mufflers is the absorption of acoustic energy by using various acoustic materials through the muffler. There are various theoretical and experimental methods employed in acoustic modeling of components which constitute the exhaust system. The “Two Dimensional Analytical Method”, the “Finite Element Method” and the “Transfer Matrix Method” are given as some examples of the theoretical methods. In the theoretical investigations, the transfer matrix method is preferred in the acoustic analyses of the exhaust systems. This method provides perfect matching results with the experimental results up to cut-off frequency in the frequency range of interest. The reason for using the transfer matrix method is that it is easier to apply as compared to the two dimensional analytical method. Additionally, the transfer matrix method employs smaller matrices than the finite element method and this makes the transfer matrix method faster and more applicable on computers. In the experimental studies, “two load method” is preferred against the “two source method” both utilizing the transfer matrix method. In the “two load method” as it is easier to change the impedance at the tailpipe rather than changing the acoustic source this method was preferred., Yüksek Lisans, M.Sc.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2002, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2002, Rotor dinamiği konusundaki çalışmaların ana amacı, rotorlardaki titreşim olayının anlaşılabilirliğinin artması böylece bu titreşimlerin kontrol edilmesi ya da yok edilmesidir. Çalışmada eksen kaçıklığına sahip rotor sistemi modellendikten sonra sistemin dinamik davranışı sonlu elemanlar yöntemi ile analiz edilmiştir. Kütle dengesizliği ve şaft eksen kaçıklığı için genelleştirilmiş kuvvet vektörü sistem denkleminde kullanılarak cevap elde edilmiştir. Yazılan bilgisayar programında sonuçlar zaman domaininde hesaplanmıştır. Sistemin cevabının karışık olması durumunda zaman domaininde ayrı titreşim bileşenlerini tanımlamak çok zor hale gelmektedir. Titreşimleri bileşenlere ayırmanın en verimli yolu titreşim genliklerinin frekansla nasıl değiştiklerini gözlemlemektir. Frekans domaininde titreşim bilgilerini görmek için, zaman domainindeki bilgileri frekans domainine çevirmek için Fourier dönüşümü yapılır. Bu çalışmada program yardımıyla çeşitli kütle dengesizliği ve eksen kaçıklıklarının sonuçları elde edilmiş, sayısal sonuçlar ve literatürdeki deneysel sonuçlar karşılaştırılmıştır., The general purpose of study of rotor dynamic is to increase understanding of rotor vibration phenomena and thus provide a means for controlling or eliminating these vibrations. In the study, after the modeling of misaligned and unbalance rotor system, the behaviour dynamic of the system is analyzed by using FEM. The solution is handled by using generalized force vector due to unbalanced mass and misalignment in the system equation. In the computer programming results is computed in the time domain. In case of complex system response, determination of the individual vibration component in time domain becomes harder. The most efficient way of dividing vibration to its component is observing the changing of amplitude with frequency. To view the vibration data in the frequency domain, the Fourier transform is needed to convert the vibration data from time domain to frequency domain. In this study, by using a computer program, that is used the algorithm explained above, the dynamic behaviour of the system for different unbalanced mass and misalignment is calculated, the numerical results are compared with experimental results in the literature., Yüksek Lisans, M.Sc.

ÜRÜN TASARIMINDA SES KALİTESİ YÖNTEMİ: ELEKTRİKLİ SÜPÜRGELER ÜZERİNDE BİR UYGULAMA ÖZET Bir ürünün sesi de estetik özellikleri, gücü, rengi, ağırlığı ve fiyatı gibi o ürüne ait bir parametredir. Ses kalitesi ayrıntılı bir teknik hedef ve/veya belirli bir işlem kapsamında ortaya çıkan sesin yeterliliği olarak tanımlanır. Ürün ses kalitesi, ürüne ait sesin yeterliliğinin bir göstergesidir. Kullanıcıların mantıksal ve duygusal durumlarının yanısıra davranış biçimlerinin doğal sonucu olarak belirlenir. Hedef ürünün sesi ile ilgili özellikleri referans alınarak gerçekleştirilen bir işitsel değerlendirme süreci sonucunda ses kalitesi belirlenir. Ses kalitesi yöntemi ürün geliştirme sürecinde gittikçe artan bir öneme sahiptir. Bir ürünün ses kalitesinin belirlenmesi, öznel ve nesnel ölçümleri içerir. Ses kalitesi yöntemi altı adımdan oluşan bir tasarım çevrimidir. İlk adımda tasarlanan ürün için hedef ses belirlenir. Daha sonra, sırasıyla ürünün değişik safhalarında prototipin, geliştirilen ürünün veya son ürünün ses kaydı yapılır, üçüncü adımda ürün sesinin analizi ve ses kalitesi metriklerinin belirlenmesi gerçekleştirilir, dördüncü adımda akustik kökenli tasanm değişiklikleri yapılır, beşinci adım sesin üzerinde değişiklikler yapılmasını içerir ve son adımda jüri testi yapılır. Böylece çevrimin başına dönülür. Jüri test sonuçlan tatmin edici olana dek bu iteratif tasanm çevrimi devam eder. Psiko-akustik araştırmaların sonucunda psiko-akustik metrikler geliştirilmiştir. Metrikler, insanın ses hakkındaki öznel değerlendirmesini nesnel ölçümlere dönüştürmeye yarar. En önemli metrikler Zwicker gürlüğü, keskinlik, kabalık, dalgalanım kuvveti, ton-gürültü oranı, spektrumdan ayrılma miktarıdır. Metrikler Zwicker gürlük analizi ve/veya FFT analizi sonucu belirlenir. Bu metriklerin dışında, belli bir ürüne veya ürün sınıfına bağlı olarak sübjektif değerlendirmeyi yansıtan ve metriklerin kombinezonu ile ifade edilen özel metrikler de geliştirilmiştir. Bu çalışmanın amacı, üzerinde sistematik gürültü denetimi yapılan bir elektrikli süpürgenin ses kalitesinde ulaşılan sonuçlannı kıyaslamalı olarak değerlendirmek, yüksek ses kalitesine sahip bir elektrikli süpürge tasarımı için gerekli tasanm ilkelerini belirlemek ve elektrikli süpürgeler için mevcut ses kalitesi metriklerinin lineeer kombinezonu olan bir `rahatsızlık metriği` geliştirmektir. Bu amaçla, elektrikli süpürgenin ana gürültü kaynaklan belirlenmiş ve çeşitli gürültü denetimi çalışmaları yapılmıştır. Daha sonra öznel yargılan belirlemek amacıyla bir jüri testi gerçekleştirilmiştir. Jüri testinde pilot çalışmayı yaptıran firmanın biri yeterli önlem alınmamış, diğeri ise gürültü denetimi uygulanmış iki süpürgesinin yanısıra, dört adet de farklı firma tarafından üretilen toplam altı adetsüpürge ele alınmıştır. Bu ürünlerin ses kalitesi farklı yaş gruplarından gelen 16 kullanıcıdan oluşan bir jüri tarafından değerlendirilmiştir. Jüri testi esnasında, 1/3 oktav bantlarında ses basıncı ve ses gücü düzeyleri ölçülmüştür ve elektrikli süpürge sesleri mikrofon yardımıyla dijital teybe kaydedilmiştir. Dijital teyp kayıtlan kullanılarak, psiko-akustik metrik değerleri, bir ses kalitesi yazılımı kullanılarak hesaplanmıştır. Daha sonra jüri testi sonuçları ile psiko- akustik metrik sonuçlan arasındaki ilişki araştırılmıştır. Yapılan incelemeler sonunda elektrikli süpürgelerin ses kalitesini, süpürge sesi düzeylerinin önemli ölçüde etkilediği görülmüştür. Bunun doğal sonucu olarak Zwicker gürlüğü metriği ile süpürge ses kalitesi arasında büyük bir ilişkinin var olduğu tespit edilmiştir. Ayrıca yüksek keskinlik değerlerinin ve süpürge sesi spektrumundaki, spektrumdan ayrılan dar band gürültülerin, dolayısıyla sesin tonal özelliklerinin süpürge sesinin rahatsızlık etkisini arttırdığı gözlenmiştir. Yapılan gürültü denetimi sonucunda, süpürge ses kalitesinin yaklaşık % 50 arttığı görülmüştür. Sonuçta uygulanan sistematik gürültü denetimi ile ürün ses kalitesinde önemli ölçüde ilerlemenin sağlandığı ortaya konmuştur. Yapılan çalışmalar sonucunda, yüksek ses kalitesine sahip bir elektrikli süpürge tasarlayabilmek için mekanik tasanmda uyulması gereken tasarım ilkelerinin neler oldukları belirlenmiştir. Bu ilkeler aşağıda özetlenmiştin. ses yalıtımı için hem ses yutumu hem ses iletim kaybı sağlayan uygun özelliklerde kompozit malzemelerin kullanımı,. motorun süpürge içinde kapalı bir kutu içine alınması ve bu kutu içinde yumuşak malzeme ile yalıtılması,. motor kutusu ile dış kabuk arasında yine yumuşak yalıtım malzemesi kullanılması,. yapısal kökenli titreşimlerin yayılmasını önlemek için süpürge kabininin yüzey alanının minimum tutulması ve dış kabuğun yeteri sertlikte olması,. akustik enerjinin yutulması için hava akışının süpürge içinde emiş gücünü azaltmadan labirent kanallardan geçirilerek dışa atılması. Son olarak objektif ve öznel değerlendirmeler arasındaki ilişki kurularak, elektrikli süpürgeler için iki farklı `rahatsızlık metriği` geliştirilmiştir. Bunlardan ilki, yukarıda sayılmış olan altı metriğide içermektedir, ikincisi ise Zwicker gürlüğüne bağlı olarak hesaplanan Zwicker gürlüğü, keskinlik, kabalık ve dalgalanım kuvvetinin bir lineer kombinezonu olarak tanımlanmıştır. Her iki özel metriğin de jüri testi sonuçlarını oldukça iyi bir biçimde yansıttığı gözlenmiştir. Bu çalışmadaki sonuçlar tek kanallı ölçüm kayıtlan ile elde edilmiştir. Bu çalışmanın ileride daha da geliştirilmesi için yapılabilecekler arasında. iki kanallı, yapay baş ve gövde kullanılarak ölçüm yapılması,. ses sinyallerini insan kulağının algılamasına benzer biçimde işleyerek yükseltebilen sinyal kuvvetlendirici ünitelerin kullanılması sayılabilir. Ses kalitesi yöntemlerinin ürün kalitesini ve pazar payını arttırıcı bir araç olarak giderek daha fazla kullanılacağı kesin olarak anlaşılmaktadır. XI SOUND QUALITY IN THE PRODUCT DESIGN: AN APPLICATION ON WET AND DRY TYPE VACUUM CLEANERS SUMMARY Product sound is one of the product parameters such as aesthetics, power, color, weight price, etc. Sound quality is defined as ` the adequacy of the sound in the context of a specific technical goal and/or task` and `Product sound quality is a descriptor of the adequacy of the sound attached to a product. It results from judgements upon the totality of auditory characteristics of the said sound-the judgements being performed with reference to the set of those desired features of the product which are apparent to the users in their actual cognitive, actional and emotional situations`. Sound quality has an increasing importance in product design. The determination of product sound quality involves both subjective and objective measurements. Sound quality evaluation is a design cycle which involves six stages and first stage is determination of the target sound for product. The others are; sound recording (prototype, design, and last product), product sound analysis and determination of sound quality metrics, acustical design changes, sound editing and, jury test. As a result of psychoacoustic research, a set of psychoacoustics metrics has been developed which correlates objective measurements with subjective properties of human perception of sound. Most important metrics are; Zwicker Loudness, sharpness, roughness, fluctuation strength, tone to noise ratio, prominence ratio. Metrics can be determined based on Zwicker Loudness analysis or on FFT Analysis. The purpose of this study is to evaluate the sound quality of a particular vacuum cleaner comparatively after applying a number of noise control techniques, to determine the rules of vacuum cleaner design which has high sound quality and to develope an annoyance metric for vacuum - cleaner noise which is a linear combination of the other sound quality metrics. For this purpose, an analysis for determination of the principal sound sources in a vacuum cleaner is made and some noise control techniques are applied on the sample vacuum cleaner. Then for determination subjective judgement, is made on jury test. In jury test a total of six unite are used, two of them being identical vacuum cleaners of the pilot company, with and without noise control application and four units being taken from different brands. Sound quality of the products are subjective evaluated by a jury composed of 16 people from different age and sex groups. During the jury test vacuum cleaner's 1/3 octave band sound pressure level and sound power levels are measured and vacuum cleaner's noises are recorded on a DAT (Digital Audio Tape) using a microphone. Psychoacoustical metrics are calculated using a sound quality software by transferring the dat tape recordings to the PC. Results of the jury test and psychoacoustic metric values are then correlated by regression analysis. XllVacuum cleaner sound quality has high correlation with sound pressure (dB A) level and sound power level and the natural results of it, vacuum cleaner sound quality has high correlation with Zwicker Loudness. Also, very high sharpness and tonal components of vacuum cleaners are disturbing the sound quality. After the application of noise control, product sound quality increased 50 %. Finally, an attempt is made to quantify the improvements achieved in the product sound quality by applying the noise control techniques. Based on the work performed in this thesis a set of design principles has been identified to obtain a high sound quality for vacuum cleaners. These principles are,. using appropriate composite materials with sound barrier and absorption characteristics for better sound insulation.. inserting the electrical motor in an enclosed space on soft rubber isolators,. minimising vacuum cleaner's external surface area and producing a stiff external shell to avoid structural borne noise,. Introducing labyrinth passages with acoustical energy absorption for the air flow through the vacuum cleaner while ensuring minimal pressure drop. Finally, two special metrics were introduced for vacuum cleaner annoyance by correlating the objective and subjective evaluations. The first metric is a linear combination of the six sound quality metrics while the second metric is a linear combination of the four metrics which are determined based on Zwicker loudness analysis. All the results obtained in this thesis are based on single channel measurements, however to improve the current results for the future work it is recommended to utilise. two channel measurements with an artificial head and torso and. make use of pre-amplifiers with suitable modifications to represent human ear perception It has been observed that the product quality enhancement will be achieved and the market share of products will be increased by further using sound quality techniques in the future. Xlll 110

ÖZET Günümüz modern teknoloji çağı olarak bilinmektedir. Gelişen bu teknoloji beraberinde hızla artan rekabetide getirmiştir. Artan hızlı nüfus sebebiyle ihtiyaçlarda artmış, bunu karşılayabilmek için üretim miktarıda ihtiyaç oranında artmıştır. Ayrıca artan hızlı nüfus sebebiyle yeni yerleşim yerleri, yeni kurulacak sanayi bölgeleri, iş merkezleri vs. gerekmektedir. Tüm bunların kurulması için gerekli temel malzeme lerin başında çimento gelmektedir. Çimento fabrikaları 24 saat sürekli üretim yapan büyük endüstriyel tesislerdir. Bu fab rikalarda makina arızası ve dolayısıyla üretim kaybı hiç istenmez. Bununla birlikte, çimento sanayi belirli özellikleri olan bir üretim alanıdır. Bunlar ; ü Çok problemli bir üretim koludur. q Çok sayıda ve çok çeşitli makina teçhizatının biraraya gelmesinden oluşmaktadır. Dolayısı ile bu konuda karşılaşılan problemlerde o nispette çok ve özellikle çeşitli olmaktadır. q Hammaddesi taş ve toprağa dayalı bir sanayidir. Böyle olunca malzemenizde 99 elementin 99 'ununda bulunması ihtimali mevcut olmaktadır ve birkaç ana elementin dışında ürününüze ne etkiler getireceğini bilmeniz mümkün değildir. Bu etkiler her fabrikada izahı güç problemlerin ortaya çıkması sonucunu doğurmaktadır. Çözümlerinde ise, uygulama seviyesinde mevcut olması normal olmayan özel bilgi ve donanım gereklidir. Bu da ancak bir mütehassıs araştırma - geliştirme biriminin oluşturulmasıyla sağlanabilir. q Yeni ve daha verimli teknolojilere daha çok ihtiyaç duyan bir sanayi koludur. Tüm bu sebepler yüzünden uygulanan bakım tekniği, verimliliği doğrudan etkileyen bir faktör olacaktır. Bu tezde en modern bakım yöntemi olarak bilinen makina performansına dayalı bakım planlaması ve uygulamalarından sözedilecek, yöntemin klasik bakım yöntemlerine göre avantajlarına değinilecek, çimento fabrikalarının çalışma prensibi ile ürün eldesini doğrudan etkileyen kritik makinaların bir sınıflandırması yapılarak, makina perfor mansına dayalı bakım tekniğinin bu kritik makinalar arasından seçilen 10 makina üzerinde uygulanması incelenecektir. Periyodik titreşim ölçümleri ile makinaların performanslarının sürekli izlenmesi arasındaki farklara değinilecek, periyodik ölçümlere dayalı performans izleme yönteminden sürekli izleme yöntemine geçiş için yapılması gereken fizibilite çalışmasına yer verilecektir. Elde edilen sonuçların değerlendirilmesiyle tez sonuçlandırılacaktır. XI SUMMARY Our century is known as `the modern technology period`. This developed technology has resulted in ever increasing competition between companies. Requirements have increased due to the increased population and production quantity has also increased by the rate of this requirement. Besides, due to the fast increased population, it is necessary to build new housing places, new industry regions etc. Cement is listed at the head of the necessary main materials to establish all these requirements. Cement plants are large industrial facilities producing continuously 24 hours. At these factories failure of machinery and loss of production can never be afforded. The cement industry is a production area with its own specialities. These are ; a) It is a production branch which has many problems, b) Production includes various machinery, c) The raw materials are obtained from stone and soil. Therefore, there is a probability that all of 99 elements are in your material. So one can't know which elements affect the production except a few main elements. These affects cause sophisticated problems. Specialized knowledge is necessary to solve these problems. This can only be provided by an expert research-development unit, d) It is a industry branch which needs new and more advanced production techno logies. Due to these reasons, applied maintenance technology will be a factor which effects directly the productivity. The subjects which will be covered in this thesis is summarised below ; - Importance of maintenance techniques and failure mechanisms, - Classical maintenance systems, - Working principles of cement factories, - Critical machines which effect the production, - Maintenance planning system depending on machine performance, - Advantages of the maintenance planning system. XllA computer-based vibration system consists of an integrated hardware and software package which uses a host computer workstation for data storage and diagnostic analysis. This system also includes a portable data collector and vibration transducer. The transducer is responsible for accurately converting mechanical vibration into an anolog electrical signal in terms of phase, frequency, and amplitude. The portable data collector temporarily stores incoming vibration signals from the transducer and processes the waveforms using Fast Fourier Transform (FFT) techniques. The data collector is capable of performing a variety of machinery measurements as either a routine survey instrument or a general purpose signal analyzer. It is designed for use so a great degree of skill is not required. A technician follows memory loaded measurement routes to the appropriate locations on the machine, attaches the transducer, and starts the measurement by pressing a single button. A typical data collector has the storage capacity for hundreds of measurement points, logging for each overall vibration level, waveform data, FFT data and additional trend or analysis parameters. Some data collectors can be used as a stand-alone analyzer capable of functioning as an ossiloscope. The data collectors also allow the user to monitor instantaneous changes in FFT signature, time waveform, synchronous peak and phase, overall vibration level, machine RPM, pressure, thickness and current. Data collectors also may perform multi-plane, multi-speed balancing using a downloadable balancing program. A few data collectors can perform advanced diagnostic functions such as synchronous time averaging, peak-hold averaging, triggered data capture, coast down, and start up capture (Bode and Nyquist display). The heart of the PC-based vibration programs is the analysis software which executes on the host computer such as a portable PC. The analysis software allows for ; 1. Creation of a machine database containing information for each machine or part to be monitored, and the path or route to take in monitoring. 2. Definition of data analysis parameters that allows for more flexibility in selecting the vibration analysis method to be used on individual machine parts. 3. Exchange of database route programs and machine data with the portable collector (downloading and uploading). 4. Scanning of data for exceptional values which may indicate machine part deficiencies or problems, and verification of the health of each machine. 5. Generation of graphical displays for ease of vibration analysis and communication with management or other interested parties. 6. Compliation of data in a historical manner for trending and archiving, and allowing for justification of program continuance. XlllOnce data has been dumped into the host computer, the PC-based PDM programs provide four important vibration analysis methods in which to evaluate health of the equipment ; 1. A trained engineer using the analysis software performs vibration signature analysis on equipment. A working knowledge of vibration analysis is essential for this method. 2. The analysis software has built-in and changeable automatic problem detectors that warn which equipment 's vibration levels are beginning to exceed industry or individual companies recommended standarts. 3. The analysis software also has built-in trending capabilities to automatically warn which equipment is showing continued increases in vibration levels. The parameters used to capture these levels may be changed to fit each equipment 's characteristics. 4. The analysis software has an automated expert system capability to diagnose problems. The expert knowledge bases are built from the combined knowledge of some of the most prominent analysts in the industry. - Advantages of the maintenance planning system are ; 1. Increased equipment availability resulting in greater return on invested capital by maximizing the running time between machine overhauls. This is one of maintenance planning system 's most important benefits since vibration analysis allows for the identification of problem equipment and also confirms which equipment does not need any scheduled maintenance. 2. Reduced maintenance costs and increased productivity by reducing the probability of catastrophic failure. Actually, studies show that maintenance planning programs have virtually eliminated surprise failures in some plants. This would be of particular importance to industries that carry their equipment to a job site. Studies also show that typical emergency repair costs for catastrophic failures average 10 times the cost of planned repairs. 3. More efficient repairs because repair quality can be checked and guaranteed. Studies show that approximately 10 percent of new parts pulled off the shelf have defects or flaws. These defects can cause the replaced component to fail, and also may result in latent damage to other machine parts. It is beneficial to check the quality of newly purchased and recently replaced parts. 4. Reduced inventory cost by reducing spare part inventory. Some plants which have started successful maintenance planning programs have been able to turn 'backup' production lines into profitable ones. The main production lines showed a significant reduction in downtime to deem a backup line unnecessary. XIV5. Improved operator and public safety because condition monitoring gives adequate lead time to enable machines to be repaired before reaching a critical condition. This is of particular importance to industries such as hospitals, where the health of the patients is directly dependent upon the quality of the hospitals diagnostic and life-sustaining equipment. 6. More efficient negotiations with equipment vendors regarding the purchase and repair of equipment. Maintenance planning programs evaluate the health of new equipment and give substantial advantages in negotiating with vendors on any repairs or redesigns that might be required. Analysis Of Vibration All rotating machinery generate vibration, the analysis of which renders valuable information about the condition of the machines. This article describes the type of vibration signals to be expected for faults in typical elements. Low Frequency Range Frequency Spectra obtained from measurements made on the motor bearings or gearbox bearings will reveal low frequency components at shaft revolution speed originating from unbalance, misalignments, bent shaft etc. At the second harmonic, i.e. twice the rotation speed, components originating from bent shaft and misalignments will also be found. In other words, by comparing the change in these components with time the appearance of such faults can be detected and diagnosed. The major problem with journal bearings is hydrodynamic instability in the system consisting of shaft / oil film / bearing housing. Oil whirl is a self - excited vibration, where the center of gravity of the shaft moves around within the clearance of the bearing at sub-synchronous speed. It arises typically at lightly loaded high speed shafts, and at frequencies between 42 % and 47 % of rotational speed. It seems to depend on the surface of the shaft and bearing. A last type of fault appearing in this low frequency area is mechanical looseness. In many cases machanical looseness will create interharmonic and subharmonic components, i.e. a `half harmonic, a `one-and-a-half harmonic, a `two-and-a-half harmonic etc. Medium Frequency Range Higher up in the frequency range, components originating from the toothmesh in the gearbox will be found and are referred to as medium frequency components. XVThey will be at frequency corresponding to rotational speed multiplied by the number of teeth on the gear, and referred to as the toothmeshing frequency. When the gearbox wears, the gear profile will gradually change due to sliding between two teeth in mesh at any point except at the pitch point. This indicates that changes due to wear in a gearbox will turn up at the second harmonic of toothmeshing frequency and, since the change is not sinusoidal, higher harmonics will be revealed as well. At incipient local fault, on the other hand, will not increase the level at toothmesh frequencies and its harmonics. Imagine a cracked tooth which is not yet broken, and will consequently not be noticed by the operation personnel. As the fault spreads out in the gearbox to cover several defective teeth, the signal will change character from the pulse-like one just described to one which has more energy and which looks more like an amplitude modulated signal. And it has high amplitude sidebands around toothmeshing frequencies and its harmonics, spaced at rotational speed. High Frequency Range An incipient fault in a rolling-element bearing will typically be a crack or corrosion pit either on the inner race, on the outher race, or on the rolling element itself. This crack will create small impulses every time one of the rolling-elements passes over it. These impulses will transmit energy to the bearing housing, which in turn will vibrate at its natural frequency (resonance frequency). So we are able to detect an incipient fault in a rolling-element bearing by monitoring the level of vibration filtered around one of the resonance frequencies of the mechanical structure. These pulses have far less energy than the low frequency and medium frequency signals, and must therefore be detected at high frequencies. XVI 256

ÖZET Günümüzde artan yaşam standartları insanları, konutlarda kullanılan beyaz eşyaların ürettiği gürültüye karşı daha duyarlı kılmaktadır. Bu nedenlerle soğutma amacıyla kullanılan cihazların, özellikle insanların dinlenme ve uyku zamanları olan gecenin geç saatlerinde aralıklarla meydana gelen çalışmalarından kaynaklanan gürültü seviyeleri kullanıcıların şikayetlerine neden olmaktadır. Soğutma amacıyla kullanılan cihazlarda kullanılan kompresörler, titreşim ve gürültünün en önemli kaynaklarından birisidir. Bu bakımdan, daha sessiz cihazların geliştirilebilmesi için kompresörlerin titreşim ve gürültü özelliklerinin iyi anlaşılmasına ihtiyaç vardır. Soğutucularda kullanılan kompresörlerin çevreye yaydıkları gürültünün azaltılabilmesi için en önemli tasarım parametresi, kompresörden muhafazaya iletilen dinamik kuvvetlerin azaltılmasıdır. Bu amaçla hazırlanacak kompresör sisteminin dinamik matematik modeli ve bu modelin çözümleri, kompresör sisteminin dinamik özelliklerinin incelenmesi ve iyileştirilmesi yönünde tasarım ve geliştirme aşamalarında kullanılabilecek etkin bir araçtır. Bu çalışmada, Türkiye'de üretilen ve çoğunlukla soğutma amacıyla kullanılan pistonlu, tek silindirli, sızdırmaz kompresörlerin dinamik davranışları incelenmektedir. Kurulan matematik modelde, bugüne kadar yapılmış olan diğer çalışmalardan farklı olarak, harekete ait diferansiyel denklemlerin katsayıları zamanın fonksiyonu olarak elde edilmiştir. Böylece kompresörün hem kalkış anlarında, hem kararlı çalışması durumunda ve hem de duruş anlamdaki dinamik davranışlarını en doğru şekilde inceleme olanağı elde edilmiştir. Birinci bölümde, çalışmanın yapılmasının önemi ve gereği üzerinde durulmuştur. Ayrıca, konuyla ilgili olarak daha önce yapılmış olan çalışmalar özetlenmiştir. İkinci bölümde, kompresör sisteminin hareket denklemleri, Lagrange Çarpanları metodu kullanılarak elde edilmiştir. Sisteme tesir eden tahrik ve yük momentleri detaylı olarak ifade edilmişlerdir. Elde edilen kompresör sisteminin denklemleri Runge-Kutta sayısal çözüm yöntemi kullanılarak çözülmüşlerdir. Üçüncü bölümde, kompresöre ait fiziksel özellikler tayin edilmiştir. Bu amaçla kullanılan, deneysel yöntemler açıklanmıştır. Ayrıca kullanılan bilgisayar programları ve yapılan analizler anlatılmıştır. Dördüncü bölümde, kompresörün akustik özelliklerinin belirlenmesine yönelik olarak yapılan deneysel analizler açıklanmıştır. Kompresörün akustik özelliklerinin iyileştirilmesi amacıyla alınabilecek önlemler tartışılmıştır. Beşinci bölümde, kompresörün kalkış, rejim hali ve duruş süreçlerindeki dinamik davranışlarının teorik ve deneysel sonuçlan birlikte ortaya konulmuştur. Çalışmanın altıncı bölümünde konuya ait değerlendirmeler yapılmıştır. Bu değerlendirmeler çerçevesinde yapılan öneriler ve gelecekte yapılması düşünülen çalışmalar yedinci bölümde sunulmuştur. SUMMARY DYNAMIC AND ACOUSTIC ANALYSIS OF A RECIPROCATING REFRIGERATION COMPRESSOR 1. INTRODUCTION The ever growing demand in the market for highly efficient, more reliable and less expensive compressors and an inherent opportunity for a competitive edge have activated the manufacturers to develop state-of-the- art analytical tools to predict, evaluate and optimize the existing as well as new designs. Fractional horsepower reciprocating compressors are commonly found in HVAC systems such as household refrigerators and air conditioners. Figure 1 shows a typical reciprocating piston compressor. Recently, noise and vibration problems are gaining increasing importance in the electrical home appliances and the trend is to manufacture lighter and higher quality goods. In general, it is a complicated process to find the generating sources and propagating paths of noise in the compressors. The compressor noise is a major contributor to the overall noise level of the domestic refrigerator. The periodic operation of the compressor and noise radiated during operation has been deemed to be a considerable problem by the consumer, especially during the quiet hours of the night. As a result, there is an increased need for understanding the noise and vibration characteristics of these compressors for the purpose of noise control. Fig.1. Reciprocating Refrigeration Compressor. V!The subject has raised a lot of interest among the researchers and plenty of publications exist covering the noise and vibration analysis of different type of compressors, [6-33]. Single cylinder reciprocating compressors are composed of three main mechanical subsets: the hermetic housing on rubber mounts, the compressor unit mounted inside the housing on an internal suspension composed of springs plus the discharge pipe and the slider-crank mechanism, which consist of a piston, a connecting rod and a crankshaft. These machines are driven by an AC motor and the start-up torque is high enough to attain the nominal speed of rotation rapidly within milliseconds. The transient start-up, steady-state and shut-down motions have to be studied carefully since the compressor design improvement implies an in-depth knowledge of the compressor unit. In addition, compressor manufacturing is characterized by mass production on several lines, therefore a method for predicting the dynamic behaviour is required to save time at the design stage, thus reducing the number of prototypes as well as costs. In a practical design of a spring suspension system for a hermetically sealed compressor, the primary considerations are the spring location and the counter balancing of the crankshaft. The proper selection and design will lead to suspension designs which minimizes the transmitted spring force. To the best of our knowledge, studies on hermetic refrigerant compressors involving theoretical and experimental investigations have not considered the dynamic behaviour during start-up and shut-down phases, except a recent publication, [32], which appeared while the present study was in progress. The aim of the present study is:. to develop the differential equations of motion,. to prepare an associated computer code and. to conduct experimental investigation on the dynamic behaviour of a single cylinder refrigerant compressor during start-up, steady-state and shut-down motions. The compressor unit is mounted on several internal mounts and is submitted to the reaction forces acting on the stator body due to the slider- crank mechanism motion. The primary objective of the compressor suspension system is to isolate the compressor motion and the force transmission to the hermetic shell surrounding the compressor. Isolation of these forces and the motion will avoid the vibration of the hermetic shell and thus avoid the transmission of vibration to the supporting structure. It is expected that the prevention of the vibrations of the shell will decrease the noise radiation from the shell. VIIThe main assumptions made in the present study can be summarizes as follows: 1. The motion of the hermetic housing is neglected. 2. Stator body, piston, connecting rod and crankshaft are assumed to be rigid. 3. The discharge line and suspension springs are massless, deformable bodies having only elastic and damping effects. 4. The effect of the discharge pressure within the discharge line is omitted. 5. All body forces due to the gravity field are omitted, i.e. vibrations about the static equilibrium position. 6. The compression and expansion processes are expressed as polytropic processes and discharge process is expressed as constant pressure process. 7. The compressor motor dynamics are approximated using the curve fitted to the measured motor torque-speed curve. 8. At the start-up instant the equivalent resistive torque acting on the crank shaft is calculated from the steady state P-V diagram. 9. At the shut-down instant the driving moment is ideally equated to zero. 2. MATHEMATICAL MODELLING The cross-sectional view and the reference frames of a typical single cylinder compressor are illustrated in Figure 2. The compressor consists of a frame, rotor, crankshaft, connecting rod, piston and mounting springs. The fixed inertial coordinate system (OXYZ) is placed at the equilibrium position of the center of the frame of the slider-crank mechanism. The motion of the moving frame, (CXcYcZc), which is connected to the center of rotation of the crank C, is calculated with respect to the inertial coordinate system (OXYZ). Y,YC° +AD(a^)TADIkDTAT(co° +ADto^) +(r° + Arbc +5°Arbc)Tmb(rc° + Arbc +S°Arbc) +(a>° + AB(D^)TABIbBTAT((û° + ABog) 4rc° + Arpc +5°Arpc)Tmp(rc° + Arpc +5c°Arpc) +o°TAIpA V +{rc° + Arrc +5°Arrc)Tmr(rc° + Arrc +S°Arrc) +(a>° + AD(orc)TADIrDTAT((o° + ADorc) +{rc° + Ar^ +52Ar^)Tmcw(rc° + Ar^ +5°Ar^) +(rc° +5°Argc)Tmk(fc° +5°ArgcJ410G0p +ai 1191p + aU292p + ali1303p + a,,14x90 +al15x61 + ai16x92 + aU7x93 + ai18y9c +ali1By91 + ai20y92 + ai21y93 + ai>22z80 +aIi23z91 +ai24z92 + ai25z93 + ai268; +ai.270o6l + ai,280O02 + ai,29ö0Ö3 + ai,3o6l + ai.3lölö2 + ^^l^ + ai.3302 + ai,34e263 +ai3SQ23 + ai36x+ ai37y + aL38z+ ai3990 + a^e, + aw182 + ai4283 + ai43p + ai44x +awsy + aueZ+aW7e0 +ai.4«9i +^4962 + ai.5o63 = f,(t)+2oeM, (5) i=1,...,8 ?'o These eight nonlinear differential equations of the compressor system can be expressed in matrix form as follows: mq + cq + kq = f(t) + 2ap (6) where introducing matrices,a. m La8, C = LC8; '1,2 '2,2 a 3.2 a 4,2 '5,2 `6,2 a 75 `8,2 '1,2 '2,2 '3,2 '45 '5,2 '6,2 '7,2 '8,2 '1.3 `2,3 `3,3 '4,3 '5,3 '6,3 '7,3 '8,3 '1,3 '2,3 '3,3 '4,3 '5,3 '6,3 '7,3 '8,3 `1,4 '2,4 `3,4 a 4,4 '5,4 '6,4 a 7,4 `8,4 1,4 '2,4 '3,4 '4,4 '5,4 '6,4 '7,4 '8,4 '1,5 '2,5 '3,5 '4,5 `5,5 a 6.5 '7,5 `8,5 '1.5 '2.5 '3.5 '4,5 '5,5 '6,5 '7,5 '8,5 `1.6 ?`2.6 '1.7 `1,8 '3,6 a 4.6 '5.6 '6,6 '7,6 '8,6 '1.6 '2.6 '3,6 '4,6 '5,6 '6,6 '7,6 '8,6 '2.7 a 3,7 '4,7 `5,7 `6,7 a 7,7 '8,7 '1,7 '2.7 '3,7 '4,7 '5,7 '6,7 '7,7 '8,7 32.8 33,8 a4,8 a5,8 36,8 a7,8 a8.8. C1,8l C2,8 C3,8 C4,8 C5,8 C6.8 C7.8 c8,8J k = and vectors q = i x y z e0 e, e, e3 IPJ o o 0 P = V e, o XIThe inertial terms of the moving parts expressed by the elements of the mass matrix of the slider-crank mechanism depend on the crankshaft position, (3, are hence functions of time. The relationship between the Euler Parameters is given by e§+e?+e!+e! = i. (7) The resulting system comprises eight differential equations plus an algebraic equation which should be solved simultaneously. The resulting system of equations can be put into the following matrix form suitable for a step-by-step numerical integration procedure: q = -m`1cq - m`1kq + m`1f(t) - 2TpTq (8) Equation (8) further can be transformed into the following form: x = Ex + Q (9) where E = 0 I -m1k -m1c-2TpT. Q = m`1f(t)J ' x = In order to apply the Ruge-Kutta-Merson Algorithm Equation (9) can be expressed as: F(x) = Ex + Q where the step-by-step solution of the vector x is given by ?1 x = x+-(k1+4k4+k5) (10) (11) The error in the numerical process at any step can be calculated using the following expression: e = ^(2k1-9k3+8k4-k5) (12) where XII> k1=-hF(x) 1 k^-hFCx+kJ 1 7 1 1 k3=-hF^x+-k1+-k k4 = 3hFlX+8kl+¥k 1 J 3 9 ^ ks = 3hFlX+2kl_2k3+6k4y 3. EXPERIMENTAL WORK Two different experiments are conducted in this study. In the first experiment sound intensity measurements are made and the noise mapping around the compressor is obtained. It is noticed that the main contribution to the noise level is due to the vibration of the bottom part of the hermetic shell which is directly excited by the forces transmitted through the suspension springs. In the second experiment, an attempt is made to verify the proposed model. For that purpose an experimental set-up is prepared. The time response analysis of the compressor within its closed housing requires a load stand unit, a multi-channel measurement system and transducers which are used in a hostile environment: electromagnetic field, aggressive oil and refrigerant gas, high tempreture and pressure. Additionally, difficulties exist in obtaining accurate knowledge of parameters such as friction, driven torque and relative pressure. Due to these difficulties not all the data relevant for predicting the dynamic model could be obtained. Therefore the comparison made is based on the order of magnitude rather than step-by-step time response. Even under such an unfavorable conditions the model yielded reasonably comparable results at start-up, shut-down and steady-state conditions. 4. THEORETICAL AND EXPERIMENTAL RESULTS Experimental and theoretical results are obtained in sturt-up, steady- state and shut-down conditions. The response of the stator is computed at the locations of the transducers. To make a comparison on the same basis experimental and theoretical results are obtained under similar conditions as far as possible. The results are summarized briefly in Table 1. The comparison between experimental and computed results show that the mathematical model and the computer code developed are reasonably satisfactory for predicting the maximum vibration amplitudes during start-up, steady-state and shut-down stages. The prediction can be improved by improving the model by taking into account the following points: XIII1. Damping effect of the oil within the closed housing of the compressor should be incorporated into the model. 2. The pressure built-up in the cylinder should be measured and included in the model during start-up. 3. Position of the crank angle as an initial condition should be accurately determined. 4. At the shut-down stage variation of the electromagnetic moment should be measured and included in the analysis. Theoretical curves for the start-up, steady-state and shut-down stages are given in the results section of the thesis. Since the model does not allocate the four effects which are itemized above, it is not expected to duplicate the time variation of the response signal obtained in the experimental analysis. At steady state measured values are systematically lower than the theoretical values. This discrepancy can be explained by the lack of sufficient damping in the theoretical model as indicated by item 1 above. Table 1. The Experimental and Theoretical Values for the Maximum Displacement of The Compressor at. 5. CONCLUSIONS The comparison between experimental and computed results show that the model and the computer code developed are reasonably satisfactory for predicting the start-up, the steady-state and shut-down motions. The code will be used by a compressor manufacturer. To improve the measurements obtained from the set-up:. pressure in the cylinder during start-up and shut-down should be accurately measured,. the crank angle at t=0 should be accurately determined,. speed as function of time should be monitored,. variation of the electromagnetic moment should be measured at the shut-down. made: To improve the mathematical model following modifications can be xiv. damping effects of the oil interaction with the stator body in the hermetic casing,. start-up and shut-down pressure variations,. variation of the electromagnetic moment should be included in the model analytically, Although the developed model has certain limitations it is expected that it will serve as a tool to compressor manufacturers. Improvements on the model are still in progress. The dynamic model and simulation of the compressor will allow the designers to predict the response of the compressor for various proposed suspension designs before such designs are built, thus reducing the time and effort needed to find improved designs. xv 153

ÖZET Yapılarda titreşim gücü dağılımı ve titreşim gücünün yapı içerisinde izlediği yol, titreşimsiz ürün tasarımı, yapılarda titreşim kontrolü ve ürün kalite kontrolü gibi konularda gün geçtikçe önem kazanmaktadır. Titreşim gücünü etkileyen parametrelerin belirlenmesi, söz konusu konularda yapılan çalışmalara oldukça büyük ölçüde katkı sağlayacaktır. Bu çalışmada, öncelikle liner yapılarda titreşim gücünün farklı analiz (Sonlu Elemanlar ve Modal Analiz) yöntemleriyle formüle edilmesi ve modellenmesi sağlanmıştır. Elde edilen modeller, endüstride üretim hattında elektrik motorundan akan aktif titreşim gücünü temel alan bir kalite konrol sisteminin ölçüm düzeneği üzerine uygulanmıştır. Göz önüne alman ölçüm düzeneğinin yapısal özellikleri deneysel çalışmalar sonucu elde edilmiştir. Ayrıca ölçüm düzeneğini belirleyen özellikler, farklı yazılım ortamlarında geliştirilen modellere girilmiştir. Yapılan çalışmalar sonucunda, liner bir yapıya girilen titreşim gücünün; yapısal özelliklere, sınır şartlarına, uygulanan kuvvetin genliğine, frekansına ve faz farkına bağlı olarak değiştiği tespit edilmiştir. Elde edilen bu parametrelerin değişmesi durumunda, yapı üzerindeki titreşim gücü dağılımının nasıl bir dağılım göstereceğine dair bilgiler elde edilmiştir. Titreşim gücünü etkileyen özellikler göz önüne alınarak, kalite kontrol amacıyla kullanılacak en uygun ölçüm düzeneğine yönelik tasarım parametreleri elde edilmiştir. Ayrıca çalışma sonucu elde edilen bilgiler, titreşimsiz ürün tasarımında ve titreşim kontrolünde de kullanılabilecektir. Kalite kontrol amacıyla titreşim gücü sinyalinin kullanılması konusunda yapılan ilk çalışmalardan olması nedeniyle oldukça önemlidir. XVI SUMMARY MECHANICAL POWER FLOW IN LUMPED AND DISTRIBUTED PARAMETER SYSTEMS: APPLICATION TO A QUALITY CONTROL MEASUREMENT SET-UP. The purpose of this study is to understand the dynamic characteristics of the mechanical power flow in linear structures so that the parameters influencing the power flow can better be understood. The ultimate goal of the study is, to obtain the parameters of the optimum measurement set-up which is used for quality control of the electrical motors of the washing machine. In order to get the design parameters of the optimum measurement set-up, the model of the active vibration power flowing from the product to the measurement set-up should be established. On the production line in industry, the functionality expected from the measurement set-up which is developed for quality control purposes, is to support the static load of the product and to maximize the active vibration power flowing through the power flow transducer which is located above the energy absorber, Figure 1. Figure. 1. The Measurement Set-up Designed For Measuring Active Power Flowing From The Electrical Motor. XVUThe proposed measurement set-up contains; T bone plate located on three air-springs so that the motor can be supported, the energy absorber which is positioned in the middle of the plate, and the power flow transducer located above the absorber in order to measure the active power flowing from the product. The sketch of the measurement set-up is shown in Figure. 1. The power flow transducer consists of two parts, first part is the accelerometer and the other part is the force transducer. These two transducers are put together and the power flow transducer is obtained. The cross correlation of the signal coming from the accelerometer and the force transducer gives the mechanical power signal at the measurement point in the structure. The cross correlation signal is used for quality control purpose. Supplied mechanical power in linear structures, using analysis methods (Modal Analysis, Finite Element Methods), is discussed theoretically. The formulation of the supplied mechanical power in lumped and continuous systems is given in Section 2. Experimental studies are performed to get the structural properties of the components of the measurement set-up in Section.3. Using the formulation derived in Section 2, the models of the supplied mechanical power in linear structures are established in Section 4. The evaluation of these models are made, and then the models are applied on the measurement set-up in Section 5. During the studies conducted, the parameters influencing the measurement technique are revealed. Therefore the design parameters of the optimum measurement set-up for a quality control system, using the mechanical power signal coming from the product, are obtained. With the three different points of view, three different models of mechanical power for linear structures are developed. Taking the measurement set-up into account as a lumped parameter system, the average value of the supplied mechanical power to the structure is obtained. The purpose of using a lumped model is to see the dependence of the power flow on the forcing frequency variation and the rigid body motion of the structure. In the second model, the modal model of the structure is formed to obtain the mechanical power at a point defined in the as function of the frequency. Using this approach the effects of the elastic body motion of the structure on the mechanical power spectrum can be obtained. In the third model the Finite Element approach is used and the mechanical power distribution in the structure is obtained at a specific frequency. Finite Element model of the mechanical power distribution in the structure is conducted for a harmonic force input. The theoretical background of three different models of mechanical power in a linear structure are discussed in Section.2. The supplied mechanical power is simply formulated as a one degree-of-freedom linear system in Section.2. 1. The XVlllequation of motion of one degree-of-freedom viscos damped system is considered in Eq.l. mx(t) + cx(t) + kx(t) = F(t) (1) Using harmonic motion assumption, the second order differential equation is solved. F(t) = Feiû5t (2) x(t) = Vei(ût (3) where F = the amplitude of the force [N], V = the amplitude of the velocity [m/s], t = time [s], co = frequency [rad/s]. If the Eq.2 and Eq.3 are put into the Eq.l, and the time averaged values of the force and the velocity are considered, the mobility of one degree of freedom system can be obtained as in Eq.4. V ceo 2 - ico (k - co m) ? = _(k-co2m)2+c2co2 where i denotes the imaginary part. (4) The mechanical power is defined as the multiplication of the time averaged values of the force and the velocity : n = t=^{FV*} (5) where XIX(...), = time averaged values, = complex conjugate, IT = mechanical power [W]. IfEq.4 is substituded into Eq. 5 then Eq.6 is obtained. ceo2 + ico(k-©2m) n-Ip (k-co m)2 +c2co: (6) In Eq.6, the real and imaginary parts are defined as the active and reactive power respectively. These two definitions are given in Eq.7 and Eq.8. 1 (k~co m) +c^co' Pn = -F2co2 `_,,2, 2 (7) Qrr lxfl (k~ro m) vn = -Fco- y--2 j-j (8) 2 (k-co m) +c2© where p n = the active power [W], Q IT = the reactive power [W]. In a multi-degree of freedom system, similiar formulation is valid, but in this case, active and reactive components of the power are expressed in terms of matrix quantities. Using the formulation derived so far, the lumped parameter model of the structure is obtained. In order to govern the modal model of the structure, the vibration modes of the structure are obtained by calculating the eigen-values and the eigen-vectors. The superposition of the vibration modes of the structure gives the modal model of the structure. Modal properties of the structure are obtained from spatial model of the structure,by using the orthogonality principle or by performing experimental modal analysis on the structure and using the curve fitting method. While the modal model of the structure giving the mobility characteristics for one degree of freedom system is given by Eq. 4, the eigen-values of the modal model is calculated in Eq. 9 and Eq. 10. The eigen-values of the modal model are complex quantities since the elements of the modal model are complex. XX^ (k.-^m.V+^c? (9) AQi= *» -*2*i (10) 1 ', (ki-û)2mi)2+©2c? X? = the real part of eigen values of modal model in i * mode, X? = the imaginary part of eigen values of modal model in ift mode. Hence, using the modal model of the structure, the mechanical power is obtained. The formulation of the mechanical power in an elastic linear structure is defined in the continuum mechanics by Eq.ll, using the continuity equation and assuming that the damping mechanism in the material is modelled by linear viscoelastic model of Kelvin type. r1 = -25ljYj=Pri+iQri (i= 1,2,3,j= 1,2,3 ) (11) where çTj. = stress tensor [N/rm], T; = mechanical power flowing unit area in i direction [W/rm], p rf = active power flowing unit area in i direction [W/rm], Q Tj ' = reactive power flowing unit area in i direction [W/rm], v j = velocity tensor [m/s]. The velocity and stress values which are defined as tensors in the continuum mechanic formulation, in Finite Element approach become matrices due to discretization. The Finite Element model of the mechanical power is developed in Section.2.3, in this derivation Eq. 11 is used. Using the harmonic forcing, at a specific frequency the distribution of the active and reactive power in the structure for three directions, is calculated. In the Finite Element approach in order to create the solid model of the structure, an eight node cotinuum element is used. This element has displacements in the three direction as the degrees-of-freedom at the nodes and capable of yielding the stresses in three directions. In order to understand the dynamic behaviour of the measurement set-up the test motor is placed on the T-bone plate and started and `Operational Deflection XXIShape` study is performed. This study is explained in Section.3.1. As a result of this study, it is seen that there is a substantial amount of motion not only in the vertical direction but also in the horizontal plane. Experimental Modal Analysis study is performed on the T bone plate with free free boundary conditions. Later, the analysis is repeated in Section.3.2 on the. assembled measurement set-up so that the effects of the other components on T bone plate can be identified. Modal properties of the all measurement set-up is obtained from Experimental Modal Analysis study in order to add them into the modal model of the measurement set-up. Mode shapes of the T bone plate in free free boundary condition are also calculated in Finite Element environment in Section. 3. 3 in order to compare the mode shapes obtained from Experimental Modal Analysis with the mode shapes obtained from Finite Element Analysis. In order to add the internal damping properties of the T bone plate in the Finite Element model, two different methods are used. The loss factor of the T bone plate in first mode is obtained in Section.3.4 by using the resonant and non-resonant methods.The average value of the two results is considered in the Finite Element model of the measurement set-up. The structural properties of the other components of the set-up, excluding the T bone plate, are provided by assuming these components as one degree of freedom elements individually and by performing mobility measurements on them. As a result of these mobility measurements, stiffness, damping and mass properties of the air spring, absorber and transducer with coupling stinger are obtained. These values are used in the three different models created for the measurement set-up in Section.5. The lumped parameter model of the structure is used for the two degree of freedom system to get the input active and reactive power at one point. Inputs of the model are, structural properties of the two degree of freedom system, degree of freedom where the force is applied, degree of freedom where the response is taken, amplitude and the frequency of the force. Outputs of the model are, mobility spectrum, active and reactive power spectrums at a point where the force is applied. First, this model is applied to a case study given in reference [8], and the model is validated, then the model is applied on the current measurement set-up. The measurement set-up is considered as the two degree of freedom system. First degree of freedom is associated to the mass of the T-bone plate and this mass is connected to the ground by the air springs, second degree of freedom is related to the mass of the absorber and this mass is grounded by the spring and dashpot of the absorber. xxnThe connection properties between these two degree of freedom are taken from the stiffness and damping of the transducer coupled with the stinger. Modal model of the measurement set-up is governed, using the mode shapes and modal properties of the measurement set-up in 0-10 kHz frequency range. In order to get the active and reactive power spectrum at a point where the force is applied, the degree of freedom where the force is applied and the degree of freedom where the response is taken, should be added to the model. An exciting force which has the unit amplitude over the frequency bands of interest is applied on all the degrees of freedom of the measurement set-up, in order to determine the point which is the most suitable for the supplied active power to the structure to be maximum. Finite Element model of the structure is formed by using the geometry, material properties, boundary condition of the structure, point where the force is applied, amplitude and the frequency of the force. Outputs of the model corresponds to these inputs are, active and reactive power distribution for three direction in the structure at the excitation frequency. First, the model is applied on an already tested structure for validation [20], then the model is applied on the test set-up. In this approach the Finite Element model of the measurement set-up is established by using the solid model of the T bone plate, the mode shapes of the plate in free free boundary condition, structural properties of the air springs and absorber, the loss factor of the T bone plate, material properties of the plate, boundary condition, the point where the force is applied, the frequency and amplitude of the force. In the excitation condition, active and reactive power distribution is calculated for the three directions in the structure at the excitation frequency. Consequently, the factors influencing the mechanical power are determined as the structural properties, boundary conditions, application point, frequency, amplitude of the force, and also the phase angle between the excitation forces. Added to this, supplied mechanical power at one point in the structure is related to the mobility properties of the structure at that point. The real part of the mobility spectrum is proportional to the active power spectrum at same point. The imaginary part of the mobility is proportional to the reactive power spectrum at same point. There is a peak in the natural frequency region of the real part of the mobility spectrum while the phase angle between the velocity and the force signal is zero at resonance. At resonance region, the supplied active power to the structure is maximum, but the reactive power is minimum. XXlllMechanical power spectrum can be modified by changing the structural properties this changes the mobility spectrum as well. The modification of the mass, stiffness and the boundary condition of the structure affect the natural frequency of the structure, and hence modify the active and reactive power spectrum of the structure. The change in the damping properties of the structure affects the amplitude of the mechanical power. If a quality control system is to be designed based on the measurement of active power signal flowing from the product in one direction at one point, then the structural properties on both ends of the point should be considered carefully because it dramatically affects the active power spectrum at this point. If it is desired that the effects of the structural properties are minimized, the natural frequency of the structure (the measurement set-up) should be away from the frequency range of interest. On the other hand, the maximum active power flowing into the structure is observed at the natural frequencies. If the natural frequencies of the structure and the excitation frequencies of the source coincide then the supplied active power flowing into the structure will be maximum, reactive power flowing in structure will be minimum at the excitation frequencies. If it is not possible to change the excitation frequencies of the source, then changing the properties of the structure and the designing a suitable mobility spectrum is necessary, depending on the excitation frequencies of the source. However, it is considerably difficult to design the mobility spectrum given the excitation frequencies. xxiv 97

ÖZET ESNEK ROTORLARDA DENGELEME PROBLEMİNİN İNCELENMESİ Bu çalışmada, endüstride geniş bir kullanım alanına sahip olan dönel makinalarm (rotorlar) yerinde dengelenmesine yönelik araştırmalar teorik ve deneysel olarak incelenmiştir. Bu maksatla öncelikle dönel makinalar ve bu makinalarm yerinde dengelenmesini sağlayacak yöntemlerin başlıcalan olan Modal Dengeleme ve Tesir Katsayıları ile Dengeleme yöntemleri tanıtılmış, daha sonra her iki yöntemin avantajlarını birleştirirken dezavantajlarını elimine eden ve son gelişmiş yöntem olan Birleşik Dengeleme Yaklaşımı adı verilen yöntemin anlatılmasına geçilmiştir. BDY ve tesir katsayıları ile dengeleme yöntemlerinin gerçek rotorlar üzerinde nasıl sonuçlar vereceğini görebilmek maksadıyla bir deney düzeneği kurulmuş ve elde edilen sonuçların başarılı bir dengeleme için yeterli olup olmadığı incelenmiştir. Bu yöntemlerin gerçek bir rotora uygulanması sırasında, mil-disk sisteminin boyutlandırılabilmesi maksadıyla kritik hızların ve mod şekillerinin hesaplanması gerekli olduğu için, bu değerleri veren ve elde mevcut olan bir bilgisayar programı çalışmamıza uygun olarak konfigüre edilmiştir. Dengeleme işlemine geçildiğinde ise, her iki yöntemin de işleyiş prosedürünü adım adım izleyen, ölçülen titreşim değerlerini giriş datalan olarak alan ve sonuçların elde edilmesini sağlayan birer bilgisayar programı hazırlanmıştır, özellikle BDY yönteminin oldukça karmaşık işlemler içermesi ve yöntemi uygulayacak olan operatörün adımlardan herhangi birini atlaması ya da yanlış uygulaması ile istenilen sonuçlara ulaşılamayacağı için, pratikte bu programı kullanacak olanlara kolaylık sağlamak maksadı ile, programda her adımdaki işlemler ve gerekli uyanlar diyaloglar yoluyla ekrana yazdınlmıştır. Uygulamalar sırasında ise elde mevcut motorla istediğimiz kritik hızlara çıkılamadığı ve mil uzunluğu deney tesisatı sebebiyle sınırlandırıldığı için, BDY yönteminin test edilmesi uzun uğraşlara rağmen mümkün olmamış, sadece Tesir Katsayılar Yöntemi test edilebilmiştir. SUMMARY THE ANALYSIS OF THE BALANCING PROBLEM IN FLEXIBLE ROTORS The mass balancing of rotating machines (referred to more briefly as rotor balancing) has been an important consideration in machinery design and manufacture for more than forty years. Reduction in weight, coupled with increased operating speeds, have substantially increased the flexibility of rotating machinery. Also, many types of machines are currently being designed to operate supercritically, and this trend is likely to continue. Consequently, reduction of mass unbalance in rotating machinery is often essential to ensure safe operation and reasonable life for such machines. The unbalance in the rotors will not only cause vibrations, but also transmit rotating forces to the bearings and to the foundation structure. The forces thus transmitted may cause damage to the machine parts and its foundation. In some cases, if the transmitted force is large enough, it might affect even the neighbouring machines and structures. For this reason, it is necessary to remove the unbalance of a rotor for its smooth runnings by adding or removing calculated correction masses. The location where the correction mass is to be added or removed can be determined simply by an experiment of statics if the rotor has only one disk or the balancing is required in one plane. Such a process is called static balancing. If the unbalance is distibuted along the length of the rotor, as it is in many cases, then a static balancing procedure cannot be used to determine the correction masses. This is because the centre of gravity of the rotor may be brought to the centre line of the shaft, but there could be an unbalanced moment left in the rotor, generating equal and opposite rotating reactions. This unbalance can be removed by dynamic tests only and hence called dynamic unbalance. This procedure is simple if the rotor is ideal or rigid. In this study rigid rotor balancing methods are studied in some detail for knowledge. If the shaft uerîects and the deflection changes with speed, as it does in the vicinity of the critical speeds, the problem of balancing is complicated. In this case, these rotors need special flexible rotor balancing procedures. Consequently, rotor balancing procedures are classified according to the type of rotor for which they are designed, rigid or flexible. When rigid rotors are balanced, the shape of the centroidal axis as a function of speed does not change. For flexible rotors, the shape of the centroidal axis, and thus the unbalance distribution, does change with speed. In general, rigid rotor balancing procedures and tools are not applicable to the balancing of flexible rotors. However, the same is not true for the current flexible rotor balancing procedures. These procedures can be classified into two groups, modal balancing and influence coefficient balancing. In the past, these two general balancing methods have competed, each with the inherent advantages and disadvantages of the other. In both cases, the disadvantages have been significant enough to prevent widespread acceptance of the balancing procedure. Consequently, most flexible rotor balancing is currently done by trial and error procedures, which are inefficient. In this study, a unified approach to the balancing of flexible rotors [1] which bridges the gap between modal and influence coefficient balancing is described. This 11new balancing method (hereafter referred to as the Unified Balancing Approach) combines the advantages of modal and influence coefficient balancing and eliminates the disadvantages of the two current procedures. At the same time, the Unified Balancing Approach provides a straightforward, practical approach to flexible rotor balancing, is well-suited to both production and field balancing. Briefly,the influence coefficient method seeks those correction masses in a predetermined set of planes which will minimize measured vibration (readings) at a series of sensors and speeds as predicted by influence coeffîcients,relating reading to mass additions.The influence coefficients are normally determined by a series of trial mass tests. The modal method seeks to balance the rotor,one mode at a time, with a set of masses specifically selected to leave previously balanced lower modes undisturbed. Sensitivity to this combination of masses as a set is determined empirically by a series of trial mass tests. Specific advantages and disadvantages, identified with each method,take on different relative importance according to the requirements of a particular balancing process.The only trial mass runs that are required at the highest balancing speed involve the combined weight sets for modal response. Individual trial mass runs are not required. Since, by design, the lower modes are not affected by a modal trial mass set, this mass set can be made large enough to assure good sensitivity at the mode being balanced. Eliminating the danger of upsetting the condition of previously balanced modes generally simplifies the balancing of higher modes. Only a basic understanding of the dynamics of the rotor system is needed to select locations for vibration sensors and balancing planes. It is not necessary to have prior knowledge of the system. Modal balancing does, of course, assume that the responsee of the rotor system is linear and that the vibration sensors are sufficiently accurate. Modal balancing sometimes requires analytical procedures for selecting the sets of trial masses used for correcting spesific modes. Not only does the requirement for this analysis complicate the balancing procedure but any inaccuracies in this analytical model reduce the effectiveness of the balancing procedure. Analytical procedures are not necessary, however, and several proponents of modal balancing employ emprical procedures to determine the modal trial mass sets. This emprical procedure has disadvantages, also. It requires substantial operator insight into the modal character of the whirl of a rotor due to unbalance, and this requirement inhibits the efficiency of production balancing techniques. Balancing a rotor to eliminate vibration as measured by a single sensor may not result in minimum vibration for the rotor as a whole, particularly when mode shapes are distorted or non-planar. Difficult may also occur as a result of vibration from other modes not previously balanced. Methods have been developed to correct this problem [7], but once again operator insight is required. As for the influence coefficient balancing : influence coefficient balancing is an entirely emprical procedure, its effectiveness limited only by the linearity of the rotor system and the resolution of the vibration sensors. Only a minimum knowledge of the rotor system is required for specifying the locations of the vibration sensors and balancing planes (for the application of correction masses). Influence coefficient balancing provides for the simultaneous balancing of more than one mode, provided that the necessary sensitivity information (influence coefficients) is available. This feature can reduce the required number of balancing runs but does require measurement of the rotor vibration at speeds near the relevant critical speeds. It may innot be safe to ruri through öne of these critical speeds to make measurements at higher speeds. The emprical nature of this procedure and the modest skills required of the operatör, along with the required calculation form, make this procedure ideally suited for computerizing and automating. Least-squares minimization of vibration data allows for balancing criteria, based on reducing the vibration for a number of points in the rotor system and for any number of modes. This feature of influence coefficient balancing can often be used to compensate for distorted ör non-planar mode shapes and the effect of other unbalanced modes. Additional data manipulation techniques, designed to compansate for measurement errors inherent in any realistic application of rotor balancing are available. Since individual trial mass runs must be made at the highest balancing speed when using influence coefficient balancing, the total number of runs required for this speeds may be limited by the response of lightly damped lower modes to these trial masses. The initial unbalance preseni for the higher modes may make it impossible to acquire sensitivity data, using a restricted trial mass, at a speed clo.se enough to a higher critical speed to produce a significant effect from this trial mass. That is, if the largest possible trial mass is small compared to the initial unbalance at the higher mode, the resolution of the sensitivity data acquired for this trial mass may be substantially reduced. When a particular mode is balanced after a number of other modes have been balanced, a large quantity of data may be required. This requirement can affect the accuracy and convenience of the balancing procedure. Due to their inherent disadvantages, neither modal nor influence coefficient balancing has gained widespread acceptance. Consequently, most practical flexible rotor balancing uses öne of a number of trial and error procedures which are generally inefficient and often ineffective. The Unifîcd Balancing Approach (UBA) is intended to incorporate the advantages of both the influence coefficient and modal balancing methods, while eliminating the disadvantages of both methods. That is, the Unified Balancing Approach uses a modal method of applying correction masses in modal sets using data derived in an emprical manner and requiring a minimum of prior knowledge of the dynamics of the rotor. Essentially, the technique involves the calculation of modal trial mass sets based on the computerized influence coefficient procedure of taking trial mass data. Generally, these modal trial mass setş are determined in such a way as to affect the mode of interest while not affecting the lower modes that have already been balanced. Hovvever, if appropriate data are available from previous tests ör from numerical predictions, a modal trial mass set can be conducted that will have no effect on any general set of modes. in general, the number of planes required for the modal trial mass set is öne more than the number of modes which must not be affected. The procedure for applying the UBA is described below and also illustrated in the flowchart in section 3.2.2. to give a clearer understanding of the process. This approach is designed to handle a number non-ideal (but not uncommon) conditions, including the occurrence of non-planar ör distorted mode shapes and the existence of measurement error and fmite vibration data resolution. A computer program is generally required to perform the necessary calculations. ivl Thc hasic proccdurc lor implcmcnting tlıc UBA is as follovvs: 1.The balancing specifıcations for the rotor system are defined. These specifıcations include the number, types (e.g., displacement, velocity, acceleration ör force) and calibration factors of the vibration sensors to be used and the specifıc sensors vvhich are most sensitive for each of the modes to be balanced; the number of balancing planes and balancing mass holes specifıcations for each plane, as well as the specifıc planes vvhich comprise each of the modal trial mass sets; the angular location of the trial masses and vvhether additional trial mass runs for reduction of measıırement errors are to be used; the numbeer of modes to be balanced and the speed at which vibration data are to be taken for each mode (these values may be changed by the operatör during the balancing procedure, if necessar). This is the only step in this balancing procedure which requires engineering-level decisions to be made. The remaihder of the steps may be handled by a skilled operatör. Thus this balancing procedure is well suited for application to production balancing, where step öne needs to be done only önce. 2.Thc rotor is run at a vcry slow spccd (to cnsure that no dynamic response is preseni), and rendings nro inken from displncement mensııring sensors only. These readings are referred to as static runout data and are due to rotor surface eccentricity and other sources of synchronous data, such as an initial bend in the rotor, vvhich are not caused by rotor mass unbalance. in general, this static runout data is independent of the rotational speed of the rotor. The true vibratory response of the rotor at other rotational speeds is found by subtracting the static runout from the measured vibration readings. If none of the sensors measure rotor displacement, this step may be by- passed. 3.The undisturbed rotor (no trial masses installed) is run, and vibration readings are taken for the mode to be balanced and for ali modes which are not to be affected by the balancing procedure, at corresponding specifıed speeds. If previously measured modal influence coefficients are available for the mode to be balanced, the balancing procedure is continued with step 6. If this is not case, but there are no uneffected modes (vvhich is usually the case when balancing the fırst rotor mode), the balancing procedure is continued with step 5. Othenvise, step 4 follovvs. 4.Individıml irial masses are installed in each of the planes to be used for balancing the mode of interest (öne at a time), and vibration readings are taken at the specifıed data speed for each of the unaffected modes. If reciprocity is used, only öne balancing plane is used here. Either öne ör two trial masses are used for each balancing plane, as specifıed in step 1. The complex modal trial mass ratios (the vector rw) are then calculated. 5.A modal trial mass set is installed (vvhich consists of a single mass, if there are no unefTeclcd modes), and vibration readings are taken at the specifled data speed for the mode to be balanced. A seeond modal Iriııl mass sel is instnlled and another set of vibration readings are taken, if so specifıed in step öne. The modal influence coefficients for the mode being balanced are then calculated, based on this modal trial mass data and the uncorrected rotor data from step 3. v6. The modal correction mass set is calculated for the mode being balanced, based on the modal influence coefficients and' the uncorrected rotor data from step 3. The operator then installs this modal correction mass set. 7. Another set of vibration readings is taken at the specified data speed for the mode being balanced to determine whether sufficient improvement of the rotor response has been achieved by this balancing procedure. The measured residual vibration is either displayed to the operator, for the. operator to evaluate, or is compared to perviously specified balancing criteria (from step 1.) If the measured residual vibration is determined to be acceptable, the balancing procedure is continued with step 9; otherwise step 8 follows, i 8. If additional improvement in the measured residual vibration is required, and if it is determined that this improvement can be achieved using the same balancing speed (using the same modal influence coefficients), the balancing procedure is continued with step 6. If, however, it is determined that a more sensitive balancing speed (nearer to the critical speed) is required, or that updated modal influence coefficients are needed (due to non-linearity in the rotor system), the balancing procedure is continued with step 5. A balancing speed nearer to the relevant critical value may also be required to isolate the mode being balanced from other modes.9. If one or more modes remain to be balanced, the mode just balanced is added to the list of unaffected modes; the next mode to be balanced is chosen, and the balancing procedure continues with step 3. 9. If one or more modes remain to be balanced, the mod just balanced is added to list of uneffected modes; the next mode to be balanced is choosen (as specified in step 1), and the balancing procedure continues with step 3. 1 0. After the last mode has been balanced, the modal influence coefficients which have been measured may be stored for later use in rebalancing the same rotor or in balancing other rotors of the same type. This completes the balancing procedure for this rotor. It should be noted that, when using measured modal influence coefficients to apply. to a class of rotors, it may be necessary to measure the modal influence coefficients for several of these rotors and statistically average these values to get a set of modal influence coefficients which can be applied with good results to a whole class of rotors. As indicated above, the UBA inherently avoids the principal disadvantages of both modal and influence coefficient balancing. However, there is a cost associated with these improvements. The UBA has some apparent advantages over both modal and influence coefficient balancing, as follows 1. No prior knowledge of the modes of vibration of the rotor to be balanced is required. The necessary information is determined emprically in the course of balancing. It is helpful if the critical speeds are known approximately. VI2. As with modal balancing, the modes can be balanced individually, while not affecting the balance of other modes that have already been balanced. 3. In particular cases where the lower modes are more lightly damped than the highter modes, the use of modal trial mass sets allows the application of substantial trial masses at the insensitive modes, while not aggravating the response of the lower modes. It should be noted that individual trial mass runs that are required for the UBA always involve modes that have already been balanced. Therefore, the vibration readings for these individual trial mass runs can always be taken at speeds close to the critical speeds so that small trial masses can be used with substantial effect, while the rotor is not prevented from going through lower critical speeds. 4 Using the UBA, a smaller number of runs is required at the highest speeds of the rotor than would be required for influence coefficient balancing with individual trial mass runs. For rotors which obey the prinnciple of reciprocity, and for which closely adjecent planes and sensors are available, the total number of individual trial mass runs can also be substantially reduced ( as discussed below ). 5. In many cases, the balancing of a particular mode begins well below the corresponding critical speed, and the response at the balancing speed is reduced to the point where, even though no additional improvement can be made (because of limited vibration pickup sensitivity), the critical speed can still not be negotiated. Using the Unified Balancing Approach, only a single trial mass run is required at the higher balancing speed. However, using influence coefficient balancing, a complete new set of trial mass runs is required at the higher speed. Thus, use of the Unified Balancing Approach substantially reduces the number of balancing runs. 6. Production balancing of a series of idendical rotors can be greatly simplified through the use of emprically determined and statistically avaraged standard modal influence coefficients. The trial mass runs become small or even nonexistence. In this study, the three principal flexible rotor balancing methods: modal balancing, influence coefficient balancing and the Unified Balancing Approach are described. Each of these methods is discussed in some detail, including analytical basis and specific implementation procedures. Then, for UBA and for influence coefficient balancing, computer programs are prepared. The program for influence coefficient balancing is applied on an experimental set-up for 4 measurement points and 3 correction planes using the influence coefficient balancing procedures, and then results are discussed in the last section. Vll 107

ÖZET Bu çalışmada değişken kalınlıklı halka plakların, radyal doğrultuda etkiyen uniform düzlem içi yükler altında, statik ve dinamik stabilitesi incelenmiştir. Bu 3maçla sonlu elemanlar metodu kullanılmıştır. Plağın sonlu eleman modeli kurulurken dairesel simetri özelliği dikkâte alınmış ve plağın sektör tipi eşparametrik elemanlardan oluşan, daire dilimi şeklindeki simetrik bir parçası modellenerek dalga yayılım tekniği ile plağın bütünü için analiz gerçekleştirilmiştir. Bu yaklaşım bilgisayar kullanımında gerek en az düzeyde hafıza gereksinimi göstermesi, gerekse hızlı sonuç alınması bakımından önemli üstünlüklere sahiptir. Eleman kalın ve ince plaklar için kullanıma uygundur. Halka plağın kalınlığı, yarıçap doğrultusunda h=hm3X(r/r0)+5 bağıntısına göre dış kenara doğru kalınlığı artarak veya h=hmax(r/ri)-*' bağıntısına göre kalınlığı azalarak değişmektedir. Yapılan çalışmada değişken kalınlıklı halka plakların statik stabilitesi incelenirken, kalınlık değişi miride ki bu iki farklı durumun, katsayısı ndaki artışın, sınır şartlarının ve delik büyüklüğünün, kritik burkulma yüküne ve burkulma modlarına etkisi araştırılmıştır. Dinamik stabilite analizinde bu faktörlerin ve sabit yükün dinamik kararsızlık bölgelerine etkisi incelenmiştir. Kullanılan modelin doğruluğunun kanıtlanabil mesi için elemanın geometrisi değiştirilerek kalınlığı lineer olarak değişen halka plakların serbest titreşimleri ve statik stabilitesi ile ilgili elde edilen sonuçlar, diğer araştı rmacıların çalışmalarıyla mukayese edilmiş ve bulunan değerlerin birbirine yakın olduğu görülmüştür. Geliştirilen bilgisayar programıyla karmaşık kalınlık geometrisine sahip dairesel plakların statik ve dinamik stabilite karakteristiklerini incelemek mümkündür. 3d STATIC AND DYNAMIC STABILITY OF VARIABLE THICKNESS ANNULAR PLATES SUMMARY Recenty, lightweight stuctures have been extensively used as fundamental structural elements in various industrial fields. Therefore the buckling and parametric resonance analysis of plates have increased in practical importance. In this study the static and dynamic stability of annular plates subject to in- plane loads is investigated. It is a well known fact that when in- plane load reaches to a critical value, the plate looses its orijinal shape and out-of- plane deformations take place. This is called static stability. The load at which buckling occurs is thus a desing criterion for the plate. The other well known fact is that a plate subject to in- plane forces generally experience forced in- plane vibrations and when exciting frequencies overlap with the natural frequencies of the plate, resonance will occur. Apart from this phenomena, for certain values of exciting frequencies an entirely different type resonance will occur in transverse direction and annular plate is said to be dynamically unstable. This type of resonance is also called parametric resonance. In the dynamic stability problem the spectrum of values or the parameters causing this unstable motion is investigated and the regions of the dynamic instability is determined. Constructions of the light structures in the present day technology makes it essential to predict these dynamically unstable regions for different structural elements In the present study variable thickness sector type isoparametric annular plate element is employed to investigate the static and dynamic stability problem. All the applications presented in this study are analyzed using the wave propagation tecniques. The advantage of this method is that only a single sub-structure is represented by a finite element model. The thickness is either increased or decreased in outward radial direction by the equations h=hmax( r/r0)+x or h=nmax(r/n)-* respectively. The effect of thickness variation, boundary conditions, hole size on the critical buckling load and buckling mode is investigated. The instability regions of annular plates with parobolic thickness are determined for wide range of exciting frequencies using Bolotin Method's with effect of static forces taken into account. The choice of the element in sector form enables the structural analysis to solve dynamic stability of annular plates with sectoral cut-outs. The sector element developed İ3 based on the Mirdlin plate theory. The element, which is shown in Figure 1, has 8 nodes, 24 degress of freedom and has variable thickness. The degress of freedom per node are the transverse displacement, and the radial and angular slopes. Numerical integration is used to evaluate the kinetic and potential energies of the element. The element yields good results for both thin and thick plate. The sector element is referred to a set of cyli ndrical coordi nates, r, 8 and z, and a set of non-dimensional curvilinear coordinates tj, Ç, Ç with the following relationship between the two spaces [26]: Xu! = 0.5q0.5(-iî+ıj2)h1 +(l-r2)]i2 +0.5(r+r2)li3 r=0.5(r3-r1)+r1+0.5(r3-r1)n 8 = -6% 6T 5- T] O: Displacerr^nt rı«te o : Geometric node (D Figure 1. Isoparametric sector element. Let Ni(^ rj), i=l, 2,, 8 be the tvo dimensional parabolic stepe functions of the isoparametric element. Also let Vj, §/, arid % be the transverse displacement, radial slope and angular slope of the element nodes, respectively. With reference to the Mindlin's plate theory, the displacement field within the element is approximated by [26] 0 sNiCtn) 0 O O -zNi&ri) Ni(tn) O O. ^i (2) where u, v, v are the deformations in the r, e, z directions respectively and shape functions of the isoparametric element are [25] Hi(tn) = /v +&»)fl +no)«o +no - *) i= 1,3,5,7 Ni(tn)=i(ı-^)(i+îio) Ni(tn)=;(i+^o)(ı-n2) i =2,6 i =4,8 (3) XillThe basic assumptions of thick plate theory as presented by Mindlin are 1 25]: 1: The deflection of the mid-surface of the plate is small in comparion with the plate thickness. 2: The transverse normal stress is negligible. 3: Normals to the mid-surface of the plate before deformation remain straight, but not necessarily normal to the midsurface, after deformation. With assumption 1 and 2, strain energy of the plate is given by [26] U=^Jv(e]T[D][c]dv (4) where V is the volume and, in cylindrical coordinates. [c]T = [err e* 2t# 2e` 2^] (5) and [D] = (6) The strain-displacement relations are Hz 2C T2 2e, 9zJ i « i-i n O 0 3_ 3r 12. r 36 J Using assumption 3, the displacement can be expressed in the form (7) (8) where y, $ are the rotations about the r and 8 axes and w is the normal displacement of the middle plain. Equations (4) to (8) together give the strain energy. XIVThe kinetic energy is given by [24] lf f'.2.2.2^ T=- pi +y +v dv (9) where u, v, w, are the velocities. The additional strain energy in bending due to in- plane stress trr, Tre, Tee is 126] lf f (b&f 3dl3d fl3df dv (10) Substituting equations (1) arid (2) into equations (4), (9) and (10) give the stiffnes, mass and stability (geometric stiffness) matrices respectively. Numerical integration using (2x2), (3x3), arid (2x2) Gaussian meshes is used. Figure 2. The in- plane loads acting on the annular plate. The in-plane loads are assumed to be acting on the annular plate along the inner and outer edges as shown in Figure 2. Under the uniform loads per unit length of P^ snd Pout respectively, the force distribution in the radial and angular directions are given by [27] XVr2r2 1 Pr2-Pjî n r2-r2(W ^r2 rg-r? Pf«=0 When a flat plate is subjected to periodic in- plane forces the governing matrix equation of motion is [MJ{Y)+[Kj{t}-[Kj{q} = 0 (12) where [M] = Mass matrices of total sturucture, [K*] = Elastic stiffness matrix of the total stucture, [Kg] = Stability (geometric stiffness) matrix of the total stucture which is a function of the axial load P. It is customary to express the periodic radial force as follows PW = Po+ptCOSCDt }o+ I {a^sin- +{^003 - (17) where {a^} arid {ft^} are the vectors independent of time. Substitution of equations ( 1 6) and ( 1 7) into equation ( 1 5) leads to eigenvalue system for the dynamic stability boundary. For the solutions with period 2T ^]-^*[K«s]±pP*[K«t]-TtMl = 0 for the solutions period T [ig-c1 {q}` ^- T~- ^ (q>. Wn-i Figure 3. Rotationary periodic structure and an identical substructure XViilThe symmetry of the problem makes it possible to apply the wave propagation technique ifa repetitive section of the plate as shown in Figure 4. is considered then it is possible to classify three groups of degrees of freedom {qi_}, {qi} and {qR} respectively. If the static and dynamic components of the in- plane force P acts in the same manner, the stability boundaries can be written as follows for repetitive structure [M]{q'} +[[K]- (oP*+pP*cos wt)[S]]{q} = {F} (27) The first approximation of the dynamic unstabity regions for the repetitive structure can be determined from [[K]-^±ipjp*[Sl-^[M]j{ii}= {F} (28) 1 Element R/4 Element Figure 4. Repetitive structure of the annular plate with 4 along the radial and 1 element along the annular direction For a repetitive structure degress of freedom and the forces acting will be expressed in vector form as follows. {«} = Ql «E (F} = 1*. A. (29) I n a fi nite cyclic structure duri rig a conti nuous wave propagation { F}=0. Also the relation between the two repetitive substructures is such that [37] {fcl-^Cfc} (30) ow- mi (31) xixwhere >ı is complex wave propagation constant. If the equations (28) and (30) are used together with the {F}=0 constraint then the dynamic stability equation for a repetitive structure becomes. [Kr* 3-?g si}M m the submatriees [Kr] and [K1] in their explicit form are as follows. 1X00] = Kll +krr +Ia3 Figure 5. Thickness variation of the annular plate for positive A S'alues