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Kesiti kademeli değişen plakların titreşimi

Kesiti kademeli değişen plakların titreşimi

##### Dosyalar

##### Tarih

1994

##### Yazarlar

Yay, Gökay

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu tezde kesiti kademeli değişen plakların titreşimi, karmaşık ve zor problemleri kabuledilebilir bir yaklaşıklıkla çözebilen ve bir sayısal analiz yöntemi olan sonlu elemanlar metodu kullanılarak bulunmuş ve hesaplanması oldukça uzun zaman alan bu problemlerin çözümünde sonlu elemanlar metodunu kullanan ANSYS paket programından yararlanılmıştır. öncelikle yapılan bu çalışmada, plak üzerindeki süreksizlik bölgeleri ; gerek büyüklük, gerek konum ve gerekse malzeme yönünden değişken tutularak genelleme yoluna gidilmiş ve konstrüktör mühendise, efektif bilgisayar kullanımı ile arzuladığı dinamik karaktere ulaşma ve optimizasyon yolu açılmak istenmiştir. Aynı zamanda tezde ANSYS programı hakkında ayrıntılı bilgi verilmesiyle kahnmayıp sonlu elemanlar yönteminin teorisi de anlatılmaya çalışılmıştır. Bu yüzden tezin 2. bölümün de sonlu elemanlar yöntemi hakkında genel bir bilgi verildikten sonra yöntemin temel ifadelerine değinilmiştir. 3. bölümde yöntem, plak eğilme durumu için ele alınarak gerekli bağıntılar oluşturulmuştur. 4. bölümde yüklerin eşdeğer noktasal gösterimi üzerinde durulmuştur. 5, bölümde ANSYS programı hakkında genel bir bilgi verildikten sonra, yapısı ve içeriği tanıtılmıştır. 6. bölümde titreşim karakteristiklerinin saptanmasında yardımcı olan ve bir yapının veya parçanın doğal frekanslanyla birlikte mod şekillerini saptamakta kullanılan aynı zamanda ANSYS programında bir çözüm yöntemi olarak yer alan Mod-Frekans Analizi (Modal Analiz) hakkında bilgi verilip teorisi açıklanmıştır. 7. bölümde ANSYS programı yardımıyla kesiti kademeü değişen plak titreşimlerinin elde edilmesi anlatılmış, aynı zamanda bu bölümde farklı durumlar için elde edilen sonuçlar tablolar ve grafikler halinde gösterildikten sonra plağın birinci mod şekli, her bir grafikteki eğriler üzerinden okunan değişik değerler için ayrı ayrı çizilmiştir. 8. ve son bölümde ise sonuçların gözden geçirilip değerlendirilmesi yapılmış ve öneriler sunulmuştur.

Finite elements are applied to a broad range of engineering problems, but they are most widely used for solid and structural calculations, this being tare in which commercial finite programs are well established in engineering practice. Such programs are available at all levels, often at quite a modest cost. As a consequence, the use of finite element models over quite a broad spectrum of design applications is now common, supplementing and refining the 'strength of materials' calculations which have dominated such analysis in the past and circumventing, in many instances, the provisions of conservatives design codes. The requirement that engineering graduates be familiar at a basic level with the finite element concept, so that they can use such programs with an awareness of their capabilities and limitations, is becoming an important component of engineering training. The term 'finite element' appears to have been coined by Clough in article published in 1960 [1]. It was used to describe a computational approach to the analysis of elastic membranes in which the continuum was divided into discrete number of small but finite subregions or 'elements'. The idea itself was not new. In fact, the notion of dividing a continuum into finite pieces had been suggested by Courant in 1943. The practical implementation of such an approach only materialized in the mid-1950s, however, with the advent of digital computation. It was Turner, Clough and others who then combined the idea of discrete elements with the 'stiffness' approach to matrix structural analysis, to produce a systematic procedure which later became known as the finite element method. An interesting account of these early developments is to be found in Clough's own commentary on the period. The popularity of the method grew rapidly in the years that followed. Within ten years of the invention of the term, more man one thousand articles dealing with finite elements had been published in the scientific literature. Two decades later, the number of articles listed in the COMPENDEX -PLUS engineering index approaches 50 000. This phenomenal rate of growth is a reflection of the degree to which the finite element concept has complemented the emerging capabilities of the digital computer. Most importantly, it has lent itself to development of multipurpose programs. In the late sixties, this gave rise to the first, commercial, finite element computer codes. These were programs, capable of solving different physical problems through changes to the input data rather than to the code itself. Such programs have expanded and proliferated in the years since and now include such xxu industry standards as NASTRAN, ABAQUS, ADINA, ANSYS, PAFEC, SAP, MARC AND EASE, to name but a few. More recently they have been joined by a new generation of PC or workstation orientated codes such as MSC / PAL, ANSYS- PC, SAP90, COSMOS / M, ALGOR and so on. Their application to practical problems of stress analysis constitutes one of the major advances in mechanical and structural design of the present era. Also contributing to rapid growth of method from the late sixties onwards was the realization that it could be applied to problems other than those of solid and structural analysis. The first suggestion that this might be the case came with work of Zienkiewicz and others in demonstrating that the method could be used for field problems involving Laplace's and Poisson's equations (steady state thermal conduction, for example, or potential flow of an inviscid fluid). These developments were accompanied by the realization, also in the early sixties, that the Galerkin approach and other 'weighted residuals' techniques could be used as a theoretical basis for applying finite elements to virtually any problem which could be expressed in terms of partial differential equations. As a consequence, current areas of application now include fluid mechanics, electromagnetic theory and so on. In addition to single applications, the method also lends itself to the solution of coupled problems involving two or more constituents: the seismic interaction of dams and reservoirs, for example, or the interaction of waves and offshore structures, acoustic-structural coupling, elasto-hydrodynamic lubrication and so on. The finite element method was well advanced by the late sixties in its capacity to solve practical two and three-dimensional problems in linear elasticity. However, it still required substantial computation - in the context of that commonly available at the time - and was accessible only to those engaged in large-scale industrial or institutional research. One of the most significant recent developments in the application of the method has arisen not so much from any specific advance in the methodology itself (new element formulations, solution algorithms etc.) but simply from the steady decrease in the effective cost of computating. This has brought finite element computation within the mainstream of engineering practice by making it accessible, for routine design analysis, to engineers with relatively modest computational facilities at their disposal. Many commercial programs now operate effectively on engineering workstation or personal computers and perform substantial analyses for a small fraction of the cost associated with such calculations a decade or so ago. This is clearly a trend which has a long way to run and one which will make the method even more accessible to the engineering profession in years to come. A side effect of the same phenomenon, the inexorable decrease in the real cost of computing, has been the use of computer graphics in the engineering environment. This in turn has driven the growth in computer aided design and manufacture (CAD/CAM). Practical implementation of the finite element method has been encouraged by such developments since the generation of finite element models can readily be integrated with the geometric modeling techniques which lie at the heart of the CAD/CAM revolution. There are consequently few CAD systems today which do xxui not provide a facility for the interchange of geometrical information with compatible finite element programs. This integration between geometric design (using a CAD system) and quantitative analysis (using a finite element code) has rendered the preparation of finite element data and the display of the results of finite element analysis, if not trivial, at least an order of magnitude less time consuming than was the case with the finite element programs of the sixties and seventies. In this thesis, ANSYS Finite Element Analysis Program was used to determine vibrations of the plate with stepped thickness. Swanson Analysis Systems, Inc. (SASI) was founded in 1970 by Dr. John Swanson to develop computer-based technology for engineering analysis. The ANSYS finite element program is the major product of SASI. It was first developed for use by the power generation industry but now meets the finite element analysis (FEA) requirements of most industries, from automotive and electronics to aerospace and chemical. The ANSYS program is a large-scale, general purpose software program recognized around the world for its many capabilities. From a handful of users in 1970, the program is now installed at more than 2000 commercial sites, more than 5500 universities, and is used by thousands of engineers worldwide. The ANSYS program allows engineers to construct computer models of structures or machine components, apply operating loads to them, and study the physical responses such as stress levels or temperature distributions. This process gives design engineering groups and firms an alternative to the multiple-prototype building, testing, and rebuilding cycle, as well as expensive product overdesign or field retrofits for underdesign. In some environments, prototype testing is undesirable or impossible. The ANSYS program has been used in several cases of this type, including biomedical aplications such as hip implants and intraocular lenses. Other representative applica tion range from a heavy-duty crane hook, to anintegrated circuit chip, to the bit- holding system of a continuous coal-mining machine. With the ANSYS program, engineers can pinpoint a potential design defect or determine the optimum geometry before a design is in production or use. For example, an engineering consulting firm applied the ANSYS design optimization capability to an elastic disk used in the clutch assembly of an automobile engine. The goals were to extend fatigue life and to achieve a uniform stress distribution in the disk, while staying within geometric and mechanical interface constraints. Trough its design optimization procedure, the program performed a series of solutions on a parametrically defined solid model of the disk, automatically adjusting selected dimensions at each solution until the optimum shape was achieved. Results showed that the difference between maximum and niinimum von Mises stress within the disk was reduced by 27 percent, maximum stress was reduced by 28 percent, and fatigue life was increased by 35 percent. ANSYS design optimization enabled the analysts to xxiv reduce the number of costly prototypes, tailor rigidity and flexibility to meet objectives, and find the proper balance in geometric modifications. Competitive companies look for ways to produce the highest quality product at the lowest cost. ANSYS FEA can help significantly by reducing design and manufacturing costc and by giving engineers added confidence in the products they design. FEA is most effective when used at the conceptual design stage, as well as later on in the manufacturing process to verify the final design before prototyping. The ANSYS program operates on super computer, mainframes, minicomputers, workstations, and 386- and 486- based PCs, and is specifically tuned to run efficiently on each system. SASI continually works with new hardware platforms and operating systems on which the ANSYS program can be run. This program include many kind of analysis. One of them is structural analysis that is probably the most common application of the finite element method. The term structural (or structure) implies not only civil engineering structures such as bridges and buildings, but also mechanical components such as pistons, machine parts, tools, and the like. Many types of structural analyses are available in the ANSYS program, and are given below. The primary unknowns (nodal degrees of freedom) calculated in a structural analysis are displacements. Other quantities, such as strains, stresses, and reaction forces, are then derived from the nodal displacements. The following types of structural analyses are possible :. Static Analysis. Modal Analysis. Harmonic Analysis. Transient Dynamic Analysis. Spectrum Analysis. Buckling Analysis In addition to the above analysis types, several special-purpose features are available, such as fracture mechanics, composites and fatigue. The majority of ANSYS element types are structural elements, ranging from simple spars and beams to more complex layered shells and large strain solids. As it is known, vibrations occur everywhere - in cars, airplanes, buildings, machine tools, home appliances, nuclear piping system, bridges, even in satellites in outer space. In most cases, vibrations are not desirable, because they can cause damage. It is important, therefore, to know the vibration characteristics of a structure or a machine component while it is being designed. Modal analysis helps in determining vibration characteristics : it calculates the natural frequencies and mode shapes of a structure. It is also used as a starting point for other, more detailed xxv dynamic analyses, such as a transient dynamic analysis or a harmonic response analysis. Modal analysis is used to determine the natural frequencies and mode shapes of a structure or component. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. They are also required if you want to do a spectrum analysis or a mode superposition harmonic or transient analysis. Modal analysis in the ANYSY program is a linear analysis. Any non- linearities, such as plasticity and contact (gap) elements, will be ignored even if they are defined. You can choose from four mode extraction methods - reduced, subspace, unsymmetric, and damped. The damped method allows you to include damping in the structure. Modal analysis can also be performed on a prestressed structure, such as a spinning turbine blade. Another useful feature is modal cyclic symmetry, which allows you to review the mode shapes of a cyclically symmetric structure by modeling just a sector of it. The procedure for a modal analysis consists of four main steps:. Build model. Apply loads and obtain the solution Expand the modes Review the results In this study, firsts of all, discontinuity regions over the plate were investigated by varying parameters such as quantity and material for several locations. The main aim of this study is to show the constructor to obtain the required dynamic characteristics of a structure by using the computer effectively. Moreover, in the thesis not only the ANSYS program was explained, but also the finite element theory that will help any user to understand structure of program, was able to be presented. Therefore, in chapter 2 after giving general information about finite element method, the basic expressions of method for structural applications are presented.. In chapter 3, the method was considered for plate bending applications, and the basic expressions were constituted. In chapter 4, equivalent nodal representation of loads was explained. In chapter 5, structure and content of ANSYS program were introduced after giving a general information on it. In chapter 6, a general knowledge about Mode-Frequency Analysis (Modal Analysis) included in ANSYS program as a solving method was given, and its theory was able to be explained. This analysis specified above helps in determining vibration characteristics, and also is used to determine the natural frequencies and mode-shape of structures or machine components. xxvi In chapter 7, obtaining vibrations of plates with stepped thickness by using ANS YS program was explained and also obtained result in different sates were shown as charts and graphics, after then first mode-shape of plate for various values that had been read on the curves in each graphics, was drawn separately. In chapter 8 that is the last chapter of this thesis, results were reviewed and generally discussed, and suggestions were presented.

Finite elements are applied to a broad range of engineering problems, but they are most widely used for solid and structural calculations, this being tare in which commercial finite programs are well established in engineering practice. Such programs are available at all levels, often at quite a modest cost. As a consequence, the use of finite element models over quite a broad spectrum of design applications is now common, supplementing and refining the 'strength of materials' calculations which have dominated such analysis in the past and circumventing, in many instances, the provisions of conservatives design codes. The requirement that engineering graduates be familiar at a basic level with the finite element concept, so that they can use such programs with an awareness of their capabilities and limitations, is becoming an important component of engineering training. The term 'finite element' appears to have been coined by Clough in article published in 1960 [1]. It was used to describe a computational approach to the analysis of elastic membranes in which the continuum was divided into discrete number of small but finite subregions or 'elements'. The idea itself was not new. In fact, the notion of dividing a continuum into finite pieces had been suggested by Courant in 1943. The practical implementation of such an approach only materialized in the mid-1950s, however, with the advent of digital computation. It was Turner, Clough and others who then combined the idea of discrete elements with the 'stiffness' approach to matrix structural analysis, to produce a systematic procedure which later became known as the finite element method. An interesting account of these early developments is to be found in Clough's own commentary on the period. The popularity of the method grew rapidly in the years that followed. Within ten years of the invention of the term, more man one thousand articles dealing with finite elements had been published in the scientific literature. Two decades later, the number of articles listed in the COMPENDEX -PLUS engineering index approaches 50 000. This phenomenal rate of growth is a reflection of the degree to which the finite element concept has complemented the emerging capabilities of the digital computer. Most importantly, it has lent itself to development of multipurpose programs. In the late sixties, this gave rise to the first, commercial, finite element computer codes. These were programs, capable of solving different physical problems through changes to the input data rather than to the code itself. Such programs have expanded and proliferated in the years since and now include such xxu industry standards as NASTRAN, ABAQUS, ADINA, ANSYS, PAFEC, SAP, MARC AND EASE, to name but a few. More recently they have been joined by a new generation of PC or workstation orientated codes such as MSC / PAL, ANSYS- PC, SAP90, COSMOS / M, ALGOR and so on. Their application to practical problems of stress analysis constitutes one of the major advances in mechanical and structural design of the present era. Also contributing to rapid growth of method from the late sixties onwards was the realization that it could be applied to problems other than those of solid and structural analysis. The first suggestion that this might be the case came with work of Zienkiewicz and others in demonstrating that the method could be used for field problems involving Laplace's and Poisson's equations (steady state thermal conduction, for example, or potential flow of an inviscid fluid). These developments were accompanied by the realization, also in the early sixties, that the Galerkin approach and other 'weighted residuals' techniques could be used as a theoretical basis for applying finite elements to virtually any problem which could be expressed in terms of partial differential equations. As a consequence, current areas of application now include fluid mechanics, electromagnetic theory and so on. In addition to single applications, the method also lends itself to the solution of coupled problems involving two or more constituents: the seismic interaction of dams and reservoirs, for example, or the interaction of waves and offshore structures, acoustic-structural coupling, elasto-hydrodynamic lubrication and so on. The finite element method was well advanced by the late sixties in its capacity to solve practical two and three-dimensional problems in linear elasticity. However, it still required substantial computation - in the context of that commonly available at the time - and was accessible only to those engaged in large-scale industrial or institutional research. One of the most significant recent developments in the application of the method has arisen not so much from any specific advance in the methodology itself (new element formulations, solution algorithms etc.) but simply from the steady decrease in the effective cost of computating. This has brought finite element computation within the mainstream of engineering practice by making it accessible, for routine design analysis, to engineers with relatively modest computational facilities at their disposal. Many commercial programs now operate effectively on engineering workstation or personal computers and perform substantial analyses for a small fraction of the cost associated with such calculations a decade or so ago. This is clearly a trend which has a long way to run and one which will make the method even more accessible to the engineering profession in years to come. A side effect of the same phenomenon, the inexorable decrease in the real cost of computing, has been the use of computer graphics in the engineering environment. This in turn has driven the growth in computer aided design and manufacture (CAD/CAM). Practical implementation of the finite element method has been encouraged by such developments since the generation of finite element models can readily be integrated with the geometric modeling techniques which lie at the heart of the CAD/CAM revolution. There are consequently few CAD systems today which do xxui not provide a facility for the interchange of geometrical information with compatible finite element programs. This integration between geometric design (using a CAD system) and quantitative analysis (using a finite element code) has rendered the preparation of finite element data and the display of the results of finite element analysis, if not trivial, at least an order of magnitude less time consuming than was the case with the finite element programs of the sixties and seventies. In this thesis, ANSYS Finite Element Analysis Program was used to determine vibrations of the plate with stepped thickness. Swanson Analysis Systems, Inc. (SASI) was founded in 1970 by Dr. John Swanson to develop computer-based technology for engineering analysis. The ANSYS finite element program is the major product of SASI. It was first developed for use by the power generation industry but now meets the finite element analysis (FEA) requirements of most industries, from automotive and electronics to aerospace and chemical. The ANSYS program is a large-scale, general purpose software program recognized around the world for its many capabilities. From a handful of users in 1970, the program is now installed at more than 2000 commercial sites, more than 5500 universities, and is used by thousands of engineers worldwide. The ANSYS program allows engineers to construct computer models of structures or machine components, apply operating loads to them, and study the physical responses such as stress levels or temperature distributions. This process gives design engineering groups and firms an alternative to the multiple-prototype building, testing, and rebuilding cycle, as well as expensive product overdesign or field retrofits for underdesign. In some environments, prototype testing is undesirable or impossible. The ANSYS program has been used in several cases of this type, including biomedical aplications such as hip implants and intraocular lenses. Other representative applica tion range from a heavy-duty crane hook, to anintegrated circuit chip, to the bit- holding system of a continuous coal-mining machine. With the ANSYS program, engineers can pinpoint a potential design defect or determine the optimum geometry before a design is in production or use. For example, an engineering consulting firm applied the ANSYS design optimization capability to an elastic disk used in the clutch assembly of an automobile engine. The goals were to extend fatigue life and to achieve a uniform stress distribution in the disk, while staying within geometric and mechanical interface constraints. Trough its design optimization procedure, the program performed a series of solutions on a parametrically defined solid model of the disk, automatically adjusting selected dimensions at each solution until the optimum shape was achieved. Results showed that the difference between maximum and niinimum von Mises stress within the disk was reduced by 27 percent, maximum stress was reduced by 28 percent, and fatigue life was increased by 35 percent. ANSYS design optimization enabled the analysts to xxiv reduce the number of costly prototypes, tailor rigidity and flexibility to meet objectives, and find the proper balance in geometric modifications. Competitive companies look for ways to produce the highest quality product at the lowest cost. ANSYS FEA can help significantly by reducing design and manufacturing costc and by giving engineers added confidence in the products they design. FEA is most effective when used at the conceptual design stage, as well as later on in the manufacturing process to verify the final design before prototyping. The ANSYS program operates on super computer, mainframes, minicomputers, workstations, and 386- and 486- based PCs, and is specifically tuned to run efficiently on each system. SASI continually works with new hardware platforms and operating systems on which the ANSYS program can be run. This program include many kind of analysis. One of them is structural analysis that is probably the most common application of the finite element method. The term structural (or structure) implies not only civil engineering structures such as bridges and buildings, but also mechanical components such as pistons, machine parts, tools, and the like. Many types of structural analyses are available in the ANSYS program, and are given below. The primary unknowns (nodal degrees of freedom) calculated in a structural analysis are displacements. Other quantities, such as strains, stresses, and reaction forces, are then derived from the nodal displacements. The following types of structural analyses are possible :. Static Analysis. Modal Analysis. Harmonic Analysis. Transient Dynamic Analysis. Spectrum Analysis. Buckling Analysis In addition to the above analysis types, several special-purpose features are available, such as fracture mechanics, composites and fatigue. The majority of ANSYS element types are structural elements, ranging from simple spars and beams to more complex layered shells and large strain solids. As it is known, vibrations occur everywhere - in cars, airplanes, buildings, machine tools, home appliances, nuclear piping system, bridges, even in satellites in outer space. In most cases, vibrations are not desirable, because they can cause damage. It is important, therefore, to know the vibration characteristics of a structure or a machine component while it is being designed. Modal analysis helps in determining vibration characteristics : it calculates the natural frequencies and mode shapes of a structure. It is also used as a starting point for other, more detailed xxv dynamic analyses, such as a transient dynamic analysis or a harmonic response analysis. Modal analysis is used to determine the natural frequencies and mode shapes of a structure or component. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. They are also required if you want to do a spectrum analysis or a mode superposition harmonic or transient analysis. Modal analysis in the ANYSY program is a linear analysis. Any non- linearities, such as plasticity and contact (gap) elements, will be ignored even if they are defined. You can choose from four mode extraction methods - reduced, subspace, unsymmetric, and damped. The damped method allows you to include damping in the structure. Modal analysis can also be performed on a prestressed structure, such as a spinning turbine blade. Another useful feature is modal cyclic symmetry, which allows you to review the mode shapes of a cyclically symmetric structure by modeling just a sector of it. The procedure for a modal analysis consists of four main steps:. Build model. Apply loads and obtain the solution Expand the modes Review the results In this study, firsts of all, discontinuity regions over the plate were investigated by varying parameters such as quantity and material for several locations. The main aim of this study is to show the constructor to obtain the required dynamic characteristics of a structure by using the computer effectively. Moreover, in the thesis not only the ANSYS program was explained, but also the finite element theory that will help any user to understand structure of program, was able to be presented. Therefore, in chapter 2 after giving general information about finite element method, the basic expressions of method for structural applications are presented.. In chapter 3, the method was considered for plate bending applications, and the basic expressions were constituted. In chapter 4, equivalent nodal representation of loads was explained. In chapter 5, structure and content of ANSYS program were introduced after giving a general information on it. In chapter 6, a general knowledge about Mode-Frequency Analysis (Modal Analysis) included in ANSYS program as a solving method was given, and its theory was able to be explained. This analysis specified above helps in determining vibration characteristics, and also is used to determine the natural frequencies and mode-shape of structures or machine components. xxvi In chapter 7, obtaining vibrations of plates with stepped thickness by using ANS YS program was explained and also obtained result in different sates were shown as charts and graphics, after then first mode-shape of plate for various values that had been read on the curves in each graphics, was drawn separately. In chapter 8 that is the last chapter of this thesis, results were reviewed and generally discussed, and suggestions were presented.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1994

##### Anahtar kelimeler

Bilgisayar programları,
Levhalar,
Titreşim,
Computer programs,
Plates,
Vibration