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Konik Sürtünme Diskli Varyatörün Sonlu Elemanlar Metodu İle Analizi

Konik Sürtünme Diskli Varyatörün Sonlu Elemanlar Metodu İle Analizi

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##### Dosyalar

##### Tarih

1998

##### Yazarlar

Yürük, Serkan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Karşılaştığımız bir çok mühendislik probleminin çözümü için analitik çözümler elde etmek çoğu zaman mümkün olmamaktadır. Analitik çözümler bir problemi basitleştirerek sonuca gitmeye çalışırlar. Mekanik tasarımda karşımıza çıkan ve matematiksel olarak tanımlanması kolay olmayan şekiller, karmaşık malzeme özellikleri, değişik sınır koşullan vb., gibi durumlar içeren problemler için yaklaşık fakat kabul edilebilir bir çözüme ihtiyaç vardır, bu da sayısal analiz yöntemleriyle mümkün olmaktadır. Bir ürünün tasarım aşamasından, üretim ve test aşamalarına kadar her alanda kullanılan bir çok sayısal yöntem mevcuttur. Sonlu elemanlar metodu, sonlu farklar metodu ve sınır elemanlar metodu mühendislik alanında en sık karşılaşılan sayısal analiz yöntemleridir. Sonlu Elemanlar Metodu kırk yılı aşkın bir süredir differansiyel denklemlerin çözümü için bilim ve mühendislik alanında kullanılan nümerik bir yöntemdir. Bu metod çok geniş sahalarda başarılı bir şekilde kullanılmakladır. Sonlu Elemanlar Metodunun yapısal analiz (Statik, Modal, Frekans, Harmonik, Dinamik, Tayf), termal analiz, elektromanyetik analiz, akışkanlar mekaniği ve akustik olmak üzere çok geniş bir yelpazede uygulama alanı bulması ve makul sonuçlar vermesi metoda olan ilgiyi arttırmış ve mühendislik uygulamalarında yaygın olarak kullanılmasına neden olmuştur. Burada nümerik yöntemlerle elde edilen çözümlerin gerçek çözümlerler için bir yaklaşım olduğu unutulmamalıdır. Gerçek çözümler ile yaklaşık çözümler arasındaki farkın derecesi çözülecek problemin yapısına ve ele alış tarzına bağlıdır. Bir probleme, sayısal analiz yöntemlerini uygulayıp, gerçeğe yakın sonuçlar elde edebilmek için bir ön çalışmanın yapılması gereklidir. Analiz yapan kişinin sayısal analiz yöntemlerinin beraberinde getirdiği kıstasları, incelenen yapının maruz kaldığı gerilme tiplerini ve sınır şartlarını çok iyi bilmesi gereklidir. Daha önce yapılan benzer çalışmaların ve bunların sonuçlarının incelenmesi bu konuda analiz yapan kişiye yardımcı olabilir. Yöntemin tam olarak anlaşılması, problemin çözümünde yapılması gerekli olan kabullerin doğru bir şekilde uygulanmasını sağlar. Sayısal bir yöntem olan ve çok yaygın olarak kullanılan sonlu elemanlar metodu çok basit problemleri çözmek için bile belir li bir programlama diline ihtiyaç duyar. Sonlu elemanlar metodunda elle çözülebilecek problemler sınırlı olduğundan daha komplike problemlerin çözümü için mutlaka bilgisayarlara ihtiyaç vardır. Bilgisayar teknolojisindeki hızlı gelişme, bilgisayarların kapasitelerinde bir artış, maliyetlerde de bir düşüş getirmiştir. Bu durum sonlu elemanlar metodunun gelişmesinde ve yaygınlaşmasında en büyük faktörlerden biri olmuştur. Günümüzde sonlu elemanlar yöntemi ile çözüm için yazılmış bir çok program mevcuttur. Bunların içinde en yaygın olarak kullanılanlardan biri de ANSYS (Analysis System) programıdır. İleriki bölümlerde sonlu elemanlar yöntemi ve ANSYS paket programı hakkında daha detaylı bilgi verilmiş ve ANSYS programı kullanılarak konik sürtünme diskli varyatörün analizi yapılmıştır.

The solution method that the engineer has available for the design of structural system can be divided into two groups, analytical and numerical. Analytical solutions to the engineering and science problems are possible only if the geometry and boundary condition of the problem are simple. In a lot of engineering problem, generally machine and structural systems consist of various structural element such as beams, links, plates, and shells or a combination of the these. Their overall geometry becomes extremely complicated and cannot be represented by a single mathematical expression. The solution problem with complex geometry, boundary condition and loading become possible only through numerical approximations. In spite of their versatility, however, numerical solutions are only approximation to the actual solution. The degree of correlation between these approximations and actual solution depend on complication of problem. Analytical solution also provides a means of checking numerical result for accuracy and convergence. In recent years, numerical methods become popular due to rapid advancements in computer technology and its availability to engineer. In these methods, Finite Element Method is the most popular method in engineering science. The roots of the method can be traced back to three separate research group; engineers, physicists and applied mathematicians. Although in principle published already, the finite element method obtained its real impetus from the independent developments carried out by engineers. Important original contributions appeared in papers by Turner and Argyris and Keksey. The name " finite element method " was presented for plane stress analysis. Since then, a large amount of research has been devoted to the technique, and a very large number of publications on the finite elementjnethod are available at presents The rapid development of the finite element method as an analysis tool essentially began with the advent of the electronic digital computer. Using the finite element method on digital computer, the numerical solution of a complex structural problem that is including algebra equation becomes possible and very effective. The basic concept of the finite element method is divide a solution region into so called finite element of several convenient shapes such as truss, triangles or rectangular, brick, tetrahedran, ring, etc. (Truss in one dimensional cases triangles or rectangularin two dimensional cases and tetrahedral or brick in three dimensional cases) The unknown field variable is expressed in term of values of the field variables at specified points called nodes. Nodes connect elements to the next elements. Nodes usually lie on the element boundaries. But some elements may have a few interior nodes. The more number of elements used to model, the XUl problem, and the better approximation to the solution obtained. Note that the data storage requirement an solution time rapidly with an increase in the number of element. The finite element method of analysis has been recognised as apower tool and has been effectively used to obtain solution in structural analysis (Static, Modal, Harmonic, Transient, Dynamic, Spectrum etc.), thermal analysis and fluid dynamic etc. One of the chief features of the finite element method is the way curved boundaries can be realistically treaded by using higher order isoparametric elements. On the other hand, it has ability to tread anisotrapic materials discontinuous and non linear structural problems. In spite of above advantages, three are some disadvantages finite element method. It is that in order to solve even very simple problems a computational approach is needed. Also solution of complex problem is needed large high-speed digital computer. Sometimes solution time goes up further if the problem is non linear. An idealized analysis procedure will bring all steps into proper order. The procedure includes initial planning, deciding if a finite element analysis is needed, doing a needed analysis, and presenting the results. Following this procedure should allow the user to perform an effective and efficient analysis. A block diagram is the next page shows the analysis step. When the engineer has created a conceptual design ready for analysis or has an existing design failure needing analysis, the first step is to define the analysis problem as clearly as possible. The definition includes the determination of what type of analysis is to be performed (static, dynamic, etc.), whether the solution will be two or three dimensional, and what the criteria will be for the analysis. The criteria for the analysis identify the important variables for evaluation. These could include the maximum stress, the average stress, the strain, and the deformation, fracture load, yield load, critical stress location, or several other factors. The most critical of these serve to guide the modelling and to guide the presentation of result. After completion of the problem definition, an approximate engineering analysis is defined. Using approximate solutions, estimate the values of critical variables identified in the problem. Based on these approximate results and the degree of confidence placed in the approximate solutions, determine if you need to do a finite element analysis. If the decision is yes, begin by developing a conceptual finite element model. The analyst developing conceptual model will lay out the geometry of the model, choose the kind of element that would be best suited, and roughly plan the mesh for the model. Examine the structure for any symmetries that exist in the geometry and loading conditions, and from those symmetries select a repeating section that will define the model geometry. Exploiting symmetry reduces the effort required to create a model as well as reducing the number of computations and increasing the numerical accuracy. For the section model choose the type of finite element formulation needed to represent correctly the structural behaviour. Also, identify any supporting element types needed for special interfaces or boundary condition. Finally sketch a rough mesh plan making an effort to lay out the element subdivisions according to expected XIV solution variations. Completion of the conceptual model will provide the information needed to choose the proper computer program to use. The Analysis Step by Step There are three stages, which describe the use of any existing finite element program. PREPROCESSOR INPUT DATA Control Data, Materials, Node and Element Definition, Boundary Conditions, Loads =s FORM ELEMENT [k] Read Element Data, Calculate Element StiShess Matrix, [k] A V FORM SYSTEM Assemble Element [k] to Form the System Stiffness Matrix, [K] APPLY DISPLACEMENT BOUNDARY CONDITION COMPUTE DISPLACEMENTS Solve the System Equations [K] {D} ={F} for the Displacements {D} = [K]-1{F} c Element File Load File Element File Load File COMPUTE STRESSES Calculate Stresses and Output Files for Postprocessor Plotting c> Displacement, Stress Files POSTPROCESSOR K Finite Element Computer Program Block Diagram XVI The preprocessing stage creates the model of the structure from inputs provided by the analyst. A preprocessor then assembles the data into a format suitable for execution by the processor in the next stage. The processor is the computer code that generates and solves the system equations. The third stage is postprocessing. The solution in numeric form is very difficult to evaluate except in the simplest cases. The postprocessor accepts the numeric solution, presents selected data, and produces graphic displays of the data that are easier to understand and evaluate. The sort history of ANSYS finite element analysis program can explained briefly, as below. Swonson Analysis System, 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. The ANSYS program is one of the most widely recognised large-scale general-purpose programs for engineering analysis available today. It is used world wide for solutions to design challenges in the aerospace, automotive, power, consumer, machinery, chemical, biomechanics, and electrical/electronics industries. Especially ANSYS\Mechanical is used design analysis program for determining displacement, stress, forces, temperatures, and pressure distributions, such as thermal and structural characteristics of printed circuit board. To support this broad range of use, the ANSYS program includes powerful capabilities such as preprocessing, solid modelling, structural, thermal, magnetic, fluid flow and coupled field analysis postprocessing, premier graphics capabilities, and design optimization. The ANSYS program is continually revised and updated to enhance existing features, to add new finite element analysis technology, and to make use of advancements in computer hardware. All of the powerpul capabilities work together to increase productivity in model building, and provide high quality analysis tools. The main idea of a lot of companies is the same. They try to produce the highest quality product at the lowest cost. The ANSYS program allows engineers to construct computer models of structure 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 lo the multiple-prototype building, testing, and rebuilding. Thus, the 4esigr±ân4 manufacturing cost reduce. The ANSYS program is organized into two basic levels: Begin level and Processor (or Routine) level. The begin level acts as a gateway into and out of the ANSYS program. It is also used for certain global program controls such as changing the jobname, clearing the database, and making copies of binary files. When you first enter the program, you will be at the Begin level. At the Processor level, several processors (routines) are available, each serving a specific purpose. For instance, the general Preprocessor (PREP7) is where you build the model, the solution processor (SOLUTION) is where you apply loads and obtain the solution, and the general postprocessor (POST1) is where you evaluate the results of solution. Enler ANSYS Exit ANSYS ANSYS Pogram Organization There are different analysis techniques in the ANSYS finite element program. Some of them are "Structural Analysis", "Thermal Analysis", "Flotran Analysis", and "Coupled Field Analysis" Structural analysis is the most common application of the finite element method in Mechanical Engineering. There are many types of structural analysis in the ANSYS program. Such as static, modal, harmonic, transient dynamic, spectrum, and buckling. A static analysis calculates the effects of steady loading conditions on a structure, while ignoring inertia and damping effects, such as those caused by time- varying loads. A static analysis can, however, include steady inertia loads (such as gravity and rotational velocity), and time-varying loads that can be approximated as static equivalent loads (such as the static equivalent wind and seismic loads commonly defined in many building codes). Static analysis is used to determine the displacements, stresses, strain, and forces in structures or components caused by loads that do not induce significant inertia and damping effects. Steady loading and response conditions are assumed; that is, the loads and the structure's response are assumed to vary slowly with respect to time. The kinds of loading that can be applied in a static analysis include:. Externally applied forces and pressures. Steady-state inertial forces (such as gravity or rotational velocity). Imposed (non-zero) displacement. Temperatures (for thermal strain). Fluences (for nuclear swelling) The procedure for a static analysis consist of three main steps: Build the model. 1. Apply loads and obtain the solution. 2. Re vi ew the resu İt s XVHl In the first step to build the model, you specify the jobname and analysis title and then use PREP7 to define the element types, element real constants, material properties, and the model geometry. These tasks are common to most analysis. In the second step, you define the analysis type and analysis option, apply loads, specify load step option and begin the finite element solution. In the last step, results from a static analysis are written to the structural results file, Jobname.RST. They consist of following data:. Primary data:. Nodal displacements (UX, UY, UZ, ROTX, ROTY, ROTZ). Derived data:. Nodal and element stresses. Nodal and element strains. Element forces. etc. You can review these results using POST1, the general postprocessor, and POST26, the time-history processor. To review results in POST1 or POST26, the database must contain the same model for which the solution was calculated. The result file (Jobname.RST) must be available. In this thesis, a computer program based on finite element method called ANSYS was used to determine stress; strain and displacement on the elements of variable speed drive mechanism (variator). The main aim of this study is to show the designer 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 the 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, structure and content of ANSYS program were explained, after giving a general information on it. In chapter 4, a general knowledge about variable speed drive with conical disks, Jil additional to, the solution are obtained for this mechanism. The results are shown as graphics. In chapter 5 that is the last chapter of thesis, result were reviewed and generally discussed and suggestion were presented.

The solution method that the engineer has available for the design of structural system can be divided into two groups, analytical and numerical. Analytical solutions to the engineering and science problems are possible only if the geometry and boundary condition of the problem are simple. In a lot of engineering problem, generally machine and structural systems consist of various structural element such as beams, links, plates, and shells or a combination of the these. Their overall geometry becomes extremely complicated and cannot be represented by a single mathematical expression. The solution problem with complex geometry, boundary condition and loading become possible only through numerical approximations. In spite of their versatility, however, numerical solutions are only approximation to the actual solution. The degree of correlation between these approximations and actual solution depend on complication of problem. Analytical solution also provides a means of checking numerical result for accuracy and convergence. In recent years, numerical methods become popular due to rapid advancements in computer technology and its availability to engineer. In these methods, Finite Element Method is the most popular method in engineering science. The roots of the method can be traced back to three separate research group; engineers, physicists and applied mathematicians. Although in principle published already, the finite element method obtained its real impetus from the independent developments carried out by engineers. Important original contributions appeared in papers by Turner and Argyris and Keksey. The name " finite element method " was presented for plane stress analysis. Since then, a large amount of research has been devoted to the technique, and a very large number of publications on the finite elementjnethod are available at presents The rapid development of the finite element method as an analysis tool essentially began with the advent of the electronic digital computer. Using the finite element method on digital computer, the numerical solution of a complex structural problem that is including algebra equation becomes possible and very effective. The basic concept of the finite element method is divide a solution region into so called finite element of several convenient shapes such as truss, triangles or rectangular, brick, tetrahedran, ring, etc. (Truss in one dimensional cases triangles or rectangularin two dimensional cases and tetrahedral or brick in three dimensional cases) The unknown field variable is expressed in term of values of the field variables at specified points called nodes. Nodes connect elements to the next elements. Nodes usually lie on the element boundaries. But some elements may have a few interior nodes. The more number of elements used to model, the XUl problem, and the better approximation to the solution obtained. Note that the data storage requirement an solution time rapidly with an increase in the number of element. The finite element method of analysis has been recognised as apower tool and has been effectively used to obtain solution in structural analysis (Static, Modal, Harmonic, Transient, Dynamic, Spectrum etc.), thermal analysis and fluid dynamic etc. One of the chief features of the finite element method is the way curved boundaries can be realistically treaded by using higher order isoparametric elements. On the other hand, it has ability to tread anisotrapic materials discontinuous and non linear structural problems. In spite of above advantages, three are some disadvantages finite element method. It is that in order to solve even very simple problems a computational approach is needed. Also solution of complex problem is needed large high-speed digital computer. Sometimes solution time goes up further if the problem is non linear. An idealized analysis procedure will bring all steps into proper order. The procedure includes initial planning, deciding if a finite element analysis is needed, doing a needed analysis, and presenting the results. Following this procedure should allow the user to perform an effective and efficient analysis. A block diagram is the next page shows the analysis step. When the engineer has created a conceptual design ready for analysis or has an existing design failure needing analysis, the first step is to define the analysis problem as clearly as possible. The definition includes the determination of what type of analysis is to be performed (static, dynamic, etc.), whether the solution will be two or three dimensional, and what the criteria will be for the analysis. The criteria for the analysis identify the important variables for evaluation. These could include the maximum stress, the average stress, the strain, and the deformation, fracture load, yield load, critical stress location, or several other factors. The most critical of these serve to guide the modelling and to guide the presentation of result. After completion of the problem definition, an approximate engineering analysis is defined. Using approximate solutions, estimate the values of critical variables identified in the problem. Based on these approximate results and the degree of confidence placed in the approximate solutions, determine if you need to do a finite element analysis. If the decision is yes, begin by developing a conceptual finite element model. The analyst developing conceptual model will lay out the geometry of the model, choose the kind of element that would be best suited, and roughly plan the mesh for the model. Examine the structure for any symmetries that exist in the geometry and loading conditions, and from those symmetries select a repeating section that will define the model geometry. Exploiting symmetry reduces the effort required to create a model as well as reducing the number of computations and increasing the numerical accuracy. For the section model choose the type of finite element formulation needed to represent correctly the structural behaviour. Also, identify any supporting element types needed for special interfaces or boundary condition. Finally sketch a rough mesh plan making an effort to lay out the element subdivisions according to expected XIV solution variations. Completion of the conceptual model will provide the information needed to choose the proper computer program to use. The Analysis Step by Step There are three stages, which describe the use of any existing finite element program. PREPROCESSOR INPUT DATA Control Data, Materials, Node and Element Definition, Boundary Conditions, Loads =s FORM ELEMENT [k] Read Element Data, Calculate Element StiShess Matrix, [k] A V FORM SYSTEM Assemble Element [k] to Form the System Stiffness Matrix, [K] APPLY DISPLACEMENT BOUNDARY CONDITION COMPUTE DISPLACEMENTS Solve the System Equations [K] {D} ={F} for the Displacements {D} = [K]-1{F} c Element File Load File Element File Load File COMPUTE STRESSES Calculate Stresses and Output Files for Postprocessor Plotting c> Displacement, Stress Files POSTPROCESSOR K Finite Element Computer Program Block Diagram XVI The preprocessing stage creates the model of the structure from inputs provided by the analyst. A preprocessor then assembles the data into a format suitable for execution by the processor in the next stage. The processor is the computer code that generates and solves the system equations. The third stage is postprocessing. The solution in numeric form is very difficult to evaluate except in the simplest cases. The postprocessor accepts the numeric solution, presents selected data, and produces graphic displays of the data that are easier to understand and evaluate. The sort history of ANSYS finite element analysis program can explained briefly, as below. Swonson Analysis System, 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. The ANSYS program is one of the most widely recognised large-scale general-purpose programs for engineering analysis available today. It is used world wide for solutions to design challenges in the aerospace, automotive, power, consumer, machinery, chemical, biomechanics, and electrical/electronics industries. Especially ANSYS\Mechanical is used design analysis program for determining displacement, stress, forces, temperatures, and pressure distributions, such as thermal and structural characteristics of printed circuit board. To support this broad range of use, the ANSYS program includes powerful capabilities such as preprocessing, solid modelling, structural, thermal, magnetic, fluid flow and coupled field analysis postprocessing, premier graphics capabilities, and design optimization. The ANSYS program is continually revised and updated to enhance existing features, to add new finite element analysis technology, and to make use of advancements in computer hardware. All of the powerpul capabilities work together to increase productivity in model building, and provide high quality analysis tools. The main idea of a lot of companies is the same. They try to produce the highest quality product at the lowest cost. The ANSYS program allows engineers to construct computer models of structure 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 lo the multiple-prototype building, testing, and rebuilding. Thus, the 4esigr±ân4 manufacturing cost reduce. The ANSYS program is organized into two basic levels: Begin level and Processor (or Routine) level. The begin level acts as a gateway into and out of the ANSYS program. It is also used for certain global program controls such as changing the jobname, clearing the database, and making copies of binary files. When you first enter the program, you will be at the Begin level. At the Processor level, several processors (routines) are available, each serving a specific purpose. For instance, the general Preprocessor (PREP7) is where you build the model, the solution processor (SOLUTION) is where you apply loads and obtain the solution, and the general postprocessor (POST1) is where you evaluate the results of solution. Enler ANSYS Exit ANSYS ANSYS Pogram Organization There are different analysis techniques in the ANSYS finite element program. Some of them are "Structural Analysis", "Thermal Analysis", "Flotran Analysis", and "Coupled Field Analysis" Structural analysis is the most common application of the finite element method in Mechanical Engineering. There are many types of structural analysis in the ANSYS program. Such as static, modal, harmonic, transient dynamic, spectrum, and buckling. A static analysis calculates the effects of steady loading conditions on a structure, while ignoring inertia and damping effects, such as those caused by time- varying loads. A static analysis can, however, include steady inertia loads (such as gravity and rotational velocity), and time-varying loads that can be approximated as static equivalent loads (such as the static equivalent wind and seismic loads commonly defined in many building codes). Static analysis is used to determine the displacements, stresses, strain, and forces in structures or components caused by loads that do not induce significant inertia and damping effects. Steady loading and response conditions are assumed; that is, the loads and the structure's response are assumed to vary slowly with respect to time. The kinds of loading that can be applied in a static analysis include:. Externally applied forces and pressures. Steady-state inertial forces (such as gravity or rotational velocity). Imposed (non-zero) displacement. Temperatures (for thermal strain). Fluences (for nuclear swelling) The procedure for a static analysis consist of three main steps: Build the model. 1. Apply loads and obtain the solution. 2. Re vi ew the resu İt s XVHl In the first step to build the model, you specify the jobname and analysis title and then use PREP7 to define the element types, element real constants, material properties, and the model geometry. These tasks are common to most analysis. In the second step, you define the analysis type and analysis option, apply loads, specify load step option and begin the finite element solution. In the last step, results from a static analysis are written to the structural results file, Jobname.RST. They consist of following data:. Primary data:. Nodal displacements (UX, UY, UZ, ROTX, ROTY, ROTZ). Derived data:. Nodal and element stresses. Nodal and element strains. Element forces. etc. You can review these results using POST1, the general postprocessor, and POST26, the time-history processor. To review results in POST1 or POST26, the database must contain the same model for which the solution was calculated. The result file (Jobname.RST) must be available. In this thesis, a computer program based on finite element method called ANSYS was used to determine stress; strain and displacement on the elements of variable speed drive mechanism (variator). The main aim of this study is to show the designer 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 the 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, structure and content of ANSYS program were explained, after giving a general information on it. In chapter 4, a general knowledge about variable speed drive with conical disks, Jil additional to, the solution are obtained for this mechanism. The results are shown as graphics. In chapter 5 that is the last chapter of thesis, result were reviewed and generally discussed and suggestion were presented.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1998

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1998

Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1998

##### Anahtar kelimeler

Sonlu elemanlar yöntemi,
Varyatör,
Finite element method,
Variator