Sonlu elemanlar yöntemi kullanılarak bilgisayar yardımıyla millerin analizi ve tasarımı /
Yazar: İSMAİL ESEN
Danışman: Y.DOÇ.DR. İSMAİL GERDEMELİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Yüksek Lisans, 1994
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Bilgisayar destekli analiz = Computer aided analysis ; Bilgisayar destekli tasarım = Computer aided design ; Mil tasarımı = Shaft design ; Sonlu elemanlar yöntemi = Finite element method

Özet:
ÖZET Dişli kutuları, pompalar, elektrik motorları gibi makina- larda mekanik güc ileten makina elamanlarının tasarımı zor ve zaman alıcıdır, özellikle takım tezgahlarının hareket iletim sistemi veya hassas çalışan aletler söz konusu olduğunda da ha ciddi analiz ve tasarım gerekir. Gerçek makina elamanları na ait mukavemet hesaplarını bilinen mukavemet yöntemleriy le hassas olarak yapmak genellikle imkansız olduğu için ba sitleştirme yapılarak çözümler elde edilir. Bu basitleştirmede yapılan hata değer olarak tasarlanan mekanik sistemim amaçlanan hassasiyetinden büyük ise, yapılan hesaplamayla amaçlanan çözü me ulaşılamayacağı açıktır. Bu tez çalışmasında Sonlu Elamanlar Yöntemini kullanarak milleri modelleyip deplasmanlar, statik ve dinamik zorlanmalar açısından kontrol edebilen Quick Basic di- inde yazılmış bir bilgisayar programı ve kullanılması anlatılmak tadır. Dönen yada dönmeyen milleri analiz etmede kullanılabi lmen bu program iki bölümden oluşmaktadır. Program, birinci bölümde mil boyunca eğim, sehim, açısal deplasmanları ve statik normal, kayma, ve asal gerilmelerini hesaplar. İkinci bölümde ise, mil üzerinde istenen kritik noktalardaki emniyet katsayılarını statik ve dinamik şartlar için kontrol ederek verilen emniyet katsayısı şartına göre gerekirse boyutlandırma yapar. Herhangi bir şekle sahip ve üzerinde bir veya daha çok yerde dış kuvvet ler (kesme kuvvetleri, eğilme momentleri, burulma momentleri) bulunan miller modellenip analiz edilebilir. Çalışmanın amacı hesaplama kolaylığı için yapılan basitleştirmeden dolayı ortaya çıkan hataları ve zaman kaybını önlemek ve tasarımcılara daha hassas hesaplar yapabilecekleri bir araç sağlamaktır. VI

Abstract:
SUMMARY The desing work of mechanical power transmission compo nents in machines such as gear boxes, elektrik motors, etc. is complicated and time consuming. Serious analysis is required when drive system of machine tools or other accurate appliances are considered. For a designer it may generally be difficult to determine real deflections and stresses for real machine elements using classical strenght of materials methods. So this designer often simplifies the systems being analysed. And this simplifica tions can result in inaccurate calculations, and there is an error in solutions. The aimed solutions cannot be obtained if the error is greater than the desired sensitivity of the machine. This study introduses a computer program which is written in Quick Basic. The program can be used to analyse rotating or non-rotating shafts, and consists of two sub program. The first sub program computes deflections and static stresses along a shaft. The second sub program computes safety factors at selected nodes and deter mines new dimemsions of sections in terms of given safety factors if it is necessary. The prime aim of this study to supply an accurate and quick calculation device for designers. VII1. THEORY The finite element method used to determine deflections an static stresses requires the shaft to be divided into a system of elements. These elements are inter-connected and form the shaft by means of nodes. A shaft can be considered as a one dimension al body and divided into a desiered quantity of simple elements. It will be explained how to divide a shaft into a system of elements as follows. 1 3- Figure 1 A shaft under a system of loading. VIIIFigure 2 The shaft divided into elements. Fugure 1 shows a shaft having non-uniform configuration, and under a system of loading. And Figure 2 illustrates the shaft divided into elements. The parts with different cross-sections must be defined as different elements because their stiffnesses are different. Since in finite element method the external and reaction forces are applied nodes, new nodes must be defined at force application points. A shaft having a complex configura tion can be simply modelled using this method. The next step is to determine the element stiffness matrix of the individual ele ment. By assambling these elemental stiffness matrices The over all stiffness matrix is obtained. IX2. FORMULATION F M / e V Figure 3 Model of cantilevered beam used to determine stiffness matrix in bending. n Ml 1» vl FE "2 M2 v? 8Î 8? Figure 4 Free body diagram of the cantilevered beamThe following can be written from Figure 3 and figure 4 F2 = F ; Fi = -F M2 = M ; Mı = -M2 -FL = -M-Fl then (1) (2) The deflection in consideration with the effect of shear can be written by using the moment-area method. v 6 = (3) 2EI EI where k is the shear factor. (k= 1.18 for round sections) The relaionship above can be arranged in matrix form as u e L3 kL + 3EI AG L2 2EI L2 2EI L EI F M (4) XIo = ı>ı - \>2 - 0] L e = 02 - eı (5) Doing matrix operations and substituting; Mi F2 M2 12AGEI AGL3+12EIkL 1 0 5L -1 0,5L L2 kEI L2 kEI 0,5L [- + - ] -0,5L [- - - ] 3 AG 6 AG 1 -0 5L 1 -0,5L \A kEI L2 kEI 0.5L [- - - ] -0,5L [- + - ] 6 AG 3 AG ul »2 e2 obtained. (6) Figure 5 External torque applied on cantilevered beam. If there is external torque applied on the cantilevered beam, it can be shown that XIITl T2 GJ L G.f GJ L GJ 4>1 4>2 (7) The torque matrix can be combined with the matrix equation (6) (8) {P«} is the external force matrix [K] is the overall stiffnegg matrix {5*} is te vector of displacement 12EIAG a = AGL3+12EIkL b = 0,5L L2 kE c = - + - 3 AG L2 kEI d = = 0,5L2 - c 6 AG XIIIJ GJ The overall stiffness matrix is formed by adding each individual matrix. Ml Ti M2 T2 F3 M3 T3 a41 a42 a43 a44+bll a45 +b12 a46 + b13 b14 a52 a52 a53 a54+b21 a55 + b22 a56 +b23 b a61 a62 a63 a64+b31 a65+ b32 a66+ b33 b b b b 41 b42 b b b44+cn b b b b +c 12 The overall stiffness matrix is diogonal and symmetrical. It has a bandwith of six elements. Since the overall sitiffhess matrix has three degrees of fredom (i.e u, 8, $ ), there must be at least three end- conditions. The more information on how to impose the end- conditions on the overall stiffness matrix can be seen in referance in- XIV3 COMPUTER PROGRAM The program was written in Quick Basic and divided into three parts, i.e., input data, main processing,and output. 3. 1 Input data First of all, the shaft to be analysed must be divided into a system of elements. Then element lenght and outer diameter for each element, and the external force application and bearing points' node numbers, elasticity and shear moduluses, and exter nal forces must be ready for data input. As the program runs, the user will be asked to input necessary inputs. If the shaft has hollow sections the inner diameters will be asked. If the shaft or beam has a section different from round secion then the second moments of area will be asked. The user can select the type of sections from Shaft Menu. See page 15. 3.2 Main processing When data inputing has finished, the program starts to run up to data output section. Firstly it forms elements' stiffness matrices. Then builts up the overall stiffness matrix by adding the element stiffness matrices. After that the overall stiffness matrix is rearra nged in terms of end-conditions. Then its inverse is formed. And new overall stiffness matrix is solved by multiplying with force matrix to obtain displacement matrix. Finally, the overall stifness matrix is rearranged and element stiffness matrices are multiplied with deflections of each elements to obtain internal forces. XV3.3 Output The main output menu gives the following options. 1. Displacement sub menu deflections; slopes; angular deflections; I. Force sub menu shear forces; bending moments; torques; stresses; 3. Drawing sub menu loadig; displacement diagrams; shear force diagrams; bending moment diagrams; 4 DESING AND FATIGUE ANALYSIS Fatigue analysis needs some data such as ultimate strengt, yielding point of material and some fatigue factors such as stress concentration, size, load, surface finish, temperature and miscellaneouB-effects factor. Dynamic and static analysis are done and if necessary new values of critical sections' diameter are computed in terms of given safaty factor. In this program Max-shear stress and Distortion energy teorem are used to determine safety factors. XVI