Çok parçalı sistemlerin analizine bir örnek: azdırma malafa sisteminin deformasyon analizi

Abstract

ÖZET Teknikte kullanılan çok parçalı sistemlerin analizi bu sistemlerin homojen olmayan yapıları ve sistemde görülen süreksizlikler dolayısıyla zor luklar gösterir. Ayrıca bu sistemlerde sistemin ve elemanların yüklenme ve sınır şartlarının tayini elemanların deformasyon sırasında birbirlerine uygu ladıkları ikinci mertebe etkiler dolayısıyla oldukça zordur. Karışık yapıları dolayısıyla teknikte kullanılan bir çok sistemin ayrıntılı analizlerinde güç lüklerle karşılaşılmaktadır. Bu çalışmada endüstride dişli çark açmada kulla nılan azdırma tezgahının malafa sisteminin matematik modelinin teşkili ve de formasyon bakımından analizi yapılmaktadır. Bu sistemde meydana gelen bozulma hallerinin daha iyi anlaşılabilmesi ve sistemin dizaynı sırasında sistem pa rametrelerinin daha uygun seçilebilmesi bakımından sistemin ayrıntılı matema tik modelinin teşkili ve analizi bir ihtiyaç halini almıştır. Azdırma malafa sistemi, azdırma bıçağı ve ayar burçlarının azdırma malafasına takılmasıyla teşkil edilen çok parçalı bir sistemdir. Malafa ucun daki somunun sıkılmasıyla sisteme «n gerilme verilip rijitliği arttırılmış tır. Klasik mekanik ve mukavemet yaklaşımları dışında sistemin etraflı ana lizinin şimdiye kadar yapılmamış olması, yapılan deneyler, ve teknikteki kul lanışı dışında sistemin davranışı hakkında bir fikre sahip olunmaması sonucu konu burada ele alınmıştır. Çalışmada çeşitli matematik analiz metodları incelenerek çok parçalı sistemlerin matematik modellerinin çıkarılması ve analizleri için ana kaide ler çıkarılıp genel bir çözüm metodu geliştirilmiştir. Buna uygun olarak sis tem için kurulan teori ve teşkil edilen matematik modelden faydalanarak sis temin deformasyon bakımından analizi yapılmıştır. Elde edilen sonuçlar yapıl mış olan deney sonuçlarıyla karş ılaşt ırılmışt ir. Giriş bölümünde çeşitli analiz metodları incelenmekte, sistemin sü rekli bir matematik modelle ifade edilemeyeceği ve matematik modelin sistem ayrıklaştırılarak sayısal olarak ifade edilmesi gerektiği sonucuna varılmak tadır. Ayrıca sistemin incelenmesinde ve matematik modelinin çıkarılmasında parçaların deformasyon sırasında meydana getireceği ikinci mertebe etkilerin ve deforme olmuş haldeki geometrik uyum ve dinamik dengelerinin göz önüne alınması gereğine işaret edilmiştir. Bu amaçla sistemin deforme olmuş haldeki geometrisini temsil eden bir geometrik analog modelin geliştirilmesi uygun görülmüştür. Sistemin süreksizlik noktaları olan burç ara kesitlerinde teşkil edilen teoride, kesit büyüklüklerinin kuvvet ve moment denge denklemlerini sağlayacak şekilde ve deforme olmuş haldeki kesit geometrisi esas alınarak hesap edilebileceği gösterilmiştir. Burada burç ve milin ön gerilmenin yeter li olduğu halde bileşik kirigler gibi beraber çalıştığı, ön gerilmenin yeter siz veya bollaşma olması halinde ise serbest kirişler gibi ayrı ayrı çalıştı ğı kabul edilerek iki ayrı teori geliştirilmiştir. İncelenen sistemde deforme olmuş denge hali önceden bilinemediğinden ve burç aralıklarının eksenel kuvvet ve eğilme momenti altındaki davranışları sistemde moment dağılışını önemli derece etkilediğinden sistem ancak iteratif olarak çözül ebilmektedir. Sistemin deforme olmuş herhangi bir andaki geomet risi için burç açıları geliştirilen geometrik analog modelden faydalanarak tespit edilmekte sonra bu hal için kesit aralıkları için geliştirilen teori-den faydalanarak sistemdeki moment dağılımı tespit edilmektedir. Bu moment dağılımı altında sistemin deformasyonu evvelki deformasyona yeteri kadar ya kın olduğunda iterasyonun yakınsadığı ve denge haline ulaşıldığı kabul edile rek cevaplar tespit edilmektedir. Çalışmada geliştirilen bilgisayar programının ana hatları ve akış şeması verilmekte, problemin yukardaki esaslara göre nasıl çözüldüğü göste rilmektedir. Ayrıca teşkil edilen analog model için kurulan teori açıklanmak ta ve bu modelin radyal ve eksenel geometrik uyum şartlarını nasıl sağladığı anlatılmaktadır. Bu çalışmada kurulan teori ve geliştirilen bilgisayar programı ile, geometrik analog model kullanarak çok parçalı bir sistemin analizinin nasıl gerçekleştirilebileceği gösterilmiştir. Daha önce yapılmış olan deney sonuç larıyla elde edilen bilgisayar sonuçları karşılaştırılarak aralarında iyi bir uyum olduğu görülmüş ve teşkil edilen teorinin doğru olduğu sonucuna varıl mıştır.

AN EXAMPLE TO THE ANALYSIS OF MULT I -PART SYSTEMS: * DEFORMATION ANALYSIS OF AN ARBOR MECHANISM. IN A BOBBING MACHINE SUM MA R Y The analysis of multipart systems used in engineering presents some difficulties due to the nonhomogenous structure of these systems and discon tinuities encountered in them. Furthermore the determination of loading and boundary conditions of the system and the elements comprising the system is rather difficult as a result of second order effects that the elements apply to each other during deformation. Some difficulties are encountered in the detailed analysis of many systems used in engineering because of their complex structures. In this work the mathematical model of the arbor system of a hobbing machine used in industry to hob gears is established and the analysis of the arbor system in view of deformation is performed. The establishment of the detailed mathematical model of the system and its analysis became a matter of necessity in order to understand thoroughly the stages of failure occuring in the system and to select appropriately the parameters during the design of the system. The arbor system is a multipart system consisting of a hob and adjustment sleeves mounted on an arbor. The rigidity of the system is increased by tightening the nut at the end of the arbor and thus the system is pretens ioned. The motives for this study are the following: Firstly, a detailed analysis of the system has not been made beyond the simplified classical mechanics and strerigth of materials methods. Secondly knowledge about the behaviour of the system rests upon a few experiments and operating experience; no theoretical investigation has been made. After investigating various mathematical methods of analysis, a general solution method is developed in this work by deriving the main priciples for analysis and determination of mathematical models of multipart systems. The deformation analysis of the system is made by the use of theory and mathematical model established in this work for the system. The results obtained are compared with the results of the experiments. This thesis contains eight sections. The first section deals with the importance of the subject and problems encountered in the analysis of multipart systems. The investigation of these problems with particular reference to the system considered in this study has been made. The second section treats various methods of analysis and it is concluded that the system cannot be represented by a continous mathematical model. Also in this chapter the problem is presented and similar systems dealing with the same problem are examined. In the third section the principles of the model are established and the numerical model is presented. The system is discretized and deformations are determined by numerical integration. Furthermore, it is pointed out that in the analysis of the system the second order effects in the mathematical model during the deformation of the parts and the second order effects of geometry and dynamic equilibrium of the deformed system must be taken into consideration. It is. concluded that this ccv. be realized only by developing a geometrical analog model of the system. In the fourth section the mid sections in between the adjacent sleeves -being the discontinuity points of the system- are examined. It is shown that the parameters of the mid sections can be calculated by satisfying equilibrium equations of iorce and iroTent and by considering the geometry of the deformed- vıı - section. Two different theories are developed for composite beams and compound beams. If the pretension of the system is adequate, the shaft and the sleeves behave as a single part like a composite beam. If the pretension of the sysetm is not enough, the shaft and the sleeves behave separately as a compound beam. The first theory can be stated as follows. In the mid section of adjacent sleeves, the upper parts of the sleeves are cdtopressed in order to carry the bending moment at the cross-section whereas the lower parts of the sleeves are separated because they are not able to carry tension forces. Mean while the shaft carries all of the tension forces and part of the bending moment. The area of contact of the sleeves changes according to the axial force at the cross-section as can be seen in Fig 1. If there is no yB,.ABj Axial Force and bending moment Figure 1- The changes in the mid section of adjecent sleeves due to axial force axial force the sleeves will contact each other only above the neutral axis. If the axial force is adequate the sleeves will have a complete contact and the stress will be distributed as if the section were full. If the sleeves and shaft act as a composite beam the position of the neutral axis at the cross- section in between the adjacent sleeves is given by where, yci B JB AB+AM (1) A ? Contact area of adjacent sleeves A : Cross sectional area of shaft y_: y coordinate of centroid of contacting section of sleeves- vıı ı - Since the lowest contacting point of the rings is below the neutral axis by the amount of xn due to the axial force the following expression can be written for the neutral axis yGn = XD + yD * (2) where yn is the y coordinate of the lowest contacting point of the sleeves and is given by yD = Ry cosa (3) Taking into account the normal strain Cq at the neutral axis as well as the curvature U) of the mid section, xn can be calculated by (4) (5) (6) The variables in the last two relations are: F : Axial force v E : Modulus of elasticity M : Moment at the mid section IG: Sectional moment oî inertia The sectional moment of inertia is given by XG ` XB + XM > (7) where Ig is the sectional moment of inertia of the contacting parts of the sleeves and I>j is the sectional moment of inertia of the shaft with respect to the neutral axis. Since all parameters at the mid section of the adjacent sleeves are a function of the angle a which determines the contacting areas of the sleeves the cross-sectional areas can be- found by determining the angle a. By equating relations (1) and (2), and noting that they must be equal, one obtains * - -yc]; - yen = ° <8> GD The angle a which satisfies the above relation (8) can be determined by iteration using regula-falsi and interval halfing methods together. Taking into account the axial equilibrium of the system the axial forces acting at the mid sections of the adjecent sleeves can be found. As can be seen in Fig. 2, the axial force Fg occurring in the system can be calculated as the sum of the pretension force Fp and the axial force Fgg. The axial force F^g is the force acting at the mid section of the adjacent sleeves which carry the highest bending moment. Thus one can write FB - Fp + FEB * (9)FB=VFB2 F8=Fv+FEB Figure 2- The axial forces acting on sleeves,r mid section can be determined by finding the axial force FEB a can write The axial force acting on any othei _ t this mid section due to bending. Therefore ont Fv - FB ` FEB (10) If the sleeves and shaft act separately as a compound beam the ;urvatures of the mid sections of the sleeves and shaft will npt be equal ;an be seen in Fig. 3. If the given pretension force of the system is not Figure 3- The deformation and curvatures at the mid section of adjecent sleeves when the sleeves and shaft act separately in bending.adequate or the system gets loose dur must be examined further in order to occuring in the system. The system ge the system cannot meet the looseness tries to meet this looseness by separ adjacent sleeves, For this reason the move to a point above the neutral axi most as can be seen in Fig. 3. The sep of the adjacent sleeves gives at the mid section. The curvature of the sh depending on the moment acting on the ing operation the resulting situation understand thoroughly the failures ts loose if the given initial tension of caused by the force F£u. The system ating at the mid sections of the lowest contacting points of the sleeves s at the mid section which separates aration of the sleeves at the mid section same time the curvature ^Pb °f tne sleeves aft section ^M c*n be determined shaft. The position of the neutral axis can be written as ^PbVb yc ^BAB +SV> (11) If there is looseness at the mid section, the sleeves will not contact each other at the neutral axis and therefore a discontinuity will result. For this reason the above relation must be expressed as (yB`yp)AB (12) If the curvature of the sleeves at the mid section (1Y,B) and shaft section C$u) are known the cross-sectional parameters can be determined by the iteration of the expression (8) that represents the equilibrium of the axial force at the mid section.But it must be taken into consideration that the relation (4) is x° `SPb as it is shown in the Fig. 4, (13) Figure 4- The cross-sectional parameters at the mid section of adjacent sleeves when the sleeves and shaft act separateily. Section. 5 deals with the transmission of moment in the system and at the mid sections of the adjacent sleeves. Since the sleeves and the shaft act separately each member will show a resistance to bending by the amount of their own moment of inertia. The moment carried by the sleeves depends on the- XI - rigidity of the mid sections at the adjacent sleeves and the distribution of the moment at the points where the rings and shaft lean against each other. In this work the transmission of moment is given by MT * FB.yB +% «B +^MEÎM as it is explained in Fig. 5. (14) Figure 5- The transmission of moment at the mid section of adjacent sleeves I The equilibrium, conditions dictate that the applied moment (Mj) and the reactant moment (Mx) be equal at the mid section: *amt- - M - M_ em t l i (15) This equation must be satisfied together with equation (8). In other words, theory set forth here regarding the mid section requires the provision of force and moment equilibrium together. Further, by the help of expression (15), a relation is established between sleeve (9b) aI*d shaft 09jj) curvature. At the mid sections, the openings between the adjacent sleeves and the shaft moments can be calculated by curvatures ^g and ^Vm which are interrelated to each other. In section 6, where the geometrical analogue model is examined, the sleeve and shaft deformations are calculated according to principles put forward in section 5. Also, the conformation of these deformations, to radial and axial geometric conformation conditions are examined. Furthermore, the conformity of the positions of the shaft and the sleeves within the fitting tolerances is set by the geometrical analogue model and the angles between sleeves at the mid sections are calculated. If the parting at the sleeve mid sections are in conformity with the values previously used, then the radial conformation condition is secured. Axial conformation condition can be expressed as follows: (F -2F )/, 1 B 2- E /leeveT.? yD(L). tan^L) (16) by assuming that the parting between sleeves for the existing axial force - will stand against the looseness in the system. The variables in equation (16) are: K` : Axial rigitidy of the system A, (L) : Total cross-sect: i opal area of the sleeves sleeveijj : Number of sleeve mid sections The left hand side of the equation expresses the looseness in the system and the right hand side of the equation shows the axial deformation and partings between each adjacent sleeve mid sections. By the help of equation (16), axial force is adjusted and thus the axial force equilibrium in the system and geometrical conformation is assured. The method of solving the problem and computer program written for this purpose is explained in Section 7. The solution of the problem is shown in this section by the flow chart for the program developed. This flow chart can be summarized as follows. Sleeve and shaft deformations can be determined by using the momentum distribution and the cross-sectional parameters calculated so that the force and momentum equilibriums are secured at mid sections. By the geometrical analogue model the conformity of shaft and sleeve positions within fitting tolerance is assured. If the partings at the sleeve mid sections are in conformity with the values previously used, then the radial conformation condition is secured. If this condition is not fulfilled, the equilibrium condition at sleeve mid sections is searched again for the newly found angles between sleeve mid sections. When the radial conformity condition is secured, axial conformity condition in the system is checked. In case of non- equilibrium, the axial force is adjusted and computations explained above are repeated. When the equilibrium conditions at each mid cross-section together with radial and axial conformity conditions are satisfied, the equilibrium of the system and deformations for this multi-part system is determined. In section 8 the experimental results found previously are compared with the computer program results obtained for the same parameters and it is observed that the deviation is between 0 % and 15 %. It was observed that the differences between the experimental and theoretical results was basically due to uncertainties in the bearings used in the experiments. Eowever in general the results are in good agreement. By the evaluation of the results, a reference to the future studies is made.