## Elastik kirişlerin elektriksel simülasyon yolu ile modellenmesi 1996
Erdem, H. Burcu
##### Yayınevi
Fen Bilimleri Enstitüsü
##### Özet
Bu tez çalışmasının amacı; Elektrik devreleri ile mekanik sistemler arasında bir paralellik kurulması yada ortak Ur noktada birleşmektir. Böyle bir benzerliği hedeflemek için öncelikle elektrik devrelerini oluşturan en az Hd uçlu yada çok uçlu elemanlara karşılık mekanik sistemler de hangi elemanların bulunduğunu, mekanik sistemler belirlendikten sonra da uç büyüklüklerinin tanımlanması gerekir. Ancak bu uç büyüklüklerinin elektrik devrelerinde kullanılan akım yada gerilim büyüklüklerinden farkı skater değil de vektörel karekterde olmalandır. Elektrik ve mekanik sistemler de kullanılan bağıntıların önemi; bazı uçlarında elektriksel değişkenler, diğerlerinde ise mekanik değişkenlerin sözkonusu olduğu "etektro-mekanik" çok uçluların gözönüne alınması halinde işlev kazanmaktadır. Bu tür çok uçluların ani güç ifadesi yazıldığında, elektrik ve mekanik güçlerin toplamına eşit olacağından, bu toplamdaki her terimin aynı birimle ölçülmesi zorunluluğu vardır. Bu çalışma da iki uçlu yaylarla oluşturulan, çok uçlu yay kutularından meydana gelen mekanik sistemlerin matematiksel modelleri ele alınmıştır. Daha sonra mekanik sistemdeki büyüklüklerin elektriksel karşılıktan yerleştirilerek ele alınan mekanik sistem elektriksel sistem şeklinde modellenmiştir. Devre analizindeki gerekli tanımlardan faydalanarak bu sistem birden fazla biçimde ifade edilmiştir. Spice programını kullanarak bulunan devrelerdeki akım değerleri ite mekanik sistemdeki denge ve moment denklemleri yardımı ile hesaplanan kuvvet değerleri nümerik olarak aynı sonuçlan vermiştir. Böylelikle de ilgili sistemler arasında paralellik kurulabilir. Daha sonra üç uçlu ideal kaldıraç.makara ve dişli kutusu ele alınarak ideal bağ elemanlarının özellikleri incelenmiştir. Çünkü bu mekanik elemanların uç denklemlerinin elektriksel sistem karşılıktan; birbirini magnetik yoldan etkileyen iki endüktansın oluşturduğu, iki kapılı bir düzenin ktealleştirilmesi ile elde edilen transformatöre aittir. Çok uçlu mekanik elemanların, özellikle üç boyutlu uzayda hareketi sözkonusu olduğunda katı cismin modellenmesi büyük önem taşımaktadır, örneğin dönen bir katı cismin hareketi incelenirken, farklı eksen takımlarının kullanılması inceleme de bazı kolaylıktan beraberinde getirmektedir. Bu eksen takımları arasındaki bağlantıyı da Ek bölümünde incelenen T dönüşüm matrisi sağlamaktadır
The target of this work is to find similarities between mechanical and electrical systems and represents the mechanical systems, as electrical circuits to find numerical solutions. For this reason; each elements of the mechanical systems should be simulated with a component of the electrical circuits. If components that use in mechanical systems are motion, it could be selected T force and V velocity variables or if it turns, selected "M(t)° moment and "w(t)" angular velocity variables. In order to write the terminal equations representing the non - linear capacitor and non-linear inductance, °q(t)" electrical charge, "(t)" magnetic flux are used. The variables used in electrical and mechanical systems are given in table. Table 1: The variables used in electrical and mechanical systems. These equations are very useful for the analysis of the electro-mechanical murti port devices which have electrical variables at some ports while having mechanical variables at the others. Materials are tested in tension, in compression, in twist, in bending or in Vcombinations of these loadings. Machines in which two or more of these teste can be performed are called universal testing machines. By reason of its simplicity, the tension test is most commonly and easily performed^ major portion of strenght theory rests on assumption gleaned from the tension test Beams may be classified according to material composition and unstrained form. In statics, the subject that is basic and preparatory to this one, there is a group of problems on frames and trusses that require determination of tension and compression forces in the component parts of these structures induced by specified loads. The process involved in the solution of such problems is one of analysis. If the problems were so changed as to necessiate calculation of tile proper sizes for the component parts in order that the structure as a whole could safely carry the given loads, then the process involved in their solution would be one of design. Analysis and design are inverse processes. On the Other hand, to design a structure is to determine, from given or assumed loads and major dimensional limitations, the appropriate sizes and forms of the component parts (sometimes also the proper material to use when there are two or more materials to chose from) in order that the structure will function effectively and safely. Thus a simple beam is one that is made of one material, is straight in the unstraine state, and is of uniform cross section. It is quite distinct from, say, a compound beam that may be composed of two or more materials (for example, a reinforced concrete beam), or from a uniformly stressed beam that may have a varying cross section, or from a curved beam such as a hook or an arch. Beams may also be classified after their manner of support A simply supported beam rests on supports that permit rotation freely. It is implied that the support is capable of reacting in one sense only, as suggested by the knife edge. If it is desired to imply the possibility of either sense for the reaction, a hinge may be depicted. A simply supported beam that extends beyond a support is called a beam with an overhang. A beam that is supported at one end only is called cantilever beam, the supported end shown embeded in a wall, is referred to as a built in end. A catilever beam is all overhang. Beams may be classified according to whether or not the reactions at the support may be determined without recourse to principles other than those of equilibrium. A statically determinate beam is one which involves reactive forces that can be uniquely solved for from the equlibrium equations alone. If equations other than those of equilibrium are needed to effect a solution for these reactive forces, then the beam involved is a statically determinate one. A beam that rests on several simple supports, called a continous beam is statically indeterminate. VIIf a beam may be subjected to different patterns of loading throught the expected usefull life of the structure, many of which cannot be accurately predicted, then it is obviously impractical to desisn it as variable cross section beam. The sensible act in such a situb'on is to design the beam with a uniform cross section whose strenght should be governed by the probable maximum bending moment or maximum shear force that could be expected. Two pointed beam that built in supported force to be bended by force P in figure 1 can be modeled by six pointed spring box and terminal equations can be shown at eq. 1. FIGURE 1: Two pointed beam that built in supported force to be bended by force P. 02{t) f<="" positive.="" exerted="" by="" restoring="" in="" that="" it="" points="" toward="" origin.="" real="" springs="" will="" obey="" eq.="" (2),="" known="" as="" hooke's="" law,="" if="" we="" do="" not="" stretch="" them="" beyond="" limited="" range,="" can="" think="" magnitude="" per="" unit="" elongation.="" thus="" very="" stiff="" have="" large="" values="" k.="" x="S*F" (3)="" may="" also="" be="" represented="" such="" condition="" s="" (k"1)="" coefficient="" flexibility="" box.="" vnthe="" quantity="" e="" young'="" modulus,="" or="" modulus="" elasticity.="" essentially="" given="" material.="" heat="" treatmant,="" cold="" working,="" alloying="" moderate="" temperature="" changes="" little="" effect="" on="" value.="" steel="" common="" use="" value="" 2.="" "mo4="" kg="" mm2="" calculations.="" t="" moment="" inertia="" with="" respect="" impartial="" axe="" (z="" axe)="" iz="JAy2dA=" bh3="" 12="" (4)="" dy="" v="" zzza="" th="" 2="" İ*2="" h="" figure="" 2:="" calculation="" analitical="" plane.="" s22="Sl1=(l/3E0," s,2="S2l=(-1/6Er)," s31="Sl3=-ab(l+b)/6EII," s!4="S24=(1/I)" sis="S25=ki/i)." s23="S32=ab(l+a)/6ELi," s34="(b/I)," 844="855=0" 835="(afl)," s4i="S42=-(1/l)." ssrki="" ij.sa^l="" l),="" 854="845=0," s33="(a.b)2/3En" s43="-(b/r)." ss3="-(afl)" possible="" represent="" an="" elastic="" beam="" which="" designed="" six="" pointed="" box,="" electrical="" circuit="" consist="" ideal="" nonideal="" transformers="" other="" elements.="" numerical="" solution="" found="" using="" pspice="" electronic="" analysis="" program.="" result="" simulation="" are="" calculated="" equilibrium="" equations="" at="" mechanical="" system="" should="" identical="" current="" obtained="" circuits.="" mathematical="" inductances="" transformer="" follows;="" vffll="[Lul,=UKU" ufdiagflv'ua10lja]="" =="" ?j="" la.="" lu="" (5)="" (6)="" ki2="" ki3="" -="" kin="" k\2="" kn...="" k2n="" kin...="" la="" (7)="" kj="" coupling="" matrix="" matrix.="" diagonal="" entries="" l="" self-inductances="" while="" off="" ones="" mutual="" inductances.="" note="" since="" u="" (5)is="" congruent="" transformation="" hence="" positive="" semidefinite="" matrix.this="" implies="" once="" fact="" that,="" \ku\="\Lu" jlıı.lu="" <="" style="margin: 0px; padding: 0px; outline: 0px;">\ The magnitude of coupling coefficients cannot exceed unity. The case where lkjjl=1 characterized by saying that coils l_ji and Ljj are perfectly coupled. Many system models require only the basic active and passive 1-portsjoinedoand 1 junctions but we sometimes also need two ports. We now consider two ideal port elements, both of which are ideal in the sense of power conservation, just as the junctions are. One of the ideal two ports is called a transformer.(TF); in fact, the ideal electric transformer is modeled by a bond graph transformer. In mechanical systems a variety of devices are represented in idealized form as transformers, including the rigid, massless lever shown in fig. 3 The transformer has two ports, and the efforts, (forces) at two ports are proportional to each other, as are flows (velocities). For the lever, if F1 and F2 are vertical forces as shown and the pivot is frictionless, by taking moments about the pivot that found the equilibrium relation, aFt=bF2 Fi=(b/a)F2 (8) DCAnother relation is found for the vertical velocities V, and V2 (assuming small angular rotation) by computing the angular velocity of the rod as, -(Vi/a)=w=(V2/b) -(b/aJV^ (9) The ratio (b/a) is called a modulus of transformer. Ifeq. (8) and (9) are multiplied together appropriately and the common factor (b/a) is canceled out, Fı.V^P^P^Fî-Va (10) A lever represented as an ideal transformer which indicates that power flowing into port 1 is always equal to power flowing out of port2. This power relation is embodied in the bond graph symbol in Rg.3, which shows the sign convention of half arrows pointing throuugh the TF symbol.Since both forces are shown pointing down in Rg.3, Eq. (8) must be written with V, a b V2 « T B F, F2 Figure 3: A lever represented as a transformer. a positive proportionality factor b/a Then Vi and V2 up Eq.(9) is written with the same proportionalityfactor (b/a). The result is that F^V, is power being supplied to the left hand end of the lever and F2.V2 is power being expended by the right-hand end of the lever on whatever is connected to it Similarly; Mathematical models of ideal roller and ideal gearbox have also same proporeties of the mathematical model of ideal transformer.Although no actual systems are truly linear, control-system design leans heavily on linear models. This is understandable, since only for linear systems is there a generally complete set of available tools for design. However We must keep in mind that what linear models tell us is possible only may be possible only to a limited extent Physical limitations or nonlinearities, often provide constraints not obvious in the mathematical model.
##### Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996
##### Anahtar kelimeler
Benzetim, Elastik kirişler, Simulation, Elastic beams