Elastik krank-biyel mekanizması titreşimlerinin sonlu elemanlar yöntemiyle incelenmesi

Abstract

Bu çalışmada amaç, elastik krank-biyel mekanizması nın titreşim karakteristiklerinin sonlu elemanlar yönte miyle incelenmesi ve sistemde yer İTİ an çeşitli,frekans par amet reler inin yine dizayn sırasında kullanılacak olan sistem parametrelerinden(literatürlerde daha önce söz konusu edilebilen çalışmalara ek olarak) piston ile si lindir arasındaki lineer yay etkileri, boyuna yöndeki yer değiştirmeler, boyuna kuvvet etkileri ve krank elas tikliği dahil analiz edilmesidir. Bu amaca bağlı kalarak, öncelikle krank biyel mekanizmasının davranışlarını yete rince ifade edecek bir matematiksel model oluşturulmuştur. Bu model oluşturulurken, krank açısal hızının sabit olduğu kabul edilmiş, piston kütlesinden dolayı krank ve biyele etkiyen normal kuvvetlerin mekanizmaya etkileri göz önüne alınmıştır. Ayrıca piston ile silindir arasındaki lineer yay etkileri de göz önüne alınmıştır. Titreşim analizi yapılırken, rijit krank elastik biyel ve elastik krank biyel olmak üzere inceleme yapılmış ve bu şekilde elde edilen titreşim karakteristikleri < titreşim öz frekans ları ve titreşim biçimleri) arasında karşılaştırma yapı larak, çeşitli parametrelerin Cpiston yay katsayısı, piston kütlesi, devir sayısı, krank açısı değişimi) etkileri incelenmiştir. Her iki karşılaştırma modeli için fortran IV programlama diliyle programlar hazırlan mış ve sonuçlar gerek tablolar, gerekse grafik şeklinde gösterilerek analiz edilmiştir

In the present work, vibration characteristics of crank -connecting rod-slider mechanism, are analysed con sidering bending and longitudinal deformations of crank and coupler, and thin beam theory, by the finite element method. First, some details about application of the fini te element method, and then, the details of the vibration analysis of the elastic crank -si i der mechanism are summa rized as follows ; General Considerations About Finite Element Method The increasing complexity of structures and sophis tication of digital computers have been instrumental in the development of new methods of analysis, particularly of the finite element method. The idea behind the finite element method is to provide a formulation which can ex ploit digital computer automation for the analysis of ir regular systems. To this end, the method regards a comp lex structure as an assemblage of finite elements, where the points are known as nodes, the entire structure is compelled to act as one entity. Although the finite element method considers conti nuous individual elements, it is in essence a discretiza tion procudure, as it expresses the displacement at any point of the continuous element in terms of finite number of displacements at the nodal points multiplied by given interpolation functions. To illustrate the idea, we refer to the one- dimensional system shown in fig. 1. The system is divided into a finite number n of elements of width 1, where n. 1 * L, and the motion of the system is defined in terms of the "nodal displacement" UiCO Ci =1, 2,..n3. The advantage of finite element method over any other method is that the equations of motion for the sys tem can be derived by first deriving the equations of mo tion for a typical finite element and then assembling the individual elements 'equations of motion. The motion at any point inside the element is obtained by means of in terpolation, where the interpolation functions are gene rally low-degree polynomials and they are same for &sr&ry element. X Fig. 1. Interpolation Functions - XI The finite element method» as practiced today, began as a method of structural analysis, being related to di rect stifness method. This direct approach may be satis factory for static problems, but encounters difficulties in handling dynamic problems, such as in vibrations of con tinuous media. Such problems are treated better by a va riational approach. In fact, the finite element method can be regarded as a special case of Rayleigh-Rltz method, although since its inception the method has acquired a li fe of its own, going well beyond the original structural appl i cat i ons. Our purpose is to present some of basic ideas invol ved in the use of the finite element method for vibration problems rather than a detailed treatment of the subject. We shall be concerned only one-dimensional elements, al though the concepts and developments presented quite ge neral and can readily be applied to two-and three-di men- si onal el emen t s. - - Derivation Of The Element Mass» Stiffness Matrix By The Direct Approach We adopt an approach by deriving the element mass and stiffness matrix as the matrix relating the nodal for ce vector. To this end, we consider a rod in axial vib ration and derive the mass and stiffness matrix for a ty pical element such as in fig. 2. We carry out the task in two steps. In the first step we derive an expression for the axial displacement of an arbitrary point inside the element in terms of the nodal displacements and in the second step we use this expression to relate the nodal displacements to the nodal forces. Although, in vibrati ons all these quantities are functions of time, for the purpose of deriving the stiffness matrix, they can all be regarded as constant in time. i Ç(t) U, Ui"* f2ct>. fta/t) A Fig. 2. Derivation Of Axial Stiffness And Mass Matrix The axial stiffness can be assumed to be constant over the element, so that the differential equation for the axial displacement UCx) is : EA 32U ax2 = O CO < x <1 D. C1D xxx - By integrating CI D twice we obtain UCxD = Cz.x + Ci C23 Where Ci and Cz are constants of integration. Using boun dary conditions in fig. 1 UCCO = Ui ; UC1Z) = Uz C3D C4Z> " Uz- Ui Ci =. ; Cz = Ui inserting the constants just obtained into eq. C2Z>, we ob- obtain the expression for the axial displacement ; U = CI --*->. Ui. + *-Uz. C5Z> We can expression kinetic and potential energy for the e- lement as ; " _ 1 f! «U.2 _ 1 fV ÖU _: Ek ~ 2"mJC -q^-) i Ep - 2-EAU -^-) C6D Using Lagrange equation C73 in eq. C5D, C6Z> we obtain C M ] and C K 3 as follows ; d f 3L dt^ Oq ÖL 7> --g- = Ffc Ck = 1,2..nZ> ; L = E^ -E C73 [ M 3 = -T2 'l.,,,.jaT1 -1! 6^1 2J ' K J TT^-i lj C8D where C M 3 is mass matrix and L K 3 is stiffness matrix The another case of particular interest is the bar in bending vibration. We propose to use the same approach as above to derive the corresponding element shown in fig. 3 I i i.u*«> itKx,t> U{*> U Fig. 3. Derivation Of Bending Stiffness And Mass Matrix For uniform bending stifness, the differential equation for the displacement UCx.tZ) is EI ox4 CO < x <1 D C9Z) Xlll - Integrating eq. C8Z> four times, we have UCxD = -|-Ci.xa+-|-C2.x2+ Ca.x + C*. C1CÖ To determine these constants, we refer to fig. 3 and write UCOZ) = Ui; UC03 = Uz; UC13 = Ua; I^ID = U* C11D where Ui. Ua are nodal displacements and Ua, U4 are nodal rotations, or nodal angular displacements. Introducina eq.ClOT into eq. CUD we obtain the expression for the ben ding displacement as follows ; U = CI ~ 3C -fo2 + 2C -f Z>a)Ui +Çx -21C J^O2 + 1C J*0 9;> u2 +C3C^D2 - 2C^^)Ua +Ç-lC^-f +1C^>U* C12> In equations C6D, C125 using lagrange eq. C73, we obtain t M j and t K 1 as follows : CM3 = ml 420 156 221 54 -131 221 41 2 131 -31 2 54 131 1S6 -221 -131 -312-221 41 2 ;CK1 = The another case is axial and bending vibration of a bar. To this end, we will use a way which called super position method. At this method we consider the bar is consisting axial and bending vibrations, so we seperate these vibrations and derive mass and stiffness matrix I", iUCt) i.lK%-t> Cx/fc)6 jr uwjiyi^ X Fig. 4. Derivation Of Axial 4 Bending Stiffness And Mass Matrix If we consider the bar as consisting only axial vibration the mass and stiffness matrix are as follows ; xiv - C140 If we consider the bar Is consisting only bending vibra tion, the mass and stiffness matrix are as follows ; C153 If we sum eq. CI 43 and C15D using superpozision method, t M 3 = C M 1+ i M ] b e £M3' ml 420 140 O O 70 O O t K 3 = t K O 156 221 O 54 -131 -31 221 41 2 O 131 2 70 O O O 54 131 140 O O 156 O -221 O -131 -31 2 O -221 41 2 ]+ C K 3 CK3 El Cl/rD O O O 121 61 C-l/rD O O -12 O 61 O 61 41 : O -61 21 : c-ı/rr o o ClXr32 O O -12 -61 O 12 -61 O 61 21 ; O -61 41 C163 C where r is rotation radius of cross section area r = I/A> - xv - Analysis Of Elastic Mechanism The finite element theory of structural analysis has been applied to modelling an elastic linkage using force method as well as displacement method. The lumped parame ter approach has also been a useful technique. The rigid-body kinematic analysis of the linkage is dependent on its input motion characteristic as well as on its geometry. Due to elastic deformations of the members, the effective link lengths as well as their resulting angu lar positions vary from those of a purely rigid linkage. If small elastic deformations are assumed» the effects of elastic deformations on the rigid-body kinematic equations are negligible. The assumption that the elastic motions of the members do not affect the rigid-body kinematics of the linkage, has been made extensively by researches. Each member of linkage is idealized by one beam element only. This, in conjuction with the assumed mode shapes, permits at most one inflection point across the length of the element and limits the number of modes can be correct ly computed. Also, one-element Idealization results in lost of higher model frequencies. While one element idea lization is used for simplicity, mul ti -el ement idealization of each member must be used for a more accurate modelling. In the force finite el ement method, the internal for ces are assumed to be unknowns and equilibrium conditions are used to generate equations. Additional equations are sometimes essential and are derived with the help of com patibility conditions. The displacement finite element met hod, on the other hand, utilizes the nodal displacements as the unknowns of the problem. The compatibility conditi ons in and among the elements are initially satisfied. The equations are expressed in terms of these nodal disp lacements utilizing the equilibrium conditions at each no dal point. The displacement finite element method formu lation is in general found to be much simpler for a larg er number of structural problems. For this reason, this latter method will be used here to develop the equations of motion. A general beam element is shown in fig. 3 in two fra mes of reference. These are fixed (O-X-Y) and the rotating (o-x-y) frames, the rotating frame of reference being such that the x-axis parallel to the beam element axis. The elastic deformations of the beam element may be complete ly described by six nodal displacements, Ui through U