Kalın Çubukların Titreşim Frekansları Ve Dinamik Stabilitesinin Sonlu Elemanlar Yöntemiyle Analizi

Abstract

Bu çalışmada Sonlu Elemanlar Yöntemi kullanılmak suretiyle kesit dönme ataletlerinin hesaba katıldığı elastik ince çubukların; dönme ataleti ve kayma şekil değiştirmelerinin hesaba katıldığı 4 ve 6 serbestlik dereceli iki ayrı kalın çubuk elemanının, değişik mesnet şartlarında titreşim analizi ve statik ve dinamik halde burkulma yükleri incelenmiştir. Çubukların titreşim analizi, ankastre-ankastre, ankastre-serbest, basit-basit mesnet hallerinde boyuna uzama da hesaplara dahil edilerek incelenmiştir. Bu analizde kesit dönme ataleti ve kayma deformasyonu hesaba katılmış ve bunların etkileri de incelenmiştir. Boyuna uzama için lineer sonlu elemanlar kullanılmıştır. Yine dönme ataleti etkisinde ankastre- serbest halde çubuğun serbest ucundaki nokta kütle etkisi de hesaplanmıştır. Bunun yanında, ele aldığımız kalın çubuklardan 4 serbestlikli için, ankastre-serbest, ankastre-ankastre, basit-basit mesnet hallerinde serbest titreşim frekansları elde edilmiştir. 6 serbestlikli için, sadece basit-basit mesnet halinde serbest titreşim frekansları elde edilmiştir. Buna ek olarak, ele aldığımız kalın çubuklardan 4 serbestlikli için, ankastre- serbest, ankastre-ankastre, basit-basit mesnet hallerinde burkulma yükleri elde edilmiştir. 6 serbestlikli için, sadece basit-basit mesnet halinde burkulma yükleri elde edilmiştir. Kalın çubukların kesit kalınlığı arttıkça değişen serbest titreşim frekansları ve bu frekanslara etki eden dönme ataleti ve kayma etkileri incelenmiştir. Kalın çubukların ince ve kalın kenarına göre serbest titreşim frekansları ile bu frekanslara etki eden dönme ataleti ile kayma etkileri incelenmiştir. Teorik çalışmalar için birisi sert plastik diğeri çelik olmak üzere iki çubuk kullanılmıştır. Çubukların elastik modülleri daha önce yapılan deneysel olarak elde edilen bir çalışmadan alınmıştır. XIV Teorik çalışmalar sonucu elde edilen çeşitli parametrelerin etkileri tablolarda gösterilerek sonuçlar yorumlanmış ve dizayn açısından önlemleri gösterilmiştir. Sonuçta, dönme atalet etkisi sebebiyle ince çubuklarda serbest titreşim frekanslarının azaldığı incelendi. P eksenel kuvveti etkisi altında, kuvvet arttıkça frekansların azaldığı incelendi. P eksenel kuvveti kritik burkulma yüküne eşit olduğunda 1. özfrekansın O'a eşit olduğu incelendi. Bunun yanında, ankastre-serbest halde nokta kütle etkisinde serbest titreşim frekanslarının azaldığı gözlendi. Dönme atalet etkisi ve kayma deformasyonları sebebiyle kalın çubuklarda serbest titreşim frekanslarında ve burkulma yüklerinde bir azalma görülmüştür. Bu azalmanın ihmal edilemeyecek derecede büyük olduğu gözlendi. Sadece dönme ataleti etkisi sebebiyle frekanslardaki azalmanın ihmal edilecek derecede küçük olduğu elde edildi. Dönme atalet etkisinin dinamik halde burkulma yüklerinde azalmaya sebep olduğu fakat statik halde burkulma yüklerine etki etmediği görülmüştür. Bunun yanında dinamik halde devir sayısı arttıkça burkulma yüklerinin azaldığı ve kritik devir sayısında burkulma yükünün 0 olduğu görüldü. Kalın çubuklarda çubuk kütlesi etkilerine dahil edilen kayma deformasyonunun, çubuğun geometrik matrisine etki eden kayma deformasyonundan ve aynı zamanda dönme atalet etkilerinden fazla olduğu görülmüştür.

The vibration frequencies and stability of rectangular cross-section uniform beams, considering shear deformation and rotatory inertia are analyzed by the finite element method. The Finite Element Model approximates a structure in two different ways. The first approximation made in finite element modelling is to separate the structure into number of discrete elements. These discrete elements are called finite elements. For each element, differential equation of motion is determined and numerically solved for whole system. In the analysis of free vibration frequencies and stability, for elastic thin beam, rotatory inertia is considered. In these analysises, two different thick beam elements are used. The first 4 degrees of freedom thick beam element has two nodes, with three degrees of freedom at each node. In the first thick beam element, Wi and w2 are transverse displacements at nodes 1 and 2, and j(T-U)dt+J* Sj(T-U)dt+j8Adt = 0 (6) ti t, where T is kinetic energy, U is potential energy and 8A is virtual work done by active forces. As a result, the equation of motion of the system can be written as: [[K]-[M]of]{8}i = 0 (7) where [M] is the mass and [K] is the stiffness matrices. For a beam undergoing transverse vibration, the potential energy can be given as: Vew=^El}[ 05f* <8> For thin beams and thick beams with four degrees of freedom, the potential energy stored in the element due to stresses created by axial force P used in the buckling problems is defined as follows: *0 ^2 VdxJ dx (9) For a beam undergoing axial vibration, the potential energy is as follows: XVlll Veu^JE/{ dur dx (10) For a beam undergoing shear deformation, the potential energy can be written as: Ve^JkGAv^d) (11) For thick beams with six degrees of freedom, the potential energy stored in the element due to stresses created by axial force P used in the buckling problems is defined as follows: Vg=^plJ(p2dx * o (Pl->P) (12) For thin beams and thick beams with four degrees of freedom, the total stiffness matrices is formed as follows: M= [0] [Ke] 2*2 eu (13) 6*6 For thick beams with six degrees of freedom, the total stiffness matrice is also formed as: Ke] kd6 1*1 eu [Ke] [Ke]1"" L ejew [Ke] Ik 1*1 eu ef3 ejew (14) 8*8 XIX In order to obtain the mass matrix, the kinetic energy which is equal to transverse bending deflection, H dw>2 Tew = 2pAjl"d7|dx (15) o For a beam undergoing axial deflection, the kinetic energy ca be given as follows: Teu=öPAJ 2" j^dt 2 dx (16) For a beam with rotatory inertia, the kinetic energy can be given as: Tewd4pljfeTdx (17) 2 o^dt The kinetic energy due to the point mass on free end of clamped-free beam is given below: 1mfdx(L)Y Tek=öm- - (18) 2 V dt The element matrices given above are combined to form the global stiffness and mass matrices according to FEM formulation. Using these matrices the system equations are formed as: [[KMMK^SJ-O (19) and solved numerically. xx The buckling loads of the beam can be calculated as follows: P = X2^ (20) Pcr is called the critical load and when P is equal to Pcr, the first free vibration frequency becomes zero. In that case, the beam takes a non-zero but static deflected shape. In the buckling problem, the critical buckling loads of an elastic structure are investigated. This corresponds to the calculations of eigenvalues of the generalized eigenvalue problem. For thin beams and thick beams with four degrees of freedom, in the case of a structure subjected to an initial stress field ax, the equation (9) must be added to the strain energy of the system. For thick beams with six degrees of freedom, when a structure subjected to initial shear stress field ixy, the equation (12) must be added to strain energy of the system. Since the geometric stiffness expression has the same form as in the linear stiffness one, it is obvious that the linear equation of motion are replaced in the prestressed case by AMfl+[K|{8},+[MI{6},=0 (21) and related to eigen value problem takes the form (In the case of harmonic motion). XIK]0 +[KJ]{8}, -rflMKSlı = 0 (22) when Q=0, we get an eigenproblem with eigenvalues h corresponding to the critical loads. XXI [Kl + XlK]fl {8},=0 (23) The eq(23) consists an eigenvalue problem in which we seek to determine the eigenvalues Xi (1=1,2 ). The eigenvalue Xmin gives the prestress state in which the system buckles. For theoretical calculations, two different beams which one is a plastic beam and the other is a metal beam are used. As a result of this study, the free vibration frequencies and buckling loads of elastic thick beams, considering rotatory inertia and shear deformation are examined by the finite element model. Both buckling and vibration analysises are implemented for different boundary conditions. These conditions are simple-simple beam, clamped-free beam and fixed- fixed beam. In vibration analysis, the free vibrations of elastic thin beam considering only rotatory inertia are obtained by numerical calculations. Because of rotatory inertia, it is observed that the free vibration frequencies decrease considerably. But this decrease is so small that it can be neglected. For an elastic thin beam, considering only rotatory inertia, the beam is subjected to an axial force P. When the axial force P increases, the free vibration frequencies of the beam decrease. When the axial force P is equal to the critical buckling load of beam, the first eigenfrequency becomes zero. The eigenfrequencies are symmetrically lowered by a compression load and increased by an axial tensional force. For thick beams, considering rotatory inertia and shear deformation, the free vibration frequencies are investigated, in conclusion, the eigenfrequencies decreased substantially. These decreases aren't neglected. Consideration of shear deformations in thick beam are made in two ways. In the first study, shear deformation regarded only in stiffness matrice [K]. In the second study, shear deformation regarded both in stiffness matrice [K] and mass matrice [M]. It is obtained that the shear deformation xxii effects due to the terms in mass matrice [M] is higher than the terms in stiffness matrice [K]. In thick beams, the vibrations are also examined in two ways, these are flapwise and edgewise vibrations. In conclusion, it is determined that shear deformation and rotatory inertia in vibrations of edgewise motion of the beam are higher than flapwise one. In the buckling analysis, the critical buckling loads of the beams considering shear deformation and rotatory inertia are obtained by the numerical analysis. In the dynamic case, the first buckling load is equal to zero, when the beam is excited by the first eigenfrequency. In the dynamic case, when the revolution speed increases, the buckling loads are lowered. In the static analysis, the buckling loads of the beams considering only rotatory inertia are constant, i.e. rotatory inertia doesn't affect the buckling loads of the beam. In the static and dynamic analysis, the buckling loads of the beams considering both shear deformation and rotatory inertia are investigated. As a result of this, it has been observed that the buckling loads of the beams decrease substantially. These decreases aren't neglected. In the present work as a comparison, X frequency parameters which are used in our thesis related to thick beams, a frequency parameters which are taken from M. İPEK's research report and associated with thick beams are also compared and a good agreement has been found.