Genel şekilli plakların serbest titreşimlerinin sonlu elemanlar kullanılarak hesabı - Şamhal, Erkan (Fen Bilimleri Enstitüsü, 1993)
Bu çalışmada ince dörtgen plakların serbest titreşimleri sayısal bir çözüm yöntemi olan Sonlu Eleman lar kullanılarak incelenmiştir. Plak denklemleri, iki boyutlu elastisite teorisi kullanılarak elde edilmiş olup klasik plak teorisi adı altında literatürde mevcuttur. Plağın ince ve kalınlık doğrultusunda ortaya çıkan çökmelerin küçük olduğu varsayılmıştır. Burada; cebrik denklem takımının; düğüm noktalarının bilinmeyen yerdeğiştirmeleri cinsinden ifade edildiği matris deplasman yaklaşımı ile beraber sonlu eleman meto du kullanılmıştır. Plak sistemi, sonlu sayıda dörtgen elemana ayrıldıktan sonra her düğüm noktasında iki adet dönme ve bir adet çökme serbestliğinin bulunduğu kabul edilmiştir. Plak eleman için çökme fonksiyonu, dördüncü dereceden bir polinom olarak elde edilmiş, iki boyutta Ç ve 7) tabi koordinatları kullanılarak her bir serbestliğe karşı gelen biçim fonksiyonları oluşturulmuştur. Her bir eleman için eleman rijitlik matrisi ve ardından sistem rijitlik matrisi elde edildikten sonra sistemin sınır şartları son elde edilen rijitlik matrisine 'Uygulanmış tır. Plağın serbest titreşimleri inceleneceğinden yük vektörünün sıfır olacağı bilinmektedir. Plağın sadece düşük birkaç modunun hesaplarla elde edilmesinin davranışın açıklanması bakımından yeterli olacağı görülmüş ve plak kütlesinin sonlu elemanların düğüm noktasında toplandığı kabul edilmiştir. Böylece kütle matrisi köşegen olarak elde edilmiştir. Hesapların yapılarak açısal frekans ve mod şekillerinin bulunması için özdeğer ve bunlara karşı gelen özvektörlerin çözülebildiği klasik-standart özdeğer yöntemi kullanılmıştır. Sözkonusu hesapların hassas biçimde ve kısa sürede yapılması.çeşitli tip ve mesnet şartlarındaki plaklara uygulanabilmesi için bir bilgisayar programı geliştiril miştir.

In this study, free vibration of thin general shaped isotropic plates is investigated by using finite elements method a well-known. A plate is a solid body, bounded by two parallel flat surfaces, whose lateral dimensions (the width and the length of a rectangular plate) are large compared to the distance (the thickness of the plate) between the flat surfaces. The theory of elastic plates is an approximation of three - dimensional elasticity theory to two - dimensions, and it can be used to determine the state of stress and deformation in plates. The approximations are simplifications that allow us only to consider deformation of the mid-plane of the plate. The plate thickness is measured in a direction normal to the mid-plane. The flexural properties of a plate depend greatly upon its thickness which is small in comparison with its other dimensions. Behaviour of plates may be classified into three groups: Thin plates with small deflections, thin plates with large deflections and thick plates. According to the criterion often applied to define a thin plate (for purposes of technical calculations ); the ratio of the thickness to the smaller span length should be less than 1/20. Here, only small deflections of thin plates are considered. Such a simplification, consistent with the magnitude of deformation, has been commonly found in plate structures. S It is assumed that plate is homogeneous and isotropic. A homogeneous material displays identical properties throughout. When the properties are the same in all directions, the material is called isotropic. Such a solution is concerned with establishing the relations of the strain, stress, displacements and finally plate equation. The small -def lection theory of bending of thin plates is known as the classical plate theory and is based on vi assumptions similar to those used in the Euler-Bernoulli theory (or the classical beam theory). The fundamental assumptions of the Kirchhof f-Love or classical plate theory are : 1) The displacements of the mid-surface are small compared with the thickness of the plate and, therefore, the slope of the deflected surface is very small and the square of the slope is negligible compared to unity. 2) The mid-plane of the plate remains unstrained, that is, the mid-plane of the plate is neutral plane after the bending. 3) Plane sections initially normal to the mid-surface remain plane and normal to the mid-surface after bending. Analogous to beams, this assumptions implies that the transverse shear strains e and e are negligible. The xz yz deflection of the plate is thus associated principally with bending strains. Consequently, the transverse normal strain e resulting from transverse loading can be neglected. 4) The transverse normal stress a is small compared with z the other stress components of the plate and, therefore, can be neglected. The finite element method can be regarded as an extension of the displacements method two and three - dimensional continuum problems, such as plates. In this method, the actual continuum is replaced by an equivalent idealized structure composed of discrete elements, referred to as finite elements, connected together a number of nodes. By assuming displacement fields or stress patterns within an element, it is possible, by the use of energy theorems, to derive as stiffness matrix relating the nodal forces to the nodal displacements of an element. The stiffness matrix of the assemblage of element is then generated. If the conditions of equilibrium are applied at every node of the idealized structure, a set of simultaneous algebraic equations can be formed, the solution of which gives all the nodal displacements, which in turn are used to determine all the internal stresses. The finite-element method may be also considered as a variational procedure in which the approximating functions are systematically derived by representing the given domain as a collection of simple subdomains. The vii method differs in two ways from the traditional variational methods in generating the algebraic equations of the problem. First, the approximating functions are often algebraic polynomials that are developed using ideas from the interpolation theory; second, the approximating functions for subdomains into which a given domain is divided. Since the approximating functions are algebraic polynomials, the computation of coefficient matrices of the algebraic eqations can be automatized on the computer. Here, after being discretized plate system into parts, it is assumed that there are three nodal displacements on every node. First, displacement term in lateral direction and the others, d and $ rotations, x y about y and x directions, respectively. Displacement function for rectangular plate is chosen fourth order polynomial as follows. 2 2 3 W = a +ax+ay+ax + a xy + a y +ax + 612 w 4 5 ' J İCT 11 J 12 J (1) The nodal displacements < 6 > are related to the e displacement within the element by means of a displacement function < W >. The latter is expressed in the following general form =CN3<6> where the braces indicate a column matrix and the matrix [ N ] is a function of position. This matrix is referred to as shape function. It is, of course, desirable that a displacement function be chosen such that the true displacement field be as closely represented as possible. When coefficients a through a in the displacement 1 " 12 r function are known, the equation having been written for W will provide the displacement at all locations in the plate. The nodal displacement can be written as follows : <<5> = [C] (3) e From the foregoing, the solution for the unknown constants is < a > = C C ]"*< 6 > (4) e Matrix [ C ] is dependent upon the coordinate dimensions of the nodal points. The displacements function may now viii be written in the form following -1 =[N3<6>=[L][C]<6> <5> e e e where [ L ] = 1,x,y,x2,xy,y2,x9,x2y,xy2,ya,x9y,xy9 (6) Strain-nodal displacement matrix t B ] can be written as follows -i IB1=<6><£> (7) e e After matrix C B ] is established, stiffness matrix of an element is obtained in the form + 1 +1 [ ke3 = t J J [ B ]T£ D ] t B 3 [J[ dÇdT? (8) -i -i Stiffness matrices obtained for each quadrilateral plate element are assembled for whole plate system. Then, the boundary conditions associated with supporting of plate system are applied on the system stiffness matrix. For the case of a free vibrating plate, the external force is zero, and the differential equation of the undamped motion becomes D v^^w + m = 0 (9) For the purpose of practical analysis, a vibrating plate often must be represented by an equivalent discrete system. In devising such a lumped-parameter substitute system which approximates the dynamic response of the real structure with satisfactory accuracy, the concepts of framework and finite element representations of elastic continua can be extended by adding the appropriate inertia terms. The convential procedure in construction of a mass matrix [ R 1 consists of simple lumping of the masses, associated with the tributary areas of the nodal points. This results in a diagonal matrix in the form ix m [ R ] = 11 O m <10) To solve free vibration of a plate using a numerical method, the expression which contains stiffness matrix §. »and [K3, angular frequency w., nodal amplitudes, matrix [ M 3 should be arranged by the form following mass ( K - b M )l. = 0 (11) Equation (11) has the form of eigenvalue problem. From the theory equation, nontrivial solutions exists determinant of the coefficient matrix is Thus, | K - «Z M | = 0 the algebraic of homogeneous only if the equal to zero. (12) Expansion of this determinant yields a polynomial order n called the characteristic equation. The n of roots t«>. I of this polynomial are the characteristic values, or eigenvalues. Substitution of these roots (one at a time) into the homogeneous equations £Eq(ll)l produces the characteristic vectors, or eigenvectors $,, within arbitrary constants. A computer program is developed so that the solutions can be obtained at a short time and precisely; and used for each plate having various supporting conditions.
Free vibrations of thin arbitrory shaped plates using finite element method
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1993
Keywords: Levhalar; Sonlu elemanlar yöntemi; Titreşim; Plates; Finite element method; Vibration