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Takviyeli dairesel silindirik kabuk yapıların serbest titreşimlerinin incelenmesi

Takviyeli dairesel silindirik kabuk yapıların serbest titreşimlerinin incelenmesi

##### Dosyalar

##### Tarih

1988

##### Yazarlar

Mecitoğlu, Zahit

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

Institute of Science and Technology

Institute of Science and Technology

##### Özet

Yapısal titreşimler hava ve uzay araçlarının yorulma dayanımı vb araç. içindeki mekanik ve elektronik aksamın sağlıklı olarak çalışması bakımından oldukça önemlidir. Yapının dinamik davranışının belirlenmesi için serbest titreşimlerin incelenmesi gerekmektedir. Bu çalışmada uçak ve roketlerin yarı-monokok gövde yapıları göz önünde bulundurularak bir takım takviye kirişleri ve çerçevelerle desteklenmiş silindirik kabuk yapının doğal -.frekansları ve mod şekilleri incelenmiştir. Kabuk yapı için birinci mertebe Love yaklaşımına dayalı ince kabuk teorisi kullanılmıştır. Hareketin varyasyonel denklemi Hamilton ilkesi yardımıyla yazılmıştır. Takviye parçalarının katkısı ortotropik malzeme yaklaşımıyla kapsama alınmıştır. Serbest titreşim probleminin çözümüne sonlu elemanlar yöntemiyle yaklaşım yapılmıştır. Bu amaçla izo- parametrik bir eleman olan sekiz düğüm noktalı semiloo-f eleman kullanılmıştır. Karakteristik matris denklem elde edilerek Sturm dizisi özellisine dayalı bisection yönte miyle özde>3erler elde edilmiştir. özvektörler tersiterasyonla bulunmuştur. Uçlarından kesme diya-f rami arıyla mesnetlenmi ş takviyeli silindirik yapıların doğal -frekansları ve mod şekilleri belirlenmiştir. Dolu dikdörtgen kesitli takviye parçalarıyla desteklenmiş bir silindirik kabuk için.frekansların takviye aralıklarıyla ve takviye parçalarının boyutlarıyla demişimi incelenmiştir. İkinci bir çalışma olarak pro-fii kesitli takviye kirişi ve çerçeveler kullanılmış ve -frekansların takviye aralıklarıyla, silindirik kabuğun boyu ile demişimi araştırılmıştır. Ayrıca üzerinde dikdörtgen şeklinde bir oyuk ve kalınlık artışı bulunan takviyeli silindirik kabukların doğal frekansları elde edilmiştir.

The importance of structural vibration to the operation o-f electronic and mechanical equipment, and to the strength o-f structural elements in a missile or aircraft, rests on the -fact that it supplies one of the environmental conditions in which these items must operate. The major adverse e-f-fects o-f vibration on these items are the shortening of life due to wearing or metal fat i que and the production of spurious actuations or signals which preclude proper performance of the component. Vibration of aircraft and missiles is produced by a complicated system of forces and dynamic loads. Some of these forces are applied to the structure of the vehicle by other solid bodies, such as the ground or the engines. Such forces are localised and are propagated mechanically through the vehicle by its structural members. Aerodynamic forces are aplied to the structure by the surrounding air and are spatially distributed over the surfaces of the vehicle. Thin shell structures are used in many branches of technology, such as aerospace engineering, building construction, mechanical. engineering, ship building, chemical engineering, and nuclear reactor engineering. The theory of shells is therefore an important, "'«hject in structural mechanics. A three dimensional linear elasticity analysis is undoubtedly the most accurate linear analysis possible, especially for very thick shells. However, even today with large-capacity, high-speed computers available, such an analysis is seldom used in structural dynamics except in the case of simple geometrical configurations such as a cylinder or sphere. In practice, linear thin shell equations are commonly used and the most of them are based on Kirchhof f -Love hypothesis E33. On the other hand, some theories were presented in order to improve Love's first approximation shell theory [11-143. Some shear theories has been developed considering to be inconsistent to omit the transverse shear strains in the Love second- approximation theory C 20-22 3. VI i Plate and shell type structures stiffened with beam type elements are extensively utilised in the constructions of aircraft, missile and ship. The analyses of stiffened shells may he conveniently divided into two categories; those which consider the stiffeners to be closely spaced in order to average or "smear" the stiffening effects over the surface of shell, and those which consider the stiffeners as discrete elements. A considerable body of literature dealing with the dynamics of stiffened and unstiffened shell structures has devoloped over last 35 years. An extensive review of the available literature until 1973 was given by Leissa C343. Some of the recent works and the detailed information related with the importance and classification of vibrations in aircrafts and missiles was also given in the introduction. The aim of this work was to investigate the free vibrations of the stiffened circular cylindrical shell structures parametrical ly. The circular cylindrical shells having a rectangular cutout and thickness discontinuity have been considered. Love's first- approximation theory is used for the shell structure. The effects of stiffeners are treated with orthotropic material approach and takig into account the eccentricity of stiffeners. The shear diaphragm end conditions are considered. The rest of the thesis has been organised as four chapters. In chapter 2» the fundamental equations of el astodynamics are derived. Problems of continuum mechanics have different but equivalent and interdependent global, local and variational formulations. All the formulations have their natural origin in global formulations established through the fundamental axioms of continuum mechanics. These axioms are expressed as conservation of mass, balance of momentum, balance of moment of momentum, conservation of energy, principle of entropy, conservation of charge, Faraday's law of induction and Ampere's law. The basic axioms are valid for all materials irrespective of their constitution. It is therefore expected that their mathematical expressions generally are not sufficient to predict uniquely the behaviour of all subtances under prescribed boundary and initial conditions» In order to fake account of the nature of different materials, some additional relations as known constitutive equations, are required..Furthermore, in the shell theory, certain assumptions must be made in order to reduce the shell geometry from three-dimensional one to a two-dimensional one. In the classical theory of small displacements of thin shells the assumptions that were made by Love may be summarised as fol lows s Thickness of the shell is small vi 1 1 compared with the other characteristic dimensions. Strains and displacements are suf f icientl y small so that, the first-order terms may be retained only in the strain- displacements relations» The theory is physically linear. fit point on the normal of the shell is on same normal of the deformed shell and its distance to middle surface of the shell remains unchanged. The transverse normal stress is small compared with other normal stress components and may be neglected. Under these assumptions and using the basic -axioms and constitutive relations we obtain the following strain energy expression for the shell structure Et Wk - - - - - t Et3 Î4 Shell constructions stiffened by means of discrete stringers and rings may be more efficient from weight standpoint than to use a sufficiently thick single- layered construction. The composite structure must be represented as a combination of the shell elements and stiff ener elements each having its own equations of motion and coupled to each other by equations of continuity. If the stiffeners are relatively closely spaced, it becomes desirable to simplify the analysis by considering them to be approximated by a stiffening sheet with certain bending, twisting and extensional properties. This representation can be accurately made for the purpose of determining free vibration frequencies and mode shapes, but not stress resultants. In order to obtain a theory that remains within the framework of the theory of the shell, we must make certain assumptions about the stiffeners. We assume that the stiffeners are concentrated along curvilinear coordinates. The stiffener dimensions are assumed as small compared to the principal radii of curvature of the shell. The normal strains in the stiffener and in the sheet are equal at their contact point. The stiffness of the stiffeners in-plane direction, perpendicular to their coordinates is equal to zero. The stiffeners carry torsional moment on account of their torsional rigidity. With these assumptions, the stiffener strain energy per unit middle surface area is, x EA a> i EA ss W«.- / \ c.1- / ^ e «» ''-' ır*r - '»n<;="" )ml="" +.="" <;="" style="margin: 0px; padding: 0px; outline: 0px;">"ai3Kla! ?- b b Eîy adding the stiff ener strain energy to the strain energy of the shell and applying the Hamilton's principle, the variational equation -for dynamical problem is obtained £W - «ST = 0 ' where W is the total strain energy of the stiffened shell per unit middle surface area and T is the kinetic energy o-f the sti-f-fened shell structure expressed as = *J eû^dv V The shear diaphgram end conditions -for a closed circular cylindricall shell o-f -finite length L, is mentioned as follows v = w = M" = NM = 0 for >i = 0, L Finite element analysis of thin shells has received considerable attention during recent years and a number of thin shell finite elements have been derived. The analysis of the problem formulated in chapter 2, is based on the semi 1 oof element developed by Irons C613. The element is non-conforming and basically adopts the well-known isoparametric S-noded parabolic model» Each node has three associated displacement components and some measure of C1 continuity is provided by the introduction of normal rotation variables of discrete points (Loof nodes) on the element periphery. The patch test is passed for plane elements with straight sides, and rigid body motions are satisfied exactly for any combinations of elements of any geometry. Kirchhoff 's assumptions of shell theory is applied as the transverse shears are constrained to be zero at selected points, chosen to be the Gaussian quadrature positions. These shear constraints are then employed to eliminate certain nodal variables. Bince the el merit does not accomodate lateral shear, it is restricted to thin shell situations. A variation in thickness in invidual elements and a thickness discontinuity between adjacent elements can be readily handled. Sharp edges, curved sides and surfaces, and multiply-connected regions are all allowed. Discretising the structure with the semi loof elements and applying the Hamilton's principle, following matrix equation is obtained -for the dynamic problems MQ + KQ = P If it. is assumed that, there is no external -force, P, and the displacements, G are harmonics, the dynamical problem reduces to the following real symmetric general eigenvalue problem to determinate the natural -frequencies and mode shapes o-f a structure CK - Q^MIQ = 0 where S"2 is the natural -frequency. This type of eigenvalue problems may be reduced to standard form using the Cholesky factorisation of M K0 = ftS20 If the mass matrix is not positive definite, the factorisation of the stiffness matrix may be used. The eigenvalues of the standard problem are determined by the method of bisection applied to the Sturm sequence. This method enables one to compute the eigenvalues lying a specified range without having to compute any other. Chapter four is devoted to the applications. The effects of the spacings and dimensions of stringer and rings on the natural frequencies of the stiffened circular cylindrical shell structures are investigated. Two different stiffened circular cylindrical shell models are used for parametric works. The first of them is stiffened by the stringers and/or rings with rectangular cross section and the other one is stiffened by the stringers- and/or rings with profile shape cross section. Both of the models used in application are discretised with 24 semiloof quadrilateral element, resulting 88 nodes and 376 degrees of freedom. The computer program has been coded in FORTRAN IV language. All the numerical computations are done on IBM 4341 digital computer. Evaluation of the results and the recommendations are presented at the last chapter. One may summarise them as follows: - Although in this work stiffeners are assumed to be "distributed" over the whole surface of the shell, the prediction of the natural frequencies is satisfactory. While the number of the elements increases, the results converge to the frequencies obtained experimental ly. - The results of the parametric works made by MODEL İ may be expressed ass The natural frequencies of the stringer stiffened shell are smaller than those of the unstiffened shell. When the stringers have been used together with the rings, it has been XI found that the -frequencies increase with the number of the stringers and are higher than those of the unstif f ened shell structure. If the number of stringers is hold to be constant, the frequencies- may exhibit increasing or decreasing -for different modes while the number of rings increases. - The results obtained MODEL 2 shows that in the stringer stiffened shell, increasing of the number of stringers causes increasing of the natural frequencies. The variations are generally linear. The changing of the frequencies with the number of rings has been found to be of different character. While the frequencies corresponding to some modes increases, some ones decreases. The variation may be also of different character.

The importance of structural vibration to the operation o-f electronic and mechanical equipment, and to the strength o-f structural elements in a missile or aircraft, rests on the -fact that it supplies one of the environmental conditions in which these items must operate. The major adverse e-f-fects o-f vibration on these items are the shortening of life due to wearing or metal fat i que and the production of spurious actuations or signals which preclude proper performance of the component. Vibration of aircraft and missiles is produced by a complicated system of forces and dynamic loads. Some of these forces are applied to the structure of the vehicle by other solid bodies, such as the ground or the engines. Such forces are localised and are propagated mechanically through the vehicle by its structural members. Aerodynamic forces are aplied to the structure by the surrounding air and are spatially distributed over the surfaces of the vehicle. Thin shell structures are used in many branches of technology, such as aerospace engineering, building construction, mechanical. engineering, ship building, chemical engineering, and nuclear reactor engineering. The theory of shells is therefore an important, "'«hject in structural mechanics. A three dimensional linear elasticity analysis is undoubtedly the most accurate linear analysis possible, especially for very thick shells. However, even today with large-capacity, high-speed computers available, such an analysis is seldom used in structural dynamics except in the case of simple geometrical configurations such as a cylinder or sphere. In practice, linear thin shell equations are commonly used and the most of them are based on Kirchhof f -Love hypothesis E33. On the other hand, some theories were presented in order to improve Love's first approximation shell theory [11-143. Some shear theories has been developed considering to be inconsistent to omit the transverse shear strains in the Love second- approximation theory C 20-22 3. VI i Plate and shell type structures stiffened with beam type elements are extensively utilised in the constructions of aircraft, missile and ship. The analyses of stiffened shells may he conveniently divided into two categories; those which consider the stiffeners to be closely spaced in order to average or "smear" the stiffening effects over the surface of shell, and those which consider the stiffeners as discrete elements. A considerable body of literature dealing with the dynamics of stiffened and unstiffened shell structures has devoloped over last 35 years. An extensive review of the available literature until 1973 was given by Leissa C343. Some of the recent works and the detailed information related with the importance and classification of vibrations in aircrafts and missiles was also given in the introduction. The aim of this work was to investigate the free vibrations of the stiffened circular cylindrical shell structures parametrical ly. The circular cylindrical shells having a rectangular cutout and thickness discontinuity have been considered. Love's first- approximation theory is used for the shell structure. The effects of stiffeners are treated with orthotropic material approach and takig into account the eccentricity of stiffeners. The shear diaphragm end conditions are considered. The rest of the thesis has been organised as four chapters. In chapter 2» the fundamental equations of el astodynamics are derived. Problems of continuum mechanics have different but equivalent and interdependent global, local and variational formulations. All the formulations have their natural origin in global formulations established through the fundamental axioms of continuum mechanics. These axioms are expressed as conservation of mass, balance of momentum, balance of moment of momentum, conservation of energy, principle of entropy, conservation of charge, Faraday's law of induction and Ampere's law. The basic axioms are valid for all materials irrespective of their constitution. It is therefore expected that their mathematical expressions generally are not sufficient to predict uniquely the behaviour of all subtances under prescribed boundary and initial conditions» In order to fake account of the nature of different materials, some additional relations as known constitutive equations, are required..Furthermore, in the shell theory, certain assumptions must be made in order to reduce the shell geometry from three-dimensional one to a two-dimensional one. In the classical theory of small displacements of thin shells the assumptions that were made by Love may be summarised as fol lows s Thickness of the shell is small vi 1 1 compared with the other characteristic dimensions. Strains and displacements are suf f icientl y small so that, the first-order terms may be retained only in the strain- displacements relations» The theory is physically linear. fit point on the normal of the shell is on same normal of the deformed shell and its distance to middle surface of the shell remains unchanged. The transverse normal stress is small compared with other normal stress components and may be neglected. Under these assumptions and using the basic -axioms and constitutive relations we obtain the following strain energy expression for the shell structure Et Wk - - - - - t Et3 Î4 Shell constructions stiffened by means of discrete stringers and rings may be more efficient from weight standpoint than to use a sufficiently thick single- layered construction. The composite structure must be represented as a combination of the shell elements and stiff ener elements each having its own equations of motion and coupled to each other by equations of continuity. If the stiffeners are relatively closely spaced, it becomes desirable to simplify the analysis by considering them to be approximated by a stiffening sheet with certain bending, twisting and extensional properties. This representation can be accurately made for the purpose of determining free vibration frequencies and mode shapes, but not stress resultants. In order to obtain a theory that remains within the framework of the theory of the shell, we must make certain assumptions about the stiffeners. We assume that the stiffeners are concentrated along curvilinear coordinates. The stiffener dimensions are assumed as small compared to the principal radii of curvature of the shell. The normal strains in the stiffener and in the sheet are equal at their contact point. The stiffness of the stiffeners in-plane direction, perpendicular to their coordinates is equal to zero. The stiffeners carry torsional moment on account of their torsional rigidity. With these assumptions, the stiffener strain energy per unit middle surface area is, x EA a> i EA ss W«.- / \ c.1- / ^ e «» ''-' ır*r - '»n<;="" )ml="" +.="" <;="" style="margin: 0px; padding: 0px; outline: 0px;">"ai3Kla! ?- b b Eîy adding the stiff ener strain energy to the strain energy of the shell and applying the Hamilton's principle, the variational equation -for dynamical problem is obtained £W - «ST = 0 ' where W is the total strain energy of the stiffened shell per unit middle surface area and T is the kinetic energy o-f the sti-f-fened shell structure expressed as = *J eû^dv V The shear diaphgram end conditions -for a closed circular cylindricall shell o-f -finite length L, is mentioned as follows v = w = M" = NM = 0 for >i = 0, L Finite element analysis of thin shells has received considerable attention during recent years and a number of thin shell finite elements have been derived. The analysis of the problem formulated in chapter 2, is based on the semi 1 oof element developed by Irons C613. The element is non-conforming and basically adopts the well-known isoparametric S-noded parabolic model» Each node has three associated displacement components and some measure of C1 continuity is provided by the introduction of normal rotation variables of discrete points (Loof nodes) on the element periphery. The patch test is passed for plane elements with straight sides, and rigid body motions are satisfied exactly for any combinations of elements of any geometry. Kirchhoff 's assumptions of shell theory is applied as the transverse shears are constrained to be zero at selected points, chosen to be the Gaussian quadrature positions. These shear constraints are then employed to eliminate certain nodal variables. Bince the el merit does not accomodate lateral shear, it is restricted to thin shell situations. A variation in thickness in invidual elements and a thickness discontinuity between adjacent elements can be readily handled. Sharp edges, curved sides and surfaces, and multiply-connected regions are all allowed. Discretising the structure with the semi loof elements and applying the Hamilton's principle, following matrix equation is obtained -for the dynamic problems MQ + KQ = P If it. is assumed that, there is no external -force, P, and the displacements, G are harmonics, the dynamical problem reduces to the following real symmetric general eigenvalue problem to determinate the natural -frequencies and mode shapes o-f a structure CK - Q^MIQ = 0 where S"2 is the natural -frequency. This type of eigenvalue problems may be reduced to standard form using the Cholesky factorisation of M K0 = ftS20 If the mass matrix is not positive definite, the factorisation of the stiffness matrix may be used. The eigenvalues of the standard problem are determined by the method of bisection applied to the Sturm sequence. This method enables one to compute the eigenvalues lying a specified range without having to compute any other. Chapter four is devoted to the applications. The effects of the spacings and dimensions of stringer and rings on the natural frequencies of the stiffened circular cylindrical shell structures are investigated. Two different stiffened circular cylindrical shell models are used for parametric works. The first of them is stiffened by the stringers and/or rings with rectangular cross section and the other one is stiffened by the stringers- and/or rings with profile shape cross section. Both of the models used in application are discretised with 24 semiloof quadrilateral element, resulting 88 nodes and 376 degrees of freedom. The computer program has been coded in FORTRAN IV language. All the numerical computations are done on IBM 4341 digital computer. Evaluation of the results and the recommendations are presented at the last chapter. One may summarise them as follows: - Although in this work stiffeners are assumed to be "distributed" over the whole surface of the shell, the prediction of the natural frequencies is satisfactory. While the number of the elements increases, the results converge to the frequencies obtained experimental ly. - The results of the parametric works made by MODEL İ may be expressed ass The natural frequencies of the stringer stiffened shell are smaller than those of the unstiffened shell. When the stringers have been used together with the rings, it has been XI found that the -frequencies increase with the number of the stringers and are higher than those of the unstif f ened shell structure. If the number of stringers is hold to be constant, the frequencies- may exhibit increasing or decreasing -for different modes while the number of rings increases. - The results obtained MODEL 2 shows that in the stringer stiffened shell, increasing of the number of stringers causes increasing of the natural frequencies. The variations are generally linear. The changing of the frequencies with the number of rings has been found to be of different character. While the frequencies corresponding to some modes increases, some ones decreases. The variation may be also of different character.

##### Açıklama

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1988

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1988

Thesis (Ph.D.) -- İstanbul Technical University, Institute of Science and Technology, 1988

##### Anahtar kelimeler

Roketler,
Serbest titreşimler,
Uçaklar,
Rockets,
Free vibrations,
Airplanes