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Sonlu elemanlar yöntemi ile titreşim analizi

Sonlu elemanlar yöntemi ile titreşim analizi

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Tan, Ayhan

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Birinci bölümde mühendislikte kullanılan yapılardan. yapıların dinamik ve statik yükler altındaki davranışlarından bahsedilmiştir. ikinci bölümde sonlu elemanlar yöntemi genel hatlarıyla anlatılmıştır. Yöntemin kısa tarihçesi, yöntemin aşamaları, ve uygulama alanları kısaca anlatılmıştır. Uçüncü bölümde Hamilton prensibi ve Langrange denklemleri genel olarak anlatılmıştır. Dördüncü bölümde kirişlerin ve eksenel elemanların enerji ifadeleri, bu elemanların hareket denklemleri ve sınır koşulları bulunmuştur. Beşinci bölümde sonlu elemanlar yöntemi kullanılarak anlatılan elemanların modları ve frekansları bulunmuştur. Ayrıca Rayleigh-Ritz yöntemi anlatılmıştır. Altıncı bölümde ise zorlanmış titreşimin ana denklemi verilmiş ve periyodik Yüklemeden bahsediliniştir. Yedinci bölümde ele alınan problemler anlatılmıştır. Sekizinci bölümde ise sonuçlar ve değerlendirmeler yapılmıştır. Ekler 'de Ansys5.0 sonlu eleman yazılımı anlatılmış programları ve komutları verilmiş, İncelenen yapılardan kirişin doğal frekansları, mod şekilleri, ve periyodik bir yük altındaki şekil değişimi incelenmiştir. Halkanın ise iç ve dış basınç altında yerdeğiştirmeleri incelenmiştir. Bu incelemelerden elde edilen sonuçlar gösterilmiştir.

in this thesis, energy functions of an axial element and beam bending ele¬ ment and axial vibration of rods, bending vibration of beams and finite element displacement method have been reviewed. in the first chapter, the importance of the computional methods and finite elements method has been introduced briefiy. And behavior of structures under load have been reviewed. in the second chapter, a brief history of finite element displacement method, fundamentals of the finite displacement method and its practice areas have been introduced. in the third chapter, Hamilton's principle and Lagrange's equations have been introduced. Equations of motion have been derived by using Hamilton's principle. Ha¬ milton's principle states that, the actual path followed by a dynamical process is such as to that the condition / 2 (6(T -U) + 8Wncdt = O (1) J ti is satisfied. That is, the integral of (T - 17) + Wnc takes an extremum value which can be shown to be a minimum. When Hamilton's principle is applied to discrete systems it can be expressed in a more convenient form. (M}{q} + [C]{q} + (K}{q} = {Q} (2) This is called Langrange's equations where {q} = column matrix of system displacements vi {q} = column matrix of system velocities [M] = square symmetric matrix of inertia coefficients [C] = square symmetric matrix of damping coefficients [A'] = square symmetric matrix of stiffness coeficients In the fourth chapter, the energy functions are derived for various structural elements. Equations of motion of any structural system can be obtained from the energy function of the system. The energy function consists of the strain energies, the dissipation function and the work done by the applied loads. In this chapter, energy functions, which consist of strain and kinetic en ergies, the dissipation function and the work by the applied loads for axial element and beam bending element have been derived. For an axial element of constant cross- sectional area A and length 2a, strain energy, kinetic energy and virtual work have been derived as follows: v=\S7EA(^?dx (3) pAü2dx (4) = / pxSudx (5) J a where 17 = the strain energy of the system T = the kinetic energy of the system 8W = virtual displacement In deriving the energy functions for a beam bending element it is assumed that the vibration occurs only in one of the principal planes of the beam. The beam, which is of length la and has a constant cross-sectional area A, strain energy, kinetic energy and work have been derived as follows: u=\rEi^dx (6) vii 6W where Ix= f y2dA (7) JA T = \ f a PAv2dx (8) ^ J a SW= f Py8vdx (9) J a in this section equation of motion and the boundary conditions of continuous structural elem.en.ts have been fornıulated using the Hamilton's principle. Equation of motion and the boundary condition for an axial element are derived as follows: EA^-/>A^+"' = ° ^ throughout the region -a < x < a, either du EA^- = Q ör 6u = O (11) dx at both ends x = -a and x = a. Equation of motion and boundary condition for a beam bending element are derived as follows: -EI'^-"AW+P' = 0 (12) throughout - a < x < a. in addition, öne from each of the following two set of condition £^=° - !=° w d3v EIZ- = Q ör u = 0 (14) must be satisfied at both ends z = - a and x = a. Note that, in this case two condition are required at each boundary. The conditions v = O and |^- = O represent the vanishing of the displacement and the slope respectively. in the fifth chapter, Rayleigh-Ritz method and finite displacement method have been introduced in details. The response of simple structures, such as uniform axial and beam bend¬ ing elements have been obtained by solving differential equations of motion viii together with the appropriate boundary conditions. in many practical situa- tions eith.er the geometrical ör material properties vary, ör it may be that the shape of the boundaries cannot be described in temas of known functions. Also, practical structures consist of an assemblage of components of diferrent types, namely, beams, plates, shells and solids. in these situations it is inapossible to obtain analytical solutions to th equations of motion which satisfy the bound¬ ary conditions. This difficulty is overcome by seeking approximate solutions which satisfy Hamilton's principle. There are a nunaber of thecniques available for determining approximate solutions to Hamilton's principle. Öne of the most widely used procedures is the Rayleigh-Ritz method, which is described in this section. A generalisation of the Rayleigh-Ritz method, known as the finite element displacement method, is introduced. The principal features of this method are described by considering rods and beams. The Rayleigh-Ritz method approximates the solution with a finite expan- sion of the form n «B(*,0=5>X*)??(*) (14) i=ı where the qf(t} are unkown functions of time, t, and the Qj(x) are prescribed functions of a;, which are linarly indepent. A set of functions are linarly inde- pendant if n ^2 j(x} = O for ali z (15) j=ı implies that aj=Q for j = 1,2,...., n (16) Each of the functions k(x)dx (20) Jo fL Kjk = / EA^'kdx (21) Jo where prime denote differentitation with respect to x. Langrange's equations is solved for the {#"}, then an approximate solution are obtained for u(x, t). Integral of Hamilton's principle involve derivatives up to order p. then the function (x) must satisfy the following criteria in order to ensure covergence of the solution. (1) Be linerly indepent. (2) Be continuous and have continuous derivatives up to order (p - 1). in this section only the cases p = l and p = 2 are considered. (3) Satisfy the geometric boundary conditions. These involve derivatives up to order (p - 1). (4) Form a complete series. A series of functions is said to be complete if the 'mean square error' vanishes in the limit, that is /L n (u - Y* jq1j}'Zdx = O (22) j=î in order to asses the convergence of the method, solutions are obtained using the sequence of functions u1, u2, u3,...., un. This sequence is called min- imising sequence. Use of a minimising sequence ensures monotonic covergence of the solution. Using functions $j(x), which form a complete series, ensures monotonic convergence to the real solution. The proof of convergence of the Rayleigh-Ritz method is based upon the proof of the convergence of the expansion of an arbitrary function by means x of an infinite series of linearly indepent functions. If polynomials are used, then use can be nıade of Weierstrass:s Aproximation Theorem. Since the func¬ tions are required to have continuous derivatives up to order (p - 1), then ali derivatives up to this order will covergence uniformly. it should be noted that in using the Rayleigh-Ritz method the equations of motion and boundary conditions will only be satisfied approximately. To determine natural frequan- cies and modes of free vibration of a structure Langrange's equations can be reduced t o (M}{qn} + (K]{qn} = O (23) Since the motion is harmonic then {$"(*)} = (An}sinwt (24) \vhere the amplitudes {An} are independent of time and w is the frequency of the vibration. These equations give [K-w2]{An} = 0 (25) This equation reperesents a set of n linear homogeneous equations in the un- knowns A", A%,..., AT. The condition that these equations should have a non- zero solution is that the determinant of the coefficients should vanish, that is det[K - w2M] = [K - w*M] = O (26) This equation can be expanded to give a polynomial of degree n in w2. This polynomial equation will have n roots u>f,w%,,..., w2n. Such roots are called eigenvalues. Since [M] is positive definite, and [K] is either positive definite ör positive semi-definite, the eigenvalues are real and either positive ör zero, are approximate values will be greater than the true frequencies of the system. Corresponding to each eigenvalues w2, there exsist a unique solution. These solution are known as the eigenvectors. The approximate shape of a vibration mode is given by n un(x} = ^MAl (2?) J=l in this section, the lower frequencies and mode shapes of the free-clamped rod and the centilever beam have been obtained by using the Rayleigh-Ritz method. The finite element approach yields an approximate analysis based upon an assumed displacement field, a stress field, ör a mixture of these within each xi element. The assumption of displacement functions is the most commonly used technique. The following steps are suffice to describe this approach: (1) The continuum is divided into a finite number of stibregions of simple geometry. (2) Key points were selected on the elements to serve as nodes, where conditions of equilibrium and compatibility are to be enforced. (3) Displacement functions are assumed within each element so that the dis placement at each generic point are depend upon nodal values. (4) Strain-displacement and stress-strain relationships are satisfied within a typical element. (5) Stiffness and equivalent nodal loads were, determined for a typical element using work or energy principles. (6) Equilibrium equation are determined, and these equations are solved. It is required that, in fact, a solution technique which is numerically sta ble, easily programmed and can be adapted to a wide range of problem types without excessive interference of the user. From a structural viewpoint the finite element method provides the most satisfactory solution technique in this category. The essence of the finite element method involves dividing the structure into a suitable number of small pieces called finite elements. The intersection of the sides of the elements occur at nodal points or nodes and the interfaces between element are called nodal lines and nodal planes. For structural problems involving static or dynamic loads it is suitable to define the behavior of the structure in terms of displacements and (or) stress. Within each of the elements it is needed to select a pattern or shape for the unknown displacement or stress. In the case of a displacement field the shape function defines the behavior of displacements within an element in terms of unknown quantities specified at the element nodes. These nodal values are known as nodal connection quantities and allow the deformation behaviour in one element to be communicated with the adjacent elements. For a specific element type (beam, rod, etc.), the shape functions are identi cal for each element. Thus a given element need only be programmed once and the computer can repeat the operations for one, general, element as often as xn cal for each element. Thus a given element need only be programmed once and the computer can repeat the operations for one, general, element as often as required. The derivation of element equations for one-dimensional structural elements have been considered in this section. These elements can be used for the analysis of bar type system, like planar trusses, beams, continuous beams, planar beams, grid systems and space frames. A rod element is a bar which can resist only axial forces and can deform only in the axial direction. It will not be able to carry transverse loads or bending moments. A beam element is a bar which can resist not only axial forces but also transverse loads and bending moments. In this section, axial vibration of rods and bending vibration of beams have been introduced in details. In the sixth section, response of a structure under periodic force has been introduced in detail. A periodic force can be represented by means of a Fourier series of harmonically varying quantities of the the form 1 °° f(t) = -ao + y^ (arcoswrt + brsinwrt) (28) 2 r=l In the seventh chapter, problems have been given. In the eight chapter, results about problems have been given. In the last section, Ansysö.O finite element program has been introduced. And its command and programs, which wrote in ANSYS were given. First the mode shapes and the natural frequencies of the beam are found. Then a periodic force applied on the same beam and its responses are obtanied. Then a periodic force applied inside and outside of a ring and its displacements are found.

in this thesis, energy functions of an axial element and beam bending ele¬ ment and axial vibration of rods, bending vibration of beams and finite element displacement method have been reviewed. in the first chapter, the importance of the computional methods and finite elements method has been introduced briefiy. And behavior of structures under load have been reviewed. in the second chapter, a brief history of finite element displacement method, fundamentals of the finite displacement method and its practice areas have been introduced. in the third chapter, Hamilton's principle and Lagrange's equations have been introduced. Equations of motion have been derived by using Hamilton's principle. Ha¬ milton's principle states that, the actual path followed by a dynamical process is such as to that the condition / 2 (6(T -U) + 8Wncdt = O (1) J ti is satisfied. That is, the integral of (T - 17) + Wnc takes an extremum value which can be shown to be a minimum. When Hamilton's principle is applied to discrete systems it can be expressed in a more convenient form. (M}{q} + [C]{q} + (K}{q} = {Q} (2) This is called Langrange's equations where {q} = column matrix of system displacements vi {q} = column matrix of system velocities [M] = square symmetric matrix of inertia coefficients [C] = square symmetric matrix of damping coefficients [A'] = square symmetric matrix of stiffness coeficients In the fourth chapter, the energy functions are derived for various structural elements. Equations of motion of any structural system can be obtained from the energy function of the system. The energy function consists of the strain energies, the dissipation function and the work done by the applied loads. In this chapter, energy functions, which consist of strain and kinetic en ergies, the dissipation function and the work by the applied loads for axial element and beam bending element have been derived. For an axial element of constant cross- sectional area A and length 2a, strain energy, kinetic energy and virtual work have been derived as follows: v=\S7EA(^?dx (3) pAü2dx (4) = / pxSudx (5) J a where 17 = the strain energy of the system T = the kinetic energy of the system 8W = virtual displacement In deriving the energy functions for a beam bending element it is assumed that the vibration occurs only in one of the principal planes of the beam. The beam, which is of length la and has a constant cross-sectional area A, strain energy, kinetic energy and work have been derived as follows: u=\rEi^dx (6) vii 6W where Ix= f y2dA (7) JA T = \ f a PAv2dx (8) ^ J a SW= f Py8vdx (9) J a in this section equation of motion and the boundary conditions of continuous structural elem.en.ts have been fornıulated using the Hamilton's principle. Equation of motion and the boundary condition for an axial element are derived as follows: EA^-/>A^+"' = ° ^ throughout the region -a < x < a, either du EA^- = Q ör 6u = O (11) dx at both ends x = -a and x = a. Equation of motion and boundary condition for a beam bending element are derived as follows: -EI'^-"AW+P' = 0 (12) throughout - a < x < a. in addition, öne from each of the following two set of condition £^=° - !=° w d3v EIZ- = Q ör u = 0 (14) must be satisfied at both ends z = - a and x = a. Note that, in this case two condition are required at each boundary. The conditions v = O and |^- = O represent the vanishing of the displacement and the slope respectively. in the fifth chapter, Rayleigh-Ritz method and finite displacement method have been introduced in details. The response of simple structures, such as uniform axial and beam bend¬ ing elements have been obtained by solving differential equations of motion viii together with the appropriate boundary conditions. in many practical situa- tions eith.er the geometrical ör material properties vary, ör it may be that the shape of the boundaries cannot be described in temas of known functions. Also, practical structures consist of an assemblage of components of diferrent types, namely, beams, plates, shells and solids. in these situations it is inapossible to obtain analytical solutions to th equations of motion which satisfy the bound¬ ary conditions. This difficulty is overcome by seeking approximate solutions which satisfy Hamilton's principle. There are a nunaber of thecniques available for determining approximate solutions to Hamilton's principle. Öne of the most widely used procedures is the Rayleigh-Ritz method, which is described in this section. A generalisation of the Rayleigh-Ritz method, known as the finite element displacement method, is introduced. The principal features of this method are described by considering rods and beams. The Rayleigh-Ritz method approximates the solution with a finite expan- sion of the form n «B(*,0=5>X*)??(*) (14) i=ı where the qf(t} are unkown functions of time, t, and the Qj(x) are prescribed functions of a;, which are linarly indepent. A set of functions are linarly inde- pendant if n ^2 j(x} = O for ali z (15) j=ı implies that aj=Q for j = 1,2,...., n (16) Each of the functions k(x)dx (20) Jo fL Kjk = / EA^'kdx (21) Jo where prime denote differentitation with respect to x. Langrange's equations is solved for the {#"}, then an approximate solution are obtained for u(x, t). Integral of Hamilton's principle involve derivatives up to order p. then the function (x) must satisfy the following criteria in order to ensure covergence of the solution. (1) Be linerly indepent. (2) Be continuous and have continuous derivatives up to order (p - 1). in this section only the cases p = l and p = 2 are considered. (3) Satisfy the geometric boundary conditions. These involve derivatives up to order (p - 1). (4) Form a complete series. A series of functions is said to be complete if the 'mean square error' vanishes in the limit, that is /L n (u - Y* jq1j}'Zdx = O (22) j=î in order to asses the convergence of the method, solutions are obtained using the sequence of functions u1, u2, u3,...., un. This sequence is called min- imising sequence. Use of a minimising sequence ensures monotonic covergence of the solution. Using functions $j(x), which form a complete series, ensures monotonic convergence to the real solution. The proof of convergence of the Rayleigh-Ritz method is based upon the proof of the convergence of the expansion of an arbitrary function by means x of an infinite series of linearly indepent functions. If polynomials are used, then use can be nıade of Weierstrass:s Aproximation Theorem. Since the func¬ tions are required to have continuous derivatives up to order (p - 1), then ali derivatives up to this order will covergence uniformly. it should be noted that in using the Rayleigh-Ritz method the equations of motion and boundary conditions will only be satisfied approximately. To determine natural frequan- cies and modes of free vibration of a structure Langrange's equations can be reduced t o (M}{qn} + (K]{qn} = O (23) Since the motion is harmonic then {$"(*)} = (An}sinwt (24) \vhere the amplitudes {An} are independent of time and w is the frequency of the vibration. These equations give [K-w2]{An} = 0 (25) This equation reperesents a set of n linear homogeneous equations in the un- knowns A", A%,..., AT. The condition that these equations should have a non- zero solution is that the determinant of the coefficients should vanish, that is det[K - w2M] = [K - w*M] = O (26) This equation can be expanded to give a polynomial of degree n in w2. This polynomial equation will have n roots u>f,w%,,..., w2n. Such roots are called eigenvalues. Since [M] is positive definite, and [K] is either positive definite ör positive semi-definite, the eigenvalues are real and either positive ör zero, are approximate values will be greater than the true frequencies of the system. Corresponding to each eigenvalues w2, there exsist a unique solution. These solution are known as the eigenvectors. The approximate shape of a vibration mode is given by n un(x} = ^MAl (2?) J=l in this section, the lower frequencies and mode shapes of the free-clamped rod and the centilever beam have been obtained by using the Rayleigh-Ritz method. The finite element approach yields an approximate analysis based upon an assumed displacement field, a stress field, ör a mixture of these within each xi element. The assumption of displacement functions is the most commonly used technique. The following steps are suffice to describe this approach: (1) The continuum is divided into a finite number of stibregions of simple geometry. (2) Key points were selected on the elements to serve as nodes, where conditions of equilibrium and compatibility are to be enforced. (3) Displacement functions are assumed within each element so that the dis placement at each generic point are depend upon nodal values. (4) Strain-displacement and stress-strain relationships are satisfied within a typical element. (5) Stiffness and equivalent nodal loads were, determined for a typical element using work or energy principles. (6) Equilibrium equation are determined, and these equations are solved. It is required that, in fact, a solution technique which is numerically sta ble, easily programmed and can be adapted to a wide range of problem types without excessive interference of the user. From a structural viewpoint the finite element method provides the most satisfactory solution technique in this category. The essence of the finite element method involves dividing the structure into a suitable number of small pieces called finite elements. The intersection of the sides of the elements occur at nodal points or nodes and the interfaces between element are called nodal lines and nodal planes. For structural problems involving static or dynamic loads it is suitable to define the behavior of the structure in terms of displacements and (or) stress. Within each of the elements it is needed to select a pattern or shape for the unknown displacement or stress. In the case of a displacement field the shape function defines the behavior of displacements within an element in terms of unknown quantities specified at the element nodes. These nodal values are known as nodal connection quantities and allow the deformation behaviour in one element to be communicated with the adjacent elements. For a specific element type (beam, rod, etc.), the shape functions are identi cal for each element. Thus a given element need only be programmed once and the computer can repeat the operations for one, general, element as often as xn cal for each element. Thus a given element need only be programmed once and the computer can repeat the operations for one, general, element as often as required. The derivation of element equations for one-dimensional structural elements have been considered in this section. These elements can be used for the analysis of bar type system, like planar trusses, beams, continuous beams, planar beams, grid systems and space frames. A rod element is a bar which can resist only axial forces and can deform only in the axial direction. It will not be able to carry transverse loads or bending moments. A beam element is a bar which can resist not only axial forces but also transverse loads and bending moments. In this section, axial vibration of rods and bending vibration of beams have been introduced in details. In the sixth section, response of a structure under periodic force has been introduced in detail. A periodic force can be represented by means of a Fourier series of harmonically varying quantities of the the form 1 °° f(t) = -ao + y^ (arcoswrt + brsinwrt) (28) 2 r=l In the seventh chapter, problems have been given. In the eight chapter, results about problems have been given. In the last section, Ansysö.O finite element program has been introduced. And its command and programs, which wrote in ANSYS were given. First the mode shapes and the natural frequencies of the beam are found. Then a periodic force applied on the same beam and its responses are obtanied. Then a periodic force applied inside and outside of a ring and its displacements are found.

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

##### Anahtar kelimeler

Sistem analizi,
Sonlu elemanlar yöntemi,
Titreşim analizi,
System analysis,
Finite element method,
Vibration analysis