Eğri eksenli düzlemsel çubukların statik ve dinamik problemlerin analitik çözümü
Analytical solutions of static and dynamic problems of planer curved beams
- Tez No: 39266
- Danışmanlar: PROF.DR. MUSTAFA SAVCI
- Tez Türü: Doktora
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Diferensiyel denklemler, Kayma deformasyonu, Statik problemler, Titreşim problemleri, Differential equations, Shear deformation, Static problems, Vibration problems
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
When these equations are studied, it is seen that these are not solved for arbitrary geometry. The fundemental matrix of this set of equations is not obtained exactly. There is no exact solutions of the equations unless the coefficients are constant. Exact solutions can be obtained only for the circular beams having constant cross-section. There are only approximate solutions for arbitrary geometry. For this reason, vibration problem of the circular beams having constant cross-sections are studied in this chapter. The first five mode shapes are given for hinged-hinged, clamped-clamped, clamped-hinged and clamped- free boundary conditions. The examples inthe literature are solved and some different results are obtained. The cause of the differences are explained. Some experimental results are given. The excellent agreement is obtained between the experimental and theoretical results. XIX
Özet (Çeviri)
represented by a space curve whose every point is coupled with a rigid orthonormal vector diad. The vectors are chosen to be perpendicular to the tangent vector of the space curve in the initial state and they represent the cross-section of the beam. In deformed configuration, these directors still remain unit and perpendicular each other because of the assumption of a rigid cross-section. But, in contrast with Kirchhoffs beam theory, the restrictions of perpendicular cross-section and inextensible arc length are removed. The displacement of the point on the space curve and the rotation of the cross-section constitute the displacement field of the beam. The change in the tangent vector from the initial state and the rate of change vector of the deformed director diad are choosen to represent the state of strain. The scalar form of the governing differential equations whose variables are vectors are written. The equations are given the case that the principal axis of the cross-section do not coincide the principal normal and bi-normal of the central line of the beam. Then the equations are re-written for the case that directions coincide. At last, these equations are simplyfied for the in-plane and out-of-plane static and dynamic problems of beams curved in a plane. The inertia effects are considered as distributed forces. The governing differential equations of the static problem are used. The balance occurs, while the inertia effects and the real forces acts on it together. Then the governing differential equations of the vibration of spatially curved and twisted beams are obtained. The solutions of governing differential equations of static problems of planar curved beams are given in the fourth chapter. Exact solutions are found for the boundary value problem with arbitrary boundary conditions. A solution procedure is described to obtain fundamental matrix for arbitrary geometry and boundary conditions. Fundamental matrix and its inverse are also obtained analytically for the parabolic beams, All the integrals are calculated and the elements of the matrix are obtained as a function of the arc angle. The solution is describe the beams having arbitrary curvature and cross- section. The concentrated force acts at arbitrary direction at any point on the beam. Some numerical examples given in the literature are solved and some different type examples are shown in this chapter. Results are given in tables and diagrams. The displacement, rotation angle, shear force, normal force and bending moments diagrams are shown. The beam having arbitrary distributed load is considered. The static problem is also solved for the beam having arbitrary distributed load. Some numerical examples are designed. Results are given in tables. In the fifth chapter, the dynamic problems of the beams are studied. The inertia effects are considered as distributed forces. The differential equations of the vibration of the planar curved beams are given. xvmWhen these equations are studied, it is seen that these are not solved for arbitrary geometry. The fundemental matrix of this set of equations is not obtained exactly. There is no exact solutions of the equations unless the coefficients are constant. Exact solutions can be obtained only for the circular beams having constant cross-section. There are only approximate solutions for arbitrary geometry. For this reason, vibration problem of the circular beams having constant cross-sections are studied in this chapter. The first five mode shapes are given for hinged-hinged, clamped-clamped, clamped-hinged and clamped- free boundary conditions. The examples inthe literature are solved and some different results are obtained. The cause of the differences are explained. Some experimental results are given. The excellent agreement is obtained between the experimental and theoretical results. XIXrepresented by a space curve whose every point is coupled with a rigid orthonormal vector diad. The vectors are chosen to be perpendicular to the tangent vector of the space curve in the initial state and they represent the cross-section of the beam. In deformed configuration, these directors still remain unit and perpendicular each other because of the assumption of a rigid cross-section. But, in contrast with Kirchhoffs beam theory, the restrictions of perpendicular cross-section and inextensible arc length are removed. The displacement of the point on the space curve and the rotation of the cross-section constitute the displacement field of the beam. The change in the tangent vector from the initial state and the rate of change vector of the deformed director diad are choosen to represent the state of strain. The scalar form of the governing differential equations whose variables are vectors are written. The equations are given the case that the principal axis of the cross-section do not coincide the principal normal and bi-normal of the central line of the beam. Then the equations are re-written for the case that directions coincide. At last, these equations are simplyfied for the in-plane and out-of-plane static and dynamic problems of beams curved in a plane. The inertia effects are considered as distributed forces. The governing differential equations of the static problem are used. The balance occurs, while the inertia effects and the real forces acts on it together. Then the governing differential equations of the vibration of spatially curved and twisted beams are obtained. The solutions of governing differential equations of static problems of planar curved beams are given in the fourth chapter. Exact solutions are found for the boundary value problem with arbitrary boundary conditions. A solution procedure is described to obtain fundamental matrix for arbitrary geometry and boundary conditions. Fundamental matrix and its inverse are also obtained analytically for the parabolic beams, All the integrals are calculated and the elements of the matrix are obtained as a function of the arc angle. The solution is describe the beams having arbitrary curvature and cross- section. The concentrated force acts at arbitrary direction at any point on the beam. Some numerical examples given in the literature are solved and some different type examples are shown in this chapter. Results are given in tables and diagrams. The displacement, rotation angle, shear force, normal force and bending moments diagrams are shown. The beam having arbitrary distributed load is considered. The static problem is also solved for the beam having arbitrary distributed load. Some numerical examples are designed. Results are given in tables. In the fifth chapter, the dynamic problems of the beams are studied. The inertia effects are considered as distributed forces. The differential equations of the vibration of the planar curved beams are given. xvmWhen these equations are studied, it is seen that these are not solved for arbitrary geometry. The fundemental matrix of this set of equations is not obtained exactly. There is no exact solutions of the equations unless the coefficients are constant. Exact solutions can be obtained only for the circular beams having constant cross-section. There are only approximate solutions for arbitrary geometry. For this reason, vibration problem of the circular beams having constant cross-sections are studied in this chapter. The first five mode shapes are given for hinged-hinged, clamped-clamped, clamped-hinged and clamped- free boundary conditions. The examples inthe literature are solved and some different results are obtained. The cause of the differences are explained. Some experimental results are given. The excellent agreement is obtained between the experimental and theoretical results. XIXrepresented by a space curve whose every point is coupled with a rigid orthonormal vector diad. The vectors are chosen to be perpendicular to the tangent vector of the space curve in the initial state and they represent the cross-section of the beam. In deformed configuration, these directors still remain unit and perpendicular each other because of the assumption of a rigid cross-section. But, in contrast with Kirchhoffs beam theory, the restrictions of perpendicular cross-section and inextensible arc length are removed. The displacement of the point on the space curve and the rotation of the cross-section constitute the displacement field of the beam. The change in the tangent vector from the initial state and the rate of change vector of the deformed director diad are choosen to represent the state of strain. The scalar form of the governing differential equations whose variables are vectors are written. The equations are given the case that the principal axis of the cross-section do not coincide the principal normal and bi-normal of the central line of the beam. Then the equations are re-written for the case that directions coincide. At last, these equations are simplyfied for the in-plane and out-of-plane static and dynamic problems of beams curved in a plane. The inertia effects are considered as distributed forces. The governing differential equations of the static problem are used. The balance occurs, while the inertia effects and the real forces acts on it together. Then the governing differential equations of the vibration of spatially curved and twisted beams are obtained. The solutions of governing differential equations of static problems of planar curved beams are given in the fourth chapter. Exact solutions are found for the boundary value problem with arbitrary boundary conditions. A solution procedure is described to obtain fundamental matrix for arbitrary geometry and boundary conditions. Fundamental matrix and its inverse are also obtained analytically for the parabolic beams, All the integrals are calculated and the elements of the matrix are obtained as a function of the arc angle. The solution is describe the beams having arbitrary curvature and cross- section. The concentrated force acts at arbitrary direction at any point on the beam. Some numerical examples given in the literature are solved and some different type examples are shown in this chapter. Results are given in tables and diagrams. The displacement, rotation angle, shear force, normal force and bending moments diagrams are shown. The beam having arbitrary distributed load is considered. The static problem is also solved for the beam having arbitrary distributed load. Some numerical examples are designed. Results are given in tables. In the fifth chapter, the dynamic problems of the beams are studied. The inertia effects are considered as distributed forces. The differential equations of the vibration of the planar curved beams are given. xvmWhen these equations are studied, it is seen that these are not solved for arbitrary geometry. The fundemental matrix of this set of equations is not obtained exactly. There is no exact solutions of the equations unless the coefficients are constant. Exact solutions can be obtained only for the circular beams having constant cross-section. There are only approximate solutions for arbitrary geometry. For this reason, vibration problem of the circular beams having constant cross-sections are studied in this chapter. The first five mode shapes are given for hinged-hinged, clamped-clamped, clamped-hinged and clamped- free boundary conditions. The examples inthe literature are solved and some different results are obtained. The cause of the differences are explained. Some experimental results are given. The excellent agreement is obtained between the experimental and theoretical results. XIX
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