Düzlem içi kuvvetler etkisindeki dikdörtgen ortotrop plakların düzlem içi ve düzlem dışı titreşimleri

Abstract

Ö Z E T Bu çalışmada ortotrop malzemeden yapılmış ince dikdörtgen plak ların düzlem içi dinamik kenar yükleri etkisindeki davranışı incelenmiştir. Problemin niteliği gereği çözüm sayısal hesap larla elde edildiğinden genel bir bilgisayar programı gelişti rilmiştir. Buna ait akış diyagramı tezin sonunda verilmiştir. Bu bilgisayar programı yardımıyla levha titreşimleri, plağın statik burkulması, burkulma sonrasında plak davranışı, statik kenar yükleri etkisinde plak titreşimleri, titreşim modları ve frekanslarının genlikle değişimi, dinamik kenar yükleri etki sinde önsehimli ve önsehimsiz plakların davranışı incelenebil- mektedir. Probiemde düzlem içi atalet kuvvetleri de gözönüne alınabilmektedir. Von-Kârmân plak varsayımlarının esas alındığı bu çalışmada bi rinci bölümde konu ile ilgili yayınlar verilmiş ve çalışmanın amacı belirtilmiştir. İkinci bölümde elastodinamikteki temel denklemler verilmiş olup bunlardan virtüel iş bağıntısı plak için yazılarak plak denklem leri ve sınır koşulları elde edilmiştir. Üçüncü bölümde koordinatlar, yer değiştirmeler ve gerilmeler değişken dönüşümü yapılarak boyutsuz hale getirilmiştir. Daha sonra boyutsuz yer değiştirmeler için iki doğrultuda Fourier serileri seçilerek Lagrange çarpanları yöntemi uygulanmıştır. Böylece sadece zamana bağlı adi diferansiyel denklem sistemi elde edilmiştir. Bu bölümün sonunda levhaların düzlem içi tit reşimleri ele alınmış olup bununla ilgili çeşitli sonuçlar verilmiştir. VVI üçüncü bölümde elde edilmiş bulunan adi diferansiyel denklem sisteminin çözümü dördüncü bölümde yapılmıştır. Burada Kirchhoff plağı hali için burkulma yükleri ve titreşim frekans ları arasındaki ilişki açıklanmıştır. Von Karman plağında bur kulma sonrası yük-yer değiştirme eğrisinin elde edilebilmesi için bir iterasyon verilmiştir. Statik veya dinamik kenar yük leri etkisinde bulunan plağın davranışı seçilen başlangıç ko şulları altında bir başlangıç değer problemi olarak ele alınmış olup Newmark yöntemi kullanılarak adım adım integrasyon uygu lanmıştır. Çeşitli yüklemeler için sayısal sonuçlar elde edilmiştir. Beşinci bölümde ise bu çalışmada elde edilen sonuçlarla ilgili bir değerlendirme sunulmuştur.

IN-PLANE AND OUT-OF-PLANE VIBRATIONS OF RECTANGULAR ORTHOTROPIC PLATES UNDER THE EFFECT OF IN-PLANE, LOADING SUMMARY The determination of natural frequencies is fundamentally important in the design of many structural elements. Thin rectangular plates are such commonly used elements. Therefore, there has been a considerable amount of work done on the vibrations of rectangular plates. If a plate is subjected to pulsating in-plane loads, the plate will generally experience forced in-plane vibrations and for certain frequencies of the pulses, in-plane resonance will take place. However, a completely different type of resonance will occur, when there is a certain relationship between the natural frequency of the out- of -plane vibration, the frequency of the pulses and their magnitudes. This problem is called the dynamic stability of plates. When the in-plane forces are static then we have the problem of plate stability which has been the subject of many investigators (References from 33 to 54 ). The most recent work dealing with dynamic in-plane forces are given in Ref. 15,55 through 69. In some of these in-plane inertia forces are included with certain limitations Ref. 15,37,38,40,41,50, 59,61,62,68. In this thesis, thin rectangular orthotropic plates are investigated under the action of in-plane forces. Assumptions for von Karman' s plate theory are used and the in-plane inertia forces are included in the analysis. Therefore, at first the in-plane vibrations of thin plates are studied and later the general case of simultaneous in-plane and out-of-plane vibrations are considered. Thus the interaction between the in-plane and out-of-plane vibrations are studied. VIIVIII In the first chapter of the thesis, the subject of the thesis and a literature survey are given and the object of the study is explained. In the second chapter, the basic equations of elastodynamics are given and the assumptions used in the study are explained. In the analysis the following support conditions are considered : i) clamped on four sides ii) simply supported on four sides iii) clamped on two opposite sides and simply supported on the remaining sides. In all these cases, sides can move freely in the plane of the plate as shown in Figure 2.3. Rigid body motion of the plate in its own plane is also taken into account as shown in Figure 2.8. The inertia forces due to this rigid body motion are included in the differential equations. Initially the plate is considered to be under the action of the static in-plane forces along the edges. Then the dynamic in-plane forces are added to the static equilibrium configuration (see Equations 2.4, 2.5 through 2.7). The dynamic edge loads are chosen of the following types a) step function, b) impulsive, c) increasing linearly and d) harmonic as shown in Figure 2.5. The loading along the edges are chosen to be uniform normal loading, trapezoidal normal loading or uniform shear loading or any linear combination of these. In this chapter, the virtual work equation (Eq, 2.38) and the equations of motion of the plate (Eq. 2.41) and the boundary conditions (Equations 2.42 through 2.50) are obtained making use of the basic equations and the assumptions. In the third chapter, by changing the variables dimensionless quantities for the coordinates, stresses and displacements are defined. The in-plane and out-of-plane displacements are expanded into two-dimensional Fourier series as shown in Equations 3.2 5 and 3.39. The force boundary conditions given by Eq. 2.50 put certain constraints on the coefficients (or the generalized coordinates) of the Fourier series as seen in Eq. 3.46. These constraints are independent of time when static buckling isIX investigated. In order to determine the generalized coordinates Lagrange multipliers are used in the analysis. As a result of this approach, a set of differential equations is obtained. This set consists of the differential equations given by Equations 3.54 through 3.56 which will yield the generalized coordinates for in-plane forced vibrations and the differential equation given by Eq. 3.57 which contains the generalized coordinates for parametrically excited out-of-plane vibrations. In the case of von Ka'rman plate there is a nonlinear interaction between these two sets of equations. These equations are written in matrix form and dimensions of the matrices are reduced by eliminating the Lagrange's multipliers. At the end of the third chapter, the in-plane vibrations of the plate is investigated and some of the results are given. In the fourth chapter the solution is obtained for the set of ordinary differential equations derived in the third chapter. Here, the relationship between the buckling loads and the frequencies of the Kirchhoff 's plate is explained. An iteration scheme is given for obtaining the load-displacement curve for the post-buckling behaviour of von Karman* s plate. Newmark's method of stepwise integration is applied to study the behavior of the plate under static and dynamic edge loadings by considering the plate as an initial value problem under the chosen initial conditions. Numerical examples are presented. The summary öf the results are given below. Case of In-Plane Vibrations : - If the in-plane inertia forces are neglected, the displacements of the plate are expressed in terms of a and an given by Eq,3.39. Otherwise, the terms containing the coefficients u.. (t) and v.. (t) defined by the linear combination of the free vibration modes (Eq.3.89) should be added to these displacements.- The edge loadings are zero in the case of free vibration modes The vibration modes can be grouped into four by considering the symmetry and anti-symmetry of these modes and the calculations are carried out according to these groups. - The convergence of the frequencies of the free vibrations is very fast depending on the number of terms in the series. - In any type of symmetry, the first frequency of the free vibrations increases with the material parameters < and decreases with v. Case of out-of-plane vibrations : - Under the assumptions of Kirchhof f ' s plate, the in-plane displacements will be the same as above - In the case of von Karman' s plate when the in-plane inertia forces are neglected and even for the static case, there are terms containing the coefficients u.. and v.. in Eq. 3.39. The large deflections involve quadratic terms in the differential equations for the in-plane displacements and stresses are expected. In this study, these expressions are expressed in terms of the series given by Eq.3.32 utilizing the coefficients h.. in Eq. 3.33. These series simplify the application of the variational method used and the satisfaction of the in-plane boundary conditions. - Parallel to the previous studies, the convergence rates for the static buckling loads and the frequencies of the out-of- plane free vibrations are very fast. - The iteration technique used for determining the post- buckling displacements also shows good convergence rate. The number of terms used in the case of in-plane series effects the rate of convergence. The norm given by Eq.4.17 increases by 6 % if the number N in Equ. 3.39 changes from 4 to 6 and by 2 % if N changes from 6 to 8. Thus the convergence rate for the out-of-plane displacements is accelerated. - The frequencies of the out-of-plane free vibrations are varied by the increase of the edge loadings as expected. ThisXI variation obtained for the material and the geometric properties of the plate given in Eq. 3.85 are presented in Fig. 4. 5. The variation of a free vibration mode and its frequency as a function of the amplitudes are obtained by giving an initial out-of-plane velocity consistent with that mode as shov/n in Fig. 4. 6. Using a stepwise integration method initial value problem is solved. There exists a relationship between the given initial velocity and the maximum value of the norm given by Eq. 4.37 in the case of Kirchhoff's plate while no such relationship exists in the case of von Karman' s plate. The C i i values of the term - c- max f obtained both for Kirchhoff's and voni -Karman' s assumptions are compared for the same initial velocity. For various initial velocities the results are 0.001 and 0.001 ; 0.1 and 0.09853 ; 2.0 and 1.037 ; 50.0 and 5.924 for the Kirchhoff's and von. Karman' s plates respectively. In other words, Kirchhoff's assumptions yield results quite different than von Karman' s assumptions as the initial velocities increase. The results obtained for the edge loadings increasing linearly with time are quite similar to the ones given in Ref.67. In this thesis, in an example the results are effected by 2 % if the in-plane inertia forces are included. In this case, since the step-wise integration method is used the time increments should be selected according to the period of vibrations. If the time increments are not taken small, then the instability of the integration appears. The results obtained for the edge loadings varying harmonically with time are similar to the ones giyen in Ref. 63. Since a perturbation analysis is used in that study, the magnitude of the harmonic loading should be selected to be small in comparison to the buckling load. Otherwise many terms should be considered in the perturbation series. In this thesis there is no limitation for the magnitude of the harmonic loading and the out-of-plane vibrations can be investigated considering multimodes.