İki serbestlikli sistemlerin lineer olmayan titreşimleri

Abstract

Bu çalışmada lineer olmayan iki serbestlik dereceli bir sistemin sönümlü ve zorlanmış titreşimleri incelenmiştir sistemde birinci yay kübik nonlineerlikli ikinci yay ve sönümler ise lineer halde alınarak çözüme gidilmiştir. Çalışma dört bölümden meydana gelmektedir. Birinci bölümde, konuyla ilgili genel tanımlamalar ve açıklam alara yer verilmiş, nonlineerlik kaynakları ve çeşitleri anla tılmış ayrıca çalışma hakkında çok kısa bir açıklama yapılmıştır. ikinci bölümde, çözüm yöntemleri anlatılmış nitel ve nicel analiz örneklerle açıklanmıştır. Lineer olmayan titreşimlerde temel çözüm yöntemi olan Pertürbasyon Yöntemi ve bunun versi yonları olan Katlı ölçek, Lindstedt-Poincare ve Krylov- Bogoliubov-Mitropolski yöntemleri konusunda bilgi verilmiştir Üçüncü bölümde, problemin tanıtımı yapılmış ve hareket denklemleri çıkarılmıştır. Hareket denklemleri Katlı ölçek yöntemiyle pertürbasyon serisine açılarak zayıf zorlama halinde asıl rezonans durumu ve kuvvetli zorlama halinde süper harmonik ve subharmonik rezonans durumları incelenmiştir Dördüncü bölümde, asıl rezonans durumu ve süper harmonik rezonans durumunda durağan titreşim için, aj ve a% genliklerinin sönümlü ve sönümsüz hal için Oj ve Ojj frekans kaymalarına, ve zamana bağımlı hal için ise aj ve &2 genliklerinin sönümlü halde zamana göre değişimleri eğriler yardımıyla incelenmiştir.

In thi3 study damped and forced vibration of a nonlinear system with two-degree of freedom has been investigated. In this system, the solution was obtained, by assuming the first spring &3 cubic nonlinear and by taking the second spring and dampings a 3 linear. The study is composed of four parts. In the first part, general definitions and explanations related with the subject have been given and, the different sources of nonlinearity have been discussed. For the different sources of nonlinearity, such as geometric nonlinearity that is due to large motion, material nonlinearity that is due to the. behavior of material properties and inertia! nonlinearity that is due to influence of a cetral-force field were explaned with examples. In the second part, the solution methods of the nonlinear systems were discussed. The Qualitative Analysis, The Quantitative Analysis and The Perturbation Method with its the most widely used versions; The Method of Miltiple Scale, The Lindsedt PoincarĞ Method and The Krylov-Bogoliubov-Mitropolski Method (K.B.M) were explaned by examples. The qualitative analysis is made by integrating any nonlinear equation and by investigating the integral curves which are called level curves on the (u,v) plane (called phase plane). This analysis shows us whether the equations of motion have a vibrational motion or periodic vibrational motion characteristic The quantitative analysis is used for obtaining approximate solution of any nonlinear equation. The Perturbation Method İ3 the most widely U3ed method for the approximate solution of nonlinear ordinary or partial differential equation, or for solution of a linear equation with variable coefficient which cannot be solved exactly. The variable parameter e«i is formed the basis of this method. It is created, by appropriat transformation, by result of nondimensionalisation or by giving the initial and boundary condition of differential equation. In the perturbation method, the desired quantities are expanded to the perturbation series with initial and boundary conditions. The coefficients of 3ame orders of parameter e are equalized to zero. For the each order of parameter c a differential equation that obtains the initially and boundary conditions is provided. The zeroth order equation is solved easily and by making use of this solution the first order solution is performed. Since difficulties after the first order, the solution is stopped at the first order. -v- Because of the simplicity in application. The Method of Multiple Scale which has three versions was used for the solution of equations. ¥e used the first version of the method of multiple scale which was generalized by Sturrock, Frieman, Nayfe and Sadri [1]. In this version, in addition to the expansion of dependent variables to the perturbation series, also the derivatives are expanded to the series. For this reason, this version is named as the derivative expansion method. Below a short expalation related with the first version is given. The displacemet x is regarded to be a function of t and e2t. The underlying idea of the method of multiple scale is to consider the expansion representing the response to be a function of multiple independent variables, or scales, instead of a single variable. One begins by introducing new independent variables according to Tn = eû t for n= 0,1,2 It follows that the derivatives with respect to t become expansions in terms of the partial derivatives with respect to the Tn according to d dT0 3 3Ti d + -+ =D0+eDi+. dt dt 3T0 dt 3Ti d2 _ = D02+2eD0Dj+e2(Di2+2DoD2) + dt2 One assumes that displacement x is represented by an expansion having the form x^eMxiCIVTi,... )+e2x2(T0,Ti,... )+e3x3(T0, T±.... ) These defined expressions have been substituded in their places in our equations. As an example problem to the Multiple Scale and Lindstedt Poincare Method, the equation u+aiu+a2u2+a3u3=0 has been solved by both methods. And the same results have been found for the frequency w and displacement u. On the other hand, in Krylov-Bogoliubov-Mitropolski Hethod (K.B.M), only the basic equations were given. In the third part, the representation of the problem has been made and the equations of motion have been formed. The Method of Multiple Scale was used for the solution of equations of motion. -vi- Spring forces vere ehoosed, having the form Fsl = knx1+ki2x13+k13x1 Fs2 - k2ix2+k23x2 as it is shown the first spring force has a nonlinear characteristic term and the second spring force has only linear characteristic terms. The externally applied forces were choosed, having the form Fi(t) = ficosQit F2