Browsing by Author "Sinir, BG"

Now showing 1 - 18 of 18
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    Buckling configurations and dynamic response of buckled Euler-Bernoulli beams with non-classical supports
    Sinir, BG; Özhan, BB; Reddy, JN
    Exact solutions of buckling configurations and vibration response of post-buckled configurations of beams with non-classical boundary conditions (e.g., elastically supported) are presented using the Euler-Bernoulli theory. The geometric nonlinearity arising from mid-plane stretching (i.e., the von Karman nonlinear strain) is considered in the formulation. The nonlinear equations are reduced to a single linear equation in terms of the transverse deflection by eliminating the axial displacement and incorporating the nonlinearity and the applied load into a constant. The resulting critical buckling loads and their associated mode shapes are obtained by solving the linearized buckling problem analytically. The buckling configurations are determined in terms of the applied axial load and the transverse deflection. The first buckled shape is the only stable equilibrium position for all boundary conditions considered. Then the pseudo-dynamic response of buckled beams is also determined analytically. Natural frequency versus buckling load and natural frequency versus amplitudes of buckling configurations are plotted for various non-classical boundary conditions.
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    A general solution procedure for nonlinear single degree of freedom systems including fractional derivatives
    Yildiz, B; Sinir, S; Sinir, BG
    This paper considers oscillations of systems with a single-degree-of-freedom (SDOF) including fractional derivatives. The system is assumed to be an unforced condition. A general solution procedure that can be effectively applied to various types of fractionally damped models, where damping is defined by a fractional derivative, in engineering and physics is proposed. The nonlinearity of the mentioned models contains not only damping but can also consist of acceleration or displacement. This study proposed a new general model that includes but not limited to modified fractional versions of the well-known linear, quadratic, Coulomb and negative damped models. The method of multiple time scales is performed to obtain approximate analytical solutions. The solution, the amplitude, and the phase in the applications are plotted for various fractional derivative parameter values. In order to confirm their validity, our results for the case of the fractional derivative parameter equal to one are compared with others available in the literature.
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    DYNAMICAL BEHAVIOR OF AN AXIALLY MOVING STRING CONSTITUTED BY A FRACTIONAL DIFFERENTIATION LAW
    Demir, DD; Sinir, BG; Bildk, N
    In this paper, the dynamical behavior of an axially moving string modeled by fractional derivative is investigated. The governing equation represented motion is solved by the method of multiple scales. Considering principal parametric resonance, the stability boundaries for string with simple supports are obtained. Numerical results indicate the effects of fractional damping on stability.
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    Application of fractional calculus in the dynamics of beams
    Demir, DD; Bildik, N; Sinir, BG
    This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.
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    Linear dynamical analysis of fractionally damped beams and rods
    Demir, DD; Bildik, N; Sinir, BG
    The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.
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    Determining Critical Load in the Multispan Beams with the Nonlinear Model
    Demir, DD; Sinir, BG; Usta, L
    The beams which are one of the most commonly used structural members are quite important for many researchers. Mathematical models determining the response of beams under external loads are concluded from elasticity theory through a series of assumptions concerning the kinematics of deformation and constitutive behavior. In this study, the derivation of the nonlinear model is introduced to determine the critical load in the multispan beams. Since the engineering practice of this kind of problems is very common, determining the critical load is quite important. For this purpose, the nonlinear mathematical model of the multispan Euler-Bernoulli beam is firstly obtained. To be able to obtain the independent of the material and the geometry, the present model are became dimensionless. Then, the critical axial load can be determined via the nonlinear solution of the governing equation.
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    The Combination Resonance Analysis for an Axially Moving String
    Demir, DD; Sinir, BG; Bildik, N
    In this paper, the vibrations of an initially stressed moving string with fractional damping are investigated. Traveling string with two modes are considered and the approximate analytical solutions are obtained by using the method of multiple scales. The stability boundaries are analytically determined. Consequently, it is found that instability appears when the frequency is close to the sum or difference of any two natural frequencies.
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    Linear vibrations of continuum with fractional derivatives
    Demir, DD; Bildik, N; Sinir, BG
    In this paper, linear vibrations of axially moving systems which are modelled by a fractional derivative are considered. The approximate analytical solution is obtained by applying the method of multiple scales. Including stability analysis, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted by graphs. It is determined that the external excitation force acting on the system has an effect on the stiffness of the system. Moreover, the general algorithm developed can be applied to many problems for linear vibrations of continuum.
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    Infinite mode analysis of a general model with external harmonic excitation
    Sinir, BG
    This study proposes a general solution procedure for infinite mode analysis. The equation of motion is written in a general form using spatial differential operators, which are suitable for perturbation techniques. The multiple time scales method is applied directly to solve the proposed equation of motion. General investigations of some resonance cases are provided, such as parametric, sum type, difference type, and a combination of sum and difference type resonances. The proposed general solution procedure is applied to one- and two-dimensional problems. The results demonstrate that this general solution procedure obtains good solutions in the dynamic analysis of beams, plates, and other structures. (C) 2014 Elsevier Inc. All rights reserved.
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    THE SOLUTION OF THE GOVERNING EQUATION OF THE BEAM ON LINEAR SPRING FOUNDATION MODELED BY A DISCONTINUITY FUNCTION
    Dönmez Demir, D; Sinir, BG; Kahraman, E
    The structural engineering researches have attracted considerable attention by many scientist for several decades. Determining the dynamical behaviors of structural elements with some discontinuous is of great importance in many engineering applications. The mentioned structures can be modelled two different ways. In the first approximation so-called the classical approach, a fourth order differential equation are written for each part of beam separated in the distinct discontinuity locations. Therefore, we obtain a system of equation containing n +1 number of the differential equation with boundary and transient conditions. Secondly, the real problem can be reformulated by only one differential equation having discontinuity function. In this study, we introduce the method of multiple scales as the solution technique. Since we encountered by the differential equation with discontinuity function in the part of order discretization during the perturbative solution, we have used a numerical technique for the solution. The mentioned technique is applied on the beam model lying on lineer spring foundation called as Winkler type foundation.
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    Pseudo-nonlinear dynamic analysis of buckled pipes
    Sinir, BG
    In this study, the post-divergence behavior of fluid-conveying pipes supported at both ends is investigated using the nonlinear equations of motion. The governing equation exhibits a cubic nonlinearity arising from mid-plane stretching. Exact solutions for post-buckling configurations of pipes with fixed-fixed, fixed-hinged, and hinged-hinged boundary conditions are investigated. The pipe is stable at its original static equilibrium position until the flow velocity becomes high enough to cause a supercritical pitchfork bifurcation, and the pipe loses stability by static divergence. In the supercritical fluid velocity regime, the equilibrium configuration becomes unstable and bifurcates into multiple equilibrium positions. To investigate the vibrations that occur in the vicinity of a buckled equilibrium position, the pseudo-nonlinear vibration problem around the first buckled configuration is solved precisely using a new solution procedure. By solving the resulting eigenvalue problem, the natural frequencies and the associated mode shapes of the pipe are calculated. The dynamic stability of the post-buckling configurations obtained in this manner is investigated. The first buckled shape is a stable equilibrium position for all boundary conditions. The buckled configurations beyond the first buckling mode are unstable equilibrium positions. The natural frequencies of the lowest vibration modes around each of the first two buckled configurations are presented. Effects of the system parameters on pipe behavior as well as the possibility of a subcritical pitchfork bifurcation are also investigated. The results show that many internal resonances might be activated among the vibration modes around the same or different buckled configurations. (c) 2012 Elsevier Ltd. All rights reserved.
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    BIFURCATION AND CHAOS OF SLIGHTLY CURVED PIPES
    Sinir, BG
    Non-linear vibrations of slightly curved pipes conveying fluid with constant velocity are investigated. The curvature is taken as an arbitrary function of the spatial variable. The initial displacement is considered due to the geometry of the pipe itself. The ends of the curved pipe are assumed to be immovable simple supports. The equations of motion of pipes are derived using Hamilton's principle and solved by Galerkin method. The bifurcation diagrams are presented for various amplitudes of the curvature function and fluid velocity. The periodic and chaotic motions have been observed in the transverse vibrations of slightly curved pipe conveying fluid.
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    THE EQUATION OF MOTION OF AXIALLY COMPOSITE BEAMS
    Dönmez Demir, D; Sinir, BG
    We present to obtain the equations of motion of the axially composite beams. The composite beams are produced from two or more different materials. In this study, the material varies along the beam axis. In other words, it is seen that the beam is made of different materials, as the beam proceeds along its axis. The material is homogeneous and the beam is formed by combining the step by step along the beam axis. The mathematical model of this problem can be presented in two different ways. In the first, a multispan beam approach is used. In this approach, the variation of each material is given as one span. The equation of motion is obtained as number of various material and four different transient conditions are written for each material alteration point. In the other, one equation is introduced. This equation contains the discontinuity function. The material variation is modeled with the discontinuity functions. Thus, two different models are obtained for only problem.
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    The analysis of nonlinear vibrations of a pipe conveying an ideal fluid
    Sinir, BG; Demir, DD
    In this study, the non-linear vibrations of fixed-fixed tensioned pipe with vanishing flexural stiffness and conveying fluid with constant velocity are considered. The fractional calculus approach is introduced in the constitutive relationship of viscoelastic material. The pipe is on fixed support and the immovable end conditions result in the extension of the pipe during vibration and hence are introduced further nonlinear terms to the equation of motion. Analytical solutions are obtained by using the method of multiple scales. Nonlinear frequencies versus the amplitude of deflection are calculated. For frequencies close to one times the natural frequency, stability of steady-state solutions is analysed. (C) 2015 Elsevier Masson SAS. All rights reserved.
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    Derivation of Governing Equations by Using Vector Approach and Comparison of Analytical Solutions of Post-buckling Behaviors of Transverse Functionally Graded Shear Deformable Beam Theories
    Sinir, BG
    In this study, the post-buckling behavior of a transverse functionally graded beam is investigated. Euler-Bernoulli beam theory (EBT) and shear deformable beam theories are taken into account in deriving the mathematical models of the beam using the vector approach. Timoshenko theory (TBT) and six different higher order beam theories (HOBT), namely Reddy, Touratier, Soldatos, Karma, Akavci and Violet, are considered as shear deformable beam theories. It has been shown that mathematical models of shear deformable beam theories can be obtained using the vector approach. There are two different models developed for shear deformable beam theories depending on normal and shear forces and moment. It is found that the model which is named as Model 2 in this study yield inappropriate results. The functionally graded materials are characterized by using power law functions. The non-dimensional integro-non-linear differential equations system is solved analytically. The critical load values calculated for EBT, TBT and HOBT theories depending on different material conditions and slenderness ratio values are presented in tables. In addition, pitchfork bifurcation diagrams are drawn showing the post-buckling behavior of the beam. In addition, the regions where the examined beam theories are valid depending on the slenderness coefficient are shown.
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    Dynamical analysis of the general beam model with singularity function
    Demir, DD; Sinir, BG; Kahraman, E
    The aim of this study is to present a general model with variable coefficients corresponding to some structural elements such as beam, string, bar, and rod. To solve general model with variable coefficients, a different solution procedure combining a method of multiple scales (MINIS) and a finite difference method (FDM) is presented in this study. This technique provides an advantage in the numerical solution of the structural element model containing any discontinuity and in its dynamical analysis by perturbation method. Furthermore, two problems including discontinuity are considered to show the accuracy of the method presented. The comparisons of the numerical results obtained from the proposed method and classical method are introduced.
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    Nonlinear free and forced vibration analyses of axially functionally graded Euler-Bernoulli beams with non-uniform cross-section
    Sinir, S; Çevik, M; Sinir, BG
    Nonlinear free and forced vibrations of axially functionally graded Euler-Bernoulli beams with non-uniform cross-section are investigated. The beam has immovable, namely clamped-clamped and pinned-pinned boundary conditions, which leads to midplane stretching in the course of vibrations. Nonlinearities occur in the system due to this stretching. Damping and forcing terms are included after nondimensionalization. The equations are solved approximately using perturbation method and mode shapes by differential quadrature method. In the linear order natural frequencies and mode shapes are computed. In the nonlinear order, some corrections arise to the linear problem; the effect of these nonlinear correction terms on natural frequency is examined and frequency response curves are drawn to show the unstable regions. In order to confirm the validity, our results are compared with others available in literature.