Browsing by Author "Boyaci H."
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Item A comparison of different versions of the method of multiple scales for partial differential equations(Academic Press, 1997) Boyaci H.; Pakdemirli M.Applications of the methods of multiple scales (a perturbation method) to partial differential systems arising in non-linear vibrations of continuous systems are considered. Two different versions of the method of multiple scales are applied to two general non-linear models. In one of the models, the small parameter (ε) multiplies an arbitrary non-linear cubic operator whereas in the other model, arbitrary quadratic and cubic non-linearities exist. The linear parts of both models are represented by arbitrary operators. General solutions are found by applying different versions of the method of multiple scales. Results of the first version (reconstitution method) and the second version (proposed by Rahman and Burton [8]) are compared for both models. From the comparisons of both methods, it is found that the second version yields better results. Applications of the general models to specific problems are also presented. A final recommendation is to use the second version of the method of multiple scales combined with the direct-perturbation method in finding steady state solutions of partial differential equations. © 1997 Academic Press Limited.Item The direct-perturbation method versus the discretization-perturbation method: Linear systems(Academic Press, 1997) Pakdemirli M.; Boyaci H.[No abstract available]Item Transverse vibrations of tensioned pipes conveying fluid with time-dependent velocity(Academic Press Ltd, 2000) Öz H.R.; Boyaci H.In this study, the transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with time-dependent velocity are investigated. Two different cases, the pipes with fixed-fixed end and fixed-sliding end conditions are considered. The time-dependent velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is derived using Hamilton's principle and solved analytically by direct application of the method of multiple scales (a perturbation technique). The natural frequencies are found. Increasing the ratio of fluid mass to the total mass per unit length increases the natural frequencies. The principal parametric resonance cases are investigated in detail. Stability boundaries are determined analytically. It is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system. When the velocity fluctuation frequency is close to zero, no instabilities are detected up to the first order of perturbation. Numerical results are presented for the first two modes.Item Vibrations of a stretched beam with non-ideal boundary conditions(Association for Scientific Research, 2001) Pakdemirli M.; Boyaci H.A simply supported Euler-Bernoulli beam with immovable end conditions is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.Item Non-linear vibrations and stability of an axially moving beam with time-dependent velocity(Elsevier Science Ltd, 2001) Öz H.R.; Pakdemirli M.; Boyaci H.Non-linear vibrations of an axially moving beam are investigated. The non-linearity is introduced by including stretching effect of the beam. The beam is moving with a time-dependent velocity, namely a harmonically varying velocity about a constant mean velocity. Approximate solutions are sought using the method of multiple scales. Depending on the variation of velocity, three distinct cases arise: (i) frequency away from zero or two times the natural frequency, (ii) frequency close to zero, (iii) frequency close to two times the natural frequency. Amplitude-dependent non-linear frequencies are derived. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.Item Effect of non-ideal boundary conditions on the vibrations of continuous systems(Academic Press, 2002) Pakdemirli M.; Boyaci H.[No abstract available]Item Non-linear vibrations of a simple-simple beam with a non-ideal support in between(Academic Press, 2003) Pakdemirli M.; Boyaci H.A simply supported Euler-Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.Item Vibrations of a simply supported beam with a non-ideal support at an intermediate point(Association for Scientific Research, 2003) Pakdemirli M.; Boyaci H.A simply supported Euler-Bernoulli beam with an intermediate support is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. Approximate analytical solution of the problem is found using the Method of Multiple Scales, a perturbation technique. Ideal and non-ideal frequency response curves are contrasted.Item Vibrations of stretched damped beams under non-ideal boundary conditions(Indian Academy of Sciences, 2006) Boyaci H.A simply supported damped Euler-Bernoulli beam with immovable end conditions are considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections and moments. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.Item Perturbative derivation and comparisons of root-finding algorithms with fourth order derivatives(Association for Scientific Research, 2007) Pakdemirli M.; Boyaci H.; Yurtsever H.A.Perturbation theory is systematically used to generate root finding algorithms with fourth order derivatives. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. Expanding Taylor series up to fourth order derivatives and taking two, three and four correction terms in the perturbation expansions, three different root finding algorithms are derived. The algorithms are contrasted numerically with each other as well as with the Newton-Raphson algorithm. It is found that the quadruple-correction-term algorithm performs better than the others. © Association for Scientific Research.Item Effects of non-ideal boundary conditions on vibrations of microbeams(2007) Ekici H.O.; Boyaci H.Effects of non-ideal boundary conditions on the vibrations of microbeams are investigated. Stretching effect as well as axial force is included along with the non-ideal boundary conditions. The Method of Multiple Time Scales (a perturbation technique) is employed to solve the non-dimensional equation of motion for subharmonic and superharmonic resonance cases. The frequencies and mode shapes obtained are compared with the ideal boundary conditions case and the differences between them are contrasted on frequency response curves. © 2007 SAGE Publications Los Angeles.Item Generation of root finding algorithms via perturbation theory and some formulas(2007) Pakdemirli M.; Boyaci H.Perturbation theory is systematically used to generate root finding algorithms. Depending on the number of correction terms in the perturbation expansion and the number of Taylor expansion terms, different root finding formulas can be generated. The way of separating the resulting equations after the perturbation expansion alters the root-finding formulas also. Well known cases such as Newton-Raphson and its second correction, namely the Householder's iteration, are derived as examples. Moreover, higher order algorithms which may or may not be the corrections of well known formulas are derived. The formulas are contrasted with each other as well as with some new algorithms obtained by modified Adomian Decomposition Method proposed in Ref. [S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887-893]. © 2006 Elsevier Inc. All rights reserved.Item A root-finding algorithm with fifth order derivatives(Association for Scientific Research, 2008) Pakdemirli M.; Boyaci H.; Yurtsever H.A.Perturbation theory is used to generate a root finding algorithm with fifth order derivatives. The algorithm is called Quintuple-Correction-Term algorithm. The new algorithm is contrasted with the previous Quadruple-Correction-Term and Triple-Correction-Term algorithms in the literature. It is found that adding a fifth correction term in the algorithm does not improve the performance. © Association for Scientific Research.Item A new perturbation algorithm with better convergence properties: Multiple scales lindstedt poincare method(Association for Scientific Research, 2009) Pakdemirli M.; Karahan M.M.F.; Boyaci H.A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed for the first time. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and variable amplitude cases. Three problems are solved: Linear damped vibration equation, classical duffing equation and damped cubic nonlinear equation. Results of Multiple Scales, new method and numerical solutions are contrasted. The proposed new method produces better results for strong nonlinearities. © Association for Scientific Research.Item A new perturbation algorithm for strongly nonlinear oscillators(2009) Pakdemirli M.; Karahan M.M.F.; Boyaci H.A new perturbation algorithm combining the Method of Multiple Scales and Lindstedt-Poincare techniques is proposed. The algorithm combines the advantages of both methods. Convergence to real solutions with large perturbation parameters can be achieved for both constant amplitude and variable amplitude cases. Three problems are solved: Linear damped vibration equation, classical duffing equation and damped cubic nonlinear equation. The new method does not violate the main assumption of perturbation series that correction terms should be much smaller than the leading terms. It is proven that for arbitrarily large perturbation parameter values, correction terms remain much smaller that the leading terms. Results of Multiple Scales, new method and numerical solutions are contrasted. The proposed new method produces much better results for strong nonlinearities.Item Nonlinear transverse vibrations of a slightly curved beam carrying a concentrated mass(2009) Özkaya E.; Sarigül M.; Boyaci H.In this study, a slightly curved Euler Bernoulli beam carrying a concentrated mass was handled. The beam was resting on an elastic foundation and simply supported at both ends. Effects of the concentrated mass on nonlinear vibrations were investigated. Sinusoidal and parabolic type functions were used as curvature functions. Equations of motion have cubic nonlinearities because of elongations during vibrations. Damping and harmonic excitation terms were added to the equations of motion. Method of multiple scales, a perturbation technique, was used for solving integro-differential equation analytically. Natural frequencies were calculated exactly for different mass ratios, mass locations, curvature functions, and linear elastic foundation coefficients. Amplitude-phase modulation equations were found by considering primary resonance case. Effects of nonlinear terms on natural frequencies were calculated. Frequency-amplitude and frequency-response graphs were plotted. Finally effects of concentrated mass and chosen curvature function on nonlinear vibrations were investigated. © 2009 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH.Item Nonlinear vibrations of axially moving beams with multiple concentrated masses Part I: Primary resonance(Techno-Press, 2010) Sarigül M.; Boyaci H.Transverse vibrations of axially moving beams with multiple concentrated masses have been investigated. It is assumed that the beam is of Euler-Bernoulli type, and both ends of it have simply supports. Concentrated masses are equally distributed on the beam. This system is formulated mathematically and then sought to find out approximately solutions of the problem. Method of multiple scales has been used. It is assumed that axial velocity of the beam is harmonically varying around a mean-constant velocity. In case of primary resonance, an analytical solution is derived. Then, the effects of both magnitude and number of the concentrated masses on nonlinear vibrations are investigated numerically in detail.Item Strength of wheat and barley stems and design of new beam/columns(2010) Deǧer G.; Pakdemirli M.; Candan F.; Akgün S.; Boyaci H.In this study, physical and mechanical properties of wheat and barley stems are examined. Transverse sections of the stems are magnified by a microscope and the material structure in the transverse sections are analysed with image processing programs. Geometric properties such as inner, outer radius, stem wall thickness and density variation of the material along the radius are measured and density variations are approximated by a mathematical model. Moment of inertia of the cross-sectional area which plays a vital role in resistance against bending and buckling is calculated approximately. Using the material density variations of the wheat (Triticum sativum L.) stems, new beam/columns are designed. Stress distributions in this new design and conventional designs of equivalent weight are compared using ANSYS program. It is found that stresses are more uniformly distributed in the new design with maximum stresses being lower than the conventional designs. © Association for Scientific Research.Item A new perturbation-iteration approach for first order differential equations(Association for Scientific Research, 2011) Pakdemirli M.; Aksoy Y.; Boyaci H.Two new perturbation-iteration algorithms for solving differential equations of first order are proposed. Variants of the algorithm are developed depending on the differential order of Taylor series expansions. The iteration algorithms are tested on a number of first order equations. Much better solutions than the regular perturbation solutions are achieved. © Association for Scientific Research.Item Forced vibrations of strongly nonlinear systems with multiple scales lindstedt poincare method(Association for Scientific Research, 2011) Pakdemirli M.; Karahan M.M.F.; Boyaci H.Forced vibrations of duffing equation with damping is considered. Recently developed Multiple Scales Lindstedt-Poincare (MSLP) technique for free vibrations is applied for the first time to the forced vibration problem in search of approximate solutions. For the case of weak and strong nonlinearities, approximate solutions of the new method are contrasted with the usual Multiple Scales (MS) method and numerical simulations. For weakly nonlinear systems, frequency response curves of both perturbation methods and numerical solutions are in good agreement. For strongly nonlinear systems however, results of MS deviate much from the MSLP method and numerical simulations, the latter two being in good agreement. Keywords- Perturbation Methods, Lindstedt Poincare method, Multiple. © Association for Scientific Research.