Browsing by Subject "Approximate analytical solutions"

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    Analytical and numerical solutions of electro-osmotically driven flow of a third grade fluid between micro-parallel plates
    (2008) Akgül M.B.; Pakdemirli M.
    Electro-osmotic flow of a third grade fluid between micro-parallel plates is considered. The equations of motion are derived and made dimensionless. Approximate analytical solutions are obtained by perturbation techniques. Constant viscosity and temperature dependent viscosity (Reynolds model) cases are treated separately. Numerical solutions of the equations are also obtained. Influences of non-Newtonian parameter, Joule heating effect, viscosity index and electro-kinetic effect on the velocity and temperature profiles are shown. Approximate and numerical solutions are contrasted. © 2008 Elsevier Ltd. All rights reserved.
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    A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case
    (2009) Burak Özhan B.; Pakdemirli M.
    Nonlinear vibrations of a general model of continuous system is considered. The model consists of arbitrary linear and cubic operators. The equation of motion is solved by the method of multiple scales (a perturbation method). The primary resonances of external excitation is analysed. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady-state solutions and their stability are discussed. Finally, the solution algorithm is applied to two different engineering problems. One of the application is the transverse vibration of an axially moving Euler-Bernoulli beam and the other is a viscoelastic beam. © 2009.
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    Approximate analytical solutions for flow of a third-grade fluid through a parallel-plate channel filled with a porous medium
    (2010) Aksoy Y.; Pakdemirli M.
    The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy's law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold's model viscosity, and Vogel's model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement. © 2009 Springer Science+Business Media B.V.
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    A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation
    (2010) Burak Özhan B.; Pakdemirli M.
    A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler-Bernoulli beam, and (2) axially moving viscoelastic beam. © 2010 Elsevier Ltd. All rights reserved.
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    A new perturbation solution for systems with strong quadratic and cubic nonlinearities
    (2010) Pakdemirli M.; Karahan M.M.F.
    The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt-Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MSmethod and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd.
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    Non-linear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports
    (2011) Baǧdatli S.M.; Öz H.R.; Özkaya E.
    In this study, nonlinear transverse vibrations of a tensioned Euler-Bernoulli beam resting on multiple supports are investigated. The immovable end conditions due to simple supports cause stretching of neutral axis and introduce cubic nonlinearity to the equations of motion. Forcing and damping effects are included in the analysis. The general arbitrary number of support case is investigated and 3, 4, and 5 support cases analyzed in detail. A perturbation technique (the method of multiple scales) is applied to the equations of motion to obtain approximate analytical solutions. 3:1 internal resonance case is also considered. Natural frequencies and mode shapes for the linear problem are found for the tensioned beam. Nonlinear frequencies are calculated; amplitude and phase modulation figures are presented for different forcing and damping cases. Frequency-response and force-response curves are drawn. Different internal resonance cases between modes are investigated. © Association for Scientific Research.