Browsing by Author "Karahan M.M.F."
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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 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.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.Item New approximate solutions for the strongly nonlinear cubic-quintic duffing oscillators(American Institute of Physics Inc., 2016) Karahan M.M.F.; Pakdemirli M.Strongly nonlinear cubic-quintic Duffing oscillator is considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions. © 2016 Author(s).Item Approximate solutions for the nonlinear third-order ordinary differential equations(Walter de Gruyter GmbH, 2017) Karahan M.M.F.A new perturbation method, multiple scales Lindstedt-Poincare (MSLP) is applied to jerk equations with cubic nonlinearities. Three different jerk equations are investigated. Approximate analytical solutions and periods are obtained using MSLP method. Both approximate analytical solutions and periods are contrasted with numerical and exact results. For the case of strong nonlinearities, obtained results are in good agreement with numerical and exact ones. © 2017 Walter de Gruyter GmbH, Berlin/Boston 2017.Item Vibration analysis of a beam on a nonlinear elastic foundation(Techno-Press, 2017) Karahan M.M.F.; Pakdemirli M.Nonlinear vibrations of an Euler-Bernoulli beam resting on a nonlinear elastic foundation are discussed. In search of approximate analytical solutions, the classical multiple scales (MS) and the multiple scales Lindstedt Poincare (MSLP) methods are used. The case of primary resonance is investigated. Amplitude and phase modulation equations are obtained. Steady state solutions are considered. Frequency response curves obtained by both methods are contrasted with each other with respect to the effect of various physical parameters. For weakly nonlinear systems, MS and MSLP solutions are in good agreement. For strong hardening nonlinearities, MSLP solutions exhibit the usual jump phenomena whereas MS solutions are not reliable producing backward curves which are unphysical. Copyright © 2017 Techno-Press, Ltd.Item Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method(Springer, 2024) Demir İ.; Karahan M.M.F.; Aktürk N.Purpose: In this article, the transverse vibration motion of a self-excited beam which is subjected to a distributed and a singular load is analyzed using the differential transformation method (DTM). Methods: The Euler–Bernoulli beam model is employed. The beam is modeled to represent the cutting tool holder motion in machining. A delayed distributed load and a vibration velocity-dependent singular load are considered as forcing. Analysis is performed for different time delays, widths of distributed load, and beam lengths in the time domain. The Laplace transform method is deployed for the stability analysis. Multi-step DTM is applied for the mathematical solution. Matlab® ddesd solutions is used for mathematical comparison. Results: When the width of the distributed load increases and then the vibration amplitude increases. An increase in the beam length causes the amplitude to increase. The vibration amplitude increases as the delay time decreases. However, the reduction of some delay values reduces the amplitude. Conclusion: The equation that expresses the variation of the distributed load width with respect to the delay time is compatible results in accordance with the experimental studies in the literature. If beam length increases, the stability region of the width of the distributed load will decrease. The effect of beam length for stability can be adjusted by changing the time delay. © Springer Nature Singapore Pte Ltd. 2023.