Browsing by Subject "Modulation equations"

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    General solution algorithm for three-to-one internal resonances of a cubic nonlinear vibration model
    (2009) Özhan B.B.; Pakdemirli M.
    A generalized nonlinear vibration model of continuous systems is considered. The model includes arbitrary linear and cubic differential and/or integral operators. Linear operators represent the linear parts and cubic operators represent the nonlinear parts of the model. The generalized equation of motion is analyzed by using the method of multiple scales (a perturbation method). Three-to-one internal resonances between natural frequencies are obtained. The amplitude and phase modulation equations are presented. Approximate solution is derived. Steady state solutions and their stability are discussed. Solution algorithm is applied to nonlinear vibration model of an axially moving Euler Bernoulli beam. Constant velocity case of axially moving beam is analyzed. Natural frequencies of beam are given for different velocity values. Steady state solutions and their stability are determined numerically. Frequency response relations are obtained. Energy transfer of one mode to another via a three-to-one internal resonance is observed. Jump phenomena of the system are shown graphically by choosing different vibration and beam parameter values.
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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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    Nonlinear transverse vibrations of a slightly curved beam resting on multiple springs
    (International Institute of Acoustics and Vibrations, 2016) Özkaya E.; Sarigül M.; Boyaci H.
    In this study, nonlinear vibrations of a slightly curved beam of arbitrary rise functions is handled in case it rests on multiple springs. The beam is simply supported on both ends and is restricted in longitudinal directions using the supports. Thus, the equations of motion have nonlinearities due to elongations during vibrations. The method of multiple scales (MMS), a perturbation technique, is used to solve the integro-differential equation analytically. Primary and 3 to 1 internal resonance cases are taken into account during steady-state vibrations. Assuming the rise functions are sinusoidal in numerical analysis, the natural frequencies are calculated exactly for different spring numbers, spring coefficients, and spring locations. Frequency-amplitude graphs and frequency-response graphs are plotted by using amplitude-phase modulation equations.