Resonant non-linear vibrations in continuous systems—I. Undamped case |
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Authors: | MJ Ablowitz A Askar AS Cakmak H Engin |
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Institution: | Princeton University, Princeton, NJ 08540, U.S.A. |
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Abstract: | A method is presented for obtaining periodic solutions to forced oscillations of non-linear systems governed by equations of the form uss?uyy?εf(u,u,y,uyy…,s) = 0. The method is presented by application to the equation uss?uyy?εu2yuyy= 0 which governs the vibrations of a soil layer that is free on the top surface and is forced harmonically at the bedrock. It is shown that unlike the ODE case (Duffing equation), the PDE requires an infinite number of periodicity conditions to correctly characterize the resonant region and these conditions lead to an infinite number of branches in the dispersion spectrum. Calculations indicate that these branches tend to an envelope curve. The uniform approach presented by Millmann and Keller is discussed in order to determine in what sense it can be viewed as an effective approximation for the fundamental mode. |
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