RESONANT NONLINEAR VIBRATIONS OF CONTINUOUS SYSTEMS .2. DAMPED AND TRANSIENT-BEHAVIOR

The method presented in Part I is extended to cover the damped and transient behavior of nonlinear systems described by equations of the form uss-uyy-εf(u,uy, uyys,...,s) = 0. The method is presented by application to the equation uss-uyy-εuyys-εu2yuyy= 0. Similar to the undamped case it is again shown that the PDE requires an infinite number of periodicity conditions to correctly characterize the resonant region. However, damping eliminates some of the branches of the amplitude-frequency spectrum of the undamped case. In fact, for ε = 0.25 all but the outermost branch disappear. A method of multiple time scales is presented for the study of the transient behavior and the stability of the branches for steady vibrations. The stability analysis yields an interior stable point in the amplitude-frequency spectrum which has no analog in the Duffing equation. Finally via the multiple scale procedure in the spirit of the early work of Zabusky and Kruskal one obtains forced Burgers and Korteweg-de Vries equations on a finite interval. © 1979.