The nonlinear normal modes of a geometrically nonlinear multi-span beam consisting of n segments, coupled by means of torsional stiffeners are examined. Assuming that the stiffeners possess large torsional stiffness, the beam displacements are decomposed into static and flexible components. It is shown that the static components are much smaller in magnitude than the flexible ones. A Galerkin approximation is subsequently employed to discretize the problem, whereby the computation of the nonlinear normal modes of the multi-span beam is reduced to the study of the periodic solutions of a set of weakly coupled, weakly nonlinear ordinary differential equations. Numerous stable and unstable, localized and non-localized nonlinear normal modes of the multi-span beam are detected. Assemblies consisting of n = 2, 3, and 4 beam segments are examined, and are found to possess stable, strongly localized nonlinear normal modes. These are free synchronous oscillations during which only one segment of the assembly vibrates with finite amplitude. As the number of periodic segments increases, the structure of the nonlinear normal modes becomes increasingly more complicated. In the multi-span beams examined, nonlinear mode localization is generated through two distinct mechanisms: through Pitchfork or Saddle-node mode bifurcations, or as the limit of a continuous mode branch when a coupling parameter tends to zero.
The paper considers the minimization of the l∞-induced norm of the closed loop in linear periodically time varying (LPTV) systems when state information is available for feedback. A state-space approach is taken and concepts of viability theory and controlled invariance are utilized. It is shown that a memoryless periodically varying nonlinear controller can be constructed to achieve near-optimal performance. The construction involves the solution of several finite linear programs and generalizes to the periodic case earlier work on linear time-invariant systems (LTI).
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