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The initial value problem for a lumped parameter model arising from design of magneto-electromechanical device with a current-carrying conductor is analyzed. The differential equation is nonlinear because it includes the magnetic force term. The analysis for the dynamic pull-in occurring in the system is presented. The pull-in threshold is given analytically in terms of model parameters. Sufficient conditions for the existence of periodic solutions are proved analytically and verified numerically. The results can be useful for understanding and design of one-degree-of-freedom models of magnetically actuated beams.

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Blossman-Myer and Burggren (Comp. Biochem. Phys. A, 155, 2010,259-263) found that the cocoon of the silkworm,Bombyx mori, while creating a tough barrier offering mechanical protection to the pupa, imposes no barrier to the diffusion of oxygen or water vapor. In view of its mechanical properties, silk cocoon is an "emperor's new clothes" for pupa. A theoretical analysis is given to explain the fascinating phenomenon by a fractal hydrodynamic model for discontinuous membrane composed of hierarchical silk cascade. The model reveals that when the fractal dimensions of the microstructure of the cocoon meet a special condition, air and water vapor can flow through the cocoon as if there were no cocoon. However, at extremes of weather, the cocoon is an excellent thermal protection for both high and cool temperatures. A better understanding of the cocoon mechanism could help the further design of bio-mimetic artificial clothes for special applications.