This paper develops an integrated model of periodic 1D structures with piezoelectric
actuators for complete active/passive control. The approach utilizes the property of
periodic structural elements that create stop and pass band regions in the frequency
spectra, predominantly in the higher frequency range. This basic property of
periodic structures is enhanced by the application of periodically placed piezoelectric
actuators, with piezo-forces as a function of displacement. Using collocated feedback
control, the piezoelectric actuators can introduce the proper force to reduce wave
propagation, both in the high and low frequency ranges. An analytical model
is developed to predict the performance of the periodic rods and beams with
piezoelectric actuators acting as controllers. For the purpose of this research,
only geometric periodicity is considered and every cell is assumed to be identical.
In the present paper we prove a unique common fixed point theorem for a family of weakly compatible self maps in non-Archimedean Menger PM-spaces without using the notion of continuity. Our result generalizes and extends some well known previous results.
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