Stabilization of Collective Motion in Synchronized, Balanced and Splay Phase Arrangements on a Desired Circle

This paper proposes a design methodology to stabilize collective circular motion of a group of N-identical agents moving at unit speed around individual circles of different radii and different centers. The collective circular motion studied in this paper is characterized by the clockwise rotation of all agents around a common circle of desired radius as well as center, which is fixed. Our interest is to achieve those collective circular motions in which the phases of the agents are arranged either in synchronized, in balanced or in splay formation. In synchronized formation, the agents and their centroid move in a common direction while in balanced formation, the movement of the agents ensures a fixed location of the centroid. The splay state is a special case of balanced formation, in which the phases are separated by multiples of 2 pi/N. We derive the feedback controls and prove the asymptotic stability of the desired collective circular motion by using Lyapunov theory and the LaSalle's Invariance principle.

Long-period modulated structure and electric-field-induced structural transformation in Na0.5Bi0.5TiO3-based lead-free piezoelectrics

Na0.5Bi0.5TiO3- based lead-free piezoelectrics exhibiting giant piezostrain are technologically interesting materials for actuator applications. The lack of clarity with regard to the structure of the nonpolar phase of this system has hindered the understanding of the structural mechanism associated with the giant piezostrain and other related phenomena. In this paper, we have investigated the structure and field-induced phase transformation behavior of a model system (0.94 - x) Na0.5Bi0.5TiO3-0.06BaTiO(3)-xK(0.5)Na(0.5)NbO(3) (0.0 <= x <= 0.025). A detailed structural analysis using neutron powder diffraction revealed that the nonpolar phase is neither cubic nor a mixture of rhombohedral (R3c) and tetragonal (P4bm) phases as commonly reported in literature but exhibits a long-period modulated structure, which is most probably of the type root 2 x root 2 x n with n = 16. Our results suggest that the giant piezoelectric strain is associated with a field-induced phase transformation of the long-period modulated structure to rhombohedral R3c structure above a critical field. We also demonstrate that the giant piezostrain is lost if the system retains a fraction of the field-induced R3c phase. A possible correlation among depolarization temperature, giant piezostrain, and its electrical fatigue behavior has also been indicated.