Nonlinear dynamics of a functionally graded piezoelectric   micro-resonator in the vicinity of the primary resonance
  This research is on the nonlinear dynamics of a two-sided electrostatically actuated capacitive micro-beam. The microresonator is composed of silicon and PZT as a piezoelectric material. PZT is functionally distributed along the height of the micro-beam according to the power law distribution. The micro-resonator is simultaneously subjected to DC piezoelectric and two-sided electrostatic actuations. The DC piezoelectric actuation leads to the generation of an axial force along the length of the micro-beam and this is used as a tuning tool to shift the primary resonance of the micro-resonator. The governing equation of the motion is derived by the minimization of the Hamiltonian and generalized to the viscously damped systems. The periodic solutions in the vicinity of the primary resonance are detected by means of the shooting method and their stability is investigated by determining the so-called Floquet exponents of the perturbed motions. The basins of attraction corresponding to three individual periodic orbits are determined. The results depict that the higher the amplitude of the periodic orbit, the smaller is the area of the attractor 
Meysam T. Chorsi, Saber Azizi, Firooz Bakhtiari-Nejad
cond-mat.mes-hall