Fig. 1: Schematic View of the harvester Governing equations and possible scenarios We use the energy methods to model the dynamics of the system. The harvester is made up of three coupled systems; the cantilever beam which is characterized by the deflection of beam, the circuit connected to the piezoelectric element and the circuit connected to the coils. The electrical circuit for harvesting is simplified to be only a resistor in order to focus on the transduction. The value of the resistive load in the piezoelectric circuit is and the value of the resistive load in the electromagnetic circuit is . The energy in various components of the entire electromechanical system is: the elastic and magnetic potential energies stored in the beam and the magnetic field ( ), the electrostatic energy stored in the piezoelectric patch ( ), the kinetic energy stored in the beam and the tip mass ( ), the magnetic potential energy between the tip and base magnets ( ), the electromagnetic energy stored in the coils ( ) and the energies dissipated by the resistors and damping of the beam. The displacement of the beam, the flux linkage across the piezoelectric element and the charge through the coils are the coordinates used for identifying the system. Following the guidelines in [19], the Lagrangian of the system is: (1) The dynamic deflection of the beam is simplified by a single mode Galerkin’s method. The deflection at each point and at a certain time relative to the base is 463
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