. The approach is a common practice in study of nonlinear vibrations of bi-stable structures [9] and is justified by center manifold reduction[20]. The static deflection shape of the beam under a unit load applied to the tip is used as the shape function. The following integrals are defined to facilitate abbreviation of formulas: (2-a) (2-b) Each of the terms in Lagrangian are related to the states as follows: (3) where are the stiffness coefficients, are the piezoelectric constants [21], are the electric field components and is the magnetic force potential. The Magnetic force is experimentally measured and is characterized as . The magnetic force potential is therefore: . We let denote the Young’s modulus of the steel substrate, the Young’s modulus of the piezoelectric patch, the area moment of inertia of the steel beam about its geometric center and stand for the area moment of inertial of the cross section of each piezoelectric patch about the center line of the steel substructure. Eq. (3) is simplified to: (4) In Eq. (4), is the flux linkage across the piezoelectric patch, is the cross-sectional area and is the z-coordinate of the centroid of the patch. The z and x-coordinates have been defined in Fig. 1. The base motion, characterized by , should be taken into account when calculating the kinetic energy .The kinetic energy is evaluated as (5) The densities of the steel substrate and the piezoelectric patch are and respectively. The total mass of the substrate and each of the piezoelectric patches are and . The cross sectional area of the substrate is and stands for the 464
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