We consider a uniform bimorph canti

We consider a uniform bimorph cantilever beam with length L under its base horizontal and vertical excitations as shown in Fig. 1.The beam is composed of a substrate and two piezoelectric layers. The piezoelectric layers are bounded by two in-plane electrodes of negligible thickness connected to the load resistance RLRL . The beam is treated as the Euler–Bernoulli beam with length L, width b, and height hb=ts+2tphb=ts+2tp , and shearing deformation and rotatory motion are neglected. tsts is the thickness of the substrate layer and tptp is the thickness of each piezoelectric layer. Setting the oxy as the inertia coordinate system, the clamped end displacements of the beam are wx(t)wx(t) and wy(t)wy(t) in the horizontal and vertical direction, respectively. The o′x′y′o′x′y′ is the base-fixed coordinate system (moving with the base). s is the coordinate along the middle plane of the beam. u(s, t) and v(s, t) are the displacements of the beam relative to the o′x′y′o′x′y′ coordinate system. u(s, t) is in the x′x′ direction and v(s, t) in the y′y′ direction. We assume that the beam is inextensible