or where the low letter indexes after a comma mean the derivation with respect to corresponding coordinates. After substituting the equations (7), (8), into (3) and (4) and ignoring the polarization gradient, we have The equation of motion of the beam are derived via Hamilton’s principle [7]: where δ ( · ) denotes the first variation, T$_{k}$ is the kinetic energy, and W$_{d}$ is the work done by the external forces and moments. The kinetic energy of the beam by using the equation (6) can be written as where I = ∫ $_{A}$z $^{2}$dA , A is the area of the beam’s cross section, V$_{b}$ is the volume of the beam, ϱ is the mass density of the nano-switch’s material, and the length of the beam is equal to l . If the beam is under the transverse force q , the work done is equal to Performing variation in (10) and using (9), (11), and (12), we obtain the equations describing the electromechanical behavior of the beam H = 1 2 c$_{11}$z $^{2}$( w $^{′′}$) 2 + 1 2 a$_{33}$ ( ϕ ′ z + ( f$_{1133}$ + d$_{31}$z ) w $^{′′}$) 2 − d$_{31}$w ′′ z a$_{33}$ ( ϕ ′ z + ( f$_{1133}$ + d$_{31}$z ) w $^{′′}$) − f$_{1133}$ w ′′ a$_{33}$ ( ϕ ′ z + ( f$_{1133}$ + d$_{31}$z ) w $^{′′}$) − ε$_{0}$ 2 E 2 z − ϕ ′ z a$_{33}$ ( ϕ ′ z + ( f$_{1133}$ + d$_{31}$z ) w $^{′′}$) (9) P$_{3}$ ≡ P$_{z}$ = 1 a$_{33}$ ( − E$_{z}$ + ( f$_{1133}$ + d$_{31}$z ) w $^{′′}$) , (8) δ ∫ t 2 t 1 ( T$_{k}$ − H + W$_{d}$ ) dt = 0 (10) T$_{k}$ = 1 2 ∫ V b ϱ { ˙ u } $^{T}${ ˙ u } dV$_{b}$ = 1 2 ∫ V b ϱ [ ˙ w 2 + z $^{2}$( ˙ w $^{′}$) $^{2}$] dV$_{b}$ = 1 2 ϱ ∫ l 0 [( ˙ w ) $^{2}$A + I ( ˙ w $^{′}$) $^{2}$] dx, (11) W$_{d}$ = ∫ l 0 qwdx. (12) w $^{IV}$( c$_{11}$I − d 2 $_{31}$I a$_{33}$ − f 2 1133 a$_{33}$ A ) − 1 a$_{33}$ ∂ 2 ∂x $_{2}$( f$_{1133}$Q el z + d$_{31}$M el z ) + ϱA ¨ w + ϱI ∂ $^{4}$w ∂x $^{2}$∂t2 = q, (13) 6