content_image/359595.jpg content_image/359594.jpg content_image/359596.jpg content_image/359597.jpg content_image/359598.jpg content_image/359599.jpg
or 

<img src='content_image/359595.jpg'>

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 

<img src='content_image/359594.jpg'>

The equation of motion of the beam are derived via Hamilton’s principle [7]: 

<img src='content_image/359596.jpg'>

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 

<img src='content_image/359597.jpg'>

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 

<img src='content_image/359598.jpg'>

Performing variation in (10) and using (9), (11), and (12), we obtain the equations describing the electromechanical behavior of the beam 

<img src='content_image/359599.jpg'>

