B. Energy of the micro-cantilever plate with concentrated masses attached

First, the energy of the plate is calculated. The potential energy of the deformation of an isotropic elastic body is

For the Poisson-Kirchhoff thin plate in this study, we have σ_z = τ_yz = τ_zx = 0, so Eq (6) can be further simplified. Substituting Eqs (3) and (5) into Eq (6) yields

where D is the bending stiffness of the plate and can be expressed as[17]

Eq (7) is the potential energy due to the deformation of the plate derived with classical mechanics. When the effect of the size of the plate is considered, the bending stiffness of the plate is changed and is represented by D′[17]

Accordingly, the potential energy due to the deformation of the plate is

Comparing Eqs (7) and (10) shows that their only difference lies in the bending stiffness.

If only the transverse bending vibration of the plate (movement in the z direction, that is w) is considered and the movements in the plane of the plate (movements in the x and y directions, that are u and v) are neglected, the kinetic energy of the plate is

After the energy of the plate is obtained, the energy of the attached concentrated masses must be derived. As stated above, it is assumed that N numbers of particles that can be taken as concentrated masses are attached to the upper (or lower) surface of the micro-cantilever plate. As the particles can be taken as concentrated masses, their sizes are negligible compared with the size of the plate, so the particles have no strain energy resulting from deformation, and only the kinetic energy of the particles needs to be considered.

The i^th particle has mass M_i and coordinates (x_i,y_i,z_i), where z_i = H/ 2 (when the particle is attached to the upper surface of the plate) or z_i = −H/ 2 (when the particle is attached to the lower surface of the plate). Its velocity is vzi=∂w(xi,yi,zi)∂t (only the movement of the particle in the direction of the transverse bending vibration of the plate is considered (or in the z direction,w)), so the total kinetic energy of N numbers of particles is

Now, the expression of the total energy of the micro-cantilever plate with N attached particles is as follows: