## 1 Introduction
Accurate simulation results can be obtained by overall modeling of the MFC and structure. However, this is hard to implement for complicated structures. With regard to the structure, the mass of the MFC slice is negligible and produces little influence on the structural mode. As a result, the MFC is equivalent to the external actuating force to be exerted onto the structure, which is a simpler and more reasonable method. [PERSON] et al. [12] arranged MFCs on vertical fin structures to deduce the MFC equivalent actuating bending moment equation and the sensing voltage equation by utilizing the classical plate theory (CPT), and accomplished the simulation and experiment of vibrational control of vertical fin structures based on negative feedback by establishing a simplified model of the vertical fin structures attached to the MFC with ABAQUS finite element software, proving that MFCs have a great controlling effect on the vertical fin structures. [PERSON] et al. [13] studied the equivalent actuating force based on the elastic deformation theory under the condition of partial abhesion of the MFC, analyzed the influence of the percentage and location of delamination on actuating performance, and verified the experimental result. [PERSON] et al. [14] studied MFC's active control over a cantilever beam structure in axial motion, calculated the actuating force of the MFC by CPT, and used the MFC as an equivalent actuating force to be exerted onto the cantilever beam structure, proving that the MFC can effectively restrain vibration of the cantilever beam structure. [PERSON] et al. [15] modeled a plate structure integrated with a piezoelectric actuator in a semi-analytical fashion, and applied the analytical equivalent force to the finite element models. The equivalent force is deduced from the bending effort balance between the equivalent loads, and is verified by the actuating experiment of composite laminates integrated with the P1 type of MFC. The experimental results were in good agreement with the simulation results. [PERSON] et al. [16] designed an active serrated microflap based on an MFC to reduce the fatigue load of wind turbine blades, obtained the actuating force of the MFC by CPT and the variation principle, and established a feedback control system suitable for the microflap by making MFC equivalent to the actuating force, indicating that this active serrated microflap can effectively reduce fatigue load. At present, the calculating methods of the MFC equivalent actuating force have mostly been concentrated on considering the bending deformation and tensile deformation of the MFC integrated plate structure, but ignored the influence of shear deformation. Aiming for piezoceramics, [PERSON] and [PERSON] [17] studied the actuating force and actuating bending moment formulas considering shear deformation of the stickup layer and provided a distribution curve of piezoceramic's actuating force to simulate the actuating performance of the piezoelectric material well. Thereafter, [PERSON] and [PERSON] [18] studied the actuating force formula of a piezoelectric material considering bending deformation, but not considering shear deformation. [PERSON] et al. [19] took uniformly polarized piezoelectric actuators with different shapes as the study objects. By analyzing their equivalent forces and wavenumber spectrum, the influence of the shape of the piezoelectric actuators on the modal response of flat plates is studied, and the general expression of equivalent force of piezoelectric actuators was deduced by the small deflection theory. The results show that the uniformly polarized piezoelectric actuator has a high order mode coupling trend regardless of the shape of the piezoelectric actuator. [PERSON] et al. [20] deduced the analytical expressions of equivalent forces of an orthotropic piezoelectric actuator based on the small deflection theory under general conditions by the Hamilton's principle and two different mathematical methods. Both methods obtained the same analytical expressions of equivalent forces.