## 1 Introduction  
As a widely used piezoelectric smart material, piezoceramics can have an actuating performance through the inverse piezoelectric effect and a sensing performance through the direct piezoelectric effect. They are often applied in areas such as structural vibration reduction, energy harvesting, and health monitoring, etc. [1; 2; 3]. Piezoceramics are characterized by their high sensitivity and good thermal stability, albeit poor flexibility. Therefore, the National Aeronautics and Space Administration (NASA) lab in Langley has developed a new type of piezoelectric smart material, macro fiber composite (MFC) [4], which is recombined with piezoceramic fiber rods with rectangular sections, an epoxy resin basal body, a copper electrode, and a polymer film. With a higher electromechanical transformation efficiency and better flexibility, MFC has wide application prospects and research value. In the vibration reduction of thin-walled structures and deformation control, MFC has received extensive attention. In the study of controlling the vibration response of thin-walled structures with MFC, its actuating force can be obtained by two methods: (i) overall modeling is made for the MFC and the structure, then voltage directly serves as the input to perform a simulating calculation for the structure; (ii) or the MFC is equivalent to the external actuating force to be exerted on the structure for the simulating calculation. With regard to the former, [PERSON] and [PERSON] [5] studied MFC's active control over thermally-induced deformation of the composite plate, performed finite element modeling of the MFC and flat plate construction with ABAQUS finite element software, obtained the actuating force of the MFC by the finite element method, proposed an active closed-loop control algorithm, and verified the feasibility of the MFC control over thermally-induced deformation. [PERSON] et al. [6] have performed overall modeling for the MFC and shell structure, and then solved the actuating force of the MFC by the variation principle and finite element discretization. On this basis, they undertook a vibration reduction simulation and experimental analysis for an end-sealing shell structure attached to the MFC with a negative velocity feedback method. Through overall modeling of the MFC and thin-walled cylindrical structure, [PERSON] et al. [7; 8] calculated the actuating force of the MFC by using the Donnell-Mushtari shell theory and Lagrange equation, as well as the variational principle and finite element discretization, respectively. Based on this, they have studied the vibration control effect of the cylindrical structure with the MFC in air and in water, which proved that the MFC can effectively reduce vibrations in both conditions. [PERSON] et al. [9] performed a finite element analysis on the MFC integrated plate structure with different fiber orientations, established the linear electromechanical coupling model of the MFC integrated plate-shell structure, analyzed the actuating force of the MFC by variational principle and finite element discretization, and performed contrastive analysis between the simulation results and the experimental results. Aiming for vibrational control of flexible structures, [PERSON] et al. [10] studied the optimal configuration of the MFC actuator, established a model for MFC integrated plate-shell structures with the finite element method, and obtained the actuating force of the MFC by the Hamilton's principle. On this basis, they provided the best location and fiber orientation for MFCs in different working conditions. [PERSON] et al. [11] have studied the aeroelastic stability of composite plates embedded within MFCs in supersonic flows, and established the dynamical equation of MFC and plate structure through virtual work principle, and gave the actuating force expression of the MFC, which indicates that both MFC lamination angles and temperature variation will produce remarkable influences on the aeroelastic stability boundary of curved composite plates in supersonic airflows.