## 2 Configuration and Operating Principle  
The configuration of the proposed framed piezoelectric actuator is depicted in Figure 1. The framed piezoelectric actuator is mainly composed of four straight beams (including two short and two long straight beams) bonded with sixteen pieces of piezoelectric plates. These piezoelectric plates are polarized along their thickness direction, and each straight beam with the glued piezoelectric plates form a piezoelectric composite beam. The two short piezoelectric composite beams are set in parallel to each other, and their two ends are perpendicularly connected with the two long piezoelectric composite beams, altogether constructing the rectangular frame structure of the piezoelectric actuator. Two projections located in the middle of the two short piezoelectric composite beams, respectively, are employed as the driving feet of the proposed framed piezoelectric actuator. The framed piezoelectric actuator, a rail, and a preload mechanism constitute a railed carrying system, as shown in Figure 1. The preload mechanism includes a pair of leaf springs, a pair of supporting wheels, and four perforated bolts. The supporting wheels locating in the middle of the leaf springs are in contact with the bottom surface of the rail. Both ends of each leaf spring are fixed on the long piezoelectric composite beam by two perforated bolts. Thus, the preload between the rail and framed piezoelectric actuator can be adjusted by altering the tension of the leaf springs. To minimize the impact of the leaf springs on the vibration characteristic of the framed piezoelectric actuator, all perforated bolts are fixed at the vibration node positions of the long piezoelectric composite beams.  
In order to generate the elliptical motion of the two driving feet, two orthogonal operating vibration modes (Modes A and B) are selected to be excited in the proposed framed piezoelectric actuator, as shown in Figure 2. In mode A, the two short piezoelectric composite beams are excited to present the first order bending vibration along the \(x\)-direction when they are applied with a sinusoidal signal, as shown in Figure 2a. In this case, the two driving feet obtain a displacement component in the \(x\)-direction. As a cosine signal is applied to the two long piezoelectric composite beams, the third order bending vibration and the first order bending vibration both along the \(y\)-direction are stimulated in the two long and short piezoelectric composite beams, respectively, as shown in Figure 2b. This is mode B. Under this circumstance, the two driving feet vibrate along the \(y\)-direction. To maximize the excitation effect of all piezoelectric plates, they are located at the peak positions of corresponding vibration mode. While two voltage signals having a temporal phase difference of \(\pi\)/2 are simultaneously applied to the two long and short piezoelectric composite beams, respectively, the two vibration modes are coupled in the framed piezoelectric actuator, leading to that elliptical trajectories moving with the same direction are produced in the two driving feet. Thus, the framed piezoelectric actuator can move on the rail by friction, achieving the carrying system function. Once the two excitation signals have a temporal phase difference of \(-\pi\)/2, the elliptical trajectories of the two driving feet will be simultaneously reversed, resulting in an opposite motion of the railed carrying system.  
Figure 1: Configuration of the proposed framed piezoelectric actuated railed carrying system.