# Single Cylinder-Type Piezoelectric Actuator with Two Active Kinematic Pairs
## 2 Description of the Actuator and Modeling Results  
Several principle design schemes of the piezoelectric actuator with two kinematic pairs are shown in Figure 1, where \(w\) is the total number of the degrees-of-freedom of moving links \(1\) and _1a_; \(2\) is the piezoelectric cylinder with special sectioning of electrodes, and \(3\) is a frame, on which additional displacements occur. Thus, the displacements of the moving link are quite evident: link \(1\) rotatesaround axes \(x\), \(y\), \(z\) and affects the translational motion in directions \(x\), \(y\) and rotation around axis \(z\) (Figure 1a); link \(1\) rotates around axis \(z\), affects translational motion in the direction of \(z\) axis and moves on the plane \(x\), \(y\) (Figure 1b); links \(1\) and \(1a\) rotate around axis x and move along it (Figure 1c); link \(1\) rotates around axes \(x\), \(y\), \(z\) and affects translational motion in the direction of \(z\) axis and rotation around the same axis (Figure 1d). The number of degrees-of-freedom could be increased: a schematic of the piezoelectric actuator with three active kinematic pairs (\(w=8\)) is shown in Figure 2. To obtain 8 degrees-of-freedom, it is necessary to solve the problem of electrode sectioning of the piezoelectric actuator with the aim to exclude the unwanted oscillations of some specific forms. Two types of motion (the sphere rotating on the cylinder and the sphere moving on the plane) can be controlled separately by:  
* controlling the forces, acting in the contact zones of the sphere, piezoelectric cylinder and plane by introducing controllable brakes with the help of permanent magnets (Figure 3) or using layers of electro-rheological suspensions, controlled by high voltage
* connecting the same frequency voltage to specific electrodes, but with a different phase (\(180^{\circ}\)) and amplitude, as in the main driving source
* using additional elements related to the position of the nodes of oscillation  
We will further analyze the piezoelectric cylinder-type actuator with two kinematic pairs as shown in Figure 4a. A triangular electrode configuration is used to excite vibrations of the actuator (Figure 4b). The brakes of active kinematic pairs are not included. The piezoelectric cylinder has the following dimensions: 28 mm (inner diameter) \(\times\) 33 mm (outer diameter) \(\times\) 20 mm (height) and is made from PZT-8 material. A steel sphere is positioned on top of the cylinder (Figure 4a). The outer surface of the cylinder is covered with triangular electrodes as shown in Figure 4b, while the inner surface of the cylinder has a single electrode and is grounded. Figure 1: General schematics of the piezoelectric actuator with two active kinematic pairs. Figure 2: The schematic of the piezoelectric actuator with three active kinematic pairs. Two electrodes on the outer surface are excited at the same time to achieve rotation of the rotor. Different voltages are applied to achieve the desired motion--larger voltage results in higher amplitude but not necessarily in the proper direction of the contact point. Therefore, improper voltages may result in elliptical trajectories of the contact points moving in opposite directions. It was chosen to apply a voltage of 100 V to the first electrode and 50 V to the second. Harmonic excitation signal was used. To determine proper resonance frequencies, the amplitudes of contact point vibration were investigated, when excitation frequency was changing in the range 0 Hz-150,000 Hz. The amplitudes of the contact point in the middle of electrode 4 (Figure 4b) are shown at Figure 5. The modal analysis shows that the required eigenform is at 88,371 Hz (Figure 6); therefore, a resonant frequency at 88,600 Hz will be further analyzed. This frequency resonates with the correct eigenform. The calculations and analysis were performed using ANSYS software. Figure 4: (**a**) Analyzed kinematic pair; (**b**) Proposed triangular electrode scheme.