The ring type piezoelectric actuator generating elliptical movement

G. Baurienė*, J. Mamcenko**, G. Kulvietis***, A. Grigoravičius****, I. Tumasonienė***** *Kaunas University of Technology, Kęstučio 27, 44312Kaunas, Lithuania, E-mail: genaovaite.baurienė@ktu.lt **Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius, LT-10223, Lithuania, E-mail: jelena.mamcenko@vgtu.lt ***Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius, LT-10223, Lithuania, E-mail: genadijus.kulvietis@vgtu.lt ****Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius, LT-10223, Lithuania, E-mail: arturas.grigoravicius@vgtu.lt *****Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius, LT-10223, Lithuania, E-mail: inga.tumasoniene@vgtu.lt


Introduction
Piezoelectric actuators have advanced features if compared to others and are widely used for different commercial applications [1][2][3][4].
The demand for new type displacement transducers that can achieve high resolution and accuracy of the driving object increases nowadays [5][6][7].
A lot of design and operating principles are investigated to transform mechanical vibrations of piezoceramic elements into elliptical movement of the contact zone of an actuator [8,9].
Elliptical movement of piezoelectric actuators fall under two typesrotary and linear.Rotary type actuators are one of the most popular because of high torque at low speed, high holding torque, quick response and simple construction.Linear type traveling wave actuators feature these advantages as well but development of these actuators is a complex problem [10][11][12].
Summarizing the following, the following types of piezoelectric actuators can be specified: traveling wave, standing wave, hybrid transducer, and multi-mode vibrations actuators.[7,13,14].
The ring shaped piezoelectric actuator generating elliptical movement is presented and analyzed in this paper.

The influence of geometric parameters on domination coefficients
Usually, for numerical analysis of piezoactuators the software such as ANSYS is used.By the algorithm of eigenvalue problem eigenfrequencies for systems are sorted in the ascending order; thereby the sequences of eigenforms change.This rule for sorting frequencies is disadvantageous when numerical analysis of multidimensional piezoactuators needs to be automated.This problem is also important for optimization, since calculations are related both to eigenfrequencies and eigenforms.If the eigenfrequency is chosen incorrectly, the piezoactuator will not function, so it is very important to numerically determine eigenforms and place them inside the eigenform matrix of the construction model [15].
Calculation of eigenfrequencies and forms for a given construction (Fig. 1) is proposed in this paper.
Then for the n th eigenfrequency the following sum can be formed: where k is the number of degrees of freedom in a node, l is the number of nodes (degrees of freedom) in the model, r is the size of the form vector for the k th coordinate, n k i A is the value of the eigenform vector for the i th element.Fig. 1 The scheme for determining rational geometric parameters Then the ratio is formed: where n k j m is the oscillation domination coefficient.The sum n k S corresponds to the oscillation energy of the n th eigenfrequency in the k th direction, and the ratio n k j m is the ratio of oscillation energies of the n th eigenfrequency in the coordinate directions of j and k.
These coefficients have to be called partial domination coefficients since they estimate energy only in two coordinate directions.The domination coefficients discussed above have the following shortcomings: Not normalized.Because of this the range of the domination coefficients calculated varies from 0 to infinity.
In the case of three dimensions, six domination coefficients result.Such a number of coefficients aggravate analysis.
To solve this problem the following algorithm is proposed: find the sum of the amplitude squares of piezoactuator oscillations in all directions of the degrees of freedom for a point, i.e., the full system energy in all directions [16,17]: where n is the eigenfrequency for a system, k is the number of degrees of freedom in a node, n ik A is the value of the ei- genform vector for the i th element.
Then the ratio is calculated [8]: where n j m is the oscillation domination coefficient corre- sponds to the n th eigenform.The index j of domination coefficients indicates in which direction the energy under investigation is the largest.j can assume such values: 1 corresponds to the x coordinate, 2y, and 3z, etc.Having calculated domination coefficients in all directions of degrees of freedom and having compared them to each other, we can determine the dominant oscillation type.The domination coefficients calculated according to formula (4) are normalized, so their limits vary from 0 to 1.It is very convenient for analyzing the influence of various parameters on domination coefficients.
To clearly determine the eigenform and its place in the eigenform matrix of the construction model, it is not enough to calculate only the oscillation domination coefficients.Domination coefficients only help to differentiate eigenforms by dominating oscillations, for example, radial, tangential, axial, etc.
Because of this an additional criterion is introduced into the process of determining eigenform, individual for each eigenform, i.e., the number of nodal points or nodal lines for the form.That depends on the dimensionality of the eigenform.During calculations the number of nodal points of beam-like and two-dimensional piezoactuators is determined by the number of sign changes in oscillation amplitude for the full length of the piezoactuator in the directions of coordinate axes.
Summarizing the algorithm for determining eigenforms of piezoactuator oscillations (Fig. 1), we can note that it is composed of two integral stages: calculating domination coefficients and determining the number of nodal points or lines of the eigenform.This algorithm is not tightly bound to multidimensional piezoactuators, so it can be successfully applied in analysing oscillations of any constructions.When solving the problems of piezoactuators dynamics for high precision microrobots where repeated calculations with higher eigenfrequencies are involved, it is proposed to modify the general algorithm introducing the stage of determining eigenforms with the help of domination coefficients [18].

Design and results of numerical modeling
Numerical modeling of piezoelectric actuator was performed to validate actuator design and operating principle through the modal analysis.
Modal analysis of piezoelectric actuator was performed to find proper resonance frequency.Material damping was assumed in the finite element model [19].
Finite element model software ANSYS 11.0 was employed for simulation and finite element model was built.
Principle scheme of the analysed piezoelectric actuator is provided in Fig. 2. Fig. 2 Principle scheme of ring shaped piezoelectric actuator, where Router radius, rinner radius, hheight PZT-8 piezoceramics was used for the ring.The polarization vector is directed along the width of the ring.The detailed properties of this material are provided in Table 1.Geometric parameters of the ring are chosen in such a way that the eigenfrequency of the 2 nd flexional form is as high as possible, since in this way its rapidity is guaranteed.
The first iteration of calculations of piezoelectric actuator was performed to find proper resonance frequency and in order to determine the same eigenform of elliptical movement with diferent inner radius.
During analysis the ring dimensions have been changed.Geometric parameter's proportions used in the finite element model for modal analysis (Fig. 3) are provided in Table 2.A more detailed analysis of domination coefficients (according to which better flexibility was examined) is provided in Tables 3, 4 and Figs. 4, 5.A more detailed analysis of model eigenfrequencies (by crossply and rotative movements) is provided in Table 5 and Fig. 6.FEM represented below reflect the best crossply and rotation movement.
The 2 nd iteration of calculations of piezoelectric actuator was performed to find proper resonance frequency and in order determine the same eigenform of elliptical movement with diferent outer radius.During analysis in the 2 nd iteration of calculations dimensions of the ring have been changed.Geometric parameter's proportions used in the finite element model for modal analysis (Fig. 7) are provided in Table 6.A more detailed analysis of domination coefficients (according to which better flexibility was examined) is provided in Tables 7, 8 and Figures 8, 9     A more detailed analysis of model eigenfrequencies (by crossply and rotative movements) is provided in Table 9 and Fig. 10.

Table 9
The eigenfrequencies (by crossply and rotative movements) in the 2 nd iteration of calculations Having compared the influence of geometric parameters on domination coefficients and eigenfrequencies in two iterations of the calculation, it can be claimed that with the help of domination coefficients the eigenform of elliptical rotation can be partially determined.
Also, during analysis the oscillation amplitude A has to remain unchanged or change unsignificantly.The resulting construction would satisfy technical characteristics of the system and be rational from a technological standpoint.

Conclusions
Results of numerical modeling and simulation of piezoelectric actuator are presented and analyzed in this paper.
Numerical modeling of piezoelectric actuator was performed to validate design and operating principle of the actuator through its modal response analysis.
While changing geometrical parameters of piezoelectric actuators the variation in the modal shape sequence has been observed.
Identification of modal shapes sequence is the necessary step in order to automate numerical experiments of multicomponent piezoelectric actuators.
In practical part modal analysis is performed, eigenform determined and eigenfrequency calculated, the size of inner and outer radius represented maximum rotation and flexibility.
In two iterations of the calculation, the best result of crossply movement was obtained in model 3 with coefficient 0.890225, which was achieved with eigenfrequency of 29191Hz (Table 7).
In two iterations of calculation, the best result of rotation movement was obtained in model 2 with coefficient 0.835137, which was achieved with eigenfrequency of 107425 Hz (Table 8).
Experimental studies confirmed that elliptical rotation oscillations were obtained on the surface of the actuator.
G. Baurienė, J. Mamcenko, G. Kulvietis, A. Grigoravičius, I. Tumasonienė THE RING TYPE PIEZOELECTRICAL ACTUATOR GENERATING ELLIPTICAL MOVEMENT S u m m a r y A design of ring type piezoelectric actuator generatig elliptical movement is proposed and analyzed in the paper.Elliptical movement is generated at the area of the actuator applying harmonic excitation signals with different phases.Numerical modeling based on finite element method was performed in order to find resonant frequencies and modal shapes of the actuator and to calculate the size of inner and outer radius, ensuring maximum rotation and bending movements under excitation scheme.Results of numerical studies are discussed.

Fig. 10
Fig.10The influence of geometric parameters on eigenfrequencies f. (by crossply and rotative movements) in the 2 nd iteration of calculations:1 -Sτ (rotative), 2 -Sφ (crossply)FEM represented below the best reflect the crossply and rotation movement.Having compared the influence of geometric parameters on domination coefficients and eigenfrequencies in two iterations of the calculation, it can be claimed that with the help of domination coefficients the eigenform of elliptical rotation can be partially determined.Also, during analysis the oscillation amplitude A has to remain unchanged or change unsignificantly.The

Table 1
Properties of the material used for modelling

Table 3
The domination coefficients of crossply movement

Table 4
The domination coefficients of rotative movement

Table 7
The domination coefficients of crossply movement in the 2 nd iteration of calculations

Table 8
The domination coefficients of rotative movement in the 2 nd iteration of calculations