The piezoelectric material used in cylindrical transducers requiring high-precision displacements indicates that accuracy depends on design and technological factors. The analyzed criteria have enabled selection of piezoelectric material for optimal mechatronic systems having maximum displacement. Experimental investigation of precision dynamic systems by means of 3D holographic visualization enabled to collect appreciably more information about the vibrating surface in comparison with traditional methods. The developed methodology of analysis of experimental data derived from 3D holographic visualization using holographic measurement stand allowed to obtain results that are indispensable for optimization of design of mechatronic systems or its constituent elements

Fedaravicius, A.

Patašiene, L.

Ragulskis, K.

Vasiliauskas, R.

JVE International

  VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1306 855. 3D holographic visualization of vibrations of cylindrical piezoceramic transducers  K. Ragulskis1, R. Vasiliauskas2, L. Patašienė3, A. Fedaravičius4 1, 3Kaunas University of Technology, Kęstučio str. 27, LT-44312 Kaunas, Lithuania 2Mykolas Romeris University, Faculty of Public Security V. Putvinskio 70, LT-4421 Kaunas, Lithuania 4Kaunas University of Technology, Institute of Defence Technologies Kęstučio str. 27, LT-44312 Kaunas, Lithuania E-mail: 1k.ragulskis@jve.lt, 2r.vasiliauskas@mruni.eu, 3laima.patasiene@ktu.lt, 4alfedar@ktu.lt (Received 10 June 2012; accepted 4 September 2012)  Abstract. The piezoelectric material used in cylindrical transducers requiring high-precision displacements indicates that accuracy depends on design and technological factors. The analyzed criteria have enabled selection of piezoelectric material for optimal mechatronic systems having maximum displacement. Experimental investigation of precision dynamic systems by means of 3D holographic visualization enabled to collect appreciably more information about the vibrating surface in comparison with traditional methods. The developed methodology of analysis of experimental data derived from 3D holographic visualization using holographic measurement stand allowed to obtain results that are indispensable for optimization of design of mechatronic systems or its constituent elements.  Keywords: piezomaterial, mechatronics systems, 3D holographic visualization, cylindrical transducers vibrations.  Introduction    Piezoceramics can be significantly more sensitive to electric and mechanical effects compared to natural crystals. In addition, it is mechanically strong, chemically inert and resistant to atmospheric effects. Piezoelectric cells can be made of various sizes and parameters which enable them to be used in any manufactured structures with 90 % efficiency. Theoretical investigation of vibrations of cylindrical transducers and dynamic analysis of their components  have indicated that increase in the loading force and initial tension decrease the harmonic components of fluctuations (Figs.1-2).    Fig. 1. Mode of vibration in a cylindrical transducer Fig. 2. Hysteresis loops corrected in cylindrical transducers vs. the feedback value  Fig. 3 shows an interferogram when a control signal is sent only to one active control cylindrical transducer of the mechatronic system. In this case, even deformation of the working part surface in the operation area of the active cylindrical transducer is observed. Slight surface deformation in the operation areas of other active piezostack emerges due to conditions of their fixing onto the surface of the working part of the mechatronic system.   855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1307 This study employs the double-exposure holographic visualization technique for the quality assessment of surface deformation. The essence of this technique is the recording of holograms of two objects (being in different conditions, initial and deformed, for instance, before and after increase of voltage) on the same layer of a light-sensitive photographic plate. Upon illumination of the hologram after two exposures with a copy of the cylindrical transducers, both of them reflected by the object surface before and after the deformation, are restored at the same time [1, 6].     Fig. 3. Construction of the transducer:  1 - holder, 2 - piezomaterial, 3 - stator, 4 - rotor Fig. 4. Dashed line: experimental investigations, continuous line: theoretical investigations  The results of their interference – the system of interferential bands is observed against the background of the surface of the object image, which provides information about changes in the object status that occurred in the period between expositions. 3D holographic visualization of vibrations of cylindrical transducers used in the experimental work has strengthened the expressions of differential equations and was used for drawing final conclusions of the investigation. The ineffective electrical energy is stored as electrostatic energy in the piezoceramic material and reverts it to the power supply in the final process of an operating cycle. The analyzed criteria have made it possible to choose the piezomaterial for an optimal construction having a maximum displacement [7, 8]. Experimental investigation of precision vibrosystems by means of holographic interferometry enables one to obtain appreciably more information about the vibrating surface in comparison with traditional methods. The paper deals with the consideration of methods for determination of the vibrational characteristics of precision mechanical systems from the holographic interferograms of linked analysis of these characteristics received by using numerical techniques based on the theories of mechanical vibrations and holographic interferometry [4, 5, 9].  Method of holographic visualization of vibrations of cylindrical transducers   When wave properties are defined by a standing wave, quantitative analysis of interferograms shall be performed in the following method. Think about point i located on a surface of piezoceramic cylinder (Fig. 5) whose spatial vibration vector may be defined by equation:  ( ) ( ) ( ) ( )ktWjtVitUtRi ˆ,ˆ,ˆ,, ϕϕϕϕ ++=                 (1) where ϕ is an angular coordinate of the analyzed point on piezoceramic cylinder; t – time; ˆˆ ˆ,  ,  i j k  – unit vectors of corresponding coordinate axis z, t, or r; W – normal constituent of spatial vibration at point i; U and V – tangential constituents. If an observation unit vector of point i is named 0K̂ , and vector opposite to an illumination unit vector of point i is named iK̂ , the sensitivity vector of point i will be defined by the following equation: kKjKiKK iritizi ˆˆˆ ++=                     (2)  855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1308 where ˆ ,  ,  i i iz t rK K K  shall be calculated from the following system of equations:  iiizK 12 coscos ϕϕ +=  1 1 2 2sin sin sin sini i i i itK θ φ θ φ= −                   (3) 1 1 2 2cos sin cos sini i i i itK θ φ θ φ= + ,  where 1 2,  i iθ θ  are angles formed by the observation and illumination vectors of point i in relation to coordinate axis r; 1 2,  i iϕ ϕ  are the corresponding angles formed by the observation and an illumination vectors in relation to coordinate axis z.   Fig. 5. Interpretation of vibrations of a standing wave in a piezoceramic transducer  A change of a light wave phase that appears due to surface vibration of a piezoceramic cylinder, when the light travels from the source to the analyzed point i located on the surface of piezoceramic cylinder and then proceeds to holographic interferogram, is defined as [3]: ( )λϕπiiiKtR ,2=Ω                      (4) where λ is a wavelength of a laser light used for fixing the interferogram; ( )tRi ,ϕ , and iK  are defined by equations (1) and (2). Scalar product of spatial vibration and sensitivity vectors is defined as:  ( ) ( ) ( ) iritiz KtWKtVKtUKR ,,, ϕϕϕ ++=                (5) It we insert this equation into the expression of the change of a light wave phase (4), we get: ( ) ( ) ( )[ ]iritizi KtWKtVKtU ,,,2 ϕϕϕλπ++=Ω               (6) where tangential constituents U and V of spatial vibration vector ( ),iR tϕ  and normal vibration constituent W at a surface point i are expressed as:  ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ),cos,,cos,,cos,000iiiiiitWtWtVtVtUtUγωϕϕβωϕϕαωϕϕ+=+=+=                  (7)   855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1309 where ( )ϕiU 0 , ( )ϕiV0 , ( )ϕiW0  are values of amplitude tangential and normal vibration constituents of spatial vibration vector ( )tRi ,ϕ  that are expressed as:  ( ) uijnuji FAU ∑=10 ϕ , ( ) vijnvji FAV ∑=10 ϕ , ( ) wijnwji FAW ∑=10 ϕ ,          (8)  where i – number of point i located on a surface of cylinder; j – number of a type of self-excited vibration of a piezoceramic cylinder; Fij – value of the amplitude of a number of self-excited vibration type j in point i; Aj – coefficient of influence of self-excited vibration type j;                 n – number of self-excited vibrations types. Characteristic function of the distribution of interference bands on a surface of piezoceramic cylinder, when holographic interferogram is fixed at time-average and harmonic vibration is present, is defined as [2]: ( ) ( )01expTx i iM i dtTΩ = Ω∫                       (9) Inserting expressions (7) into the equation (6) we get the value of iΩ  that should be inserted into characteristic function of band distribution (9). We obtain the following:  ( ) ( ) dtttiTMTix ∫ Ω−Ω=Ω021 sincos2exp1ωωλπ,              (10)  where: ( ) iiriiitiiizi KWKVKU γϕβϕαϕ cos)(cos)(cos 0001 ++=Ω          (11) ( ) iiriiitiiizi KWKVKU γϕβϕαϕ sin)(sin)(sin 0002 ++=Ω           (12) After the use of the equation [1] the equation (10) will look like: ( ) ( )  += 2122210 ΩΩΩ nJM ix ,                   (13) where J0 – Bessel function of the first kind of order zero. Using (7), (11), (12) and (13) values of point i located on the surface of piezoceramic cylinder, we will obtain the following equation of the distribution of the interference bands on the surface of vibrating piezoceramic cylinder:  211121112sinsinsincoscoscos4 ∑+∑+∑+ ∑+∑+∑=iiznuijujiitnvijvjiirnwijwjiiznuijujiitnvijvjiirnwijwjiKFAKFAKFAKFAKFAKFAαβγαβγπλΩ      (14)  The equation (14) is commonly used for identification of spatial vibration coordinates, when the object is analyzed in a method of time-average. In the equation, we know the parameter iΩ , which is obtained for the centre points of dark interference bands of interferograms with the help of the following equation (1):  ( )( )ππΩ25.0125.025.0−+−=ppp,                  (15)  where p is the order number of an interference band (in a holographic interferogram), in the centre of which point i is located; the order number is calculated from the brightest nodular band.  855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1310 Coefficients ,  ,  i i ir t zK K K  of the equation (14) are obtained from the equation (3), using optical scheme parameters obtained during the experiment of fixing holographic interferogram. Coefficients ,  ,  w v uij ij ijF F F  are calculated using analytical expressions of self-excited vibration types related to geometrical shapes of transducers and conditions of their fixing in various structures. In an attempt to calculate coefficients ,  ,  w v uj j jA A A  and angles ,  ,  i i iα β γ  we need to minimize the equation formed on a basis of the equation (14) in a method described [2]:  22211121114sinsinsincoscoscos− ∑+∑+∑− ∑+∑+∑=πλΩαβγαβγiiiznuijujiitnvijvjiirnwijwjiiznuijujiitnvijvjiirnwijwjiKFAKFAKFAKFAKFAKFAF      (16)  Function Fi shall be differentiated with respect to each unknown component of vibration. In respect of wjA  we obtain the following:  ( )iirwijiiznuijujiitnvijvjiirnwijwjiirwijiiznuijujiitnvijvjiirnwijwjwjiijKFKFAKFAKFAKFKFAKFAKFAAFGγαβγγαβγsinsinsinsin2coscoscoscos2111111 ∑+∑+∑×− ∑+∑+∑=∂∂=(17)  Analogous expressions will be obtained in respect to other unknown components of vibration, i.e. ,  v uj jA A .  In respect of the unknown iγ :  ( ) ( )iirnwijwjiiznuijujiitnvijvjiirnwijwjiirnwijwjiiznuijujiitnvijvjiirnwijwjiiinKFAKFAKFAKFAKFAKFAKFAKFAFGγαβγγαβγγcossinsinsinsincoscoscos21111111131∑ ∑+∑+∑+−∑ ∑+∑+∑=∂∂=+(18)  Analogous expressions will be obtained in respect to the other unknowns iα  and iβ . In such a way q number of points evenly located on a deformed (due to vibration) surface of piezoceramic cylinder is analyzed. According to the method described above, the calculation of coefficients ,  ,  i i ir t zK K K  and ,  ,  w v uij ij ijF F F  of the equation (14) is followed by the formation of the matrix ( )iG . If q is the number of points analyzed and n is the number of types of self-excited vibration of piezoceramic cylinder obtained in result of the resolution of vibration at point i on a surface of piezoceramic cylinder in respect to corresponding coordination axis r, t and z, then matrix ( )iG  will feature a dimension q×3(2n+3). Solutions of a non-linear equation (14) will be obtained in a way of iteration. We will have to find out the following vector of unknowns:  [ ]γβα ,,,,....,,,....,,,....,, 212121 WnWWVnVVUnUUV AAAAAAAAAN = ,          (19)  that will ensure minimum value of correction vector defined from the following expression:  { } [ ] { }PTR 1−= ,                      (20)   855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1311 where [ ]T  and { }P  are obtained from:  ( ) ( )ijqiijGGT ∑==111 , where j = 1, 2, …, (3+3n)               (21) ( )ijqiijGFP ∑−==1, where j = 1, 2, …, (3+3n)                (22)  Thus, performing the analysis of holographic interferograms in transducers with excited vibrations of a standing wave by means of the described approach, we will obtain several holographic interferograms from various illumination and observation angles. We select q number of points evenly located in the centers of interference bands on the surface of transducers. Sensitivity vector projections ˆ ,  ,  i i iz t rK K K  are defined for each point i ( )1,  2,  ,  i q= …  selected in a way of an experiment. ,  ,  w v uij ij ijF F F  of each point i selected are calculated using analytical expressions of the first n self-excited vibrations related to geometrical shape and fixing conditions of transducers. iΩ  is defined for each selected point i located in the centre of dark interference bands. These values shall be inserted into the equation (14). For the calculation of coefficients ,  ,  w v uj j jA A A  and angles ,  ,  i i iα β γ  equation (16) shall be transformed to vector ( )iG  and, using (20) and (21), minimum value correction vector is formed.   Fig. 6. Waves formed on holographic interferogram of a surface of piezoceramic cylinder when parameters of excitation and claims are optimal: a) angle of light for fixed hologram 800; b) angle of light for fixed hologram 400   Fig. 7. Waves formed on holographic interferogram of a surface of piezoceramic cylinder when parameters of excitation and claims are not optimal: a) angle of light for fixed hologram 800; b) angle of light for fixed hologram 400  855. 3D HOLOGRAPHIC VISUALIZATION OF VIBRATIONS OF CYLINDRICAL PIEZOCERAMIC TRANSDUCERS. K. RAGULSKIS, R. VASILIAUSKAS, L. PATAŠIENĖ, A. FEDARAVIČIUS     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2012. VOLUME 14, ISSUE 3. ISSN 1392-8716 1312  Fig. 8. Theoretical investigation of the radial oscillation amplitude of a surface of piezoceramic cylinder with application of interferograms in Figs. 6-7  Conclusion  The analyzed criteria enabled selection of the piezomaterial for optimal construction exhibiting maximum displacement. The automatic control has been determined to affect the correction of the hysteresis loop thus leading to reduction of displacement error up to 0.2 %. Performing the analysis of holographic interferograms in transducers with excited standing wave vibrations by means of the reported method allowed formation of several holographic interferograms from various illumination and observation angles. Experimental investigation of cylindrical transducers makes it possible to determine optimal initial tension force, the dependence of displacement of free cylindrical transducers in terms of constructional and technological parameters. The experimental results are easy to access and they are applied for development of tools for loosening rigid tightenings and eliminating corrosion impurities in machining and tool adjustment, medical devices, optical systems as well as items used in criminology for adjustment of various elements.  References  [1] G. Baurienė, A. Fedaravičius, L. Patašienė, K. Ragulskis Investigation of combined piezostacks and their application in optical systems. Proceedings of the International Conference “Vibroengineering – 2004”, Kaunas, 2004, p. 60-63. [2] Tonin R. F., Bies D. A. General theory of time-average holography for the study of three-dimensional vibrations at a single frequency. J. Optical Soc. of America, 1978, Vol. 68, p. 924-931. [3] Ch. West Holographic Interferometry. Мoscow: Мir, 1982, 504 p. [4] Palevičius A.,Vasiliauskas R. Application of laser interferometry and holographic methods for analysis and design of vibrational systems. Vibroengineering, 1998, p. 358-381. [5] K. Ragulskis, A. Palevičius, R. Vasiliauskas Calculation of projections of the sensitivity vector on the basis of the photos of holographic interferograms in the process of decoding of spatial vibrations. Vibroengineering, Vol. 3(56), 1986, p. 133-141. [6] Patašienė Laima, Vasiliauskas Raimondas, Fedaravičius Algimantas Holographic interferometry method for determination of layer piezostack parameters. Ultrasound, Kaunas: Technologija, 2007, Vol. 1(62), p. 23-25. [7] Baurienė Genovaitė, Fedaravičius Algimantas, Patašienė Laima Dynamics of piezomaterials used in the optical systems. Journal of Electroceramics, Dordrecht: Springer, 2008, Vol. 20, No. 3-4,           p. 313-316. [8] Patašienė Laima, Ragulskis Kazimieras, Fedaravičius Algimantas Dynamics processes in exciting vibration of piezoactuators. Rare Attractors and Rare Phenomena in Nonlinear Dynamics: Proceedings of the 2nd International Symposium RA'11 held in Riga-Jūrmala, Latvia, 16-20 May 2011, Riga: RTU, 2011, p. 142-145. [9] R. Vasiliauskas, A. Bubulis, L. Patašienė, K. Ragulskis Investigation of wave processes in circular transformers of motion by using the method of holographic interferometry. Herald of Machine Building, Мoscow, 1981, No. 1, p. 16-18. 

3D holographic visualization of vibrations of cylindrical piezoceramic transducers

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3D holographic visualization of vibrations of cylindrical piezoceramic transducers

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