589. Investigation of motion control of piezoelectric unimorph for laser shutter systems
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1 589. Investigation of motion control of piezoelectric unimorph for laser shutter systems A. Buulis 1, V. Jurenas 2, S. Navickaite, V. Rugaityte 4 Kaunas University of Technology, Kaunas, Lithuania algimantas.uulis@ktu.lt 1 vytautas.jurenas@ktu.lt 2 sigita.navickaite@stud.ktu.lt viktorija.rugaityte@stud.ktu.lt 4 (Received Septemer 21; accepted 9 Decemer 21) Astract. This paper presents the design and testing of a resonance frequency-tunale piezoelectric unimorph chopper and shutter that employ a magnetic force technique and magnetorheological fluid (MRF). This technique enaled to increase the frequency of the resonance up to 11% of the untuned resonant frequency. A piezoelectric unimorph cantilever with a natural frequency of 126 Hz is used as the laser eam chopper or shutter, which is successfully tuned in a frequency range of Hz therey enaling continuous control of the laser eam over the entire frequency range tested. A theoretical model ased on variale magnetic field strength and MRF damping is presented. The magnetic force and MRF applied for damping of transient virations of the piezoelectric unimorph shutter have een experimentally determined. Keywords: virations, imorph, piezo materials, laser shutter, position control, magneto rheological fluids. Introduction Nowadays laser technologies are widely used in modern equipment. Laser shutting or chopping systems are used for laser eam control. Laser eam goes without disturances through laser shutter system if the shutter is opened, ut if the shutter is fully or partially closed, laser eam power is fully or partially locked. Lately in development of high-tech products there is oserved an increasing tendency to apply smart materials or materials with controllale properties (piezoactive and shape-memory materials, suspensions with controllale rheology electro- or magneto-rheological fluids (ERF and MRF), artificial muscles, etc.). In this paper authors investigate position control of piezoelectric unimorph for laser shutting system using MRF and magnetic forces. Most laser eam choppers or shutters developed to date are single resonance frequency ased, and while recent orts have een made to roaden the frequency range of laser eam choppers, what is lacking is a roust tunale frequency technique of the chopper. Shutters differ from choppers in that they are not limited to a simple periodic on-off cycle ut will follow an aritrary, varying pattern of openings and closings. Optical shutters are useful for low frequency chopping, particularly when slow or non-periodic ehavior is needed. 5
2 The scheme of the investigated laser eam shutter and its working principle Fig. 1. The scheme of the piezoelectric laser shutting system: L p length of piezoelectric plate, L length of piezoelectric unimorph 1, C length of special plate 2 for the laser eam 5 shutting, d distance etween the end of unimorph 1 and permanent magnet 4, B m thickness of permanent magnet 4, - MRF Investigated piezoelectric laser eam shutter consists of piezoelectric unimorph actuator 1 and special ferromagnetic plate 2 for the laser eam shutting. Its design is illustrated in Fig. 1. Bending deformations of unimorph 1 can e actuated y the piezoelectric ect. The direction of deformation and deformation rate depend on materials used in actuator, polarization direction and electric field that depend on supply voltage. Laser eam 5 is locked if electricity does not influence piezoactuator, ut in the case when actuator is powered, unimorph actuator is ended y and laser eam 5 goes without ostruction through laser shutter system [2]. The motion of the unimorph 1 is controlled using MRF. Permanent magnet 4 is used in order to generate magnetic force and magnetic field that changes viscosity of MRF. Magnetic force and MRF that increase the stiffness and damps virations of the cantilever unimorph 1 depends on distance etween the tip of unimorph and permanent magnet 4. In the present approach a magnetic force is used to alter the overall stiffness of the piezo unimorph cantilever. This enales one to increase the overall stiffness of the device using magnetic force and to change the natural frequency of the device. Here, attractive magnetic force is used to shift the eam natural frequency to higher frequencies with respect to the original resonant frequency (with no magnetic force present). Unimorph and MRF used in design of investigated laser eam shutting system A standard rectangular shape unimorph actuator under activation is illustrated in Fig. 1. The actuator consists of a single piezoelectric layer, with length L p, onded to a purely elastic layer, with length L. Ferromagnetic steel is chosen for the elastic layer. When a voltage U is applied across the thickness of the piezoelectric layer, longitudinal and transverse strain is induced. The elastic layer opposes the transverse strain which leads to a ending deformation [1]. The piezoelectric material used for the unimorph is PZT of type PZ26 (Noliac company). Ferromagnetic steel is used as elastic layer. The properties of materials are listed in Tale 1. 54
3 Tale 1. Properties of the PZT material Pz26 and steel Property Values of PZT Values of steel Unit Density kg m Dielectric constant 1 Coupling factor.47 d 1 charge constant m V 1 Young s modulus Pa Mechanical quality factor 1 The outer surfaces of the piezo unimorph are plated with a layer of nickel electrodes, approximately 2 µm thick. The unimorph shown in Fig. 1. is driven into ending viration y applying an AC voltage U across the electrodes, whose peak to peak amplitude is 6 V. Geometrical parameters of the laser shutting system with the piezoelectric unimorph and permanent magnet (Neodymium magnet, flux density B=1,2 T) are shown in Tale 2. Tale 2. Geometric parameters of the unimorph L, mm L p, mm B m, mm C, mm d, mm In this paper authors investigate position control of piezoelectric unimorph for laser shutting system using MRF and magnetic forces. Tale. Specification of MRF used in experimental study [] MRF MRF-14CG MRF-122EG Appearance Dark Gray Liquid Dark Gray Liquid Viscosity, 4 C (14 F),.28 ±.7.42 ±.2 Calculated as slope 8-12 sec -1 Density g/cm Solids Content y Weight, % Flash Point, C ( F) >15 (>2) >15 (>2) Operating Temperature, C ( F) -4 to +1 (-4 to +266) -4 to +1 (-4 to +266) Theoretical analysis For ective design of the unimorphs it is crucial to understand their coupled electromechanical ehavior through application of mathematical modeling [4,5]. There are two crucial parameters descriing a piezoelectric unimorph actuator: Blocking force F : representing the external force to e applied so that no displacement occurs when an external voltage U is applied. Free displacement δ dc : representing the displacement caused y an applied voltage U when no external force is exerted at the tip of the actuator. A third fundamental parameter, mechanical stiffness, is then readily derived: k = F /δ dc. Theoretical formulas that link locking force and free displacement to geometrical dimensions as well as piezoelectric and mechanical properties of the actuator can e found in [6]: 55
4 2 L AB( B+ 1) δ dc = d U 2 1 hpd hpwab( B+ 1) F = d1u 4s p( AB+ 1) F WhpD k = = δ dc 4s pl ( BA+ 1) (1) where A = s p /s s = Ys/Yp, B = hs/hp, C = ρs/ρp, D = A 2 B 4 + 2A(2B + B 2 + 2B ) + 1. Here, L is the unimorph length, W is the width, U is the applied voltage, d 1 is the transverse piezoelectric coicient. Suscripts p and s refer to, respectively, the piezoelectric layer and to the steel one. s p and s s are the elastic compliances, h p and h s are the thicknesses, Y p and Y s are the Young s moduli, and ρ p and ρ s are the densities of the piezoelectric and steel layers respectively. To calculate the amplitude of the cantilever deflection in the presence of the magnetic coupling force in z direction, we use a modified version of the standard spring mass model. The model parameters are shown schematically in Fig. 2. F p (t) z(t) k k mag m e m Fig. 2. Lumped model of the laser shutting system The cantilever is represented y a one-dimensional spring mass system, where m is the ective mass of the cantilever eam, k is the spring constant (cantilever eam stiffness), k mag - stiffness introduced via application of the magnetic force. The cantilever deflection, z(t), is driven y a piezoelectric force F p, which excites sinusoidal ending oscillations of the cantilever with an angular frequency ω. Electrical and mechanical damping are descried y the coicients e and m respectively and are determined experimentally. To minimize the complexity of the prolem, the magnetic coupling force F mag is considered to e onedimensional, acting only in the z-direction, and the magnetic force in the axial direction is ignored (justification for this approximation is provided in [7]). In general, the magnetic coupling force F mag is a complicated non-linear function of the deflection z and the magnet/cantilever tip separation distance d. As descried in [7], the experimentally determined magnetic force values are fit to an empirically determined analytical expression for F mag (z, d): F mag(z, d) az = (2) 4 + cz 56 where a,, and c are experimentally otained parameters [7].
5 When peak to peak ending amplitude of the cantilever is smaller than the radius of the magnet, the F mag can e written as: F mag = az/ = k mag z, () Where k mag stiffness introduced via application of the magnetic force F mag. In the case of no magnetic force, the ective stiffness of a multilayered (through the thickness) cantilevered eam can e written as [8]: W k = (4) 4L ( n E h ) n1 n2 nieihi + i= 1 j= 1 j j j where W is the width of the eam, n 1 and n 2 are the numers of piezoelectric and elastic layers, respectively, E i and h i are the Young s modulus and height of each piezoelectric layer, and E j and h j are the Young s modulus and height of each elastic layer. The ective mass of a multilayered cantilever eam with a tip mass can e approximated as: m n1 n2 = mt +, 2L( ni ihi + n jρ jh j ) i= 1 j= 1 ρ (5) where ρ i and ρ j are the densities of the piezoelectric and elastic layer and m t is the tip mass. From this the corresponding natural frequency of the multilayered cantilevered eam is given as: k ω = (6) m In the case of the magnetic force, one can sustitute equation (7) into k = k +k mag to calculate the required applied magnetic stiffness to tune the virating eam from an original resonance frequency ω to the desired tuned resonance frequency ω t : 2 2 k = m ω k = m ( ω ω ) (7) mag t 1 The cantilever deflection z(t) can then e determined y solving the differential equation for a one-dimensional forced harmonic oscillator. According to equations (-7) the equation can e written as:... k Fp( t ) z( t ) + z( t ) + z( t ) = (8) m m m where = e + m total ective damping of the system; k = k +k mag - total ective stiffness of the system, F p - piezoelectric force can e calculated y equation (1) as F. The induced stiffness from the applied magnetic force and MRF also induces a certain amount of additional damping within the device. It is oserved that the damping increases with the increase in induced stiffness required for resonance frequency tuning. The damping is determined y first tuning the resonance frequency via the magnetic stiffness approach, and then performing a ump test to otain an amplitude decay plot with time. From this amplitude decay plot the damping is calculated using the relationship: 57
6 1 z1 = ln( ) (9) 2π z 2 where z 1 and z 2 are consecutive amplitudes. The piezoelectric cantilever can e approximated with 1 degree-of-freedom harmonic oscillator (Fig. 2), whose response to a modulated applied force is characterized y its transfer function: z( t ) Fp( t ) = (1) m ( ω ω ) + ω ω tan α = (11) 2 2 ω ω where ω is the applied force frequency, ω the oscillator resonant frequency and the system damping factor. Equation (1) gives the cantilever oscillation amplitude when a modulated force F p (t) is applied. Equation (11) gives the phase lag of the cantilever oscillation with respect to the applied force. Experiments of piezoelectric unimorph motion control for laser eam shutter The experimental setup is illustrated in Fig.. It consists of: 1 piezoelectric unimorph actuator, 2 permanent magnet, ferromagnetic plate for laser eam shutting, 4 laser sensor head LK-82G with controller LK-G1PV, 5 laser eam of laser sensor head, 6 analog digital converter (ADC) PicoScope-424, 7 computer, 8 MRF. Fig.. Layout of the experimental setup for the evaluation of dynamics of the piezoelectric unimorph. A non-contact laser displacement device Keyence LK-82G was used for measuring frequency response of the piezoelectric unimorph actuator. Permanent Niodymium magnet with the residual flux density B=1,2 T is applied for tuning resonance frequency of the cantilevered unimorph. The change of the separation distance d in range from 1 mm to 15 mm etween a tip of cantilever and magnet was performed y means of the precision micro-screw drive. For the additional improve the ect of the magnetic force, two types of the MR fluids (MRF-122EG, MRF-14CG) was placed in the separation gap etween the cantilever tip and magnet. A sinusoidal voltage U =6 V is used to excite the piezoelectric unimorph resonant frequencies of the ending virations. As a result of experiments we have curves of resonance frequency responses versus separation distance d etween tip of the cantilever eam and the cylindrical 58
7 magnet and the corresponding applied magnetic force shown in Fig. 4 a. The resonant amplitudes of the cantilever at the corresponding resonance frequencies and applied MRF are presented in Fig. 4. Frequency, Hz d, mm Amplitude, mkm Resonance frequency, Hz a) ) Fig. 4. Resonace characteristics of the cantilever; a - resonance frequency of the resonant cantilever versus separation distance d etween tip of the cantilever eam and the cylindrical magnet; amplitude of the cantilever virations at the resonant frequencies : - without MRF; - with MRF-122EG ; with MRF -14CG,8 18,7 16 Magnetic flux, T,6,5,4,,2,1 Amplitude, mkm d, mm d, mm a) ) Fig. 5. Magnetic flux (a) and amplitude of the cantilever at the resonance frequencies () versus separation distance d etween tip of the cantilever eam and the cylindrical magnet: - without MRF; - with MRF-122EG ; with MRF -14CG Magnetic flux in the gap etween the cantilever ferromagnetic tip and permanent magnet is measured with magnetic flux meter SH1-8 and its dependence on distance d etween tip of the cantilever eam and the cylindrical magnet is given in Fig. 5 a. The amplitude of the cantilever at the resonance frequencies versus separation distance d etween tip of the cantilever eam and the cylindrical magnet is provided in Fig. 5. The results indicate that in the cases of high 59
8 density MRF-14CG the increase of the viscosity and the viration damping at the small d values eliminate the ect of the resonance frequency tunaility. MRF-122EG can e applied as a damper for the suppression of the shutter-induced transient virations, when piezoelectric unimorph shutter is driven y TTL impulse. A magnetic force technique (without MRF) enaled to increase the frequency of the resonance up to 11% of the untuned resonant frequency. A piezoelectric unimorph cantilever with a natural frequency of 126 Hz is used as the laser eam chopper or shutter, which is successfully tuned in a frequency range of Hz to enale a continuous control of the laser over the entire frequency range. Conclusions This paper reports the design and testing of a resonance frequency-tunale piezoelectric unimorph chopper and shutter, which operation is ased on magnetic force technique and application of magnetorheological fluid. A methodology is presented for the analysis and design optimization of piezoelectric unimorph actuator used for laser eam shutting systems. The magnetically-coupled piezoelectric unimorph cantilever motion is analyzed y means of a simple harmonic oscillator model with a non-linear magnetic force term. The results reveal that magnetic coupling can e used to increase the frequency of the resonance up to 11% of the untuned resonant frequency. Magnetic forces and MRF damping ect of transient virations of the piezoelectric unimorph shutter were experimentally evaluated. Acknowledgment This research was funded y a grant AUT-9/21 and TPA-2/21 from the Research Council of Lithuania References [1] Sitti M, Campolo D, Yan J, Fearing R. Development of PZT and PZN-PT Based Unimorph Actuators for Micromechanical Flapping Mechanisms Constant magnet motors. USA, 21. [2] Donoso A, Sigmund O. Optimization of piezoelectric imorph actuators with active damping for static and dynamic loads, E.T.S. Ingenieros Industriales Universidad de Castilla - La Mancha 171 Ciudad Real, March 27. [] Riesch C, Reichel E, Keplinger F, Jakoy B. Characterizing Virating Cantilevers for Liquid Viscosity and Density Sensing, Hindawi Pulishing CorporationJournal of SensorsVolume 28, Article ID 69762, 9 pages doi:1.1155/28/69762 [4] Smits G, Ballato A. Dynamic admittance matrix of piezoelectric cantilever imorphs J. Microelectromech. Syst.(1994), P [5] Tao W, Paul I. Dynamic peak amplitude analysis and onding layer ects of piezoelectric imorph cantilevers, Smart Mater. Struct. 1 (24) 2 21 [6] Q. Wang, X. Du, B. Xu, and L. Cross. Electromechanical coupling and output iciency of piezoelectric ending actuators IEEE Trans. on Ultrasonics, Ferro. and Freq. Control, vol.46, pp , May [7] Lin J-T, Barclay Lee, Alphenaar B. The magnetic coupling of piezoelectric cantilever for enhanced energy harvesting iciencys. Smart Mater. Struct. 19 (21) [8] Vinod R. Challa, Prasad M. G., Yong Shi and Frank T. Fisher. A viration energy harvesting device with idirectional resonance frequency tunaility. Smart Mater. Struct. 17 (28)
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