UDK 519.61/.64:620.179.1:681.586.773 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 49(1)95(2015) AMPLITUDE-FREQUENCY RESPONSE OF AN ALUMINIUM CANTILEVER BEAM DETERMINED WITH PIEZOELECTRIC TRANSDUCERS AMPLITUDNO-FREKVENČNI ODZIV KONZOLNEGA NOSILCA IZ ALUMINIJA, UGOTOVLJEN S PIEZOELEKTRIČNIMI PRETVORNIKI Zuzana Lašova1, Robert Zemčik2 1University of West Bohemia in Pilsen, Department of Mechanics, Univerzitni 22, 306 14, Plzen, Czech Republic 2European Centre of Excellence NTIS - New Technologies for Information Society, Faculty of Applied Sciences, University of West Bohemia, Univerzitm'22, 306 14, Plzen, Czech Republic zlasova@kme.zcu.cz Prejem rokopisa - received: 2013-10-16; sprejem za objavo - accepted for publication: 2014-02-13 This work is focused on the creation of an appropriate finite-element model of an aluminum cantilever beam using a pair of piezoelectric patch transducers. Thanks to the reversible behavior of the piezoelectric effect each patch transducer can represent either an actuator or a sensor. For a precise prediction of the amplitude values in the numerical simulations each transducer is calibrated before being attached to the beam with strain gauges. From these experiments piezoelectric properties of each piezoelectric patch are obtained. The cantilever beam is actuated with a voltage signal applied to one of the patches. The signal is a linear chirp (sine wave with a swept frequency) with a sufficient range to affect the selected natural frequencies. The time response of the beam from the piezoelectric sensor and, alternatively, from the laser position sensor is transformed with the T algorithm to obtain the characteristics of the time-frequency domain (spectrogram). The finite-element model of the ST cantilever beam with the piezoelectric patches was created using 3D solid structural and piezoelectric bricks in Ansys. The time response of the model to the chirp voltage signal was determined with a transient analysis. The amplitude/frequency characteristics are compared with the experimental results. Keywords: piezoelectric materials, frequency spectrum, finite-element analysis To delo obravnava izdelavo primernega modela z metodo končnih elementov konzolnega nosilca iz aluminija z uporabo para piezoelektričnih pretvornikov v obliki obliža. Zaradi reverzibilnega vedenja piezoelektričnega pojava je lahko vsak obližast pretvornik aktuator ali senzor. Za natančno napovedovanje vrednosti amplitude pri numeričnih simulacijah je bil vsak pretvornik pred namestitvijo na nosilec kalibriran z napetostnimi lističi. Iz teh preizkusov so dobljene piezoelektrične lastnosti vsakega piezoelektričnega obliža. Konzolni nosilec je bil aktiviran s signalom električne napetosti, uporabljene na enem od obližev. Signal je linearno cvrčanje (sinus s šablonirano frekvenco) s primernim območjem, da se vpliva na izbrane naravne frekvence. Časovni odziv nosilca iz piezoelektričnega senzorja in alternativno s položaja laserskega senzorja je pretvorjen s STFT-algo-ritmom, da se dobi značilnosti vedenja čas - frekvenca (spektrogram). Izdelan je bil model s končnimi elementi konzolnega nosilca s piezoelektričnimi obliži. Z uporabo 3D strukturnih in piezoelektričnih opek v Ansys in z uporabo končnih elementov je bil izdelan model konzolnega nosilca s piezoelektričnimi obliži. Časovni odziv modela na cvrčeč signal napetosti je bil določen s prehodno analizo. Značilnosti amplitude in frekvence so primerjane z eksperimentalnimi rezultati. Ključne besede: piezoelektrični material, spekter frekvenc, analiza končnih elementov 1 INTRODUCTION luating its response to specific actuating signals. One way of establishing damage is by detecting the changes Automatic detectiom of impacts amd hiddem defects im in a structure's modal properties (particularly the basic structures (denoted as structural health monitoring, natural frequency),3 providing the global information on SHM) is the present-day trend in the non-destructive its state. Another way, denoted as the pitch-catch techtestimg. The data obtaimed from the semsors applied to a nique, uses the scattering of stress waves when the structure are tramsmitted to the comtrol system, which actuating signal approaches a structural defect.4 evaluates the state of the structure and responses appro- The sensors and actuators in SHM systems are made priately (e.g, it enables the warning system). Especially ,, • , , , , the structures made of composite materials affected by of smart materials that have a capability to comvert vari- hidden defects such as fibre debonding or delamimatioml ous kimds of emergy' e.g.' piezoelectric materials (Ro- call for an integration of the novel SHM methods. chelle salt, tourmaline or artificially produced ceramics) Two approaches are usually distinguished in SHM: respond to a mechanical deformation by generating elec- the passive and active ones. In a passive system "natural" tric voltage amd vice versa. Therefore, piezoelectric impulses like impacts or crack propagation create stress materials can be used as both sensors and actuators. waves that are sensed by a grid of sensors. The source This work is focused on creating a reliable FE model can be then localized and reconstructed.2 In an active of a piezoelectric transducer used in SHM. The model is SHM system the health of a structure is assessed by eva- then tested in the case when two patches are applied to a cantilever beam. An appropriate FE model of a piezoelectric patch is the key part for designing a future SHM system. 2 MODEL OF THE PIEZOELECTRIC MATERIAL The piezoelectric material can be described with a set of constitutive equations: r ^ 1 rc -eT ir £ 1 L £ J ß J (1) where o [Pa] is stress vector 6x1, C [Pa] is the matrix of elastic coefficients 6x6, £ is strain vector 6x1, D [C/m2] is electric displacement vector 3x1, E [V/m] is electric-field intensity vector 3x1, ß is dielectric matrix 3 x 3 (with electric permittivity constants on its diagonal) and e [C/m2] is piezoelectric stress matrix 3x6: e = 0 0 0 0 e15 0 0 0 0 e24 0 0 e31 e32 e33 0 0 0 (2) The rows of matrix e denote the direction of the electric field and the columns refer to the strain components, e.g., constant e32 of the piezoelectric actuator quantifies strain £2 induced by the electric field in transversal direction 3. In the producer's datasheet piezoelectric strain matrix d is listed instead of e. These matrices are related using the stiffness matrix: e T = CdT (3) ^=1 r 1 V 21 V 31 ! E. E2 E 3 1 V.2 1 V 32 E 2 E 3 V 23 1 E2 E 3 ! 0 0 0 0 0 0 1 0 1 G 23 0 L 0 0 0 0 0 0 0 0 0 0 1 0 G13 1 0 G„ (4) where E1 is the Young's modulus in the direction perpendicular to the plane of isotropy, while E2 = E3 are the Young's moduli in the plane of isotropy. G12 = G13 are the shear moduli in the planes perpendicular to the plane of isotropy, G23 = G32 are the shear moduli in the plane of isotropy and are defined with: E2 E3 2 3 (5) G 23 = 2(1+v 23) 2(1+v 32) Poisson's ratios V12 = V13 give a measure of the extension (compression) in the plane of isotropy due to the extension (compression) in the direction perpendicular to this plane and vice versa. v23 = v32 are the Poisson's ratios in the plane of isotropy. 3 DETERMINATION OF PIEZOELECTRIC COEFFICIENTS OF THE PATCHES The piezoelectric patches used in the experiments are DuraAct P-876.A12 with a layer of piezoelectric ceramic (type PIC-255 5) with the dimensions of 50 mm x 30 mm x 0.2 mm. The top and bottom surfaces of the ceramic are silvered and connected to the soldering pads. Due to the high fragility the ceramic is embedded in a protective polymeric foil with the dimensions of 61 mm x 35 mm x 0.5 mm. The mechanical and electrical properties of the constituent materials are presented in Tables 1 and 2. Material properties of the piezoelectric ceramic were determined by the producer.5 Table 1: Material properties Tabela 1: Lastnosti materiala A transversally isotropic material model is considered for the piezoelectric ceramic with the main direction of anisotropy identical with the direction of polarity. Stiffness matrix C is obtained with the inversion of compliance matrix S: Units PZT Foil Al Young's modulus E [GPa] See Tab.2 3 68 Poisson's ratio V [ - ] 0.3 0.3 Density P [kg/m3] 7800 1580 2777 Relative electric permittivity ß1/ß0* [ - ] 1650 - - [ - ] 1650 [ - ] 1750 Piezoelectric coefficients e31,e32 [C/m] 6.4 _ _ e33 [C/m] -20.5 _ _ * ßo = 8.85418 x 10 12 F/m is vacuum permittivity Table 2: Elastic parameters of piezoelectric ceramic PIC-255 Tabela 2: Parametri elastičnosti piezoelektrične keramike PIC-255 E1 [GPa] 62.1 E2 = E3 [GPa] 48.3 V12 = V13 [ - ] 0.34 V23 = V32 [ - ] 0.34 G12 = G13 [GPa] 23.1 G23 = G32 [GPa] 17.9 Although geometric and material properties of the piezoelectric patches are supposed to be similar for one type and production set, various results with differences up to 20 % were obtained using two patches of the same type and set. For this reason the piezoelectric properties of the patches used in following experiment need to be determined. In the first experiment two pairs of strain gauges (HBM rosettes 6/650 RY91) were glued on the two 0 Figure 1: Piezoelectric patch with the applied strain gauges, one on the front side and one on the back side Slika 1: Piezoelektrični obliž z napetostnimi lističi: eden spredaj in eden zadaj piezoelectric patches (Figure 1), one on each side of a patch, to eliminate the influence of the minor patch curvature. The strain response to the applied static-electric voltage was measured and these values are presented in Table 3. The difference between these two patches loaded with 100 V was 14.4 %. Table 3: Measured strains of the two patches (loaded with 100 V) Tabela 3: Izmerjene napetosti dveh obližev (obremenjenih s 100 V) Patch 1 /sensor 2 / actuator £11 by SG 1 10.27 x 10-5 11.74 x 10-5 £11 by SG 2 9.38 x 10-5 11.23 x 10-5 £11 - averaged 9.83 x 10-5 11.49 x 10-5 The piezoelectric-matrix coefficients were identified using a FE model in Ansys v.14. The model was created using 3D hexagonal elements: 20-node piezoelectric bricks (SOLID 226) for PZT and 20-node structural bricks (SOLID 186) for the protective foil. The piezoelectric elements have an additional degree of freedom for the electric potential in each node. One layer of these elements was used, which was sufficient for an approximation of the electric potential across the thickness of a patch. The nodes of the piezoelectric elements in the top and bottom surfaces are represented by the silver electrodes, where the relevant electric potential is applied. The model parameter ^31 was optimized to match the experimental data. Coefficients 632 and 633 were calculated as 632 = 631 and 633 was chosen to maintain the mutual ratio of 633/631 ~ -2.5. The resulting values are presented in Table 4. Table 4: Identified piezoelectric coefficients of the two patches Tabela 4: Ugotovljeni piezoelektrični koeficienti pri dveh obližih Patch 1 /sensor 2 / actuator 631, 632 [C/m] 7.1 8.3 633 [C/m] -17.8 -20.8 4 FREQUENCY RESPONSE OF THE CANTILEVER BEAM The calibrated patches were glued to an aluminium beam with the dimensions of 1000 mm x 30 mm x 3 mm, each on one side. The beam was clamped at 100 mm of its length and 10 mm from the patches. The patches were connected to the National Instruments data acquisition system (NI CompactDAQ) supplied with an actuating module NI 9215, sensing module NI 9236 and an amplifier. The beam was actuated by one of the patches and the vibration was sensed by another patch and laser position sensor OptoNCDT. The experimental set-up is presented in Figure 2. The actuating signal was a linear chirp (sine wave with a linearly swept frequency) defined with the following equation: x(t) = X • sinj^^ 0 + 2n k 2 /0 + 21 1 (6) where X is the amplitude, is the initial phase, /0 is the initial frequency, k is the chirp rate defined by: /1 — /0 k=- t. (7) where /1 denotes the final frequency and ti is the final time. Table 5: Parameters of the actuating signal Tabela 5: Parametri vzbujevalnega signala Amplitude X [V] 75 Initial phase [rad] 0 Initial frequency /0 [Hz] 0 Final frequency /1 [Hz] 100 Final time t1 [s] 20 Sampling frequency /s [Hz] 50000 The properties of the actuating chirp signal are presented in Table 5. The range of frequencies was chosen to contain the lowest natural frequencies of the beam for the relevant two out-of-plane bending modes. The time-voltage responses of the piezoelectric sensor and laser sensor were recorded (Figure 3). A short-time Fourier transform (STFT) was performed to obtain a spectrogram (Figure 4). The length of the Hanning window in STFT was set to 105 samples providing a frequency precision of 0.5 Hz. The two lowest natural Figure 2: Experimental set-up Slika 2: Eksperimentalni sestav Figure 3: Time response of the cantilever beam measured with PZT Slika 3: Časovni odziv konzolnega nosilca, izmerjen s PZT Figure 4: Spectrogram of the cantilever beam, the two lowest natural frequencies are marked with arrows Slika 4: Spektrogram konzolnega nosilca; puščici prikazujeta dve najnižji naravni frekvenci frequency was 25 kHz. The deflection of the beam tip was measured with the laser sensor and compared with the results of the FE static analysis (Table 7). The difference between the experiment and FE was 2.4 %. Table 7: Comparison of the experimental and numerical results Tabela 7: Primerjava eksperimentalnih in numeričnih rezultatov Beam-tip deflection Uz Experiment 0.206 mm FEA 0.201 mm Difference 2.4 % 6 CONCLUSION A numerical model of a piezoelectric transducer was created in Ansys using three-dimensional piezoelectric and structural finite elements. The piezoelectric coefficients of each patch were calibrated using strain gauges and it was found that they differ by 14 %. The FE model of the piezoelectric transducer was tested on a problem of bending an aluminum cantilever beam. The two lowest natural frequencies were determined experimentally and compared with the results of the FE modal analysis with sufficient match for the given frequency precision. The amplitude of deflection of the beam loaded with the low-frequency voltage was measured with the laser sensor and compared to the result of the FE static analysis with a difference of 2.4 %. This numerical model proved to be suitable for designing the SHM systems based on the change in the structure's natural frequencies. The reliability of the model will be further tested for the case of transient problems such as those used in the pitch-catch SHM systems. frequencies can be detected in the spectrogram. In Table 6 the experimental results are compared with the numerically calculated natural frequencies. Table 6: Comparison of natural frequencies Tabela 6: Primerjava naravnih frekvenc FEM [Hz] Experiment [Hz] 1s' 9.9 10.0 2nd 60.5 60.0 5 AMPLITUDE RESPONSE OF THE CANTILEVER BEAM TO THE HARMONIC LOADING To obtain the amplitude response the structure was loaded with a harmonic sine wave with a low frequency (1 Hz) with a duration of 40 s to approach steady oscillations. The amplitude was 100 V and the sampling Acknowledgement The work was supported by projects SGS-2013-036 and GA P101/11/0288. 7 REFERENCES 1V. Laš, R. 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