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Keywords = Langevin transducers

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16 pages, 1350 KiB  
Review
Advances in Langevin Piezoelectric Transducer Designs for Broadband Ultrasonic Transmitter Applications
by Jinwook Kim, Jinwoo Kim and Juwon Kang
Actuators 2025, 14(7), 355; https://doi.org/10.3390/act14070355 - 19 Jul 2025
Viewed by 261
Abstract
Langevin ultrasonic transducers, also known as Tonpilz transducers, are widely used in high-power ultrasonic applications, including underwater sonar arrays, ultrasonic cleaning, and sonication devices. Traditionally designed for narrowband operation centered on a fundamental longitudinal resonance mode, their performance has been limited by structural [...] Read more.
Langevin ultrasonic transducers, also known as Tonpilz transducers, are widely used in high-power ultrasonic applications, including underwater sonar arrays, ultrasonic cleaning, and sonication devices. Traditionally designed for narrowband operation centered on a fundamental longitudinal resonance mode, their performance has been limited by structural constraints that tie resonance frequency to overall transducer length and mass. However, technical demands in biomedical, industrial, and underwater technologies have driven the development of broadband Langevin transducers capable of operating over wider frequency ranges. Lower frequencies are desirable for deep penetration and cavitation effects, while higher frequencies offer improved resolution and directivity. Recent design innovations have focused on modifications to the three key components of the transducer: the head mass, piezoelectric drive stack, and tail mass. Techniques such as integrating flexural or edge-resonance modes, adopting piezocomposite stacks, and tailoring structural geometry have shown promising improvements in bandwidth and transmitting efficiency. This review examines broadband Langevin transducer designs over the past three decades, offering detailed insights into design strategies for future development of high-power broadband ultrasonic transducers. Full article
(This article belongs to the Section Control Systems)
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16 pages, 3334 KiB  
Article
Lead-Free Ceramics in Prestressed Ultrasonic Transducers
by Claus Scheidemann, Peter Bornmann, Walter Littmann and Tobias Hemsel
Actuators 2025, 14(2), 55; https://doi.org/10.3390/act14020055 - 25 Jan 2025
Cited by 1 | Viewed by 1111
Abstract
Today’s ultrasonic transducers find broad application in diverse technology branches and most often cannot be replaced by other actuators. They are typically based on lead-containing piezoelectric ceramics. These should be replaced for environmental and health issues by lead-free alternatives. Multiple material alternatives are [...] Read more.
Today’s ultrasonic transducers find broad application in diverse technology branches and most often cannot be replaced by other actuators. They are typically based on lead-containing piezoelectric ceramics. These should be replaced for environmental and health issues by lead-free alternatives. Multiple material alternatives are already known, but there is a lack of information about their technological readiness level. To fill this gap, a small series of prestressed longitudinally vibrating transducers was set up with a standard PZT material and two lead-free variants within this study. The entire process for building the transducers is documented: characteristics of individual ring ceramics, burn-in results, and free vibration and characteristics under load are shown. The main result is that the investigated lead-free materials are ready to use within ultrasonic bolted Langevin transducers (BLTs) for medium-power applications, when the geometrical setup of the transducer is adopted. Since lead-free ceramics need higher voltages to achieve the same power level, the driving electronics or the mechanical setup must be altered specifically for each material. Lower self-heating of the lead-free materials might be attractive for heat-sensitive processes. Full article
(This article belongs to the Special Issue Piezoelectric Ultrasonic Actuators and Motors)
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18 pages, 10683 KiB  
Article
Optimal Design and Operation of an Ultrasonic Driving System for Algae Removal Considering Underwater Environment Load
by Changdae Joo and Taekue Kim
Sensors 2025, 25(2), 542; https://doi.org/10.3390/s25020542 - 18 Jan 2025
Viewed by 1016
Abstract
This study investigates the optimal design and operation of an underwater ultrasonic system for algae removal, focusing on the electromechanical load of Langevin-type piezoelectric transducers. These piezoelectric transducers, which operate in underwater environments, exhibit variations in electrical–mechanical impedance due to practical environmental factors, [...] Read more.
This study investigates the optimal design and operation of an underwater ultrasonic system for algae removal, focusing on the electromechanical load of Langevin-type piezoelectric transducers. These piezoelectric transducers, which operate in underwater environments, exhibit variations in electrical–mechanical impedance due to practical environmental factors, such as waterproof molding structures or variations in pressure and flow rates depending on the water depth. To address these challenges, we modeled the underwater load conditions using the finite element method and analyzed the impedance characteristics of the piezoelectric transducer under realistic environmental conditions. Based on this analysis, we developed an ultrasound-driven system capable of efficient output control by incorporating the impedance characteristics of the transducer under load variations and subaquatic conditions. This study proposes analytical and experimental methods for modeling and analyzing practical ultrasound-driven systems for algae removal. Full article
(This article belongs to the Section Physical Sensors)
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11 pages, 9049 KiB  
Article
Ultrasonic Enhancement of Chondrogenesis in Mesenchymal Stem Cells by Bolt-Clamped Langevin Transducers
by Jinhyuk Kim, Hyuncheol Bae, Hyuk-Soo Han and Jungwoo Lee
Micromachines 2023, 14(1), 202; https://doi.org/10.3390/mi14010202 - 13 Jan 2023
Cited by 4 | Viewed by 2200
Abstract
We recently investigated the design and fabrication of Langevin-type transducers for therapeutic ultrasound. Effect of ultrasonic energy arising from the transducer on biological tissue was examined. In this study, the transducer was set to radiate acoustic energy to mesenchymal stem cells (MSCs) for [...] Read more.
We recently investigated the design and fabrication of Langevin-type transducers for therapeutic ultrasound. Effect of ultrasonic energy arising from the transducer on biological tissue was examined. In this study, the transducer was set to radiate acoustic energy to mesenchymal stem cells (MSCs) for inducing differentiation into cartilage tissue. The average chondrogenic ratio in area was 20.82% in the control group, for which no external stimulation was given. Shear stress was applied to MSCs as the contrast group, which resulted in 42.66% on average with a 25.92% minimum rate; acoustic pressure from the flat tip causing transient cavitation enhanced chondrogenesis up to 52.96%. For the round tip excited by 20 Vpp, the maximum differentiation value of 69.43% was found, since it delivered relatively high acoustic pressure to MSCs. Hence, the results from this study indicate that ultrasound pressure at the kPa level can enhance MSC chondrogenesis compared to the tens of kHz range by Langevin transducers. Full article
(This article belongs to the Special Issue Feature Papers of Micromachines in Biology and Biomedicine 2022)
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10 pages, 3442 KiB  
Article
High-Frequency Ultrasonic Radiator Power Increase by Means of Summation of Vibrations of Symmetrically Arranged Langevin Transducers
by Vladimir N. Khmelev, Andrey V. Shalunov, Victor A. Nesterov and Alexander S. Bochenkov
Symmetry 2023, 15(1), 208; https://doi.org/10.3390/sym15010208 - 11 Jan 2023
Viewed by 1772
Abstract
The article focuses on the research of processes of summation of ultrasonic vibrations of individual Langevin transducers symmetrically arranged on a common summator. This design could be used to solve the problem of increasing the power of a high-frequency ultrasonic radiator by means [...] Read more.
The article focuses on the research of processes of summation of ultrasonic vibrations of individual Langevin transducers symmetrically arranged on a common summator. This design could be used to solve the problem of increasing the power of a high-frequency ultrasonic radiator by means of the summation of the vibrations of individual Langevin transducers. The results of the presented theoretical and experimental investigations supported the possibility of summation of symmetrical diametrical vibrations when they are transformed into longitudinal vibrations. As a result, a new design scheme for high-frequency radiator construction was developed that provides symmetry and uniformity of vibration amplitude distribution of piezoceramic elements. This made it possible to eliminate the damage of piezoceramic elements. The article establishes that an acoustic power (continuous mode) of 1450 W at an efficiency of 78% is achieved when the vibrations from 9 to 11 of symmetrically arranged Langevin transducers at a resonant frequency of 30.05 kHz with an amplitude of 26 µm are summed. The use of this ultrasonic high-frequency radiator with increased power will provide intensification of processes in various areas of industry. Full article
(This article belongs to the Special Issue Symmetry Methods in Mechanics of Materials)
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17 pages, 5034 KiB  
Article
Modeling and Measurement of an Ultrasound Power Delivery System for Charging Implantable Devices Using an AlN-Based pMUT as Receiver
by Antonino Proto, Libor Rufer, Skandar Basrour and Marek Penhaker
Micromachines 2022, 13(12), 2127; https://doi.org/10.3390/mi13122127 - 1 Dec 2022
Cited by 18 | Viewed by 4009
Abstract
Ultrasound power delivery can be considered a convenient technique for charging implantable medical devices. In this work, an intra-body system has been modeled to characterize the phenomenon of ultrasound power transmission. The proposed system comprises a Langevin transducer as transmitter and an AlN-based [...] Read more.
Ultrasound power delivery can be considered a convenient technique for charging implantable medical devices. In this work, an intra-body system has been modeled to characterize the phenomenon of ultrasound power transmission. The proposed system comprises a Langevin transducer as transmitter and an AlN-based square piezoelectric micro-machined ultrasonic transducer as receiver. The medium layers, in which elastic waves propagate, were made by polydimethylsiloxane to mimic human tissue and stainless steel to replace the case of the implantable device. To characterize the behavior of the transducers, measurements of impedance and phase, velocity and displacement, and acoustic pressure field were carried out in the experimental activity. Then, voltage and power output were measured to analyze the performance of the ultrasound power delivery system. For a root mean square voltage input of approximately 35 V, the power density resulted in 21.6 µW cm−2. Such a result corresponds to the data obtained with simulation through a one-dimensional lumped parameter transmission line model. The methodology proposed to develop the ultrasound power delivery (UPD) system, as well as the use of non-toxic materials for the fabrication of the intra-body elements, are a valid design approach to raise awareness of using wireless power transfer techniques for charging implantable devices. Full article
(This article belongs to the Special Issue Piezoelectric Ultrasound Transducer for Biomedical Applications)
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14 pages, 5819 KiB  
Article
Acoustic and Thermal Characterization of Therapeutic Ultrasonic Langevin Transducers under Continuous- and Pulsed Wave Excitations
by Jinhyuk Kim and Jungwoo Lee
Sensors 2022, 22(22), 9006; https://doi.org/10.3390/s22229006 - 21 Nov 2022
Cited by 6 | Viewed by 2636
Abstract
We previously conducted an empirical study on Langevin type transducers in medical use by examining the heat effect on porcine tissue. For maximum acoustic output, the transducer was activated by a continuous sinusoidal wave. In this work, pulsed waves with various duty factors [...] Read more.
We previously conducted an empirical study on Langevin type transducers in medical use by examining the heat effect on porcine tissue. For maximum acoustic output, the transducer was activated by a continuous sinusoidal wave. In this work, pulsed waves with various duty factors were applied to our transducer model in order to examine their effect on functionality. Acoustic power, electro-acoustic conversion efficiency, acoustic pressure, thermal effect on porcine tissue and bovine muscle, and heat generation in the transducer were investigated under various input conditions. For example, the results of applying a continuous wave of 200 VPP and a pulse wave of 70% duty factor with the same amplitude to the transducer were compared. It was found that continuous waves generated 9.79 W of acoustic power, 6.40% energy efficiency, and 24.84 kPa acoustic pressure. In pulsed excitation, the corresponding values were 9.04 W, 8.44%, and 24.7 kPa, respectively. The maximum temperature increases in bovine muscle are reported to be 83.0 °C and 89.5 °C for each waveform, whereas these values were 102.5 °C and 84.5 °C in fatty porcine tissue. Moreover, the heat generation around the transducer was monitored under continuous and pulsed modes and was found to be 51.3 °C and 50.4 °C. This shows that pulsed excitation gives rise to less thermal influence on the transducer. As a result, it is demonstrated that a transducer triggered by pulsed waves improves the energy efficiency and provides sufficient thermal impact on biological tissues by selecting proper electrical excitation types. Full article
(This article belongs to the Special Issue Ultrasound-Based Sensors for Physical Therapy Applications)
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12 pages, 5714 KiB  
Article
Acoustic Power Measurement and Thermal Bioeffect Evaluation of Therapeutic Langevin Transducers
by Jinhyuk Kim and Jungwoo Lee
Sensors 2022, 22(2), 624; https://doi.org/10.3390/s22020624 - 14 Jan 2022
Cited by 3 | Viewed by 3177
Abstract
We recently proposed an analytical design method of Langevin transducers for therapeutic ultrasound treatment by conducting parametric study to estimate the effect of compression force on resonance characteristics. In this study, experimental investigations were further performed under various electrical conditions to observe the [...] Read more.
We recently proposed an analytical design method of Langevin transducers for therapeutic ultrasound treatment by conducting parametric study to estimate the effect of compression force on resonance characteristics. In this study, experimental investigations were further performed under various electrical conditions to observe the acoustic power of the fully equipped transducer and to assess its heat-related bioeffect. Thermal index (TI) tests were carried out to examine temperature rise and thermal damage induced by the acoustic energy in fatty porcine tissue. Acoustic power emission, TI values, temperature characteristics, and depth/size of thermal ablation were measured as a function of transducer’s driving voltage. By exciting the transducer with 300 Vpp sinusoidal continuous waveform, for instance, the average power was 23.1 W and its corresponding TI was 4.1, less than the 6 specified by the Food and Drug Administration (FDA) guideline. The maximum temperature and the depth of the affected site were 74.5 °C and 19 mm, respectively. It is shown that thermal ablation is likely to be more affected by steep heat surge for a short duration rather than by slow temperature rise over time. Hence, the results demonstrate the capability of our ultrasonic transducer intended for therapeutic procedures by safely interrogating soft tissue and yet delivering enough energy to thermally stimulate the tissue in depth. Full article
(This article belongs to the Collection Medical Applications of Sensor Systems and Devices)
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13 pages, 3557 KiB  
Article
Theoretical and Empirical Verification of Electrical Impedance Matching Method for High-Power Transducers
by Jungwoo Lee and Jinhyuk Kim
Electronics 2022, 11(2), 194; https://doi.org/10.3390/electronics11020194 - 9 Jan 2022
Cited by 2 | Viewed by 4236
Abstract
In our prior study, a systematic approach was used to devise Langevin transducers for high-power applications where the energy efficiency was not considered in the design criteria. In this paper, the impedance matching methods are thus proposed to evaluate what matching topology is [...] Read more.
In our prior study, a systematic approach was used to devise Langevin transducers for high-power applications where the energy efficiency was not considered in the design criteria. In this paper, the impedance matching methods are thus proposed to evaluate what matching topology is appropriate for their use. Both the series inductor scheme and low pass filter composed of a series inductor and shunt capacitor are examined as matching circuits. According to MATLAB simulation, the resonance frequency is seen at 36.79 kHz due to a series L circuit, and its associated impedance is reduced by 70.45% from that of its non-matching condition. The measured resonance frequency is 36.77 kHz and the corresponding impedance is decreased by 59.52%. Furthermore, the acoustic pressure is measured to determine the effect of the matching circuit on the transducer’s actual behavior. The transducer with a series L circuit shows more efficient matching results, 2.28 kPa of positive acoustic pressure is emitted without matching and 3.35 kPa is emitted with a series L element, respectively. As a result, this study demonstrates how to evaluate the influence of matching circuits by using our customized approach rather than commercial SPICE programs, as well as how to experimentally verify the acoustic behavior of high-power Langevin transducers. Full article
(This article belongs to the Section Industrial Electronics)
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10 pages, 4038 KiB  
Article
Core–Shell Droplet Generation Device Using a Flexural Bolt-Clamped Langevin-Type Ultrasonic Transducer
by Kentaro Omori, Nozomu Fujimoto, Takefumi Kanda, Shuichi Wakimoto and Norihisa Seno
Actuators 2021, 10(3), 55; https://doi.org/10.3390/act10030055 - 9 Mar 2021
Cited by 4 | Viewed by 3054
Abstract
Droplets with a core–shell structure formed from two immiscible liquids are used in various industrial field owing to their useful physical and chemical characteristics. Efficient generation of uniform core–shell droplets plays an important role in terms of productivity. In this study, monodisperse core-shell [...] Read more.
Droplets with a core–shell structure formed from two immiscible liquids are used in various industrial field owing to their useful physical and chemical characteristics. Efficient generation of uniform core–shell droplets plays an important role in terms of productivity. In this study, monodisperse core-shell droplets were efficiently generated using a flexural bolt-clamped Langevin-type transducer and two micropore plates. Water and silicone oil were used as core and shell phases, respectively, to form core–shell droplets in air. When the applied pressure of the core phase, the applied pressure of the shell phase, and the vibration velocity in the micropore were 200 kPa, 150 kPa, and 8.2 mm/s, respectively, the average diameter and coefficient of variation of the droplets were 207.7 μm and 1.6%, respectively. A production rate of 29,000 core–shell droplets per second was achieved. This result shows that the developed device is effective for generating monodisperse core–shell droplets. Full article
(This article belongs to the Section Miniaturized and Micro Actuators)
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19 pages, 6374 KiB  
Article
Development of an Accurate Resonant Frequency Controlled Wire Ultrasound Surgical Instrument
by Jungsuk Kim, Kyeongjin Kim, Sun-Ho Choe and Hojong Choi
Sensors 2020, 20(11), 3059; https://doi.org/10.3390/s20113059 - 28 May 2020
Cited by 16 | Viewed by 5280
Abstract
Our developed wire ultrasound surgical instrument comprises a bolt-clamped Langevin ultrasonic transducer (BLUT) fabricated by PMN-PZT single crystal material due to high mechanical quality factor and electromechanical coupling coefficient, a waveguide in the handheld instrument, and a generator instrument. To ensure high performance [...] Read more.
Our developed wire ultrasound surgical instrument comprises a bolt-clamped Langevin ultrasonic transducer (BLUT) fabricated by PMN-PZT single crystal material due to high mechanical quality factor and electromechanical coupling coefficient, a waveguide in the handheld instrument, and a generator instrument. To ensure high performance of wire ultrasound surgical instruments, the BLUT should vibrate at an accurate frequency because the BLUT’s frequency influences hemostasis and the effects of incisions on blood vessels and tissues. Therefore, we implemented a BLUT with a waveguide in the handheld instrument using a developed assembly jig process with impedance and network analyzers that can accurately control the compression force using a digital torque wrench. A generator instrument having a main control circuit with a low error rate, that is, an output frequency error rate within ±0.5% and an output voltage error rate within ±1.6%, was developed to generate the accurate frequency of the BLUT in the handheld instrument. In addition, a matching circuit between the BLUT and generator instrument with a network analyzer was developed to transfer displacement vibration efficiently from the handheld instrument to the end of the waveguide. Using the matching circuit, the measured S-parameter value of the generator instrument using a network analyzer was −24.3 dB at the resonant frequency. Thus, our proposed scheme can improve the vibration amplitude and accuracy of frequency control of the wire ultrasound surgical instrument due to developed PMN-PZT material and assembly jig process. Full article
(This article belongs to the Special Issue Ultrasonic Sensors 2019–2020)
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9 pages, 1529 KiB  
Article
Parametric Study of Bolt Clamping Effect on Resonance Characteristics of Langevin Transducers with Lumped Circuit Models
by Jinhyuk Kim and Jungwoo Lee
Sensors 2020, 20(7), 1952; https://doi.org/10.3390/s20071952 - 31 Mar 2020
Cited by 19 | Viewed by 5254
Abstract
We recently proposed a numerical model using equivalent circuit models to analyze the resonance characteristics of Langevin transducers and design them in a systematic manner. However, no pre-load torque biased by a metal bolt was considered in the model. Here, a parametric study [...] Read more.
We recently proposed a numerical model using equivalent circuit models to analyze the resonance characteristics of Langevin transducers and design them in a systematic manner. However, no pre-load torque biased by a metal bolt was considered in the model. Here, a parametric study is, therefore, carried out to reveal how model parameters are adapted to incorporate the pre-compression effect into our existing model. Analytical results are compared with corresponding experimental data, particularly regarding the input electrical impedance and effective electromechanical coupling coefficient for the transducer at resonance modes. The frequency response of input impedance is presented as a function of torque, both theoretically and experimentally. For 10.0 N·m bias, for instance, both resonance and anti-resonance frequencies are calculated as 38.64 kHz and 39.78 kHz, while these are measured as 38.62 kHz and 39.77 kHz by the impedance analyzer. The impedance difference between these cases is 14 Ω at resonance and 9 kΩ at anti-resonance, while the coupling coefficients in both cases become 0.238 and 0.239, respectively. Hence, these test results are closely matched with their theoretical values. Consequently, this study provides a quantitative guideline that specifies the pre-loading condition of bolt clamps with proper parameter settings to predict the intended resonance characteristics of Langevin transducers. Full article
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13 pages, 25125 KiB  
Article
Experimental Setup for Dynamic Analysis of Micro- and Nano-Mechanical Systems in Vacuum, Gas, and Liquid
by Bram van den Brink, Farbod Alijani and Murali Krishna Ghatkesar
Micromachines 2019, 10(3), 162; https://doi.org/10.3390/mi10030162 - 26 Feb 2019
Cited by 3 | Viewed by 3794
Abstract
An experimental setup to perform dynamic analysis of a micro- and nano-mechanical system in vacuum, gas, and liquid is presented. The setup mainly consists of a piezoelectric excitation part and the chamber that can be either evacuated for vacuum, or filled with gas [...] Read more.
An experimental setup to perform dynamic analysis of a micro- and nano-mechanical system in vacuum, gas, and liquid is presented. The setup mainly consists of a piezoelectric excitation part and the chamber that can be either evacuated for vacuum, or filled with gas or water. The design of the piezoelectric actuator was based on a Langevin transducer. The chamber is made out of materials that can sustain: vacuum, variety of gases and different types of liquids (mild acids, alkalies, common alcohols and oils). All the experiments were performed on commercial cantilevers used for contact and tapping mode Atomic Force Microscopy (AFM) with stiffness 0.2 N/m and 48 N/m, respectively, in vacuum, air and water. The performance of the setup was evaluated by comparing the measured actuator response to a finite element model. The frequency responses of the two AFM cantilevers measured were compared to analytical equations. A vacuum level of 0.6 mbar was obtained. The setup has a bandwidth of 10–550 kHz in vacuum and air, and a bandwidth of 50–550 kHz in liquid. The dynamic responses of the cantilevers show good agreement with theory in all media. Full article
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10 pages, 3154 KiB  
Article
Spontaneous Brillouin Scattering Spectrum and Coherent Brillouin Gain in Optical Fibers
by Vincent Laude and Jean-Charles Beugnot
Appl. Sci. 2018, 8(6), 907; https://doi.org/10.3390/app8060907 - 1 Jun 2018
Cited by 5 | Viewed by 5197
Abstract
Brillouin light scattering describes the diffraction of light waves by acoustic phonons, originating from random thermal fluctuations inside a transparent body, or by coherent acoustic waves, generated by a transducer or from the interference of two frequency-detuned optical waves. In experiments with optical [...] Read more.
Brillouin light scattering describes the diffraction of light waves by acoustic phonons, originating from random thermal fluctuations inside a transparent body, or by coherent acoustic waves, generated by a transducer or from the interference of two frequency-detuned optical waves. In experiments with optical fibers, it is generally found that the spontaneous Brillouin spectrum has the same frequency dependence as the coherent Brillouin gain. We examine the origin of this similarity between apparently different physical situations. We specifically solve the elastodynamic equation, giving displacements inside the body, under a stochastic Langevin excitation and in response to a coherent optical force. It is emphasized that phase matching is responsible for temporal and spatial frequency-domain filtering of the excitation, leading in either case to the excitation of a Lorentzian frequency response solely determined by elastic loss. Full article
(This article belongs to the Special Issue Brillouin Scattering and Optomechanics)
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12 pages, 761 KiB  
Article
Nonlinear Dynamic Modeling of Langevin-Type Piezoelectric Transducers
by Nicolás Peréz Alvarez, Andrea Cardoni, Niccolo Cerisola, Enrique Riera, Marco Aurélio Brizzotti Andrade and Julio Cezar Adamowski
Actuators 2015, 4(4), 255-266; https://doi.org/10.3390/act4040255 - 18 Nov 2015
Cited by 8 | Viewed by 8484
Abstract
Langevin transducers are employed in several applications, such as power ultrasound systems, naval hydrophones, and high-displacement actuators. Nonlinear effects can influence their performance, especially at high vibration amplitude levels. These nonlinear effects produce variations in the resonant frequency, harmonics of the excitation frequency, [...] Read more.
Langevin transducers are employed in several applications, such as power ultrasound systems, naval hydrophones, and high-displacement actuators. Nonlinear effects can influence their performance, especially at high vibration amplitude levels. These nonlinear effects produce variations in the resonant frequency, harmonics of the excitation frequency, in addition to loss of symmetry in the frequency response and “frequency domain hysteresis”. In this context, this paper presents a simplified nonlinear dynamic model of power ultrasound transducers requiring only two parameters for simulating the most relevant nonlinear effects. One parameter reproduces the changes in the resonance frequency and the other introduces the dependence of the frequency response on the history of the system. The piezoelectric constitutive equations are extended by a linear dependence of the elastic constant on the mechanical displacement amplitude. For introducing the frequency hysteresis, the elastic constant is computed by combining the current value of the mechanical amplitude with the previous state amplitude. The model developed in this work is applied for predicting the dynamic responses of a 26 kHz ultrasonic transducer. The comparison of theoretical and experimental responses, obtained at several input voltages around the tuned frequency, shows a good agreement, indicating that the model can accurately describe the transducer nonlinear behavior. Full article
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