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High Frequency Flexural Ultrasonic Transducers


This research is a fellowship project funded by the Engineering and Physical Sciences Research Council (EPSRC), grant number EP/N025393/1, led by Prof Steve Dixon, with Dr Andrew Feeney and Dr Lei Kang as Research Fellows, and Jonathan Harrington providing technical expertise. The project period is 2016 - 2021.

Project Goal and Overview

Presently, flexural ultrasonic transducers are designed and operated exclusively in ambient atmospheric conditions. They are used up to around 50 kHz, where the wavelengths in fluids are relatively long, thus reducing measurement resolution. Flexural ultrasonic transducers which can operate effectively at higher frequencies are therefore desirable, but also to withstand higher pressure and temperature levels associated with more hostile environments encountered in many industrial and aerospace applications. The major objective of this research is the development of such devices, known as high frequency flexural ultrasonic transducers (HiFFUTs). These HiFFUTs should be able to operate with high efficiency in both liquid and gas environments. General target operating conditions with respect to pressure are outlined in Table 1, and relating to temperature in Table 2.

We analyse both commercial and custom-made transducers, and we fabricate our own devices within our group, to meet the requirements of the project. We use the latest vibration characterisation techniques such as sensor systems in transmit-receive mode, acoustic microphones, and laser Doppler vibrometry, to assess the performance of all of our devices, and are working with industry partners, detailed below, for qualification in practical application.

The Flexural Ultrasonic Transducer

The flexural ultrasonic transducer is a uni-morph device, composed of a piezoelectric driver material in the form of a ceramic disc bonded to a metal cap. Two transducer configurations are shown in Figure 1, where the principle of operation is centred on the vibration of the piezoelectric element which is coupled to the metal cap through a bonding agent, subsequently causing bending of the metal cap, with a significant level of vibration amplification. The vibration response of the cap can be considered analogous to standard plate modes of vibration, convenient for the numerical modelling of flexural transducer behaviour. One significant advantage of flexural transducers over more conventional ultrasonic transducers is that since the vibrating metal cap is coupled to the loading medium, the effective impedance is much lower than the acoustic impedance of the cap material. This enables the transducer to couple efficiently to low impedance media, and operate with lower input voltage. Flexural transducers have been utilised for a variety of applications, including in proximity sensing environments such as car-parking systems, and for non-destructive evaluation.


Figure 1: Schematics of classical flexural ultrasonic transducers, which have traditionally been employed in proximity sensing and industrial metrology. Two possible configurations are shown, highlighting freedom in cap geometry, but both relying on the same operating principle. HiFFUTs are a distinct evolution of the flexural ultrasonic transducer.

The principal modes of vibration of a flexural transducer are shown in Figure 2, which exhibits two axisymmetric modes, termed (0,0) and (1,0), and two asymmetric modes, designated as (0,1) and (1,1). Higher order modes exist, such as the (2,0) axisymmetric mode.

Figure 2: Plate modes of vibration.

Figure 2: Plate modes of vibration, used as analytical representations of the vibration modes of the flexural transducer.

The cap membrane is considered as a thin plate which is clamped around the circumferential edge. The transverse displacement of the cap membrane plate can be determined through the following differential equation:


The rigidity of the cap membrane can be calculated using the following relationship, where E is the Young's modulus, h is the cap membrane thickness, and υ is the Poisson's ratio:


The angular modal frequency can hence be determined using the following equation, where λ is a Bessel function constant relating to the mode shape, a is the cap membrane radius, D is the rigidity of the cap membrane, and ρ is the density of the cap membrane material. To convert the ω expression to frequency, this entire relationship should be divided by 2π.


Characterisation of Transducer Performance

The dynamic characterisation of flexural ultrasonic transducers has been extensively studied during the course of this project so far, and progress in HiFFUT development has already been made, including for high temperature environments towards 200°C. Downloads and publications are available at the bottom of this web-page, but some key outcomes are depicted in the following figures.


Figure 3: Vibration response of a flexural ultrasonic transducer, discretised into three zones. This response is the result of a burst sine excitation, a commonly applied signal in the operation of these transducers.


Figure 4: The vibration response of a flexural ultrasonic transducer in the initial response zone (see Figure 3), compared with the analog model fit computed from the derived equation shown below. This is real laser Doppler vibrometry (LDV) data compared with the response computed using the characteristic equation, for (a) a resonance 40 kHz drive frequency, and (b) an off-resonance 44 kHz drive frequency.


In this equation, the F magnitudes relate to the resonant amplitude of the transducer, E is the forced vibration amplitude, and α is related to the system damping.


Figure 5: Nonlinearity in the dynamic response of a flexural ultrasonic transducer operating at steady-state, showing resonance frequency drift with increasing excitation voltage.


Figure 6: Nonlinearity in the dynamic response of a flexural ultrasonic transducer in ring-down, demonstrating that even in the absence of forced excitation, nonlinearity remains detectable.


Figure 7: Measurement of high frequency and higher order vibration modes of an aluminium flexural ultrasonic trandsducer in air, at a distance of 0.5 metres with a burst-sine input of 10 VP-P and 150 cycles. This data was published in IEEE Sensors Journal in 2018. Note the difference in time scales for clarity.

HiFFUT Design Tool

This section provides an expedient and rapid estimator for basic HiFFUT design and operating parameters, for the fundamental axisymmetric operating modes of vibration, and for a selection of standard cap materials. This estimator tool is constructed from the underlying mathematics of the vibration modes for an edge-clamped plate, which are summarised above, to generate the operating frequency for a designated mode of vibration, accounting for membrane rigidity in each case. The higher the rigidity, the less compliant the membrane, and therefore less efficient for propagating ultrasound energy. This is important to consider in the design of HiFFUTs. This tool is strictly an estimator and therefore should be used as a guideline only. This is because the vibration characteristics and dynamic performance of a HiFFUT will not precisely match that of an edge-clamped plate, being a more complex physical system. For accurate HiFFUT design, finite element methods are strongly recommended.

To use this tool, three membrane parameters are required as a minimum, comprising the diameter, thickness, and material, where a selection of standard HiFFUT cap materials has been built into the tool for convenience. A key feature of this tool is its ability to generate modal frequencies based on custom material properties. To enter a custom material, simply enter its material properties in the empty fields, and select "Custom", which is located under the HiFFUT Membrane Material drop-down menu. Note that when specifying the membrane diameter, this does not include the side-wall geometry. Once the relevant tool fields are populated, the modal frequencies for the first three axisymmetric modes of vibration can be determined by selecting "Calculate".

The estimator is provided below, followed by the supplementary information important for the practical design and fabrication of HiFFUTs. Properties for selected cap membrane materials are shown in Table 3 for reference.

Estimator of HiFFUT Operating Frequency

HiFFUT Membrane Diameter (mm):
HiFFUT Membrane Thickness (mm):
HiFFUT Membrane Material:
Density (kg/m3):
Young's Modulus (GPa):
Poisson's Ratio:
Axisymmetric Mode of Vibration:
Modal Frequency (kHz):

The key design considerations for HiFFUTs are:

  • Environmental conditions, in particular pressure and temperature. This should be determined prior to any design process. The type of fluid will also be important to define, for example air, oil, or gas.
  • Frequency and mode of operation. The operating frequency can be selected prior to HiFFUT design using the tool above, by manipulating the cap membrane material and dimensions. One solution may be to exploit higher order vibration modes, such as the (1,0) or (2,0) modes rather than the (0,0) mode, in the generation of high frequencies.
  • Amplitude of sound propagation. The achievable vibration amplitude of the HiFFUT will depend on the cap material and physical dimensions of the HiFFUT and its components. A sufficiently robust transducer design will be required, to ensure the ability to operate the transducer at high amplitudes.
  • Losses and efficiency. The thickness of the HiFFUT cap and the diameter of the radiating membrane affect the resonance frequency. The mechanisms by which these parameters affect resonance frequency have been reported, but less is known regarding how they affect transducer efficiency. The design and simulation of transducer performance should consider ceramic dielectric loss, in addition to internal mechanical loss. The electromechanical coupling coefficient is a useful property for assessing HiFFUT performance.
  • Cap radiating membrane material, and associated geometry. The elastic properties of the cap material affect the vibration response of the HiFFUT, where the Youngs modulus, Poissons ratio, and density of the material are most signicant. There is flexibility in the choice of cap material, the most popular being stainless steel, aluminium and titanium, as provided in the design tool. The material and the cap dimensions will affect the vibration behaviour of the HiFFUT, but the vibration motion predominantly relies on how the membrane layer is defined, where the side-wall of the cap provides far less influence. Higher pressure environments may necessitate thicker membranes, depending on the design, but the cost of this is reduced compliance, and hence a lower efficiency. As the cap membrane diameter is increased, the magnitudes of the modal frequencies generally reduce. A compromise between cap membrane diameter and resonance frequency is hence required. A larger cap diameter enables a higher amplitude of vibration to be achieved for a specific excitation signal, but for a lower resonance frequency. However, a cap membrane diameter which is too small will make transducer fabrication difficult, for example with respect to soldering electrode connections.
  • Driver material, and associated geometry. High pressure and temperature levels restrict the types of active driver which can be efficiently used in HiFFUTs. Strategies for shielding piezoelectric ceramics should be considered, or the use of alternative forms of active driver materials. Furthermore, the thickness of the driver element, such as the piezoelectric ceramic, should be lower than that of the cap, to ensure sufficient flexure in operation in the generation of the axisymmetric modes of vibration. However, the cap cannot be of a thickness which restricts the bending motion to generate the mode shapes.
  • Mechanical coupling. The problems surrounding the bonding of flextensional and cymbal type transducers has been extensively reported. Transducers can be very sensitive to bond layer inhomogeneities, differences in deposited thickness, and the type of bonding agent used. The method of bond layer deposition, and any pre-treatment such as degassing and surface preparation, should be considered. Alternatively, an entirely different coupling mechanism may be an appropriate solution.
  • Backing material. The choice of backing material, and backing layer thickness, both depend on the dimensions of the cap material and HiFFUT dimensions.

Acoustic Levitation

Acoustic levitation has been achieved using flexural ultrasonic transducers operating at 40 kHz, with a custom portable, hand-held system design as shown below. Our levitation system can be oriented in different ways, from vertical through to horizontal, maintaining the levitation of the particles. A video of our acoustic levitation system in action can be found by clicking here.


Industrial Collaboration

This research is being conducted with the invaluable support and partnership of key industry organisations.

Cygnus Instruments



Detectronic Limited



Dynoptic Systems Ltd



EES Research Ltd


EES Research Ltd is a UK owned Research & Development company with over 25 years experience developing process measurement and control instrumentation for a diverse range of industries. Our expertise includes; ultrasonic, capacitance and radar techniques for level and flow measurement. EES Research has experience in many areas including; Water, Mining, Petrochemical, Transport, Information systems, Communications systems and many others.


Research and Development, Process Control Instrumentation, Process Instrumentation, Process Measurement Instrumentation

Dr Noel Kerr, Managing Director
Email: noel at eesresearch dot com
Telephone: +44(0)7971 273000
Headquarters: EES Research Ltd, Lochiel, Llaneilian Rd, Amlwch Port, Anglesey, LL68 9HU, United Kingdom

FLEXIM Instruments Ltd.



Katronic Technologies Ltd


Katronic develop and manufacture portable and fixed installation ultrasonic flowmeters with the intention that the measurement of flow should be quick, easy and straightforward. Katronic ultrasonic flowmeters can measure on pipes of all standard acoustically conductive materials over a large diameter range and are suitable for process liquids from pure water to chemicals and effluents.

Rob Turner, Engineering Manager


National Nuclear Laboratory



Documents and Downloads

Key documents, publications and resources are provided in this section for download. In addition, documents relating to the regular project meetings with the industrial partners have been made available. Raw data sets associated with each journal publication are made available through a link provided in each article.

Key Publications

  1. A. Feeney, L. Kang, and S. Dixon, "High frequency measurement of ultrasound using flexural ultrasonic transducers," Early Online Publication, IEEE Sensors Journal, 2018.
  2. A. Feeney, L. Kang, and S. Dixon, "Nonlinearity in the dynamic response of flexural ultrasonic transducers," IEEE Sensors Letters, vol. 2, no. 1, pp. 1-4, 2018.
  3. A. Feeney, L. Kang, G. Rowlands, and S. Dixon, "The dynamic performance of flexural ultrasonic transducers," Sensors, vol. 18, no. 1, 270, pp. 1-14, 2018.
  4. A. Feeney, L. Kang, G. Rowlands, and S. Dixon, "HiFFUTs for high temperature ultrasound," Proceedings of Meetings on Acoustics, vol. 32, no. 1, 045003, 2017.
  5. A. Feeney, L. Kang, G. Rowlands, and S. Dixon, "Dynamic characteristics of flexural ultrasonic transducers," Proceedings of Meetings on Acoustics, vol. 32, no. 1, 045002, 2017.
  6. L. Kang, A. Feeney, R. Su, D. Lines, A. Jäger, H. Wang, Y. Arnaudov, S.N. Ramadas, M. Kupnik, and S. Dixon, "Two-dimensional flexural ultrasonic phased array for flow measurement," Proceedings of the 2017 IEEE International Ultrasonics Symposium, Washington D.C., USA, September 2017.
  7. S. Dixon, L. Kang, M. Ginestier, C. Wells, G. Rowlands, and A. Feeney, "The electro-mechanical behaviour of flexural ultrasonic transducers," Applied Physics Letters, vol. 110, no. 22, p.223502, 2017.
  8. T.J.R. Eriksson, S.N. Ramadas, and S.M. Dixon, "Experimental and simulation characterisation of flexural vibration modes in unimorph ultrasound transducers," Ultrasonics, vol. 65, pp. 242-248, 2016.
  9. T.J.R. Eriksson, S.M. Dixon, and S.N. Ramadas, "Metal cap flexural transducers for air-coupled ultrasonics," In 41st Annual Review of Progress in Quantitative Nondestructive Evaluation: Volume 34, AIP Publishing, vol. 1650, no. 1, pp. 1287-1291, 2015.
  10. T.J.R. Eriksson, S.M. Dixon, and S.N. Ramadas, "Flexural mode metal cap transducer design for specific frequency air coupled ultrasound generation," In 2013 IEEE International Ultrasonics Symposium (IUS), pp. 1602-1605, 2013.

Professional Engagement and Outreach Activities

We demonstrate our research, and other ultrasound-related science, to both our peers and the general public, on a regular basis. A selection of these activities are shown below.



Involvement and engagement with industry organisations.


Demonstrating our research to delegates from Harbin Institute of Technology in November 2017.


Demonstration of research at the 3rd Flow Measurement Institute Conference in July 2017.


Presenting at the RCNDE Early Career Researcher Symposium in June 2017.


Presenting HiFFUT research to the International Congress on Ultrasonics in Honolulu, Hawaii, in December 2017.



STEM outreach at the XMaS Science Gala in January 2018.


EPSRC Grants on the Web information:

The information on this web-page is managed and regularly updated by Dr Andrew Feeney.