Designing a Hollow Langevin Transducer for Ultrasonic Coring

A number of architectures for a hollow Langevin ultrasonic transducer are proposed and evaluated. One of these is optimised by finite element modelling and is then manufactured and analysed experimentally. The preload on the transducer ceramics is increased and the effect on the performance is measured. At maximum preload the results of an experimental modal analysis are used to determine the natural frequency and response of both the operating longitudinal mode and unwanted bending modes. The performance of the hollow transducer is compared to a solid commercial transducer containing the same volume of piezoceramic material. The efficiency is shown to be comparable. Higher ultrasonic displacement amplitudes are achieved with the hollow transducer although a lower Q-factor is found.

frequency is reached a response spectrum is measured and the Q-factor of the transducer is calculated. Further addition of preload is then shown to increase the magnitude and quality factor of the response at the operating frequency. Finally, the hollow transducer is contrasted with a traditional transducer of similar size, to determine its performance with reference to its solid competitors.
External bolting. An external cylinder is unlikely to be practical because it would impede access to the piezoceramics. It would also have to be, effectively, a thin-walled shell. Thin-walled structures, which are inherently non-stiff, are liable to exhibit detrimental transverse modes at ultrasonic frequencies. Furthermore, the structure would preclude the use of all but the thinnest screwthreads for prestressing purposes, and would be extremely vulnerable to fatigue.
The alternative external-bolting approach, consisting of a series of columns around the circumference of the transducer, permits access to the ceramics but would require that the ceramics themselves, which are commercially produced as rings, be shaped to fit around the columns. The number of columns and non-standard ceramics would be difficult, slow and expensive to both manufacture and assemble. The stress applied through each bolt would also have to be equalized to ensure even compressive load on the ceramic stack, as uneven prestress is known to be associated with performance inefficiencies [4].
Internal bolting. Modifying the central bolt, such that it becomes a hollow structure, appears to be a more attractive approach because the walls of the bolt remain sufficiently thick to accommodate screwthreads for assembly purposes. It also permits the use of standard piezoceramic discs and facilitates the even distribution of prestress across the piezoceramic stack. In the absence of other factors, this architecture is preferred for the targeted coring application.

Design of the Hollow Transducer
In common with several conventional transducers, end masses of steel and aluminium are used in this design to maximise longitudinal vibration output by taking advantage of their different stiffness-to-density ratios. A 90Ti/6V/4Al bolt is used because of its high quality factor and fatigue resistance, and PZT-8 ceramics driven by copper electrodes provide the vibration excitation.
The thickness of the bolt wall is set at 3 mm, and the volume of PZT-8 is required to be close to 5x10 -5 m 3 to match the volume of ceramic contained in a commercial transducer known to be effective for rock-coring operations. A separation of 2 mm is maintained between the central bolt and the electrodes and the interstitial space is filled with dielectric material to prevent internal shorts. A preload screw is provided at one end for final assembly and the other is threaded to accept a half-inch UNF stud. Although a system utilising a hollow transducer would be hollow throughout, the transducer is to be further tested using a hollow horn designed to connect to the commercial transducer and so the central stud connection is required.
The design frequency of the hollow transducer is 20 kHz for operation in the first longitudinal mode. To achieve this, an FE model of the hollow transducer is created in Abaqus, with reference to the fixed design parameters described above. The non-fixed parameters, such as the lengths of the end-masses, are then adjusted to tune the operating frequency. Finally the oscillatory stress in the central bolt is evaluated, assuming a peak-to-peak displacement of the transducer of 10 µm and is found to be well below the fatigue limit. The resulting design is presented in Fig. 1.

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Advances in Experimental Mechanics VII The grid spacing in the photograph is 5 mm and the external wiring to the electrodes is apparent.

Application of Preload
Clamps are offered up to flats ground on the bolt, front mass and back mass (see Fig. 2(a)), whilst the preload nut is tightened to increase the preload on the stack. This allows preload to be applied to the stack without any torsion which could damage the electrodes. Whilst the actual preload is almost impossible to predict due to the complexities of the transducer structure [5,6], it can be found in practice by measuring the charge on the piezoceramics during the prestressing process [7]. It is known that increasing the preload will tend to increase the first longitudinal resonance frequency of a bolted Langevin transducer as well as the associated Q-factor, and it is reasonable to expect that the same will be true for hollow transducers. The preload is increased from the minimum required for solid assembly (bias 1) to the point at which no further changes in the frequency of the first longitudinal mode appear to be achievable (bias 4). To measure the prestress applied in this condition, a sample of PZT-8 was subjected to compression testing. After ten seconds at a range of applied pressures, the voltage across the faces was recorded to form a look-up table which was then used to deduce the preload in the transducer after tightening. The minimum preload required to stabilise the frequency behaviour of the device was found to be 10 MPa. Fig. 2(b) shows the impedance analysis, and the progression from bias 1 to bias 4 shows how the results of the repeated impedance analyses are affected by increasing preload.

Broadband Results
An experimental modal analysis of the transducer is performed at bias 1 using a 3D laser Doppler vibrometer, where the responsive modes, marked A, B, C and D in Fig. 3(a), are extracted during excitation by a white noise signal between 0 and 40 kHz. The frequencies of these modes, which are respectively first bending, second bending coupled with first longitudinal, third bending and first radial, are indicated on the frequency response function (FRF) shown in Fig. 3(a). Fig. 3(b) shows a result of the second experimental modal analysis, where a much higher preload (bias 4) has been applied. The responsive modes, which are first bending, second bending, first longitudinal and third bending and are marked E, F, G and H respectively in the figure, are extracted.  Initially, the operating mode (first longitudinal) of the transducer is excited at 10.3 kHz and is coupled with the second bending mode. Increasing the preload causes the operating mode to decouple from the bending mode, increase in strength by almost an order of magnitude and move towards the 20 kHz value predicted by the finite element model. This final trend is consistent with the behaviour of conventional bolted Langevin transducers [8]. Increasing prestress also causes the frequencies of the radial and bending modes to increase, due to the stiffening of the structure as microscopic voids between the elements are progressively removed. This process can take place due to cold flow of the soft copper electrodes into surface imperfections [9]. As prestress increases from bias 1 to bias 4, the modal frequencies approach the values predicted by the finite element model, as shown in Table 1. However, increasing prestress beyond 10 MPa has a negligible effect on the frequencies of the modes studied.

Modal Analysis
The mode shapes, as predicted by FE and measured by EMA are presented in Table. The transducer was self-excited by a random signal.

Improvements to the Design
The process of preloading the transducer demonstrated that the application of significant preload requires very considerable torque. The 10 MPa loading needed to stabilise the transducer required approximately 50 Nm and so the preload nut was redesigned to facilitate the use of a hexagonal socket. FE analysis indicated that this new nut has a negligible influence on the natural frequency of the transducer, but it permitted the preload on the piezoceramic stack to be increased to 20 MPa. Although some sources recommend preloads as high as 40 MPa [10], the thin-walled structure of the central bolt makes high loads difficult to apply safely. Fig. 4(a) shows the response of the longitudinal operating mode (+/-2 kHz range) in bias condition 4, compared to the response of a solid commercial transducer containing the same volume of piezoceramic. In both cases a sinusoidal excitation of 40 V (peak-to-peak) has been applied. Fig.  4(b) shows the same measurements after the new nut has permitted doubling of the applied preload.  The transducer resonance is 10% below the design frequency and the presence of a bending mode near the tuned frequency requires careful driving for effective use. However, the additional preload beyond the point of resonance stabilisation increased the amplitude response of the hollow transducer from 1.8 to 3.2 µm and increased the Q-factor to a level comparable to that of the commercial transducer. These disparities are to a large extent the result of high tolerance requirements for the manufacture of the hollow transducer. The wall-thickness of the central bolt is just 3 mm, and so a small Applied Mechanics and Materials Vols. 24-25 manufacturing error can have a significant effect on the stiffness of the structure, whereas the same error on a solid bolt has a negligible effect. The hollow transducer is effective in delivering the required ultrasonic amplitude to a coring device and will form the basis of the ongoing design of a completely hollow ultrasonic coring system.

Conclusions
The design process leading to a prototype of a hollow Langevin transducer has found that the frequency of the operating longitudinal mode increases and tends towards the FE predictions as the mechanical preload is increased, until frequency stabilisation occurs. Thereafter the results indicate that increased preload continues to improve the mechanical quality factor of the transducer and therefore its ultrasonic vibration amplitude.
In terms of mechanical efficiency, the performance of the prototype hollow transducer was comparable to that of a commercial transducer containing the same volume of piezoceramic. The peak output at 40 V peak-to-peak excitation was over 3 µm compared to just over 2 µm for the commercial model. However, the Q-factor was significantly lower and an unwanted bending mode close to the main resonance remains a cause for concern.
Technical issues highlighted by the investigation centre on the design of the central bolt. Extremely tight manufacturing tolerances are required to ensure that its mechanical stiffness is correct, and the thread which receives the preload nut should be shallow yet extend over the greatest possible axial distance.
The hollow transducer also requires a long stack of piezoelectric elements if a significant volume of ceramic is to be incorporated. This, coupled with the low stiffness of the hollow bolt, places a constraint on the maximum operating frequency. Unwanted bending modes also appear to be a more significant problem than is the case for traditional solid bolted Langevin transducers. However, the long stack does lend itself to higher ultrasonic vibration amplitudes because equivalent percentage strains translate to higher absolute displacement.