244 S. Chen et al. As one of the first steps, this work aims to gain a comprehensive understanding of acoustic field generated from ultrasonic transducers numerically and experimentally in order to monitor and control the acoustic radiation pressure and force imparted to a structure. In this paper, a numerical boundary element model based on the Rayleigh Integral and boundary element method was first established. In this model, the transducer surface was radially discretized into 5401 source elements. Then, test setups and experimental methods for measuring vibrations and acoustic FRFs were described. After that, the vibration FRFs were examined and velocity profile of the surface of an ultrasonic transducer at certain frequencies were measured, the velocity profiles were linearly interpolated, and mapped to the source elements representing the transducer surface and was then used to calculate acoustic beam profiles by using the previous built boundary element model. The acoustic FRFs in four plane slices before the transducer were also tested and acoustic pressure profiles were then derived for these four planes. Comparison between the simulation and experimental results was made and a good agreement was found in terms of both beam profile and pressure amplitude. 23.2 Theoretical Background To simulate the acoustic field generated by ultrasonic transducers, a boundary element model based on the Rayleigh Integral and boundary element method was built. As shown in Fig. 23.1, the surface of the test transducer used in this work was discretized into a circular element at the center and 30 annular sections each with 180 source elements. In total, there are 5401 source elements and to obtain a better approximation, each source element was set in a way that its dimensions were smaller than the wavelength of the sound wave. For example, wavelengthœDc/f, where c is the speed of sound in a particular medium at certain temperature and f is the wave frequency. If f D50 kHz, œD 6.8mm. The sound pressure of a field point p(x,y,z) produced by one source element can be represented by [13] p.x; y; z/ D j! U.x’; y’/ej.!t kr/ 2 r ds (23.1) where, r is the distance between the field point and the center of the source element, ds is the area of that element source, and r Dq.x x’/2 C.y y’/2 Cz2 (23.2) ds Drs drs dphi (23.3) The overall pressure at a field point is a superposition of from all of the source elements and can be formulated by P.x; y; z/ DX l Xm p.x; y; z/ (23.4) p(x,y,z) x y z si(x’,y’,z’,dx’,dy’) r phi ds dphi rs drs ds=rs*drs*dphi Fig. 23.1 Schematic of the boundary element model used to simulate the acoustic field generated by ultrasonic transducer
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