Abstract
Ultrasonic piezoelectric transducers used in classical nondestructive testing are producing in general longitudinal vibrations in the MHz range. A simple mechanical model of these transducers would be very useful for wave propagation numerical simulations, avoiding the existing complicated models in which the real components of the transducer are modeled by finite elements. The classical model for longitudinal vibrations is not adequate because the generated longitudinal wave is not dispersive, the velocity being the same at any frequency. We have adopted the Rayleigh-Bishop model, which avoids these limitations, even if it is not converging to the first but to the second exact longitudinal mode in an elastic rod, as obtained from the complicated Pochhammer-Chree equations. Since real transducers have significant vibrations damping, we have introduced a damping term in the Rayleigh-Bishop model, increasing the imaginary part and keeping almost identical real part of the wavenumber. Common transducers produce amplitude modulated signals, completely attenuated after several periods. This can be modeled by two close frequencies, producing a “beat” phenomenon, superposed on the high damping. For this reason, we introduce a two-rod Rayleigh-Bishop model with damping. Agreement with measured normal velocity on the transducer free surface is encouraging for continuation of the research.
Highlights
Longitudinal vibrations of short rods are the basic phenomenon involved in most ultrasonic transducers, since shear wave transducers are seldom used
Ultrasonic piezoelectric transducers used in classical nondestructive testing are producing in general longitudinal vibrations in the MHz range
A simple mechanical model of these transducers would be very useful for wave propagation numerical simulations, avoiding the existing complicated models in which the real components of the transducer are modeled by finite elements
Summary
Longitudinal vibrations of short rods are the basic phenomenon involved in most ultrasonic transducers, since shear wave transducers are seldom used. The classical theory is generally accepted due to its simplicity and is called a unimodal theory since only the fundamental symmetric guided mode is approximated It is broadly used for the design of low frequency waveguides such as ultrasonic transducers and mechanical filters. The work of Kocbach [9] presents an important basis for the first step of the design process of piezoelectric transducers This has been done through an analysis of the radiated sound field, response functions, and vibration of the basic parts of piezoelectric disk transducers (the piezoelectric disk and the front layer) and an analysis of the influence of varying geometry and material parameters of the transducer on these quantities. The real transducers produce an amplitude modulated signal, completely attenuated after several periods Such behavior can be modeled by two close valued frequencies, producing a “beat” phenomenon, superposed on high damping. Agreement with measured normal velocity, shown in the fifth section, is encouraging
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