Resonant Ultrasound Spectroscopy

Pablo Nieto, winter semester 2019/20

Resonant Ultrasound Spectroscopy (RUS) uses normal modes of elastic bodies to estimate material properties such as elastic moduli E. This technique that fills an experimental gap between low-frequency stress-strain methods and ultrasonic time-delay methods. The resonance spectra of the sample are obtained by inducing it to vibration and measuring the response. From these spectra and through a mathematical model, it is possible to determine first the normal-mode frequencies and from these the material elastic properties. RUS is commonly used in geology in order to study rock physics.^[2]

Fundamentals of RUS

RUS is based in the relation between elastic parameters of a body and its normal-modes of vibration. One can analytically calculate the normal-modes in terms of the elastic parameters, size and shape. Nevertheless, for this method it is the inverse calculus that is of interest. We desire to estimate elastic parameters by the measured resonance frequencies.

During an experiment, the natural frequencies of the sample are calculated by measuring the amplitude of the sample vibration given a frequency-variant input signal. Once the desired frequency sweep is completed, the measure is yet not finished. A mathematical model is necessary in order to obtain the desired elastic parameters.

The evaluation of the data obtained by the measurement is divided in two necessary calculations. First, the forward problem, the formulation of the analytical problem that defines the sample normal-modes of vibration based on its elastic parameters, size and shape. Second, the inverse problem, the nonlinear algorithm that estimates the elastic parameters based on the measured natural frequencies.^[1]

The forward problem

The first step of the problem consists on applying an energy minimization method in order to find the relation between displacements and natural frequencies. The method used is the Lagrangian minimization. The vibration modes obtained at this point are linearly independent, but necessarily not normal between them. In this step, the complexity of the problem is significantly reduced given the condition that the sample surface is free of loads.

In the second step the goal is to transform the obtained vibration methods to normal modes. The method used is called Rayleigh-Ritz variational method and its purpose is to redefine the problem as a matrix diagonalization problem. The new vibration modes obtained are normal modes of vibration.^[1]

The inverse problem

The inverse calculation of the forward problem solution has not an analytical solution and thus a computational iterative method is necessary. Specifically developed for RUS computer software is used to find the solution in this case.

The problem is solved through the minimization of an error function that makes an initial approximation of the elastic parameters, compares it to the measured sample results and redefines afterwards the elastic parameters. This process is carried iteratively until the expected and actual results are both within a given tolerance interval. This part of the solution is the most time consuming due to the number of operations needed until the problem converges.^[1]

Measurement method

The final objective of a measurement is to estimate the elastic parameters of the sample. Nevertheless, elastic parameters are approximated mathematically in means of the method described in the previous section. Therefore, the goal of the measurement is to determine the resonance frequencies of the sample, that will constitute afterwards the input parameter for the algorithm.

Natural frequencies estimation

The sample is held between two piezoelectric transducers, one working as a transmitter and the other as a receiver. As the measurement starts, the transmitter generates a vibration of constant amplitude and variable frequency over time. At the same time, the receiver reads the signal obtained at the other end of the sample. Once the excitation frequency sweep is completely performed, resonance frequencies can be estimated through the study of the signal amplitude captured by the receiver.

Two parameters can be estimated for each resonance. First, natural frequencies fn, which are approximated by the resonance frequencies. The lighter the sample loading by the transducers is, the more accurate is the approximation. For loads slightly greater than the sample weight, accuracy is around 10%. Second, a parameter called quality factor Q is also calculated by the division between fn and the peak width at half power points. This parameter gives information about elastic dissipation.

Measuring resonance frequencies has a major benefit. Strong coupling is not necessary, since the sample acts as a natural amplifier with gain Q. This gain normally has an order of magnitude of 103. This means that the excitation energy can be very low, loading the sample extremely little.^[1]

Elastic parameters determination

In Figure 2, a representation of the whole logical process behind the RUS method is shown. First, the sample is excited in the test rig in order to measure its natural frequencies. Second, the inverse problem is solved in order to find a set of elastic constants that suits the result obtained in the first step.

Samples

Sample geometry plays a very important part in the solution of the forward problem, since free vibration directly depend on geometry. For this reason, normally rectangular parallelepiped, spherical and cylindrical samples have been used. The more complex the sample geometry is, the heavier the mathematical calculation will be. Rectangular parallelepiped samples are the most commonly used. The reason is closely related to the interest of loading the sample as little as possible. Edges provide elastically weak anchoring points, producing little loadings. In this way, point contact is granted, fulfilling requirement of free of load surfaces.

As for sample size, the values vary substantially. With a minimal size around the hundreds of microns, samples are normally between 1 mm and 1 cm. Samples must also be free of internal defects and have no certain grain orientation. Surfaces must be polished and parallel. Finally, once prepared, their density is precisely calculated.^[1]

Literature

1. Mainard, Julian: Resonant Ultrasound Spectroscopy. Physics Today 49, 1, 26 (1996).
2. Zadler, B., Le Rousseau, J., Scales, J., Smith, M.: Resonant Ultrasound Spectroscopy: theory and application. Geophys. J. Int. (2004) 156, 154–169