# The Sonic Resonance Method and the Impulse Excitation Technique: A Comparison Study
## 1 Introduction
The driven vibrations are the base of the sonic resonance method (SRM). 2. Free vibrations excited by the mechanical impulse. The free vibrations are the base of the impulse excitation technique (IET). Historically, the first technique was SRM [10; 12]. The equipment used consists of a tunable oscillator with an amplifier connected to an exciter as the source of driven vibrations. The sample is suspended in its nodal points, vibrations are registered using a sensor connected to a preamplifier, and the sensor's output signal is observed. The frequency at which the output signal reaches the maximum value is the resonant frequency. This method can be automated, for example, if the RC oscillator works in a sweeping regime [13; 14] or the sample is permanently kept in the resonant vibration with the help of a voltage-controlled oscillator in a feedback loop which contains the sensor [15]. [PERSON] et al. [16] presented an apparatus to measure elastic properties and the internal friction of materials. Their apparatus excited the sample fixed in the nodal points of the fundamental vibration mode via a light mechanical impact. The response includes many transient frequencies that rapidly die out, and thus there is a natural filtering action leaving the main fundamental resonant vibrations as the only detected signal. Then the apparatus performed a software-based analysis of the resulting vibration, i.e., IET was used. The resonant frequency of the sample was determined and Young's modulus was calculated. Similar techniques were also described in [11; 17; 18; 19; 20]. The sample vibration is captured by a microphone or a piezo electrical sensor and subsequently analyzed using the fast Fourier transformation (FFT). The result of the FFT is a frequency spectrum of the sample vibrations, where the resonant frequency of the fundamental mode of the vibrations can be found. Another way to extract the resonant frequency from the measured signal is based on measurement of the duration of a selected number of cycles by counting zero-crossings and determining the cycle period. Its duration is directly proportional to the reciprocal value of the resonant frequency. The free vibrations are naturally suppressed by internal processes in the sample. Therefore, a coefficient of the internal damping (internal friction) can be determined. The mechanical impulse can be realized manually with a hammer if the measurement is conducted at room temperature. The impulse can be generated at an elevated temperature by steel or ceramic balls which fall on the sample, and by an electromagnetic impact. In high-temperature measurements, IET can be designed as non-contact, whereas SRM needs two thin wire suspensions located at the antinodal points or at the ends of the sample. These suspensions are a drawback of SRM--they are often the source of spurious resonances, and their strength is limited at high temperatures. Consequently, IET is more reliable at high temperatures. Both methods, SRM and IET, have the same theoretical basis, which is an equation of the flexural vibration of the rectangular beam or rod with a circular cross-section, and their material is homogenous and isotropic [10,21]. A derivation of the equation of the flexural vibration can be found, for example, in [9,10,21,22,23]. The relationship for Young's modulus \(E\) derived from this simplified equation has a form:  
\[E=\left(K\frac{l^{2}f_{0}}{d}\right)^{2}\rho T, \tag{1}\]  
where \(f_{0}\) is the resonant frequency of the fundamental mode, \(\rho\) is the material bulk density, \(l\) is the length of the sample, and \(d\) is the diameter of the cylindrical sample or thickness of the prismatic sample in the direction of vibration. If \(l/d>20\), the correction coefficient \(T=1\). If \(l/d<20\), the influence of rotary inertia and shear forces have to be taken into account to obtain correct values of Young's modulus. Two ways are possible: (1) The use of the very complex frequency equation for the so-called Timoshenko beam.