# Application of Generalized S-Transform in the Measurement of Dynamic Elastic Modulus
## 3 Dynamic Elastic Modulus Measurement of Rigid Materials  
In the measurement of dynamic elastic modulus of concrete and other hard materials, the measurement process is simplified as the measurement of resonance frequency of the specimen; that is, the calculation process of peak value of resonance power-spectrum. Due to the structural differences and the existence of gaps in each specimen, there are several resonance frequencies, which are called the resonance frequency bandwidth, while in mechanical measurements, the dynamic elastic modulus measurement only focuses on the resonance peak frequency in the low range (\(f_{\mathrm{max}}\leq 20\) kHz). From the definition of [PERSON]'s law: in the range of the elastic deformation, the deformation is proportional to the force, and its proportional coefficient is called elastic modulus. Depending on the direction of the applied force, the dynamic elastic modulus can be divided into transverse dynamic elastic modulus and longitudinal dynamic elastic modulus [34]. In construction engineering, the elastic modulus of concrete is mainly measured in transverse direction. In China's current \"Test Code for Hydraulic Concrete\", the dynamic elastic modulus test is formulated with reference to ASTM c215 [35; 36]. The calculation formula for the transverse dynamic elastic modulus of a square specimen is as follows:  
\[E_{\mathrm{d}}=0.9465\frac{Gf_{f}^{2}L^{3}T_{1}}{bh^{3}} \tag{10}\]  
where \(E_{\mathrm{d}}\) is the transverse dynamic elastic modulus in Pa; \(G\) is the mass of the specimen in g; \(f_{f}\) is the main resonance frequency of the specimen; \(L\), \(b\) and \(h\) correspond to the length, width and height of the specimen in mm; \(T_{1}\) is the correction factor obtained by the side length and Poisson's ratio of the specimen [37]. In actual measurement, Poisson's ratio \(\mu\) is generally set as an empirical value [38; 39]. Before conducting the measurement, the concrete sample is positioned on a stable platform, and polyurethane foam pads are strategically placed at its center to minimize the effects of vibrations. The impulser is directed towards the middle of the sample, while the transducer is located in front of the side edge. Both the impulser and transducer rods make light contact with the surface of the specimen, and couplant is applied to the tips of the rods. The impulser comprises a high-power loudspeaker enclosed in a rigid cylinder, while the transducer is constructed of piezoelectric ceramic. The structure of the measurement system is shown in Figure 3.  
The control program of the measurement system is implemented on the TMS320F280048 32-bit floating-point digital signal processor (DSP), or digital signal controller, which offers a favorable balance between computational speed and cost, enabling the widespread deployment of such embedded devices. In the program, a 16-bit pulse modulation signal from the ePWM module is used as the signal source; first, the clock signal from the system is divided to 10 KHz, and finally modifying the period register to change the frequency of the excitation signal of the impulser. The amplitude of the excitation signal is adjusted by a programmable op-amp, and then sent to the impulser through power amplification. In the measurement, when the rod of the impulser strikes the specimen, the concrete specimen is forced to vibrate. When the frequency of the excitation source is close to the intrinsic frequency of the specimen, resonance occurs and the amplitude of the forced vibration of the specimen reaches its peak. The mechanical wave generated by the forced vibration is converted into a voltage signal by the transducer, which is directly connected to DSP and amplified by the internal PGA for 24 times, then filtered out the high frequency (>20 kHz) noise by RC low-pass filter, and finally sent to ADC module.