## 4 Results and Discussion
### Natural Frequency Analysis  
Fast Fourier transformations (FTT) were utilized to obtain the damping characteristics and natural frequency of the system in the frequency domain. The piezoelectric accelerometers were attached at various depths (\(H\) = 20 cm, 25 cm, 30 cm, 35 cm, and 40 cm) to analyse the dynamic responses of the confined granular fill. The arrangement of the accelerometer is shown in Figure 3. The confined granular fill was subjected to vibrations of various frequencies (\(f\) = 10-50 Hz). The peak amplitudes of the acceleration, velocity, and displacement response at various excitation frequencies are shown in Table 3 for both vertical and transverse vibrations.  
The peak response (acceleration, velocity, and displacement) of the confined granular fill at the same position varied directly as the excitation frequency. A similar observation was made by [PERSON] et al. (2018) for the dynamic response of the subgrade at various exciting frequencies. The peak response of the confined granular at centre point is greater than the point to the right; due to the boundary effect. The geometric law was followed to understand the relationship between stress, strain, and displacement. The displacement spectrum for various excitation frequencies characterizes the vibrations generated at different depths.  
Figure 5 shows the displacement response spectrum of the system at various excitation frequencies (\(f\) = 10-50 Hz) at varying depths of the confined granular fill. This spectrum was obtained using the \(FFT\) analyser and accelerometers. The purpose of obtaining the system's response spectrum is to analyse the damping loss factor and natural frequency of the system. The displacement has a maximum at the natural frequency for the excitation frequencies of 10-30 Hz. In this case, the peak is at 31 Hz. For high frequencies, the peak displacement tends towards 40 Hz. At high frequencies, the confined system will behave as a rigid body, so the mass just follows the upper layer motion. Figure 5E depicts that the peak displacement of the vibration was obtained at the 40 cm thickness, whereas for low frequencies (10-30 Hz), the maximum displacement obtained was closer to 30 cm. The displacement response also represents the intensity of the vibration at a deeper depth to understand the effect of vibration on the \(PZT\) patches embedded at various thicknesses of the granular fill.  
\begin{table}
\begin{tabular}{c c c} \hline
**Properties** & **Symbol** & **Magnitude** \\ \hline Dielectric constant & \(\frac{c_{\mathrm{th}}^{\mathrm{T}}}{z_{\mathrm{e}}}\) & 3270 \\ Dielectric loss & tan\(\delta\) & 0.016 \\ Electromechanical coupling factor & \(k_{31}\) & 0.52 \\ Piezoelectric charge constant (pC N\({}^{-1}\)) & \(d_{31}\) & \(-275\) \\ Piezoelectric voltage constant (10\({}^{-3}\) Vm N\({}^{-1}\)) & \(g_{31}\) & \(-9.5\) \\ Young’s modulus (GPa) & Y & 61 \\ Unit weight (kNm\({}^{-3}\)) & Y & 7.2 \\ \hline \end{tabular}
\end{table}
Table 2: Properties of the \(PZT\) patch.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Input Excitation** & & & & & & \\
**Frequency (Hz)** & & & & & & \\ \hline
**Acceleration** & **Velocity** & **Displacement** & **Acceleration** & **Velocity** & **Displacement** \\
**(\(\mu\)m/s\({}^{2}\))** & **(\(\mu\)m/s)** & **(\(\mu\)m)** & **(\(\mu\)m/s\({}^{2}\))** & **(\(\mu\)m/s)** & **(\(\mu\)m)** \\ \hline
10 & 2579 & 52 & 0.537 & 1243 & 20 & 0.290 \\
20 & 913.26 & 0.759 & 0.007 & 77.128 & 1 & 0.008 \\
30 & 272,000 & 1441 & 5.90 & 815,000 & 4389 & 18.33 \\
40 & 103,000 & 233 & 0.93 & 57,920 & 299 & 1.19 \\
50 & 2102 & 46 & 0.023 & 1766 & 45 & 0.018 \\ \hline \end{tabular}
\end{table}
Table 3: Peak amplitude of vibration in vertical and transverse direction for confined granular fill.  
Figure 5: _Cont._  
Figure 5: _Cont_.  
The natural frequency and damping loss factor were analysed using the peak pick modal parameter method from the frequency response function (FRF) of the experimental data. Table 4 represents the damping loss factors at various excitation frequencies.