## 3 Results and Discussion  
For different glucose concentrations (0 wt.%, 5 wt.%, 10 wt.% and 15 wt.%), the average spectrum curves of the transmission loss obtained by three experiments are as shown in Figure 7a, which shows the increase of resonant frequency with the increase of glucose concentration. Gaussian fitting was used to get the smooth curve (Figure 7b), by which the value of resonant frequency can be identified as shown in Table 2. Analytical results are also listed in Table 2 for comparison. The relative errors between analytical results and experimental results varied from 0.09% to 1.89%, which showed quite close results. For practical use, it is better to carry out the calibration test for the practical structure and analyte concentrations, and the analytical values are just used as a design reference. Another obvious phenomenon observed is that the Q factor of the proposed sensing device is quite low, which may come from the surface roughness of the 3D-printed structure [22; 23]. Another reason for the low Q factor is the imperfect sound reflection of the 3D-printed walls, which means that losses of acoustic power occur during the acoustic propagation. This work focuses on the location of the peak but not the magnitude of the peak, so the effect of surface roughness on the Q factor is not discussed in detail. In further studies, we will try to bring the analytical and numerical model much closer to the experimental system. We will further try to develop a more consummate experimental system for more reliable results of high quality. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Glucose Concentration** & **0 wt.\%** & **5 wt.\%** & **10 wt.\%** & **15 wt.\%** \\ \hline Analytical results (A) & 328.5 Hz & 332.1 Hz & 336.6 Hz & 342.1 Hz \\ Experiment results (B) & 326.5 Hz & 332.9 Hz & 336.3 Hz & 348.7 Hz \\ Standard deviations (C) & 1.13 & 1.21 & 1.15 & 1.32 \\ Relative error (\(|\)A \(-\) B/B \(*\) 100\%) & 0.61\% & 0.24\% & 0.09\% & 1.89\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Resonant frequencies obtained by analytical and experiment approaches. Figure 6: Schematic of the measurement principle and the experiment setup. Figure 7: Spectrum of transmission loss obtained by experiment. (**a**) Original results (average curves); (**b**) fitting results. In research on the resonant-type liquid density sensor [24], the absolute sensitivity of a sensor is defined as  
\[S_{a}=\frac{\partial f}{\partial\rho_{s}} \tag{10}\]  
and the relative sensitivity is defined as  
\[S_{r}=\frac{\partial f}{\partial\rho_{s}}\frac{\rho_{s0}}{f_{0}} \tag{11}\]  
where \(\partial f\) is the change of resonant frequency, \(\partial\rho_{s}\) is the density variation of test solution, \(f_{0}\) and \(\rho_{s0}\) are the fiducial values. For our proposed sensing devices, the absolute sensitivity obtained by experimental results is \(S_{a}=0.29\;\mathrm{Hz}/(\mathrm{kg}/\mathrm{m}^{3})\) near the resonant frequency 328.5 Hz, and the relative sensitivity is \(S_{r}=0.88\). For the cantilevered sensors, which are based on the added mass effect of the test solution, the relative sensitivity is theoretically lower than 0.5 [25]. For practical biosensor applications, a test sample of about 10.5 mL in the present structure design is a rather large amount. For a tiny bit of solution, Table 3 shows the estimated results in the case of the test sample, obtained by analytical results. When the amount of the test sample is 0.1 mL after device miniaturization, the resonant frequency goes to 3377.4 Hz and the corresponding absolute sensitivity is \(3.02\;\mathrm{Hz}/(\mathrm{kg}/\mathrm{m}^{3})\), which is higher than the previous micro-cantilever sensors at a similar frequency range. The frequency is relatively low, which locates it in the feasible range of conventional acoustic pressure sensors. When the amount of the test sample goes to 0.01 mL, the frequency goes to 28.5 kHz, but meanwhile, the sensitivity is about two times higher than that of the trapezoid cantilever beam using the 2 nd mode.