## 3 Results and Discussion
This was followed by repeated tests with FBG facing downwards. The forced vibration was detected although the sensor was flipped in the z-direction. frequencies in the same region as the FBG. The acoustic sources (phone and speaker) used in the experiments act as strong sources of forced vibrations. Thus, the theory and analysis of the sensor may depend on the forced vibrations coupled to the flexible beam and its stiffness. To support the above-mentioned concept, the response of the sensor can be analyzed using the principle of forced vibrations [16]. The acoustic force is assumed to be a time-varying tone of frequency \(\omega_{f}\) at a given amplitude \(F_{0}\), which can be described as:  
Figure 3:4  
The acoustic wave propagates through the base plate or beam 1, before the vibration is coupled to the sensing beam. The steady state solution [16] for such a forced vibration is given by \(x\) in the equation below. In this case, an undamped solution is considered as there was a continuous source of the forced vibrations (speaker) \(F\). Figure 3:5  
Figure 3:6  
Figure 3:7  
Figure 3:8  
where the moment of inertia can be described as:  
Figure 3:9  
and the stiffness is:  
Figure 2:0  
The following parameters were used for the simulation (\(E=\)70 \(\times\) 10\({}^{9}\) N/m\({}^{2}\), \(L=\) 0.47 m, \(b=\) 0.015 m and \(h=\) 0.0003 m). The value of \(f\) in \(\omega\) was taken as 10 Hz for all cases. The steady state waves for 150 Hz and 200 Hz were calculated for two different values of stiffness. As shown after substituting Equations (6) and (7) in Equation (5), \(h\) and \(L\) have a significant role to play in determining the amplitude of the steady state solution. Hence, a high stiffness results in low amplitude and this low amplitude does not allow for sufficient flexion of the FBG to sense the vibrations at high frequencies. Thus, a beam of low stiffness is more sensitive. These calculated data were compared with experimental results from case 5 (Figure 4). The experimental and simulated data were compared using the same number of samples (4000) in Figure 2:1 The experimental data shows the main peak at the speaker frequency. Smaller peaks are present at multiples of the main frequency, which shows the presence of other modes. For 150 Hz, the amplitude was sufficiently strong to generate oscillations (near 10 Hz) in the freely suspended end of the beam. The simulated data were obtained from the FFT of the steady state waves. The value of \(F_{0}\) has been normalized to obtain the closest fit (0.2 N). The simulation was conducted separately for 150 Hz and 200 Hz. To validate the assumption that the sensor can be used to detect vibrations caused by dynamic loads, the sensor has also been tested (case 6) on a three-phase motor (55 kW) of 1475 rpm. Without any prior preparation of surfaces, the sensing beam was propped on the motor housing (Figure 14) to measure the vibrations generated by the operation of the motor. Figure 15 shows the frequency spectrum of this sensor. A peak was identified at 24.8 Hz, which is due to the rotation of the motor at 1475 m\({}^{-1}\). Another peak was noted at 173.4 Hz due to the 7 th harmonic, which is often observed in motor vibration [17]. Multiple tests show that the peak at 173.4 Hz was present in all data. Figure 2:2 Sensor beam 2 used to detect vibrational frequencies of a motor, which was placed in contact with the motor housing. The contact area was near the location of the FBG. Figure 2:3 FFT data of experimental measurements at 150 Hz and 200 Hz, with sensing beam mounted on plate as shown in Figure 2:4 The simulated data are FFT results of the sinusoidal waves at 150 Hz and 200 Hz obtained from the analytical solution. \(F_{0}\) is normalized to 0.2 N in both simulations for the best fit of the amplitude. These results show that the FBG is an effective vibration sensor and is easy to implement. However, the key property determining the FBG capability to detect vibrations is its ability to expand and contract.