# Trapezoidal Cantilevers with Irregular Cross-Sections for Energy Harvesting Systems
## 4 Experimental Investigation of Modified Trapezoidal Cantilevers  
The experimental investigation was performed in order to validate the results of numerical investigation. Electrical and mechanical characteristics of the prototypes were measured and analyzed. Prototype cantilevers with the optimal dimensions were made (Figure 13). The material properties are given in Table 2. Moreover, plastic clamping structures were designed and manufactured for each cantilever. The resonant frequency of the first bending mode of the prototypes was measured employing the vibrometer POLYTEC PSV 500 (Polytec, Dexter St., MI, USA). The shapes of the vibrations are presented in Figure 14, and resonant frequencies are given in Table 3.  
It can be noticed that the values of the measured resonant frequencies and natural frequencies obtained during the numerical investigation are similar; however, the highest difference is up to  
Figure 14: Shapes of the cantilever vibrations: (**a**) conventional cantilever beam, \(\omega_{a}\) 70.05 Hz; (**b**) cantilever with cylindrical gaps, \(\omega_{b}\) 79.68 Hz; (**c**) cantilever with trapezoidal gaps, \(\omega_{c}\) 101.25 Hz. Figure 12: Comparison of the cantilever beams: (**1**) voltage density; (**2**) current density; (**3**) power density; (**a**) conventional cantilever; (**b**) cantilever with cylindrical gaps; (**c**) cantilever with trapezoidal gaps. Figure 13: Prototypes of the cantilevers: (**a**) view of the assembled system; (**b**) front view; (**c**) side view. 17.69%. Furthermore, it can be seen that the measured resonant frequencies have lower values than natural frequencies. The errors mainly occur because of the simplified numerical model and high vibration damping in the plastic clamping frame used for prototype clamping. Summarizing this part of the investigation, it can be concluded that the calculated natural frequencies and measured resonant frequencies have acceptable differences. The resonant frequencies are lower in comparison with natural ones; therefore, the lowest resonant frequencies could have a positive influence on the electrical characteristics of the cantilevers. Lower resonant frequency ensures greater cantilever tip displacements and results in a higher strain value and better electrical characteristics. The next step of experimental study was to investigate the cantilever response to the different excitations of the base and measure electrical output densities versus resistance load. An experimental setup was built for this purpose. The principle scheme of the setup is given in Figure 15. It consists of two optical displacement sensors used for displacement control at the base and tip of the cantilever; a special plastic clamping frame was used to attach the cantilever to an electromagnetic shaker; a computer was used for displacement data recording and analysis; a function generator and a power amplifier were employed for driving the electromagnetic shaker; a variable resistive load was used as the electrical load of the cantilever; a micro current probe was used for low current measurements; an oscilloscope was used for control and recording of the current and voltage values. The measured cantilever at base excitation response characteristics show the dependency of the ratio \(z_{Base}/L_{B}\) to the tip displacement of each cantilever (Figure 16a). Displacements of the base and tip were measured by optical sensors (Figure 15). The resistance load during this stage of the investigation was equal to the input resistance of the oscilloscope; i.e., 10 M\(\Omega\), so displacement characteristics were measured in open circuit conditions. Excitation frequencies were set to the resonant frequency for each cantilever. The characteristics of the open circuit voltage density versus tip displacement were investigated at the same conditions for each cantilever. The results are given in Figure 16b. The ratio between base displacement and the width of the base was chosen as a critical parameter in order to compare cantilevers.