## 7 Conclusions and Discussion  
Based on the conducted research, it can be concluded that predicting the effectiveness of piezoelectric transducers based on simulations of the elements to which they are attached is possible with satisfactory accuracy. Simulation allows estimating the resonant frequency and vibration amplitude of the studied element. However, computer simulations might involve idealized operating conditions and a lack of consideration for the influence of vibrations and noises not directly originating from the simulated object, which can be significant beyond the resonant zone of the system under consideration. In experimental studies, it is difficult to ensure ideal mounting conditions for the tested system, leading to differences in obtained results compared to simulation outcomes for real systems. To mitigate these discrepancies, it is essential to maximize the rigidity of laboratory setup elements, the motor, and mounting components since their vibrations adversely affect the obtained results. Additionally, attention should be paid to ensuring that the resonance frequencies of the laboratory setup elements are as far removed from the measurement range as possible to minimize additional disturbances. There is also a possibility to refine the material, load, and boundary conditions in the simulation to better reflect the real experimental setup. However, these factors are challenging to directly simulate without precise information on material properties and boundary conditions that may vary in real-world experiments. Simulation parameters can be adjusted based on the material properties, loads, and boundary conditions observed in the experimental setup. By doing  
Figure 9: Comparison between experimental and simulation results. so, discrepancies can be reduced and bring the simulation results closer to the actual experimental outcomes. The assumption of homogeneity in the MFC (macro-fiber composite) used in the proposed mathematical framework simplifies the analysis, but it is important to note that this assumption may not fully represent real-world conditions. Potential stress concentrations due to fibre distribution and the interactions at the interfaces, particularly between the epoxy and PZT layers, can significantly affect the mechanical and electrical performance of the energy harvesting system. These factors could lead to localized variations in material properties, which may impact the overall efficiency of the system. While these effects have not been incorporated into the current model, they are recognized as important considerations for further research. Future studies will aim to refine the model by incorporating these inhomogeneities, allowing for a more accurate representation of the system's behaviour under real-world operating conditions. The current study primarily focuses on the operational range of 7-15 Hz for the MFC piezoelectric transducers, based on the specific characteristics of the system and the intended application, which involves vibrations induced by moving vehicles in a wheeled motion. While this frequency range provides valuable insights into the system's performance, it is recognized that these results may have limited applicability to a wider range of operational conditions, including higher or lower frequencies and varying load conditions. The proposed mathematical framework, while effective within this specific range, can be extended to other frequencies and load conditions through further research and experimental validation. Future work will aim to explore these broader operating conditions to improve the general applicability and versatility of the model in different engineering applications. Developed mathematical models integrated with computer-aided design tools and utilizing the finite element method to determine the dynamic characteristics of the studied system enable effective support in its design and manufacturing, aimed at maximizing energy efficiency under assumed operational parameters, especially in terms of excitation frequency.