## 3 Results
### Comparison of FEA and Analytical Models  
By comparing the predictions of our analytical model with the results of ANSYS FEA, we demonstrate our model's accuracy and reliability. To compare the predictions of the analytical method with those of ANSYS and identify possible disparities, both techniques were used to determine the FRFs for the same slitted beams with a layer of PZT attached. With this dual approach, the dynamic behaviour of the beams in different configurations can be thoroughly examined, not only validating the analytical model but possibly offering valuable information for future piezoelectric energy harvester development. A case study to investigate how various structural alterations affect the functionality of piezoelectric energy harvesters as predicted by our analytical model and ANSYS was carried out. By analyzing resonant frequencies, damping ratios, and total frequency response, we can determine how to best optimize the design for energy harvesting applications. Figure 3: Frequency response functions of regular and slitted beams with PZT layer with tip mass (ANSYS results). Figure 2: FEA results for slitted beams under a range of mode shapes. This work used PZT-5A, a piezoelectric material that possesses both high mechanical strength and a robust piezoelectric response. [PERSON]'s modulus for PZT-5A is 1.2035 \(\times\) 10\({}^{11}\) Pa, giving it the rigidity required for the energy harvesting application, guaranteeing that the composite beam has sufficient stiffness to endure applied loads. Its density, which adds to the total mass of the beam structure, is 7750 kg/m\({}^{3}\), which is a good match for the beam. PZT-5A's damping coefficient, 1.2433 \(\times\) 10\({}^{-5}\), is an essential factor in determining how much energy is dissipated during dynamic loading and guarantees uniform energy dissipation properties across the beam. This case study calculated the FRFs of both regular and slitted beams using an analytical approach. Utilizing the given parameters and attributes, we could analytically predict the beam's dynamic behaviour and resulting frequency response. This study thoroughly examined how structural changes affect the performance of piezoelectric energy harvesters and offers insightful advice for improving their design and use in various applications. Figure 4 shows the frequency response of two beam configurations, namely a regular beam and slitted beam, using analytical method. The \(x\)-axis shows the frequency, and the \(y\)-axis, the power magnitude. The natural frequency of the regular beam is roughly 28.9 Hz, and the slitted beam had a mass added at only one tip; there are two resonance peaks at 25.8 and 42.1 Hz, respectively, illustrating that the presence of a slit and tip mass cause the slitted beam to have two resonance frequencies and a sharp dip in response at about 35 Hz. The first peak shows the basic bending mode; the second peak comes from mode coupling brought about by the slits and tip mass design, therefore changing the beam's stiffness. By greatly extending the working frequency range, these peaks help the energy harvester to gather energy from several vibrational sources. Especially in situations with different vibrational frequencies, the larger frequency response and mode coupling clearly offer a benefit by improving the energy collecting efficiency. The slit and tip mass provide complicated dynamic behaviour, which improves performance at some frequencies but causes sharp dips or anti-resonances at others. In order to optimize the design of piezoelectric energy harvesters for specific applications, these results imply that structural adjustments, such as the addition of a slit and tip mass, might substantially affect their frequency response and performance. Specifically, slitted beams present a viable way to enhance dynamic response and energy collection. The beams' resonance properties can be significantly changed by adjusting slit parameters such as slit width and length.