Modal Analysis Computational Analysis

72 The 3 rd and 4 th vibration modes happen at about the same frequency as the one with longer length, at 3168.9 Hz and 6890.6 Hz respectively. The magnitudes of the stress are also similar between the two structures, which show that the width of the free- standing structure does not significantly improve or reduce the resonant frequency. The simulation results verify that, by attaching a proof mass at the tip of a cantilever, the resonant frequency can be reduced while increasing the magnitude of stress induced on the structure, as shown in Figure 3-28. When a tungsten proof mass with dimensions of 2 mm × 9 mm × 1 mm and weight 0.35 g is attached, the first three vibration modes were reduced to 54.2 Hz, 837.2 Hz and 1386 Hz respectively. The stress distributions are similar to those without proof mass at fundamental resonant frequency but the magnitude of the induced stress is greatly increased. It is interesting to notice that, at 2 nd resonant vibration mode, the stress distribution is concentrated on the flat beam and almost no stress is developed on the S -beam. This shows that in order to optimise electrical energy generation, piezoelectric material must be printed along the length of the cantilever and not just concentrated on the end of the clamped area. The effect of proof mass distribution on the cantilever is significant at higher frequency mode as shown in Figure 3-30. In the simulation, a similar mass of 0.35 g but different dimensions of 2 mm × 3 mm × 3 mm was used. The fundamental resonant frequency of the cantilever with the distribution of mass focused on the centre of the cantilever tip is slightly lower than that spread across the width of the cantilever, at 53.1 Hz. The resonant frequency differences between these two settings become obvious when excited to higher frequency modes. The 2 nd and 3 rd resonant frequency modes happen at 662.5 Hz and 1034 Hz respectively. The simulation results show that, at fundamental resonant frequency, the stress distribution of a cantilever is concentrated on the anchor area between the base and the free-standing structure, therefore, the structure has to be reinforced in this area. In order to generate optimum electrical output, piezoelectric materials have to be present through the length of the cantilever, as the maximum stress distribution is more toward the end of the cantilever at higher resonant frequency modes. 73 Figure 3-24: Contour plot of stress distribution of a cantilever with dimension as shown in Table 3-3 under fundamental a, 2 nd order b, 3 rd order c and 4 th order d vibration modes. Figure 3-25: Diagram of maximum stress as a function of base excitation frequency for a cantilever having the dimension as shown in Table 3-3 Maxi m u m Str es s Pa Frequency Hz a b c d 74 Figure 3-26: Contour plot of stress distribution of a cantilever beam having a width of 18 mm under fundamental a, 2 nd order b, 3 rd order c and 4 th order d vibration modes. Figure 3-27: Diagram of maximum stress as a function of base excitation frequency for a cantilever having a width of 18 mm. Maxi m u m Str es s Pa Frequency Hz c d a b 75 Figure 3-28: Cantilever with full-width-coverage tungsten proof mass; contour plot of stress distribution of a cantilever beam attached with the proof mass for fundamental a, 2 nd order b and 3 rd order c vibration modes. Figure 3-29: Diagram of maximum stress as a function of base excitation frequency for a cantilever attached with full-width-coverage proof mass. Maxi m u m Str es s Pa Frequency Hz c a b 76 Figure 3-30: Contour plot of stress distribution of a cantilever beam attached with full-width- coverage proof mass for fundamental a, 2 nd order b, 3 rd order c and 4 th order d vibration modes. Figure 3-31: Diagram of maximum stress as a function of base excitation frequency for a cantilever attached with full-width-coverage proof mass. Maxi m u m Str es s Pa Frequency Hz a b c 77

3.6.2 Comparison with Calculation Results

The simulation results of the multilayer composite structure are compared with a single layer PZT with similar thickness of 100 µm. Figure 3-32 a verifies that the natural frequency of a cantilever is inversely proportional to the square of the cantilever length. The flexural rigidity of the beam changes as AgPd electrodes are added. A change in stiffness directly affects the frequency of the beam’s vibrations. The natural frequency difference between the composite structure and single layer structure becomes less significant when the length of the structure increases. The calculation results, based on a composite structure according to equation 3-14, shows a slight difference compared to the simulation results. For a cantilever length of 5 mm, the calculated natural frequency is 2.19 kHz, while the simulated natural frequencies for the composite and single structure are 2.48 kHz and 1.71 kHz. The stress, deflection and acceleration on the tip of the cantilever are directional responses as a resultant from the base excitation, as shown in Figure 3-23. The y- direction indicates translation motion while x-direction indicates longitudinal elongation motion. As the effect on z-direction is minimal compared to x- and y- directions it is therefore ignored. Figure 3-32 b shows that the acceleration at the tip of the cantilever for both composite structure and single material structures are almost similar. This is because the resonant frequency of the composite cantilever increases while the deflection decreases compared to a single material structure, and therefore produces a constant acceleration. Both the composite and single material structure are accelerated by a factor of about 200 compared to their base excitation levels, for a cantilever length of 5 mm. The acceleration level decreases to a factor of 130 when the cantilever length increases to 20 mm. The difference between calculation and simulation results is significant for a shorter cantilever. This is because the calculation results are based on a straight and flat cantilever model, whereas the simulation results are based on elevated cantilever model. Hence, at a shorter length the S -beam of the simulation model plays a significant role in determining the tip acceleration, which is not considered in the theoretical model. 78 The deflection of the cantilever can be estimated from the y -direction deformation from the ANSYS simulation results. The deflection difference between composite and single material structures is significant when the length of the structure increases. These simulation results verify the fact that as the length of a cantilever structure increases, the resonant frequency decreases. Once the resonant frequency is reduced, the cantilever would experience a greater magnitude of deflection at a constant acceleration level. For a cantilever of length 20 mm, a single material structure produces as much as three times the magnitude of deflection produced by a composite structure as shown in Figure 3-32 c. This shows that the electrode layers which are stiffer than PZT play an important role in reducing the deformation of the structure when excited to its resonance. As both of the structures were excited with the same excitation level, the maximum stresses on x -direction for both structures are similar, as shown in Figure 3-32 d. These simulation results show that a material with higher elastic modulus can be added on the outer layer of the composite structure in order to protect the more fragile and brittle piezoelectric material from overstress at the centre of the composite structure, since the stress increases with the distance from the neutral axis to the centroid of the material. From the ANSYS simulation results for a single material structure consists of PZT and a multilayer structure consists of PZT and AgPd electrodes, it can be concluded that the natural frequency and the maximum deflection of a cantilever structure depends on the elasticity of the individual layer. The theoretical calculation results for a composite structure are in a good agreement with the ANSYS simulation results for a composite structure. This verifies that the model developed in section 3.4 is reasonable good to be used to estimate the performance of a free-standing cantilever, therefore will be used in the following chapter.