PEH system with internal resonance perform better for the system parametersν = 0.001, Γ = 1 at Rl= 1 kΩ.
• It may be further noted that by adjusting the location of the attached mass the frequency of the system can be tuned and other internal resonance conditions can be utilized for harvesting more energy.
7.3.2 PEH under combination parametric and 1:3 internal resonance
In this section, specific conclusions related to an energy harvester consisting of a harmonically base excited vertical cantilever beam with piezoelectric path and at- tached mass excited at a frequency equal to sum of first two modal frequencies is highlighted. The first modal frequencies are nearly in the ratio of 1:3.
• The instability region is found to vary between the nondimensional frequencyφrange of 3.95 to 4.1, with increase in damping and decrease in load resistance. The critical amplitude of base excitation below which the harvester is ineffective is determined and found to increase with damping and load resistance. Hence to harvest energy in a wide range of frequency one should use low damping and low load resistance. It may be noted that the trivial state looses its stability due to Hopf bifurcation. But in case of principal parametric resonance it loses its stability by pitchfork bifurcation.
• From the frequency response plots its is evident that more voltage can be generated by operating the harvester at a frequency greater than sum of the first two mode frequencies (that is when the system is positively tuned). In the studied system a maximum voltage of 20 V and power of 200 mW are obtained. One may increase these values by suitably changing the system parameters.
• Both saddle node and Hopf bifurcations are observed in the multibranched nontrivial state.
• It is observed that the voltage and power increase significantly asRl increases which shows that the load resistance plays an important role in achieving an optimum power output. Here at φ= 4.5, a voltage of 0.77, 3.8, 7.77 and 15.3 V with corre- sponding power of 5.9, 29.5, 58.9 and 117 mW is achievable as the load resistance increases fromRl= 0.1 kΩ to Rl = 2 kΩ.
• With increase in damping the nontrivial states at a frequency below the sum of two modal frequencies (negatively detuned) disappears. Hence no energy can be generated in the region of frequency.
• With decrease in amplitude of base excitation the PEH system becomes ineffective for φ < 4. It is operative only for φ > 4. Hence to generate voltage for a wider range of frequency the amplitude of base excitation is moderately high.
• A significant change in the topology of the nontrivial branch of solution is observed when the amplitude of base excitation and the damping are very high (Γ = 10, ν= 0.01). Power is generated only for narrow bandwidth of frequency. Hence one should properly choose the system parameters to obtain the required voltage and power.
• It has been shown that by suitably changing the mass ratio µ, length of the beam, position of the attached mass β, Youngs modulus of the beam and thickness of the beam, one may obtain 1:3 internal resonance condition. It is clearly observed that for 0.2 < β <0.27 and 0.3 < µ < 0.4 one may find the first two modal frequencies in the ratio of 1:3 giving rise to internal resonance.
7.3.3 PEH under parametric excitation: A theoretical and experimental investigation
In this section, the specific conclusions obtained for the experimentally studied base excited PEH system is presented.
• A slider-crank mechanism is developed for providing base excitation to the PEH system which has a variable stroke length ls between 13 mm to 22 mm. Here a maximum frequency of 25 Hz can be obtained to excite the harvester near twice the principal parametric resonance condition.
• The experimentally obtained damping ratio and the fundamental frequency of the system are found to be 0.0134 and 3.6 Hz, respectively. These values are found to be in good agreement with the analytical results. Hence one may use the developed analytical formulation for determining these parameters for any other sets of system parameters.
• The supercritical pitchfork bifurcation is observed at 6.76 Hz, which is below twice the natural frequency of the system. At this point the onset of harvesting takes place. With increase in the external frequency of excitation the output voltage
increases up to the saddle node bifurcation point, beyond which one may expect a catastrophic failure due to jump down phenomenon. Hence one may operate the PEH system between the supercritical and subcritical pitchfork bifurcation point which is well below the saddle node bifurcation point.
• One may achieve a maximum power of nearly 22.7 mW around a frequency of excitation of 7.25 Hz (near saddle node bifurcation point) for external load resistance of 100 kΩ. This experimental results closely matches with the analytical findings.
• Near the theoretically observed saddle node bifurcation point, few hairline cracks are observed in the piezoelectric patch (PZT-5H) after few cycles of operation. This is because the patch is placed near to the base (spanned from 5.7 mm to 10.7 mm length from the base), where the bending moment is maximum.
• It is observed that cracks are developed in the beam near to its fixed end due to maximum stress, which leads to the breaking of the beam. Hence to improve the life of the PEH system, it should be operated at a frequency lower than the saddle node frequency. Also design modification may be carried out to strengthen the base part by using torsional springs and stoppers.
• It is observed that the load resistance influences the power output and at a certain value ofRl, which is 800 kΩ in the present case, the output power gets maximum.
The average value of voltage amplitude is found to be 54.72 V.
• As the position of attached mass increases, the threshold frequency at which beam starts transverse vibration reduces. By suitably positioning the mass, one may adjust the natural frequency of the system. Hence the resonance condition can also be tuned to regulate the output voltage. Here it may be noted that magnetic materials are used as attached mass, so they can be shifted to suitable position easily.
• The normalized power density (power/(volume×acceleration2)) for the present PEH system is found to be 15.9.
• The PEH system with the MFC patch is also investigated experimentally under the parametric excitation. It is observed that the maximum power of 2.25 µW is achieved for system parameters m = 36 gm, β = 0.233 and load resistance of 500 kΩ. Also the frequency of excitation decreases asm increases.
• As the attached mass varies from 9 gm to 45 gm while keeping its position and load
the threshold frequency at which supercritical pitchfork vibration occurs decreases from 5.2 Hz to 4.5 Hz (13.5% decrease).
• An internal resonance of 1:2 is observed in FFT of voltage time response, Which enhances the output voltage and power of the PEH system.
• As the position of the attached massβ increases, a percentage increase of 58.3 % is observed in the output voltage.
• As the MFC patch positions bring nearer to the fixed end of the beam the output power increases to nearly 25 µW.
• When the PEH system is excited at the frequency near to the sum of the first two natural frequencies (4.3 Hz + 8.3 Hz ≈13 Hz) two major frequency peaks are observed with having nearly 1:3 frequency ratio.
• For an external frequency more than 13 Hz, the FFT of voltage time response for the PEH system shows multiple frequency components. Here, due to modal interactions between several participating modes, synchronous and sub-synchronous frequencies are observed which are generally observed in rotor bearing system.
7.3.4 Dynamics of galloping based piezoelectric energy har- vester
Here the specific conclusions related to the galloping based PEH system (GPEH) is described.
• Reduction of 84.6% is observed when the position of the bluff bodyβ is kept at 0.88 as compared to the beam without a bluff body.
• PEH with D-shaped bluff body as the wind speed increases from 4 m/s to 5 m/s, a 39% increase in voltage, and a 90% increase in output power is observed at load resistance of 300 kΩ. Hence by adjusting the position of the bluff body, one may control the resonance conditions and hence the output voltage.
• As the bluff body placed close to the fixed end, the output voltage increases for PEH with D-shaped bluff body. The critical speed at which the PEH system starts to vibrate and generate power is known as the onset speed of galloping. It is observed that the galloping speed does not vary much by changing the position of the bluff body. At higher speed (U = 7 m/s), a variation of 43.8% in voltage is observed.
Hence though the onset of galloping does not depend on the positioning of the bluff body, the voltage can be controlled by varying the position of the bluff body at a higher speed.
• An optimal load resistance is observed at which the output power is found to be maximum.
• GPEH with the triangular-shaped bluff body (GPEH-TSB), as β decreases, the voltage variation with load resistance follows a similar trajectory up to 600 kΩ.
Beyond Rl>600 kΩ, the GPEH with lower β performs better.
• For GPEH-TSB, the onset of galloping changes for different positions of the bluff body.
• The onset speed of galloping is invariant for GPEH with D-shaped bluff body but varies for GPEH with the triangular shaped bluff body with respect to the bluff body position along the axis of the vertical cantilever beam.
• GPEH-DSB performs better than that of GPEH-TSB before 800 kΩ. The maximum voltage variation of 38.6% at 500 kΩ is observed.
• It is observed that the onset of galloping wind speed is nearly similar at higher β value for both GPEH-DSB and GPEH-TSB systems, but it is found to be lower for GPEH-TSB as compared to GPEH-DSB for lowerβ values.
• The ceramic PZT-5H piezoelectric patch, which is considered for the analysis, is found to be safe in the wind tunnel experiments, even in higher wind speeds beyond the 12 m/s. Here the piezoelectric patch (of length 5 cm) is placed at a distance of 3.3 cm from the fixed end.
• The maximum power of 2.95 mW is achieved with the triangular-shaped bluff body (GPEH-TSB) for parametersβ = 0.88,U = 7 m/s andRl= 1000 kΩ. Also in case of GPEH with time varying wind flow condition a maximum power of 13 mW can be achieved with system parametersβ = 0.75, U = 7 m/s, Rl = 700 kΩ and u0 = 0.1U.