In this research work, four nonlinear problems related to piezoelectric energy har- vester systems have been investigated. In the first work, a PEH system that con- sists of a vertical cantilever beam with an attached mass at an arbitrary position and piezoelectric patches covering the whole length of the beam is studied. The attached mass is placed in such a way that the second mode frequency is three times the first mode frequency giving rise to 1:3 internal resonance condition. Due to large trans- verse deformation and rotary inertia of the attached mass, the system exhibits cubic
and inertial nonlinearities. Here, the base excitation is given in such a way that the system undergoes principal parametric resonance conditions, i.e., the frequency of external excitation is twice the natural frequency of the system. The steady state output voltage and power of the system are obtained for various system parameters such as load resistance, amplitude and frequency of excitation and damping ratio.
The parametric instability regions with different load resistance and damping ratios are also studied. The critical system parameters below which the system will always remain stable are determined. So the system has to be operated above this critical value for harvesting energy. The time and frequency response plots are obtained for various system parameters which will help the designer to obtain the amplitude and range of frequencies of external excitation where the PEH system can operate.
In the second work, a similar system is studied under combination parametric res- onance of sum type. Here, the excitation frequency is nearly equal to the sum of the first two modal frequencies of the system. The fixed point response and the stability of the system are investigated. The instability regions for different system parameters are analysed. The nonlinear frequency responses of the system are stud- ied for different system parameters like load resistance, amplitude and frequency of excitation and damping ratio. Unlike the operating range of frequencies for the harvesters available in the literature here, by employing 1:3 internal resonance and principal parametric and combination parametric resonance, it has been shown that the harvester can operate in a wide range of frequency which is available in ambient condition.
It has been demonstrated that there exists a range of system parameters such as length and thickness of the beam, position of the attached mass, mass ratio and Young’s modulus of the substrate and piezoelectric patch, which can be varied to obtain 1:3 internal resonance.
In the third problem, an experimental investigation has been attempted to under- stand the system dynamics and challenges posed by the system due to large trans- verse vibrations. A slider-crank mechanism based shaker is designed and developed in-house, to provide harmonic base excitation. The system is excited at principal parametric resonance condition and the output voltage has been determined using oscilloscope. These experimental results have been compared with the theoretical findings and they are found to be in good agreement.
In the fourth work galloping based harvester is investigated both theoretically and
experimentally. Here, a vertical cantilever beam with unimorph PZT patch and an attached bluff body near its free end is considered as the PEH system. Taking both D type and triangular bluff bodies, experiments are performed in the wind tunnel to study voltage time response across a load resistance under uniform flow condition. The effect of the position of bluff bodies on voltage and power output are also studied. Theoretical investigation have been carried out considering both uniform and time varying wind speed.
For the first two problems, two mode approximations have been used as internal resonance conditions of 1:3 is considered. In the last two work single mode approx- imation has been used. The temporal equations are further reduced by using the method of multiple scales to obtain the approximate solution. The steady state responses and their stability and bifurcations have also been studied to know the operating range of system parameters for the PEH.
In the present work the nonlinear dynamics of PEH system is studied under the prin- cipal parametric and combination parametric resonance conditions along with inter- nal resonance condition. Due to the existence of cubic nonlinearity, the harvester system is capable of harvesting energy not only in principal parametric resonance condition (external excitation (Ω) twice that of fundamental frequency (ω) of the system, i.e. Ω≈2ω) but also at subharmonic (Ω≈ 3ω), superharmonic (Ω≈ ω/3) and combination parametric resonance of sum and difference type (Ω ≈ ωm ±ωn, here m= 1, 2 and n = 1, 2 ) as well.
Based on the theoretical and experimental findings the parameters affecting the power output are the position of the piezoelectric patch, number of patches and optimum value of external load resistance. Patch position nearer to the fixed end of the beam is advantageous and also more patches will help to increase the power.
Proper selection of load resistance by impedance matching procedure can be utilised to maximize the output power as well.
As the developed analytical solutions are found to be in good agreement with the experimental findings hence these analytical solutions can be used for the parametric studies for a wide range of system parameters for newly developed base excited or PEH systems without conducting expensive experiments. In the next section, the specific conclusions related to different PEH systems are highlighted.