In most of the literature base excited cantilever beam with piezoelectric patches is considered for analysis as an energy harvester. The beam is generally modelled as an Euler-Bernoulli beam. While in most cases small curvature is considered in some
cases moderately large curvature is taken which gives rise to geometric nonlinear- ity in the system. In some studies tip mass is considered to enhance the bending effect to generate higher output voltage and power. The governing coupled elec- tromechanical equation of motion of such systems is either that of a forced with/or parametrically excited system with or without nonlinearity. Most of the linear har- vesting system operates near the resonance condition. As the ambient vibration has a broad bandwidth of frequency so these linear harvesters are not suitable for producing energy in a wider range of frequency. So in such cases one may go for nonlinear forced vibration based harvester where multiple resonance conditions such as simple resonance, superharmonic and subharmonic resonance conditions. Fur- ther one may increase the bandwidth by using the concept of parametric excitation where principal parametric and combination parametric resonance conditions can be explored. As these phenomena are not yet explored, hence there is need to study the dynamics of such systems for enhanced performance of the harvester. While in most of the cases, only single mode approximation is taken for the analysis, few researchers also studied systems with more number of modes. In some cases these modes are having integer relations giving rise to internal resonance conditions of 1:1 and 1:2. Such internal resonance conditions can be achieved by attaching a mass at arbitrary position in the cantilever beam. While the dynamics of the cantilever beam without piezoelectric patch having attached mass either at the tip or at ar- bitrary position have been widely studied similar studies have not been carried out for piezoelectric embedded cantilever beam for energy harvesting purpose. Hence there is a need to carry out the nonlinear dynamic analysis of the PEH system with different types of internal resonance conditions (e.g., 1:3 internal resonance).
While most of the experimental work focused on PEH with forced vibration with simple resonance conditions, very few experimental studies have been carried out on parametric excitation based PEH. So there is a need to carry out experiments to validate the theoretical findings. The position of the attached mass is chosen in such a way that the ratio between the first two frequencies is commensurable, which leads to the existence of internal resonance of 1:3 between first two frequencies of the beam system. Hence a two mode approximation is adopted here for the dynamic analysis. Due to modal interaction, energy transfer between modes takes place. The system considered in the analysis is inspired by the work of Zavodney and Nayfeh [182] and Dwivedy and Kar [27]. The present system is compact as compared to the
systems where internal resonance between multiple beam elements are considered.
All these factors lead to the development of complex and coupled nonlinear equa- tions of motion. Also the existence of geometric, inertial nonlinearity along with 1:3 internal resonance brings the possibility of multiple resonance conditions other than primary resonance, i.e., principal parametric, combination parametric, subharmonic and superharmonic resonances.
The researchers separately explored the PEH system under parametric excitation and systems involving internal resonance. However, the system with parametric excitation and internal resonance brings rich dynamical responses i.e. quasiperiodic and chaotic [12, 27, 236]. This combination has not been explored to date for energy scavenging. The parametric excitation can be of the type of principal parametric and combination parametric.
The governing spatio-temporal equation of motion is determined, which is reduced to its temporal form by using generalized Galerkin’s method. The method of multiple scales (MMS) is used to solve the resulting equations of motion to obtain the steady state response and voltage. Here the system is analyzed under principal parametric and combination parametric resonance conditions. By utilizing parametric excita- tion and internal resonance conditions here, an attempt has been made to harvest the vibration energy for a wide range of frequencies. To perform experimental anal- ysis, an in-house shaker is designed and developed to provide base excitation to the system. Experimental studies are performed to understand the dynamics along with challenges posed by parametrically excited system on the voltage and power output across a load resistance.
Finally, the present system configuration is also studied under the effect of flow induced instability, which is the phenomena of galloping. The energy transduction requires an attachment of the bluff body with a slender structure. In the present study, two bluff bodies having D and triangular sections are considered for dynamic analysis. In most of the literature, only steady wind conditions are explored for energy transduction purposes. However in the present work the effect of both steady and unsteady wind speeds is investigated on the dynamics of the system. The mass of the bluff body is also taken into account and since the beam is in inverted position the effect of gravity affects the dynamics of the system. In all the previous studies the bluff body is placed at the top of the slender structure. While in the present work
the mathematical modeling and the experiment, both accommodated the variation in the position of the bluff body along the beam. By this variation the frequency of the system can be adjusted. The effect on the dynamics of the bluff body position is investigated. A comparative study between D and triangular section bluff body is also presented. The parametric study involves the effect of the variations of steady wind speed, unsteady wind speed component and external load resistance. The experiments of the harvester under steady state wind conditions are also performed in the wind tunnel facility. The output voltage and power are measured for different system parameters. The unsteady wind state is modeled by a simplistic model of harmonic variations in the steady state wind condition. Such kind of model can also be employed to the systems exposed to the wake galloping phenomena. Also, it is notable that the unsteady wind component brings the parametric instability into the system caused by galloping.
The present investigation explores the piezoelectric energy transduction under two kinds of known instabilities viz., parametric excitation and galloping phenomena which, in general are to be avoided. This work may find applications in health mon- itoring of the machine components and civil structures like bridges and high rise buildings. The present system may be attached to vehicles under undulated roads and train in motion as well. The galloping based PEH system (GPEH) can be em- ployed in remote locations also. Also, both excitation forces of parametric excitation and galloping are available in bridges, which can be explored for harvesting. The objective of the present work is mentioned in the following section.