In the first two studies, the response and stability of the system under primary parametric and combination parametric resonance conditions with 1:3 internal resonance have been investigated. The effect of the location of bluff bodies on voltage and power is investigated.
Introduction
In the case of forced vibrations, simple resonance occurs when the external frequency is almost equal to one of the modal or natural frequencies. The parametric resonance occurs when the external frequency Ω is almost equal to the sum of the modal frequencies, i.e. Ω = ω ±ω.
Thesis Organization
In Chapter 5, the dynamic behavior of the piezoelectric energy harvesting system is experimentally investigated under the main parametric resonance condition. The objective of this chapter is to review the literature related to piezoelectric based energy harvesters.
Applications of PEH
The survey is divided into various aspects related to Piezoelectric Energy Harvester (PEH) Systems such as potential applications, PEH Systems, overview works, environmental energy sources, piezoelectric materials, mathematical modeling, linear and nonlinear energy harvesters , gallop-based PEH systems, miscellaneous PEH systems and electrical circuits discussed in section 2.2 to 2.9. The technical gap that motivates carrying out the current research work is discussed in section 2.10.
PEH Systems
Base excited PEH systems
- Linear PEH systems
- Nonlinear PEH systems
A hybrid bistable piezoelectric beam energy harvester with internal resonance due to moving magnet is presented by Yang and Towfighian [68]. A cantilever beam with a tip mass is considered an energy harvester by Rezaei et al.
Flow induced PEH systems
Another flow-induced phenomenon such as vortex-induced vibration (VIV) is also explored for energy harvesting in the work of Akaydin et al. Various shapes of bluff bodies (Abdelkefi et al. [1]) have been explored for better performance of the gallop-based PEH (GPEH) system.
Review Works in PEH Systems
A detailed overview of the use of nonlinearity in PEH can be found in the work of Daqaq et al. They presented an in-depth study on the effect of nonlinearity (inherent or induced) in VEHs (vibrational energy harvesters).
Ambient Energy Sources
It is important for a harvesting device that its natural frequencies match or are close to the excitation frequency of the source for better energy transfer. One has to decide the transduction mechanism to harvest such available unutilized and wasted energies.
Piezoelectric Material
The fundamental frequencies of the systems considered in this work are in the range from 1 to 16 Hz. In the next part, the types of piezoelectric materials and their properties are discussed.
Mathematical Modeling and Solution Strategies
Forcing in PEH systems
- Direct excitation
- Parametrically excited PEH systems
- Stochastic excitation
- Flow induced excitation
- Simultaneous excitation
A small parametric excitation can produce a large response when the excitation frequency deviates from the main frequency of the system. 206] investigated a parametrically excited PEH system, where the rotation of the blades induces magnetic force on the beam-based harvester.
Explored resonance conditions
- Primary and secondary resonance
- Internal resonance
To improve the steady state bandwidth limit, an axially loaded energy harvester with 1:2 internal resonance is studied by Jiang et al. The efficiency of PEH systems is increased by the application of internal resonance Chunbo et al.
Solution strategies
The dynamics of the system is analyzed for the 1:2 internal resonance which is due to the nonlinear coupling between the cantilever beam and the pendulum. Chen and Jiang [244] proposed a new possibility to improve energy harvesting by tuning harvesters to 1:2 internal resonance.
Miscellaneous PEH Systems
It can also provide information about the forced frequency harmonics response [15]. The MMS perturbation technique is widely used for finding closed-form solutions of harvesters.
Electrical Circuits
So far, rechargeable batteries and supercapacitors have been found to be suitable as energy storage units for PEH systems. For a more basic understanding, especially related to electrical circuits, one can turn to the work of Ottman et al.
Summary and Identification of Research Gaps
Most literature only investigates stable wind conditions for energy transduction purposes. However, the current work investigates the effect of both stable and unstable wind speeds on the dynamics of the system.
Objectives of the Present Work
Harvester experiments under steady wind conditions are also conducted in the wind tunnel facility. As discussed in the previous chapter, most energy harvesters consist of a base-excited cantilever beam with piezoelectric patches with or without a tip mass.
Mathematical Modeling
After differentiating the moment equation with respect to 's' twice, we get which governs the equation of motion in the following form. The boundary conditions for this cantilever beam system are expressed in the following form.
Solution of Temporal Equation
The eigenvalues of [J] will define the stability and bifurcation of the fixed point response of the system. The stability and bifurcation of the system's fixed point response is characterized by the eigenvalues of the Jacobian matrix.
Results and Discussion
To verify the instability region of the system, the time responses are plotted (Fig. 3.3) corresponding to 3 different values of φ (points o1, o2 and o3), as shown in. So according to user's voltage and current generation requirements, the system can operate.
Summary
In section 4.2, the mathematical modeling is performed for the derivation of the governing nonlinear equation of motion which is solved using the method of multiple scales as presented in section 4.3. In section 4.5, parametric studies were performed to determine the system parameters for achieving 1:3 internal resonance.
Modeling and Governing Equation
The electric displacement vector (D) in terms of axial stress (σp) and generated electric field (Ee) is defined in this case by the following constitution. 4.7) the circuit equation becomes the following. It should be noted that unlike the previous chapter, where a momentum-based approach was adopted to find the equation of motion of the system, here Hamilton's principle is used to find the governing equations.
Solution of the Temporal Equation
Furthermore to find the stability of the obtained fixed point responses, the above reduced modulation equations Eq. The stability and bifurcation of the fixed point response of the system can be obtained by finding the eigenvalue value of the Jacobian [J].
Results and Discussions
Further, as the frequency φ increases, the non-trivial response remains stable and an increase in amplitude, voltage and power is observed. Both voltage and power increase and the critical value Γ moves towards the higher side with Rl.
On Achieving the Internal Resonance of 1:3
However, as the thickness of the substrate (ts) increases, the curve shifts to the lower frequency ratio side. As the distance of attached mass (β) from the fixed point increases, the curve shifts to the lower frequency ratio side and around β ≈ 0.27 the frequency ratio switches to the higher ratio side as shown in Fig.
Summary
Then the frequency response curves are plotted for the same parameters to obtain the harvest voltage and power. A piezoelectric patch is attached near the fixed end of the cantilever beam along with an attached mass positioned at an arbitrary location.
Design and Development of the Experimental Setup
Shaker Design
Kinematic Analysis
Dynamic Analysis
The experimentally obtained acceleration characteristic of the holder (without attachment of the beam system) is shown in figure. The acceleration time response (after filtering) is obtained at a certain speed of the DC motor by using the accelerometer.
Analytical Model
The non-dimensional nonlinear coupled electromechanical temporal equations of motion of the system are expressed in the following equations. The coefficients appear in equations. 5.2) the time-varying forcing term appears as a parameter, which is a typical feature of parametrically excited systems that provide energy input to the system.
Perturbation Analysis
Nef f also indicates the total effect due to cubic (α) and inertial (β, γ) nonlinear terms in the equation of motion of the beam. The non-trivial (a6= 0) frequency response of the reduced equations can be calculated using the following frequency response expression.
Results and Discussion
5.6(d) is plotted in Fig 5.6(e) and the time response of the input acceleration along with FFT of the filtered signal is plotted in Fig. 5.9(a) shows the time response of the voltage is shown for Rl= 25 kΩ, one can reach an average voltage of 17 V in this case for the frequency of excitation f = 9.85 Hz.
Experimental Findings with MFC Patch
It can be noted that the frequency of 2.5 Hz is the same as the frequency of the first mode of the system. The effect of substrate length and MFC patch position is also studied and plotted in Fig.
Summary
While harmonic-based excited piezoelectric energy harvester was considered in the previous chapters, in the present chapter wind-based energy harvester is studied. In section 6.3, the solution of the temporal equations is solved using the method of multiple scales.
Mathematical Modelling
Steady and unsteady wind flow
Fluctuations in wind speed due to vortex shedding can also be modeled as harmonic variation where the oscillation frequency is the vortex shedding frequency. Here, in the present study, the unsteady effect in the transverse direction is neglected due to its negligible effect.
Solution using by the method of multiple scales
Here cc is the complex conjugate of the preceding terms and A(T1) is an unknown complex function that will be determined later. 6.24) the following secular terms are obtained which must be eliminated for finite amplitude response of the system. Considering only the main parametric resonance condition, that is, when the frequency of the external excitation Ω is almost twice that of the natural frequency of the system ω, the proximity of external excitation is expressed as φ = 2 +σ and the following secular terms are obtained. 6.26) reduces to the following equation.
Results and Discussion
The determination of the free vibration characteristics
Using the logarithmic decrement method, we can find the damping ratio of the GPEH system. The experiment is repeated to determine the free vibration responses for three different positions of the roof body along the length of the beam.
Experimental study of GPEH in wind tunnel
- GPEH with D-shaped bluff body (GPEH-DSB)
- GPEH with Triangular-shape bluff body (GPEH-TSB)132
As the wind speed increases, the output voltage and the corresponding power increase for all values of β. It is observed that as the load resistance increases, the output voltage and the corresponding power increase.
Theoretical study
- GPEH with constant wind flow
- GPEH with time varying wind flow
The effect of steady wind speed U on output voltage and power for a triangular shaped bluff body is shown in Fig. The effect of variation of u0 and the position of triangle shape bluff body β is observed in Fig.
Summary
In this chapter, the conclusions regarding two different types of piezoelectric based energy harvesters (PEH) have been presented. In the second type, gallop-based piezoelectric energy harvesters due to flow-induced vibrations have been investigated.
General Conclusions
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 nonlinear frequency responses of the system are studied for various system parameters such as load resistance, amplitude and excitation frequency and damping ratio.
Specific Conclusions
PEH under principal parametric and 1:3 internal resonance
It can further be 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 used to obtain more energy.
PEH under combination parametric and 1:3 internal resonance 151
An internal resonance of 1:2 is observed in the FFT of the voltage time response, which improves the output voltage and power of the PEH system. As the position of the attached mass β increases, a percentage increase of 58.3% in the output voltage is observed.
Dynamics of galloping based piezoelectric energy harvester
Thus, although the onset of galloping does not depend on the positioning of the rock body, the tension can be controlled by varying the position of the rock body at a higher speed. For GPEH-TSB, the onset of galloping changes for different positions of the bluff body.
Scope of Future Work
In the current system, the internal resonance conditions are not considered in the gallop-based energy harvester. According to Euler-Bernoulli beam theory, the curvature κ(s, t) is defined as follows. C.3) The non-conservative virtual work doneWnc involves the work done in dissipating energy by viscous force effect and electric forces in moving charges by piezoelectric.
