Microelectronic devices are primarily used in sensing, actuation and data trans- mission purpose. These devices are basically battery powered and hence required recharging/replacement after power utilization, which may be difficult in remote lo- cations. The recent advancement in the development of microelectronic systems [2]
has reduced the power consumption of these devices significantly [3]. Now, the focus is to power these devices by harvesting energy from the ambient vibration for its energy requirements. This task has been driving researchers around the globe for the last two decades. These self-sustained systems may be capable of surviving even in remote areas on their own.
One solution to this end is to extract power from potential ambient sources of energy like solar, wind and vibration. Sources like solar, wind and water provide macro- scale energy in the range of few kW to GW. The energy requirement of low power electronic devices is less than a Watt (< 1W). In this regard, the micro-scale en- ergy harvesting is required to power such devices which are in the range of µW to mW. Available kinetic energy in the ambient sources is sufficient to power such de- vices. Vibrational energy is available almost everywhere in the ambient, e.g., human motions, vibration of machine components, vibration of locomotives, bridges, wind induced phenomena (i.e., galloping and vortex induced vibration) and ocean waves.
PZT / MFC / PVDF / ..
tip mass substrate
fixed end
x y
y(t)
Figure 1.1: Conventional beam based piezoelectric energy harvesting (PEH) system
Several transduction mechanisms [4] viz., electromagnetic, electrostatic, piezoelec- tric and electrochemical are exploited to convert the available energy into electricity.
The combination of these transduction mechanisms, e.g., electromagnetic and piezo- electric, are also explored [5, 6]. Among them, the piezoelectric based transduction mechanism has advantage over others because of its energy extracting capabilities over a wide range of available frequencies, structural simplicity, easy to fabricate with less peripheral components, easy to integrate with microelectronic circuits and having higher energy density [7, 8], which makes it most suitable for micro-scale applications.
Piezoelectric materials [9] are employed extensively in sensing, actuation and en- ergy harvesting purposes. With these materials, the vibrational mechanical energy can be converted into electricity. These materials are compact and having quick responses within very short period of time. Vibrational energy can be converted into electrical energy by using structural elements such as beams, plates, shells or combination of these basic elements embedded with piezoelectric patches. A beam structure or a combination of several beam structures are generally used for energy harvesting purpose due to their simplicity and size. Apart from the traditional PZT (lead zirconate titanate) materials, wide ranges of flexible piezoelectric materials are also available which can be integrated with wearable devices for sensing purposes.
Figure 1.1 shows a conventional beam based piezoelectric energy harvester where the ambient vibration is used to harvest the energy.
In general, linear vibration based piezoelectric energy harvester (PEH) systems work over a short range of bandwidth near the resonance frequency. Energy transduction reduces sharply for mistuned (away from resonance) linear harvesting systems. To
address this issue, researchers are focusing on tapping the rich dynamics associated with the response of the systems with inherent or induced nonlinearity [10]. Nonlin- ear systems display behaviours such as bifurcation, chaos, internal resonance. The multi-branched fixed point response of a nonlinear system may be periodic, quasi- periodic and chaotic. Energy harvesters based on nonlinear vibration have potential to extract energy over a wide range of available frequencies. The source of nonlin- earity can be of geometric or material in nature, and it can also arise due to external excitation and constraints such as impact, friction, backlash and freeplay [11].
Nonlinear stretching or large curvature causes the geometric nonlinearity and con- centrated or distributed mass cause inertial type nonlinearity in the system [12].
Nonlinear stress-strain relation leads to material nonlinearity and stretching leads to nonlinear strain-displacement relation, and large curvature leads to nonlinear curvature-displacement relation. Other sources of nonlinearity are external forces acting on a system. The external force may be gravitational, magnetic or fluid forces. Nonlinear behaviour also comes from constraints in the system. The perfor- mance of structures associated with the energy harvesting systems can be improved by exploiting some characteristics of the nonlinear dynamics. [13].
The governing equations of motion of nonlinear PEH systems are represented by cou- pled electromechanical partial (spatio-temporal) differential equations [14]. Apart from numerical methods, some analytical methods are also used for solving these nonlinear equations. These analytical methods include the method of harmonic balance and perturbation methods such as method of averaging, Linstedt-Poincar`e technique and method of multiple scales [13]. The method of multiple scales is a commonly used technique that provides transient and steady-state responses. The reduced equation obtained in this method can be used to study the stability of the steady-state solutions. It can also give information related to the response of the harmonics of the forcing frequency [15].
The commonly used source of energy for the vibration based energy harvesters are external excitations [16–18]. These excitations may be forced, parametric or com- bination of both. In the case of forced vibration simple resonance occurs when the external frequency nearly equals any of the modal or natural frequencies. The para- metric resonance occurs when the external frequency Ω nearly equals to the sum of the modal frequencies, i.e., Ω = ω ±ω . When ω = ω , the external frequency
is equal to twice the natural frequency and the resonance condition is known as principal parametric resonance condition. When the excitation frequency coincides with combination (sum or difference) of natural frequencies [19] the combination parametric occurs. The characteristic of a typical parametrically excited system is that a time-varying term appears as the coefficient of the response in the governing equation of motion [20]. Parametric resonance may lead to a large amplitude vibra- tion. In few studies [21, 22], the PEH system is investigated under the parametric excitation.
The interaction of flexible structures with the fluid flow may cause the self excitation of the system, such as galloping [23]. In this phenomenon the aerodynamic lift forces are generated by a small transverse motion of the structure. It is self excited flow induced instability where the aerodynamic lift forces act in the transverse direction to that of the flow. The flow energy transfers to the structure by the damping-type forces. For effective energy transfer a bluff body is usually attached to the elastic structure with piezoelectric patches to extract the wind energy [24].
Another physical phenomenon that occurs in nonlinear system is internal resonance (IR) [25]. When the natural frequencies of a system are commensurable then internal resonance occurs. For example, the internal resonance of 1 : 3 may occur when the second mode frequency is 3 times the first mode frequency. The internal resonance causes mode coupling between participating modes and leads to rich dynamical responses [26] due to energy transfer between the participating modes. The system with parametric excitation and internal resonance brings many critical phenomena.
The dynamic response in such cases may be fixed point, periodic, quasiperiodic or chaotic [27] which has not yet been explored for energy harvesting purposes.
Applications of PEH devices involves the condition or health monitoring of many systems such as tire pressure, bridge, building, aircraft and railway. In most of the structural health monitoring systems, wireless sensor nodes (WSN) or networks are used for data transfer which can be powered by integrated PEH systems. Energy involves in human activities [28] are also sufficient to power wearable or implantable biomedical monitoring sensors (i.e. pacemaker, pressure, tissue stimulation) [29].
In recent times, several miniature energy harvesting devices from walking [30], limb motion and elbow and knee joints have been developed.
There exist vast potential fields of application for smart and self-sustained PEH sys- tems. Recent progress in PEHs suggests that there exist strong commercial viability apart from the usual use of these devices. Due to the above mentioned potential ad- vantages of piezoelectric based energy harvester, in the present work, an attempt has been made to study the nonlinear dynamics of such systems to harvest energy from wide range of frequency utilizing the concept of nonlinear parametric and internal resonance conditions. A similar cantilever beam based system is also explored with galloping instability conditions for harvesting energy. The structure of the thesis is outlined in the following section.