Exploiting the Principal Parametric Resonance of an RLC Circuit for Vibratory Energy Harvesting

The use of ambient energy sources to independently power small electronic devices, a process commonly known as energy harvesting, has recently become a focus of research due to advances in low-power electronic applications. A particular class of energy harvesting devices, known as vibratory energy harvesters (VEHs), utilizes low-level vibrations present in numerous natural and man-made environments to generate electrical energy for electronic devices. This work investigates the use of a new technique to harvest energy from ambient vibrations by parametrically exciting a resonance condition of the electric current in a nonlinear oscillating circuit. To accomplish this parametric resonance phenomenon, we consider an electromechanical coupling device, an oscillating cantilever beam with a ferromagnetic tip mass, which changes the permeability of an iron-alloy cored inductor coil to produce a harmonically-varying modulation of the inductance. Such a type of harvester possesses the potential to generate large amplitude System response that is not limited by the linear damping of the system, as is the case with directly-excited systems, but rather whose behavior is governed by the nonlinearity of the system. In order to study the ability of such an energy harvesting system to generate electricity when subject to external vibrations, we develop a second-order differential equation to model the theoretical dynamic behavior of a parametrically-driven nonlinear circuit. Due to the complexity of the nonlinear and harmonically-varying components of the governing equation, we use the Method of Multiple Scales to derive an approximate analytical solution for the steady-state current response and output power of the circuit near the principal parametric resonant frequency. We show that the relationship of parameter modulation depth and load resistance characterize the bandwidth of the response and define a critical forcing threshold, below which no energy is harvested. The harvested power is maximized when the load resistance is half of the maximum load resistance at which the critical threshold is still achieved for a given forcing level. We also demonstrate the need for nonlinear damping in the system to attenuate the growth of the response to a physically attainable level. We show the dependence of the natural frequency of the circuit on the parametric forcing parameter, which can lead detuning of the system at different forcing levels. An experimental set up is developed to test the assertions presented by the analytical model. Numerous parameter constraints are balanced in the experimental design in order to be able to achieve the critical forcing threshold necessary for exciting the parametric resonance condition. The frequency response behavior of the electrical current and load power in the circuit is observed by varying the natural frequency of the system, which is compared against the variation of forcing frequency presented in the theoretical section. The beam is excited at its natural frequency of 85.8 Hz across input accelerations ranging from 1:1g – 1:5g. A maximum output power of 28.67 mW across an 8 Ω resistance is achieved at an input acceleration of 1:5g. The behavior of the experimental data is in good agreement with the findings of the theoretical model with respect to the bandwidth, nonlinear behavior, and sensitivity to forcing and damping parameters. The analytical model under predicts the peak power measured experimentally, but the general trend is well modeled. Furthermore, several key observations are noted during the experimental procedures, notably the effects of eddy current damping on the behavior of the response and the development of quasiperiodic solutions near the saddle node bifurcation point

DYNAMICAL CHAOS IN 6 ï† -RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN AN AMPLITUDE MODULATED FORCE.

Chaotic behavior of6 ï† -Rayleigh oscillator with three wells is investigated. The method of multiple scale method is usedto solve the system up to 3rd order approximation. Effect of parameters is studied numerically; all resonance cases arestudied numerically to obtain the worst case. Stability of the system is investigated using both phas

Modeling of autoresonant control of a parametrically excited screen machine

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.Modelling of nonlinear dynamic response of a screen machine described by the nonlinear coupled differential equations and excited by the system of autoresonant control is presented. The displacement signal of the screen is fed to the screen excitation directly by means of positive feedback. Negative feedback is used to fix the level of screen amplitude response within the expected range. The screen is anticipated to vibrate with a parametric resonance and the excitation, stabilization and control response of the system are studied in the stable mode. Autoresonant control is thoroughly investigated and output tracking is reported. The control developed provides the possibility of self-tuning and self-adaptation mechanisms that allow the screen machine to maintain a parametric resonant mode of oscillation under a wide range of uncertainty of mass and viscosit

New methodology for optimal placement of piezoelectric sensor/actuator pairs for active vibration control of flexible structures

This paper describes a computationally efficient method to determine optimal locations of sensor/actuator (s/a) pairs for active vibration reduction of a flexible structure. Previous studies have tackled this problem using heuristic optimization techniques achieved with numerous combinations of s/a locations and converging on a suboptimal or optimal solution after multithousands of generations. This is computationally expensive and directly proportional to the number of sensors, actuators, possible locations on structures, and the number of modes required to be suppressed (control variables). The current work takes a simplified approach of modeling a structure with sensors at all locations, subjecting it to external excitation force or structure base excitation in various modes of interest and noting the locations of n sensors giving the largest average percentage sensor effectiveness. The percentage sensor effectiveness is measured by dividing all sensor output voltage over the maximum for each mode using time and frequency domain analysis. The methodology was implemented for dynamically symmetric and asymmetric structures under external force and structure base excitations to find the optimal distribution based on time and frequency responses analysis. It was found that the optimized sensor locations agreed well with the published results for a cantilever plate, while with very much reduced computational effort and higher effectiveness. Furthermore, it was found that collocated s/a pairs placed in these locations offered very effective active vibration reduction for the structure considered

Advanced Model for Fast Assessment of Piezoelectric Micro Energy Harvesters

The purpose of this work is to present recent advances in modeling and design of piezoelectric energy harvesters, in the framework of micro-electro-mechanical systems (MEMS). More specifically, the case of inertial energy harvesting is considered, in the sense that the kinetic energy due to environmental vibration is transformed into electrical energy by means of piezoelectric transduction. The execution of numerical analyses is greatly important in order to predict the actual behavior of MEMS devices and to carry out the optimization process. In the common practice, the results are obtained by means of burdensome 3D finite element analyses (FEA). The case of beams could be treated by applying 1D models, which can enormously reduce the computational burden with obvious benefits in the case of repeated analyses. Unfortunately, the presence of piezoelectric coupling may entail some serious issues in view of its intrinsically three-dimensional behavior. In this paper, a refined, yet simple, model is proposed with the objective of retaining the Euler-Bernoulli beam model, with the inclusion of effects connected to the actual three-dimensional shape of the device. The proposed model is adopted to evaluate the performances of realistic harvesters, both in the case of harmonic excitation and for impulsive loads

Nonlinear Model Updating in Structural Dynamics

Identification of nonlinear structural dynamics has received a significant attention during last decades. Yet, there are many aspects of the identification methods of nonlinear structural models to be improved. The main objective of this study is to introduce novel identification approaches for nonlinear structures. The first step in identifying nonlinear structural elements is to detect their exact location. Hence, the first section of this study focuses on the localization of nonlinear elements in structural dynamics utilizing base excitation measured data. To this end, a localization approach is used to find the location of nonlinear electromagnetic restoring force applied to the tip of a cantilever beam.Inferring the exact location of nonlinear elements, identification methods are utilized to identify and characterize the mathematical model of nonlinear structures. However, various sources of noise and error may affect the accuracy of the identified model. Therefore, in the second part of the thesis, the effect of various sources of inaccuracy on the results of nonlinear model identification is investigated. It is shown that measurement noise, expansion error, modelling error, and neglecting the effect of higher harmonics may lead to an erroneously identified model.An optimization-based framework for the identification of nonlinear systems is proposed in this work in order to avoid the bottlenecks mentioned above. The introduced method is applied to a test rig composed of a base-excited cantilever beam subjected to an electromagnetic force at the tip. According to the nonlinear response of the system, four different functions are assumed as candidate models for the unknown nonlinear electromagnetic force. The measured response is compared with the reconstructed response using various models and the most appropriate mathematical model is selected.Utilizing optimization-based identification method to characterize complex mathematical models with large number of unknown parameters would be computationally expensive. Therefore, this study introduces a harmonic-balance-based parameter estimation method for the identification of nonlinear structures in the presence of multi-harmonic response and force. For this purpose, a method with two different approaches are introduced: Analytical Harmonic-Balance-based (AHB) approach and the Alternating Frequency/Time approach using Harmonic Balance (AFTHB). The method is applied to five simulated examples of nonlinear systems to highlight different features of the method. The method can be applied to all forms of both smooth and non-smooth nonlinear functions. The computational cost is relatively low since a dictionary of candidate basis functions is avoided. The results illustrate that neglecting higher harmonics, particularly in systems with multi-harmonic response and force, may lead to an inaccurate identified model. The AFTHB approach benefits from including significant harmonics of the response and force. Applying this method leads to accurate algebraic equations for each harmonic, including the effect of higher harmonics without truncated error. In the last part of this study, the AFTHB method is applied to two experimental case studies and identifies the nonlinear mathematical model of the structures. The first case is composed of a cantilever beam with a nonlinear electromagnetic restoring force applied to the tip which is excited by a multi-harmonic external force. In the second experimental case study, a configuration of linear springs applies a geometric nonlinear restoring force to the tip of a cantilever beam resulting in internal resonance in the dynamics of the system. The good performance of the AFTHB approach in estimating the unknown parameters of the structure is illustrated by the results of identification

Energy harvesting using porous piezoelectric beam with impacts

An analytical model of impact energy harvester consisting of a cantilever beam with integrated piezoelectric patches and a ball is developed in this paper. The material chosen to extract the energy is porous PZT, a composite material made of two phases: air and PZT. This material offers good control of the capacitance and the stiffness of the resultant composite material and expands the design space for the harvester. The cantilever beam is modelled using a single degree-of-freedom approximation, and a load resistor is used to represent the external circuit. The response of the energy harvester and the power output is obtained for harmonic base excitation, and the effect of excitation frequency, boundary distance, load resistance and porosity of the PZT material. The results highlight the potential for the impact harvester and motivate further studies to optimize the harvester

Experimental study for structural damage identification with incomplete measurements

Finite element formulation for the equation of motion should be used for identification of local damage of a structure at the element level. However, the finite element model of a structure involves a large number of degrees of freedom and requires a large number of sensor measurements. To avoid measuring vibration responses, which are difficult to obtain (e.g. rotational accelerations), and to reduce the required number of sensors, a reduced-order finite element formulation along with the adaptive sequential nonlinear least square estimation technique is proposed in this paper to identify local damages of structures. To verify the applicability and effectiveness of the proposed approach, two series of damage detection experiments were conducted using scaled cantilever beams. One series of experimental tests were conducted for the detection of constant damages. In this test series, different damage severities were simulated by drilling different number of circular holes with different sizes in a particular element of a cantilever beam. Another series of experimental tests were conducted to verify the online damage tracking capability of the proposed approach. In this test series, a stiffness element device was installed in a particular element of another cantilever beam to simulate the abrupt stiffness reduction of that element during the test. Experimental results demonstrate that the proposed reduced-order finite element model along with the adaptive sequential nonlinear least square estimation technique is effective and accurate in detection of structural damages, including the damage location and severity using only a limited number of sensors