Rajarathinam M - Academia.edu (original) (raw)

Papers by Rajarathinam M

Parametric amplification is realized by adding parametric excitation to externally driven near-re... more Parametric amplification is realized by adding parametric excitation to externally driven near-resonant oscillations. The effect of specific cubic nonlinearities on the parametrically amplified steady-state vibrations and gain is investigated theoretically. Here, gain is defined as the ratio of steady-state vibration amplitude of the directly and parametrically excited system, to vibration amplitude of the directly excited only system. The nonlinear effect of midplane stretching is compared to the effects of nonlinear inertia and curvature. An approximate analytical expression for the vibration amplitude is derived. For a given small level of transverse displacement for both the cantilever and doubly clamped beam, the effect of midplane stretching is dominant compared to those caused by nonlinear inertia and curvature. It was found that the beam slenderness ratio can be used as an effective design parameter for parametric amplifiers.

Cantilevered beams with piezoceramic layers have been frequently used as piezoelectric vibration ... more Cantilevered beams with piezoceramic layers have been frequently used as piezoelectric vibration energy harvesters in the past five years. The literature includes several single degree-of-freedom models, a few approximate distributed parameter models and even some incorrect approaches for predicting the electromechanical behavior of these harvesters. In this paper, we present the exact analytical solution of a cantilevered piezoelectric energy harvester with Euler-Bernoulli beam assumptions. The excitation of the harvester is assumed to be due to its base motion in the form of translation in the transverse direction with small rotation, and it is not restricted to be harmonic in time. The resulting expressions for the coupled mechanical response and the electrical outputs are then reduced for the particular case of harmonic behavior in time and closed-form exact expressions are obtained. Simple expressions for the coupled mechanical response, voltage, current, and power outputs are also presented for excitations around the modal frequencies. Finally, the model proposed is used in a parametric case study for a unimorph harvester, and important characteristics of the coupled distributed parameter system, such as short circuit and open circuit behaviors, are investigated in detail. Modal electromechanical coupling and dependence of the electrical outputs on the locations of the electrodes are also discussed with examples.

This work investigates the nonlinear behavior of a representative parametrically amplified macros... more This work investigates the nonlinear behavior of a representative parametrically amplified macroscale structure. Specifically, the effort examines the effects of structural and inertial nonlinearities on the near-resonant response of a base-excited, flexible cantilever beam driven by a combined (simultaneously parametric and direct) excitation. The prototypical structure is modeled using classical energy methods and key response metrics are analyzed through the use of the method of averaging. A series of experimental investigations are performed to validate analytically predicted behaviors. The work demonstrates that with the proper selection of various system parameters, both vibration amplification and attenuation can be efficiently achieved. This work provides a baseline understanding of the effect of nonlinearities on parametrically excited systems and is expected to guide future work on micro/nanoscale systems, where parametric excitations arise quite naturally.

Cantilevered beams with piezoceramic (PZT) layers are the most commonly investigated type of vibr... more Cantilevered beams with piezoceramic (PZT) layers are the most commonly investigated type of vibration energy harvesters. A frequently used modeling approach is the single-degree-of-freedom (SDOF) modeling of the harvester beam as it allows simple expressions for the electrical outputs. In the literature, since the base excitation on the harvester beam is assumed to be harmonic, the well known SDOF relation is employed for mathematical modeling. In this study, it is shown that the commonly accepted SDOF harmonic base excitation relation may yield highly inaccurate results for predicting the motion of cantilevered beams and bars. First, the response of a cantilevered Euler-Bernoulli beam to general base excitation given in terms of translation and small rotation is reviewed where more sophisticated damping models are considered. Then, the error in the SDOF model is shown and correction factors are derived for improving the SDOF harmonic base excitation model both for transverse and longitudinal vibrations. The formal way of treating the components of mechanical damping is also discussed. After deriving simple expressions for the electrical outputs of the PZT in open-circuit conditions, relevance of the electrical outputs to vibration mode shapes and the electrode locations is investigated and the issue of strain nodes is addressed.

Design considerations for piezoelectric-based energy harvesters for MEMS-scale sensors are presen... more Design considerations for piezoelectric-based energy harvesters for MEMS-scale sensors are presented, including a review of past work. Harvested ambient vibration energy can satisfy power needs of advanced MEMS-scale autonomous sensors for numerous applications, e.g., structural health monitoring. Coupled 1-D and modal (beam structure) electromechanical models are presented to predict performance, especially power, from measured low-level ambient vibration sources. Models are validated by comparison to prior published results and tests of a MEMS-scale device. A non-optimized prototype low-level ambient MEMS harvester producing 30 µW/cm 3 is designed and modeled. A MEMS fabrication process for the prototype device is presented based on past work. 121 122 N. E. duToit et al.

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Downloadable at: http://esag.harvard.edu/rice/e0\_Solid\_Mechanics\_94\_10.pdf TABLE OF CONTENTS prov... more Downloadable at: http://esag.harvard.edu/rice/e0_Solid_Mechanics_94_10.pdf TABLE OF CONTENTS provided on last three pages, pp. 87-89 elastic-plastic. Permanent deformation of a sort that depends mainly on time of exposure to a stress, and that tends to increase significantly with time of exposure, is called viscous or creep deformation and materials which exhibit that, as well as tendencies for elastic response, are called viscoelastic solids (or sometimes visco-plastic solids when we focus more on the permanent strain than on the tendency for partial recovery of strain upon unloading).

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under paramet... more A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler-Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester's behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.

The problem of controlling the vibration of a transversely excited cantilever beam with tip mass ... more The problem of controlling the vibration of a transversely excited cantilever beam with tip mass is analyzed within the framework of the Euler-Bernoulli beam theory. A sinusoidally varying transverse excitation is applied at the left end of the cantilever beam, while a payload is attached to the free end of the beam. An active control of the transverse vibration based on cubic velocity is studied. Here, cubic velocity feedback law is proposed as a devise to suppress the vibration of the system subjected to primary and subharmonic resonance conditions. Method of multiple scales as one of the perturbation technique is used to reduce the second-order temporal equation into a set of two first-order differential equations that govern the time variation of the amplitude and phase of the response. Then the stability and bifurcation of the system is investigated. Frequency-response curves are obtained numerically for primary and subharmonic resonance conditions for different values of controller gain. The numerical results portrayed that a significant amount of vibration reduction can be obtained actively by using a suitable value of controller gain. The response obtained using method of multiple scales is compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. Numerical simulation for amplitude is also obtained by integrating the equation of motion in the frequency range between 1 and 3. The developed results can be extensively used to suppress the vibration of a transversely excited cantilever beam with tip mass or similar systems actively.

As a kind of base excitation, shaking table is often used to test the dynamic characteristics of ... more As a kind of base excitation, shaking table is often used to test the dynamic characteristics of structures. However, the prediction of response to base excitation hasn't been solved effectively, which limits the further research on the test and analysis method with respect to base movement. This article is based on a cantilever beam and focuses on its response prediction under sinusoidal base excitation. By moment and force equilibrium equations, an analytical model is built for this cantilever beam, and then a method to predict dynamic response at base excitation is proposed. Finally, the method is used to solve the vibration response distributions of the cantilever beam at base excitation. Correctness of this method is also proved by comparing the result with experimental data.

The nonlinear nonplanar steady-state responses of cantilever beams to direct and parametric harmo... more The nonlinear nonplanar steady-state responses of cantilever beams to direct and parametric harmonic excitations are investigated using perturbation techniques. Modal interactions between the bending-bending and bending-bending-twisting motions are studied. Using a variational formulation, we obtained the governing equations of motion and associated boundary conditions for monoclinic composite and isotropic metallic inextensional beams. The method of multiple scales is applied either to the governing system of equations and associated boundary conditions or to the Lagrangian and virtual-work term to determine the modulation equations that govern the slow dynamics of the responses. These equations are shown to exhibit symmetry properties, reflecting the conservative nature of the beams in the absence of damping.

Methods are described for calculation of natural frequencies and mode shapes of a cantilever beam... more Methods are described for calculation of natural frequencies and mode shapes of a cantilever beam with a base excitation and tip mass whose centre of gravity does not coincide with the point of attachment. Exact expressions for natural frequencies and mode shapes are derived. Some typical results are presented.

An analysis of natural frequency and dynamic response of a cantilever beam subjected to base moti... more An analysis of natural frequency and dynamic response of a cantilever beam subjected to base motion is presented. The analysis is extended to a case in which the cantilever is mounted on a composite support. Typical numerical results for the natural frequency and the deflection of the beam are given for both cases to illustrate the attenuation of motion by the composite mount.

Generally deterministic responses are expected when deterministic system are subjected to determi... more Generally deterministic responses are expected when deterministic system are subjected to deterministic excitations, and random excitations are expected when the inputs are random processes. However, it has been shown that many nonlinear systems with or without deterministic excitations under certain conditions produce seemingly random responses. Such bounded nonperiodic, apparently random motions have been called "chaotic motions." Fourier analysis of these responses shows a broad spectrum of frequencies in spite of the fact that the excitations are of a single frequency. In this section, a few examples of chaotic response of simple dynamical systems are presented.

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