An Overview of Homogenization

Homogenization of partial differential equations is relatively a new area and has tremendous applications in various branches of engineering sciences like: material science,porous media, study of vibrations of thin structures, composite materials to name a few. Though the material scientists and others had reasonable idea about the homogenization process, it was lacking a good mathematical theory till early seventies. The first proper mathematical procedure was developed in the seventies and later in the last 30 years or so it has flourished in various ways both application wise and mathematically. This is not a full survey article and on the other hand we will not be concentrating on a specialized problem. Indeed, we do indicate certain specialized problems of our interest without much details and that is not the main theme of the article. I plan to give an introductory presentation with the aim of catering to a wider audience. We go through few examples to understand homogenization procedure in a general perspective together with applications. We also present various mathematical techniques available and if possible some details about some of the techniques. A possible definition of homogenization would be that it is a process of understanding a heterogeneous (in-homogeneous) media, where the heterogeneties are at the microscopic level, like in composite materials, by a homogeneous media. In other words, one would like to obtain a homogeneous description of a highly oscillating in-homogeneous media. We also present other generalizations to non linear problems, porous media and so on. Finally, we will like to see a closely related issue of optimal bounds which itself is an independent area of research.

Study of terahertz wave propagation properties in nanoplates with surface and small-scale effects

In macroscopic and even microscopic structural elements, surface effects can be neglected and classical theories are sufficient. As the structural size decreases towards the nanoscale regime, the surface-to-bulk energy ratio increases and surface effects must be taken into account. In the present work, the terahertz wave dispersion characteristics of a nanoplate are studied with consideration of the surface effects as well as the nonlocal small-scale effects. Nonlocal elasticity theory of plate is used to derive the general differential equation based on equilibrium approach to include those scale effects. Scale and surface property dependent wave characteristic equations are obtained via spectral analysis. For the present study the material properties of an anodic alumina with crystallographic of < 111 > direction are considered. The present analysis shows that the effect of surface properties on the flexural waves of nanoplates is more significant. It can be found that the flexural wavenumbers with surface effects are high as compared to that without surface effects. The scale effects show that the wavenumbers of the flexural wave become highly non-linear and tend to infinite at certain frequency. After that frequency the wave will not propagate and the corresponding wave velocities tend to zero at that frequency (escape frequency). The effects of surface stresses on the wave propagation properties of nanoplate are also captured in the present work. (C) 2012 Elsevier Ltd. All rights reserved.

An approach for assessing the accuracy of accelerometer responses during axial impact tests

The primary objective of the present study is to show that for the most common configuration of an impactor system, the accelerometer cannot exactly reproduce the dynamic response of a specimen subjected to impact loading. An equivalent Lumped Parameter Model (LPM) of the given impactor set-up has been formulated for assessing the accuracy of an accelerometer mounted in a drop-weight impactor set-up for an axially loaded specimen. A specimen under the impact loading is represented by a non-linear spring of varying stiffness, while the accelerometer is assumed to behave in a linear manner due to its high stiffness. Specimens made of steel, aluminium and fibre-reinforced composite (FRC) are used in the present study. Assuming the force-displacement response obtained in an actual impact test to be the true behaviour of the test specimen, a suitable numerical approach has been used to solve the governing non-linear differential equations of a three degrees-of-freedom (DOF) system in a piece-wise linear manner. The numerical solution of the governing differential equations following an explicit time integration scheme yields an excellent reproduction of the mechanical behaviour of the specimen, consequently confirming the accuracy of the numerical approach. However, the spring representing the accelerometer predicts a response that qualitatively matches the assumed force-displacement response of the test specimen with a perceptibly lower magnitude of load.

Epoch extraction based on integrated linear prediction residual using plosion index

Epoch is defined as the instant of significant excitation within a pitch period of voiced speech. Epoch extraction continues to attract the interest of researchers because of its significance in speech analysis. Existing high performance epoch extraction algorithms require either dynamic programming techniques or a priori information of the average pitch period. An algorithm without such requirements is proposed based on integrated linear prediction residual (ILPR) which resembles the voice source signal. Half wave rectified and negated ILPR (or Hilbert transform of ILPR) is used as the pre-processed signal. A new non-linear temporal measure named the plosion index (PI) has been proposed for detecting `transients' in speech signal. An extension of PI, called the dynamic plosion index (DPI) is applied on pre-processed signal to estimate the epochs. The proposed DPI algorithm is validated using six large databases which provide simultaneous EGG recordings. Creaky and singing voice samples are also analyzed. The algorithm has been tested for its robustness in the presence of additive white and babble noise and on simulated telephone quality speech. The performance of the DPI algorithm is found to be comparable or better than five state-of-the-art techniques for the experiments considered.

Non-singular terminal sliding mode guidance and control with terminal angle constraints for non-maneuvering targets

In this paper guidance laws to intercept stationary and constant velocity targets at a desired impact angle, based on sliding mode control theory, are proposed. The desired impact angle, which is defined in terms of a desired line-of-sight (LOS) angle, is achieved in finite time by selecting the missile's lateral acceleration (latax) to enforce non-singular terminal sliding mode on a switching surface designed using this desired LOS angle and based on non-linear engagement dynamics. Numerical simulation results are presented to validate the proposed guidance laws for different initial engagement geometries and impact angles.

Study on ride comfort assessment of multiple occupants using lumped parameter analysis

Growing consumer expectations continue to fuel further advancements in vehicle ride comfort analysis including development of a comprehensive tool capable of aiding the understanding of ride comfort. To date, most of the work on biodynamic responses of human body in the context of ride comfort mainly concentrates on driver or a designated occupant and therefore leaves the scope for further work on ride comfort analysis covering a larger number of occupants with detailed modeling of their body segments. In the present study, governing equations of a 13-DOF (degrees-of-freedom) lumped parameter model (LPM) of a full car with seats (7-DOF without seats) and a 7-DOF occupant model, a linear version of an earlier non-linear occupant model, are presented. One or more occupant models can be coupled with the vehicle model resulting into a maximum of 48-DOF LPM for a car with five occupants. These multi-occupant models can be formulated in a modular manner and solved efficiently using MATLAB/SIMULINK for a given transient road input. The vehicle model and the occupant model are independently verified by favorably comparing computed dynamic responses with published data. A number of cases with different dispositions of occupants in a small car are analyzed using the current modular approach thereby underscoring its potential for efficient ride quality assessment and design of suspension systems.

Evaluation of the accuracy of an accelerometer response generated by axial impact loading

The basic objective in the present study is to show that for the most common configuration of an impactor system, an accelerometer cannot exactly reproduce the dynamic response of a specimen subject to impact loading. Assessment of the accelerometer mounted in a drop-weight impactor setup for an axially loaded specimen is done with the aid of an equivalent lumped parameter model (LPM) of the setup. A steel hat-type specimen under the impact loading is represented as a non-linear spring of varying stiffness, while the accelerometer is assumed to behave in a linear manner due to its high stiffness. A suitable numerical approach has been used to solve the non-linear governing equations for a 3 degrees-of-freedom system in a piece-wise linear manner. The numerical solution following an explicit time integration scheme is used to yield an excellent reproduction of the mechanical behavior of the specimen thereby confirming the accuracy of the numerical approach. The spring representing the accelerometer, however, predicts a response that qualitatively matches the assumed load–displacement response of the test specimen with a perceptibly lower magnitude of load.

COMPUTATIONAL MODELING OF A MULTI-LAYERED PIEZO-COMPOSITE BEAM MADE UP OF MFC

The Variational Asymptotic Method (VAM) is used for modeling a coupled non-linear electromechanical problem finding applications in aircrafts and Micro Aerial Vehicle (MAV) development. VAM coupled with geometrically exact kinematics forms a powerful tool for analyzing a complex nonlinear phenomena as shown previously by many in the literature 3 - 7] for various challenging problems like modeling of an initially twisted helicopter rotor blades, matrix crack propagation in a composite, modeling of hyper elastic plates and various multi-physics problems. The problem consists of design and analysis of a piezocomposite laminate applied with electrical voltage(s) which can induce direct and planar distributed shear stresses and strains in the structure. The deformations are large and conventional beam theories are inappropriate for the analysis. The behavior of an elastic body is completely understood by its energy. This energy must be integrated over the cross-sectional area to obtain the 1-D behavior as is typical in a beam analysis. VAM can be used efficiently to approximate 3-D strain energy as closely as possible. To perform this simplification, VAM makes use of thickness to width, width to length, width multiplied by initial twist and strain as small parameters embedded in the problem definition and provides a way to approach the exact solution asymptotically. In this work, above mentioned electromechanical problem is modeled using VAM which breaks down the 3-D elasticity problem into two parts, namely a 2-D non-linear cross-sectional analysis and a 1-D non-linear analysis, along the reference curve. The recovery relations obtained as a by-product in the cross-sectional analysis earlier are used to obtain 3-D stresses, displacements and velocity contours. The piezo-composite laminate which is chosen for an initial phase of computational modeling is made up of commercially available Macro Fiber Composites (MFCs) stacked together in an arbitrary lay-up and applied with electrical voltages for actuation. The expressions of sectional forces and moments as obtained from cross-sectional analysis in closed-form show the electro-mechanical coupling and relative contribution of electric field in individual layers of the piezo-composite laminate. The spatial and temporal constitutive law as obtained from the cross-sectional analysis are substituted into 1-D fully intrinsic, geometrically exact equilibrium equations of motion and 1-D intrinsic kinematical equations to solve for all 1-D generalized variables as function of time and an along the reference curve co-ordinate, x(1).