Approximate Analytical Technique Research Articles

Determination of the amplitude-frequency for strongly nonlinear oscillator by two approximate analytical techniques

In this paper, we investigate two of the analytical approximate techniques, energy balance method and amplitude-frequency formulation, and these approximate techniques are applied to solve the strongly nonlinear differential equation of a mass attached to the center of a stretched elastic wire. We present a comparative study between the energy balance method and amplitude-frequency formulation with exact solution. The approximate results reveal that these methods are very effective and convenient for determining the frequencies of nonlinear dynamical systems.

Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method

Application of Optimal Homotopy Asymptotic Method (OHAM), a new analytic approximate technique for treatment of Falkner-Skan equations with heat transfer, has been applied in this work. To see the efficiency of the method, we consider Falkner-Skan equations with heat transfer. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature as finite difference (N. S. Asaithambi, 1997) and shooting method (Cebeci and Keller, 1971). The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

Nonlinear Hydroelastic Waves beneath a Floating Ice Sheet in a Fluid of Finite Depth

The nonlinear hydroelastic waves propagating beneath an infinite ice sheet floating on an inviscid fluid of finite depth are investigated analytically. The approximate series solutions for the velocity potential and the wave surface elevation are derived, respectively, by an analytic approximation technique named homotopy analysis method (HAM) and are presented for the second-order components. Also, homotopy squared residual technique is employed to guarantee the convergence of the series solutions. The present formulas, different from the perturbation solutions, are highly accurate and uniformly valid without assuming that these nonlinear partial differential equations (PDEs) have small parameters necessarily. It is noted that the effects of water depth, the ice sheet thickness, and Young’s modulus are analytically expressed in detail. We find that, in different water depths, the hydroelastic waves traveling beneath the thickest ice sheet always contain the largest wave energy. While with an increasing thickness of the sheet, the wave elevation tends to be smoothened at the crest and be sharpened at the trough. The larger Young’s modulus of the sheet also causes analogous effects. The results obtained show that the thickness and Young’s modulus of the floating ice sheet all greatly affect the wave energy and wave profile in different water depths.

Variational principle for Zakharov–Shabat equations in two-dimensions

We study the corresponding scattering problem for Zakharov and Shabat compatible differential equations in two-dimensions, the representation for a solution of the nonlinear Schrödinger equation is formulated as a variational problem in two-dimensions. We extend the derivation to the variational principle for the Zakharov and Shabat equations in one-dimension. We also developed an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameters in Zakharov and Shabat equations for a given pulse-shaped potential, which is equivalent to the physically important problem of finding the soliton content of the given initial pulse. Using a trial function in a rectangular box we find the functional integral. The general case for the two box potential can be obtained on the basis of a different ansatz where we approximate the Jost function by polynomials of order n instead of a piecewise linear function. We also demonstrated that the simplest version of the variational approximation, based on trial functions with one, two and n-free parameters respectively, and treated analytically.

Slotted ALOHA performance for FU-FB in frequency selective fading environment

Slotted ALOHA is a simple and straightforward random multiple access technique, which has been used extensively in data and cellular networks as the protocol for random access. The complexity of state space-based analysis methods for finite user finite buffer systems increases exponentially with buffer size and number of users. The presence of multipath frequency selective fading channel further adds to the complexity, making the analysis practically intractable. This paper uses an approximate analysis technique called tagged user analysis TUA to analyze the performance parameters of slotted ALOHA over multipath and frequency selective fading channels for finite user finite buffer systems. In TUA, the steady state system performance is evaluated from the analysis of a single user. Moreover, the state flow graph of TUA has just four states, thus reducing the complexity of the analysis. Simulation results confirm the validity of the TUA analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Analytical solution for evaluating the thermal performance and efficiency of convective–radiative straight fins with various profiles and considering all non-linearities

If the temperature change from the base to the tip of the fin is large, then for an accurate prediction of the performance of the fin, the model must incorporate the variation of thermal conductivity with temperature. Also, the heat exchange with the surrounding air is associated with the heat transfer coefficient. It has been experimentally proven that the heat transfer coefficient may not be uniform and there are large differences in the heat transfer coefficient both from fin base to fin tip and along the fin in the direction of the airflow. Same as two parameters mentioned above, the surface emissivity and internal heat generation within the fins may vary with temperature. This paper analytically investigates the temperature distribution within convective–radiative straight fins with simultaneous variation of thermal conductivity, heat transfer coefficient, surface emissivity and internal heat generation with temperature. Also, convection and radiation sink temperatures were assumed to be non-zero. Four different longitudinal fins such as rectangular, triangular, convex and exponential were considered. The calculations are carried out using the well-known approximate analytical differential transformation technique. Fin efficiency and the effects of some physically applicable parameters in this problem such as thermal conductivity, convection–conduction parameter, radiation–conduction parameter, convection sink temperature and radiation sink temperature were studied comparatively. Because a broad range of governing parameters are investigated, the results should be useful in a number of engineering applications.

Analytical approximate technique for strongly nonlinear oscillators problem arising in engineering

In this paper, a novel method called generalized of the variational iteration method is presented to obtain an approximate analytical solution for strong nonlinear oscillators problem associated in engineering phenomena. This approach resulted in the frequency of the motion as a function of the amplitude of oscillation. It is determined that the method works very well for the whole range of parameters and an excellent agreement is demonstrated and discussed between the approximate frequencies and the exact one. The most significant features of this method are its simplicity and excellent accuracy for the whole range of oscillation amplitude values. Also, the results reveal that this technique is very effective and convenient for solving conservative oscillatory systems with complex nonlinearities. Results obtained by the proposed method are compared with Energy Balance Method (EBM) and exact solution showed that, contrary to EBM, simply one or two iterations are enough for obtaining highly accurate results.

Frequency response curves of a spindle bearings system with non-linear and non-smooth terms

This article presents results of vibrations of spindle system subjected to non-linear and non-smooth ball bearing stiffness. The averaging method, an approximate analytical technique, is applied directly to the non-linear and non-smooth differential equations of motion which ultimately leads to approximate analytical expressions for the frequency response relations of the system. The effects of ball bearing clearance and spindle structural damping on the frequency response curves are investigated. Hardening-type non-linear behaviours are observed in the ranges of base excitation frequency ω near the first, second and third natural frequencies. Crossing and breaking frequency response curves are also observed due to the non-linear and non-smooth properties of ball bearing stiffness. These solutions are clear indications that significant non-linear and non-smooth phenomena are of fundamental importance in the study of spindle bearing system. They also provide direct insight into the relationship between the system parameters and vibration characteristics of the system.

Software Bertillonage

Deployed software systems are typically composed of many pieces, not all of which may have been created by the main development team. Often, the provenance of included components--such as external libraries or cloned source code--is not clearly stated, and this uncertainty can introduce technical and ethical concerns that make it difficult for system owners and other stakeholders to manage their software assets. In this work, we motivate the need for the recovery of the provenance of software entities by a broad set of techniques that could include signature matching, source code fact extraction, software clone detection, call flow graph matching, string matching, historical analyses, and other techniques. We liken our provenance goals to that of Bertillonage, a simple and approximate forensic analysis technique based on bio-metrics that was developed in 19th century France before the advent of fingerprints. As an example, we have developed a fast, simple, and approximate technique called anchored signature matching for identifying the source origin of binary libraries within a given Java application. This technique involves a type of structured signature matching performed against a database of candidates drawn from the Maven2 repository, a 275 GB collection of open source Java libraries. To show the approach is both valid and effective, we conducted an empirical study on 945 jars from the Debian GNU/Linux distribution, as well as an industrial case study on 81 jars from an e-commerce application.

An Optimal Approach to Study the Nonlinear Behaviour of a Rotating Electrical Machine

We introduce a novel analytical approximate technique, called Optimal Variational Method (OVM), to investigate the nonlinear behaviour of a rotating electrical machine modelled as an oscillator with cubic elastic restoring force and time variable coefficients. The proposed procedure involves the presence of some initially unknown convergence‐control parameters whose values are later optimally determined. Comparisons between the obtained results and exact ones reveal that OVM is very effective and convenient providing highly accurate results.

A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

A closed form solution for nonlinear oscillators frequencies using amplitude-frequency formulation

Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of a class of randomly excited nonlinear oscillators is developed. Specifically, combining the concepts of statistical linearization and of stochastic averaging the evolution of the response amplitude of oscillators with nonlinear damping is captured by a first-order stochastic differential equation (SDE). This equation has nonlinear drift but constant diffusion coefficients. This convenient feature of the SDE along with the concept of the Wiener path integral is utilized in conjunction with a variational formulation to derive an approximate closed form solution for the response amplitude PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes. In this manner, an analytical Wiener path integral based technique is developed for treating certain stochastic dynamics problems for the first time. Further, the technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

Theory of hybrid photorefractive plasmonic liquid crystal cells

We use the theory of optical waveguides to study analytically the voltage-dependent response of a surface plasmon polariton (SPP) at the interface between a photorefractive liquid crystal cell and a semi-infinite gold layer. For sufficiently large electric fields the alignment of the liquid crystal can be calculated analytically. The resulting correction to the SPP dispersion relation is then determined in terms of the applied field and the liquid crystal surface alignment relative to the SPP propagation direction. The approximate analytic techniques developed here are shown to be accurate when compared to rigorous diffraction theory and experimental measurements. The approximate equations are a powerful tool of general application. They can be used to study SPP propagation at the interface between a metal and any nonhomogeneous or anisotropic dielectric and are also applicable to self-assembled monolayers and biosensing applications.

Analytical techniques for analysis of fully developed laminar flow through rectangular channels

Approximate analytical techniques are established to determine the velocity distribution for laminar fluid flow through straight rectangular channels. The separation of variables is adopted for the exact analytical solution. The closed form solution for the temperature distribution in the flow field is demonstrated and the Nusselt number is evaluated. The result determined by different techniques has been presented in a comparative way to easily understand the accuracy level of predictions.

Pressure in MHD/Brinkman flow past a stretching sheet

This paper deals with the pressure in a steady two-dimensional/axisymmetric MHD/Brinkman flow of an incompressible viscous electrically conducting fluid over a flat stretching sheet. The stretching rate of the two-dimensional case is assumed as double the stretching rate of the axisymmetric case in order to compare them by means of a unified scale. A recently proposed approximate analytic technique Kumaran et al. [1] was used to recover an exact solution of the two-dimensional case and developed an approximate analytical solution of the axisymmetric case by Kumaran and Tamizharasi [2]. In this paper, the pressure distribution of the MHD and the porous medium cases are plotted and compared. In the MHD case, the pressure distribution is finite. Also, the pressure distribution is totally different in the outer boundary layer for the two-dimensional and the axisymmetric cases.

Back-of-the-envelope calculations in friction stir welding – Velocities, peak temperature, torque, and hardness

Given the complexity and resource requirements of numerical models of friction stir welding (FSW), well-tested analytical models of materials flow, peak temperatures, torque, and weld properties are needed. Here an approximate analytical technique for the calculation of three-dimensional material flow during FSW is proposed considering the motion of an incompressible fluid induced by a solid rotating disk. The accuracy of the calculations is examined for the welding of three alloys. For the estimation of peak temperatures, the accuracy of an existing dimensionless correlation is improved using a large volume of recently published data. The improved correlation is tested against experimental data for three aluminum alloys. It is shown that the torque can be calculated analytically from the yield stress using estimated peak temperatures. An approximate relation between the hardness of the thermomechanically affected zone and the chemical composition of the aluminum alloys is proposed.

Low-Profile, Highly-Selective, Dual-Band Frequency Selective Surfaces With Closely Spaced Bands of Operation

We present a new design of a dual-band frequency selective surface (FSS) with closely spaced bands of operation and a highly-selective frequency response at each band. A multi-stage design procedure is also proposed for the design and synthesis of this class of frequency selective surfaces. The design procedure is based on synthesizing the desired device from its equivalent circuit parameter values. An approximate analytical technique is provided, which can be used to determine the values of the equivalent circuit parameters of this dual-band device from the basic system level characteristics of its transfer function including the center frequencies of operations of its two bands of operation and the bandwidth at each of these bands. The use of this design procedure is described in detail by a design example which follows the proposed design procedure step by step; the validity of the procedure is verified using full-wave EM simulations and experimental characterization of a fabricated prototype of the proposed device. Experimental characterizations of this device show that it has a stable frequency response as a function of angle of incidence for both the transverse electric (TE) and the transverse magnetic (TM) polarizations. This stability is attributed to the extremely small overall thickness of the structure as well as its small unit cell dimensions.

A Sensitivity Analysis of Nuclear Assembly Extinction Probabilities: Application to Reevaluated Fission Multiplicity Data

We implement direct and approximate local sensitivity analysis techniques within the context of stochastic point kinetics neglecting delayed neutrons and external neutron sources. After reviewing the derivation of certain probabilities that the neutron population in a nuclear assembly is exactly zero [probabilities of extinction (POEs)], we consider their dependence on physical data. We subsequently focus on fission number distribution dependence and draw comparisons between two different data sets. As various POEs are dependent upon these data through the solution of a nonlinear ordinary differential equation, local sensitivity analysis provides a useful means through which to assess the effects of data reevaluation. We first conduct this analysis generally (though approximately) using Gâteaux-derivative methodology. Following the generalized developments, exact and approximate results for 235U are presented with a discussion concerning important consequences related to criticality safety.

Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

This paper is concerned with analytical treatment of non-linear oscillations of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. An analytic approximate technique, namely Optimal Homotopy Asymptotic Method (OHAM) is employed for this purpose. It is proved that OHAM provide accurate solutions for large amplitudes and large modal constants in the considered nonlinear equations, when other classical methods fail. Our procedure provides us with a convenient way to optimally control the convergence of solution, such that the accuracy is always guaranteed. An excellent agreement of the approximate frequencies and periodic solutions with the numerical results and published results has been demonstrated. Two examples are given and the results reveal that this procedure is very effective, simple and accurate. This paper demonstrates the general validity and the great potential of the OHAM for solving strongly nonlinear problems.