Approximate Analytical Method Research Articles

Acoustic Scattering Models from Rough Surfaces: A Brief Review and Recent Advances

This paper proposes a brief review of acoustic wave scattering models from rough surfaces. This review is intended to provide an up-to-date survey of the analytical approximate or semi-analytical methods that are encountered in acoustic scattering from random rough surfaces. Thus, this review focuses only on the scattering of acoustic waves and does not deal with the transmission through a rough interface of waves within a solid material. The main used approximations are classified here into two types: the two historical approximations (Kirchhoff approximation and the perturbation theory) and some sound propagation models more suitable for grazing observation angles on rough surfaces, such as the small slope approximation, the integral equation method and the parabolic equation. The use of the existing approximations in the scientific literature and their validity are highlighted. Rough surfaces with Gaussian height distribution are usually considered in the models hypotheses. Rather few comparisons between models and measurements have been found in the literature. Some new criteria have been recently determined for the validity of the Kirchhoff approximation, which is one of the most used models, owing to its implementation simplicity.

Modeling and experiment of vibro-impact vibration energy harvester based on a partial interlayer-separated piezoelectric beam

Piezoelectric-based energy harvesting techniques offer a promising way to transform vibration energy into electric energy. However, many vibration energy harvesters (VEH) can only work under narrow bandwidths and limited high frequencies to restrict their working performance. In this paper, a vibro-impact piezoelectric VEH is proposed, where a partial interlayer-separated piezoelectric beam is designed to improve the voltage output and frequency bandwidth of the VEH. First, the mechanism of the proposed VEH is introduced and the electromechanical model is derived based on the Euler-Bernoulli beam theory and vibro-impact dynamic model. Voltage-frequency responses are then obtained by using an approximate analytical method. In addition, the effect of partial interlayer-separated piezoelectric beams on the energy harvesting performance is investigated numerically. A parametric study is performed to investigate the influence of system parameters on the voltage output in terms of bandwidth and magnitude. Finally, the theoretical solutions are validated by experimental results, the voltage output of the proposed VEH is higher than the non-impact type. The maximum output power of the proposed VEH is about 12 times more than that of the conventional one under a 0.2 g acceleration. Due to the good agreement of the variation trend between the theoretical values and experiment results, the proposed partial interlayer-separated beam VEH can be used for a further optimization of the vibration energy harvester.

Normal Mode Analysis for Connected Plate Structure Using Efficient Mode Polynomials with Component Mode Synthesis

In the engine room and stern adjacent to the main excitation force of the ship, there are many fuel and fresh water tank structures required for ship operation which are always exposed to vibrations. Therefore, it is necessary to review the anti-vibration design to prevent such vibration problems at the design stage, and for this reason, although commercial finite element analysis (FEA) programs are widely used, approximate analysis methods are still developed and used because of the limited time until modeling and analysis results are obtained. Until now, only known elastic boundary conditions have been used in many studies using approximate analysis methods used to calculate natural vibrations for beams or plates. However, many local structures, such as tank edges and equipment foundations, consist of connected structures and it is very difficult to find suitable elastic boundary conditions. Vibration analysis of many local structures in ships, such as tanks and supports for equipment, can be simplified by breaking them up into smaller subsystems which are related through geometrical conditions and natural conditions at junctions. In this study, polynomials for simple support and fixed support were proposed to represent each subsystem and a polynomial to be applied to the plate constituting the tank was proposed by combining them. Until now, there have been many studies on single beams or single plates for approximate analysis. However, there was no research on this to the extent that no reference material could be found for the connected structure. The proposed method has been applied to tanks which are bounded by bulkhead and a deck. The results of this study shows good agreements with those obtained by the FEA Software (Patran/Nastran).

The interactions between multiple arbitrarily orientated inhomogeneities with thermo-porous eigenstrains and its applications in geothermal resources

The thermo-poroelastic model for the fluid migration and heat transport around reservoir may be employed to deal with many geophysical problems including underground resources and bounding rock system, etc. In this paper, interactions between multiple ellipsoidal inhomogeneities with arbitrary orientations and eigenstrains caused by the change of pore fluid pressure, localized heating and cooling are investigated. On the basis of equivalent inclusion method (EIM), an approximate analytical method is presented to evaluate the full displacement, strain and stress fields for arbitrarily distributed inhomogeneities. This method is an effective and straightforward way to study the elastic fields of multiple inhomogeneities, since it avoids increasing the computational complexity. The illustrative benchmark studies are served to demonstrate the accuracy of the present algorithm. Furthermore, applications in geostructures for the elastic distributions of arbitrarily orientated inhomogeneities subjected to the thermo-porous eigenstrains are provided, and the closed-form solutions for the degenerate case of spheroidal inclusions for both the prolate and oblate shapes, spherical and elliptical cylindrical inclusion are presented.

Design and Modeling of Soft Continuum Manipulators Using Parallel Asymmetric Combination of Fiber-Reinforced Elastomers

Abstract Soft continuum arms (SCAs) have a large workspace, dexterity, and adaptability, but at the cost of complex design construction highlighted by concatenating several serial segments. In this paper, we propose a new design architecture for SCAs composed of a parallel combination of pneumatic actuators. The BR2 SCA featured in this work is asymmetric as it combines one soft bending (B) actuator and two soft rotating (R2) actuators as opposed to state of art symmetric architectures that adopt bending segments. Spatial deformation is obtained by combining the bending and rotating feature of the individual actuators. This paper also formulates an approximate forward analysis method based on Kirchhoff rod equations to predict the spatial deformation under external loads with an accuracy less than 9% of the SCAs length. In addition, the model also takes into account the “coupling effect” inherent to the asymmetric parallel combination, where actuating rotating actuator attenuates the bending performance and vice versa. Consequently, this work also refines the design of the SCA that minimizes the coupling effect. A detailed performance study of the refined BR2 manipulator on a swiveling base demonstrates larger workspace and higher dexterity when compared with state of art single section SCAs. The performance of the design is validated through different tasks like obstacle avoidance, pick and place task, and whole arm grasping. These performance attributes surpass any other single segment soft module and is a potential building block for constructing customized SCAs.

Thermodynamics of ultrastrongly coupled light-matter systems

We study the thermodynamic properties of a system of two-level dipoles that are coupled ultrastrongly to a single cavity mode. By using exact numerical and approximate analytical methods, we evaluate the free energy of this system at arbitrary interaction strengths and discuss strong-coupling modifications of derivative quantities such as the specific heat or the electric susceptibility. From this analysis we identify the lowest-order cavity-induced corrections to those quantities in the collective ultrastrong coupling regime and show that for even stronger interactions the presence of a single cavity mode can strongly modify extensive thermodynamic quantities of a large ensemble of dipoles. In this non-perturbative coupling regime we also observe a significant shift of the ferroelectric phase transition temperature and a characteristic broadening and collapse of the black-body spectrum of the cavity mode. Apart from a purely fundamental interest, these general insights will be important for identifying potential applications of ultrastrong-coupling effects, for example, in the field of quantum chemistry or for realizing quantum thermal machines.

A method for constructing exact solutions of nonlinear delay PDEs

We present a new method for constructing exact solutions of nonlinear delay PDEs using special solutions of simpler auxiliary PDEs without delay. The application of the method is demonstrated on nonlinear reaction–diffusion and wave-type equations with delay that include variable coefficients and arbitrary functions. New generalized traveling-wave solutions and functional separable solutions in implicit form are obtained. Results of the paper can be used to formulate test problems intended to verify the adequacy and evaluate the accuracy of numerical and approximate analytical methods of solving the corresponding nonlinear initial-boundary value problems with delay.

An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations

An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations

Introduction and Optimization of a Novel Nonlinear Model for the Free Vibration of Warp-knitted Spacer Fabrics

The aim of this study is to present a nonlinear model that can describe the free vibration damping behavior of 3D knitted fabrics. A fabric is considered as a single degree of freedom model which consists of a nonlinear spring paralleled with a dashpot both in series with a mass. The differential equations of the system motion are solved using the multiple scales method. Damped frequency ωd and damping coefficient $$\widehat\mu $$ are determined using the approximate analytical method, the Imperialistic Competitive Algorithm (ICA) and linear methods to estimate the system response. The obtained system response is then compared with experimental results. The results of the study show that, in comparison with analytical solutions and linear models, ICA can predict the behavior of spacer fabrics more accurately. It is also found that the difference between experimental and predicted models is explained by errors of 8.3 %, 13.7 % and 16.2 % for ICA, approximate analytical and linear model respectively.

Optimal Homotopy Asymptotic Method-Least Square for Solving Nonlinear Fractional-Order Gradient-Based Dynamic System from an Optimization Problem

In this paper, we consider an approximate analytical method of optimal homotopy asymptotic method-least square (OHAM-LS) to obtain a solution of nonlinear fractional-order gradient-based dynamic system (FOGBDS) generated from nonlinear programming (NLP) optimization problems. The problem is formulated in a class of nonlinear fractional differential equations, (FDEs) and the solutions of the equations, modelled with a conformable fractional derivative (CFD) of the steepest descent approach, are considered to find the minimizing point of the problem. The formulation extends the integer solution of optimization problems to an arbitrary-order solution. We exhibit that OHAM-LS enables us to determine the convergence domain of the series solution obtained by initiating convergence-control parameter Cj′s. Three illustrative examples were included to show the effectiveness and importance of the proposed techniques.

Analytical solutions of the concentration of sugar and glucose for enzymatic hydrolysis process

In this paper, we explains the approximate analytical solution for mathematical and kinetic modeling aspects for enzymatic hydrolysis. The approximate analytical method is presented by using Homotopy Analysis Method (HAM) to solve the non linear differential equations with a Michaelis-Menten kinetic law. We attempt the approximation with the help of biofuels like methanol, glycerol and biodiesel. Also explains with Numerical example.

Theoretical modeling of the vertical stiffness of a rolling lobe air spring.

In order to study the characteristics of a rolling lobe air spring, a vertical stiffness analytical model is constructed based on thermodynamics and hydrodynamics. The merit of this vertical stiffness analytical model is that an analytical solution of geometric parameters is obtained by an approximate analytic method. Meanwhile, experimental tests are carried out to verify the accuracy of the vertical stiffness analytical model. The vertical stiffness analytical model can be used to qualitatively analyze the influence of geometric parameters on the vertical stiffness characteristics of a rolling lobe air spring. Therefore, the relationship between geometric parameters and the vertical stiffness characteristics is analyzed based on the proposed model. The conclusions show that the vertical stiffness analytical model can well predict the mechanical characteristics of a rolling lobe air spring and provide guidance for parameter design and vehicle ride comfort improvement.

The stressed­strained state of a rod at crystallization considering the mutual influence of temperature and mechanical fields

This paper reports a solution to the problem of determining the motion law of the crystallization front and the thermomechanical state of a two-phase rod for the case of mutual influence of the temperature and mechanical fields. An approximate analytical method has been used to solve the problem, combined with the method of successive intervals and a Gibbs variation principle. This method should indicate what is more beneficial to nature under the assigned external influences ‒ to change the temperature of the fixed element of a body or to transfer this element from one aggregate state to another. It is this approach that has made it possible, through the defined motion law of an interphase boundary, to take into consideration the effect of temperature on the tense-deformed state in the body, and vice versa. The ratios have been obtained to define the motion law of an interphase boundary, the temperature field, and the tense-deformed state in the rod. The results are shown in the form of charts of temperature and stress dependence on time and a coordinate. An analysis of the results shows that changes in the conditions of heat exchange with the environment and geometric dimensions exert a decisive influence on the crystallization process, and, consequently, on temperature and mechanical fields. The principal result is the constructed approximate analytical method and an algorithm for solving the problem on thermoviscoelasticity for growing bodies (bodies with a moving boundary) in the presence of a phase transition considering the heat exchange with the environment. Based on the method developed, the motion law of an interphase boundary, a temperature field, and the tense-deformed state are determined while solving the so-called quasi-related problem of thermoviscoelasticity. An approximate analytical solution has been obtained, which could be used by research and design organizations in modeling various technological processes in machine building, metallurgy, rocket and space technology, and construction

Langer Modification, Quantization Condition and Barrier Penetration in Quantum Mechanics

The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, the so-called uniform asymptotic approximations, which is an analytical approximation method to calculate the wave functions of the Schrödinger-like equations, and it is applicable to various problems, including cases with poles (singularities) and multiple turning points. A distinguished feature of the method is that in each order of the approximations the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schrödinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well. Possible applications of the method to other fields are also discussed.

Solving the cubic complex Ginzburg-Laundau equation by Homotopy analysis method

Objectives: This paper obtains the series solution of the cubic complex Ginzburg-Laundau equation, by means of homotopy analysis method(HAM). Methods: In addition to the homotopy analysis method, homotopy perturbation and Adomian decomposition methods are applied to determine approximation solution of the cubic complex Ginzburg-Laundau equation and advantage of using HAM. Also a theorem is proved to guarantee the convergence of the HAM to solve this equation. Findings: Three examples are solved to illustrate the efficiency of the proposed method, this method is compared with other analytical approximate methods such as homotopy perturbation method (HPM)and Adomiam decomposition method(ADM) and it can be seen that these methods have the same results for this equation. Application: Homotopy analysis method as a reliable and valid scheme can be used to work out the cubic complex Ginzburg-Laundau equation which is nonlinear partial differential equation. Keywords: Homotopy analysis method; Ginzburg-Laundau

Approximate analytical method for finding eigenvalues of Sturm-Liouville problem with generalized boundary condition of the third kind

Approximate analytical method for finding eigenvalues of Sturm-Liouville problem with generalized boundary condition of the third kind

In-plane non-linear elastic stability of arches subjected to multi-pattern distributed load

Abstract This paper proposes a numerical and an approximate analytical method for the in-plane non-linear elastic stability of arches under the multi-pattern distributed load. There are six key parts in this paper. Firstly, the approximate analytical solution for funicular axis of arches subjected to the multi-pattern distributed load is derived from the approximations of linear elastic bending moment and horizontal reaction force in the arch end. Secondly, the numerical solution of the in-plane non-linear elastic equilibrium equation is solved by using the shooting method and the bisection method simultaneously. Thirdly, the approximate analytical solutions for the in-plane non-linear elastic equilibrium are derived according to the approximate analytical solution for funicular arch axis obtained from the first part, the simplified strain-displacement expression in Cartesian coordinate system and the virtual work principle. Fourthly, a key parameter is proposed to transform the approximate analytical solutions into the corresponding equations of catenary and parabolic arches. Fifthly, the in-plane non-linear elastic symmetric and asymmetric buckling of arches under the multi-pattern distributed load is derived analytically. Lastly, the in-plane non-linear elastic buckling behaviors of arches subjected to multi-pattern distributed load are deduced based on the obtained analytical solutions. The multi-pattern distributed load with the uniformly distributed load along the span and the uniformly distributed load along the arch is selected as example to verify the proposed method. Comparisons with numerical solutions demonstrate that the proposed approximate analytical solutions for the funicular arch axis and linear elastic horizontal reaction force agree well with the results of Runge-Kutta method and the proposed approximate buckling predictions have sufficient accuracy compared with the results of finite element method in different rise-to-span ratios, relative slenderness and arch axis parameters.

Competition between long time excitation and fading of thermoluminescence (TL) and optically stimulated luminescence (OSL)

In the present work, we consider the filling and thermal emptying of traps and centers in a simple one-trap-one-recombination-center model in a small system such as a quartz grain in nature, when both the excitation and thermal release of electrons are very slow. Due to the nature of the very slow process, Monte Carlo simulations appear to be a very appropriate method. However, in parallel, we have applied an approximate analytical method and found practically the same results although the Monte Carlo results showed some small fluctuations due to the statistical nature of the procedure. This is in line with the experimental results which are also expected to have statistical fluctuations. The main result found is that after a long enough time, measured in hundreds or thousands years or more, the filling of the trap reaches a plateau which, depending on the parameters, may be very significantly smaller than the concentration of the trap in question. This equilibrium value is the same if we start from very low, e.g. zero concentration or very high, above the equilibrium value. This plateau level depends strongly on the relevant parameters. However, comparing simulations with activation energies of 1.2 eV and 1.3 eV shows strong dependence of the plateau level on the energy. Similarly, we can expect strong dependence on the temperature at which the sample is held. The results reached here and shown in Figs. 1–4 are based on the simplest OTOR model, but similar results of approaching a plateau level which are not due to the saturation of traps may occur in more complex systems as is demonstrated by simulations based on the Bailey model for quartz which includes several traps and centers.

Nonlinear dynamic characteristics of symmetric rectangular honeycomb sandwich thin panel

In the present work, nonlinear dynamics of symmetric rectangular honeycomb sandwich thin panel is investigated. The nonlinear governing equations of the thin panel are derived by using Hamilton’s principle and Reddy’s third-order shear deformation theory. After that, the Galerkin method is used to discretize the equation of motion, resulting in a set of ordinary differential equations with respect to time. The ordinary differential equations are solved analytically by utilizing the homotopy analysis method. The influence of structural parameters to the nonlinear frequencies of the symmetric rectangular honeycomb sandwich panel with simply supported boundaries along all four edges is discussed by using the analytic approximation method. Our findings demonstrate that the nonlinear frequency ratio decreases first and then increases with the increase of the width-to-length ratio and thickness-to-length ratio. When the width-to-length ratio is greater than 10, the nonlinear frequency ratio remains almost unchanged with the further increase of width-to-length ratio.

Mathematical Modeling and Characteristic Analysis of the Vertical Stiffness for Railway Vehicle Air Spring System

Establishing a correct and reliable vertical stiffness model has an important significance on reproducing the characteristics of an air spring system. In this paper, a dynamic vertical stiffness model is developed based on thermodynamics and fluid dynamics, and geometric parameters are identified by an approximate analytical method. Meanwhile, experimental tests are performed to verify the accuracy and reliability of the proposed model. Furthermore, the impact of geometric parameters on the vertical stiffness characteristics is discussed through a sensitivity analysis. The conclusions show that the dynamic vertical stiffness model can well characterize the dynamic characteristics of the air spring system, which provides a theoretical basis for the optimal design of air spring parameters and the study of mechanical properties.