Journal of Applied and Computational Mechanics
Volume:8 Issue: 4, Autumn 2022

Twin screw machines can be used as an expander to recover the lost power in various processes causing pressure energy loss. Twin screw expanders (TSEs) have caught the attention of many researchers due to low capital, maintenance and operation costs, long lifespan and their application in two-phase fluids. However, substantial efforts required to enhance their performance. This research describes the optimization of the profile of a TSE with 4-6 lobe configuration - using surrogate-based modeling (SBM). To do so, based on the in-house code developed within FORTRAN, a TSE profile is designed and validated against available data. Then, a mathematical model is developed viaof experiments (DOE). Next, the effects of four main profile parameters are investigated on the expander performance in the entire design space. Finally, an optimized combination of parameters is offered using a multi-objective genetic algorithm. 3D computational fluid dynamics (CFD) results show that the optimized profile had more than 7% exergy efficiency compared to the base profile.

The present paper analyzes free convective heat and mass transfer of non-conducting micropolar fluid flow over an infinitely inclined moving porous plate in the presence of heat source and chemical reaction. Moreover, the effect of thermal radiation is also taken care of in the same study. The present investigation is relevant to the fabrication system in industries corresponding to the materials composed with high-temperature. Similarity technique is adopted with similarity variable to transform the non-dimensional form of the partial differential equations into ordinary differential equations. To get the approximate solution of these transformed complex nonlinear set of ODEs we have employed fourth order Runge–Kutta method in conjunction with shooting technique. The validation of the present result as well as critical issues is addressed in the discussion section refereeing to the previously published work as a particular case. The behavior of physical parameters governs the flow phenomena are displayed via graphs and tables.

The flow mechanism and entropy production of a bi-convective, magnetized, radiative nano-liquid flow for an inverted cone considering temperature-sensitive water properties is accomplished numerically. The functional nanomaterial comprises Copper, Alumina in the base liquid, water. The mathematical equations representing the system's physical characteristics are solved numerically by adopting a robust numerical approach for indulgencing non-similar solutions to understand numerous parameters' effect on temperature, velocity, salient gradients, and entropy production. The investigation summarizes that buoyancy force and injection heighten the velocity, and suction, particle percentage, radiation elevate the heat transfer. At the same time, the radiation and Brinkman number enhance the entropy generation. It is also detected from this investigation that the magnetic effect shows dual behaviour in entropy generation.

This study presents the empirical comparison between the wing root and wingtip corrugation patterns of dragonfly wing in the newly-built wind-tunnel at the IAUN. The main objective of the research is to investigate the effect of wingtip and wing root corrugations on aerodynamic forces and the flow physics around the cross-sections at Re=10000 and the angle of attack of 0° to 30°. For this aim, two cross-sections are extracted from wing root (first cross-section) and wingtip (second cross-section). The first cross-section has corrugations with higher density than the second cross-section. The comparison of lift coefficients obtained from pressure distribution and force measurement indicates an acceptable agreement between the results. Also, Particle Image Velocity (PIV) technique is used to measure the velocity field. The results show that all corrugation patterns do not have positive effects on the aerodynamic forces. The second cross-section can generate considerable aerodynamic forces compared to the first cross-section. At α=25°, the lift coefficient generated by the second cross-section is 90% and 25% higher than that of the first cross-section and the flat plate, respectively. Based on results, corrugations in the wing root's vicinity have a crucial role in the solidity of insect wings; however, corrugations in the wing tip's vicinity play a vital role in generating adequate aerodynamic forces. The comparison conducted in the current research reveals the second cross-section is an appropriate replacement for the flat plate in MAVs due to generating more essential forces for flight.

Characterization of the mechanical properties of soft biological tissues is a fundamental issue in a variety of medical applications. As such, constitutive modeling of biological tissues that serves to establish a relationship between the kinematic variables has been used to formulate the tissue’s mechanical response under various loading conditions. However, the validation of the developed analytical and numerical models is accompanied by a length of computation time. Hence, the need for new advantageous methods like artificial intelligence (AI), aiming at minimizing the computation time for real-time applications such as in robotic-assisted surgery, sounds crucial. In this study, at first, the mechanical nonlinear characteristics of human ureter were obtained from planar biaxial test data, in which the examined specimens were simultaneously loaded along their circumferential and longitudinal directions. To do so, the biaxial stress-strain data was used to fit the well-known Fung and Holzapfel-Delfino hyperelastic functions using the genetic optimization algorithm. Next, the potential of Artificial Neural Networks (ANN), as an alternative method for prediction of the mechanical response of the tissue was evaluated such that, multilayer perceptron feedforward neural network with different architectures was designed and implemented and then, trained with the same experimental data. The results showed both approaches were practically able to predict the ureter nonlinearity and in particular, the ANN model can follow up the tissue nonlinearity during the entire loading phase in both low and high strain amplitudes (RMSE<0.02). Such results confirmed that neural networks can be a reliable alternative for modeling the nonlinear mechanical behavior of soft biological tissues.

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

In the current study, a comparative analysis of two-dimensional heat transfer by the free convective flow of non-Newtonian Casson and Carreau fluid in electro-conductive polymer on the outside surface of a horizontal circular cylinder under slip and radial magnetic field effects is regarded. The Casson and Carreau fluid model formulation were first developed for the problem of the boundary layer of the horizontal circular cylinder and by using non-similarity transformations, the combined governing partial differential equations are translated into ordinary differential equations. The differential equations obtained are resolved by the Keller Box Method (KBM). The impact of the key parameters, the rate of heat transfer and skin friction is evaluated through graphs and tables. The result reveals that an increase in magnetic number decreases the velocity field of both Casson and Carreau fluid also Casson fluid is higher values when compared to Carreau fluid in variation of magnetic number.

Artificial Neural Networks (ANN) are designed to evaluate the pull-in voltage of MEMS switches. The mathematical model of a micro-switch subjected to electrostatic force is preliminarily illustrated to get the relevant equations providing static deflection and pull-in voltage. Adopting the Step-by-Step Linearization Method together with a Galerkin-based reduced order model, numerical results in terms of pull-in voltage are obtained to be employed in the training process of ANN. Then, feed forward back propagation ANNs are designed and a learning process based on the Levenberg-Marquardt method is performed. The ability of designed neural networks to determine pull-in voltage have been compared with previous results presented in experimental and theoretical studies and it has been shown that the presented method has a good ability to approximate the threshold voltage of micro switch. Furthermore, the geometric and physical effect of the micro-switch on the pull-in voltage was also examined using these designed networks and relevant findings were provided.

The prime objective of the current investigation is to explore the variation of viscosity and thermal conductivity impacts on MHD convective flow over a moving non-isothermal vertical plate in presence of the viscous-dissipative heat and thermal-radiation. The compatible transformation of similarity are employed to obtain the non-linear ODE with the appropriate boundary conditions from the governing equations and the numerical solution of the boundary value problem so obtained are solved via MATLAB bvp4c solver. Naturally, the fluid viscosity and thermal-conductivity may vary from liquid to metal with temperatures and therefore, the impact of viscosity and thermal-conductivity in this investigation is quite significant. The physical parameters along with several influences on momentum, temperature, and concentration are explicated and portrayed with graphs. In addition, the velocity, temperature and concentration gradients at the surface are evaluated and displayed in tabular form. A decent agreement is found in the present outcomes with previously issued work. Furthermore, it is found that the growth of the thermal-radiation increases the gas temperature. The present study is useful for various industrial applications like metal and polymer extrusion, continuous casting, cooling process, nuclear plant and many more.

This theoretical study analyses the effects of geometrical and fluid parameters on the flow metrics in the Hagen-Poiseuille and plane-Poiseuille flows of Herschel-Bulkley fluid through porous medium which is considered as (i) single pipe/single channel and (ii) multi–pipes/multi-channels when the distribution of pores size in the flow medium are represented by each one of the four probability density functions: (i) Uniform distribution, (ii) Linear distribution of Type-I, (iii) Linear distribution of Type-II and (iv) Quadratic distribution. It is found that in Hagen-Poiseuille and plane-Poiseuille flows, Buckingham-Reiner function increases linearly when the pressure gradient increases in the range 1 - 2.5 and then it ascends slowly with the raise of pressure gradient in the range 2.5 - 5.In all of the four kinds of pores size distribution, the fluid’s mean velocity, flow medium’s porosity and permeability are substantially higher in Hagen-Poiseuille fluid rheology than in plane-Poiseuille fluid rheology and, these flow quantitiesascend considerably with the raise of pipe radius/channel width and a reverse characteristic is noted for these rheological measures when the power law index parameter increases.The flow medium’s porosity decreases rapidly when the period of the pipes/channels distribution rises from 1 to 2 and it drops very slowly when the period of the pipes/channels rises from 2 to 11.

The flow and heat transfer of a novel type of functional phase change nanofluids, nano-‎encapsulated phase change suspensions, is investigated in the present study using a deep neural ‎networks framework. A deep neural network was used to learn the natural convection flow and ‎heat transfer of the phase change nanofluid in an enclosure. A dataset of flow and heat transfer ‎samples containing 3290 samples of the flow field and temperature distributions was used to ‎train the deep neural network. The design variables were fusion temperature of nanoparticles, ‎Stefan number, and Rayleigh number. The results showed that the proposed combination of a ‎feed-forward neural network and a convolutional neural network as a deep neural network could ‎robustly learn the complex physics of flow and heat transfer of phase change nanofluids. The ‎trained neural network could estimate the flow and heat transfer without iterative and costly ‎numerical computations. The present neural network framework is a promising tool for the design ‎and prediction of complex physical systems‎.

A new theoretical tri-hybrid nanofluid model for enhancing the heat transfer is presented in this article. This model explains the method to obtain a better heat conductor than the hybrid nanofluid. The tri-hybrid nanofluid is formed by suspending three types of nanoparticles with different physical and chemical bonds into a base fluid. In this study, the nanoparticles TiO2, Al2O3 and SiO2 are suspended into water thus forming the combination TiO2-SiO2-Al2O3-H2O. This combination helps in decomposing harmful substances, environmental purification and other appliances that requires cooling. The properties of tri-hybrid nanofluid such as Density, Viscosity, Thermal Conductivity, Electrical Conductivity and Specific Heat capacitance are defined mathematically in this article. The system of equations that governs the flow and temperature of the fluid are converted to ordinary differential equations and are solved using RKF-45 method. The results are discussed through graphs and it is observed that the tri-hybrid nanofluid has a better thermal conductivity than the hybrid nanofluid.

Large scale of hydrogen storage is needed to balance the energy supply-demand fluctuation issues. Among few of the large scale storage systems, depleted oil and gas wells are widely employed. The construction of wellbore is normally in cylindrical shape and formed by layers of cement, casing and formation. As failure of wellbore is costly, proper structural integrity assessment is essential. In this article, an analytical solution derived based on recursive algorithm for estimating the thermomechanical stresses across the wellbore structure was proposed and verified. The temperature and stresses distribution results obtained from proposed analytical solution were compared with numerical results and they were found in good agreement. The percentage of difference was observed to be less than 0.1%. Besides that, a comparison study was performed on two, four and six layers wellbore structure. It was observed that four and six layers structure can produce much lower tangential tensile stress on the steel casing of the wellbore.

The nonlinear deformation and vibrations of beams are frequently encountered as a typical example of structural analysis as well as a mathematical problem. There have been many methods and techniques for the approximate and exact solutions of nonlinear differential equations arising from the nonlinear phenomena of elastic beam structures. One method is particularly more powerful and flexible is proposed recently as the extended Rayleigh-Ritz method (ERRM) by adding the temporal variable as another dimension of deformation formulation but eliminated through the integration over a period of vibrations. Such a procedure leads to a simple, elegant, and powerful method for the approximate solutions of nonlinear vibration and deformation problems in dynamics and structural analysis. By utilizing the usual displacement function of beams, the nonlinear vibration frequencies of Euler-Bernoulli and Timoshenko beams are obtained with the same accuracy as from other approximate solutions.

The study aims to compare the characteristics of the moment-inertial schemes of two aircrafts – a mainline aircraft of a normal aerodynamic scheme and an aircraft made according to the flying wing scheme, to improve their flight performance. The study uses the method of successive approximations using relative masses (when determining m0), the formula of A. Mozhaisky, an artificial method consisting of the layout of the aircraft oriented to the virtual center of mass. Design studies at the modern level of scientific and technical development have confirmed the relevance of using the proposed methods of forming a moment-inertial appearance for promising long-haul aircraft of large passenger capacity.

In this work, the computational study of Lüders phenomenon is addressed. The material for investigation is low-carbon steel demonstrating the yield point phenomenon when pulled in tension. Modeling of samples loading is carried out in the framework of three-dimensional finite-difference method. Judging by the literature review, there is a lack of papers thoroughly addressing the curves of dependences of Lüders elongation and front propagation velocity on parameters of up-down-up constitutive equation. This work fills this gap. It is shown that the difference between the true upper and lower yield stresses, and strain hardening factor have a strong impact on the duration of the yield plateau stage and ratio of front propagation velocity vf to loading velocity vl. The results of computational study complement the experimental data presented in available literature.

Hybrid composites reinforced by two types of fibres e.g., carbon and glass have been widely researched. These two types of fibres are distinct in density and cost. An optimisation study on the carbon and glass fibre reinforced hybrid composite in three-point bending is presented in this paper. Both unidirectional and multi-directional hybrid composites are studied. The objectives are minimising the cost and weight with the flexural strength being the constraint. The three-point bending is simulated by a Finite Element Analysis based model, and optimisation is done with non-dominated sorting GA-II (NSGA-II). This optimisation approach can be extended to hybrid composites reinforced by other types of fibres.

In this paper, we propose a spectral approach to model finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. The constructed constitutive equations depend on spectral invariants, where each one has a clear physical meaning and hence are attractive for use in experiment and analysis. With the use of spectral invariants, we easily obtain the number of independent invariants and the number of invariants in the corresponding minimal integrity or irreducible basis. The proposed finite strain energy prototypes are consistent with their infinitesimal strain energy function counterparts. Some results for pure bending of a slab, and the extension and torsion of solid cylinder, that could be useful for experiments and numerical validations, are given. The proposed model could be used to obtain numerical results via modification of some computational methods found in the literature.

Thin-walled shell structures are widely used in various fields of engineering, and the process of their deformation is essentially non-linear, which significantly complicates their analysis. The author presents a new mathematical model of shell structure deformation under dynamic loading, taking into account geometrical nonlinearity, transverse shears, and stiffeners. A distinctive feature of the model is the use of the refined discrete method to account for stiffeners, suggested by the author earlier. The method has previously been used only in static problems. It uses adjusting normalizing factors and makes it possible to obtain the most correct critical load values. A technique for numerically investigation of the process of deformation of such structures under dynamic loading is based on the L.V. Kantorovich method and Rosenbrock method for solution of a stiff ODE system. The proposed approach, in contrast to commercial software based on FEM, makes it possible to study in detail the supercritical deformation of structures, to identify patterns of influence of individual parts of the mathematical model on critical loads. Calculations for shallow doubly curved shells and cylindrical panels are provided. It is shown how the number of stiffeners affects the resulting buckling load values. A comparison with the classic discrete method to account for stiffeners is performed. Phase portraits of the system are given for all problems considered.

The main purpose of this study is to deal with a thermoelastic medium containing a spherical cavity within the framework of partial elastic thermal diffusion theory based on the Atangana-Baleanu operator which is characterized by the presence of a non-local single kernel. The chemical potential of the adjacent cavity is taken as a time-dependent function. The governing equations are represented and solved in the Laplace transform domain and the numerical solutions to the Laplace inversion are obtained to address the problem in the physical domain. In the physical field, the expansion of the Fourier approach is also used to obtain the numerical solutions and the stress-strain behavior of the studied medium is graphically illustrated.

In this paper, we present a mathematical model based on the new higher shear deformation plate theory to investigate the thermo-elastic response of carbon nanotube reinforced composites (CNTRC) cross-ply laminated plates under temperature loading. Functionally graded distributions (FG) and uniform distribution (UD) of carbon nanotube reinforcement material are examined. A higher-order deformation plate that contains only four unknowns is utilized together with the principle of virtual displacement to derive the governing equations of CNTRC cross-ply laminated plates with simply supported edge conditions. Subsequently, Navier’s solution is proposed for simply supported cross-ply CNTR composite laminated plates subject to linear, nonlinear and combined variations in temperature through plate thickness. The analytical model was validated by comparing the obtained primary outcomes with those available in the literature. The numerical results of present simple analytical model are presented to show the influence of the CNT volume fraction, laminated composite structure, side to thickness and aspect ratio on the thermal stresses and deflection of the CNTRC cross-ply laminated plates.

This paper summarized some common grading curves of ballast layers and found that the content of 16-32 mm ballast particles ("middle-size particles" in this paper) had a significant effect on the direct shear performance of the ballast layer. In this paper, the direct shear tests of the ballast layers with different contents of middle-size particles were reproduced using the discrete element method (DEM). Two different compactions of the ballast samples were used, and the reasons for the changes of shear strength of the ballast layers with different size distributions were analyzed from macroscopic and microscopic perspectives. The results showed that the strengthening effect of the ballast due insertion of middle-particles could only be observed for normally compacted ballast, whereas the same insertion with fully compacted ballast would decrease the shear strengths properties. The fully compacted ballast is subjected to the dilation. The reason of the strengthening effect for the normally compacted ballast were the contraction and dilation processes. Insertion of the middle-size particles up to 20-30% at most increase the dilation processes. Thus, the results show that the ballast layers with conventional narrow particle size distribution (narrow PSD) have higher shear strength than wide range particle size distribution (wide range PSD) if the ballast is good fully compacted. Additionally, it should be noted that the number of small particles will increase during the lifecycle of the ballast layer due to corner brakeage and the external contamination. Moreover, the drainage aspects of the wide range PSD should be considered. Therefore, the excessive insertion of middle-size particles is not justified.

This paper proposes a hyperbolic heat transport model for a homogeneously perfused biological tissue irradiated by a laser beam. In particular, involving two local energy equations, one for the blood vessel and the other for the tissue, a non-Fourier-like heat equation is introduced and solved analytically using the Laplace transform method. The generalized hyperbolic model obtained is reduced to Pennes' heat transport equation in case the thermal delay time is zero and the solution obtained is in accordance with the numerical and experimental data existing in the literature. In addition, the achieved results also show that the effects of thermal relaxation and blood perfusion on temperature distribution are similar; indeed the highest temperature is expected when the delay time tR increases during tissue cooling. Finally, the consequences of the change in the values of the physical parameters characterizing the model are described and the effect of thermal relaxation on the temperature profile in the tissue during and after laser application is investigated.

The mechanical properties of a polycarbonate matrix composite with glass fiber reinforcements used for the manufacture of a multistage centrifugal pump impeller are researched in this article. The material properties are modelled using DIGIMAT (The Material Modelling Platform) to determine the strain resistance of the composite with different proportions of reinforcements. The Tsai–Hill failure criterion is used to determine the strength in all cases. The results have been verified by physical testing to determine the influence of the shape and mass proportion of reinforcements on its mechanical properties. The strength of the manufactured part is correlated to technological factors using the MARC MENTAT solver, and the most and the least favorable combinations of these factors are determined.

This article aims to investigate the free vibration of axial and bi-directional functionally graded (2D-FG) two-dimensional plane stress strip by using finite element method. The rule of mixture based on Vogit model is proposed to describe the change in the volume fractions of metal and ceramics constituents. The materials are graded continuously and smoothly in both axial and thickness directions according to the power law formula. Two-dimensional plane stress constitutive equations are proposed to describe the stress and strain across the beam domain. Finite element model using ANSYS software is developed to discretize the spatial domain of strip and modal solution is exploited to evaluate the eigenvalues (natural frequencies) and mode shapes of 2D FG strip beam. The effects of materials gradation in axial and bi-directional and boundary conditions on the natural frequencies are investigated. The proposed model can be used in design and analysis of 2D-FG structures manufactured from two different constituents and selecting the optimum gradation parameter based on the natural frequency’s constraints, such as naval, nuclear and aerospace structures.

Hot stamping involves deforming a heated blank to form components with increased mechanical strength. More recently, warm stamping procedures have been researched. The forming occurs at lower temperatures to improve process efficiency. The process is non-linear and inefficient to solve using finite element simulations and surrogate models. This paper presents the use of dimension-reduced neural networks (DR-NNs) for predicting temperature distribution in FEM warm stamping simulations. Dimensionality reduction methods transformed the input space, consisting of assembly, material, and thermal features, to a set of principal components used as input to the neural networks. The DR-NNs are compared against a standalone neural network and show improvements in terms of lower computational time, error, and prediction uncertainty.

This paper presents a numerical study of a solar air collector aiming at analyzing the influence of several geometrical parameters on the heat transfer mechanisms with minimum losses. The laminar airflow in the collector undergoes a sudden or gradual narrowing at the absorber in its path. The finite volume method is used to solve the conservative equations of the fluid flow in the system. The results for these two narrowing models, at different positions and heights, show an improvement in heat transfer and a reduction in friction, especially in the case of gradual narrowing. Both narrowing models reduce the recirculation zones and thus increase the fluid velocity (1.25 to 2.50 times the reference velocity), leading to a gain in pressure drop compared to the perpendicular shoulder case. This solution also increased system efficiency (22.41% to 50.12% for the inclined shoulder case, 21.83% to 48.86% for the perpendicular case, and 20.81% to 38.66% for the simple case).

This paper deals with the pull-in instability of cantilever nano-switches subjected to electrostatic and intermolecular forces in the framework of the two-phase nonlocal theory of elasticity. The problem is governed by a nonlinear integro-differential equation accounting for the external forces and nonlocal effects. Assuming the Helmholtz kernel in the constitutive equation, we reduce the original integro-differential equation to a sixth-order differential one and derive a pair of additional boundary conditions. Aiming to obtain a closed-form solution of the boundary-value problem and to estimate the critical intermolecular forces and pull-in voltage, we approximate the resultant lateral force by a linear or quadratic function of the axial coordinate. The pull-in behavior of a freestanding nanocantilever as well as its instability under application of a critical voltage versus the local model fraction are examined within two models of the load distribution. It is shown that the critical voltages calculated in the framework of the two-phase nonlocal theory of elasticity are in very good agreement with the available data of atomistic simulation.

This study deals with numerical simulation of two-dimensional and three-dimensional natural convection in a closed differentially heated cube filled with radiatively participating medium. To examine fluid flow and heat transfer, hybrid mesomacroscopic model was developed. Rosseland radiation model was used to determine radiative heat flux. The effect of the Rayleigh number and radiation parameter on temperature, flow pattern and mean convective Nusselt numbers was discussed in detail. It was found that thermal radiation reduced the convective heat transfer rate by around 50% when radiation parameter is increased from 0 to 4. One-cellular quasi two-dimensional flow pattern was formed when taking into account Rosseland radiation. An oblique thermal stratification was formed as the radiation was enhanced. 2D and 3D models under consideration reproduced the same values of temperatures whereas a discrepancy was revealed in velocity components. Convective mean Nusselt numbers were in a very good agreement for both pure finite difference and hybrid lattice Boltzmann simulations with an error less than 5%. Volumetric radiation lowers the time needed to reach steady-state solution by around 60% when Rayleigh number is equal to 105. Numerical performance of hybrid lattice Boltzmann method was more than 7 times higher than conventional voriticity-vector potential formulation.

Detuning such as slight variations among unit cells often occurs in the construction of periodic structures. In order to study this kind of structures, a supercell is usually introduced, which requires tremendous amounts of computer memory and time. To simplify the calculation of detuning periodic pile barriers, the sensitivity analysis of geometric parameters to the first attenuation zone is conducted. A formula for calculating the equivalent radius of detuning periodic pile barriers is proposed. Modification by the BP neural network is completed in order to get a more accurate attenuation zone. The effectiveness and convenience of the equivalent radius formula are verified by studying the dynamic performances of detuning periodic pile barriers. The influence of the extent of detuning on the accuracy of the attenuation zone is discussed and some suggestions are given. This study shows that the attenuation zone of detuning periodic pile barriers can be easily obtained by using the formula established in this paper, which provides a simple way to find the dynamic property of this kind of barriers.