Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Stability analysis of intermittently jointed rock slopes based on the stepped failure mode

In practical engineering,due to the noncontinuity characteristics of joints in rock slopes,in addition to plane failure,stepped sliding failure may occur for intermittently jointed rock slopes.Especially for intermittently bedding jointed rock slopes,the correlation and difference in strength parameters between joints and rock bridges,along with the various failure modes and intermittency of rock bridges,contribute to the complexity of stepped failure modes and the unpredictability of failure regions.Based on the upper-bound limit analysis method and multi-sliders step-path failure mode,considering the shear and tensile failure of rock bridges and the weakened relationship between the strength parameters of rock bridges and jointed surfaces,by introducing the modified M-C failure criterion and the formula for calculating the energy consumption of tensile failure of rock bridges,two failure mechanisms are constructed to obtain the safety factor(F_(s))of intermittently jointed rock slopes.The sequential quadratic programming method is used to obtain the optimal upper-bound solution for F_(s).The influence of multiple key parameters(slope height H,horizontal distance L,Slope angleβ,shear strength parameters of the rock bridgeφr and cr,Dimensionless parameter u,weakening coefficients of the internal friction angle and cohesion between the rock bridges and joint surfaces Kφand Kc)on the stability analysis of intermittently jointed rock slopes under the shear failure mode of rock bridges as well as under the tensile failure mode is also explored.The reliability of the failure mechanisms is verified by comparative analysis with theoretical results,numerical results,and landslide cases,and the variation rules of F_(s)with each key parameter are obtained.The results show that F_(s) varies linearly withφr and cr of the rock bridge and with K_(φ)and K_(c),whereas F_(s)changes nonlinearly with H and L.In particular,with the increase in Kφand Kc,Fs increases by approximately 52.78%and 171.02%on average,respectively.For rock bridge tensile failure,F_(s) shows a nonlinearly positive correlation withφr,cr,Kφand Kc.In particular,with the increase in Kφand Kc,Fs increases by approximately 13%and 61.69%on average,respectively.Fs decreases rapidly with increasing slope gradientβand decreasing dimensionless parameterμ.When Kφand Kc are both less than 1.0,the stepped sliding surface occurs more easily than the plane failure surface,especially in the case of tensile failure of the rock bridge.In addition,rock slopes with higher strength parameters,taller heights,and greater weakening coefficients are prone to rock bridge tension failure with lower Fs,and more attention should be given to the occurrence of such accidents in actual engineering.

Shaking table comparative test and associated study of a stepped wall-frame structure

A new structural system called a stepped wall-frame structure is proposed in this study to solve the bottom yielding problem of RC frames, which widely occurred during previous earthquakes such as thc Wcnchuan and Yushu earthquakes in China. A 1/5 scale ordinary RC frame model and a stepped wall-frame model were subjected to shake table motions together to study the seismic behavior of the new structural system. This paper presents the dynamic characteristics, the seismic responses and the failure and collapse mechanism of the two models under low, moderate and high intensity shaking. The test results and further analysis demonstrate that the seismic performance of stepped wall-frame structures is superior to ordinary RC frames in terms of the well-controlled deformation pattern and more uniformly distributed damage. The stepped wall can effectively suppress the bottom yielding mechanism, and is simple, economical and practical tbr engineering practice.

Novel imaging methods of stepped frequency radar based on compressed sensing

The theory of compressed sensing （CS） provides a new chance to reduce the data acquisition time and improve the data usage factor of the stepped frequency radar system. In light of the sparsity of radar target reflectivity, two imaging methods based on CS, termed the CS-based 2D joint imaging algorithm and the CS-based 2D decoupled imaging algorithm, are proposed. These methods incorporate the coherent mixing operation into the sparse dictionary, and take random measurements in both range and azimuth directions to get high resolution radar images, thus can remarkably reduce the data rate and simplify the hardware design of the radar system while maintaining imaging quality. Ex- periments from both simulated data and measured data in the anechoic chamber show that the proposed imaging methods can get more focused images than the traditional fast Fourier trans- form method. Wherein the joint algorithm has stronger robustness and can provide clearer inverse synthetic aperture radar images, while the decoupled algorithm is computationally more efficient but has slightly degraded imaging quality, which can be improved by increasing measurements or using a robuster recovery algorithm nevertheless.

Three-to-one internal resonance in multiple stepped beam systems

In this study, the vibrations of multiple stepped beams with cubic nonlinearities are considered. A three-to-one internal resonance case is investigated for the system. A general approximate solution to the problem is found using the method of multiple scales （a perturbation technique）. The modulation equations of the amplitudes and the phases are derived for two modes. These equations are utilized to determine steady state solutions and their stabilities. It is assumed that the external forcing frequency is close to the lower frequency. For the numeric part of the study, the three-to-one ratio in natural frequencies is investigated. These values are observed to be between the first and second natural frequencies in the cases of the clamped-clamped and clamped-pinned supports, and between the second and third natural frequencies in the case of the pinned-pinned support. Finally, a numeric algorithm is used to solve the three-to-one internal resonance. The first mode is externally excited for the clamped-clamped and clamped-pinned supports, and the second mode is externally excited for the pinned-pinned support. Then, the amplitudes of the first and second modes are investigated when the first mode is externally excited. The amplitudes of the second and third modes are investigated when the second mode is externally excited. The force-response, damping-response, and .frequency- response curves are plotted for the internal resonance modes of vibrations. The stability analysis is carried out for these plots.

3D Rigid-Plastic Finite Element Analysis for Skew Rolling Process of the Stepped Part

Based on rigid-plastic finite element method, a skew rolling process of stepped part is simulated. Considering nodesaving and effective remeshing, the tetrahedron solid elements are used to discrete workpiece. The workpiece material adopts rigid-plastic model, where the flow stress is function of effective strain, effective strain rate and temperature.The thermomechanical coupling is considered in the simulation. To model the spinning workpiece undergoing plastic deformation, a novel solution is presented and applied in this paper. The stress state in the workpiece and forming characteristic of skew rolling are analyzed. The forming load, including roller torque and forces in three directions are predicted. The above analyses are helpful to understanding of forming mechanisms and improving of process and die design.

Finite element analysis and experiment on multi-wedge cross wedge rolling for asymmetric stepped shaft of C45

A rigid-plastic finite element method(FEM) simulation model for a multi-wedge cross wedge rolling(MCWR) was developed to analyze an asymmetric stepped shaft. To evaluate the MCWR process and better understand its deformation characteristics, the material flowing mechanisms, temperature distributions, strain and rolling force were analyzed. The correctness of the finite element simulation is experimentally verified. Numerical simulations and experiments led to the following conclusions: when α=36° and β=7.5°, the quality of the work piece can be significantly improved. Finally, the development of the asymmetric stepped shaft is applied to industrial production.

Adsorption and Vibration of O on the Pt Stepped Surfaces

The 5-parameter Morse potential （5-MP） of interaction between an adatom and metal surface cluster is put forward and performed on O-Pt stepped surface systems. We have primarily studied O/Pt（331）, O/Pt（221） and O/Pt（553） systems and drawn some conclusions from the calculated results in good agreement with experimental and theoretical information. The O atom adsorbs stably near the step, so the step is the active place in the reaction. There are two types of atomic states, β1 and β2, in the adsorbed systems, corresponding to the three-fold states and four-fold long-bridged state behind the step respectively. Moreover, the length of （Ⅲ） terrace affects weakly the properties of all critical points in these systems.

Scattering of Oblique Water Waves by Two Unequal Surface-Piercing Vertical Thin Plates with Stepped Bottom Topography

Based on linear water-wave theory, this study investigated the scattering of oblique incident water waves by two unequal surface-piercing thin vertical rigid plates with stepped bottom topography. By using the matched eigenfunction expansion method and a least square approach, the analytical solutions are sought for the established boundary value problem. The effects of the incidence angle, location of step, depth ratio of deep to shallow waters,and column width between two plates, on the reflection coefficients, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates, are numerically examined under a variety of wave conditions. The results show that the existence of the stepped bottom between two plates considerably impacts the hydrodynamic performances of the present system. It is found that the effect of stepped bottom on the reflection coefficient of the present two-plate structure is evident only with waves of the low dimensionless frequency.Moreover, the influence of the step location on the hydrodynamic performance of the present two-plate structure is slight if the step is placed in between the two plates.

Analysis and application of wavelet transform in target imaging and detection of stepped frequency MMW radar

The imaging and target detection methods for stepped frequency signal based on the wavelet transform and its power spectrum are investigated. Not only an imaging and target detection algorithm for stepped frequency signal based on the wavelet transform, but also its respective power spectrum are proposed. The multisampling property of stepped frequency signal is studied and wavelet transform is well suited for analyzing the signal. After multisampling property of stepped frequency signal being studied, it is shown that the wavelet transform is appropriate to analyze the signal. Based on the theory, the wavelet power spectrum analysis is applied to obtain the target range profile and the binary wavelet transform is used to perform target detection. To determine a suitable wavelet scaling for imaging of range profile's MMW radar, the distance resolution ΔR technique is proposed. The effectiveness of this new method is evaluated using simulated noisy radar signal. Results show that the proposed method can bring out the exactness and low computational complexity of this method.

Effect of Shape of the Absorber Surface on the Performance of Stepped Type Solar Still

In this work, we have selected three number of stepped type solar stills of the same overall dimensions 620 mm (W) × 808 mm (L) but with different absorber surface areas due to the variation in the shape of the basin surface. The other design parameters like depth of water, thickness of glass cover, insulation thickness, type of condensing cover, absorbing material provided over the basin, and angle of inclination of the still were kept constant to study the effect of shape of the absorber surface over the distillate yield obtained. The shape of the absorber surface provided in the basins of solar stills A, E and F was flat, convex and concave respectively. When the convex and concave type stepped solar stills are used, the average daily water production has been found to be 56.60% and 29.24% higher than that of flat type stepped solar still respectively. Also an economic analysis was made. The payback period of flat type, convex type and concave type stepped solar still is 823 days, 525 days and 637 days respectively. Thus, the convex type solar still gives the returns within the least possible time as compared to other two types of stepped solar stills. The laboratory tests were conducted to test the quality of water after distillation. The tests indicate that the quality of water in terms of pH, electrical conductivity, total hardness, TDS, Alkalinity, Nitrates etc. is well within the desirable limits as prescribed by WHO for Indian specific conditions.

Grating lobes suppression method for stepped frequency GB-SAR system

A grating lobes suppression method for chirp-subpulse stepped frequency（CSSF） signals is proposed, which is applied to deformation monitoring using the ground based synthetic aperture radar（GB-SAR） system. This method is based on accurate estimation and correction of the phase and amplitude error along two dimensions（range and azimuth）, i.e., the error estimation inside the subpulse（in-subpulse error） and across the stepped frequency subpulses（cross-subpulse error） of transmitted CSSF signals. Validated both with simulated data and experimental data recorded in the deformation monitoring campaign, it can be seen that the method as well as the relative conclusions can be fully and effectively applied to most of the stepped frequency systems.

Air Entrainment and Pressure Fields over Stepped Spillway in Skimming Flow Regime

This paper deals with some aspects of the air entrainment process along the chute of spillway and study of pressure fluctuations. The experimental study has been carried out using stepped spillway model located in the campus of Government College of Engineering, Amravati (India). It is observed that air concentration is increasing with discharge as well as with number of step. Air concentration is increasing along the length of spillway. It is also observed that the bottom mean air concentration increases with step height in the upstream reach of stepped spillway, which is prone to cavitation. The pressure profiles exhibit a wavy pattern down the stepped chute and pressure on each step increases with ratio of critical depth to step height (yc/h).

Numerical investigation of air-entrainment in skimming flow over stepped spillways

As a widely used flood energy dissipator, the stepped spillway can significantly dissipate the kinetic or hydraulic energy due to the air-entrainment in skimming flow over the steps. The free-surface aeration involves the sharp deformation of the free surface and the complex turbulent shear flows. In this study, the volume of fluid （VOF）, mixture, and Eulerian methods are utilized to simulate the air-entrainment by coupling with the Reynolds-averaged Navier-Stokes/large eddy simulation （RANS/LES） turbulence models. The free surface deformation, air volume fraction, pressure, and velocity are compared for the three different numerical methods. Only the Eulerian＋RANS method fails to capture the free-surface aeration. The air volume fraction predicted by the VOF＋LES method best matches the experimental measurement, while the mixture＋LES method predicts the inception point of the air entrainment more accurately.

Numerical Simulation of Flow over Stepped Spillways with Varying Step-Angle

In the present study,the flow over the stepped spillway was numerically investigated by using Flow3D model.The effect of step angle on different properties of Nappe flow regime such as the water surface profile,location of free-surface aeration inception,Froude number at the spillway’s toe,and pressure,flow velocity,air concentration and cavitation index were evaluated.The realizable k–εwas applied as the turbulence model,and Volume of Fluid(VOF)model was used to determine the free surface flow profiles of the spillway.The model was verified using experimental data.In order to investigate the different characteristics of Nappe flow regime,17 numerical runs was designed,in which,four step angles,four flow discharge were considered to investigate the flow characteristics over the stepped spillway.The results indicated that the numerical model is well suited with the experimental data over the stepped spillway(RMSE=0.147 and ARE=6.9%).In addition,with increasing the step angles,the aeration inception point is generally moved downstream.By increasing the step angles from zero to 10 degrees,the Froude number does not change significantly,however,at the angle of 15 degrees,the Froude number decreases by about 42 percent.

China's Technologies for Exploration and Mining of Uranium Deposits Have Stepped into the World's Top Team

On June 2011, the Beijing Research Institute of Uranium Geology, an Institute within the China National Nuclear Corporation （CNNC）, and the Department of Geology ＆ Mining CNNC carried out a project Scientific Drilling.for Deep Metallogenic Research in the Xiangshan Large Uranium Ore Field. A year later on June 21, the project, representing the first scientific drilling of China＇s uranium, was officially launched, and successfully completed on June 17, 2013. A 2818.88 m of drilling depth has broken the previous record 1200 m of exploration depth.

Computational investigation of hydraulic performance variation with geometry in gabion stepped spillways

Over recent years, there has been a clear increase in the frequency of reported flooding events around the world. Gabion structures offer one means of flood mitigation in dam spillways. These types of structures provide an additional challenge to the computational modeller in that flow through the porous gabions must be simulated. We have used a computational model to investigate the flow over gabion stepped spillways. The model was first validated against published experimental results. Then, gabion stepped spillways with four different step geometries were tested under the same conditions in order to facilitate inter-comparisons and to choose the best option in terms of energy dissipation. The results show that normal gabion steps can dissipate more energy than overlap, inclined, and pooled steps. An intensive set of tests with varying slope, stone size, and porosity were undertaken. The location of the inception point and the water depth at this point obtained from this study were compared with those from existing formulae. Two new empirical equations have been derived, on the basis of a regression analysis, to provide improved results for gabion stepped spillways.

RADIAL VIBRATIONS OF AXISYMMETRICALLY LOADED STEPPED PRESSURE VESSELS

Differential equations of free/forced radial vibrations of axisymmetrically loaded stepped pressure vessels are established by using singular functions. Furthermore, their general solutions are solved, the expression of vibration mode function and frequency equations on usual supports are derived with W operator and the forced response of such vessels are calculated.

Enhanced photoluminescence of monolayer MoS_(2)on stepped gold structure

Different Mo S_(2)/Au heterostructures can play an important role in tuning the photoluminescence(PL)and optoelectrical properties of monolayer Mo S_(2).Previous studies of PL of Mo S_(2)/Au heterostructures were mainly limited to the PL enhancement by using different Au nanostructures and PL quenching of monolayer Mo S_(2)on flat Au surfaces.Here,we demonstrate the enhanced excitonic PL emissions of monolayer Mo S_(2)/Au heterostructures on Si/Si O_(2)substrates.By transferring the continuous monolayer Mo S_(2)onto a stepped Au structure consisting of 60-nm and 100-nm Au films,the Mo S_(2)/Au-60 and Mo S_(2)/Au-100 heterostructures exhibit enhanced PL emissions,each with a blue-shifted PL peak in comparison with the Mo S_(2)/Si O_(2).Furthermore,the PL intensity of Mo S_(2)/Au-60 is about twice larger than that of Mo S_(2)/Au-100.The different enhanced excitonic PL emissions in Mo S_(2)/Au heterostructures can be attributed to the different charge transfer effects modified by the stepped Au structure.This work may provide an insight into the excitonic PL and charge transfer effect of Mo S_(2)on Au film and yield novel phenomena in Mo S_(2)/Au heterostructures for further study of PL tuning and optoelectrical properties.

VIBRATIONS OF STEPPED ONE-WAY THIN RECTANGULAR PLATES SUBJECTED TO IN-PLANE TENSILE/ COMPRESSIVE FORCE IN Y-DIRECTION ON WINKLER'S FOUNDATION

Differential equations of free/forced vibrations of n_step one_way thin rectangular plates subjected to in_plane tensile/compressive force in y_direction on Winkler's foundation are established by using singular functions, their general solutions solved for, expression of vibration mode function and frequency equation on usual supports derived with W operator. Influence functions for various cases deduced here may also be used to solve problems of static buckling or stability for beams and plates in relevant circumstances.