Search results for: Engesser-Timoshenko beam theory - Piezoelectric layer.

Authors: Gökhan Yüksel, Serdar Akça, İlker Kalkan

Abstract:

The present study is an attempt to demonstrate the significant levels of contribution of the moment-resisting beam-column connections with side plates to the earthquake behavior of special steel moment frames. To this end, the moment-curvature relationships of a regular beam-column connection and its SidePlate counterpart were determined with the help of finite element analyses. The connection stiffness and deformability values from these finite element analyses were used in the linear time-history analyses of an example structural steel frame under three different seismic excitations. The top-story lateral drift, base shear, and overturning moment values in two orthogonal directions were obtained from these time-history analyses and compared to each other. The results revealed the improvements in the system response with the use of SidePlate connections. The paper ends with crucial recommendations for the plan and design of further studies on this very topic.

Keywords: Seismic detailing, special moment frame, steel structures, beam-column connection, earthquake-resistant design.

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Authors: S. Alih, A. Khelil

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In the present work, behavior of inoxydable steel as reinforcement bar in composite concrete is being investigated. The bar-concrete adherence in reinforced concrete (RC) beam is studied and focus is made on the tension stiffening parameter. This study highlighted an approach to observe this interaction behavior in bending test instead of direct tension as per reported in many references. The approach resembles actual loading condition of the structural RC beam. The tension stiffening properties are then applied to numerical finite element analysis (FEA) to verify their correlation with laboratory results. Comparison with laboratory shows a good correlation between the two. The experimental settings is able to determine tension stiffening parameters in RC beam and the modeling strategies made in ABAQUS can closely represent the actual condition. Tension stiffening model used can represent the interaction properties between inoxydable steel and concrete.

Keywords: Inoxydable steel, Finite element modeling, Reinforced concrete beam, Tension-stiffening.

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Authors: Su-Young Jeong, Won-Kee Hong, Seon-Chee Park, Gyun-Taek Lim, Eric Kim

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Researchers investigate arious strategies to develop composite beams and maximize the structural advantages. This study attempted to conduct experiments and analysis of changes in the neutral axis of positive moments of a Green Beam. Strain compatibility analysis was used, and its efficiency was demonstrated by comparing experimental and analytical values. In the comparison of neutral axis, the difference between experimental and analytical values was found to range from 8.8~26.2%. It was determined that strain compatibility analysis can be useful for predicting the behaviors of composite beams, with the ability to predict the behavior of not only the elastic location of the composite member, but also of the plastic location

Keywords: Composite beam, Strain compatibility, Neutral axis, Green Beam

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Authors: Chong Zhang, Mu-Xuan Tao

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In several earthquakes, numerous reinforced concrete (RC) frames subjected to seismic excitation demonstrated a collapse pattern characterized by column hinges, though designed according to the Strong-Column-Weak-Beam (S-C-W-B) criteria. The effect of biaxial seismic excitation on the disparity between design and actual performance is carefully investigated in this article. First, a modified load contour method is proposed to derive a closed-form equation of biaxial bending moment strength, which is verified by numerical and experimental tests. Afterwards, a group of time history analyses of a simple frame modeled by fiber beam-column elements subjected to biaxial seismic excitation are conducted to verify that the current S-C-W-B criteria are not adequate to prevent the occurrence of column hinges. A biaxial over-strength factor is developed based on the proposed equation, and the reinforcement of columns is appropriately amplified with this factor to prevent the occurrence of column hinges under biaxial excitation, which is proved to be effective by another group of time history analyses.

Keywords: Biaxial bending moment strength, biaxial seismic excitation, fiber beam-column model, load contour method, strong-column-weak-beam.

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Authors: K. Kalaiselvan, A. Elango, N.M. Nagarajan, N. Mathiazhagan

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Dissimilar joining of Titanium and Aluminum thin sheets has potential applications in aerospace and automobile industry which can reduce weight and cost and improve strength, corrosion resistance and high temperature properties. However successful welding of Titanium/Aluminium sheets is of challenge due to differences in physical, chemical and metallurgical properties between the two. This paper describes research results of Laser Beam Welding (LBW) of Ti/Al thin sheets in which many researchers have recently performed and critically reviewed from different perspectives. Also some of notable works in the field of laser welding with changes in mechanical properties, crack propagation, diffusion behavior, chemical potential, interfacial reaction and the microstructure are reported.

Keywords: Laser Beam Welding (LBW), Mechanical properties, Titanium and Aluminium thin sheets.

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Authors: Barenten Suciu, Yuya Nakamoto

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Power generation effect, obtained by inserting a piezo- electric sheet in the backlash clearance of a circular arc helical gear, is evaluated. Such type of screw gear is preferred since, in comparison with the involute tooth profile, the circular arc profile leads to reduced stress-concentration effects, and improved life of the piezoelectric film. Firstly, geometry of the circular arc helical gear, and properties of the piezoelectric sheet are presented. Then, description of the test-rig, consisted of a right-hand thread gear meshing with a left-hand thread gear, and the voltage measurement procedure are given. After creating the tridimensional (3D) model of the meshing gears in SolidWorks, they are 3D-printed in acrylonitrile butadiene styrene (ABS) resin. Variation of the generated voltage versus time, during a meshing cycle of the circular arc helical gear, is measured for various values of the center distance. Then, the change of the maximal, minimal, and peak-to-peak voltage versus the center distance is illustrated. Optimal center distance of the gear, to achieve voltage maximization, is found and its significance is discussed. Such results prove that the contact pressure of the meshing gears can be measured, and also, the electrical power can be generated by employing the proposed technique.

Keywords: Power generation, circular arc helical gear, piezo- electric sheet, contact problem, optimal center distance.

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Authors: R. G. Ramírez-Chavarría, C. Sánchez-Pérez, V. Argueta-Díaz

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In this paper we present the design of an optical device based on a Herriott multi-pass cell fabricated on a small sized acrylic slab for heat flux measurements using the deflection of a laser beam propagating inside the cell. The beam deflection is produced by the heat flux conducted to the acrylic slab due to a gradient in the refractive index. The use of a long path cell as the sensitive element in this measurement device, gives the possibility of high sensitivity within a small size device. We present the optical design as well as some experimental results in order to validate the device’s operation principle.

Keywords: Heat flux, herriott cell, optical beam deflection, thermal conductivity.

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Authors: Isao Tomita

Abstract:

We develop a periodically-poled LiNbO3 (PPLN) device for highly-efficient third-harmonic generation (THG), where the THG efficiency is enhanced with a cavity. THG can usually be produced via χ(3)-nonlinear materials by optical pumping with very high pump-power. Instead, we here propose THG by moderate-power pumping through a specially-designed PPLN device containing only χ(2)-nonlinearity, where sum-frequency generation in the χ(2) process is employed for the mixing of a pump beam and a second-harmonic-generation (SHG) beam produced from the pump beam. The cavity is designed to increase the SHG power with dichroic mirrors attached to both ends of the device that perfectly reflect the SHG beam back to the device and yet let the pump and THG beams pass through the mirrors. This brings about a THG-power enhancement because of THG power proportional to the enhanced SHG power. We examine the THG-efficiency dependence on the mirror reflectance and show that very high THG-efficiency is obtained at moderate pump-power when compared with that of a cavity-free PPLN device.

Keywords: Cavity, periodically-poled LiNbO3, sum-frequency generation, third-harmonic generation.

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Authors: Halil Tanyer Eyyuboğu

Abstract:

For a generalized Hermite sinosiodal / hyperbolic Gaussian beam passing through an ABCD system with a finite aperture, the propagation properties are derived using the Collins integral. The results are obtained in the form of intensity graphs indicating that previously demonstrated rules of reciprocity are applicable, while the existence of the aperture accelerates this transformation.

Keywords: Optical communications, Hermite-Gaussian beams, ABCD system.

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Authors: Ali Ghobadi, Seyed Mohammad Javadi, Behnam Rahimi

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The present study is concerned with the effect of exciting boundary layer on cooling process in a gas-turbine blades. The cooling process is numerically investigated. Observations show cooling the first row of moving or stable blades leads to increase their life-time. Results show that minimum temperature in cooling line with exciting boundary layer is lower than without exciting. Using block in cooling line of turbines' blade causes flow pattern and stability in boundary layer changed that causes increase in heat transfer coefficient. Results show at the location of block, temperature of turbines' blade is significantly decreased. The k-ε turbulence model is used.

Keywords: Cooling, Exciting Boundary Layer, Heat Transfer, Turbine Blade.

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Authors: Ajay Gupta, Arvind Bhardwaj, Arun Kanda

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Dr Eliyahu Goldratt has done the pioneering work in the development of Theory of Constraints. Since then, many more researchers around the globe are working to enhance this body of knowledge. In this paper, an attempt has been made to compile the salient features of this theory from the work done by Goldratt and other researchers. This paper will provide a good starting point to the potential researchers interested to work in Theory of Constraints. The paper will also help the practicing managers by clarifying their concepts on the theory and will facilitate its successful implementation in their working areas.

Keywords: Drum-Buffer-Rope, Goldratt, ProductionScheduling, Theory of Constraints.

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Authors: F. Achouri, K. Chouicha

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The main parameters affecting the workability are the water content, particle size, and the total surface of the grains, as long as the mixing water begins by wetting the surface of the grains and then fills the voids between the grains to form entrapped water, the quantity of water remaining is called free water. The aim of this study is to undertake a fractal approach through the relationship between the concrete formulation parameters and workability. To develop this approach a series of concrete taken from the literature was investigated by varying formulation parameters such as G/S, the quantity of cement C and the quantity of water W. We also call another model as the model of water layer thickness and model of paste layer thickness to judge their relevance, hence the following results: the relevance of the water layer thickness model is considered as a relevant when there is a variation in the water quantity. The model of the paste layer thickness is only applicable if we considered that the paste is made with the grain value Dmax = 2.85: value from which we see a stability of the model.

Keywords: Concrete, fractal method, paste layer thickness, water layer thickness, workability.

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Authors: Yannick Verbelen, Tim Dekegel, Ann Peeters, Klara Stinders, Niek Blondeel, Sam De Winne, An Braeken, Abdellah Touhafi

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Matching an embedded electronic application with a cantilever vibration energy harvester remains a difficult endeavour due to the large number of factors influencing the output power. In the presented work, complementary balanced energy harvester parametrization is used as a methodology for simplification of harvester integration in electronic applications. This is achieved by a dual approach consisting of an adaptation of the general parametrization methodology in conjunction with a straight forward harvester benchmarking strategy. For this purpose, the design and implementation of a suitable user friendly cantilever energy harvester benchmarking platform is discussed. Its effectiveness is demonstrated by applying the methodology to a commercially available Mide V21BL vibration energy harvester, with excitation amplitude and frequency as variables.

Keywords: Energy harvesting, vibrations, piezoelectric transducers, embedded systems, harvester parametrization.

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Authors: R. Kumaravelan, V. C. Sathish Gandhi, S. Ramesh, M. Venkatesan

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In the product design and development process, the prototyping or model making is one of the important step to finalize a product which helps in conceptualization of a design. Rapid Prototyping (RP) is layer-by-layer material deposition started during early 1980s with the enormous growth in Computer Aided Design and Manufacturing (CAD/CAM) technologies. The edges and surfaces of a complex solid model and their information are used for defining a product which is further manufactured as a finished product by CNC machining. This paper provides a better platform for researchers, new learners and product manufacturers for various applications of RP models. Subsequently it creates awareness among the peoples of recently developing RP method of manufacturing in product design, developments and its applications.

Keywords: Prototyping, layer-by-layer, CAD/CAM, product design.

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Authors: Mahmoud Zarrini, R.N. Pralhad

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In this chapter, we have studied Variation of velocity in incompressible fluid over a moving surface. The boundary layer equations are on a fixed or continuously moving flat plate in the same or opposite direction to the free stream with suction and injection. The boundary layer equations are transferred from partial differential equations to ordinary differential equations. Numerical solutions are obtained by using Runge-Kutta and Shooting methods. We have found numerical solution to velocity and skin friction coefficient.

Keywords: Boundary layer, continuously moving surface, shooting method, skin friction coefficient.

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Authors: Partha Sarathi Majee, S. Bhattacharyya

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Migration of a core-shell soft particle under the influence of an external electric field in an electrolyte solution is studied numerically. The soft particle is coated with a positively charged polyelectrolyte layer (PEL) and the rigid core is having a uniform surface charge density. The Darcy-Brinkman extended Navier-Stokes equations are solved for the motion of the ionized fluid, the non-linear Nernst-Planck equations for the ion transport and the Poisson equation for the electric potential. A pressure correction based iterative algorithm is adopted for numerical computations. The effects of convection on double layer polarization (DLP) and diffusion dominated counter ions penetration are investigated for a wide range of Debye layer thickness, PEL fixed surface charge density, and permeability of the PEL. Our results show that when the Debye layer is in order of the particle size, the DLP effect is significant and produces a reduction in electrophoretic mobility. However, the double layer polarization effect is negligible for a thin Debye layer or low permeable cases. The point of zero mobility and the existence of mobility reversal depending on the electrolyte concentration are also presented.

Keywords: Debye length, double layer polarization, electrophoresis, mobility reversal, soft particle.

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Authors: Haleem K. Hussain, Liu Gui Wei, Zhang Lian Zhen, Yongxue Li

Abstract:

It is necessary to evaluate the bridges conditions and strengthen bridges or parts of them. The reinforcement necessary due to some reasons can be summarized as: First, a changing in use of bridge could produce internal forces in a part of structural which exceed the existing cross-sectional capacity. Second, bridges may also need reinforcement because damage due to external factors which reduced the cross-sectional resistance to external loads. One of other factors could listed here its misdesign in some details, like safety of bridge or part of its.This article identify the design demands of Qing Shan bridge located in is in Heilongjiang Province He gang - Nen Jiang Road 303 provincial highway, Wudalianchi area, China, is an important bridge in the urban areas. The investigation program was include the observation and evaluate the damage in T- section concrete beams , prestressed concrete box girder bridges section in additional evaluate the whole state of bridge includes the pier , abutments , bridge decks, wings , bearing and capping beam, joints, ........etc. The test results show that the bridges in general structural condition are good. T beam span No 10 were observed, crack extended upward along the ribbed T beam, and continue to the T beam flange. Crack width varying between 0.1mm to 0.4mm, the maximum about 0.4mm. The bridge needs to be improved flexural bending strength especially at for T beam section.

Keywords: Field investigation, prestressed concrete box girder, maintenance, Qing Shan Bridge

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Authors: Abo-El-Nour N. Abd-Alla, Fatimah A. Alsheikh

Abstract:

This study deals with the phenomena of reflection and transmission (refraction) of qSV-waves, for an incident of quasi transverse vertically waves, at a plane interface of two semi-infinite piezoelectric elastic media under the influence of the initial stresses. The relations governing the reflection and transmission coefficients of these reflected waves for various suitable boundary conditions are derived. We have shown analytically that reflection and transmission coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as will as the initial stresses presented in the media. The numerical calculations of the reflection and transmission amplitude ratios for different values of initial stresses have been carried out by computer for different materials as examples and the results are given in the form of graphs. Finally, some of particular cases are considered.

Keywords: Quasi plane vertical transverse waves, reflection and transmission coefficients, initial stresses, PZT-5H Ceramic, Aluminum Nitride (AlN), Piezoelectricity

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Authors: R. Ziaie Moayed, V. Rashidian, E. Izadi

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This paper utilizes a finite element analysis to study the bearing capacity of ring footings on a two-layered soil. The upper layer, that the footing is placed on it, is soft clay and the underneath layer is a cohesionless sand. For modeling soils, Mohr–Coulomb plastic yield criterion is employed. The effects of two factors, the clay layer thickness and the ratio of internal radius of the ring footing to external radius of the ring, have been analyzed. It is found that the bearing capacity decreases as the value of ri / ro increases. Although, as the clay layer thickness increases the bearing capacity was alleviated gradually.

Keywords: Bearing capacity, Ring footing, Two-layered soil

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Authors: N. Zhang, J. S. Kuang, S. Mogili

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To investigate seismic performance of beam-column knee joints, four full-scale reinforced concrete beam-column knee joints, which were fabricated to simulate those in as-built RC frame buildings designed to ACI 318-14 and ACI-ASCE 352R-02, were tested under reversed cyclic loading. In the experimental programme, particular emphasis was given to the effect of horizontal reinforcement (in format of inverted U-shape bars) on the shear strength and ductility capacity of knee joints. Test results are compared with those predicted by four seismic design codes, including ACI 318-14, EC8, NZS3101 and GB50010. It is seen that the current design codes of practice cannot accurately predict the shear strength of seismically designed knee joints.

Keywords: Large-scale tests, RC beam-column knee joints, seismic performance, shear strength.

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Authors: Akash K. Walunj, Dipendu Bhunia, Samarth Gupta, Prabhat Gupta

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Coupled shear walls consist of two shear walls connected intermittently by beams along the height. The behavior of coupled shear walls is mainly governed by the coupling beams. The coupling beams are designed for ductile inelastic behavior in order to dissipate energy. The base of the shear walls may be designed for elastic or ductile inelastic behavior. The amount of energy dissipation depends on the yield moment capacity and plastic rotation capacity of the coupling beams. In this paper, an analytical model of coupling beam was developed to calculate the rotations and moment capacities of coupling beam with conventional reinforcement.

Keywords: Design studies, computational model(s), case study/studies, modeling, coupling beam.

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Authors: N. Siva Shanmugam, G. Buvanashekaran, K. Sankaranarayanasamy

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In this research work, investigations are carried out on Continuous Wave (CW) Nd:YAG laser welding system after preliminary experimentation to understand the influencing parameters associated with laser welding of AISI 304. The experimental procedure involves a series of laser welding trials on AISI 304 stainless steel sheets with various combinations of process parameters like beam power, beam incident angle and beam incident angle. An industrial 2 kW CW Nd:YAG laser system, available at Welding Research Institute (WRI), BHEL Tiruchirappalli, is used for conducting the welding trials for this research. After proper tuning of laser beam, laser welding experiments are conducted on AISI 304 grade sheets to evaluate the influence of various input parameters on weld bead geometry i.e. bead width (BW) and depth of penetration (DOP). From the laser welding results, it is noticed that the beam power and welding speed are the two influencing parameters on depth and width of the bead. Three dimensional finite element simulation of high density heat source have been performed for laser welding technique using finite element code ANSYS for predicting the temperature profile of laser beam heat source on AISI 304 stainless steel sheets. The temperature dependent material properties for AISI 304 stainless steel are taken into account in the simulation, which has a great influence in computing the temperature profiles. The latent heat of fusion is considered by the thermal enthalpy of material for calculation of phase transition problem. A Gaussian distribution of heat flux using a moving heat source with a conical shape is used for analyzing the temperature profiles. Experimental and simulated values for weld bead profiles are analyzed for stainless steel material for different beam power, welding speed and beam incident angle. The results obtained from the simulation are compared with those from the experimental data and it is observed that the results of numerical analysis (FEM) are in good agreement with experimental results, with an overall percentage of error estimated to be within ±6%.

Keywords: Laser welding, Butt weld, 304 SS, FEM.

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Authors: Ainon Syazana Ab. Hamid, Seripah Awang Kechil, Ahmad Sukri Abd. Aziz

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The Marangoni convective instability in a horizontal fluid layer with the insoluble surfactant and nondeformable free surface is investigated. The surface tension at the free surface is linearly dependent on the temperature and concentration gradients. At the bottom surface, the temperature conditions of uniform temperature and uniform heat flux are considered. By linear stability theory, the exact analytical solutions for the steady Marangoni convection are derived and the marginal curves are plotted. The effects of surfactant or elasticity number, Lewis number and Biot number on the marginal Marangoni instability are assessed. The surfactant concentration gradients and the heat transfer mechanism at the free surface have stabilizing effects while the Lewis number destabilizes fluid system. The fluid system with uniform temperature condition at the bottom boundary is more stable than the fluid layer that is subjected to uniform heat flux at the bottom boundary.

Keywords: Analytical solutions, Marangoni Instability, Nondeformable free surface, Surfactant.

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Authors: Kyung-Suk Choi, Hyung-Joon Kim

Abstract:

Modular structural systems are constructed using a method that they are assembled with prefabricated unit modular frames on-site. This provides a benefit that can significantly reduce building construction time. The structural design is usually carried out under the assumption that their load-carrying mechanism is similar to that of traditional steel moment-resisting systems. However, both systems are different in terms of beam-column connection details which may strongly influence the lateral structural behavior. Specially, the presence of access holes in a beam-column joint of a unit modular frame could cause undesirable failure during strong earthquakes. Therefore, this study carried out finite element analyses (FEMs) of unit modular frames to investigate the cyclic behavior of beam-column joints with the access holes. Analysis results show that the unit modular frames present stable cyclic response with large deformation capacities and their joints are classified into semi-rigid connections even if there are access holes.

Keywords: Unit modular frame, steel moment connection, nonlinear analytical model, moment-rotation relation, access holes.

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Authors: Mohd Sazli Saad, Hishamuddin Jamaluddin, Intan Zaurah Mat Darus

Abstract:

This paper presents the development of an active vibration control using direct adaptive controller to suppress the vibration of a flexible beam system. The controller is realized based on linear parametric form. Differential evolution optimisation algorithm is used to optimize the controller using single objective function by minimizing the mean square error of the observed vibration signal. Furthermore, an alternative approach is developed to systematically search for the best controller model structure together with it parameter values. The performance of the control scheme is presented and analysed in both time and frequency domain. Simulation results demonstrate that the proposed scheme is able to suppress the unwanted vibration effectively.

Keywords: flexible beam, finite difference method, active vibration control, differential evolution, direct adaptive controller

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Authors: Jin-Wei Liang, Hung-Yi Chen, Lung Lin

Abstract:

In this paper feedforward controller is designed to eliminate nonlinear hysteresis behaviors of a piezoelectric stack actuator (PSA) driven system. The control design is based on inverse Prandtl-Ishlinskii (P-I) hysteresis model identified using particle swarm optimization (PSO) technique. Based on the identified P-I model, both the inverse P-I hysteresis model and feedforward controller can be determined. Experimental results obtained using the inverse P-I feedforward control are compared with their counterparts using hysteresis estimates obtained from the identified Bouc-Wen model. Effectiveness of the proposed feedforward control scheme is demonstrated. To improve control performance feedback compensation using traditional PID scheme is adopted to integrate with the feedforward controller. 

Keywords: The Bouc-Wen hysteresis model, Particle swarm optimization, Prandtl-Ishlinskii model.

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Authors: K. Kalaiselvan, A. Elango

Abstract:

To achieve reliable welds with minimum distortion for the fabrication of components in aerospace industry laser beam welding is attempted. Laser welding can provide a significant benefit for the welding of Titanium and Aluminium thin sheet alloys of its precision and rapid processing capability. For laser welding, pulse shape, energy, duration, repetition rate and peak power are the most important parameters that influence directly the quality of welds. In this experimental work for joining 1mm thick TI6AL4V and AA2024 alloy and JK600 Nd:YAG pulsed laser units used. The distortions at different welding power and speed of titanium and aluminium thin sheet alloys are investigated. Test results reveal that increase in welding speed increases distortion in weldment

Keywords: Laser Beam Welding, Titanium, Aluminium alloy sheets and distortion.

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Authors: Haiyan Wang

Abstract:

Constuctivism learning theory lays emphasis on the learners' active learning, such as learning initiative, sociality and context. By analyzing the relationship between constructivism learning theory and learner autonomy, this paper explores how to cultivate learners' learner autonomy under the guidance of constructivism learning theory.

Keywords: Constructivism learning theory, learner autonomy, relationship, cultivation.

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Authors: Nora Taghavi, Ali Sadr

Abstract:

System identification is the process of creating models of dynamic process from input- output signals. The aim of system identification can be identified as “ to find a model with adjustable parameters and then to adjust them so that the predicted output matches the measured output". This paper presents a method of modeling and simulating with system identification to achieve the maximum fitness for transformation function. First by using optimized KLM equivalent circuit for PVDF piezoelectric transducer and assuming different inputs including: sinuside, step and sum of sinusides, get the outputs, then by using system identification toolbox in MATLAB, we estimate the transformation function from inputs and outputs resulted in last program. Then compare the fitness of transformation function resulted from using ARX,OE(Output- Error) and BJ(Box-Jenkins) models in system identification toolbox and primary transformation function form KLM equivalent circuit.

Keywords: PVDF modeling, ARX, BJ(Box-Jenkins), OE(Output-Error), System Identification.

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Authors: S.H. Mirtalaie, M.A. Hajabasi

Abstract:

In this paper, an analytical approach is used to study the coupled lateral-torsional vibrations of laminated composite beam. It is known that in such structures due to the fibers orientation in various layers, any lateral displacement will produce a twisting moment. This phenomenon is modeled by the bending-twisting material coupling rigidity and its main feature is the coupling of lateral and torsional vibrations. In addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies. Then, the governing differential equations are derived using the Hamilton-s principle and the mathematical model matches the Timoshenko beam model when neglecting the effect of bending-twisting rigidity. The equations of motion which form a system of three coupled PDEs are solved analytically to study the free vibrations of the beam in lateral and rotational modes due to the bending, as well as the torsional mode caused by twisting. The analytic solution is carried out in three steps: 1) assuming synchronous motion for the kinematic variables which are the lateral, rotational and torsional displacements, 2) solving the ensuing eigenvalue problem which contains three coupled second order ODEs and 3) imposing different boundary conditions related to combinations of simply, clamped and free end conditions. The resulting natural frequencies and mode shapes are compared with similar results in the literature and good agreement is achieved.

Keywords: Free vibration, laminated composite beam, material coupling, state space.

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