Search results for: motion editing.

Authors: Il Young Song, Du Yong Kim, Vladimir Shin

Abstract:

This paper reports on a receding horizon filtering for mobile robot systems with cross-correlated sensor noises and uncertainties. Also, the effect of uncertain parameters in the state of the tracking error model performance is considered. A distributed fusion receding horizon filter is proposed. The distributed fusion filtering algorithm represents the optimal linear combination of the local filters under the minimum mean square error criterion. The derivation of the error cross-covariances between the local receding horizon filters is the key of this paper. Simulation results of the tracking mobile robot-s motion demonstrate high accuracy and computational efficiency of the distributed fusion receding horizon filter.

Keywords: Distributed fusion, fusion formula, Kalman filter, multisensor, receding horizon, wheeled mobile robot

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Authors: Shuenn-Yih Chang, Chiu-Li Huang

Abstract:

Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.

Keywords: Dynamic analysis, high frequency, integration method, overshoot, weak instability.

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Authors: M. Amelsakhi, A. Sohrabi-Bidar, A. Shareghi

Abstract:

One of the most important issues about the structural damages caused by earthquake is the evaluating of the spectral response of the site on which the construction is built. This fact has demonstrated during many earlier earthquakes and many researchers’ reports have concerned with it. According to these reports, features of the site materials and geometry of the ground surface are considered the main factors. This study concentrates on the specific form of topographies like hills. Assessing of spectral responses of different points on the hills and beside demonstrates considerable differences between 1D and 2D methods of geotechnical analyses. A general trend of amplifications on the top of the hills and de-amplifications near the toe of the hills has been appeared within the acceleration, velocity and displacement response spectrums of horizontal motion. Evaluating of spectral responses of different sizes of the hills revealed that as much as the hill-size enlarges differences between spectral responses of 1D and 2D analyses transfers to longer range of periods and becomes wider.

Keywords: Topography effect, Amplification ratio, Response spectrum.

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Authors: Leonardo Herrera, Shield B. Lin, Stephen J. Montgomery-Smith, Ziraguen O. Williams

Abstract:

A one-directional dynamic model of a Stewart Platform was developed to assist NASA in analyzing the dynamic response in spacecraft docking operations. A simplified mechanical drawing was created, capturing the physical structure's main features. A simplified schematic diagram was developed into a lumped mass model from the mechanical drawing. Three differential equations were derived according to the schematic diagram. A Simulink diagram was created using MATLAB to represent the three equations. System parameters, including spring constants and masses, are derived in detail from the physical system. The model can be used for further analysis via computer simulation in predicting dynamic response in its main docking direction, i.e., up-and-down motion.

Keywords: Stewart platform, docking operation, spacecraft, spring constant.

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Authors: Hossain Aziz, Rinku Mukherjee

Abstract:

A potential flow model is used to study the unsteady flow past two airfoils in configuration, each of which is suddenly set into motion. The airfoil bound vortices are modeled using lumped vortex elements and the wake behind the airfoil is modeled by discrete vortices. This consists of solving a steady state flow problem at each time-step where unsteadiness is incorporated through the “zero normal flow on a solid surface" boundary condition at every time instant. Additionally, along with the “zero normal flow on a solid surface" boundary condition Kelvin-s condition is used to compute the strength of the latest wake vortex shed from the trailing edge of the airfoil. Location of the wake vortices is updated at each time-step to get the wake shape at each time instant. Results are presented to show the effect of airfoil-airfoil interaction and airfoil-wake interaction on the aerodynamic characteristics of each airfoil.

Keywords: Aerodynamics, Airfoils, Configuration, Unsteady.

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Authors: Mário C. Ricci

Abstract:

An alternative iterative computational procedure is proposed for internal normal ball loads calculation in statically loaded single-row, angular-contact ball bearings, subjected to a known thrust load, which is applied in the inner ring at the geometric bearing center line. An accurate method for curvature radii at contacts with inner and outer raceways in the direction of the motion is used. Numerical aspects of the iterative procedure are discussed. Numerical examples results for a 218 angular-contact ball bearing have been compared with those from the literature. Twenty figures are presented showing the geometrical features, the behavior of the convergence variables and the following parameters as functions of the thrust load: normal ball loads, contact angle, distance between curvature centers, and normal ball and axial deflections.

Keywords: Ball, Bearing, Static, Load, Iterative, Numerical, Method.

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Authors: Ege Can Onal, Enver Ersen, Meltem Elitas

Abstract:

Laparoscopy is a surgical operation, well known as keyhole surgery. The operation is performed through small holes, hence, scars of a patient become much smaller, patients can recover in a short time and the hospital stay becomes shorter in comparison to an open surgery. Several tools are used at laparoscopic operations; among them, the laparoscope has a crucial role. It provides the vision during the operation, which will be the main focus in here. Since the operation area is very small, motion of the surgical tools might be limited in laparoscopic operations compared to traditional surgeries. To overcome this limitation, most of the laparoscopic tools have become more precise, dexterous, multi-functional or automated. Here, we present a robotic-assisted laparoscope that is controlled with pedals directly by a surgeon. Thus, the movement of the laparoscope might be controlled better, so there will not be a need to calibrate the camera during the operation. The need for an assistant that controls the movement of the laparoscope will be eliminated. The duration of the laparoscopic operation might be shorter since the surgeon will directly operate the camera.

Keywords: Laparoscope, laparoscopy, low-cost, minimally invasive surgery, robotic-assisted surgery.

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Authors: A.R.Tahmasebi Birgani, M.Hosseinjani Zamenjani, M.R.Isvandzibaei

Abstract:

In the present work, study of the vibration of thin cylindrical shells made of a functionally gradient material (FGM) composed of stainless steel and nickel is presented. Material properties are graded in the thickness direction of the shell according to volume fraction power law distribution. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of boundary conditions on the natural frequencies of the FG cylindrical shell. The study is carried out using third order shear deformation shell theory. The governing equations of motion of FG cylindrical shells are derived based on shear deformation theory. Results are presented on the frequency characteristics, influence of constituent volume fractions and the effects of free-free boundary conditions.

Keywords: Vibration, FGM, Cylindrical shell.

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Authors: Y. Sabzehmeidani, M. Mailah, M. Hussein, A. R. Tavakolpour

Abstract:

In this paper, a worm-like micro robot designed for inpipe application with intelligent active force control (AFC) capability is modelled and simulated. The motion of the micro robot is based on an impact drive mechanism (IDM) that is actuated using piezoelectric device. The trajectory tracking performance of the modelled micro robot is initially experimented via a conventional proportionalintegral- derivative (PID) controller in which the dynamic response of the robot system subjected to different input excitations is investigated. Subsequently, a robust intelligent method known as active force control with fuzzy logic (AFCFL) is later incorporated into the PID scheme to enhance the system performance by compensating the unwanted disturbances due to the interaction of the robot with its environment. Results show that the proposed AFCFL scheme is far superior than the PID control counterpart in terms of the system-s tracking capability in the wake of the disturbances.

Keywords: Active Force Control, Micro Robot, Fuzzy Logic, In-pipe Application.

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Authors: Y. A. Adla, R. Soubra, M. Kasab, M. O. Diab, A. Chkeir

Abstract:

Over the past several years, researchers have shown a great interest in assessing the mobility of elderly people to measure their functional status. Usually, such an assessment is done by conducting tests that require the subject to walk a certain distance, turn around, and finally sit back down. Consequently, this study aims to provide an at home monitoring system to assess the patient’s status continuously. Thus, we proposed a technique to automatically detect when a subject sits down while walking at home. In this study, we utilized a Doppler radar system to capture the motion of the subjects. More than 20 features were extracted from the radar signals out of which 11 were chosen based on their Intraclass Correlation Coefficient (ICC > 0.75). Accordingly, the sequential floating forward selection wrapper was applied to further narrow down the final feature vector. Finally, five features were introduced to the Linear Discriminant Analysis classifier and an accuracy of 93.75% was achieved as well as a precision and recall of 95% and 90% respectively.

Keywords: Doppler radar system, stand-to-sit phase, TUG test, machine learning, classification

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Authors: Ahmed A. D. Sarhan, S. R. Besharaty, Javad Akbaria, M. Hamdi

Abstract:

The capability of CNC gantry milling machines in manufacturing long components has caused the expanded use of such machines. On the other hand, the machines’ gantry rigidity can reduce under severe loads or vibration during operation. Indeed, the quality of machining is dependent on the machine’s dynamic behavior throughout the operating process. For this reason, these types of machines have always been used widely and are not efficient. Therefore, they can usually be employed for rough machining and may not produce adequate surface finishing. In this paper, a CNC gantry milling machine with the potential to produce good surface finish has been designed and analyzed. The lowest natural frequency of this machine is 202 Hz corresponding to 12000 rpm at all motion amplitudes with a full range of suitable frequency responses. Meanwhile, the maximum deformation under dead loads for the gantry machine is 0.565*m, indicating that this machine tool is capable of producing higher product quality.

Keywords: Finite element, frequency response, gantry design, gantry machine, static and dynamic analysis.

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Authors: Mohammad Moeini, Mehrdad Ghyabi, Kiarash Mohtasham Dolatshahi

Abstract:

This paper investigates seismic soil-pile interaction using the Beam on Nonlinear Winkler Foundation (BNWF) approach. Three soil types are considered to cover all the possible responses, as well as nonlinear site response analysis using finite element method in OpenSees platform. Excitations at each elevation that are output of the site response analysis are used as the input excitation to the soil pile system implementing multi-support excitation method. Spectral intensities of acceleration show that the extent of the response in sand is more severe than that of clay, in addition, increasing the PGA of ground strong motion will affect the sandy soil more, in comparison with clayey medium, which is an indicator of the sensitivity of soil-pile systems in sandy soil.

Keywords: Beam on nonlinear Winkler foundation method, multi-support excitation, nonlinear site response analysis, seismic soil-pile interaction.

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Authors: Seyed Mohyeddin Ghodratian, Mehdi Ghassemieh, Mohammad Khanmohammadi

Abstract:

Excessive ductility demand on shorter piers is a common problem for irregular bridges subjected to strong ground motion. Various techniques have been developed to reduce the likelihood of collapse of bridge due to failure of shorter piers. This paper presents the new approach to improve the seismic behavior of such bridges using Nitinol shape memory alloys (SMAs). Superelastic SMAs have the ability to remain elastic under very large deformation due to martensitic transformation. This unique property leads to enhanced performance of controlled bridge compared with the performance of the reference bridge. To evaluate the effectiveness of the devices, nonlinear time history analysis is performed on a RC single column bent highway bridge using a suite of representative ground motions. The results show that this method is very effective in limiting the ductility demand of shorter pier.

Keywords: bridge, ductility demand, irregularity, shape memory alloy

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Authors: M. Yoneda

Abstract:

It is known that stationary human occupants act as dynamic mass-spring-damper systems and can change the modal properties of civil engineering structures. This paper describes the full scale measurement to explain the tuned mass damper effects of stationary people on structural damping of footbridge with center span length of 33 m. A human body can be represented by a lumped system consisting of masses, springs, and dashpots. Complex eigenvalue calculation is also conducted by using ISO5982:1981 human model (two degree of freedom system). Based on experimental and analytical results for the footbridge with the stationary people in the standing position, it is demonstrated that stationary people behave as a tuned mass damper and that ISO5982:1981 human model can explain the structural damping characteristics measured in the field.

Keywords: Dynamic interaction, footbridge, stationary people, structural damping.

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Authors: V. Girondel, L. Bonnaud, A. Caplier, M. Rombaut

Abstract:

This paper presents various classifiers results from a system that can automatically recognize four different static human body postures in video sequences. The considered postures are standing, sitting, squatting, and lying. The three classifiers considered are a naïve one and two based on the belief theory. The belief theory-based classifiers use either a classic or restricted plausibility criterion to make a decision after data fusion. The data come from the people 2D segmentation and from their face localization. Measurements consist in distances relative to a reference posture. The efficiency and the limits of the different classifiers on the recognition system are highlighted thanks to the analysis of a great number of results. This system allows real-time processing.

Keywords: Belief theory, classifiers comparison, data fusion, human motion analysis, real-time processing, static posture recognition.

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Authors: A. L. Qureshi, A. A. Mahessar, A. Baloch

Abstract:

Flood wave propagation in river channel flow can be enunciated by nonlinear equations of motion for unsteady flow. It is difficult to find analytical solution of these non-linear equations. Hence, in this paper verification of the finite element model has been carried out against available numerical predictions and field data. The results of the model indicate a good matching with both Preissmann scheme and HEC-RAS model for a river reach of 29km at both sites (15km from upstream and at downstream end) for discharge hydrographs. It also has an agreeable comparison with the Preissemann scheme for the flow depth (stage) hydrographs. The proposed model has also been applying to forecast daily discharges at 400km downstream in the Indus River from Sukkur barrage of Sindh, Pakistan, which demonstrates accurate model predictions with observed the daily discharges. Hence, this model may be utilized for flood warnings in advance.

Keywords: Finite Element Method, Flood Forecasting, HEC-RAS, Indus river.

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Authors: Alireza Fereidooni, Kamran Behdinan, Zouheir Fawaz

Abstract:

The integral form of equations of motion of composite beams subjected to varying time loads are discretized using a developed finite element model. The model consists of a straight five node twenty-two degrees of freedom beam element. The stability analysis of the beams is studied by solving the matrix form characteristic equations of the system. The principle of virtual work and the first order shear deformation theory are employed to analyze the beams with large deformation and small strains. The regions of dynamic instability of the beam are determined by solving the obtained Mathieu form of differential equations. The effects of nonconservative loads, shear stiffness, and damping parameters on stability and response of the beams are examined. Several numerical calculations are presented to compare the results with data reported by other researchers.

Keywords: Finite element beam model, Composite Beams, stability analysis

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Authors: Miroslav Popelka, Jakub Nožička

Abstract:

In this paper we deal with using Lego Mindstorms in simulation of robotic systems with respect to cost reduction. Lego Mindstorms kit contains broad variety of hardware components which are required to simulate, program and test the robotics systems in practice. Algorithm programming went in development environment supplied together with Lego kit as in programming language C# as well. Algorithm following the line, which we dealt with in this paper, uses theoretical findings from area of controlling circuits. PID controller has been chosen as controlling circuit whose individual components were experimentally adjusted for optimal motion of robot tracking the line. Data which are determined to process by algorithm are collected by sensors which scan the interface between black and white surfaces followed by robot. Based on discovered facts Lego Mindstorms can be considered for low-cost and capable kit to simulate real robotics systems.

Keywords: LEGO Mindstorms, PID controller, low-cost robotics systems, line follower, sensors, programming language C#, EV3 Home Edition Software.

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Authors: Dabiri Y, Najarian S, Eslami M R., Zahedi S, Moser D, Shirzad E, Allami M

Abstract:

The effect of muscle loss due to transfemoral amputation, on energy expenditure of hip joint and individual residual muscles was simulated. During swing phase of gait, with each muscle as an ideal force generator, the lower extremity was modeled as a two-degree of freedom linkage, for which hip and knee were joints. According to results, muscle loss will not lead to higher energy expenditure of hip joint, as long as other parameters of limb remain unaffected. This finding maybe due to the role of biarticular muscles in hip and knee joints motion. Moreover, if hip flexors are removed from the residual limb, residual flexors, and if hip extensors are removed, residual extensors will do more work. In line with the common practice in transfemoral amputation, this result demonstrates during transfemoral amputation, it is important to maintain the length of residual limb as much as possible.

Keywords: Amputation Level, Simulation, Transfemoral Amputee.

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Authors: Z. Veselý, M. Honner, J. Mach

Abstract:

The aim of the performed work is to establish the 2D and 3D model of direct unsteady task of sample heat treatment by moving source employing computer model on the basis of finite element method. Complex boundary condition on heat loaded sample surface is the essential feature of the task. Computer model describes heat treatment of the sample during heat source movement over the sample surface. It is started from 2D task of sample cross section as a basic model. Possibilities of extension from 2D to 3D task are discussed. The effect of the addition of third model dimension on temperature distribution in the sample is showed. Comparison of various model parameters on the sample temperatures is observed. Influence of heat source motion on the depth of material heat treatment is shown for several velocities of the movement. Presented computer model is prepared for the utilization in laser treatment of machine parts.

Keywords: Computer simulation, unsteady model, heat treatment, complex boundary condition, moving heat source.

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Authors: Tokuei Sako, Paul-Antoine Hervieux

Abstract:

The energy-level structure of a pair of electron and positron confined in a quasi-one-dimensional nano-scale potential well has been investigated focusing on its trend in the small limit of confinement strength ω, namely, the Wigner molecular regime. An anisotropic Gaussian-type basis functions supplemented by high angular momentum functions as large as l = 19 has been used to obtain reliable full configuration interaction (FCI) wave functions. The resultant energy spectrum shows a band structure characterized by ω for the large ω regime whereas for the small ω regime it shows an energy-level pattern dominated by excitation into the in-phase motion of the two particles. The observed trend has been rationalized on the basis of the nodal patterns of the FCI wave functions. 

Keywords: Confined systems, positron, wave function, Wigner molecule, quantum dots.

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Authors: B. Bagheri Nakhjavanlo, T. S. Ellis, P.Raoofi, Sh.ziari

Abstract:

This paper presents an application of level sets for the segmentation of abdominal and thoracic aortic aneurysms in CTA datasets. An important challenge in reliably detecting aortic is the need to overcome problems associated with intensity inhomogeneities. Level sets are part of an important class of methods that utilize partial differential equations (PDEs) and have been extensively applied in image segmentation. A kernel function in the level set formulation aids the suppression of noise in the extracted regions of interest and then guides the motion of the evolving contour for the detection of weak boundaries. The speed of curve evolution has been significantly improved with a resulting decrease in segmentation time compared with previous implementations of level sets, and are shown to be more effective than other approaches in coping with intensity inhomogeneities. We have applied the Courant Friedrichs Levy (CFL) condition as stability criterion for our algorithm.

Keywords: Image segmentation, Level-sets, Local fitting binary, Thoracic aorta.

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Authors: Alaa A. Osman, Amgad M. Bayoumy, Ismail El baialy, Osama E. Abdellatif, Essam E. Khallil

Abstract:

In this paper, a numerical simulation of a finned store separating from a wing-pylon configuration has been studied and validated. A dynamic unstructured tetrahedral mesh approach is accomplished by using three grid sizes to numerically solving the discretized three dimensional, inviscid and compressible Euler equations. The method used for computations of separation of an external store assuming quasi-steady flow condition. Computations of quasi-steady flow have been directly coupled to a six degree-offreedom (6DOF) rigid-body motion code to generate store trajectories. The pressure coefficients at four different angular cuts and time histories of various trajectory parameters and wing pressure distribution during the store separation are compared for every grid size with published experimental data.

Keywords: CFD Modelling, Quasi-steady Flow, Moving-body Trajectories, Transonic Store Separation, Moving-body Trajectories.

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Authors: M. Najafi, S. Bab, F. Rahimi Dehgolan

Abstract:

The motion of an axially moving beam with rotating prismatic joint with a tip mass on the end is analyzed to investigate the nonlinear vibration and dynamic stability of the beam. The beam is moving with a harmonic axially and rotating velocity about a constant mean velocity. A time-dependent partial differential equation and boundary conditions with the aid of the Hamilton principle are derived to describe the beam lateral deflection. After the partial differential equation is discretized by the Galerkin method, the method of multiple scales is applied to obtain analytical solutions. Frequency response curves are plotted for the super harmonic resonances of the first and the second modes. The effects of non-linear term and mean velocity are investigated on the steady state response of the axially moving beam. The results are validated with numerical simulations.

Keywords: Axially moving beam, Galerkin method, non-linear vibration, super harmonic resonances.

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Authors: Matko Šaric, Hrvoje Dujmic, Vladan Papic, Nikola Rožic

Abstract:

Detection of player identity is challenging task in sport video content analysis. In case of soccer video player number recognition is effective and precise solution. Jersey numbers can be considered as scene text and difficulties in localization and recognition appear due to variations in orientation, size, illumination, motion etc. This paper proposed new method for player number localization and recognition. By observing hue, saturation and value for 50 different jersey examples we noticed that most often combination of low and high saturated pixels is used to separate number and jersey region. Image segmentation method based on this observation is introduced. Then, novel method for player number localization based on internal contours is proposed. False number candidates are filtered using area and aspect ratio. Before OCR processing extracted numbers are enhanced using image smoothing and rotation normalization.

Keywords: player number, soccer video, HSV color space

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Authors: J. Ranjbarn, A. Alibeigloo

Abstract:

In this paper, we present an analytical method for analysis of nano-scale spherical shell subjected to thermo-mechanical shocks based on nonlocal elasticity theory. Thermo-mechanical properties of nano shpere is assumed to be temperature dependent. Governing partial differential equation of motion is solved analytically by using Laplace transform for time domain and power series for spacial domain. The results in Laplace domain is transferred to time domain by employing the fast inverse Laplace transform (FLIT) method. Accuracy of present approach is assessed by comparing the the numerical results with the results of published work in literature. Furtheremore, the effects of non-local parameter and wall thickness on the dynamic characteristics of the nano-sphere are studied.

Keywords: Nano-scale spherical shell, nonlocal elasticity theory, thermomechanical shock.

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Authors: A. Farshidianfar, P. Oliazadeh

Abstract:

In this paper the vibration behaviors of a structure equipped with a tuned liquid column damper (TLCD) under a harmonic type of earthquake loading are studied. However, due to inherent nonlinear liquid damping, it is no doubt that a great deal of computational effort is required to search the optimum parameters of the TLCD, numerically. Therefore by linearization the equation of motion of the single degree of freedom structure equipped with the TLCD, the closed form solutions of the TLCD-structure system are derived. To find the reliability of the analytical method, the results have been compared with other researcher and have good agreement. Further, the effects of optimal design parameters such as length ratio and mass ratio on the performance of the TLCD for controlling the responses of a structure are investigated by using the harmonic type of earthquake excitation. Finally, the Citicorp Center which has a very flexible structure is used as an example to illustrate the design procedure for the TLCD under the earthquake excitation.

Keywords: Closed form solution, Earthquake excitation, TLCD.

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Authors: A. Armin, R. Fotouhi, W. Szyszkowski

Abstract:

In this paper a 3-D finite element (FE) investigation of soil-blade interaction is described. The effects of blade’s shape and rake angle are examined both numerically and experimentally. The soil is considered as an elastic-plastic granular material with non-associated Drucker-Prager material model. Contact elements with different properties are used to mimic soil-blade sliding and soil-soil cutting phenomena. A separation criterion is presented and a procedure to evaluate the forces acting on the blade is given and discussed in detail. Experimental results were derived from tests using soil bin facility and instruments at the University of Saskatchewan. During motion of the blade, load cells collect data and send them to a computer. The measured forces using load cells had noisy signals which are needed to be filtered. The FE results are compared with experimental results for verification. This technique can be used in blade shape optimization and design of more complicated blade’s shape.

Keywords: Finite element analysis, soil-blade contact modeling, blade force, experimental results.

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Authors: I. Algul, G. Akgun, H. Kurtaran

Abstract:

Dynamic analysis of composite doubly curved panels with variable thickness subjected to different pulse types using Generalized Differential Quadrature method (GDQ) is presented in this study. Panels with variable thickness are used in the construction of aerospace and marine industry. Giving variable thickness to panels can allow the designer to get optimum structural efficiency. For this reason, estimating the response of variable thickness panels is very important to design more reliable structures under dynamic loads. Dynamic equations for composite panels with variable thickness are obtained using virtual work principle. Partial derivatives in the equation of motion are expressed with GDQ and Newmark average acceleration scheme is used for temporal discretization. Several examples are used to highlight the effectiveness of the proposed method. Results are compared with finite element method. Effects of taper ratios, boundary conditions and loading type on the response of composite panel are investigated.

Keywords: Generalized differential quadrature method, doubly curved panels, laminated composite materials, small displacement.

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Authors: Hamioud Saida, Khalfallah Salah

Abstract:

In this study, a spectral element method (SEM) is employed to predict the free vibration of a Euler-Bernoulli beam resting on a Winkler foundation with elastically restrained ends. The formulation of the dynamic stiffness matrix has been established by solving the differential equation of motion which was transformed to frequency domain. Non-dimensional natural frequencies and shape modes are obtained by solving the partial differential equations, numerically. Numerical comparisons and examples are performed to show the effectiveness of the SEM and to investigate the effects of various parameters, such as the springs at the boundaries and the elastic foundation parameter on the vibration frequencies. The obtained results demonstrate that the present method can also be applied to solve the more general problem of the dynamic analysis of structures with higher order precision.

Keywords: Elastically supported Euler-Bernoulli beam, free-vibration, spectral element method, Winkler foundation.

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