Search results for: Finite Klingenberg plane

Authors: T.C. Manjunath, B. Bandyopadhyay

Abstract:

This paper features the modeling and design of a Fast Output Sampling (FOS) Feedback control technique for the Active Vibration Control (AVC) of a smart flexible aluminium cantilever beam for a Single Input Single Output (SISO) case. Controllers are designed for the beam by bonding patches of piezoelectric layer as sensor / actuator to the master structure at different locations along the length of the beam by retaining the first 2 dominant vibratory modes. The entire structure is modeled in state space form using the concept of piezoelectric theory, Euler-Bernoulli beam theory, Finite Element Method (FEM) and the state space techniques by dividing the structure into 3, 4, 5 finite elements, thus giving rise to three types of systems, viz., system 1 (beam divided into 3 finite elements), system 2 (4 finite elements), system 3 (5 finite elements). The effect of placing the sensor / actuator at various locations along the length of the beam for all the 3 types of systems considered is observed and the conclusions are drawn for the best performance and for the smallest magnitude of the control input required to control the vibrations of the beam. Simulations are performed in MATLAB. The open loop responses, closed loop responses and the tip displacements with and without the controller are obtained and the performance of the proposed smart system is evaluated for vibration control.

Keywords: Smart structure, Finite element method, State spacemodel, Euler-Bernoulli theory, SISO model, Fast output sampling, Vibration control, LMI

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Authors: Abhishek Priyam, Prabha Chand

Abstract:

A theoretical investigation on the effect of collector aspect ratio on the thermal performance of wavy finned absorber solar air heaters has been performed. For the constant collector area, the various performance parameters have been calculated for plane and wavy finned solar air heaters. It has been found that the performance of wavy finned solar air heater improved with the increase in the collector aspect ratio. The performance of wavy finned solar air heater has been found 30 percent higher than those of plane solar air heater. The obtained results for wavy fin solar air heaters are compared with the available experimental data of most common type solar air heaters.

Keywords: Wavy fin, aspect ratio, solar air heater, thermal efficiency, collector efficiency factor, temperature rise.

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Authors: Jalal Khan, Daniyal Ali Sehrai, Shakeel Ahmad

Abstract:

This paper presents the design and performance evaluation of multiband metamaterial based antenna operating in the 3.6 GHz (4G), 14.33 GHz, and 28.86 GHz (5G) frequency bands, for future mobile and handheld devices. The radiating element of the proposed design is made up of a conductive material supported by a 1.524 mm thicker Rogers-4003 substrate, having a relative dielectric constant and loss tangent of 3.55 and 0.0027, respectively. The substrate is backed by truncated ground plane. The future mobile communication system is based on higher frequencies, which are highly affected by the atmospheric conditions. Therefore, to overcome the path loss problem, essential enhancements and improvements must be made in the overall performance of the antenna. The traditional ground plane does not provide the in-phase reflection and surface wave suppression due to which side and back lobes are produced. This will affect the antenna performance in terms of gain and efficiency. To enhance the overall performance of the antenna, a metamaterial acting as a high impedance surface (HIS) is used as a reflector in the proposed design. The simulated gain of the metamaterial based antenna is enhanced from {2.76-6.47, 4.83-6.71 and 7.52-7.73} dB at 3.6, 14.33 and 28.89 GHz, respectively relative to the gain of the antenna backed by a traditional ground plane. The proposed antenna radiated efficiently with a radiated efficiency (>85 %) in all the three frequency bands with and without metamaterial surface. The total volume of the antenna is (L x W x h=45 x 40 x 1.524) mm³. The antenna can be potentially used for wireless handheld devices and mobile terminal. All the simulations have been performed using the Computer Simulation Technology (CST) software.

Keywords: Multiband, fourth generation (4G), fifth generation (5G), metamaterial, CST MWS.

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Authors: S. Bennoud, M. Zergoug

Abstract:

The numerical simulation of electromagnetic interactions is still a challenging problem, especially in problems that result in fully three dimensional mathematical models.

The goal of this work is to use mathematical modeling to characterize the reliability and capacity of eddy current technique to detect and characterize defects embedded in aeronautical in-service pieces.

The finite element method is used for describing the eddy current technique in a mathematical model by the prediction of the eddy current interaction with defects. However, this model is an approximation of the full Maxwell equations.

In this study, the analysis of the problem is based on a three dimensional finite element model that computes directly the electromagnetic field distortions due to defects.

Keywords: Eddy current, Finite element method, Non destructive testing, Numerical simulations.

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Authors: U. Prasopchingchana

Abstract:

The temperature distribution and the heat transfer rates through a multi-layer door of a furnace were investigated. The inside of the door was in contact with hot air and the other side of the door was in contact with room air. Radiation heat transfer from the walls of the furnace to the door and the door to the surrounding area was included in the problem. This work is a two dimensional steady state problem. The Churchill and Chu correlation was used to find local convection heat transfer coefficients at the surfaces of the furnace door. The thermophysical properties of air were the functions of the temperatures. Polynomial curve fitting for the fluid properties were carried out. Finite difference method was used to discretize for conduction heat transfer within the furnace door. The Gauss-Seidel Iteration was employed to compute the temperature distribution in the door. The temperature distribution in the horizontal mid plane of the furnace door in a two dimensional problem agrees with the one dimensional problem. The local convection heat transfer coefficients at the inside and outside surfaces of the furnace door are exhibited.

Keywords: Conduction, heat transfer, multi-layer door, natural convection

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Authors: Juhee Lee, Yongjun Lee

Abstract:

In designing a low-energy-consuming buildings, the heat transfer through a large glass or wall becomes critical. Multiple layers of the window glasses and walls are employed for the high insulation. The gravity driven air flow between window glasses or wall layers is a natural heat convection phenomenon being a key of the heat transfer. For the first step of the natural heat transfer analysis, in this study the development and application of a finite volume method for the numerical computation of viscous incompressible flows is presented. It will become a part of the natural convection analysis with high-order scheme, multi-grid method, and dual-time step in the future. A finite volume method based on a fully-implicit second-order is used to discretize and solve the fluid flow on unstructured grids composed of arbitrary-shaped cells. The integrations of the governing equation are discretised in the finite volume manner using a collocated arrangement of variables. The convergence of the SIMPLE segregated algorithm for the solution of the coupled nonlinear algebraic equations is accelerated by using a sparse matrix solver such as BiCGSTAB. The method used in the present study is verified by applying it to some flows for which either the numerical solution is known or the solution can be obtained using another numerical technique available in the other researches. The accuracy of the method is assessed through the grid refinement.

Keywords: Finite volume method, fluid flow, laminar flow, unstructured grid.

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Authors: Abuzar.Abazari, Saeed. Ziaei-Rad, Hoseein. Dalayeli

Abstract:

Collision is considered as a time-depended nonlinear dynamic phenomenon. The majority of researchers have focused on deriving the resultant damage of the ship collisions via analytical, experimental, and finite element methods.In this paper, first, the force-penetration curve of a head collision on a container ship with rigid barrier based on Yang and Pedersen-s methods for internal mechanic section is studied. Next, the obtained results from different analytical methods are compared with each others. Then, through a simulation of the container ship collision in Ansys Ls-Dyna, results from finite element approach are compared with analytical methods and the source of errors is discussed. Finally, the effects of parameters such as velocity, and angle of collision on the forcepenetration curve are investigated.

Keywords: Ship collision, Force-penetration curve, Damage

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Authors: Thomas Jin-Chee Liu

Abstract:

In this paper, the thermo-electro-structural coupledfield in a cracked metal plate is studied using the finite element analysis. From the computational results, the compressive stresses reveal near the crack tip. This conclusion agrees with the past reference. Furthermore, the compressive condition can retard and stop the crack growth during the Joule heating process.

Keywords: Compressive stress, crack tip, Joule heating, finite element.

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Authors: Hamid Maidat, Khedidja Bouhadef, Djamel Eddine Ameziani, Azzedine Abdedou

Abstract:

This work consists of a numerical simulation of convective heat transfer in a vertical plane channel filled with a heat generating porous medium, in the absence of local thermal equilibrium. The walls are maintained to a constant temperature and the inlet velocity is uniform. The dynamic range is described by the Darcy-Brinkman model and the thermal field by two energy equations model. A dimensionless formulation is developed for performing a parametric study based on certain dimensionless groups such as, the Biot interstitial number, the thermal conductivity ratio and the volumetric heat generation, q '''. The governing equations are solved using the finite volume method, gave rise to a multitude of results concerning in particular the thermal field in the porous channel and the existence or not of the local thermal equilibrium.

Keywords: Mixed convection, porous medium, power generation, local thermal non equilibrium model.

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Authors: Maziar Ramezani, Zaidi Mohd Ripin

Abstract:

In a metal forming process, the friction between the material and the tools influences the process by modifying the stress distribution of the workpiece. This frictional behaviour is often taken into account by using a constant coefficient of friction in the finite element simulations of sheet metal forming processes. However, friction coefficient varies in time and space with many parameters. The Stribeck friction model is investigated in this study to predict springback behaviour of AA6061-T4 sheets during V-bending process. The coefficient of friction in Stribeck curve depends on sliding velocity and contact pressure. The plane-strain bending process is simulated in ABAQUS/Standard. We compared the computed punch load-stroke curves and springback related to the constant coefficient of friction with the defined friction model. The results clearly showed that the new friction model provides better agreement between experiments and results of numerical simulations. The influence of friction models on stress distribution in the workpiece is also studied numerically

Keywords: Friction model, Stress distribution, V-bending.

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Authors: S. Deelers, S. Auwatanamongkol

Abstract:

In this paper, we propose an algorithm to compute initial cluster centers for K-means clustering. Data in a cell is partitioned using a cutting plane that divides cell in two smaller cells. The plane is perpendicular to the data axis with the highest variance and is designed to reduce the sum squared errors of the two cells as much as possible, while at the same time keep the two cells far apart as possible. Cells are partitioned one at a time until the number of cells equals to the predefined number of clusters, K. The centers of the K cells become the initial cluster centers for K-means. The experimental results suggest that the proposed algorithm is effective, converge to better clustering results than those of the random initialization method. The research also indicated the proposed algorithm would greatly improve the likelihood of every cluster containing some data in it.

Keywords: Clustering algorithm, K-means algorithm, Datapartitioning, Initial cluster centers.

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Authors: Djamal Hamadi, Oussama Temami, Abdallah Zatar, Sifeddine Abderrahmani

Abstract:

The analysis and design of thin shell structures is a topic of interest in a variety of engineering applications. In structural mechanics problems the analyst seeks to determine the distribution of stresses throughout the structure to be designed. It is also necessary to calculate the displacements of certain points of the structure to ensure that specified allowable values are not exceeded. In this paper a comparative study between displacement and strain based finite elements applied to the analysis of some thin shell structures is presented. The results obtained from some examples show the efficiency and the performance of the strain based approach compared to the well known displacement formulation.

Keywords: Displacement formulation, Finite elements, Strain based approach, Shell structures.

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Authors: Ali Raad Hassan

Abstract:

Contact stress analysis between two spur gear teeth was considered in different contact positions, representing a pair of mating gears during rotation. A programme has been developed to plot a pair of teeth in contact. This programme was run for each 3° of pinion rotation from the first location of contact to the last location of contact to produce 10 cases. Each case was represented a sequence position of contact between these two teeth. The programme gives graphic results for the profiles of these teeth in each position and location of contact during rotation. Finite element models were made for these cases and stress analysis was done. The results were presented and finite element analysis results were compared with theoretical calculations, wherever available.

Keywords: Contact stress, Spur gear, Contact ratio, Finite elements

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Authors: Hamed Sabahno

Abstract:

In this paper, an inventory model with finite and constant replenishment rate, price dependant demand rate, time value of money and inflation, finite time horizon, lead time and exponential deterioration rate and with the objective of maximizing the present worth of the total system profit is developed. Using a dynamic programming based solution algorithm, the optimal sequence of the cycles can be found and also different optimal selling prices, optimal order quantities and optimal maximum inventories can be obtained for the cycles with unequal lengths, which have never been done before for this model. Also, a numerical example is used to show accuracy of the solution procedure.

Keywords: Deteriorating items, Dynamic programming, Finitereplenishment rate, Inventory control, Operation Research.

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Authors: L. Sandu, F. Topalâ, S. Porojan

Abstract:

Knowledge of factors, which influence stress and its distribution, is of key importance to the successful production of durable restorations. One of this is the marginal geometry. The objective of this study was to evaluate, by finite element analysis (FEA), the influence of different marginal designs on the stress distribution in teeth prepared for cast metal crowns. Five margin designs were taken into consideration: shoulderless, chamfer, shoulder, sloped shoulder and shoulder with bevel. For each kind of preparation three dimensional finite element analyses were initiated. Maximal equivalent stresses were calculated and stress patterns were represented in order to compare the marginal designs. Within the limitation of this study, the shoulder and beveled shoulder margin preparations of the teeth are preferred for cast metal crowns from biomechanical point of view.

Keywords: finite element, marginal geometry, metal crown

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Authors: Me. Sistaninia, G.H.Farrahi, Ma. Sistaninia

Abstract:

Thermoelastic temperature, displacement, and stress in heat transfer during laser surface hardening are solved in Eulerian formulation. In Eulerian formulations the heat flux is fixed in space and the workpiece is moved through a control volume. In the case of uniform velocity and uniform heat flux distribution, the Eulerian formulations leads to a steady-state problem, while the Lagrangian formulations remains transient. In Eulerian formulations the reduction to a steady-state problem increases the computational efficiency. In this study also an analytical solution is developed for an uncoupled transient heat conduction equation in which a plane slab is heated by a laser beam. The thermal result of the numerical model is compared with the result of this analytical model. Comparing the results shows numerical solution for uncoupled equations are in good agreement with the analytical solution.

Keywords: Coupled thermoelasticity, Finite element, Laser surface hardening, Eulerian formulation.

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Authors: Rajendra K. Ray, Kim Dan Nguyen

Abstract:

An unstructured finite volume numerical model is presented here for simulating shallow-water flows with wetting and drying fronts. The model is based on the Green-s theorem in combination with Chorin-s projection method. A 2nd-order upwind scheme coupled with a Least Square technique is used to handle convection terms. An Wetting and drying treatment is used in the present model to ensures the total mass conservation. To test it-s capacity and reliability, the present model is used to solve the Parabolic Bowl problem. We compare our numerical solutions with the corresponding analytical and existing standard numerical results. Excellent agreements are found in all the cases.

Keywords: Finite volume method, Projection method, Shallow water, Unstructured grid, wetting/drying fronts.

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Authors: N. Boutra, N. Ravot, J. Benoit, O. Picon

Abstract:

Submarine power cables used for offshore wind farms electric energy distribution and transmission are subject to numerous threats. Some of the risks are associated with transport, installation and operating in harsh marine environment. This paper describes the feasibility of an electromagnetic low frequency sensing technique for submarine power cable failure prediction. The impact of a structural damage shape and material variability on the induced electric field is evaluated. The analysis is performed by modeling the cable using the finite element method, we use sensitivity analysis in order to identify the main damage characteristics affecting electric field variation. Lastly, we discuss the results obtained.

Keywords: Electromagnetism, defect, finite element method, sensitivity analysis, submarine power cables.

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Authors: G. H. Majzoob, B. Sharifi Hamadani

Abstract:

Mapping between local and global coordinates is an important issue in finite element method, as all calculations are performed in local coordinates. The concern arises when subparametric are used, in which the shape functions of the field variable and the geometry of the element are not the same. This is particularly the case for C* elements in which the extra degrees of freedoms added to the nodes make the elements sub-parametric. In the present work, transformation matrix for C1* (an 8-noded hexahedron element with 12 degrees of freedom at each node) is obtained using equivalent C0 elements (with the same number of degrees of freedom). The convergence rate of 8-noded C1* element is nearly equal to its equivalent C0 element, while it consumes less CPU time with respect to the C0 element. The existence of derivative degrees of freedom at the nodes of C1* element along with excellent convergence makes it superior compared with it equivalent C0 element.

Keywords: Mapping, Finite element method, C* elements, Convergence, C0 elements.

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Authors: Sukrit Shankar, Chetana Shanta Patsa, Jaydev Sharma

Abstract:

Fractional Fourier Transform is a generalization of the classical Fourier Transform which is often symbolized as the rotation in time- frequency plane. Similar to the product of time and frequency span which provides the Uncertainty Principle for the classical Fourier domain, there has not been till date an Uncertainty Principle for the Fractional Fourier domain for a generalized class of finite energy signals. Though the lower bound for the product of time and Fractional Fourier span is derived for the real signals, a tighter lower bound for a general class of signals is of practical importance, especially for the analysis of signals containing chirps. We hence formulate a mathematical derivation that gives the lower bound of time and Fractional Fourier span product. The relation proves to be utmost importance in taking the Fractional Fourier Transform with adaptive time and Fractional span resolutions for a varied class of complex signals.

Keywords: Fractional Fourier Transform, uncertainty principle, Fractional Fourier Span, amplitude, phase.

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Authors: Abdulrahman A. Alghamdi, Paul M. Mummery, Mohammad A. Sheikh

Abstract:

In this paper, 3D image based composite unit cell is constructed from high resolution tomographic images. Through-thickness thermal diffusivity and in-plane Young’s modulus are predicted for the composite unit cell. The accuracy of the image based composite unit cell is tested by comparing its results with the experimental results obtained from laser flash and tensile test. The FE predictions are in close agreement with experimental results. Through-thickness thermal diffusivity and in-plane Young’s modulus of a virgin C/C composite are predicted by replacing the properties of air (porosity) with the properties of carbon matrix. The effect of porosity was found to be more profound on thermal diffusivity than young’s modulus.

Keywords: Porosity, C/C composite, image based FE modelling, CMC.

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Authors: F. Rezaie Moghaddam, J. Amani, T. Rezaie Moghaddam

Abstract:

Many computational techniques were applied to solution of heat conduction problem. Those techniques were the finite difference (FD), finite element (FE) and recently meshless methods. FE is commonly used in solution of equation of heat conduction problem based on the summation of stiffness matrix of elements and the solution of the final system of equations. Because of summation process of finite element, convergence rate was decreased. Hence in the present paper Cellular Automata (CA) approach is presented for the solution of heat conduction problem. Each cell considered as a fixed point in a regular grid lead to the solution of a system of equations is substituted by discrete systems of equations with small dimensions. Results show that CA can be used for solution of heat conduction problem.

Keywords: Heat conduction, Cellular automata, convergencerate, discrete system.

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Authors: Sh. Minapoor, S. Ajeli, M. Javadi Toghchi

Abstract:

Non-crimp 3D orthogonal fabric composite is one of the textile-based composite materials that are rapidly developing light-weight engineering materials. The present paper focuses on geometric and micromechanical modeling of non-crimp 3D orthogonal carbon fabric and composites reinforced with it for aerospace applications. In this research meso-finite element (FE) modeling employs for stress analysis in different load conditions. Since mechanical testing of expensive textile carbon composites with specific application isn't affordable, simulation composite in a virtual environment is a helpful way to investigate its mechanical properties in different conditions.

Keywords: 3D orthogonal woven composite, Aerospace applications, Finite element method, Mechanical properties.

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Authors: Zhi-Jun Lu, Qi Lu, Meng Wu, Qian Xiang, Jun Gu

Abstract:

The analysis of perforated steel members is a 3D problem in nature, therefore the traditional analytical expressions for the ultimate load of thin-walled steel sections cannot be used for the perforated steel member design. In this study, finite element method (FEM) and artificial neural network (ANN) were used to simulate the process of stub column tests based on specific codes. Results show that compared with those of the FEM model, the ultimate load predictions obtained from ANN technique were much closer to those obtained from the physical experiments. The ANN model for the solving the hard problem of complex steel perforated sections is very promising.

Keywords: Artificial neural network, finite element method, perforated sections, thin-walled steel, ultimate load.

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Authors: Pawan S. Nagda, Purnank S. Bhatt, Mit K. Shah

Abstract:

Earing defect in drawing process is highly undesirable not only because it adds on an additional trimming operation but also because the uneven material flow demands extra care. The objective of this work is to study the earing problem in the Deep Drawing of circular cup and to optimize the blank shape to reduce the earing. A finite element model is developed for 3-D numerical simulation of cup forming process in ABAQUS. Extra-deep-drawing (EDD) steel sheet has been used for simulation. Properties and tool design parameters were used as input for simulation. Earing was observed in the simulated cup and it was measured at various angles with respect to rolling direction. To reduce the earing defect initial blank shape was modified with the help of anisotropy coefficient. Modified blanks showed notable reduction in earing.

Keywords: Finite element simulation, deep drawing, earing, anisotropy.

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Authors: Raphael Zanella, Todd A. Oliver, Karl W. Schulz

Abstract:

This work deals with the finite element approximation of axisymmetric compressible flows with swirl velocity. We are interested in problems where the flow, while weakly dependent on the azimuthal coordinate, may have a strong azimuthal velocity component. We describe the approximation of the compressible Navier-Stokes equations with H1-conformal spaces of axisymmetric functions. The weak formulation is implemented in a C++ solver with explicit time marching. The code is first verified with a convergence test on a manufactured solution. The verification is completed by comparing the numerical and analytical solutions in a Poiseuille flow case and a Taylor-Couette flow case. The code is finally applied to the problem of a swirling subsonic air flow in a plasma torch geometry.

Keywords: Axisymmetric problem, compressible Navier- Stokes equations, continuous finite elements, swirling flow.

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Authors: Kyung-Su Na, Ji-Hwan Kim

Abstract:

Stress analysis of functionally graded composite plates composed of ceramic, functionally graded material and metal layers is investigated using 3-D finite element method. In FGM layer, material properties are assumed to be varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and metal. The 3-D finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties in the thickness direction. Numerical results are compared for three types of materials. In the analysis, the tensile and the compressive stresses are summarized for various FGM thickness ratios, volume fraction distributions, geometric parameters and mechanical loads.

Keywords: Functionally graded materials, Stress analysis, 3-D finite element method

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Authors: Hong Seung Kim, Chang Hoi Kim, Lili Yue

Abstract:

Single-phase, high band gap energy Zn_0.5Mg_0.5O films were grown under oxygen pressure, using pulse laser deposition with a Zn_0.5Mg_0.5O target. Structural characterization studies revealed that the crystal structures of the Zn_X-1Mg_XO films could be controlled via changes in the oxygen pressure. TEM analysis showed that the thickness of the deposited Zn_1-xMg_xO thin films was 50–75 nm. As the oxygen pressure increased, we found that one axis of the crystals did not show a very significant increase in the crystallization compared with that observed at low oxygen pressure. The X-ray diffraction peak intensity for the hexagonal-ZnMgO (002) plane increased relative to that for the cubic-ZnMgO (111) plane. The corresponding c-axis of the h-ZnMgO lattice constant increased from 5.141 to 5.148 Å, and the a-axis of the c-ZnMgO lattice constant decreased from 4.255 to 4.250 Å. EDX analysis showed that the Mg content in the mixed-phase ZnMgO films decreased significantly, from 54.25 to 46.96 at.%. As the oxygen pressure was increased from 100 to 150 mTorr, the absorption edge red-shifted from 3.96 to 3.81 eV; however, a film grown at the highest oxygen pressure tested here (200 mTorr).

Keywords: MgO, UV LED, ZnMgO, ZnO.

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Authors: Young-Jun You, Youn-Ju Jeong, Min-Su Park, Du-Ho Lee

Abstract:

Due to a lot of limited conditions, a large box type floating structure is inevitably constructed by connecting many pontoons. When a floating structure is made with concrete, concrete shear key with saw-teeth shape is often used to carry shear force. Match casting for the shear key and precise construction on a sea are very important for making separated two pontoons as one body but those are not easy work and may increase construction time and cost. To solve this problem, one-way shear key is studied in this paper for a connected part where there is some difference between upward and downward shear force. It has only one inclined plane and can resist shear force in one direction. Big shear force is resisted by concrete which forms an inclined plane and small shear force is resisted by steel bar. This system can reduce manufacturing cost of individual pontoon and construction time and cost for constructing a floating structure on a sea. In this paper, the feasibility study about one-way shear key system is performed by comparing with design example.

Keywords: Connection, floating container terminal, pontoon, pre-stressing, shear key.

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Authors: Haijie Li, Xuping Zhang

Abstract:

This paper presents a computationally efficient method for the modeling of robot manipulators with flexible links and joints. This approach combines the Discrete Time Transfer Matrix Method with the Finite Segment Method, in which the flexible links are discretized by a number of rigid segments connected by torsion springs; and the flexibility of joints are modeled by torsion springs. The proposed method avoids the global dynamics and has the advantage of modeling non-uniform manipulators. Experiments and simulations of a single-link flexible manipulator are conducted for verifying the proposed methodologies. The simulations of a three-link robot arm with links and joints flexibility are also performed.

Keywords: Flexible manipulator, transfer matrix method, linearization, finite segment method.

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