Search results for: Timoshenko beam theory

Authors: A. R. Nezamabadi, M. Karami Khorramabadi

Abstract:

This paper presents the elastic buckling of homogeneous beams with a pair of piezoelectric layers surface bonded on both sides of the beams. The displacement field of beam is assumed based on the Engesser-Timoshenko beam theory. Applying the Hamilton's principle, the equilibrium equation is established. The influences of applied voltage, dimensionless geometrical parameter and piezoelectric thickness on the critical buckling load of beam are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Mechanical Buckling, Engesser-Timoshenko beam theory - Piezoelectric layer.

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Authors: M. Karami Khorramabadi, A. R. Nezamabadi

Abstract:

This paper studies mechanical buckling of functionally graded beams subjected to axial compressive load that is simply supported at both ends lies on a continuous elastic foundation. The displacement field of beam is assumed based on Engesser-Timoshenko beam theory. Applying the Hamilton's principle, the equilibrium equation is established. The influences of dimensionless geometrical parameter, functionally graded index and foundation coefficient on the critical buckling load of beam are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Mechanical Buckling, Functionally graded beam- Engesser-Timoshenko beam theory

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Authors: M. Karami Khorramabadi, A. R. Nezamabadi

Abstract:

This paper studies free vibration of functionally graded beams Subjected to Axial Load that is simply supported at both ends lies on a continuous elastic foundation. The displacement field of beam is assumed based on Engesser-Timoshenko beam theory. The Young's modulus of beam is assumed to be graded continuously across the beam thickness. Applying the Hamilton's principle, the governing equation is established. Resulting equation is solved using the Euler's Equation. The effects of the constituent volume fractions and foundation coefficient on the vibration frequency are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Functionally Graded Beam, Free Vibration, Elastic Foundation, Engesser-Timoshenko Beam Theory.

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Authors: T.C. Manjunath, B. Bandyopadhyay

Abstract:

Active Vibration Control (AVC) is an important problem in structures. One of the ways to tackle this problem is to make the structure smart, adaptive and self-controlling. The objective of active vibration control is to reduce the vibration of a system by automatic modification of the system-s structural response. This paper features the modeling and design of a Periodic Output Feedback (POF) control technique for the active vibration control of a flexible Timoshenko cantilever beam for a multivariable case with 2 inputs and 2 outputs by retaining the first 2 dominant vibratory modes using the smart structure concept. The entire structure is modeled in state space form using the concept of piezoelectric theory, Timoshenko beam theory, Finite Element Method (FEM) and the state space techniques. Simulations are performed in MATLAB. The effect of placing the sensor / actuator at 2 finite element locations along the length of the beam is observed. The open loop responses, closed loop responses and the tip displacements with and without the controller are obtained and the performance of the smart system is evaluated for active vibration control.

Keywords: Smart structure, Timoshenko theory, Euler-Bernoulli theory, Periodic output feedback control, Finite Element Method, State space model, Vibration control, Multivariable system, Linear Matrix Inequality

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Authors: T. C. Manjunath, B. Bandyopadhyay

Abstract:

This paper deals with the design of a periodic output feedback controller for a flexible beam structure modeled with Timoshenko beam theory, Finite Element Method, State space methods and embedded piezoelectrics concept. The first 3 modes are considered in modeling the beam. The main objective of this work is to control the vibrations of the beam when subjected to an external force. Shear piezoelectric sensors and actuators are embedded into the top and bottom layers of a flexible aluminum beam structure, thus making it intelligent and self-adaptive. The composite beam is divided into 5 finite elements and the control actuator is placed at finite element position 1, whereas the sensor is varied from position 2 to 5, i.e., from the nearby fixed end to the free end. 4 state space SISO models are thus developed. Periodic Output Feedback (POF) Controllers are designed for the 4 SISO models of the same plant to control the flexural vibrations. The effect of placing the sensor at different locations on the beam is observed and the performance of the controller is evaluated for vibration control. Conclusions are finally drawn.

Keywords: Smart structure, Timoshenko beam theory, Periodic output feedback control, Finite Element Method, State space model, SISO, Embedded sensors and actuators, Vibration control.

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Authors: T. C. Manjunath, B. Bandyopadhyay

Abstract:

Active vibration control is an important problem in structures. The objective of active vibration control is to reduce the vibrations of a system by automatic modification of the system-s structural response. In this paper, the modeling and design of a fast output sampling feedback controller for a smart flexible beam system embedded with shear sensors and actuators for SISO system using Timoshenko beam theory is proposed. FEM theory, Timoshenko beam theory and the state space techniques are used to model the aluminum cantilever beam. For the SISO case, the beam is divided into 5 finite elements and the control actuator is placed at finite element position 1, whereas the sensor is varied from position 2 to 5, i.e., from the nearby fixed end to the free end. Controllers are designed using FOS method and the performance of the designed FOS controller is evaluated for vibration control for 4 SISO models of the same plant. The effect of placing the sensor at different locations on the beam is observed and the performance of the controller is evaluated for vibration control. Some of the limitations of the Euler-Bernoulli theory such as the neglection of shear and axial displacement are being considered here, thus giving rise to an accurate beam model. Embedded shear sensors and actuators have been considered in this paper instead of the surface mounted sensors and actuators for vibration suppression because of lot of advantages. In controlling the vibration modes, the first three dominant modes of vibration of the system are considered.

Keywords: Smart structure, Timoshenko beam theory, Fast output sampling feedback control, Finite Element Method, State space model, SISO, Vibration control, LMI

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Authors: Akif Kutlu, Merve Ermis, Nihal Eratlı, Mehmet H. Omurtag

Abstract:

The objective of this study is to investigate the forced vibration analysis of a planar curved beam lying on elastic foundation by using the mixed finite element method. The finite element formulation is based on the Timoshenko beam theory. In order to solve the problems in frequency domain, the element matrices of two nodded curvilinear elements are transformed into Laplace space. The results are transformed back to the time domain by the well-known numerical Modified Durbin’s transformation algorithm. First, the presented finite element formulation is verified through the forced vibration analysis of a planar curved Timoshenko beam resting on Winkler foundation and the finite element results are compared with the results available in the literature. Then, the forced vibration analysis of a planar curved beam resting on Winkler-Pasternak foundation is conducted.

Keywords: Curved beam, dynamic analysis, elastic foundation, finite element method.

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Authors: Gholam Reza Koochaki

Abstract:

This work presents the highly accurate numerical calculation of the natural frequencies for functionally graded beams with simply supported boundary conditions. The Timoshenko first order shear deformation beam theory and the higher order shear deformation beam theory of Reddy have been applied to the functionally graded beams analysis. The material property gradient is assumed to be in the thickness direction. The Hamilton-s principle is utilized to obtain the dynamic equations of functionally graded beams. The influences of the volume fraction index and thickness-to-length ratio on the fundamental frequencies are discussed. Comparison of the numerical results for the homogeneous beam with Euler-Bernoulli beam theory results show that the derived model is satisfactory.

Keywords: Functionally graded beam, Free vibration, Hamilton's principle.

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Authors: T.C. Manjunath, Student Member, B. Bandyopadhyay

Abstract:

This paper features the mathematical modeling of a single input single output based Timoshenko smart beam. Further, this mathematical model is used to design a multirate output feedback based discrete sliding mode controller using Bartoszewicz law to suppress the flexural vibrations. The first 2 dominant vibratory modes is retained. Here, an application of the discrete sliding mode control in smart systems is presented. The algorithm uses a fast output sampling based sliding mode control strategy that would avoid the use of switching in the control input and hence avoids chattering. This method does not need the measurement of the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement.

Keywords: Smart structure, Timoshenko beam theory, Discretesliding mode control, Bartoszewicz law, Finite Element Method, State space model, Vibration control, Mathematical model, SISO.

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Authors: Mohammad Hadi Jalali, Behrooz Shahriari, Mostafa Ghayour, Saeed Ziaei-Rad, Shahram Yousefi

Abstract:

Most flexible rotors can be considered as beam-like structures. In many cases, rotors are modeled as one-dimensional bodies, made basically of beam-like shafts with rigid bodies attached to them. This approach is typical of rotor dynamics, both analytical and numerical, and several rotor dynamic codes, based on the finite element method, follow this trend. In this paper, a finite element model based on Timoshenko beam elements is utilized to analyze the lateral dynamic behavior of a certain rotor-bearing system in operating conditions.

Keywords: Finite element method, Operational deflection shape, Timoshenko beam elements, Unbalance response.

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Authors: Desmond Adair, Kairat Ismailov, Martin Jaeger

Abstract:

Modelling of Timoshenko beams on elastic foundations has been widely used in the analysis of buildings, geotechnical problems, and, railway and aerospace structures. For the elastic foundation, the most widely used models are one-parameter mechanical models or two-parameter models to include continuity and cohesion of typical foundations, with the two-parameter usually considered the better of the two. Knowledge of free vibration characteristics of beams on an elastic foundation is considered necessary for optimal design solutions in many engineering applications, and in this work, the efficient and accurate variational iteration method is developed and used to calculate natural frequencies of a Timoshenko beam on a two-parameter foundation. The variational iteration method is a technique capable of dealing with some linear and non-linear problems in an easy and efficient way. The calculations are compared with those using a finite-element method and other analytical solutions, and it is shown that the results are accurate and are obtained efficiently. It is found that the effect of the presence of the two-parameter foundation is to increase the beam’s natural frequencies and this is thought to be because of the shear-layer stiffness, which has an effect on the elastic stiffness. By setting the two-parameter model’s stiffness parameter to zero, it is possible to obtain a one-parameter foundation model, and so, comparison between the two foundation models is also made.

Keywords: Timoshenko beam, variational iteration method, two-parameter elastic foundation model.

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Authors: Tayeb Bensattalah, Mohamed Zidour, Mohamed Ait Amar Meziane, Tahar Hassaine Daouadji, Abdelouahed Tounsi

Abstract:

In this paper, the Differential Transform Method (DTM) is employed to predict and to analysis the non-local critical buckling loads of carbon nanotubes with various end conditions and the non-local Timoshenko beam described by single differential equation. The equation differential of buckling of the nanobeams is derived via a non-local theory and the solution for non-local critical buckling loads is finding by the DTM. The DTM is introduced briefly. It can easily be applied to linear or nonlinear problems and it reduces the size of computational work. Influence of boundary conditions, the chirality of carbon nanotube and aspect ratio on non-local critical buckling loads are studied and discussed. Effects of nonlocal parameter, ratios L/d, the chirality of single-walled carbon nanotube, as well as the boundary conditions on buckling of CNT are investigated.

Keywords: Boundary conditions, buckling, non-local, the differential transform method.

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Authors: Babak Safaei, A. M. Fattahi

Abstract:

In the present study we have investigated axial buckling characteristics of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Various types of beam theories including Euler-Bernoulli beam theory, Timoshenko beam theory and Reddy beam theory were used to analyze the buckling behavior of carbon nanotube-reinforced composite beams. Generalized differential quadrature (GDQ) method was utilized to discretize the governing differential equations along with four commonly used boundary conditions. The material properties of the nanocomposite beams were obtained using molecular dynamic (MD) simulation corresponding to both short-(10,10) SWCNT and long- (10,10) SWCNT composites which were embedded by amorphous polyethylene matrix. Then the results obtained directly from MD simulations were matched with those calculated by the mixture rule to extract appropriate values of carbon nanotube efficiency parameters accounting for the scale-dependent material properties. The selected numerical results were presented to indicate the influences of nanotube volume fractions and end supports on the critical axial buckling loads of nanocomposite beams relevant to long- and short-nanotube composites.

Keywords: Nanocomposites, molecular dynamics simulation, axial buckling, generalized differential quadrature (GDQ).

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Authors: M. M. Shokrieh, A. Karamnejad

Abstract:

This paper deals with a numerical analysis of the transient response of composite beams with strain rate dependent mechanical properties by use of a finite difference method. The equations of motion based on Timoshenko beam theory are derived. The geometric nonlinearity effects are taken into account with von Kármán large deflection theory. The finite difference method in conjunction with Newmark average acceleration method is applied to solve the differential equations. A modified progressive damage model which accounts for strain rate effects is developed based on the material property degradation rules and modified Hashin-type failure criteria and added to the finite difference model. The components of the model are implemented into a computer code in Mathematica 6. Glass/epoxy laminated composite beams with constant and strain rate dependent mechanical properties under dynamic load are analyzed. Effects of strain rate on dynamic response of the beam for various stacking sequences, load and boundary conditions are investigated.

Keywords: Composite beam, Finite difference method, Progressive damage modeling, Strain rate.

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Authors: Nasiri Ahmadullah, Shimozato Tetsuhiro, Masayuki Tai

Abstract:

This paper presents the step-by-step procedure for using Elastic Theory to calculate the internal stresses in composite bridge girders prestressed by the Preflexing Technology, called Prebeam in Japan and Preflex beam worldwide. Elastic Theory approaches preflex beams the same way as it does the conventional composite girders. Since preflex beam undergoes different stages of construction, calculations are made using different sectional and material properties. Stresses are calculated in every stage using the properties of the specific section. Stress accumulation gives the available stress in a section of interest. Concrete presence in the section implies prestress loss due to creep and shrinkage, however; more work is required to be done in this field. In addition to the graphical presentation of this application, this paper further discusses important notes of graphical comparison between the results of an experimental-only research carried out on a preflex beam, with the results of simulation based on the elastic theory approach, for an identical beam using Finite Element Modeling (FEM) by the author.

Keywords: Composite girder, elastic theory, preflex beam, prestressing.

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Authors: Vikas Rastogi, Vipan Kumar, Loveleen Kumar Bhagi

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The present work deals with the structural analysis of turbine blades and modeling of turbine blades. A common failure mode for turbine machines is high cycle of fatigue of compressor and turbine blades due to high dynamic stresses caused by blade vibration and resonance within the operation range of the machinery. In this work, proper damping system will be analyzed to reduce the vibrating blade. The main focus of the work is the modeling of under platform damper to evaluate the dynamic analysis of turbine-blade vibrations. The system is analyzed using Bond graph technique. Bond graph is one of the most convenient ways to represent a system from the physical aspect in foreground. It has advantage of putting together multi-energy domains of a system in a single representation in a unified manner. The bond graph model of dry friction damper is simulated on SYMBOLS-shakti® software. In this work, the blades are modeled as Timoshenko beam. Blade Vibrations under different working conditions are being analyzed numerically.

Keywords: Turbine blade vibrations, Friction dampers, Timoshenko Beam, Bond graph modeling.

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Authors: M. Karami Khorramabadi, A. R. Nezamabadi

Abstract:

Stability of functionally graded beams with piezoelectric layers subjected to axial compressive load that is simply supported at both ends is studied in this paper. The displacement field of beam is assumed based on first order shear deformation beam theory. Applying the Hamilton's principle, the governing equation is established. The influences of applied voltage, dimensionless geometrical parameter, functionally graded index and piezoelectric thickness on the critical buckling load of beam are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Stability, Functionally graded beam, First order shear deformation theory, Piezoelectric layer.

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Authors: Gholamhosein Khosravi, Mohammad Azadi, Hamidreza Ghezavati

Abstract:

In this paper, vibration of a nonlinear composite beam is analyzed and then an active controller is used to control the vibrations of the system. The beam is resting on a Winkler-Pasternak elastic foundation. The composite beam is reinforced by single walled carbon nanotubes. Using the rule of mixture, the material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are determined. The beam is cantilever and the free end of the beam is under follower force. Piezoelectric layers are attached to the both sides of the beam to control vibrations as sensors and actuators. The governing equations of the FG-CNTRC beam are derived based on Euler-Bernoulli beam theory Lagrange- Rayleigh-Ritz method. The simulation results are presented and the effects of some parameters on stability of the beam are analyzed.

Keywords: Carbon nanotubes, vibration control, piezoelectric layers, elastic foundation.

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Authors: A. R. Nezamabadi, M. Karami Khorramabadi

Abstract:

This paper studies stability of homogeneous beams with piezoelectric layers subjected to axial load that is simply supported at both ends lies on a continuous elastic foundation. The displacement field of beam is assumed based on first order shear deformation beam theory. Applying the Hamilton's principle, the governing equation is established. The influences of applied voltage, dimensionless geometrical parameter and foundation coefficient on the stability of beam are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.

Keywords: Stability, Homogeneous beam- Piezoelectric layer

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Authors: O. Miraliyari

Abstract:

This article attempts to analyze functionally graded beam thermal buckling along with piezoelectric layers applying based on the third order shearing deformation theory considering various boundary conditions. The beam properties are assumed to vary continuously from the lower surface to the upper surface of the beam. The equilibrium equations are derived using the total potential energy equations, Euler equations, piezoelectric material constitutive equations and third order shear deformation theory assumptions. In order to fulfill such an aim, at first functionally graded beam with piezoelectric layers applying the third order shearing deformation theory along with clamped -clamped boundary conditions are thoroughly analyzed, and then following making sure of the correctness of all the equations, the very same beam is analyzed with piezoelectric layers through simply-simply and simply-clamped boundary conditions. In this article buckling critical temperature for functionally graded beam is derived in two different ways, without piezoelectric layer and with piezoelectric layer and the results are compared together. Finally, all the conclusions obtained will be compared and contrasted with the same samples in the same and distinguished conditions through tables and charts. It would be noteworthy that in this article, the software MAPLE has been applied in order to do the numeral calculations.

Keywords: Thermal buckling, functionally graded beam, piezoelectric layer, various boundary conditions.

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Authors: M. Ghanbari, S. Hossainpour, G. Rezazadeh

Abstract:

In this paper, longitudinal vibration of a micro-beam in micro-scale fluid media has been investigated. The proposed mathematical model for this study is made up of a micro-beam and a micro-plate at its free end. An AC voltage is applied to the pair of piezoelectric layers on the upper and lower surfaces of the micro-beam in order to actuate it longitudinally. The whole structure is bounded between two fixed plates on its upper and lower surfaces. The micro-gap between the structure and the fixed plates is filled with fluid. Fluids behave differently in micro-scale than macro, so the fluid field in the gap has been modeled based on micro-polar theory. The coupled governing equations of motion of the micro-beam and the micro-scale fluid field have been derived. Due to having non-homogenous boundary conditions, derived equations have been transformed to an enhanced form with homogenous boundary conditions. Using Galerkin-based reduced order model, the enhanced equations have been discretized over the beam and fluid domains and solve simultaneously in order to obtain force response of the micro-beam. Effects of micro-polar parameters of the fluid as characteristic length scale, coupling parameter and surface parameter on the response of the micro-beam have been studied.

Keywords: Micro-polar theory, Galerkin method, MEMS, micro-fluid.

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Authors: M. Seguini, D. Nedjar

Abstract:

An accuracy nonlinear analysis of a deep beam resting on elastic perfectly plastic soil is carried out in this study. In fact, a nonlinear finite element modeling for large deflection and moderate rotation of Euler-Bernoulli beam resting on linear and nonlinear random soil is investigated. The geometric nonlinear analysis of the beam is based on the theory of von Kàrmàn, where the Newton-Raphson incremental iteration method is implemented in a Matlab code to solve the nonlinear equation of the soil-beam interaction system. However, two analyses (deterministic and probabilistic) are proposed to verify the accuracy and the efficiency of the proposed model where the theory of the local average based on the Monte Carlo approach is used to analyze the effect of the spatial variability of the soil properties on the nonlinear beam response. The effect of six main parameters are investigated: the external load, the length of a beam, the coefficient of subgrade reaction of the soil, the Young’s modulus of the beam, the coefficient of variation and the correlation length of the soil’s coefficient of subgrade reaction. A comparison between the beam resting on linear and nonlinear soil models is presented for different beam’s length and external load. Numerical results have been obtained for the combination of the geometric nonlinearity of beam and material nonlinearity of random soil. This comparison highlighted the need of including the material nonlinearity and spatial variability of the soil in the geometric nonlinear analysis, when the beam undergoes large deflections.

Keywords: Finite element method, geometric nonlinearity, material nonlinearity, soil-structure interaction, spatial variability.

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Authors: A. Ghanbari Mardasi, N. Wu, C. Wu

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In this study, a spatial wavelet-based crack localization technique for a thick beam is presented. Wavelet scale in spatial wavelet transformation is optimized to enhance crack detection sensitivity. A windowing function is also employed to erase the edge effect of the wavelet transformation, which enables the method to detect and localize cracks near the beam/measurement boundaries. Theoretical model and vibration analysis considering the crack effect are first proposed and performed in MATLAB based on the Timoshenko beam model. Gabor wavelet family is applied to the beam vibration mode shapes derived from the theoretical beam model to magnify the crack effect so as to locate the crack. Relative wavelet coefficient is obtained for sensitivity analysis by comparing the coefficient values at different positions of the beam with the lowest value in the intact area of the beam. Afterward, the optimal wavelet scale corresponding to the highest relative wavelet coefficient at the crack position is obtained for each vibration mode, through numerical simulations. The same procedure is performed for cracks with different sizes and positions in order to find the optimal scale range for the Gabor wavelet family. Finally, Hanning window is applied to different vibration mode shapes in order to overcome the edge effect problem of wavelet transformation and its effect on the localization of crack close to the measurement boundaries. Comparison of the wavelet coefficients distribution of windowed and initial mode shapes demonstrates that window function eases the identification of the cracks close to the boundaries.

Keywords: Edge effect, scale optimization, small crack locating, spatial wavelet.

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

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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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Authors: Sh. Mirhosseini, Y. Haghighatfar, M. Sedighi

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Forced vibration problem of a delaminated beam made of fiber metal laminates is studied in this paper. Firstly, a delamination is considered to divide the beam into four sections. The classic beam theory is assumed to dominate each section. The layers on two sides of the delamination are constrained to have the same deflection. This hypothesis approves the conditions of compatibility as well. Consequently, dynamic response of the beam is obtained by the means of differential transform method (DTM). In order to verify the correctness of the results, a model is constructed using commercial software ABAQUS 6.14. A linear spring with constant stiffness takes the effect of contact between delaminated layers into account. The attained semi-analytical outcomes are in great agreement with finite element analysis.

Keywords: Delamination, forced vibration, finite element modelling, natural frequency.

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Authors: H. Rong, X. C. Yin, J. Yang, Y. N. Shen

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Based on Rayleigh beam theory, the sub-impacts of a free-free beam struck horizontally by a round-nosed rigid mass is simulated by the finite difference method and the impact-separation conditions. In order to obtain the sub-impact force, a uniaxial compression elastic-plastic contact model is employed to analyze the local deformation field on contact zone. It is found that the horizontal impact is a complicated process including the elastic plastic sub-impacts in sequence. There are two sub-zones of sub-impact. In addition, it found that the elastic energy of the free-free beam is more suitable for the Poisson collision hypothesis to explain compression and recovery processes.

Keywords: beam, sub-impact, elastic-plastic deformation, finite difference method.

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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: Yuwaraj M. Ghugal, Ajay G. Dahake

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A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick cantilever isotropic beams are considered for the numerical studies to demonstrate the efficiency of the. Results obtained are discussed critically with those of other theories.

Keywords: Trigonometric shear deformation, thick beam, flexure, principle of virtual work, equilibrium equations, stress.

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Authors: Yuwaraj M. Ghugal, Ajay G. Dahake

Abstract:

A trigonometric shear deformation theory for flexure of thick beams, taking into account transverse shear deformation effects, is developed. The number of variables in the present theory is same as that in the first order shear deformation theory. The sinusoidal function is used in displacement field in terms of thickness coordinate to represent the shear deformation effects. The noteworthy feature of this theory is that the transverse shear stresses can be obtained directly from the use of constitutive relations with excellent accuracy, satisfying the shear stress free conditions on the top and bottom surfaces of the beam. Hence, the theory obviates the need of shear correction factor. Governing differential equations and boundary conditions are obtained by using the principle of virtual work. The thick simply supported isotropic beams are considered for the numerical studies to demonstrate the efficiency of the results obtained is discussed critically with those of other theories.

Keywords: Trigonometric shear deformation, thick beam, flexure, principle of virtual work, equilibrium equations, stress.

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Authors: Faruk Fırat Çalım, Nurullah Karaca, Hakan Tacettin Türker

Abstract:

In this study, out-of-plane free vibrations of a circular rods is investigated theoretically. The governing equations for naturally twisted and curved spatial rods are obtained using Timoshenko beam theory and rewritten for circular rods. Effects of the axial and shear deformations are considered in the formulations. Ordinary differential equations in scalar form are solved analytically by using transfer matrix method. The circular rods of the mass matrix are obtained by using straight rod of consistent mass matrix. Free vibrations frequencies obtained by solving eigenvalue problem. A computer program coded in MATHEMATICA language is prepared. Circular beams are analyzed through various examples for free vibrations analysis. Results are compared with ANSYS results based on finite element method and available in the literature.

Keywords: Circular rod, Out-of-plane free vibration analysis, Transfer Matrix Method.

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