Clamped-free Boundary Conditions Research Articles

Finite element investigation of multi-walled carbon nanotubes as mass sensors

Using the finite element method, multi-walled carbon nanotubes-based mass sensors are studied. The effects of different parameters such as number of walls of nanotube, nanotube diameter and nanotube length on the mass sensibility are explored. It is shown that the maximum sensitivity of nanotubes under clamped-free boundary conditions occurs when the mass is located at the farthest location from the clamped edge. Comparing the results of nanotubes with different walls, it is observed that single-walled carbon nanotubes are the best mass sensors. Moreover, it is represented that for large masses connected to the nanotubes, the mass sensor can only identify that an external mass added to it and the mass value is not identifiable.

Pyroeffects on Magneto-Electro-Elastic Sensor patch subjected to thermal load

The magneto-electro-elastic (MEE) material under thermal environment exhibits pyroelectric and pyromagnetic coefficients resulting in pyroelectric and pyromagnetic effects. The pyroelectric and pyromagnetic effects on the behavior of multiphase MEE sensors bonded on top surface of a mild steel beam under thermal environment is presented in this paper. The aim of the study is to find out how samples having different volume fractions of the multiphase MEE composite behave in sensor applications. This is studied at optimal location on the beam, where the maximum electric and magnetic potentials are induced due to pyroelectric and pyromagnetic effects under clamped-free and clamped-clamped boundary conditions. The sensor which is bonded on the top surface of the beam is modeled using 8-node brick element. The MEE sensor bonded on mild steel beam is subjected to uniform temperature rise of 50K. It is assumed that beam and sensor is perfectly bonded to each other. The maximum pyroelectric and pyromagnetic effects on electric and magnetic potentials are observed when volume fraction is vf=0.2. The boundary conditions significantly influence the pyroelectric and pyromagnetic effects on electric and magnetic potentials.

Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics

This study investigates the axial vibration of carbon nanotubes (CNTs) embedded in an elastic medium using scale dependent doublet mechanics (DM) theory. Governing equations and all boundary conditions of CNTs are derived based on the variational principle. Free vibration frequencies are obtained and compared with the classical elasticity results for clamped-clamped (C-C) and clamped-free (C-F) boundary conditions. The effect of elastic medium stiffness, nanorod length and doublet separation distance on the axial vibration is examined. It is obtained that important differences exist between vibration frequencies predicted by classical elasticity theory and DM. DM theory can be used in the nano length scale design of structures.

Free vibration of functionally graded thin beams made of saturated porous materials

This study presents free vibration of beam made of porous material. The mechanical properties of the beam is variable in the thickness direction and the beam is investigated in three situations: poro/nonlinear nonsymmetric distribution, poro/nonlinear symmetric distribution, and poro/monotonous distribution. First, the governing equations of porous beam are derived using principle of virtual work based on Euler-Bernoulli theory. Then, the effect of pores compressibility on natural frequencies of the beam is studied by considering clampedclamped, clamped-free and hinged-hinged boundary conditions. Moreover, the results are compared with homogeneous beam with the same boundary conditions. Finally, the effects of poroelastic parameters such as pores compressibility, coefficients of porosity and mass on natural frequencies has been considered separately and simultaneously.

Eringen’s Stress Gradient Model for Bending of Nonlocal Beams

This paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.

Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field

This paper deals with nonlocal divergence and flutter instability analysis of carbon nanotubes (CNTs) conveying fluid embedded in an elastic foundation under magnetic field. Nonlocal constitutive equations of Eringen and Euler–Bernoulli beam theory are used in the formulations. Also, the foundation is described by the Winkler and Pasternak models. The governing equation of motion and boundary conditions are derived using extended Hamilton’s variational principle. The extended Galerkin’s approach is adopted to reduce the partial differential equation governing the dynamics of the CNTs to a system of coupled ordinary differential equations. In the present study, four different boundary conditions are considered, namely the pined–pined (P–P), clamped–pined (C–P), clamped–clamped (C–C) and clamped–free (C–F). A detailed parametric study is conducted to elucidate the effects of the nonlocal effect, longitudinal magnetic field, elastic Winkler and Pasternak foundations and geometrically boundary conditions on the instability characteristic of CNTs. It was observed that the only instability type for the investigated CNT with clamped–free boundary condition (cantilever) is flutter, while CNT conveying fluid with both ends supported loses its stability by divergence first and then by flutter with increase in fluid velocity. It was also found that the magnetic field and the Winkler and Pasternak foundations increase the stiffness of the system. Therefore, flutter instability region is enlarged significantly due to the existence of springs, shear foundations and magnetic field. Also, results show that the nonlocal parameter has a prominent effect on the stability behavior of CNTs, in which increasing nonlocal parameter results in the decrease in stability region. Furthermore, it was shown that the stability behavior of CNT is strongly affected by different boundary conditions. Finally, the validity of the present analysis is confirmed by comparing the results with those obtained from the literature.

Free vibration analysis of axially layered functionally graded short beams using experimental and finite element methods

AbstractFree vibration behavior of short beams made of axially layered functionally graded material (FGM) was investigated experimentally and numerically. Beams, which have gradation of the material properties in the axial direction, are fabricated by powder metallurgy technique using different weight fractions of aluminum and silicon carbide powders. In order to determine elasticity modulus of axially layered functionally graded (FG) beams, homogeneous beams containing different weight fractions of Al (aluminum) and SiC (silicon carbide) are produced, and these homogeneous beams are subjected to tensile tests. Density of each homogeneous layer is also calculated experimentally. After determination of the mechanical properties of each layer of the FG beams, they are modeled in a finite element program (ANSYS) according to Timoshenko beam theory, and free vibration analyses are performed. Fundamental frequencies of the axially layered FG beams produced are also calculated experimentally. FG beams with clamped-free boundary conditions are considered. Layers of the axially FG beams are considered to have symmetric configurations. Effect of the change in weight fractions of SiC particles and sorting order of layers to fundamental frequency of the beam is investigated. Experimental results obtained are compared with numerical results.

Analytical solution of piezoelectric energy harvester patch for various thin multilayer composite beams

The study of piezoelectric cantilever beam as vibration based energy harvesting has been emerged over the past two decade. This paper presents distributed parameter electroelastic modeling of various multilayer composite beams with piezoelectric energy harvesters. This analytical model of beam with piezoceramic patch is developed base on classical laminate theory for the clamped-free boundary condition. For piezoceramic patch, two thin PZT-5A are symmetrically bonded on composite beams. The closed-form steady state response for coupled electrical output and structural vibration are obtained under harmonic time base excitation. The base motion vibration is applied in the form of translation in a transverse direction. The electrical output and vibration response are derived and generalized base on analytical electroelastic frequency response functions (FRFs) relating to base excitation for different layout composite laminate. Experiments are conducted to verify the first two natural frequencies for all specimens without active element layers. According to effect of damping coefficient on analytical electroelastic modeling, the parameter of damping ratio is given by experimental test. Finally, the effect of various composite laminated layout with piezoelectric patch harvester on power generated are discussed in details as well.

Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam

This paper is proposed to study the free vibration of a rotating Timoshenko nanobeam based on the nonlocal theory considering thermal and surface elasticity effects. The governing equations and the related boundary conditions are derived using the Hamilton’s principle. In order to solve the problem, generalized differential quadrature method is applied to discretize the governing differential equations corresponding to clamped–simply and clamped–free boundary conditions. In this article, the influences of some parameters such as nonlocal parameter, angular velocity, thickness of the nanobeam, and thermal and surface elasticity effects on the free vibration of the rotating nanobeam are investigated, and the results are compared for different boundary conditions. The results show that the surface effect and the nonlocal parameter and the temperature changes have significant roles, and they should not be ignored in the vibrational study of rotating nanobeams. Also, the angular velocity and the hub radius have more significant roles than temperature change effects on the nondimensional frequency. It is found that the nonlocal parameter behavior and the temperature change behavior on the frequency are different in the first mode for the rotating cantilever nanobeam.

Nonlinear free vibrations of centrifugally stiffened uniform beams at high angular velocity

In this paper, we study the bending nonlinear free vibrations of a centrifugally stiffened beam with uniform cross-section and constant angular velocity. The nonlinear intrinsic equations of motion used here are geometrically exact and specific to beams exhibiting large amplitude displacements and rotations associated with small strains. Based on the Timoshenko beam model, these equations are derived from Hamilton׳s principle, in which the warping is considered. All coupling terms are considered including Coriolis terms. The studied beams are isotropic with clamped-free boundary conditions. By combining the Galerkin method with the harmonic balance method, the equations of motion are converted into a quadratic function treated with a continuation method: the Asymptotic Numerical Method, where the generalized displacement vector is presented as a series expansion. While analysing the effect of the angular velocity, we determine the amplitude versus frequency variations which are plotted as backbone curves. Considering the first lagging and flapping modes, the changes in beam behaviour from hardening to softening are investigated and identified as a function of the angular velocity and the effect of shear. Particular attention is paid to high angular velocities for both Euler–Bernoulli and Timoshenko beams and the natural frequencies so obtained are compared with the results available in the literature.

Experimental and theoretical investigation on transverse vibration of delaminated cross-ply composite beams

Delamination is a major damage mode in laminated composite structures which causes reduction in stiffness and strength and affects their vibration characteristics. This paper deals with the effects of delamination size and its thickness-wise and lengthwise location on the vibration characteristics of cross-ply laminated composite beams. Free and constrained mode models are introduced and compared in the analytical and finite element methods for the first three modes and a comprehensive discussion among these results is done. To verify the results, modal tests were carried out on the delaminated specimens. Unlike the available experimental research, the proportion of delamination size to the beam length (a/L) is relatively small (i.e., a/L=0.20, 0.10 and 0.05). Moreover, these experiments are focused on the effect of the axial location of delamination on the first three natural frequencies. All results are considered under both the clamped-free and clamped-clamped boundary conditions. Finally, some interesting relationships are presented between the frequencies reduction and their corresponding mode shapes, which can be useful for delamination detection.

Buckling of Timoshenko beams composed of two-dimensional functionally graded material (2D-FGM) having different boundary conditions

In the present paper, two-dimensional functionally graded materials (2D-FGMs) are presented for the first time to investigate the buckling of beams with different boundary conditions. It is assumed that the material properties of the beam vary in both axial and thickness directions according to the power-law form. Basis on the Timoshenko beam theory (TBT), the critical buckling load of 2D-FG beams is obtained using the Ritz method. In order to obtain buckling load, the trial functions for axial, transverse deflections and rotation of the cross-sections are expressed in polynomial forms. Clamped–clamped (CC), Clamped-simple (CS), simple–simple (SS) and Clamped-Free (CF) boundary conditions are considered. The boundary conditions are satisfied by adding auxiliary functions to the displacement functions. At the same time, buckling load of 2D-FG beam is calculated for Euler–Bernoulli beam theory for comparison purposes. Some numerical results are provided to examine the effects of the material gradation, shear deformation (or aspect ratio) and different boundary conditions on the buckling behavior of 2D-FG beams.

Aeroelastic characteristics of magneto-rheological fluid sandwich beams in supersonic airflow

Supersonic aeroelastic instability of a three-layered sandwich beam of rectangular cross section with an adaptive magneto-rheological fluid (MRF) core layer is investigated. The panel is excited by an airflow along it’s longitudinal direction. The problem formulation is based on classical beam theory for the face layers, magnetic field dependent complex modulus approach for viscoelastic material model and the linear first-order piston theory for aerodynamic pressure. The classical Hamilton’s principle and the assumed mode method are used to set up the equations of motion. The validity of the derived formulation is confirmed through comparison with the available results in the literature. The effects of applied magnetic field, core layer thickness and constraining layer thickness on the critical aerodynamic pressure are studied. The onset of instability in terms of the critical value of the nondimensional aerodynamic pressure for the sandwich beam is calculated using the p-method scheme. Simply supported, clamped–clamped and clamped-free boundary conditions are considered. The results show that the magnetic field intensity and thickness ratios have significant effects on the instability bounds.

Stability analysis of a piezoelectrically actuated micro-pipe conveying fluid

This paper presents the stability analysis of a fluid-conveying micro-pipe axially loaded with a pair of piezoelectric layers located at its top and bottom surfaces. Based on Euler–Bernoulli beam theory, the governing equations of the system are derived by applying Hamilton’s variational principle. Galerkin projection technique is used to extract the frequency equations. Taking into account clamped-free boundary conditions with and without intermediate support, stability of the system is investigated to demonstrate the influence of flow velocity as well as the voltage of the piezoelectric layers on the flow-induced flutter instability. It is shown that imposing voltage difference to piezoelectric layers can significantly suppress the effect of fluid flow on vibrational frequencies and thus extend the stable margins. Moreover, effects of the intermediate support on the stability of the system are examined and it is shown that for some particular range of system configuration, the instability type may change from flutter to divergence.

Elastic Rods and Shear Beams with Random Field Properties under Random Field Loads: Fractal and Hurst Effects

AbstractResponses of elastic rods and shear beams with random field properties and also possibly under random field forcing are studied for random fields with linear, Matern, Cauchy, and Dagum covariances. The latter two allow decoupling of the fractal dimension and Hurst effect. The authors find second order characteristics of the beam displacement under clamped–free boundary conditions. Overall, for a given variance, the variance of the output is strongest for linear, then Matern, then Cauchy, and, finally, Dagum forcing. This is interesting and counterituitive because the Dagum and Cauchy models grasp the fractal characteristics and, additionally, the Hurst effect. In a number of (simpler) cases the results may be obtained in explicit analytical forms, but as Cauchy and Dagum models are introduced, one has to resort to numerics.

A layer-wise MITC9 finite element for the free-vibration analysis of plates with piezo-patches

The present article considers the free-vibration analysis of plate structures with piezoelectric patches by means of a plate finite element with variable through-the-thickness layer-wise kinematic. The refined models used are derived from Carrera’s Unified Formulation (CUF) and they permit the vibration modes along the thickness to be accurately described. The finite-element method is employed and the plate element implemented has nine nodes, and the mixed interpolation of tensorial component (MITC) method is used to contrast the membrane and shear locking phenomenon. The related governing equations are derived from the principle of virtual displacement, extended to the analysis of electromechanical problems. An isotropic plate with piezoelectric patches is analyzed, with clamped-free boundary conditions and subjected to open- and short-circuit configurations. The results, obtained with different theories, are compared with the higher-order type solutions given in the literature. The conclusion is reached that the plate element based on the CUF is more suitable and efficient compared to the classical models in the study of multilayered structures embedding piezo-patches.

Atomistic Simulation on Buckling Behavior of Monolayer Graphene

The buckling behavior of monolayer graphene sheets with simple-supported, clamped-free and clamped-clamped boundary conditions is investigated by the atomic-scale finite method (AFEM). The initial static equilibrium state of monolayer graphene sheet is obtained in the simulation as a waved configuration which is close to the real graphene observed in experiments. With the increase of compressive displacement, the force displays three stages: linear increasing, nonlinear increasing and decreasing slowly after a sudden drop. Different from the prediction by classical theory, the critical buckling loads of graphene sheets with different boundary conditions are similar, which is attributed to the initial waved configuration of the monolayer graphene sheets.

Nonlinear modelling and dynamic analysis of cracked Timoshenko functionally graded beams based on neutral surface approach

The present work accounts Timoshenko beam theory followed by Ritz approximation and an iterative technique to deal nonlinear free vibration problems of cracked Functionally Graded Material (FGM) beams based on neutral surface location. Using neutral surface as a reference rather than the midsurface reduces the complexity of nonlinear problems. It is assumed that crack always remains open. Analysis is carried out for clamped-clamped and clamped-free boundary conditions. Nonlinear frequencies and mode shapes corresponding to first three mode of vibration are obtained for the first time for different crack parameters, amplitudes of vibration and material indexes. The accuracy of the present solution is verified by comparing some of the obtained results with existing solutions. It can be concluded that present results are not only accurate but the methodology is very simple and easy to perform.

Size-dependent dynamic stability analysis of microbeams actuated by piezoelectric voltage based on strain gradient elasticity theory

In the current paper, dynamic stability analysis of microbeams subjected to piezoelectric voltage is presented in which the microbeam is integrated with piezoelectric layers on the lower and upper surfaces. Both of the flutter and divergence instabilities of microbeams with clamped-clamped and clamped-free boundary conditions are predicted corresponding to various values of applied voltage. To take size effect into account, the classical Timoshenko beam theory in conjunction with strain gradient elasticity theory is utilized to develop nonclassical beam model containing three additional internal length scale parameters. By using Hamilton’s principle, the higher-order governing differential equations and associated boundary conditions are derived. Afterward, generalized differential quadrature method is employed to discretize the size-dependent governing differential equations along with clamped-clamped and clamped-free end supports. The critical piezoelectric voltages corresponding to various values dimensionless length scale parameter are evaluated and compared with those predicted by the classical beam theory. It is revealed that in the case of clamped-free boundary conditions, the both of flutter and divergence instabilities occur. However, for the clamped-clamped microbeams, only divergence instability takes place.

The influence of size-dependent shear deformation on mechanical behavior of microstructures-dependent beam based on modified couple stress theory

In this investigation, a parametric study is performed to explore the influence of size-dependent shear deformation on static bending, buckling and free vibration behavior of microbeams based on modified couple stress classical and first shear deformation beam models. It is indicated that the influence of size-dependent shear deformation on mechanical behavior of the microbeams has an ascending trend with respect to dimensionless material length scale parameter. Moreover, the sensitivity of size-dependent shear deformation to dimensionless gyration radius and Poisson ratio has an ascending and a descending trend, respectively. The results show that the size-dependent shear deformation has the highest influence on the mechanical behavior of the microbeam with clamped–clamped boundary conditions followed by clamped–pined, pined–pined and clamped–free boundary conditions. As an application to micro electro mechanical systems (MEMS), the effect of size-dependent shear deformation on static pull-in voltage of an electrostatic microbridge is studied.