Cantilever End Conditions Research Articles

Flexural-torsional analysis of steel beam structures using ANSYS

The theories for uniform torsion and non-uniform torsion are studied. The aim of the paper is to study the behaviour of the beams of different sections for combined bending and torsion under various loadings and support conditions. The comparative study of the finite element for beams with different geometry sections of solid and hollow circular sections, rectangular section and symmetric I sections subjected to concentric and eccentric loading conditions with simply supported and cantilever end conditions is presented using ANSYS 18.1 workbench. The length of the beam is kept constant as 4 m. The transverse and axial deformation, the equivalent stresses and strains of beams were studied along with the mode of failure and effect of bending and torsion. It was observed that concentric loading resulted in transverse deflections while the torsional moment loading causes in-plane deformation in form of twisting of the cross-section with maximum stress generation and the eccentric loading results in combined transverse and in-plane axial deformation. From the result obtained it can be said that open sections has lesser efficiency for torsional and eccentric type loading as higher stresses and axial deflections were observed as compared to the hollow circular and rectangular sections.

Enhancement of vibrational characteristics of rotating CFRP composite tapered plates reinforced by graphene nanoparticles

AbstractIn the present study, the dynamic behavior of rotating carbon fiber reinforced polymer (CFRP) composite tapered plates reinforced with graphene nanoparticles has been investigated using a finite element (FE) formulation. The mechanical properties of CFRP laminates containing 0–0.5 wt% graphene nanoparticles were assessed using ASTM standard tests. The experimental dynamic analysis was conducted on CFRP and graphene reinforced CFRP composites under cantilever end conditions. The first‐order shear deformation theory (FSDT)‐based FE model was developed using MATLAB software to obtain the natural frequencies of the graphene reinforced CFRP plates. The validity of the FE results is compared with literature, and it shows excellent agreement for composite structures. Furthermore, parametric analysis is performed to explore the influence of wt% of graphene, rotating speed, setting angle, and ply thickness on the dynamic responses of graphene reinforced CFRP tapered plates.Highlights Effects of graphene reinforcement in CFRP composites are investigated. The mechanical characteristics were evaluated using ASTM standards. FSDT is employed to model the rotating tapered graphene‐CFRP plates. Vibration behavior of graphene‐CFRP composites is studied experimentally. Effects of various factors on the vibrations of the CFRP plates are studied.

Static and dynamic characteristics of jute/glass fiber reinforced hybrid composites

The free vibration and bending characteristics of hybrid jute composite beams are examined in the current paper using the finite element method (FEM). The hybrid jute composite beam contains jute fiber and glass fiber reinforced polymer composites. The hybrid composites are manufactured via a vacuum-assisted hand layup method. Experimental vibration analysis was performed on composite beams to predict natural frequencies under cantilever end conditions. The higher order theory based finite element formulation was developed and are solved using MATLAB® software to obtain the natural frequencies of the jute and hybrid jute composite beams. The validity of the FEM results (natural frequency, deflection and stress) is compared with the literature, and it shows very good agreement for composites. This study also examines the impact of stacking order, end condition, and ply orientation on the vibration and bending properties of hybrid jute composite beams.

On Effects of Shear Deformation on the Static Pull-in Instability Behaviour of Narrow Rectangular Timoshenko Microbeams

MEMS devices utilize electrostatics as preferred actuation method. The accurate determination of pull-in instability parameters (i.e., pull-in voltage and pull-in displacement) of such devices is critical for their correct design. It should be noted that similar to parallel plate capacitors, the electrostatic force between the surface of the deformable microbeam and stationary ground is non-linear in nature. Hence the analysis associated with MEMS devices is always inherently non-linear. In the literature, these devices have been majorly analyzed as Bernoulli-Euler microbeams with cantilever or clamped-clamped beam end conditions. However, Dileesh et al. (doi: 10.1115/ESDA2012-82536) have studied the static and dynamic pull-in instability behavior of slender cantilever microbeams by developing a six-nodded spectral finite element based on the Timoshenko beam theory (TBT-SFE). They have demonstrated the accuracy of the TBT-SFE by comparing their results with corresponding results of COMSOL-based three-dimensional finite element simulations. In addition, effects of shear deformation also start to play significant role as the beam thickness-to-length ratio increases. In this paper, authors have developed the TBT-SFE based on the work by Dileesh et al. for the case of statics. However, unlike Dileesh et al. where they have developed a six-nodded TBT-SFE, authors have investigated the best combination of number of nodes per element and total number of elements to carry out the study. For this purpose, authors have first calculated results of the maximum beam transverse displacement, for a shear deformable propped-cantilever microbeam under the action of uniformly distributed transverse load, obtained by utilizing the developed TBT-SFE with different combinations of number of nodes per element and total number of elements. These results are then compared with corresponding analytical results available in the literature to arrive at the best combination of number of nodes per element and total number of elements for the electrostatic-elastic analysis. In the second step, the finalized TBT-SFE is utilized to determine static pull-in instability parameters of narrow microbeams with various fixity conditions and beam thickness-to-length ratios. This study highlights the importance of transverse shear effects on pull-in instability parameters of Timoshenko microbeams.

Effect of material on fluid elastic instability of parallel triangular tube array subjected to water cross

The effect of tube material and structure on fluid elastic instability is examined experimentally using a parallel triangular finned tube array with a P/D ratio of 1.78. Flow-induced vibration is a common cause of failure in shell and tube heat exchangers, and it may result in substantial harm to the heat exchanger as well as significant financial loss. There are different mechanisms responsible for flow induced vibration out of which fluid elastic instability and vortex shedding are the most severe due to their sudden occurrence and high amplitude of vibration. Different parameters affect the tube vibration when the tube array is subjected to water crossflow. In this paper effect of tube material and fin are examined for fluid elastic instability. Experimentation was performed to measure demanding velocity at fluid elastic instability for plain and finned tubes for steel, aluminum and copper material. A fully flexible tube array with a cantilever end condition and with a constant pitch ratio subjected to water cross flow was considered. Testing was done with a gradually increasing water flow rate from 5 m3/hr and increases up to 75 m3/hr to obtain fluid elastic instability. The relationship between the critical velocity at fluid elastic instability and the mass damping parameter was investigated using Connor's equation. The amplitude response of vibration concerning the change in velocity shows that the addition of fins, as well as the material change, greatly affects the fluid elastic instability. A small peak before the occurrence of fluid elastic instability was observed. This is maybe due to vortex shedding which further required to be verified using the Strouhal number.

Hearing distributed mass in nanobeam resonators

One-dimensional vibrating nanostructures show remarkable performance in detecting small adherent masses added to a referential configuration. The mass sensing principle is based on measuring the resonant frequency shifts caused by the unknown attached masses. In spite of its important application in several fields, few studies have been devoted to this inverse eigenvalue problem. In this paper we have developed a distributed mass reconstruction method for initially uniform nanobeams based on measurements of the first lower resonant frequencies of the free bending vibration. Two main inverse problems are addressed. In the first problem, the mass variation is determined by using the first lower eigenfrequencies of a supported nanobeam, under the a priori assumption that the mass variation has support contained in half of the axis interval. In the second problem, we show that the a priori assumption can be removed, provided that the spectral input data include an additional set of first lower eigenfrequencies belonging to a second spectrum associated to different end conditions. The nanobeam is modelled using the modified strain gradient elasticity accounting for size effects. The reconstruction is based on an iterative procedure which takes advantage of a closed-form solution when the mass change is small, and shows to be convergent under this assumption and for smooth mass variation. The accuracy of the reconstruction deteriorates in presence of discontinuous mass variation. For these cases, a constrained least-squares optimization filtering shows to be very effective to reduce the spurious oscillations around the target coefficient. Numerical simulations show that the identification method performs well even for not necessarily small mass changes and it is stable in presence of errors on the data. An experimental validation of the method has provided encouraging results, despite the fact that only the first four eigenfrequencies under cantilever end conditions were used.

Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory

In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyze the slip boundary conditions between the fluid-flow and the nanotube's wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton's principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied.

Mass detection in nanobeams from bending resonant frequency shifts

Abstract Nanobeams are frequently used as vibration based-sensors to detect mass changes caused, for instance, by attachment of foreign atoms/molecules or chemical/molecular absorption. This paper deals with the bending vibration of a uniform nanobeam carrying a single point mass (direct problem) as well as the identification of the attached mass (inverse problem). The nanobeam is described using the modified strain energy theory adapted to the Euler-Bernoulli beam model, and the natural vibration frequencies have been obtained. Under the assumption of small intensity of the concentrated mass, a solution of the inverse problem based on the measurement of the mass-induced shifts in the first two eigenfrequencies is proposed. Both the cases of simply supported and cantilever end conditions are discussed in detail. The theoretical method is verified by numerical simulation and numerical tests agree well with analytical results.

Crack identification in rods and beams under uncertain boundary conditions

This paper deals with the inverse problem of identifying a crack in a rod in axial vibration with partially unknown end conditions from a minimum number of resonant frequency variations. It is assumed that the crack is small and is modelled by an elastic spring acting along the rod axis. A first set of results concerns a uniform bar with both ends restrained by means of elastic springs having unknown flexibility. Under the hypothesis that the flexibility caused by the crack is small and of the same order of the flexibility of the elastic end constraints, it is shown that the inverse problem can be formulated in terms of the variations of the first three natural frequencies measured from the undamaged bar under ideal condition of fixed ends. It is proved that knowledge of this set of eigenfrequency variations can uniquely determine the overall flexibility induced by the end conditions, and the position (up to symmetry) and severity of the crack, by means of closed form expressions. The identification method can be also applied to axial vibrations of uniform cantilevers with elastically restrained end condition, and to transversely vibrating uniform beams either under elastic transverse support at both ends or under cantilever end conditions. The method was verified by numerical simulation and, in the case of the cantilever in bending vibration, by experimental data. Numerical analysis allowed to study in detail some singular situations occurring in the mathematical formulation of the inverse problem and to test the robustness of the method to errors on the data.

Experimental Investigation on Vortex Shedding and Fluid Elastic Instability in Finned Tube Arrays Subjected to Water Cross Flow

The paper presents results of an experimental study on fluid elastic instability and vortex shedding in plain and finned arrays exposed to water cross flow. The parallel triangular array with cantilever end condition is considered for the study. Pitch ratios considered are 2.1 and 2.6 while fin densities considered are 4 fpi (fins per inch) and 10 fpi. The results for critical velocity at instability for two finned tube arrays are presented. Apart from results on fluid elastic vibration behavior, extensive results on vortex shedding are also presented to study the phenomenon in tube arrays subjected to water cross flow. The test parameters measured are water flow rate, natural frequency, and vibration amplitudes of the tubes. The datum case results were first obtained by testing plain arrays with pitch ratios 2.1 and 2.6. This was then followed by experiments with finned arrays with pitch ratios 2.1 and 2.6, and each with two different fin densities. The higher pitch ratios typical of chemical process industries resulted in the delayed instability threshold due to weak hydrodynamic coupling between the neighboring tubes. The results indicated that finned arrays are more stable in water cross flow compared to plain arrays. The Strouhal numbers corresponding to small peaks observed before fluid elastic instability are computed and compared with the expected ones according to Owen's hypothesis. It was concluded that peaks observed are attributed to vortex shedding observed for all the arrays tested in water.

Crack identification in elastically restrained vibrating rods

In this paper we consider the problem of identifying an open crack in a longitudinally vibrating rod with smooth variable profile by minimal eigenfrequency data. The crack is assumed to be open during vibration and it is modelled by an elastic spring acting along the rod axis. Most, if not all, the results available in the literature for this inverse problem refer to ideal end conditions, that is the rod is either under free or supported end conditions. As an example of almost optimal result, it is known that the knowledge of the fundamental (positive) natural frequency of the rod under free-free and cantilever end conditions allows for the unique determination of the crack, without any restriction on the damage severity. In this paper we show that the analysis of the analogous crack identification problem for rods under elastically restrained end conditions leads to different results and that, in general, the knowledge of the fundamental frequency belonging to two spectra associated to different end conditions is not sufficient for the uniqueness of the solution. The method we used to solve the inverse problem is of constructive type and it is based on general properties of the eigenfrequencies as functions of the position and severity of the crack. The identification procedure has been tested numerically on rods under various damage scenarios. Numerical results agree well with the theory, even in presence of noisy input data.

The Effect of Fin Pitch on Fluid Elastic Instability of Tube Arrays Subjected to Cross Flow of Water

Failure of tubes in shell and tube exchangers is attributed to flow induced vibrations of such tubes. There are different excitations mechanisms due to which flow induced vibration occurs and among such mechanisms, fluid elastic instability is the most prominent one as it causes the most violent vibrations and may lead to rapid tube failures within short time. Fluid elastic instability is the fluid–structure interaction phenomenon which occurs when energy input by the fluid force exceeds energy expended in damping. This point is referred as instability threshold and corresponding velocity is referred as critical velocity. Once flow velocity exceeds critical flow velocity, the vibration amplitude increases very rapidly with flow velocity. An experimental program is carried out to determine the critical velocity at instability for plain and finned tube arrays subjected to cross flow of water. The tube array geometry is parallel triangular with cantilever end condition and pitch ratios considered are 2.6 and 2.1. The objective of research is to determine the effect of increase in pitch ratio on instability threshold for plain tube arrays and to assess the effect of addition of fins as well as increase in fin density on instability threshold for finned tube arrays. Plain tube array with two different pitch ratios; 2.1 and 2.6 and finned tube arrays with same pitch ratio; 2.6 but with two different fin pitches; such as fine (10 fpi) and coarse (4 fpi) are considered for the experimentation. Connors’ equation that relates critical velocity at instability to different parameters, on which instability depends, has been used as the basis for analysis and the concept of effective diameter is used for the present investigation. The modal parameters are first suitably modified using natural frequency reduction setup that is already designed and developed to reduce natural frequency and hence to achieve experimental simulation of fluid elastic instability within the limited flow capacity of the pump. The tests are carried out first on plain tube arrays to establish the same as the datum case and results are compared to known results of plain tube arrays and hence the quality of the test rig is also assessed. The fluid elastic vibration tests are then carried out on finned tube arrays with coarse and fine fin pitches and effects of fins and fin pitch on instability threshold are shown. The vibration response of the tube is recorded for each gradually increasing flow rates of water till instability point is reached. The parameters at the instability are then presented in terms of dimensionless parameters to compare them with published results. It is concluded that, arrays with higher pitch ratios are unstable at comparatively higher flow velocities and instability threshold for finned tube arrays is delayed due to addition of the fins. Further, it is concluded that, instability threshold for finned tube arrays with fine fin pitch is delayed compared to coarse fin pitch and hence for increased fin density, instability threshold is delayed. The experimental results in terms of critical velocities obtained for different tube arrays subjected to water cross flow will serve as the base flow rates for air–water cross flow experiments to be conducted in the next phase.

Investigation of the Elastic Modulus Effect on Free Vibration Characteristics of Fiber Reinforced Polymer Composite Beams Made of Various Fundamental Shapes

Woven fiber reinforced polymer composites plays major role in structural applications. Structures subjected to dynamic situations, this leads to increased attention in finding characteristic behaviour/performance of FRP composites beams under vibrating conditions. The objective of this work is to understand and investigate the influence of cross sectional shapes of beams on successive vibration resonance frequencies and its damping performance under cantilever end conditions. The beams were fabricated by hand lay-up method in two different modulus material and cross sections like I and channel; maintaining uniform cross sectional area and moment of inertia apart from length of the beams using woven carbon fibre. The investigation was performed by impulse frequency response method using standard test procedure on low modulus Glass/epoxy composite beams and compared with high modulus Carbon/epoxy beams. Frequency response curve of I and channel shaped beams were compared for their damping performance. The modal frequencies of all fabricated beams were obtained using MEScopeVES® software and modal resonant frequencies, loss factor were compared against shapes.

Dynamic behaviour of non-uniform Bernoulli-Euler beams subjected to concentrated loads travelling at varying velocities.

This paper investigates the dynamics behaviour of non-uniform Bernoulli-Euler beams subjected to concentrated loads ravelling at variable velocities. The solution technique is based on the Generalized Galerkin Method and the use of the generating function of the Bessel function type. The results show that, for all the illustrative examples considered, for the same natural frequency, the critical speed for the system consisting of a non-uniform beam traversed by a force moving at a non-uniform velocity is greater than that of the corresponding moving mass problem. It was also found that, for fixed axial force, an increase in foundation moduli reduces the response amplitudes of the dynamical system. Furthermore, it was shown that the transverse-displacement amplitude of a clamped-clamped non-uniform Bernoulli-Euler beam traversed by a load moving at variable velocities is lower than that of the cantilever. The response amplitude of the same dynamical systems which is simply supported is higher than those which consist of clamped-clamped or clamped-free (Cantilever) end conditions. Finally, an increase in the values of foundation moduli and axial force reduces the critical speed for all variants of the boundary conditions Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 79-102

Coupled Efficient Zigzag Finite Element Analysis of Piezoelectric Hybrid Beams under Thermal Loads

A new one-dimensional beam finite element is developed for hybrid piezoelectric beams under thermal load, using a coupled efficient layerwise (zigzag) theory developed recently by the authors. The theory accounts for the layerwise variations of the axial and the transverse displacements while keeping the number of displacement variables independent of the number layers. The beam element has two nodes with four mechanical and a variable number of electric potential degrees of freedom at each node. In the thickness direction, the thermal and the electric fields are approximated as piecewise linear across an arbitrary number of sublayers in a layer. Cubic Hermite interpolation is used for the deflection and electric potentials at the sublayers and linear interpolation is used for the axial displacement and the shear rotation. The thermal field is computed using a consistent six-noded thermal finite element with a quadratic interpolation along longitudinal direction and a linear interpolation along thickness direction. The formulation is validated by comparing the results with the Navier-type solution of the zigzag theory for simply-supported hybrid beams. The element is free from shear locking. The accuracy of the zigzag theory is established by comparing the results of hybrid composite and sandwich beams with the two-dimensional finite element results using ABAQUS for cantilever and clamped-clamped end conditions under different thermal loads. The control of thermal deflection by the application of actuation potential is illustrated. The effects of electric boundary conditions and the pyroelectric effect on the response are discussed.

Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam

An exact dynamic stiffness matrix for a twisted Timoshenko beam is developed in this paper in order to investigate its free vibration characteristics. First the governing differential equations of motion and the associated natural boundary conditions of a twisted Timoshenko beam undergoing free natural vibration are derived using Hamilton's principle. The inclusion of a given pretwist together with the effects of shear deformation and rotatory inertia, gives rise in free vibration to four coupled second order partial differential equations of motion involving bending displacements and bending rotations in two planes. For harmonic oscillation these four partial differential equations are combined into an eighth order ordinary differential equation, which is identically satisfied by all components of bending displacements and bending rotations. This difficult task has become possible only with the help of symbolic computation. Next the exact solution of the differential equation is obtained in completely general form in terms of eight arbitrary constants. This is followed by application of boundary conditions for displacements and forces. The procedure leads to the formation of the dynamics stiffness matrix of the twisted Timoshenko beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to compute the natural frequencies and mode shapes of a twisted Timoshenko beam with cantilever end condition. The exact results from the present theory are compared with numerically simulated results using simpler theories, and some conclusions are drawn.

Free vibration analysis of a twisted beam using the dynamic stiffness method

An exact dynamics stiffness matrix is developed and subsequently used for free vibration analysis of a twisted beam whose flexural displacements are coupled in two planes. First the governing differential equations of motion of the twisted beam undergoing free natural vibration are derived using Hamilton's principle. Next the general solutions of these equations are obtained when the oscillatory motion of the beam is harmonic. This is followed by application of boundary conditions for displacements and forces, which essentially leads to the formation of the dynamics stiffness matrix of the twisted beam relating harmonically varying forces with harmonically varying displacements at its ends. The resulting dynamic stiffness matrix is used in connection with the Wittrick–Williams algorithm to compute natural frequencies and mode shapes of a twisted beam with cantilever end condition. These are compared with previously published results to confirm the accuracy of the method, and some conclusions are drawn.

Frequency equation and mode shape formulae for composite Timoshenko beams

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the composite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional rotations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinantal form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are available.

USE OF GENERALIZED MASS IN THE INTERPRETATION OF DYNAMIC RESPONSE OF BENDING–TORSION COUPLED BEAMS

The interpretation of mode shapes and dynamic response of bending-torsion coupled beams is assessed by using the concept of generalized mass. In the first part of this investigation, the free vibratory motion of bending–torsion coupled beams is studied in detail. The conventional method of interpreting the normal modes of vibration consisting of bending displacements and torsional rotations is shown to be inadequate and replaced by an alternative method which is focussed on the constituent parts of the generalized mass arising from bending and torsional displacements. Basically, the generalized mass in a particular mode is identified and examined in terms of bending, torsion and bending–torsion coupling effects. It is demonstrated that the contribution of individual components in the expression of the generalized mass of a normal mode is a much better indicator in characterizing a coupled mode. It is also shown that the usually adopted criteria of plotting bending displacement and torsional rotations to describe a coupled mode can be deceptive and misleading. In the second part of the investigation, attention is focussed on the dynamic response characteristics of bending–torsion coupled beams when subjected to random bending or torsional loads. A normal mode approach is used to establish the total response. The input random excitation is assumed to be stationary and ergodic so that with the linearity assumption, the output spectrum of the response is obtained by using the frequency response function. The contribution of each normal mode to the overall response is isolated. Particular emphasis is placed on bending-induced torsional response and torsion-induced bending response. A number of case studies involving different types of bending–torsion coupled beams with Cantilever end conditions are presented. The limitations of existing methods of modal interpretation are highlighted, and an insight into the mode selection for response analysis is provided.

Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element

In this article, a dynamic finite element formulation for the free vibration analysis of axially loaded bending-torsion coupled beams is presented. Based on the Euler–Bernoulli and St. Venant beam theories, the exact solutions of the differential equations governing the uncoupled vibrations of an axially loaded uniform beam are found. Then, employing these solutions as basis functions, the analytical expressions for uncoupled bending and torsional dynamic shape functions are derived. Exploiting the principle of virtual work, together with the variable approximations based on the resulting shape functions, leads to a single frequency dependent element matrix which has both mass and stiffness properties. The application of the theory is demonstrated by an illustrative example of a bending-torsion coupled beam with cantilever end conditions, for which the influence of axial force on the natural frequencies is studied. The correctness of the theory is confirmed by the published results and numerical checks.