Beam Geometrical Parameters Research Articles

Multimode Analysis of Geometrically Nonlinear Transverse Free and Forced Vibrations of Tapered Beams

In this study, the geometrically nonlinear free and forced vibrations of tapered beams are investigated on the basis of the Euler-Bernoulli beam theory and the von Karman geometric nonlinearity assumptions. The aim of the analysis is to determine tapered beams nonlinear frequencies and modes, and the associated stress distributions by means of bending moment diagrams. The linear problem is first solved. Then, to tackle the nonlinear problem, the displacement function is expanded as a series of the linear modes and the discrete expressions for the strain and kinetic energies are derived. Assuming a point excitation of the beam, the algebraic nonlinear system obtained based on Hamilton’s principle is solved by an approximate method. The effect of the geometric nonlinearity in both the free and forced cases was illustrated and discussed and the effect of the variation of various tapered beam geometrical parameters. The effect of varying the excitation level was also examined.

Deployable Structures Based on Buckling of Curved Beams Upon a Rotational Input

AbstractInspired by the recent success of buckling‐induced reconfigurable structures, a new class of deployable systems that harness buckling of curved beams upon a rotational input is proposed. First, experimental and numerical methods are combined to investigate the influence of the beam's geometric parameters on its non‐linear response. Then, it is shown that a wide range of deployable architectures can be realized by combining curved beams. Finally, the proposed principles are used to build deployable furniture such as tables and lamp shades that are flat/compact for transportation and storage, require simple or no assembly, and can be expanded by applying a simple rotational input.

In-plane responses of multilayer FG-GPLRC curved beams in thermal environment under moving load

As a first attempt, the transient in-plane responses of multilayer functionally graded graphene platelet-reinforced composite (FG-GPLRC) curved beams in thermal environment under a concentrated moving load are investigated. The motion equations are derived based on the first-order shear deformation theory by considering the influences of the initial thermal stresses. The Chebyshev–Ritz method together with Newmark’s time integration scheme is employed to solve the equations of motion with different sets of boundary conditions. In this regard, the Chebyshev polynomials in conjunction with suitable boundary functions are used to construct the admissible functions of the field variables. Each layer is built up from an isotropic homogeneous polymer matrix reinforced by uniformly distributed and randomly oriented graphene platelets (GPLs). The multilayer FG-GPLRC curved beams are composed of a sufficient number of GPL-reinforced layers to create gradation without abrupt change in their material properties along the beam thickness direction. After validating the approach, the effects of moving load velocity, the GPL weight fraction and through-the-thickness distributions, total number of GPLRC curved beam layers, the curved beam geometric parameters, thermal environment and edge boundary conditions on the dynamic behavior of FG-GPLRC curved beams subjected to a moving load are studied.

Out-of-plane vibration of laminated FG-GPLRC curved beams with piezoelectric layers

Abstract As a first endeavour, the out-of-plane vibration characteristics of laminated functionally graded graphene platelets reinforced composite (FG-GPLRC) curved beams bonded by piezoelectric layers are investigated. The displacement components are approximated through the beam thickness direction based on the first-order shear deformation theory (FSDT). Accordingly, the shear deformation and rotary inertia effects due to both the torsional and flexural deformations are considered. The effective mechanical properties of the nanocomposite layers are estimated using the modified Halpin-Tsai model. The governing equations are derived by employing Hamilton's principle, which are discretized in the spatial domain using the differential quadrature method (DQM). After validating the approach, some useful results are provided which can be used for future researches. In this regards, the effects of graphene platelets (GPLs) distribution patterns, GPLs weight fraction and dimensions, number of GPLs reinforced layers, piezoelectric layer thickness, the curved beam geometric parameters and boundary conditions on the vibrational characteristics of the laminated FG-GPLRC curved beams embedded in piezoelectric layers are studied.

Static buckling and vibration analysis of continuously graded ceramic-metal beams using a refined higher order shear deformation theory

Purpose The purpose of this paper is to study the static buckling and free vibration of continuously graded ceramic-metal beams by employing a refined higher-order shear deformation, which is also the primary goal of this paper. Design/methodology/approach The proposed model is able to catch both the microstructural and shear deformation impacts without employing any shear correction factors, due to the realistic distribution of transverse shear stresses. The material properties are supposed to vary across the thickness direction in a graded form and are estimated by a power-law model. The equations of motion and related boundary conditions are extracted using Hamilton’s principle and then resolved by analytical solutions for calculating the critical buckling loads and natural frequencies. Findings The obtained results are checked and compared with those of other theories that exist in the literature. At last, a parametric study is provided to exhibit the influence of different parameters such as the power-law index, beam geometrical parameters, modulus ratio and axial load on the dynamic and buckling characteristics of FG beams. Originality/value Searching in the literature and to the best of the authors’ knowledge, there are limited works that consider the coupled effect between the vibration and the axial load of FG beams based on new four-variable refined beam theory. In comparison with a beam model, the number of unknown variables resulting is only four in the general cases, as against five in the case of other shear deformation theories. The actual model represents a real distribution of transverse shear effects besides a parabolic arrangement of the transverse shear strains over the thickness of the beam, so it is needless to use of any shear correction factors.

Magneto-electro-elastic analysis of piezoelectric–flexoelectric nanobeams rested on silica aerogel foundation

This article explores that the study on bending of magneto-electric-elastic nanobeams relies on nonlocal elasticity theory. The Vlasov’s model foundation utilizes the silica aerogel foundation. The guiding expressions of nonlocal nanobeams in the considered framework are used extensively and where parabolic third-order beam theory is achieved after using Hamilton’s principle. Parametric work is introduced to scrutinize the influence of the magneto-electro-mechanical loadings, nonlocal parameter, and aspect ratio on the deflection characteristics of nanobeams. It is noticed that the boundary conditions, nonlocal parameter, and beam geometrical parameters have significant effects on dimensionless deflection of nanoscale beams.

Correlation of process parameters and properties of TiO2 films grown by ion beam sputter deposition from a ceramic target

The correlation between process parameters and properties of TiO2 films grown by ion beam sputter deposition from a ceramic target was investigated. TiO2 films were grown under systematic variation of ion beam parameters (ion species, ion energy) and geometrical parameters (ion incidence angle, polar emission angle) and characterized with respect to film thickness, growth rate, structural properties, surface topography, composition, optical properties, and mass density. Systematic variations of film properties with the scattering geometry, namely the scattering angle, have been revealed. There are also considerable differences in film properties when changing the process gas from Ar to Xe. Similar systematics were reported for TiO2 films grown by reactive ion beam sputter deposition from a metal target [C. Bundesmann et al., Appl. Surf. Sci. 421, 331 (2017)]. However, there are some deviations from the previously reported data, for instance, in growth rate, mass density and optical properties.

Low velocity impact analysis of an axially functionally graded carbon nanotube reinforced cantilever beam

In this article, a new analytical model is presented for low velocity impact on an axially functionally graded (AFG) carbon nanotube (CNT) reinforced cantilever beam. In order to estimate the material properties of the composite media, the refined rule of mixtures approach containing the efficiency parameters is employed. Based on the third order shear deformation beam theory, the beam deformation is formulated and the contact force is obtained with the aid of the nonlinear Hertz contact law. To derive the equations of motion, energy method with combining the polynomial Ritz method and generalized Lagrange equations are employed. To validate the accuracy of the analytical model, the convergence and comparison studies are carried out with those are reported in the literature in the special cases and simulating the process with ABAQUS software. Afterward, the influences of CNTs distribution, CNTs volume fraction, impact position, impactor initial velocity, and beam geometrical parameters on the contact force and beam deflection at the impact position are studied in detail. POLYM. COMPOS., 39:E969–E983, 2018. © 2017 Society of Plastics Engineers

Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.

Latching in bistable electrostatically actuated curved micro beams

Curved beams subjected to transverse force may exhibit latching phenomena, namely remain in the buckled configuration under zero force. In this study we investigate the latching in bistable electrostatically actuated initially curved and prestressed beams. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam as base functions. Criteria for the existence of latching are derived in terms of the beam geometric parameters and axial load. Two conditions are formulated: The necessary criterion establishes the appearance of latching in the case of a symmetric snap-through, and the sufficient condition assures the existence of latching in the presence of symmetry breaking. The results provided by the RO model are compared to results obtained by direct numerical analyses. Furthermore, the dynamic behavior of curved beams prone to latching, and actuated by a suddenly applied electrostatic force of finite duration, is numerically studied. The dependence of the response on the loading parameters and the damping is presented through maps. These maps indicate the conditions required for trapping the beam at a secondary stable latching point, which is either accessible or inaccessible under quasi-static actuation. In addition, dynamic release of a latched beam is presented without the usage of an additional electrode, usually used for generating an opposite quasi-static release force.

Ultralow frequency acoustic bandgap and vibration energy recovery in tetragonal folding beam phononic crystal

This paper investigates ultralow frequency acoustic properties and energy recovery of tetragonal folding beam phononic crystal (TFBPC) and its complementary structure. The dispersion curve relationships, transmission spectra and displacement fields of the eigenmodes are studied with FEA in detail. Compared with the traditional three layer phononic crystal (PC) structure, this structure proposed in this paper not only unfold bandgaps (BGs) in lower frequency range (below 300 Hz), but also has lighter weight because of beam structural cracks. We analyze the relevant physical mechanism behind this phenomenon, and discuss the effects of the tetragonal folding beam geometric parameters on band structure maps. FEM proves that the multi-cell structures with different arrangements have different acoustic BGs when compared with single cell structure. Harmonic frequency response and piezoelectric properties of TFBPC are specifically analyzed. The results confirm that this structure does have the recovery ability for low frequency vibration energy in environment. These conclusions in this paper could be indispensable to PC practical applications such as BG tuning and could be applied in portable devices, wireless sensor, micro-electro mechanical systems which can recycle energy from vibration environment as its own energy supply.

Finite Element Formulation for the Large-Amplitude Vibrations of FG Beams

Abstract On the basis of Euler-Bernoulli beam theory, the large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The von Karman type nonlinear strain-displacement relationship is employed where the ends of the beam are constrained to move axially. The material properties are assumed to be graded in the thickness direction according to the power-law and sigmoid distributions. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law index, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible.

Topology optimization of piezoelectric materials and application to the cantilever beams for vibration energy harvesting

A new design analysis method based on FEM and a topology optimization for piezoelectric materials was developed for the unimorph cantilevered energy harvesters which can produce the maximum electric power outputs. The optimum topology of a piezoelectric material layer on the harvesting beam has been calculated by considering natural frequencies of beams, electromechanical couplings of piezoelectric materials, tip masses and MMA (method of moving asymptotes). The piezoelectric coefficients such as elasticity, capacitance and piezoelectric coupling were interpolated by element density variables in the topology optimization. The optimum design method was verified by vibration tests and measuring voltage outputs of the harvester and a good correlation between two results has been obtained. The effects of beam geometric parameters and several piezoelectric materials (PZT, PVDF, PMN-PT and piezoelectric fiber composites) on power generation were also investigated. The beam with PMN-PT generated the largest voltage and the next largest is PZT, MFC and PVDF, respectively.

Dynamic response of a delaminated composite beam with general lay-ups based on the first-order shear deformation theory

The dynamic response analysis of a delaminated composite beam with a general lay-up traversed under an arbitrary moving/non-moving force is presented. By employing the energy method and introducing a new finite element, the global mass and stiffness matrices for a Laminated Composite Beam (LCB) of Timoshenko type are derived in which the material couplings (bending–tension, bending–twist, and tension–twist couplings) with the Poisson’s effect are considered. In deriving the governing equation the non-penetration condition is imposed by employing the method of Lagrange multipliers. Out of a self-developed finite element program, the natural frequencies and time response of such LCB are obtained. To check on the accuracy of the derived equation and hence, developed program, the obtained results are compared with the results from other available references out of which very good agreements are observed. Finally, by changing the laminate’s lay-ups, beam geometrical parameters and type of external force, the LCB’s natural frequencies and time responses are obtained as a primary base for the structural design.

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.

On the postbuckling and free vibrations of FG Timoshenko beams

The postbuckling behavior of functionally graded beams is investigated by means of an exact solution method. The Von-Karman type nonlinear strain–displacement relationships are employed. The effects of the transverse shear deformation and rotary inertia are also included based upon the Timoshenko beam theory. After writing the kinetic and potential energy functionals, the governing equations of motion including the axial, transverse deflections and also the cross sectional rotation are derived using the Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. Neglecting the inplane inertia, the three equations of motion are reduced to two nonlinear partial–integral–differential equations in terms of the transverse mid-plane deflection and the cross sectional rotation. FG beams are considered to have fixed–fixed, fixed–hinged, and hinged–hinged end conditions. A closed-form solution is achieved for the postbuckling deformation as a function of the exerted axial load which is beyond the critical buckling load. In order to study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first buckled configuration of a hinged–hinged FG beam. This leads to a characteristic equation whose eigenvalues are the natural frequencies and the corresponding eigenvectors also determine the mode shapes. The influences of power-law exponent, some commonly used boundary conditions and beam geometrical parameters on the static deflection and free vibration frequencies are studied. A comparison of the present results with those obtained via Euler–Bernoulli beam theory clarifies the overestimation of the frequencies by the later one.

Characterisation of steels with microdefects using a laser interferometry technique

Low microdefect level is essential for ensuring the performance of material. Early detection of microdefect areas allows the estimation of the residual life of constructions and prevents catastrophic failures of components. Ultrasound laser interferometry provides a unique inspection method that enables the study and visualisation of the processes of elastic wave interaction with microdefects embedded in solids, in a non-contact way. In the present paper, the distribution of acoustic fields on the surface of the metal disks obtained was investigated with the use of a laser Doppler interferometer. Specimens containing micropores, cut from the bent section of a steam pipeline that had been subjected to long-term operation at a heat-electric generating station, were used for the studies. Various methods of data digital processing have been applied to trace changes in ultrasound beam energy distribution and geometrical parameters of ultrasound beam projection in case of the presence of microdefects.

Collapse of truss core sandwich beams in 3-point bending

Sandwich beams, comprising a truss core and either solid or triangulated face-sheets, have been investment cast in an aluminium–silicon alloy and in silicon brass. The macroscopic effective stiffness and strength of the triangulated face-sheets and tetrahedral core are estimated by idealising them as pin-jointed assemblies; tests show that this approximation is adequate. Next, the collapse responses of these sandwich beams in 3-point bending are measured. Collapse is by four competing mechanisms: face-yield, face-wrinkling, indentation and core shear, with the active collapse mode dependent upon the beam geometry and yield strain of the material. Upper bound expressions for the collapse loads are given in terms of the effective properties of the faces and core of the sandwich beam; these upper bounds are in good agreement with the measured beam response, and are used to construct collapse mechanism maps with beam geometrical parameters as the axes. The maps are useful for selecting sandwich beams of minimum weight for a given structural load index. The optimisation reveals that truss core sandwich beams are significantly lighter than the competing concept of sandwich beams with a metallic foam core.

The plastic collapse of sandwich beams with a metallic foam core

Plastic collapse modes of sandwich beams have been investigated experimentally and theoretically for the case of an aluminium alloy foam with cold-worked aluminium face sheets. Plastic collapse is by three competing mechanisms: face yield, indentation and core shear, with the active mechanism depending upon the choice of geometry and material properties. The collapse loads, as predicted by simple upper bound solutions for a rigid, ideally plastic beam, and by more refined finite element calculations are generally in good agreement with the measured strengths. However, a thickness effect of the foam core on the collapse strength is observed for collapse by core shear: the shear strength of the core increases with diminishing core thickness in relation to the cell size. Limit load solutions are used to construct collapse maps, with the beam geometrical parameters as axes. Upon displaying the collapse load for each collapse mechanism, the regimes of dominance of each mechanism and the associate mass of the beam are determined. The map is then used in optimal design by minimising the beam weight for a given structural load index.

4524687 Adjustable carriage drive mechanism: Henry J Bubley assigned to American Screen Printing Equipment Company

Curved beams subjected to transverse force may exhibit latching phenomena, namely remain in the buckled configuration under zero force. In this study we investigate the latching in bistable electrostatically actuated initially curved and prestressed beams. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam as base functions. Criteria for the existence of latching are derived in terms of the beam geometric parameters and axial load. Two conditions are formulated: The necessary criterion establishes the appearance of latching in the case of a symmetric snap-through, and the sufficient condition assures the existence of latching in the presence of symmetry breaking. The results provided by the RO model are compared to results obtained by direct numerical analyses. Furthermore, the dynamic behavior of curved beams prone to latching, and actuated by a suddenly applied electrostatic force of finite duration, is numerically studied. The dependence of the response on the loading parameters and the damping is presented through maps. These maps indicate the conditions required for trapping the beam at a secondary stable latching point, which is either accessible or inaccessible under quasi-static actuation. In addition, dynamic release of a latched beam is presented without the usage of an additional electrode, usually used for generating an opposite quasi-static release force.