Dynamic stability of Timoshenko beams resting on an elastic foundation

A finite element model is developed for the stability analysis of a Timoshenko beam resting on an elastic foundation and subjected to periodic axial loads. The effect of an elastic foundation on the natural frequencies and static buckling loads of hinged-hinged and fixed-free Timoshenko beams is investigated. The results obtained for a Bernoulli-Euler beam which is a special case of the present analysis show excellent agreement with the available results obtained by other analytical methods. The regions of dynamic instability are determined for different values of the elastic foundation constant. As the elastic foundation constant increases the regions of dynamic instability are shifted away from the vertical axis and the width of these regions is decreased, thus making the beam less sensitive to periodic forces.     

Analysis of radial nonlocal effect on the structural response of carbon nanotubes

In this Letter, finite element model is developed to study the effect of nonlocal parameter in the radial structural response of carbon nanotubes. Timoshenko beam model is employed. The influence of nonlocal parameter in the radial direction due to interaction of atoms is defined as the radial nonlocal effect. It is found that there is significant influence of radial nonlocal effect on the structural response of the carbon nanotubes.     

Exact wave-based Bloch analysis procedure for investigating wave propagation in two-dimensional periodic lattices

This paper presents an exact, wave-based approach for determining Bloch waves in two-dimensional periodic lattices. This is in contrast to existing methods which employ approximate approaches (e.g., finite difference, Ritz, finite element, or plane wave expansion methods) to compute Bloch waves in general two-dimensional lattices. The analysis combines the recently introduced wave-based vibration analysis technique with specialized Bloch boundary conditions developed herein. Timoshenko beams with axial extension are used in modeling the lattice members. The Bloch boundary conditions incorporate a propagation constant capturing Bloch wave propagation in a single direction, but applied to all wave directions propagating in the lattice members. This results in a unique and properly posed Bloch analysis. Results are generated for the simple problem of a periodic bi-material beam, and then for the more complex examples of square, diamond, and hexagonal honeycomb lattices. The bi-material beam clearly introduces the concepts, but also allows the Bloch wave mode to be explored using insight from the technique. The square, diamond, and hexagonal honeycomb lattices illustrate application of the developed technique to two-dimensional periodic lattices, and allow comparison to a finite element approach. Differences are noted in the predicted dispersion curves, and therefore band gaps, which are attributed to the exact procedure more-faithfully modeling the finite nature of lattice connection points. The exact method also differs from approximate methods in that the same number of solution degrees of freedom is needed to resolve low frequency, and arbitrarily high frequency, dispersion branches. These advantageous features may make the method attractive to researchers studying dispersion characteristics, band gap behavior, and energy propagation in two-dimensional periodic lattices.     

The response of a beam to a transient pressure wave load

The dynamic behaviour of beam structures under pressure waves is investigated. The propagation of the bending waves under a moving single load is first studied for three types of beam: a Bernoulli-Euler beam, a beam with shear deflection and a Timoshenko beam. Then the responses of the Bernoulli-Euler and the Timoshenko beam are studied under moving pressure wave excitation. The results are presented as dynamic amplification factors (DAF). The influence of the load parameters (load shape, propagation speed, pressure wave duration, etc.) and the beam parameters (slenderness, damping, etc.) is discussed. The load shape (symmetrical, asymmetrical) and the propagation speed strongly influence the response. The results are compared with available approximate solutions for the corresponding lumped element, single degree of freedom model of the structure.     

Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory

This Letter develops a model that analyzes the resonant frequency of the chiral single-walled carbon nanotubes (SWCNTs) subjected to a thermal vibration by using Timoshenko beam model, including the effect of rotary inertia and shear deformation. The analytical solution is derived and the frequency equation is obtained. The results based on the beam model show that the frequency increases with decreasing the nanotube aspect ratio of length to diameter. In addition, the frequency obtained by Timoshenko beam model is lower than that calculated by Euler beam model. As the nanotube aspect ratio of length to diameter decreased, the discrepancy is more obvious. Furthermore, as the effect of thermal vibration increases, the frequency for chiral SWCNTs decreases.     

COUPLED WAVES ON A PERIODICALLY SUPPORTED TIMOSHENKO BEAM

A mathematical model is presented for the propagation of structural waves on an infinitely long, periodically supported Timoshenko beam. The wave types that can exist on the beam are bending waves with displacements in the horizontal and vertical directions, compressional waves and torsional waves. These waves are affected by the periodic supports in two ways: their dispersion relation spectra show passing and stopping bands, and coupling of the different wave types tends to occur. The model in this paper could represent a railway track where the beam represents the rail and an appropriately chosen support type represents the pad/sleeper/ballast system of a railway track. Hamilton's principle is used to calculate the Green function matrix of the free Timoshenko beam without supports. The supports are incorporated into the model by combining the Green function matrix with the superposition principle. Bloch's theorem is applied to describe the periodicity of the supports. This leads to polynomials with several solutions for the Bloch wave number. These solutions are obtained numerically for different combinations of wave types. Two support types are examined in detail: mass supports and spring supports. More complex support types, such as mass/spring systems, can be incorporated easily into the model.     

Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures

Scale effect on static deformation of micro- and nano-rods or tubes is revealed through nonlocal Euler–Bernoulli beam theory and Timoshenko beam theory. Explicit solutions for static deformation of such structures with standard boundary conditions are derived. Results show that the scale effect would not manifest itself for micro-structures with length of the order of micro-meters, however, will be noticeable for nano-structures in their static responses. In addition, the shear effect is evident for nano-structures, especially for carbon nanotubes in most of current references, indicating the importance of applying higher-order beam theory in static analysis of nano-structures.     

Fabrication of large-area 3D photonic crystals using a holographic optical element

A holographic technique for fabricating 3D photonic crystal is presented. The key element in the fabrication system is a holographic optical element (HOE) consisting of three gratings. Used in combination with a mask, the HOE can generate four beams under single illuminating beam, and 3D lattice structures can be formed by the interference of the four beams. Holographic approach is used to make HOE, so large area lattice structures can be fabricated. Numerical simulations indicate that beam intensity ratio of central beam to outer beam is one of the factors that affects the structures fabricated in photoresist, and high diffraction efficiency of the gratings in HOE is favorable when using cw laser with relatively low power as light source. Experimental results show clear 3D lattice structures fabricated using the HOE, verifying the effectiveness of the technique.     

A finite element for the vibration analysis of Timoshenko beams

A Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation. The element properties are based on a coupled displacement field; the lateral deflection is interpolated as a quintic polynomial function and the cross-sectional rotation is linked to the deflection by specifying satisfaction of the governing differential equation of moment equilibrium in the absence of the rotary inertia term. Numerical results confirm that this procedure does not preclude convergence to true Timoshenko theory solutions since rotary inertia is included in lumped form at element ends. The new Timoshenko beam element has good convergence characteristics and where comparison can be made in numerical studies it is shown to be generally more efficient than previous elements.     

Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model

The free vibration of a shear deformable beam with multiple open edge cracks is studied using a lattice spring model (LSM). The beam is supported by a so-called two-parameter elastic foundation, where normal and shear foundation stiffnesses are considered. Through application of Timoshenko beam theory, the effects of transverse shear deformation and rotary inertia are taken into account. In the LSM, the beam is discretised into a one-dimensional assembly of segments interacting via rotational and shear springs. These springs represent the flexural and shear stiffnesses of the beam. The supporting action of the elastic foundation is described also by means of normal and shear springs acting on the centres of the segments. The relationship between stiffnesses of the springs and the elastic properties of the one-dimensional structure are identified by comparing the homogenised equations of motion of the discrete system and Timoshenko beam theory.     

Study of Some Factors Affecting Thermodynamic Properties of the Insulating Phase of the Periodic Anderson Lattice Model

Within the framework of slave-boson mean-field theory, we study the thermodynamic properties of the periodic Anderson lattice model with half-filled conduction band and one 4f electron at each primitive cell and the degeneracy N_d = 2. It is found that after taking into account the direct nearest-neighbor f-f hopping, such a periodic Anderson lattice model can exhibit both an insulating ground state and a heavy-fermion metal ground state depending on the value of the bare f energy level E_f, the hybridization matrix element V, and the direct f-f hopping strength δ. This is unlike the case neglecting the direct f-f hopping, in which such a periodic Anderson lattice model will predict an insulating ground state only.     

Assessment of Timoshenko Beam Models for Vibrational Behavior of Single-Walled Carbon Nanotubes Using Molecular Dynamics

In this paper, we study the flexural vibration behavior of single-walled carbon nanotubes (SWCNTs) for the assessment of Timoshenko beam models. Extensive molecular dynamics (MD) simulations based on second-generation reactive empirical bond-order (REBO) potential and Timoshenko beam modeling are performed to determine the vibration frequencies for SWCNTs with various length-to-diameter ratios, boundary conditions, chiral angles and initial strain. The effectiveness of the local and nonlocal Timoshenko beam models in the vibration analysis is assessed using the vibration frequencies of MD simulations as the benchmark. It is shown herein that the Timoshenko beam models with properly chosen parameters are applicable for the vibration analysis of SWCNTs. The simulation results show that the fundamental frequencies are independent of the chiral angles, but the chirality has an appreciable effect on higher vibration frequencies. The SWCNTs is very sensitive to the initial strain even if the strain is extremely small.     

Single-beam analysis of damaged beams: Comparison using Euler–Bernoulli and Timoshenko beam theory

A new general formulation that is applicable to the damaged, linear elastic structures ‘unified framework’ is used to obtain analytical expressions for natural frequencies and mode shapes. The term mode shapes is used to mean the displacement modes, the section rotation modes, the sectional bending strain modes and sectional shear strain modes. The formulation is applicable to damaged elastic self-adjoint systems. The formulation has two unique aspects: First, the theory is mathematically rigorous since no assumptions are made regarding the physical behavior at a damage location, therefore there is no need to substitute the damage with a hypothetical elastic element such as a spring. Since the beam is not divided at the damage location, rather than an 8 by 8, only a 4 by 4 matrix is solved to obtain the natural frequencies and mode shapes. Second, the inertia effects due to damage which have till now been neglected by researchers are accounted for. The formulation uses a geometric damage model, perturbation of mode shapes and natural frequencies, and a modal superposition technique to obtain and solve the governing differential equation. Timoshenko beam theory is then taken as an example, and its results are compared with results using Euler–Bernoulli beam theory and finite element models. The range of applicability of the two theories is ascertained for damage characteristics such as depth and extent of damage and beam characteristics such as slenderness ratio and Poisson?s ratio. The paper considers rectangular notch like non-propagating damage as an example of the damage.     

Free vibrations of a mono-coupled periodic system

Vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method. The frequency equation for the whole system is shown to be obtained in terms of the eigenvalues, or their natural logarithms, which are often called “propagation constants”, of the transfer matrix for a single periodic subsystem. In case of a mono-coupled system this frequency equation may be solved graphically by using the propagation constant curve, thereby saving a great deal of computational effort. Two types of mono-coupled systems are considered as numerical examples: a spring-mass oscillating system and a continuous Timoshenko beam resting on regularly spaced knife-edge supports. Depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method, the numerical solutions become either exact or approximate.     

Observation of second-band vortex solitons in 2D photonic lattices

We demonstrate second-band bright vortex-array solitons in photonic lattices. This constitutes the first experimental observation of higher-band solitons in any 2D periodic system. These solitons possess complex intensity and phase structures, yet they can be excited by a simple highly localized vortex-ring beam. Finally, we show that the linear diffraction of such beams exhibits preferential transport along the lattice axes.     

Numerical and experimental analysis of the vibration and band-gap properties of elastic beams with periodically variable cross sections

The band-gap properties of non-uniform periodic beams are analyzed using numerical and experimental methods. The flexural wave equations are established based on the Euler–Bernoulli and Timoshenko beam theories. The beams with periodically variable cross sections are investigated. The transfer matrix method is used to explore the dynamic behaviors of the periodic beams, that is, the natural frequencies of the finite periodic beams with different cross-section ratios between the adjacent sub-cells and the band-gaps of the infinite periodic beams based on the Bloch theory. The validity and accuracy of the band-gaps acquired by the present method are verified by comparing the results with those obtained from the finite element method and the vibration experiments. The effects of the different lengths of adjacent sub-cells on the band-gap properties are then investigated. The research results and conclusions should be useful in the study of vibration control applications.     

Free vibrations of skirt supported pressure vessels — an engineering application of timoshenko beam theory

Timoshenko beam theory is applied to the study of the free vibrations of skirt supported pressure vessels in this paper; such systems are used in the process and power generation industries as well as aboard nuclear powered vessels. It is shown that the analysis is not significantly more complicated than the analysis of skirt-vessel combinations by Euler-Bernoulli beam theory. This latter analysis is provided in an appendix. Two sets of boundary conditions are considered: namely, the cases of (a) a cantilevered system and (b) a fixed-pinned system. The first two natural frequencies of nine typical cases are calculated and compared with the corresponding results obtained from Euler-Bernoulli beam theory. The numerical differences are significant so that if a beam theory is adequate to model the system, it is clear that Timoshenko beam theory is the appropriate one to use. In addition, the first two mode shapes for a particular case are presented for comparison with the corresponding mode shapes predicted by Euler-Bernoulli beam theory. Finally, some comments on the modeling and analysis of specific, real systems are made. It is emphasized that the purpose of the paper is to demonstrate that Timoshenko beam theory is not unduly difficult to apply to problems of engineering interest when a beam theory model is suitable.     

Numerical investigation of elastic modes of propagation in helical waveguides

Steel multi-wire cables are widely employed in civil engineering. They are usually made of a straight core and one layer of helical wires. In order to detect material degradation, nondestructive evaluation methods based on ultrasonics are one of the most promising techniques. However, their use is complicated by the lack of accurate cable models. As a first step, the goal of this paper is to propose a numerical method for the study of elastic guided waves inside a single helical wire. A finite element (FE) technique is used based on the theory of wave propagation inside periodic structures. This method avoids the tedious writing of equilibrium equations in a curvilinear coordinate system yielding translational invariance along the helix centerline. Besides, no specific programming is needed inside a conventional FE code because it can be implemented as a postprocessing step of stiffness, mass and damping matrices. The convergence and accuracy of the proposed method are assessed by comparing FE results with Pochhammer-Chree solutions for the infinite isotropic cylinder. Dispersion curves for a typical helical waveguide are then obtained. In the low-frequency range, results are validated with a helical Timoshenko beam model. Some significant differences with the cylinder are observed.     

On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring–mass systems. First, the coefficient matrices for an intermediate pinned support, an intermediate concentrated element, left- and right-end support of a Timoshenko beam are derived. Next, the overall coefficient matrix for the whole structural system is obtained using the numerical assembly technique of the finite element method. Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.     

Plane wave propagation and frequency band gaps in periodic plates and cylindrical shells with and without heavy fluid loading

Free plane wave propagation in infinitely long periodic elastic structures with and without heavy fluid loading is considered. The structures comprise continuous elements of two different types connected in an alternating sequence. In the absence of fluid loading, an exact solution which describes wave motion in each unboundedly extended element is obtained analytically as a superposition of all propagating and evanescent waves, continuity conditions at the interfaces between elements are formulated and standard Floquet theory is applied to set up a characteristic determinant. An efficient algorithm to compute Bloch parameters (propagation constants) as a function of the excitation frequency is suggested and the location of band gaps is studied as a function of non-dimensional parameters of the structure's composition. In the case of heavy fluid loading, an infinitely large number of propagating or evanescent waves exist in each unboundedly extended elasto-acoustic element of a periodic structure. Wave motion in each element is then presented in the form of a modal decomposition with a finite number of terms retained in these expansions and the accuracy of such an approximation is assessed. A generalized algorithm is used to compute Bloch parameters for a periodic structure with heavy fluid loading as a function of the excitation frequency and, similarly to the previous case, the location of band gaps is studied.