Exact closed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory

A general procedure for the determination of the natural frequencies and buckling load for a set of beam system under compressive axial loading is investigated using Timoshenko and high-order shear deformation theory. It is assumed that the set beams of the system are simply supported and continuously joined by a Winkler elastic layer. The model of beam includes the effects of axial loading, shear deformation and rotary inertia. In the special case of identical beams, explicit expressions for the natural frequencies and the critical buckling load are determined using a trigonometric method. The influences of the compressive axial loading and the number of beams in the system on the natural frequencies and the critical buckling load are discussed. These results are of considerable practical interest and have wide application in engineering practice of frameworks.     

The second frequency spectrum of Timoshenko beams

A second spectrum of frequencies was reported in early analytical work on the vibrations of Timoshenko beams. However, in subsequent finite element modelling this phenomenon was either ignored or not definitively classified and recorded. In fact, from a recent finite element analysis with a high precision element it was even concluded that there is no separate second spectrum of frequencies except for the special case of hinged-hinged beams and it was asserted that previous investigators had misinterpreted some frequencies thus introducing the notion of second frequencies. In this paper, a simple linear beam element with independent displacement fields and reduced integration to eliminate shear locking is used and enables one to detect the second spectrum accurately. Guidelines are provided which help to identify and classify the frequencies into two separate spectra.     

A new rectangular beam theory

A new theory for beams of rectangular cross-section which includes warping of the cross-sections is presented in the present work. By satisfying the shear-free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required. It is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in Timoshenko beam theory provided that for the Timoshenko equation the shear coefficient is taken to be $\text{5}\text{6}$; this value lies within the range of values, 0·822–0·870, appearing in the literature for the beam of rectangular cross-section. Results for two typical static examples are given for both the new theory and Timoshenko beam theory. These results are compared with the solutions of the comparable problems in the linear theory of elasticity. For the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while Timoshenko beam theory underestimates the shear correction term by 20%. For the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution.     

An exact frequency equation in closed form for Timoshenko beam clampled at both ends

The author has discovered several errors which are not typographical in the frequency equations for a Timoshenko beam clamped at both ends by Huang who presented the frequency equations and normal mode equations for all six common types of simple, finite beams in closed form for the first time. The exact frequency equations in closed form for Timoshenko beams clamped at both ends are derived based on his analysis. And then in order to justify the amended solutions of Huang, two versions of the closed form exact method and the Ritz method are applied. The frequency equations by the previous researcher present frequencies for only the flexural modes, while the closed form exact method and the Ritz method give ones for the thickness–shear modes as well as the bending modes. The purpose of the present study is to reveal the errors, correct them, and give some numerical results.     

Free vibrations of a laminated beam by a microstructure theory

The free vibrations of a laminated beam are considered within the framework of a theory that models the composite beam as a macrohomogeneous beam with microstructure. The beams are assumed to consist of several parallel alternating layers of two homogeneous, isotropic elastic materials. The system of three coupled partial differential equations is solved exactly, and attention is devoted to the determination of natural frequencies of vibration of laminated beams with (i) hinged-hinged ends and (ii) clamped-clamped ends. For the sake of comparison, the same boundary value problems are also solved within the framework of the so-called effective modulus theory, which treats the composite as a transversely isotropic and “fictitiously” homogeneous Timoshenko beam, with effective moduli and density. For relatively long beams, i.e., in the low frequency range, the natural frequencies obtained from the two theories are in excellent agreement, but as the depth-to-length ratio, ζ, increases the microstructure frequencies are observed to be much lower than the effective modulus frequencies, the magnitude of the effect becoming more pronounced with increasing mode number n.     

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.     

Dynamic coefficient of a two-layered thick beam with imperfect bonding

This paper is concerned with the impulse response of a two-layered prestressed beam with flexible bonding resting on an elastic foundation. Two dissimilar layers of the beam are assumed to bend according to the Timoshenko beam theory. Numerical results are given for simply supported beams subjected to a uniformly distributed step load. The dynamic coefficient (= dynamical maximum moment/static moment in the layer) is calculated as a function of the shear bond stiffness (ranging from zero to infinity) for several values of the axial prestress and the foundation stiffness parameters. The results are also compared with those from the Euler beam theory.     

含损伤黏弹性阻尼加筋层合板声功率和指向性变异

该文旨在研究损伤位置和程度对自由阻尼加筋层合板声辐射功率和指向性的影响.基于Mindlin和Timoshenko梁理论,建立了自由阻尼层合板-梁组合结构有限元模型.数值求解四边简支边界条件自由阻尼加筋层合板振动响应,继而通过Rayleigh积分计算加筋层合板辐射声功率和指向性.将计算得到的前4阶模态固有频率、声辐射功率...     

Eigenvalue analysis of Timoshenko beams and axisymmetric Mindlin plates by the pseudospectral method

A study of the free vibration of Timoshenko beams and axisymmetric Mindlin plates is presented. The analysis is based on the Chebyshev pseudospectral method, which has been widely used in the solution of fluid mechanics problems. Clamped, simply supported, free and sliding boundary conditions of Timoshenko beams are treated, and numerical results are presented for different thickness-to-length ratios. Eigenvalues of the axisymmetric vibration of Mindlin plates with clamped, simply supported and free boundary conditions are presented for various thickness-to-radius ratios.     

MODELLING OF TRANSVERSE VIBRATION OF SHORT BEAMS FOR CRACK DETECTION AND MEASUREMENT OF CRACK EXTENSION

The method of detection of location of crack in beams based on frequency measurements is extended here to short beams, taking into account the effects of shear deformation and rotational inertia through the Timoshenko beam theory and representing the crack by a rotational spring. Methods for solving both forward (determination of frequencies of beams knowing the crack parameters) and inverse (determination of crack location knowing the natural frequencies) problems are included. A method to estimate crack extension from a change in the first natural frequency is presented. Both numerical and experimental studies are given to demonstrate the accuracy of the methods. The accuracy of the results is quite encouraging.     

Vibrations of double-nanotube systems with mislocation via a newly developed van der Waals model

This study deals with transverse vibrations of two adjacent-parallel-mislocated single-walled carbon nanotubes (SWCNTs) under various end conditions. These tubes interact with each other and their surrounding medium through the intertube van der Waals (vdW) forces, and existing bonds between their atoms and those of the elastic medium. The elastic energy of such forces due to the deflections of nanotubes is appropriately modeled by defining a vdW force density function. In the previous works, vdW forces between two identical tubes were idealized by a uniform form of this function. The newly introduced function enables us to investigate the influences of both intertube free distance and longitudinal mislocation on the natural transverse frequencies of the nanosystem which consists of two dissimilar tubes. Such crucial issues have not been addressed yet, even for simply supported tubes. Using nonlocal Timoshenko and higher-order beam theories as well as Hamilton's principle, the strong form of the equations of motion is established. Seeking for an explicit solution to these integro-partial differential equations is a very problematic task. Thereby, an energy-based method in conjunction with an efficient meshfree method is proposed and the nonlocal frequencies of the elastically embedded nanosystem are determined. For simply supported nanosystems, the predicted first five frequencies of the proposed model are checked with those of assumed mode method, and a reasonably good agreement is achieved. Through various studies, the roles of the tube's length ratio, intertube free space, mislocation, small-scale effect, slenderness ratio, radius of SWCNTs, and elastic constants of the elastic matrix on the natural frequencies of the nanosystem with various end conditions are explained. The limitations of the nonlocal Timoshenko beam theory are also addressed. This work can be considered as a vital step towards better realizing of a more complex system that consists of vertically aligned SWCNTs of various lengths.     

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.     

The tones of the kalimba (African thumb piano)

The acoustic spectrum of the kalimba (African thumb piano) is measured and analyzed for tonal structure. The frequency f(1) of the fundamental tone of each tine (key) is investigated in relation to the frequencies of its two dominant overtones, f(2) and f(3). These frequencies are identified as the first three modes of transverse vibration of a beam of rectangular cross section. As is typical for vibrating-beam instruments, the overtone sequence is inharmonic, that is, the sequence f(1), f(2), f(3),[ellipsis (horizontal)] is unevenly spaced and the frequency ratios f(2)/f(1) and f(3)/f(1) are not integers. The kalimba tines are modeled by applying the Euler-Bernoulli beam equation with one end clamped, the other end free, and an intermediate point (the bridge) simply supported. Unlike the cases of free-free and clamped-free beams, it is found that the clamped-supported-free frequency ratios f(2)/f(1) and f(3)/f(1) are not fixed values, but depend uniquely upon where the bridge supports and subdivides the tine. The model solution is more thoroughly investigated analytically for the special case in which the beam segment ratio is unity, which has some analytic solutions. Numerically computed mode frequencies agree well with acoustic measurements, validating the model. Mode shapes are computed for the first three modes of a typical tine.     

Free vibration of FGM Timoshenko beams with through-width delamination

Free vibration of functionally graded beams with a through-width delamination is investigated.It is assumed that the material property is varied in the thickness direction as power law functions and a single through-width delamination is located parallel to the beam axis.The beam is subdivided into three regions and four elements.Governing equations of the beam segments are derived based on the Timoshenko beam theory and the assumption of‘constrained mode’.By using the differential quadrature element method to solve the eigenvalue problem of ordinary differential equations governing the free vibration,numerical results for the natural frequencies of the beam are obtained.Natural frequencies of delaminated FGM beam with clamped ends are presented.Effects of parameters of the material gradients,the size and location of delamination on the natural frequency are examined in detail.     

Assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories

Dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of Eringen. To this end, the nanotube structure is modeled by an equivalent continuum structure (ECS) according to the nonlocal Euler-Bernoulli, Timoshenko and higher order beam theories. The nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established. Analytical solutions of the problem are presented for simply supported boundary conditions. The explicit expressions of the critical velocities of the nonlocal beams are derived. Furthermore, the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ECS are examined through various numerical simulations. The role of the scale effect parameter, the slenderness ratio of the ECS and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail. The results show the importance of using nonlocal shear deformable beam theories, particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity.     

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.     

Spectral element modeling for extended Timoshenko beams

Periodic lattice structures such as the large space lattice structures and carbon nanotubes may take the extension-transverse shear-bending coupled vibrations, which can be well represented by the extended Timoshenko beam theory. In this paper, the spectrally formulated finite element model (simply, spectral element model) has been developed for extended Timoshenko beams and applied to some typical periodic lattice structures such as the armchair carbon nanotube, the periodic plane truss, and the periodic space lattice beam.     

Pitch change during reverberant decay

The natural frequencies and mode shapes of a number of box beams are calculated by using the finite element displacement method. The structures are considered as assemblages of plates, and in general it is necessary to consider both the in-plane and transverse motion of the plates. A method of representing these two types of motion in the analysis of the vibrations of box beams is presented. A number of box beams of varying sectional parameters are analysed as systems of plates and the results compared with the predictions of Euler and Timoshenko beam theories. The comparisons show that for short beams constructed of thin plates, the new method can successfully represent the localized plate deformations, which cannot be described by beam theory.     

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

This paper studies free vibration of axially functionally graded beams with non-uniform cross-section. A novel and simple approach is presented to solve natural frequencies of free vibration of beams with variable flexural rigidity and mass density. For various end supports including simply supported, clamped, and free ends, we transform the governing equation with varying coefficients to Fredholm integral equations. Natural frequencies can be determined by requiring that the resulting Fredholm integral equation has a non-trivial solution. Our method has fast convergence and obtained numerical results have high accuracy. The effectiveness of the method is confirmed by comparing numerical results with those available for tapered beams of linearly variable width or depth and graded beams of special polynomial non-homogeneity. Moreover, fundamental frequencies of a graded beam combined of aluminum and zirconia as two constituent phases under typical end supports are evaluated for axially varying material properties. The effects of the geometrical and gradient parameters are elucidated. The present results are of benefit to optimum design of non-homogeneous tapered beam structures.     

Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory

A nonlocal Euler–Bernoulli elastic beam model is developed for the vibration and instability of tubular micro- and nano-beams conveying fluid using the theory of nonlocal elasticity. Based on the Newtonian method, the equation of motion is derived, in which the effect of small length scale is incorporated. With this nonlocal beam model, the natural frequencies and critical flow velocities for the case of simply supported system and for the case of cantilevered system are obtained. The effect of small length scale (i.e., the nonlocal parameter) on the properties of vibrations is discussed. It is demonstrated that the natural frequencies are generally decreased with increasing values of nonlocal parameter, both for the supported and cantilevered systems. More significantly, the effect of small length scale on the critical flow velocities is visible for fluid-conveying beams with nano-scale length; however, this effect may be neglected for micro-beams conveying fluid.