1,360,575 research outputs found
The second spectrum of Timoshenko beam theory - Further assessment
No full textA review of contributions and views on the second spectrum of Timoshenko beam theory (TBT) over the past two decades, together with some new results, are presented. It is shown that the Timoshenko frequency equation factorises not solely for hinged–hinged end conditions, as is often claimed, but also for guided–guided and guided–hinged; these new cases may be regarded as portions of a multi-span hinged–hinged beam. A higher-derivative Lagrangian that leads directly to the well-known fourth-order Timoshenko beam equation is reviewed. A simple relationship between the so-called Ostrogradski energy and the mechanical energy is derived for hinged–hinged end conditions. It is shown that the Ostrogradski energy is positive for the first spectrum but negative for the second; within some branches of physics, this would be sufficient to conclude that the second spectrum is “unphysical”. A numerical example presented by Levinson and Cooke is re-examined using both TBT and exact plane stress elastodynamic theory. Agreement is excellent for the first spectrum. However, the second spectrum predictions are not in consistent agreement with any single mode of vibration. For long wavelength it is very close to the second asymmetric mode, but as wavelength shortens, it becomes closer to the second symmetric, then the third asymmetric modes. The conclusion remains unchanged: the second spectrum predictions of TBT should be disregarded
On the valid frequency range of Timoshenko beam theory
No full textThe frequency equation of Timoshenko beam theory factorises for hinged-hinged end conditions, leading to a first and second spectrum of natural frequencies; the latter is largely inaccurate and can be isolated and disregarded. For the majority of other end conditions, when the frequency equation does not factorise, one may think in terms of pseudo-second spectrum contributions arising when evanescent waves become propagating above the cut-off frequency , and it is conjectured that these may have a corrupting effect on the frequency predictions. Comparisons with measured and simulated frequencies lead to the conclusion that Timoshenko predictions above the cut-off frequency should be disregarded for those end conditions for which the frequency equation does not factorise
Size effect in the bending of a Timoshenko nanobeam
No full textThe size effect should be considered due to the large ratio of surface area to volume when the characteristic length of a beam lies in the nanoscale. The size effect in the bending of a Timoshenko nanobeam is investigated in this paper based on a recently developed elastic theory for nanomaterials in which only the bulk surface energy density and the surface relaxation parameter are involved as independent parameters to characterize the surface property of nanomaterials. In contrast to the Euler nanobeams and the classical Timoshenko beam not only the size effect but also the shear deformation effect in Timoshenko nanobeams is included. Closed-form solutions of the deflection and the effective elastic modulus for both a fixed-fixed Timoshenko nanobeam and a cantilevered one are achieved. Comparing to the classical solution of Timoshenko beams the size effect is obviously significant in Timoshenko nanobeams. The shear deformation effect in nanobeams cannot be neglected in contrast to the solution of Euler-Bernoulli nanobeams when the aspect ratio of a nanobeam is relatively small. Furthermore the size effect exhibits different influences on the bending behavior of nanobeams with different boundary conditions. A nanobeam with a fixed-fixed boundary would be stiffened while a cantilevered one is softened by the size effect compared to the classical solution. All the findings are consistent with the existing experimental measurement. The results in this paper should be very useful for the precision design of nanobeam-based devices
Helical structure of the waves propagating in a spinning Timoshenko beam
No full textThe aim of the paper is to study the cause of a frequency-splitting phenomenon that occurs in a spinning Timoshenko beam. The associated changes in the structure of the progressive waves are investigated to shed light on the relationship between the wave motion in a spinning beam and the whirling of a shaft. The main result is that travelling bending waves in a beam spinning about its central axis have the topological structure of a revolving helix traced by the centroidal axis with right-handed or left-handed chirality. Each beam element behaves like a gyroscopic disc in precession being rotated at the wave frequency with anticlockwise or clockwise helicity. The gyroscopic effect is identified as the cause of the frequency splitting and is shown to induce a coupling between two interacting travelling waves lying in mutually orthogonal planes. Two revolving waves travelling in the same direction in space appear, one at a higher and one at a lower frequency compared with the pre-split frequency value. With reference to a given spinning speed, taken as clockwise, the higher one revolves clockwise and the lower one has anticlockwise helicity, each wave being represented by a characteristic four-component vector wavefunction.Two factors are identified as important, the shear-deformation factor q and the gyroscopic-coupling phase factor ?. The q-factor is related to the wavenumber and the geometric shape of the helical wave. The ?-factor is related to the wave helicity and has two values, +?/2 and ??/2 corresponding to the anticlockwise and clockwise helicity, respectively. The frequency-splitting phenomenon is addressed by analogy with other physical phenomena such as the Jeffcott whirling shaft and the property of the local energy equality of a travelling wave. The relationship between Euler's formula and the present result relating to the helical properties of the waves is also explored
Detecting vibration frequencies and shapes for timoshenko beam model: A comparison between available eigensolvers
No full textThe Timoshenko beam model [1], which takes into account both rotary inertia and shear deformation, is often considered a more suitable tool for analyzing beam structures than the simpler Euler-Bernoulli one, which disregards such effects. However, the dynamics of Timoshenko beams is still an active area of research, because its spectrum consists of two branches, see, among others, [2], which, above a transition frequency, usually occurring between the first vibration modes, appear to be strictly entangled. This effect creates a challenging tasks for eigenvalue solvers, as it has been already shown for the case of simply-supported Timoshenko beams, where a closed form solution is available for both frequency equations and vibration modes (which come out to be simple sinusoids). Contrarily to what happens for Euler-Bernoulli beams in the case of Timoshenko beams isogeometric elements produce better results than standard finite elements, but do not allow to reproduce exactly the vibration shape beyond some limited part of the spectrum. For instance, by using 5th order NURBS, and 500 degrees of freedom (DOFs), only the first 270 modal shapes are reproduced accurately, while the others present not-acceptable errors, as it has been shown in [3]. In this paper special attention is devoted to the numerical algorithms for solving the eigenvalue problem, using as an example a simply-supported Timoshenko beam under free vibrations conditions. The results obtained using the QZ algorithm, QR algorithm and the more general Lanczos and Arnoldi iteration methods are compared and discussed in order to determine their comparative performances
A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams
No full textThis paper presents a nonlocal sinusoidal shear deformation beam theory for the bending, buckling, and vibration of nanobeams. The present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion as well as the boundary conditions of the beam are derived using Hamilton’s principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported beam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory. The comparison firmly establishes that the present beam theory can accurately predict the bending, buckling, and vibration responses of short nanobeams where the small scale and transverse shear deformation effects are significant
A contact problem for a thermoelastic Timoshenko beam
No full textIn this paper, a dynamic contact problem between a Timoshenko beam and two rigid obstacles is considered.
Thermal effects are also taken into account and the contact is modeled using the classical Signorini condition. The global
existence in time of solutions is found by considering related penalized problems, proving some a priori estimates and passing
to the limit. An exponential decay property is also showed
Modeling of comb polymers with a high branching density
No full textBranched macromolecules are currently of great scientific interest. They are formed by a backbone carrying many closely spaced linear arms, and their most important property consists on their backbone stiffness due to the large branching density. From this property, one can reasonably expect lyotropic behaviour of these systems in solution and adsorbed on a surface. These molecules have a cylindrical shape and a characteristic size that ranges from a few nanometers (radius of their circular section) to hundreds of nanometers (contour length). Little is known at present about the nature of the backbone stiffness and its dependence on the monomers excluded volume and on the stereochemical constraints. The availability of high-performance computers allowed us to apply the Metropolis Monte Carlo algorithm to a coarse-grained model to describe bottle-brushes in a diluted solution or adsorbed on a surface. Accurate results are obtained for the value of the Flory exponent, the persistence length and the distribution functions of the distances between the monomers
A stress-driven local-nonlocal mixture model for Timoshenko nano-beams
No full textA well-posed stress-driven mixture is proposed for Timoshenko nano-beams. The model is a convex combination of local and nonlocal phases and circumvents some problems of ill-posedness emerged in strain-driven Eringen-like formulations for structures of nanotechnological interest. The nonlocal part of the mixture is the integral convolution between stress field and a bi-exponential averaging kernel function characterized by a scale parameter. The stress-driven mixture is equivalent to a differential problem equipped with constitutive boundary conditions involving bending and shear fields. Closed-form solutions of Timoshenko nano-beams for selected boundary and loading conditions are established by an effective analytical strategy. The numerical results exhibit a stiffening behavior in terms of scale parameter
A dynamic-stiffness approach for damped locally-resonant Timoshenko beams
No full textA dynamic-stiffness matrix approach is presented for locally-resonant Timoshenko beams with a periodic array of viscously-damped multi-degree-of-freedom resonators. First, an exact dynamic condensation of the resonator degrees of freedom is pursued, expressing the resonator reaction forces in terms of the deflections of the attachment points via pertinent frequency-dependent stiffness terms. On this basis, a direct integration deriving from the theory of generalized functions provides the exact dynamic-stiffness matrix of the beam, whose size is 4 × 4 for any the number of resonators and degrees of freedom within the resonators. The dynamic-stiffness matrix is used to calculate the complex eigenvalues of the beam by a recently introduced contour-integral algorithm, without missing anyone. Further, it provides the transmittance for an insight into the elastic wave attenuation properties of the beam
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