324,770 research outputs found
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
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
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
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
Timoshenko N. Modelling of electric arc furnace off-gas removal system. / N. Timoshenko, A. Semko, S. Timoshenko. Ironmaking and Steelmaking, vol. 41, № 4, 2014. Р. 257-261.
No full textA mathematical model of an electric arc furnace off-gas removal system, aimed at improving the
energy efficiency of the smelting process, was created and verified by methods of physical
modelling and numerical simulations. For an industrial-scale furnace, equipped with annular
distributing off-gas suction system, the possibility to reduce fugitive emissions by 40%, air inflow
by 24%, total emissions by 13% and the melting dust removal by 33% was shown. The obtained
data will allow, according to estimations, specific energy consumption to reduce by at least 16 –
20kW ht -1 and increase the environmental performance of the furnace
Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions
No full textIn this paper, the free vibrations of a wide range of tapered Timoshenko beams are investigated. The cross section of the beam varies continuously and the variation is described by a power function of the coordinate along the neutral axis of the beam. The static Timoshenko beam functions, which are the complete solutions of a tapered Timoshenko beam under a Taylor series of static load, are developed, respectively, as the basis functions of the flexural displacement and the angle of rotation due to bending. The Rayleigh-Ritz method is applied to derive the eigenfrequency equation of the tapered Timoshenko beam. Unlike conventional basis functions which are independent of the cross-sectional variation of the beam, these static Timoshenko beam functions vary in accordance with the cross-sectional variation of the beam so that higher accuracy and more rapid convergence have been obtained. Some numerical results are presented for both truncated and sharp-ended Timoshenko beams. On the basis of convergence study and comparison with available results in the literature it is shown that the first few eigenfrequencies can be given with quite good accuracy by using a small number of terms of the static Timoshenko beam functions. Finally, some valuable results are presented systematically.link_to_subscribed_fulltex
Analysis of non-homogeneous Timoshenko beams with generalized damping distributions
No full textThis paper presents a study on the effects of generalized damping distributions on non-homogeneous Timoshenko beams. On the basis of some fundamentals of modal analysis for damped continuous systems applied to the particular case of the Timoshenko beam model, the eigenproblem is solved by applying a method combining a state-space representation with a transfer matrix technique, yielding closed-form expressions for the eigenfunctions. After validation by means of numerical examples using the finite element method, response functions in both the time and the frequency domain are discussed and compared according to different damping distribution
Fractional visco-elastic Timoshenko beam deflection via single equation
No full textThis paper deals with the response determination of a visco-elastic Timoshenko beam under static loading condition and taking into account fractional calculus. In particular, the fractional derivative terms arise from representing constitutive behavior of the visco-elastic material. Further, taking advantages of the Mellin transform method recently developed for the solution of fractional differential equation, the problem of fractional Timoshenko beam model is assessed in time domain without invoking the Laplace-transforms as usual. Further, solution provided by the Mellin transform procedure will be compared with classical Central Difference scheme one, based on the Grunwald-Letnikov approximation of the fractional derivative. Moreover, Timoshenko beam response is generally evaluated by solving a couple of differential equations. In this paper, expressing the equation of the elastic curve just through a single relation, a more general procedure, which allows the determination of the beam response for any load condition and type of constraints, is developed. © 2015 John Wile
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