On the two frequency spectra of Timoshenko beams

Levinson, M.; Cooke, David W.

doi:10.1016/0022-460x(82)90480-1

Cited by 75 publications

(51 citation statements)

References 11 publications

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“…We now extend the calculations presented by Levinson and Cooke [4]; the beam has length L ¼ 0:5 m, depth h ¼ 0:125 m, Young's modulus E ¼ 210 Â 10 9 N=m 2 , density r ¼ 7850 kg=m 3 , Poisson's ratio n ¼ 0:3, and unit thickness consistent with plane stress conditions. First, note that the Euler-Bernoulli prediction for the fundamental frequency ðn ¼ 1Þ is o 1 ¼ 7368 rad=s which is some 9.5% in excess of the exact value, so for this short beam a Timoshenko model should always be used.…”

Section: Numerical Examplementioning

confidence: 55%

“…Following the notation and sign conventions of Levinson and Cooke [4], with the exception that the shear coefficient is written as k rather than k 2 , and the transverse displacement is written as v rather than w, the coupled equations of free vibration of a uniform Timoshenko beam are…”

Section: Timoshenko Beam Theory and Factorisation Of The Frequency Ementioning

confidence: 99%

“…First, note that the Euler-Bernoulli prediction for the fundamental frequency ðn ¼ 1Þ is o 1 ¼ 7368 rad=s which is some 9.5% in excess of the exact value, so for this short beam a Timoshenko model should always be used. Two values of the shear coefficient are employed in the comparison: k ¼ 5 6 as in [4], and k ¼ 5ð1 þ nÞ=ð6 þ 5nÞ. The exact plane stress theory is presented in Appendix A.…”

Section: Numerical Examplementioning

confidence: 99%

“…[2] for wave propagation in an infinite beam of circular cross-section, is extended to standing waves in a thin rectangular (plane stress) beam of finite length, having hinged-hinged end conditions; the example chosen is that considered by Levinson and Cooke [4]. For the first spectrum, it is seen that agreement between the plane stress elastodynamic prediction and TBT1 is better than À0.4% and +0.55% for the first 20 natural frequencies, when one uses the shear coefficient k ¼ 5ð1 þ nÞ=ð6 þ 5nÞ.…”

Section: Introductionmentioning

confidence: 99%

“…The second issue, whether the second spectrum exists for beams having other than hinged-hinged end conditions, is described by Levinson and Cooke [4], and further discussed by Prathap [5]. Levinson and Cooke, along with many other researchers, were of the view that a second spectrum could exist only for hinged-hinged ends, when the frequency equation factorises, but regarded the two spectrum terminology as ''tenuous'' and, ''at best, an aid to computation''.…”

Section: Introductionmentioning

confidence: 99%

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The second spectrum of Timoshenko beam theory—Further assessment

Stephen

2006

Journal of Sound and Vibration

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A 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. r

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Section: Numerical Examplementioning

confidence: 55%

Section: Timoshenko Beam Theory and Factorisation Of The Frequency Ementioning

confidence: 99%

Section: Numerical Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The second spectrum of Timoshenko beam theory—Further assessment

Stephen

2006

Journal of Sound and Vibration

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Three‐dimensional solid‐to‐beam transition elements for structural dynamics analysis

Gmür

Kauten

1993

Numerical Meth Engineering

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SUMMARYComplex structural components such as those encountered in many industrial applications may generally be considered as being composed of shell-or beam-like portions linked to three-dimensional solid continua. When discretized into finite elements, these structures present geometrical and mathematical difficulties at the connections between the different element types since the nodal degrees of freedom allocated to the solid, shell and beam elements are incompatible with each other. The development of specific and reliable transition finite elements is, thus, of outstanding practical importance. This paper presents efficient Co compatible transition elements with a variable number of nodes for modelling solid to beam junctions. Based upon the standard isoparametric solid and beam formulations, the current approach includes the properties of both solids and beams, verifies the basic continuity, smoothness and completeness criteria inherent in the finite element convergence requirements, and avoids the shear locking phenomenon typical of Co elements by using a strain-projection method. Several numerical examples which compare this formulation to analytical and experimental solutions are provided in order to show the applicability and efficiency of this approach.

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Asymptotic behavior of weakly dissipative Bresse‐Timoshenko system on influence of the second spectrum of frequency

2018

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In the present work, we consider a dissipative Bresse‐Timoshenko type system, which is free of physical anomaly know as second spectrum of frequency according important observations made by Elishakoff et al., and we establish a new result of exponential decay. We prove that frictional damping acting on vertical displacement of this system is exponentially stable regardless the equality between velocities of wave propagation. This result is new and completely different from ones obtained by Almeida Júnior et al. for the Timoshenko system with the same damping acting on vertical displacement. Our approach is strongly inspired in a recent result due to Almeida Júnior and Ramos where the authors justified, from physical point of view, a classical and pioneering result due to Soufyane. That is, the dissipative mechanism of frictional type acting on angular rotation of the Timoshenko system truncates the consequences of the second spectrum when the velocities of wave propagation are equal. Also, for dissipative Bresse‐Timoshenko systems which are free of the second spectrum, the exponential decay occurs regardless of any condition on coefficients of system. Based on the new results, it is demonstrated that the classical Bresse‐Timoshenko beam theory, as an overcomplicated theory, requires damping terms to eliminate the second spectrum.

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On the two frequency spectra of Timoshenko beams

Cited by 75 publications

References 11 publications

The second spectrum of Timoshenko beam theory—Further assessment

The second spectrum of Timoshenko beam theory—Further assessment

Three‐dimensional solid‐to‐beam transition elements for structural dynamics analysis

Asymptotic behavior of weakly dissipative Bresse‐Timoshenko system on influence of the second spectrum of frequency

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