The second spectrum of Timoshenko beam theory—Further assessment

Abstract

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

On the valid frequency range of Timoshenko beam theory

“…Similarly in Ref. [2], for the thin rectangular cross-section, the TBT2 prediction did not provide consistent agreement with any single mode of vibration; at long wavelength it was very close to the second exact asymmetric mode (as one would wish), but as wavelength shortens it became closer to the second symmetric (extensional) mode, then the third asymmetric mode. In both studies it was concluded that the second spectrum predictions of Timoshenko beam theory should be disregarded.…”

“…This allows one to think in terms of a first and second spectrum of natural frequencies, or first and second branches of a wave propagation dispersion diagram. It has been shown recently [2] that the transcendental frequency equation also factorises for guided-guided and guided-hinged conditions; these new end combinations can be regarded as portions of a multi-span hinged-hinged beam. Below the second spectrum cut-off frequency given by o co ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi kAG=rI p , TBT predicts both propagating (TBT1) waves, and evanescent (TBT2) waves associated with the hyperbolic spatial functions.…”

“…In Ref. [2] comparison was made between ''exact'' plane stress predictions for a standing wave in a short hinged-hinged beam of thin rectangular cross-section. The agreement between the TBT1 predictions and the exact lower branch asymmetric mode was within the range À0:4% to+0.54% for the first 20 modes of vibration when using the shear coefficient k ¼ 5 1 þ n ð Þ= 6 þ 5n ð Þ.…”

Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode

“…Note that the relatively poor accuracy of the second spectrum with respect to detailed theories cannot be a measure of its physical meaningfulness (contrary to the suggestion of Stephen (2006) and Stephen & Puchegger (2006)). …”

“…Levinson & Cook (1982), Stephen (1982) and Nesterenko (1993) did realize the mathematical existence of these roots (clues coming from the quadratic equation (2.5)) but were unsure of their physical meaning. Since (i) the first root in equation (2.7) can be obtained by incorrectly treating the problem as one of regular perturbation while the second root cannot be, (ii) the second spectrum is less accurate with respect to three-dimensional theories, (iii) the so-called Ostrogradsky energy is negative (Nesterenko 1993), and (iv) only those frequencies 'which turn into frequencies of the EB equation' should be regarded as physical (Nesterenko 1993) when e and s approach zero, the second spectrum has been rejected (Levinson & Cook 1982;Stephen 1982Stephen , 2006Chervyakov & Nesterenko 1993;Nesterenko 1993;Stephen & Puchegger 2006) as being non-physical and a mathematical artefact of the correction to the EB field equation. There are several fallacies in these objections which will be discussed in the following.…”