Helical structure of the waves propagating in a spinning Timoshenko beam

Helical structure of the waves propagating in a spinning Timoshenko beam

The 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.

helical wave, propagating wave, four-component vector wavefunction, gyroscopic effect, timoshenko beam, euler's formula

3913-3934

Chan, K.T.

7d33b162-af28-4970-9080-3de063cec990

Stephen, N.G.

af39d0e9-b190-421d-86fe-28b793d5bca3

Reid, S.R.

87f18c46-f3d6-43cf-b43b-cf4d49a0f688

2005

Chan, K.T.

7d33b162-af28-4970-9080-3de063cec990

Stephen, N.G.

af39d0e9-b190-421d-86fe-28b793d5bca3

Reid, S.R.

87f18c46-f3d6-43cf-b43b-cf4d49a0f688

Chan, K.T., Stephen, N.G. and Reid, S.R.
(2005)
Helical structure of the waves propagating in a spinning Timoshenko beam.
*Proceedings of the Royal Society A*, 461 (2064), .
(doi:10.1098/rspa.2005.1524).

## Abstract

The 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.

Text

** Chan_05.pdf
- Accepted Manuscript**
## More information

Published date: 2005

Keywords:
helical wave, propagating wave, four-component vector wavefunction, gyroscopic effect, timoshenko beam, euler's formula

## Identifiers

Local EPrints ID: 23445

URI: http://eprints.soton.ac.uk/id/eprint/23445

ISSN: 1364-5021

PURE UUID: d6d3d067-526d-494f-bb71-4bbf9d271718

## Catalogue record

Date deposited: 17 Mar 2006

Last modified: 08 Jan 2022 09:52

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## Contributors

Author:
K.T. Chan

Author:
S.R. Reid

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