### Abstract

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 theta. The q-factor is related to the wavenumber and the geometric shape of the helical wave. The theta-factor is related to the wave helicity and has two values, + pi/2 and -pi/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.

Original language | English |
---|---|

Pages (from-to) | 3913-3934 |

Number of pages | 21 |

Journal | Proceedings of the Royal Society of London - A Mathematical and Physical Sciences |

Volume | 461 |

DOIs | |

Publication status | Published - 2005 |

### Keywords

- helical wave
- propagating wave
- four-component vector wavefunction
- gyroscopic effect
- Timoshenko beam
- Euler's formula
- SHAFT
- VIBRATION
- AXIS

### Cite this

*Proceedings of the Royal Society of London - A Mathematical and Physical Sciences*,

*461*, 3913-3934. https://doi.org/10.1098/rspa.2005.1524

**Helical structure of travelling waves in a spinning Timoshenko beam.** / Reid, Stephen R; Stephen, N. R.; Wang, W. Q.; Chan, K. T.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society of London - A Mathematical and Physical Sciences*, vol. 461, pp. 3913-3934. https://doi.org/10.1098/rspa.2005.1524

}

TY - JOUR

T1 - Helical structure of travelling waves in a spinning Timoshenko beam

AU - Reid, Stephen R

AU - Stephen, N. R.

AU - Wang, W. Q.

AU - Chan, K. T.

PY - 2005

Y1 - 2005

N2 - 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 theta. The q-factor is related to the wavenumber and the geometric shape of the helical wave. The theta-factor is related to the wave helicity and has two values, + pi/2 and -pi/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.

AB - 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 theta. The q-factor is related to the wavenumber and the geometric shape of the helical wave. The theta-factor is related to the wave helicity and has two values, + pi/2 and -pi/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.

KW - helical wave

KW - propagating wave

KW - four-component vector wavefunction

KW - gyroscopic effect

KW - Timoshenko beam

KW - Euler's formula

KW - SHAFT

KW - VIBRATION

KW - AXIS

U2 - 10.1098/rspa.2005.1524

DO - 10.1098/rspa.2005.1524

M3 - Article

VL - 461

SP - 3913

EP - 3934

JO - Proceedings of the Royal Society of London - A Mathematical and Physical Sciences

JF - Proceedings of the Royal Society of London - A Mathematical and Physical Sciences

SN - 0080-4630

ER -