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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.<br/>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

Topics:
TA, QC

Year: 2005

OAI identifier:
oai:eprints.soton.ac.uk:23445

Provided by:
e-Prints Soton

Downloaded from
http://dx.doi.org/10.1098/rspa.2005.1524

- (1995). Advanced engineering dynamicsi. 2nd edn,
- (2005). Chan and others 3934
- (1998). Elastic wave motions in an axially strained, inﬁnitely long rotating Timoshenko shaft.
- (1995). Elastic wave propagation in a Timoshenko beam spinning about its longitudinal axis.
- (1985). Modal balancing of ﬂexible shaft without trial weights.
- (2002). Superposed standing waves in a Timoshenko beam.
- (1961). The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions.
- (1919). The lateral vibration of loaded shafts in the neighbourhood of a whirling speed—the effect of want of balance.
- (1980). The use and interpretation of vibration measurements.
- (1959). The vibration of rotating shafts.
- (1968). Vibration and balancing of ﬂexible shafts.
- (1984). Vibrations and waves in physics. 2nd edn,
- (1998). Wave reﬂection and transmission in an axially strained, rotating Timoshenko shaft.

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