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25 April 2024 |
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Twisting Somersault | | Holger R. Dullin
; William Tong
; | | Date: |
24 Oct 2015 | | Abstract: | A complete description of twisting somersaults is given using a reduction to
a time-dependent Euler equation for non-rigid body dynamics. The central idea
is that after reduction the twisting motion is apparent in a body frame, while
the somersaulting (rotation about the fixed angular momentum vector in space)
is recovered by a combination of dynamic and geometric phase. In the simplest
"kick-model" the number of somersaults $m$ and the number of twists $n$ are
obtained through a rational rotation number $W = m/n$ of a (rigid) Euler top.
This rotation number is obtained by a slight modification of Montgomery’s
formula [9] for how much the rigid body has rotated. Using the full model with
shape changes that take a realistic time we then derive the master
twisting-somersault formula: An exact formula that relates the airborne time of
the diver, the time spent in various stages of the dive, the numbers $m$ and
$n$, the energy in the stages, and the angular momentum by extending a
geometric phase formula due to Cabrera [3]. Numerical simulations for various
dives agree perfectly with this formula where realistic parameters are taken
from actual observations. | | Source: | arXiv, 1510.8046 | | Services: | Forum | Review | PDF | Favorites | |
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