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A MODEL FOR THE SIMULATION OF VISCERAL MASS DISPLACEMENT IN
DROP JUMPING
Istok Colja and Vojko Strojnik
Faculty of Sport, University of Ljubljana, Ljubljana, Siovenia
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INTRODUCTION
Drop jumps may be mathematically modelied as a mass-spring-damper
system with a sufficient accuracy (Aurin and Zatsiorsky, 1984). It was stated that a
mass of internal viscera does not affect a maximum jumping height and a frequency
of jumping. Minetti and Belli (1994) found visceral mass, which presents 14% of the
total body mass, oscillating in an opposite phase than a musculo-skeletal mass
during hopping and thus significantly influencing the jumping height and frequency
of jumping, but also an energy consumption as weil. The aim of the present study
was to assess an influence of the visceral mass on jumping height in a single drop
jump by mathematical modelling.
METHODS
The model (Fig. 1) consisted of two masses connected by aspring and
damper, where mass m2 presented the visceral mass.
Elastic module K2 and damping mod­
ule B2 defined an attachment of m2 to
the other parts of the body. The model
was described with two differential
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equations:
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of the human body, AO
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where m1 'presented the mass of the
K,
external container, x 1 and X2 vertical
displacements of m 1 and m2 from a
position of equilibrium, B2 the damping
coefficient, K l and K2 the stiffness co­
efficients.
Numerical solution was per­
Figure 1 Scheme of mass-spring­
formed by MATLAB (The MathWorks
damper system
Inc.). The vertical displacement of the
centre of gravity of both masses (CG) was calculated by varying K2 and B2 sys­
tematically at constant K l . The value for K l was taken from Aurin and Zatsiorsky,
1984).
Each jump was subdivided into two phases, a contact phase and aerial
phase. The maximum jumping height was defined as the apex of the trajectory of
CG in aerial phase, calculated by formula:
H max
585
= v; 1(2 * g)
where Hmax was
the maximum height of CG, vA was velocity of CG at the instant of take-off, 9 was
gravitational acceleration.
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RESULTS
Dependence of jumping height on K2 and B2 is presented in Figure 2.
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Figure 2 Jumping height (H) depending on Figure 3 MI is veJocity ofmass m J , M2 is
K2 and 82
velocity of mass m 2 relatively to ml> at
K2 = 0 and 8 2 = 550. VerticaJ dotted line de­
notes the instant of take-off.
It became c1ear that the main response of the system was attained by in­
creasing B2 fram 0 to 600 Ns/m. Afterwards increased B2 didn't influence the jump­
ing height significantly
Figures 3 and 4 show two typical examples of m, and m2 velocity during the
contact phase. The velocity of m2 is presented relatively to the movement of m, in
both figures. The first example represents condition in which m2 oscillated in a
phase shift with m, and where vertical displacement according to m, may be up to 8
cm (Minetti and Belli (1994». With increasing stiffness of the system, the phase
shift between m 2 and m 1 become smaller and corresponding jumping height in­
creased as weIl. The second example represents condition when both masses, m,
and m2 oscillated in parallel. When the m2 is delayed for 0.032 s as in the first ex­
ample, the jumping height was 0.388 m. When the oscillation became almost paral­
lel (Ll.t=0.002 s), the jumping height reached 0.431 m.
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as presented in Fig
(Blickhan, R. 1989)
control of visceral
greater K2 and 82
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CONClUSION
With increasing K2 and B2 , the
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observed system became more stift. It
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seems that damping was a crucial factor
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for determining the maximal jumping
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height, because at constant B2 the in­
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creasing of K2 did not influence the
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changes in jumping height a lot, except
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at very low B2 . Increasing stiftness of the
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system smaller phase shift between m1
TIME (s)
and m2 as weil as the oscillation ampli­
Figure 4 MI is velocity ofmass m l ,M2 is tude of m . Both of them contributing to
2
velocity of mass m2 reIatively to ml> at
the higher velocity at the end of contact
K 2 = 3600 in B 2 = 2200. Vertical dotted line phase. With sufficiently high B and K ,
2
2
denotes the instant of take-off.
m2 would oscillate strictly in parallel with
m1 and would no Ionger influence the
jumping height (Aurin and Zatsiorsky,
1984). In that case the observed system
as presented in Figure 1, could be considered simple as a mass-spring system
(Blickhan, R. 1989). Results of the present ?tudy indicating the importance for a
control of visceral mass movement for maximising the result. In practice, the
greater K2 and B2 can be achieved by increased abdominal pressure.
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REFERENCES
Aurin, AS., ZatsiorsKy, V.M. (1984) Biomechanical characteristics of hu­
man ankle-joint muscles. Eur. J. Appl. Physiology 52: 400-406.
Minetti, AE., Beili, G. (1994) A model for the estimation of visceral mass
displacement in periodic movements. J. Biomechanics 27: 97-101.
Blickhan, R. (1989) The spring-mass model for runing and hoping. J.
Biomechanics 22: 1217-1227
587