Biomechanics of leg swing and its effect on multi-turn jump performance in figure skating Текст научной статьи по специальности «Медицинские технологии»

BIOMECHANICS OF LEG SWING AND ITS INFLUENCE ON PERFORMANCE OF MULTI-TURN JUMPS IN FIGURE SKATING V.I. Vinogradova, professor, Dr.Hab.

Moscow state university of mechanical engineering (MAMI), Moscow Key words: figure skating, leg swing.

Introduction. The supervision of the implementation of multi-turn jumps in figure skating shows that it is accompanied by swings of the leg free from the support. However, the observations are not useful to answer the question about the role of this action in maintaining athlete's balance when jumping and particularly in the increase of the number of turns in jumps, which depends on jump height, flight time in a jump and initial rotational speed of a figure skater when he jumps off the ice. The qualitative biomechanics, based on the observations, does not quantify the actions of a figure skater to improve the technique of performance of single figure skating elements and jumps in particular. The fundamentals of jump biomechanics in figure skating are considered in the book [1], where a figure skater is modeled using anthropomorphic mechanisms of varying complexity. The quantitative analysis promotes the science-based determination of the individualization of elite athletes' training. It helps to identify the ways of enhancement of the technique of performance of multi-jumps. However, it does not cover swing of the leg free from support, and therefore there is no quantitative assessment of leg swing to increased jump height, and hence the flight time in a jump, which affects the number of turns in it. The influence of leg swing on the increase of the initial rotational speed of a figure skater jumping is not considered. The author seeks to make up for an obvious flaw in the organization of the foundations of the biomechanics of jump performance.

The algorithm of the approximate quantitative assessment of the impact of leg swing on the increase of

the number of turns in a jump is based on the simple anthropomorphic mechanisms and laws of

theoretical mechanics [2]. The approximate estimation algorithm is based on the formulated assumptions.

Clearing some of them helps to specify an approximate estimate, improving the algorithm.

Theory. A jump of a figure skater, whose weight isP, is performed by pushing with a figure skater's support leg on the ice, that is, impulse force S.

Let us assume that the vector of impulse force S is directed along the normal to the ice surface and occurs at the constant speed. The impulse force is determined by the expression

S = nPAt, (1)

where P - figure skater's weight, n - coefficient of overload of figure skater when pushing, At - finite operating time interval of strength of pushing nP. If the initial operating instant time of a push is the origin, then

S = nPt, (2)

where t - operating time of strength of pushing.

Using the theorem of change of momentum using impulse force, we write

mv = S,

where v - speed of a figure skater and, consequently, his center of mass, at the moment of jumping off the ice, m - figure skater's mass. So, considering (2), we get

nPt

v =-= ngt, (3)

m

where g - acceleration of gravity.

Using the initial speed v in a jump a figure skater (his center of mass) moves along the normal to the ice surface to the height h, when his speed becomes zero. In this case the potential energy of figure skater's weight is accumulated.

According to the law of conservation of mechanical energy it can be presented as follows

mv2

= -Ph . (4)

mv2

So, h = —2p • Considering (3), we get

2

2/2

h = - gn-L. (5)

2

Thus, we got a formula to determine jump height without a leg swing, just by pushing with a support leg. It is known [1] that the time of flight of a figure skater in a jump is determined by the formula

T = 2 2h . (6)

V g

Considering (5), we determine the flight time of a figure skater in a jump owing to impulse force S of push with a support leg

T = 2nt. (7)

Next, we define how leg swing influences the increase of height of a figure skater's jump. We make a number of assumptions. Let the leg be modeled by the uniform beam with the length l, which is connected with the figure skater's body with a hinge and without friction. Assume that swing (rotation of the beam along the hinge axis) is made using internal forces for the anthropomorphic mechanism in the plane passing through the symmetry axis of a figure skater, which is directed along the normal to the ice surface. Hereat, ( = const. Then aH=p /t, wherep-turning angle of leg (beam) at swing, t- turn (swing) time of leg. Assume that when p = n /2 the rotation of the beam (leg swing) stops immediately.

Assume that a figure skater jumps off the ice surface with a leg swing to the height h without pushing with a support leg. According to the law of conservation of mechanical energy the energy of beam rotation (leg swing) becomes equal to the potential energy of a figure skater when lifting his center of

mass to the height h .

1 2 —

2 J(H = -Ph. (8)

_ j (2 i

Thus, h = —2pr ,where JH = 3mHl2 - moment of inertia of a uniform beam (foot) relative to the hinge

axis, mH - beam (foot) weight, l - its length.

_ 1 m 1 m

Then h =--- (l2. We introduce the notation a =---. As a result we obtain

6 g m 6g m

h = a(Hl2. (9)

If we admit that a « const for different figure skaters, then jump height h using only leg swing quadratically (very much) depends on the speed ( of leg swing and length l.

Certainly the obtained result should be interesting for a trainer and an athlete when organizing an individual training process to increase the number of turns in a jump.

The flight time T of a figure skater with the help of a leg swing, modelled with a uniform beam, is defined by the formula (6). If we assume that the height h is determined by the formula (9), then

T=(10)

g V 3 m

If we use the formula (5) to determine jump height only by pushing with a support leg on the ice, and the formula (9) - using only leg swing, the maximum contribution of leg swing to the jump height can be determined.

Ац = 100% = (1 - ^¿2")100%. (11)

h gn t

If a swing-up leg is modeled uisng a uniform cone with its top directed to the ice surface and the base connected with the body with a hinge and without friction, the moment of inertia relative to the hinge axis is defined in the following way

JH = J x2dm или JH = pJ x2dV,

(V) (V)

m

where p = - density of a uniform cone with the mass mH and volume V.

1 2

If the cone height is equal to leg length / of a figure skater, and Rc - radius of the cone base, V = -

the cone volume. So, p = -—H-.

nR2/

Then

Jh=% J x2dV. (12)

nRJ (V)

The hinge of the cone-shaped leg is in the center of the base of the direct cone. Its center is taken as the origin of the axis x, directed along the axis of the cone to its top, the ice surface. At a random distance along the axis x from the origin, along the normal to the axis of two infinitely close dx planes we cut out the elementary blunted cone. Hereby, the larger base of the elementary blunted cone has a radius r + dr, less - r . The elementary volume dV of the blunted cone is determined by the common formula

dV = [(r + dr)2 + r2 + (r + dr)r ].

Keeping this expression for the values of only the first order of vanishing, we obtain

dV = nr2dx, где r = (/ - x)tgP. Тогда dV = ntgfi(/2 - 2/x + x2)dx. Substituting this expression for

the elementary volume dV in (12) and integrating, we obtain

JH = mf. (13)

For leg swing, which is modelled with a uniform cone, using the law of conservation of mechanical energy (8), we obtain

— 1 m

hK = b/V, where b = — . (14)

H 20g m

Thus, for a cone-shaped leg

*Z* 100%% = (1 - ^

h gn t

Considering (14), like before (10), we get

А K = 100% = (1--2V)100%. (15)

T = \ 1 mvJ. (16)

gV 10 m

Proceeding from the observations of jump execution in figure skating, leg swing is performed in a plane which is inclined at the angles to the ice surface. Consequently, the angular velocity vector o>H of

rotation of the leg model is factorized into a vector, which increases the jump height and a vector which increases the rotation speed of a figure skater in a jump relative to its axis, namely the initial rotation speed of a figure skater in a jump.

Denote the part of the angular velocity o>H of leg swing, which increases jump height h - a>Hh, and its another part which increases the rotation speed of a figure skater in a jump - a>HW. Then

Wh = WH sin^ Whw=WH cosa

Hereby, in the formulas (9) and (14) instead of o>H one should keep in minda>Hh, i.e. hц = aa2Hhl2 u hK = ba2Hhl2.

Owing to the results (9) and (14), the dependence of jump height due to leg swing, approaching the cylindrical or cone shapes, can be determined: hц / hK = a / b = 3,3. If a leg has a cylindrical shape, its swing is 3,3 times more effective if it is close to the cone shape.

We estimate the influence of the rotational speed a>HW of leg swing on the rotational speed w of a figure

skater around its axis in a jump, that is the number of turns. Assume that the rotational speed of a figure skater in a jump is created only with leg swing. Assume that the rotational energy of leg swing goes into rotation of a figure skater after a tuck position. We are to model a figure skater after a tuck position with the help of a uniform cylinder with the weight P of a figure skater. According to the law of conservation of mechanical energy it can be presented as

1 2 1 2

2jHK»sina) = -Jw . (17)

1 2

If a leg is modelled with the help of a uniform beam, JH = 3mHl. For a uniform cylinder - J = 0,5mR ,

where R - radius of cylinder - models of a figure skater. It is obvious that

HW

sin a. (18)

Assume that a leg is modelled using a uniform cone. Hereby, its moment of inertia relative to the hinge is defined by the formula (13).

Using the law of conservation of mechanical energy (17) for the conical model of leg we'll obtain

/1 mH l .

WK =Jt — — ®Hwsina (19)

V 5 m R

The results (18), (19) enable to estimate the difference in rotational speed of a figure skater, which is created by leg swing of different shapes: w / o>K = 1,83 . Leg, close to a cylinder in shape, reports angular

speed to a figure skater in a jump, which is almost twice higher than the same speed with a cone-shaped leg.

The formulas (18), (19) can be used to determine the rotational speed of a figure skater in a jump as a result of leg swing and considering the formulas from the book [1] - to determine the incremental rotational speed of a figure skater as a result of leg swing.

Experiment. The formulas can be used to individualize training programs for elite athletes via the experimental determination of some parameters using easy techniques.

1. One needs scales to determine figure skater's weight P . Free standing dynamometer is used to define push strength impulse of a figure skater in a jump to determine an overload n.

2. One needs a time identifier to measure one-impulse time and leg swing time. Leg rotational speed

is estimated as follows: w = n/2.

t

3. Anthropometric characteristics m, mH, I, R, RK, ( and other ones are approximate [1].

The individualization of the training process can be scientifically justified using the findings owing to simple experiments. Conclusion. The theoretical findings and the simple algorithm of the individual experiment help a trainer

and an athlete to design a science-based individual program of training multi-turn jumps for elite athletes

in view of the leg swing performance.

References

1. Vinogradova, V.I. The basics of jump biomechanics in figure skating / V.I. Vinogradova. -Moscow: Sovetsky sport, 2012. (In Russian)

2. Targ, S.M. Theoretical mechanics: a short course / S.M. Targ. - Moscow: Nauka, 1968. - 478 P. (In Russian)

Author's contacts: hom. 8(499) 308 06 94. mob. 8 915 224 87 91.