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Russian Journal of Biomechanics
THE INFLUENCE OF AERODYNAMIC FORCES ON THE MOVEMENT OF
SPORTSMEN AND SPORT BALLS
R.N. Rudakov*, Yu.l. Nyashin*, A.R. Podgayets*, A.F. Lisovski**, S.A. Miheeva***
* Department of Theoretical Mechanics, Perm State Technical University, 29a, Komsomolsky prospect, 614600, Perm, Russia, e-mail: [email protected]
** Tchaikovsky State Institute of Physical Culture, 67, Lenin street, 617762 Tchaikovsky, Perm Region, Russia *** Department of Physical Culture, Perm State Technical University, 29a, Komsomolsky prospect, 614600, Perm, Russia
Abstract: This paper is devoted to research of movement of sports balls and sportsmen in some kinds of sports where essential influence on movement is rendered with aerodynamic forces. Mathematical modelling of table tennis and volleyball is considered. From comparison of calculated and experimental data the aerodynamic parameter is found for alpine skiing. The technique of determination of aerodynamic coefficients of drag and lift in ski jumping is developed.
Key words: aerodynamic forces, table tennis, volleyball, alpine skiing, ski jumping
Some kinds of sports (table tennis, volleyball, alpine skiing, and ski jumping) are considered. Aerodynamic forces have a material effect on movement of a ball in the two first kinds of sports and movement of sportsmen in the two mountain skiing kinds of sports. The problem of mathematical modelling in these kinds of sports was regarded in our previous works [1-8]. The results of new researches are submitted in the present work. In a problem about table tennis interaction of a ball with a table and racket is taken into account. Mathematical modelling of the game of two players is carried out, one of which makes rolling beats and the other - trim beats. Features of game - higher velocity of racket movement is revealed for «rolling» style of play and higher velocity of the sportsman's movement across the ground for «trim» style of play. So-called power service is investigated in volleyball. Power service is a service at which the sportsman strikes a ball in a phase of a jump above his side of the ground. The influence of coordinates of a point of impact - its height and distance from the border of the ground - on efficiency of power service is shown. Videotape recordings of power service are analysed.
Aerodynamic coefficients of alpine skier are needed for mathematical modelling of alpine skiing. Theoretical solution of the problem of airflow around the skier is extremely difficult, and experiment in the wind tunnel demands big material inputs. We have applied a way of indirect determination of drag of the sportsman that is development of work [6]. The profile of the alpine skiing route "Snowflake" (Tchaikovsky city) was measured. The duration of downhill skiing of 4 sportsmen in 3 poses (high, average and low) was measured, too. The technique of least squares was used to get the equation of the route's curve. The mathematical model of «smooth» alpine skiing is worked out with respect to the force of friction between skis and snow and aerodynamic forces. The aerodynamic parameter that determines drag
1. Introduction
force was determined for given sliding friction factor from the comparison of theoretical and experimental duration of downhill skiing. The greatest difficulties arise when determining aerodynamic coefficients of aerodynamic forces of drag and lift during the flight phase of a ski jump. Known experimental data correspond to a ski jump style of the 60-s. Indirect definition of aerodynamic coefficients with the use of integral characteristics of a jump -flight time and distance - gives insufficient information on their dependence on attack angle of skis and the sportsman. And, at last, mathematical modelling of airflow around the ski jumper is possible only for rather simple models of skier-on-skis system. This work develops the idea of determining the aerodynamic coefficients from the equations of movement of the center of gravity of moving object by processing of the videotape recordings of the volleyball ball flight [9]. The aerodynamic coefficients of drag and lift forces are found from the differential equations of movement of the center of gravity of skier-on-skis system in this paper. We need to know components of velocity and acceleration for every moment of time to use this method.
The technique of research of sports movements offered in work and results of researches can be applied for perfection of performance of these movements in the considered kinds of sports.
2. Mathematical modelling of table tennis
Flat trajectories of ball's movement during table tennis are considered. The purpose of research was to reveal, what should be the movement of the racket that the ball has got on the opponent side with the maximal velocity of movement of the center of gravity.
The following forces are applied to the ball during the flight phase: gravity P = mg,
drag force R = -0.5pScD v v and lift Q = 0.5pcL v , where m is mass of the ball, p
|cox v
is air density, S is area of midlength section, cD and cL are aerodynamic coefficients of drag force and lift force, v is velocity of center of gravity, co is angular velocity of the ball around the axis that passes its center. Aerodynamic coefficients cD and cL for ball are known from experiments [10]. According to these data cD and cL can be considered constant for studied range of ball velocities. If the ball is not rotating or the angular velocity is small we can assume the following values of aerodynamic coefficients: cD= 0.45, cL~ 0. For big angular
velocities (co > 1.5—, r is radius of the ball) aerodynamic coefficients are cD= 0.6, cL= 0.35.
Rise of lift force is caused by so-called Magnus effect. When forces of viscosity rotate the ball, the air is involved in orbital movement that leads to breakdown of airflow symmetry and appearance of lateral force.
In the reference frame shown in Fig. 1 differential equation of movement of the center of gravity have the following kind
mx = -kcD v x +
OJX V
kcL(&x v)
my =-mg-kcDv y +-—, (2.1)
COX V
1 2 /~2 2 &=-p7ir , V = +y ,
where dot denotes time derivative. Initial conditions must be set at every part of the trajectory
W] \y
0 O w2
Fig. 1. Trajectories of the ball received with rolling (solid line) and trim (dotted line).
Corresponding racket velocities are Wj and w2 .
t = o: X = x0, y = y0,
x = x0, y = y0, io = to0. (2.2.)
Change of angular velocity during the flight phase was not taken into account.
Typical example of play is examined. One player (left player) makes only rolling beats and the other (right player) makes only trim beats (see Fig. 1). In both cases vector of angular velocity to is opposite to Oz axis. In that case equations (2.1) are simplified
Qx=kcLvy, Qy=-kcL\x (2.3)
(x > 0 for the left player, so Q < 0 and force of lift is directed down; x< 0 for the right
player and Qy > 0, so lift is directed up). The term «lift» does not show the direction of this
aerodynamic force for «rolling» play: the force Q actually lowers the trajectory.
The following data were used for calculations: mass of the ball m = 0.0023 kg, radius of the ball r =0.019 m, acceleration of gravity g =9.8 m/s2, coefficient of restitution for impact interaction of the ball and the racket and the tennis table is 0.8 and coefficient of sliding friction of the ball and table surface is 0.2. These coefficients were found experimentally. The size of the table and height of the net are standard.
The algorithm of play is following. The player with rolling serves receives flying ball in the point where the trajectory of the ball crosses vertical line I (see Fig. 1). Then optimization problem has been solved: what initial velocity vq the player should impart the ball to make the ball fall on the opponent's side with maximal velocity with constraint on the height of the trajectory when x = 0; y>hs + r, where hs is the height of the net. This problem is reduced to reiterated solution of Cauchy problem (2.1) - (2.2) by the method of step by step integration. Varied parameters are scalar of vector v0 and angle a between v0 and the horizontal. Then racket velocity w, is found that impart velocity v0 to the ball. This problem is solved with help of theorems of momentum conservation and conservation of the moment around the axis passing the center of gravity during impact. For optimal trajectory the point where the ball hits the opponent's side of the table and velocity of hit are found. Analysis of impact the ball and the table determines velocity of the ball after the impact what gives initial conditions for solution of Cauchy problem (2.1) - (2.2) and finding consequent movement of the ball. The right player receives the ball in the point, where the ball crosses the line II (see Fig. 1). Algorithm of finding racket velocity and ball movement for that player is described earlier.
Ball's angular velocity appears to be higher than 1.5 v/r at every phase of the game, so Magnus effect has been always taken into account in the flight phase, that is there was assumed that cD- 0.6, cL = 0.35. The most interesting result is the following: racket velocity
of player with rolling beats is almost 2 times higher that the racket velocity of the player with trim beats (w, =19.1 m/s, w2 = 10.6 m/s). Angles of these velocities to the horizontal are correspondingly equal aj =22.8°, a2 =27.4°. Another result that is well known to the sportsmen was found: the «trim player» has to move a lot on his ground. The ball is received on the distance of 4 meters from the table in the example above.
3. Power service in volleyball
The most effective modern service in volleyball is called power service and represents the service in the jump above the ground that impart the ball big initial velocity. Examination of the game shows that each player chooses different point for the hit. Some hit the ball right above the border of the ground and some make a long jump forward. We investigated the influence of place of the impact on effectiveness of power service. The effectiveness criterion is maximum value of velocity of the ball when it touches the opponent's ground (v^).
The coordinate origin is chosen under the net (see Fig. 2). The point where ball's flight begins has position data xMq = /, yMo - h. We investigated the range of parameters
/ = 0 + 2 m, h = 2 + 3.5 m.
The movement of the ball is described by equations (2.1) with boundary conditions (2.2). Rotation of the ball is not taken into account (co = 0), so aerodynamic coefficients have the following values: cD = 0.5, cL = 0. Parameters of the ball are: m - 021 kg, r = 0.105 m. Length of the ground is 18 m, height of the net is 2.43 m (see Fig. 2). When we consider the ball to be a material particle we take into account the radius of the ball in the constraint on height of the flight in the point x =9 m
x = 9 m: y >2.535 m. (3.1)
Control points / =0, 1, 2 m, h = 2, 2.5, 3, 3.5 m were taken for analysis. For each pair of parameters the optimization of power service was held with restraint (3.1). Numerical value of initial velocity v0 and its angle to the horizontal a were found, for which the ball
touches the opponent's ground with maximum velocity. Parabolic service (service with big angles to the horizontal) has not been taken into account, although it can produce higher value of \k than power service, provided the roof of sport hall is sufficiently high. But parabolic flight takes much time and the opponent can easily prepare to receive the ball. For each pair of parameters the Cauchy problem (2.1) - (2.2) was solved. Varied parameters were v0 and a.
Optimization results were appeared to be exceptionally interesting. We found than increase of the forward jump distance I practically does not affect optimal velocity v k.
Moreover, for small heights (h = 2 + 2.5 m) \k even drops a little. Only from the height h = 3.5 m that is above the reach of the most high humans, velocity \k begins to increase with increase of /. The highest and the lowest values of optimal velocity \k was found for 1 = 2 m for different values of heights of the service h: for h = 3.5 m \k =18.8 m/s, and h - 2 m gives \k = 12.5 m/s. The results mentioned do not mean that the sportsman does not need to jump forward. It was found that increase of 1 for ft <3.5 m leads to substantial decrease of initial velocity v0 corresponding the optimal value of vk. For example, for
h - 2.5 m the change of / from 0 to 2 m leads to decrease of v0 from 25.1 m/s to 22.3 m/s, but the terminal velocity \k decreases only by 0.5 m/s, changing from 14.1 m/s to 13.6 m/s. Increase of the height of the service h always leads to increase of optimal value of velocity \k that certainly demands increase of initial velocity, too. Tables 1-4 show accordingly initial
velocity v0, angle of initial velocity to the horizontal a, terminal velocity vk and flight time t that are found from optimization problem.
Table 1. Initial velocity v0 in m/s.
\ /, m h, m\ 0 1 2
2.0 23.2 21.5 20.4
2.5 25.1 23.6 22.3
3.0 27.1 26.9 25.7
3.5 31.3 30.5 32.5
Table 3. Terminal velocity v^ in m/s.
Table 2. Angle of initial velocity to the
îorizontal a in degrees.
x\ /, m h, m 0 1 2
2.0 9.6 10.4 10.3
2,5 5.4 5.4 5.3
3.0 1.8 0.6 0.0
3.5 -2.7 -3.5 -5.9
Table 4. Flight time t in s.
m h, m 0 1 2
2.0 13.1 12.7 12.5
2.5 14.1 13.8 13.6
3.0 15.1 15.4 15.3
3.5 17.1 17.2 18.8
\ /, m h, m 0 1 2
2.0 1.1 1.1 1.1
2,5 1.0 1.0 1.0
3.0 0.9 0.9 0.9
3.5 0.8 0.8 0.7
The results of given study of power service can be of use to trainers. They can help trainers to choose the strategy of perfection of performance of power service in dependence on physical data of specific sportsman. Analysis of videotape recording of power services of highly qualified players (1-st sport grade) shows that they are far from optimal. The angle of ball departure during the service is much higher than one given in Table 2 and thus initial velocities do not reach optimal values.
Fig. 3. Forces affecting of alpine skier during downhill skiing.
4. Determining aerodynamic parameter in alpine skiing
Results of research presented below form a preparation step for building of the mathematical model of movement of alpine skier in a special slalom. Aerodynamic forces depend on pose of a sportsman and can be determined during smooth downhill skiing, without turns. Knowing of aerodynamic forces, we can find friction forces that appear during cut-in of skis edging to snow on turns by similar method described below.
The profile of part 230 m long of alpine skiing route "Snowflake" (Tchaikovsky city) was measured. Fig. 3 gives general view of this part. It represents a curved line in section, the equation of this line can be written in the form (p = cp(/), where cp is angle between tangent to the line at any its point M and the horizontal and / is a curvilinear coordinate of point M, with coordinate origin at the beginning of the route part (/ = O, M ),
The angle tp was measured every 10 m; 24 measurements were made (see Table 5).
The route is represented by a kinked curve consisting of 23 strait pieces. This kinked line was approximated by a smooth curve with the help of least-squares method. The best approximation was found to be a circle
cp = C0 +C,/,
C0=T^(32.8±0.9) rad, (4.1)
q =-£-(-0.110 ±0.007) rad/m.
1 oO
Radius of the circle r = 4^=520 m.
d(.p
The experiment was carried out to determine the time of downhill skiing by this route by 4 alpine skiers. Each of them skied 3 times in different poses. The results of this experiment are presented in Table 6.
Let us create a mathematical model of the downhill skiing along the noted part of the route. Alpine skier is affected by the following forces: gravity P, aerodynamic forces of drag R and lift Q, normal reaction of support N and friction force F (F =/N), where / is a friction coefficient for skis and snow.
Differential equations of skier's movement in natural coordinates look as follows:
Psincp-F-i?, (4.2)
dt
Tab e 5. Measurements of t le profi e of alpine skiing route.
/, m 0 10 20 30 40 50 60 70 80 90 100 110
30 33 31 28 28 28 27 26 26 23 21 21
/, m 120 130 140 150 160 170 180 190 200 210 220 230
q>.° 21 20 16 15 14 13 13 12 11 10 9 7
Table 6. Time (in seconds) of passing the measured route part by alpine skiers.
Sportsman, Height, Weight, Time t, s
№ m kg High pose Middle pose Low pose
1 1.69 65 13.25 12.75 12.25
2 1.75 60 13.25 13.10 12.25
3 1.65 60 13.25 13.11 12.34
4 1.70 65 13.28 13.29 12.35
2
=-P cos y + Q + N, (4.3)
v-r£. (4.4,
With regard to earlier equations for P, R, Q equations (4.2) - (4.3) can be rewritten v^ = rg(sincp-/coscp)-[/ + ^^v2, (4.5)
where ji = S(c D - fcL) is aerodynamic parameter. Initial condition for this problem is
<p = C0: v = 0. (4.6)
From exact solution of Cauchy problem (4.5) - (4.6) velocity as a function of angle cp can be found
v = A/(v4sincp + 5cos(p + C(-2i(p)),
r ro . _ 26-/ 2bf-1 yl sin (p0 + 2? cos C0 „ b = f + A = B = -2rg-±--, C =--^ v 0 . (4.7)
2m 46 +1 4b +1 exPv~ 2bC0)
Equation (4.4) gives formula for time t of passing the route part
Vt 1
/ = ^0=^0. <p*=C0 +230C,. (4.8)
<Po
This definite integral can be calculated with the help of approximate formulas. For each sportsman the time of downhill skiing for different initial parameter jj. was calculated for known value of friction coefficient (/=0.05). The sought quantity \i is that, for which the calculated time coincides with experimental one. The results of calculations of aerodynamic parameters are presented in Table 7. The method described does not give a possibility to determine aerodynamic coefficients cD and cL separately. But knowing the aerodynamic parameter p, is sufficient for mathematical modeling of alpine skiing. Change of
friction coefficient / has a little effect on the value of p. Estimation shows that change of / by 0.02 will lead to change in aerodynamic parameter no more than by 1 %.
As Table 7 shows, the aerodynamic parameter p weakly depends on the height of the sportsman and his weight in given limits. Taking average of table values, we can assume that aerodynamic parameter p for high, middle and low poses is 0.52, 0.48, 0.35, respectively. This method of finding the aerodynamic parameter p can be used in teaching and training of alpine skiers.
Table 7. Aerodynamic parameter p for 4 alpine skiers for friction coefficient / =0.05.
Sportsman № p, m2
High pose Middle pose Low pose
1 0.54 0.45 0.35
2 0.50 0.47 0.33
3 0.50 0.47 0.34
4 0.54 0.54 0.37
5. Method of determining aerodynamic coefficients during a ski jump
Different aspects of mathematical modelling of a ski jump have been researched in the papers [1-3, 7-8, 11-13]. Earlier authors relied on experimentally found aerodynamic coefficients for ski-jumping technique of 60-s years of XX century. Authors of later papers tried to find aerodynamic coefficients theoretically on the basis of simple models of airflow around ski-jumper. The drawback of these methods is that they did not allow to find the absolute value of aerodynamic coefficients although they described their dependence on attack angle rather well. Integral characteristics of the ski jump - flight distance and time - of calculated and experimental trajectories obtained from videotape recording were compared for complete determination of aerodynamic coefficients. Here we suggest a new method of
processing experimental data by local characteristics of a jump that allows to find the dependence of aerodynamic coefficients of drag and lift on attack angle.
Let us consider the flight phase of a ski jump and regard a skier-on-skis system as a material particle. The particle is affected by gravity P = mg, force of drag R = -O.SpSt^ vv, lift force Q = 0.5 pScL v[v x k], where k is an ort of Oz axis (see Fig. 4).
Trajectory of the flight lies in the plane Oxy. Differential equations of skier's movement are
mx = -~pScD\x + ^pScL vj>,
my = mg-lpScD\y-^pScL\x, (5.1)
(.2 .2V2 v = [x +y J .
The paper [9] shows, how a processing the videotape recording of volleyball ball flight (with cL= 0) can give a coefficient cD from equation (5.1). The results presented in that work agree with earlier experimental data on dependence of cD on velocity of the ball's movement. Moreover, the authors of cited paper confirmed the existence of so-called crisis of resistance for big velocities when the increase of velocity leads to abrupt drop of cD and then increase. It is possible to find coefficients cD and cL from (5.1) using the same method. It is only necessary to know dependencies x(t) and y(t). These dependencies can be obtained from a professional processing a professional videotape recording a ski jump.
We suggest a bit more physically founded method of solution of the problem. The trajectories of ski-jumper's flight are considered in natural reference system Mmb (see Fig.
4)
maT = mg sin p - pScD v2,
man = mg cos (3 -1 pScL w2, (5.2)
where aT, an are tangential and normal accelerations of ski-jumper, (3 is angle of velocity to the horizontal coordinate axis Ox. Equations (5.2) give coefficients cD and cL multiplied by midlength section area S
ScD=-^-(-aT+gsinp), pv
ScL=^-(-an+g cost). (5.3)
pv
Experimental values aT, an and p should be found at sufficiently small intervals of time At in the flight phase. This could be done by frame-by-frame processing videotape recording a ski jump. We can determine the way A/, passed by the ski-jumper during time period At by the use of two consequent frames (/, / +1). The easiest way to do it is to look at marker on the sportsman's costume. Average velocity during time period At is
Average tangential acceleration is
v/+r
<5.5)
Normal acceleration is defined by change in direction of vector of ski-jumper's velocity. Let the interval A/,- along which average velocity during time period At is directed have an angle p, to the horizontal. This angle is convenient to found by vertical component Ay,- of interval A/,-
P,=arcsin^-. (5.6)
лГ * (5.7)
Average normal acceleration during small period of time is
v/sin^, -p,.) At
On every i frame we need to measure attack angle a, - the angle between skies and the horizontal. It is assumed that the ski-jumper lies between skis, which is typical for new style of ski jump called V-style. So equations (5.3) taking into account (5.4) - (5.7) give dependencies of aerodynamic coefficients on attack angle. This allows to build a mathematical model of a ski jump that has optimal distance and safety of sportsman's landing.
Measurement errors are inevitable in videotape frames processing, so it is desirable to build a spline-interpolation of A/,(?), Ayt(t) for a whole jump. Scales of measured values
can be found from skis length or using special markers on the sportsman's costume. The TV camera should stand still and strictly aflat.
Suggested method of indirect determining skier-on-skis system aerodynamic coefficients is universal, rather simple and can be used in trainer's practice.
Conclusions
The questions of mathematical modelling in several kinds of sports are examined. The method of determining aerodynamic coefficients in alpine skiing and ski jumping is designed. The results of this work are of theoretical and practical value and can be used in training and coaching of sportsmen.
Acknowledgments
The authors thank M.A. Osipenko and A.V. Kamenskih for their valuable help in numerical calculations.
References
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ВЛИЯНИЕ АЭРОДИНАМИЧЕСКИХ СИЛ НА ДВИЖЕНИЕ СПОРТСМЕНОВ И СПОРТИВНЫХ СНАРЯДОВ
Р.Н. Рудаков, Ю.И. Няшин, А.Р. Подгаец, А.Ф, Лисовский, С.А. Михеева
(Пермь, Чайковский; Россия)
Рассмотрено несколько видов спорта - настольный теннис, волейбол, скоростной спуск на лыжах, прыжки с трамплина. На движение мяча в двух первых видах спорта и движение спортсменов в горнолыжных видах спорта существенное влияние оказывают аэродинамические силы. В задаче об игре в настольный теннис учтено взаимодействие мяча со столом и ракеткой. Проведено математическое моделирование игры двух игроков, один из которых играет накатом, другой -подрезкой. Выявлены особенности игры - большая скорость движения ракетки при игре накатом, а при игре подрезкой спортсмену приходится больше перемещаться по площадке. При игре в волейбол исследована так называемая силовая подача, при которой спортсмен ударяет по мячу в фазе прыжка в зоне своей площадки. Показано влияние координат точки удара (её высоты и расстояния от линии площадки по горизонтали) на эффективность силовой подачи. Проанализированы видеозаписи силовой подачи.
Применен способ косвенного определения аэродинамического сопротивления движению спортсмена. Был снят профиль горы на горнолыжной трассе «Снежинка» г. Чайковского, а также экспериментально определены длительности спуска 4-х спортсменов в 3-х стойках: высокой, средней и низкой. Методом наименьших квадратов получено уравнение профиля горы. Составлена математическая модель движения лыжника при «гладком» спуске (без поворотов) с учётом силы трения лыж о снег и аэродинамических сил. При известном коэффициенте трения скольжения был найден определяющий силу сопротивления воздуха аэродинамический параметр по сопоставлению времени спуска, найденного в численном моделировании, с экспериментальными данными. Наибольшие трудности возникают при определении аэродинамических коэффициентов лобового сопротивления и подъёмной силы в фазе полёта при прыжке спортсмена с трамплина. Известные экспериментальные данные относятся к технике прыжка на лыжах 60-х годов. Косвенное определение
аэродинамических коэффициентов по интегральным характеристикам прыжка -дальности и времени полёта даёт недостаточную информацию, об их зависимости от угла атаки лыж и спортсмена. И, наконец, математическое моделирование обтекания лыжника потоком воздуха возможно лишь на достаточно простых моделях системы лыжник-лыжи. В настоящей работе из дифференциальных уравнений движения центра масс системы лыжник-лыжи получены аэродинамические коэффициенты лобового сопротивления и подъёмной силы. Для этого в каждый момент времени надо знать компоненты скорости и ускорения центра масс системы.
Предложенная в работе методика исследования спортивных движений и результаты исследований могут быть применены для совершенствования техники выполнения этих движений в рассмотренных видах спорта. Библ. 13.
Ключевые слова: аэродинамические силы, настольный теннис, волейбол, скоростной спуск на лыжах, прыжки с трамплина
Received 17 December 2000
