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Optimal flight technique for V-style ski jumping
K. Seo*, M. Murakami† and K. Yoshida†
*Faculty of Education, Yamagata University, Japan
†
Institute of Engineering Mechanics and Systems, University of Tsukuba, Japan
Abstract
This paper describes the results of a numerical study to determine the optimal flight technique
for V-style ski jumping. We attempt to answer the question of how a jumper should fly in order
to increase their flight distance in a V-style posture. The index of performance in this optimization study is the horizontal flight distance, which is mathematically equivalent to the total flight
distance, and the control parameters are the ski-opening angle and the angle of forward lean that
the jumper employs (the body–ski angle). The flight trajectory is simulated on the basis of an
aerodynamic database constructed from wind tunnel test data. It is found that the ski-opening
angle should be increased in the first half of the flight, and then maintained at a constant value
during the rest of the jump. Optimal control of the ski-opening angle when there is either a
headwind or tailwind is also discussed. It is found that the jumper needs to control the skiopening angle over a wider range in the case of a headwind than in the case of a tailwind. The
jumper’s skill in controlling the ski-opening angle is very important for increasing the flight
distance, especially in the case of a tailwind.
Keywords: angle of forward lean, headwind, optimization study, ski jumping, ski-opening angle
Introduction
There have been several research studies that have
looked into optimizing the flight of ski jumpers. A
simple optimization study was made (Remizov, 1984)
in which the angle of attack was chosen as a control
parameter for the cases where the pitching moment
was ignored and the two skis were held parallel. Jin et
al. (1995) carried out a wind tunnel test using a scale
model (at 40%) for three different jumping postures,
the parallel style, the V-style and the flat V-style. In VCorrespondence address:
Kazuya Seo
Faculty of Education, Yamagata University,
1-4-12 Kojirakawa,
Yamagata 990-8560, Japan
Tel: +81-23-628-4350
Fax +81-23-628-4450
E-mail: [email protected]
© 2004 isea
Sports Engineering (2004) 7, 00–00
style flight, the angle between the body and the skis
was fixed at 30°. The flat V-style was named after the
posture in which the body is situated in the plane that
also includes the two skis. They pointed out that it was
advantageous to adopt the flat V-style in the first half
of flight and then to transfer to the V-style during the
remainder of the flight. Yoshida (1998) carried out an
optimization study on the basis of the aerodynamic
data published by Watanabe & Watanabe (1993). The
angle of the forward lean was chosen as a control
parameter, under conditions where the ski-opening
angle was held constant at 36°. They took account of
the pitching moment as well as the drag and the lift in
the equation system that they used to characterize the
flight dynamics.
Our study is divided into two parts. We first
obtained a body of experimental data to construct an
‘aerodynamic database’, which has already been
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Optimal flight technique for V-style ski jumping
Seo et al.
reported in our recent paper (Seo et al., 2004). This
second part is devoted to a study in which we attempt
to optimize the distance of flight of a ski jumper by
utilizing the aerodynamic database. We also intend to
demonstrate the practical validity of the database in
the present study. The primary objective of this study
is to give an answer to the question of how to control
the ski-opening angle from the start of the jump to the
finish to achieve the longest flight distance. The skiopening angle and the angle of forward lean are
controlled to achieve optimization. We also discuss
how a jumper should adjust the ski-opening angle in
the case of a headwind or tailwind in order to best
increase the flight distance.
present model, the whole body is assumed to be
situated on a single plane, which means that the waist
bend angle is zero. The flight path from take-off
(t = 0) to a specified time t is given by:
X =
Y =
∫
∫
t
Uj cos βj dt
t
Uj sin βj dt
If t is equal to the landing time tf, this will give the
landing position. The relative velocity Ur and the
relative flight angle βr are defined by:
Ur = √(Uj cos βj – Uw)2 + (Uj sin βj )2
Uw , β j & β r < 0
X
Uw
φ
βj
βr
Jumping hill
Uj
Ur
Y
Figure 1 Co-ordinate system and definition of angles of βj, βr and
φ. U is the velocity of the centre of gravity. The suffixes j, r and w
indicate the motion of the jumper with respect to the ground and
to air, respectively, and the wind.
2
βr = tan–1
(
Uj sin βj
Uj cos βj – Uw
)
(3)
(4)
The equations of motion in the directions parallel and
normal to the flight path are expressed respectively by:
m
mUr
δUr
= mg sin βr – D
δt
(5)
δβr
= mg cos βr – L
δt
(6)
Here, D and L represent the drag and the lift, and m
and g are the mass of the body–ski combination and
the acceleration due to gravity. The equation of the
pitching moment around the center of gravity of the
body–ski combination is
I δω = M
δt
Y
(2)
0
Basic equations
It is assumed that the motion of the body–ski combination occurs in a fixed vertical plane, as shown in
Fig. 1. The origin is defined as the take off point,
while the X-axis is in the horizontal direction and the
Y-axis is in the vertical direction. The velocity of the
jumper (the body–ski combination) is expressed by Uj,
the wind velocity by Uw, and the relative velocity with
respect to the air by Ur . For simplicity, the wind is
assumed to blow horizontally. Since a tailwind is
defined as a positive wind velocity Uw, Uw in Fig.1
(headwind) is defined as being negative. The flight
angles βj and βr are the angles of Uj and Ur, both with
respect to the X-axis. The flight angles βj and βr in
Fig.1 are also negative. The angle between the plane
of the body and the X-axis is expressed by φ. In the
(1)
0
ω =
δφ
δt
(7)
(8)
Here, I is the moment of inertia of the body–ski combination and ω is the angular velocity of the body
angle φ. A definition of the characteristic parameters
of the body–ski combination is given in Fig. 2. The
angle of attack, which is the angle between the skis and
the direction of Ur, is defined as α. The ski-opening
angle is denoted by λ. The aerodynamic drag, lift and
moment in Eqns. (5) through (7) are proportional to
Ur2, which were all obtained during wind tunnel tests
as functions of α, θ and λ (Seo et al., 2004). The wind
Sports Engineering (2004) 7, 00–00 © 2004 isea
Seo et al.
λ
M
L
φ
D
θ
α
mg
Uw
βj
βr
Uj
Uw , β j & β r < 0
Figure 2 Definition of characteristic parameters.
tunnel data were acquired for α at 5° intervals between
0° and 50°, and for θ at intervals of 10° between 0° and
40°, respectively. The ski-opening angle λ was set at
one of 0°, 10° and 25°. The torso and legs of the
model were always set in a straight line. The tails of
the skis were always in contact on the inner edges.
The relationship between the four angles (φ, α, βr and
θ) can be expressed by Eqn. (9):
φ = α + βr + θ
(9)
Here, βr is negative, as shown in Figs. 1 and 2.
Optimization
Eqns. (1), (2), (5) through (8) are converted into a
system of simultaneous differential equations for the
optimization calculation:
.
x =
.
x1
.
x2
.
x3
.
x4
.
x5
.
x6
Uj cos βj
Uj sin βj
–g sin βr – D/m
=
(L/M – g cos βr) /Ur
ω
M/I
(10)
Here the state variables as a function of time can be
given as follows:
x =
x1
x2
x3
x4
x5
x6
© 2004 isea
=
X
Y
Ur
βr
φ
ω
Sports Engineering (2004) 7, 00–00
(11)
Optimal flight technique for V-style ski jumping
It has been reported that initial take-off conditions,
such as the take-off speed and the direction of takeoff, can greatly affect the subsequent flight
(Bruggemann et al., 2002). However, since there are
no experimental aerodynamic data for the high values
of θ that correspond to the posture in the initial flight
phase immediately after take-off, the calculation must
begin at 0.4 s after a take-off in the present computation. It has been reported that it takes about 0.4 s after
take-off to transfer to a stable flight regime
(Kobayakawa & Kondo, 1985). Therefore, it is
assumed that the take-off speed is 25 ms–1 in the
direction of the take-off slope, to which is added
2.3 ms–1 due to the jumper’s take-off action at an angle
of 35° upward from the direction of the slope at
t = 0 s, and that no aerodynamic force acts on the
jumper in the gravity field during the initial 0.4 s. By
considering ballistic flight in a vacuum for 0.4 s, a set
of initial conditions for 0.4 s after take-off can be
given for the Okurayama hill at Sapporo, as follows:
x(0.4) =
10.7 m
– 2.3 m
27.8 ms–1
–16.2
20.0°
– 20.0° s–1
(12)
The problem of determining optimal control conditions requires that we calculate the time variation of
the controlling parameters. The angle of forward lean
θ and the ski-opening angle λ were chosen as the
control parameters in order to achieve the longest
flight distance for a given set of initial values, x1 (0.4)
through x6 (0.4). For the landing condition, the height
of the jumper should be equal to that of the jumping
hill, as given by the expression for the height y
between the centre of gravity of the jumper and the
jumping hill:
ψ = G(x1(tf )) – x2(tf ) = 0
(13)
where Y = G(X) is the profile of the Okurayama
jumping hill at Sapporo. This landing condition leads
to a slightly longer flight distance than occurs in
reality because the centre of gravity of the jumper is
situated above the landing slope by several tens of centimetres in realistic cases. The index of performance J
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Optimal flight technique for V-style ski jumping
Seo et al.
for the maximum flight distance is defined as:
J(θ, λ) = –x1(tf ) +
1
c ψ2
2
(14)
The goal of the optimization technique is to maximize
x1(tf ). In the optimization calculation, J is minimized
instead of maximizing x1(tf ). The coefficient c is a
penalty coefficient, which is defined as c = 104 in this
calculation to assist the numerical optimization
technique. If ψ2 is not small, J cannot be minimized
because of the large value of c. It should be added that
though it is x1(tf ) that is optimized in the computation,
the procedure is mathematically equivalent to optimizing the flight distance. A numerical solution is
obtained by applying the conjugate gradient method,
the details of which appears in the references (Kanoh,
1988).
The optimization of the ski-opening angle
Angles (°) & Angular velocity (° s –1)
The optimal solution of the ski-opening angle λ for
the longest flight distance is shown in Fig. 3, as well as
other time variations of α, βj, φ and the angular
velocity ω. The initial condition is given by Eqn. (12).
It is assumed that the angle of forward lean θ is kept at
a constant value of 7° under the zero wind condition.
The flight distance is about 139 m, which is 8 m
longer than that of a fixed posture at λ = 25° and
θ = 7°. It is found that a jumper should increase λ in
the early stages of the flight phase, and then keep a
constant value of around 26°. The angle of attack α
weakly oscillates around the trim angle due to the
features of the pitching moment data, as mentioned in
our previous paper (Seo et al., 2004). Although the skiopening angle λ only varies slightly (except in the early
stages), an oscillating variation can be recognized,
with peaks that almost correspond with those of α.
There is a trough at around 2.6 s and two peaks at
around 1.5 s and 3.5 s. It is found that the trough in λ
allows the maximum value of α to prevent stalling. On
the other hand, λ peaks around the minima of α,
resulting in more lift being achieved. It seems that the
ski-opening angle λ and the angle of attack α are complementary to each other. It is necessary to have a
.
negative angular velocity ω(= φ) at the beginning, i.e.
a nose-down angular velocity to decrease in φ, in order
to make the drag small in the early stages. The body
angle φ decreases in the early stages because of the
negative value of ω, so α also decreases slightly. The
flight trajectory and seven body postures that were
identified in the case of the optimized solution are
shown in Fig. 4, where the profile of the Okurayama
jumping hill is also shown. The body posture is drawn
at equal time intervals. The first ski-opening angle is
slightly smaller than the others, which are almost
independent of time.
Fig. 5 shows the time variations of the aerodynamic
forces for the optimized solution. The drag area SD is
defined as the drag divided by the dynamic pressure
40
30
λ
20
α
10
θ
0
φ
–10
ω
–20
–30
βj
–40
0
1
2
3
Time (s)
4
5
Figure 3 Optimized time variations of angles (α, βj, φ, λ) and the
angular velocity ω. The angle of forward lean θ is assumed to
maintain a constant value of 7°.
4
Okurayama jumping hill
Figure 4 Optimized flight trajectory and body postures through the
control of λ in the case of the Okurayama Jumping Hill.
Sports Engineering (2004) 7, 00–00 © 2004 isea
Seo et al.
0.8
0.06
Optimal flight technique for V-style ski jumping
35
SL
0.04
–1
30
SD
0.4
0.02
λ (°)
0
QM / m3
SD & SL / m 2
0.6
Uw = –1 ms
1
25
QM
0.2
0
0
1
2
3
4
5
0
20
-0.02
15
Time /s
Figure 5 Optimized time variations of the drag area SD, the lift area
SL and the moment volume QM. The angle of forward lean θ is
assumed to maintain a constant value of 7°.
0.5ρUr2. The lift area SL and the moment volume QM
are defined in the same manner (Seo et al., 2004). It
can be seen that the time variations of SD and SL are
similar to that of α. This is because these aerodynamic
forces are monotonous functions of α when α is
smaller than the stall angle. The lift area SL must be
large in the latter half of flight while keeping λ large to
achieve longer flight distances. The drag area SD
should be small in the early stages to reduce drag. The
time variation of α results from the stable pitching
oscillation around the trim angle of attack due to the
functional form of QM(α), as discussed in our previous
paper (Seo et al., 2004).
Three optimal control results are shown in Fig. 6
for the ski-opening angle λ under three different wind
conditions to determine the effect of wind on the
flight distance . The angle of forward lean θ is still
assumed to have a constant value of 7°. The optimized
flight distance results are about 142, 139 and 138 m
for Uw = –1, 0 and 1 ms–1, respectively. The headwind
produces a longer flight distance. The differences
between the optimized flight distance and that of a
fixed posture at λ = 25° and θ = 7° that is not
different from the optimized posture are 7, 8 and
11 m, respectively for Uw = –1, 0 and 1 ms–1, as shown
in Table 1. The jumper’s skill in controlling the skiopening angle is more important in the case of a
tailwind. In the case of a headwind, the jumper should
control λ such that it is smaller than in other cases in
the first half of the jump in order to make the drag
© 2004 isea
Sports Engineering (2004) 7, 00–00
0
1
2
3
4
5
Time (s)
Figure 6 Optimized time variations of the ski-opening angle λ. The
wind speeds of 1, 0 and –1 ms–1 are denoted by Uw = –1, 0 and
1 ms–1, respectively. The angle of forward lean θ is assumed to
remain constant at 7°.
Table 1 Flight distance comparison.
Uw /ms–1
–1
(Headwind)
0
1
(Tailwind)
Optimized flight
distance /m
Flight distance
under the
condition of a
fixed posture
(λ = 25°
& θ = 7°) /m
Difference of the
flight distance /m
142
135
7
139
131
8
138
127
11
small, and then make λ large in the second half to
produce greater lift. Since the aerodynamic forces are
proportional to the square of the relative velocity with
respect to air, the largest forces act on the jumper in
the case of the headwind. It is necessary for a jumper
to control λ over a wider range (about 12°) in the case
of a headwind. In the case of a tailwind, λ decreases
slightly during a flight, with a control range of λ being
just 3°.
The optimization of the angle of forward lean
Fig. 7 shows the optimal solution of the angle of
forward lean θ during the course of a jump, as well as
the time variations of α, βj, φ and the angular velocity
ω. It is assumed that the ski-opening angle λ is fixed at
a constant value of 25° under the zero wind condition.
5
Seo et al.
40
10
30
α
20
λ
9
8
θ
ω
φ
10
0
θ (°)
Angles (°) & Angular velocity (° s –1)
Optimal flight technique for V-style ski jumping
-10
–1
0
6
–1
-20
5
-30
-40
βj
0
1
2
3
4
5
Time (s)
Figure 7 Optimized time variations of angles (α, βj , φ, θ) and of
angular velocity ω. The ski-opening angle λ is assumed to remain
constant at 25°.
It is practically impossible for a jumper to precisely
control θ in the flight phase in the manner shown in
Fig. 7, because the jumper is connected to the skis by
toe bindings at the toe and connecting chords at the
heel. The jumper can only adjust the length of the
connecting chords at the heels, which determine the
minimum θ. The optimum constant value of θ could
be found by using this kind of thought process. It is
found that a jumper should rapidly decrease the value
of θ at the beginning of the jump, and then maintain it
at about 6° from 1 s into the jump until the finish.
Therefore, the best constant value of θ is about 7° on
average. The flight distance is about 132 m, which is
7 m shorter than the result of using an optimized value
of λ under the zero wind condition. Therefore, optimizing control of θ is not as effective as optimizing
control of λ. Both the angle of forward lean θ and the
angle of attack a should be small at the beginning,
which both contribute greatly to reducing drag.
Although variations in θ are rather small after 1 s into
the jump to the finish, the weak peaks of θ seem to
almost correspond to those of α, just like the relationship between λ and α that is presented in Fig. 3.
Fig. 8 shows the optimized variation of θ for three
different wind speeds. Qualitatively there is no major
difference in the time variation between these three
conditions. In the case of the tailwind, where
Uw = 1 ms–1, the jumper should make θ a little bit
larger than in the case of a headwind, where Uw = –1.
Since the relative velocity with respect to air is smaller
6
Uw =1 ms
7
4
0
1
2
3
Time (s)
4
5
Figure 8 Optimized time variations of the angle of forward lean θ.
The ski-opening angle λ is assumed to remain constant at 25°.
The headwind of 1 ms–1 is denoted by Uw = –1, the tailwind of
1 ms–1 is denoted by Uw = 1, and the zero wind condition is
given by Uw = 0.
in the case of a tailwind, it is necessary to make θ
larger in order to maximize the lift in the latter half of
the jump.
Influence of headwind on the flight distance
The flight distance data shown in Fig. 9 is obtained by
solving Eqn. (10) using the Runge-Kutta method for a
fixed posture under the initial condition given by
Eqn. (12). Fig. 9 shows the effect of the wind Uw on
the flight distance. The angle of forward lean θ and
the ski-opening angle λ are assumed to take constant
values of θ = 5, 7, 10.5° and λ = 25°, respectively. It
can be seen that the stronger headwind yields the
longer flight distance for the range of tailwinds, and
vice versa. This is because lift is increased in the case
of a headwind. The wind speed in a competition is
normally less than 4 ms–1. If the wind speed is less than
1 ms–1, which is the typical condition in most competitions, jumpers should adjust the angle of forward lean
to about 7°. The flight distance would be about 135 m
for Uw = –1, 131 m for Uw = 0, and about 127 m for
Uw = 1 in the case of θ = 7°. Therefore, the relative
gain (or loss) in flight distance is equivalent to 4 m for
a wind speed of –1 (+1) ms–1. This can be summarized
by saying that optimization brought about by the
jumper’s skill is greater than having a headwind speed
of 1 ms–1 to increase the flight distance, as shown in
Table 1. There are inflection points in the flight
Sports Engineering (2004) 7, 00–00 © 2004 isea
Seo et al.
Acknowledgements
150
140
Flight distance (m)
130
θ
10.5°
120
110
100
7°
90
Tailwind
Headwind
5°
80
70
–5
–4
–3
–2
–1
U
0
w
1
2
3
4
5
(ms –1)
Figure 9 Flight distance as a function of wind speed Uw. The angle
of forward lean θ and the ski-opening angle λ are assumed to
remain constant at θ = 5, 7, 10.5° and λ = 25°, respectively.
distance curves at around Uw = 3 for θ = 5° and
Uw = 4 for θ = 7°. The reason why the slope of the
curve becomes milder for larger tailwind speeds is
because a large tailwind pushes the jumper forward,
even though the lift becomes smaller.
Conclusions
From the results of the optimal solution to the
question posed in the introduction, we can now
summarize the key factors to achieving longer flight
during a jump.
1 The ski-opening angle should be smaller in the
first half of the jump to reduce drag, and then it
should be kept around a constant value in the
second half of the jump to maximize lift.
2 In increasing flight distance, optimal control of
the ski-opening angle is more important than
control of the angle of forward lean.
3 A jumper’s skill in controlling the ski-opening
angle is more important in the case of a tailwind
than in the case of a headwind.
4 A jumper needs to control λ over a wider range in
a headwind.
5 The angle of forward lean should be maintained
at around 7° during the flight if the wind speed is
less than 1 ms–1.
© 2004 isea
Optimal flight technique for V-style ski jumping
Sports Engineering (2004) 7, 00–00
We would like to thank Prof. Gary A. Williams,
Department of Physics and Astronomy, UCLA for
fruitful discussions and suggestions. We also would
like to thank Kaori Ota, Faculty of Education,
Yamagata University, for her support. This work is
supported by the Descente and Ishimoto Memorial
Foundation for the Promotion of Sports Science and
the Ministry of Education, Science, Sports and
Culture, Grant-in-Aid for Scientific Research
(13750840, 2001 & 15700404, 2003).
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