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Flight style optimization in ski jumping on normal, large, and ski flying hills
Journal of Biomechanics 47 (2014) 716–722
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Flight style optimization in ski jumping on normal, large, and ski flying hills Alexander Jung a, Manfred Staat a, Wolfram Müller b,n a b
Institute of Bioengineering, Aachen University of Applied Sciences, Heinrich-Mußmann-Straße 1, 52428 Jülich, Germany Medical University of Graz, Institute of Biophysics, Harrachgasse 21/4, 8010 Graz, Austria
art ic l e i nf o
a b s t r a c t
Article history: Accepted 18 November 2013
In V-style ski jumping, aerodynamic forces are predominant performance factors and athletes have to solve difficult optimization problems in parts of a second in order to obtain their jump length maximum and to keep the flight stable. Here, a comprehensive set of wind tunnel data was used for optimization studies based on Pontryagin′s minimum principle with both the angle of attack α and the body-ski angle β as controls. Various combinations of the constraints αmax and βmin(t) were analyzed in order to compare different optimization strategies. For the computer simulation studies, the Olympic hill profiles in Esto-Sadok, Russia (HS 106 m, HS 140 m), and in Harrachov, Czech Republic, host of the Ski Flying World Championships 2014 (HS 205 m) were used. It is of high importance for ski jumping practice that various aerodynamic strategies, i.e. combinations of α- and β-time courses, can lead to similar jump lengths which enables athletes to win competitions using individual aerodynamic strategies. Optimization results also show that aerodynamic behavior has to be different at different hill sizes (HS). Optimized time courses of α and β using reduced drag and lift areas in order to mimic recent equipment regulations differed only in a negligible way. This indicates that optimization results presented here are not very sensitive to minor changes of the aerodynamic equipment features when similar jump length are obtained by using adequately higher in-run velocities. However, wind tunnel measurements with athletes including take-off and transition to stabilized flight, flight, and landing behavior would enable a more detailed understanding of individual flight style optimization. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Ski jumping Optimal control Computer simulation Aerodynamics
1. Introduction The aerodynamic strategy of the athlete has become a predominant performance factor in ski jumping since the “V-style” was introduced by Jan Boklöv in 1985 (Müller, 2009). In ski flying, jump lengths far beyond 200 m have been exceeded many times, the present world record being 246.5 m (J.R. Evensen, Norway, 2011). At a given hill, jump length determinants are the in-run velocity, the take-off velocity perpendicular to the ramp, the weight, and the aerodynamic forces which depend on the equipment and on the flight style (Straumann, 1927; König, 1952; Denoth et al. 1987; Schwameder and Müller, 1995; Müller et al., 1995, 1996). The take-off phase is crucial (Virmavirta et al., 2001; Vaverka et al., 1997; Müller, 2009), but not only in terms of linear momentum: it is the angular momentum which enables the athlete to obtain an advantageous flight position as soon as possible (Arndt et al., 1995; Schwameder and Müller, 1995; Vaverka et al., 1997; Virmavirta et al., 2005, 2009). The body and
n
Corresponding author. Tel.: þ 43 316 380 3913. E-mail address: [email protected] (W. Müller).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.021
equipment configuration throughout the flight determines the drag and lift areas and thereby substantially the jump length (Müller et al., 1995, 1996; Müller, 2009; Schmölzer and Müller, 2002, 2005). The time courses of lift and drag areas can largely be influenced by the athlete by altering the angle of attack α and the body-ski angle β (Fig. 1). Remizov (1984) was the first to publish an optimization study. He applied Pontryagin's minimum principle (Pontryagin et al., 1962) to a simplified model of a ski jumper, using α(t) as the control function. The body-ski angle β(t) is at least as important as α(t) for obtaining a large jump length (Müller, 2009). Both time functions are also essential for stabilizing the flight as they determine the pitching moment together with the angle of the skis to each other (V-angle), which varies during the stabilized flight phase. Individual mean values ranged between about 251 and 401 (Müller et al. 1996; Schmölzer and Müller, 2002, 2005; Müller, 2009). The hip angle γ should be about 1601 as drag is low and lift is at its maximum in this position (Schmölzer and Müller, 2005). During the Olympic Games 2002, Schmölzer and Müller (2005) investigated individual flight styles. They found distinct jump length maximization strategies between the Olympic medalists,
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Fig. 1. Flight position of a ski jumper: angle of attack (α), body-ski angle(β), hip angle (γ), and angle of the skis to each other (V).
although jump lengths were close to each other which implies that there exist several optima which may result in similar jump lengths. Besides the choice of D- and L-functions by means of respective flight positions to maximize jump length, athletes also have to consider their individual aerodynamic features and biomechanical, sensory-motor and mental abilities, and, most importantly: flights have to be kept stable (Müller et al., 1996, Schmölzer and Müller, 2005). Athletes need to solve these extremely difficult optimization problems in “real time”. Here, a comprehensive set of wind tunnel data obtained with a reduced scale model (Meile et al., 2006) which was calibrated with measurement results performed in a large scale wind tunnel (Schmölzer and Müller, 2002) was used for optimization studies based on Pontryagin′s minimum principle with both α and β as controls. Various combinations of the constraints αmax and βmin(t) were analyzed for comparison of different optimization strategies. The Olympic hill profiles in Esto-Sadok, Russia (HS 106 m and HS 140 m), and Harrachov, Czech Republic, host of the Ski Flying World Championships 2014 (HS 205 m), were used. In order to anticipate reduced lift (L) and drag areas (D) caused by tighter and thinner jumping suits which are compulsory due to recent FIS ski jumping equipment regulations, optimization studies were also made using reduced L and D values.
2. Methods 2.1. Computer model of a ski jumper´s flight phase The inrun velocity ν0 as well as the take-off velocity perpendicular to the ramp are initial conditions for the flight phase. During the flight the gravitational force Fg ¼ mg (m¼ 72 kg, g ¼ 9.81 ms 2), the aerodynamic forces drag F d ¼ ρ=2 Dw2 and lift F l ¼ ρ=2 Lw2 determine the flight path of a ski jumper′s center of gravity (Fig. 2). The relative wind vector w is the sum of the external wind vw and the velocity v of the ski jumper: w¼vw v. We assume a plane problem with no crosswind so that w¼ (wx 0 wz)T. The air density is a function of the air pressure with ρ ¼ (p)/(RT), where T is the 1 absolute temperature and R ¼ 288:3 JK 1 kg the gas constant. The air density is ρ ¼ 1:15 kgm 3 (ICAO norm-atmosphere) at 650 m above sea level (Esto-Sadok: 600 m, Harrachov: 700 m). D ¼cdA and L ¼ clA are the drag and lift areas which can be measured in a wind tunnel. As introduced by Müller et al. (1995, 1996) tabulated functions D ¼D(t) and L ¼ L(t) based on field analyses and on wind tunnel measurements were used to consider the changing postures and angles of attack during the flight. Meile et al. (2006) performed comprehensive series of wind tunnel measurements (crosssection area: 2.00 1.46 m2) with a reduced scale model (1:(2)1/2) of a ski jumper. Range of measurements for α:20–451, for β:0–201. The hip angle γ and the V-angle were held constant at the advantageous angles 1601 and 351, respectively (Schmölzer and Müller, 2002). Data were calibrated by comparisons to measurements obtained with a 1:1 ski jumper model (height: 1.78 m, ski length: 2.60 m) in a large scale wind tunnel (cross-section area: 5 5 m2) (Schmölzer and Müller, 2002). Calibration factors fd for drag and fl for lift areas are 2.219 (SD ¼ 0.046) and 2.244 (SD ¼0.050), respectively. Both values are higher than 2.000 because in the large scale wind tunnel a highly realistic model of a ski jumping in full gear was used, whereas the reduced scale model was composed of simplified geometries. The data for D(α,β) and L(α,β) were fitted to bicubic polynomials:
Fig. 2. Flight path of a ski jumper and hill profile. Hill parameters for Esto-Sadok HS 106 m (No. 383/RUS 4), HS 140 m (No. 384/RUS 5), and Harrachov HS 205 m (FLY5/ CZE) can be found in the FIS Certificates of Jumping Hills. Hill profiles were modeled piecewise. v0 is the in-run velocity and vp0 the velocity perpendicular to the ramp. θ is the angle of the relative wind vector w and φ is the flight path angle. Lðα; βÞ ¼ b0 þ b1 α þ b2 β þ b3 α2 þ b4 β þ b5 α3 2
þ b6 β þ b7 αβ þ b8 α2 β þ b9 αβ : 3
2
This resulted in maximum errors of 1.64 10 2 m2 for D(α,β) and 1.27 10 2 m2 for L(α,β), respectively. The coefficients ak and bk are given in Appendix A. Straumann (1927) was the first to publish the equations of motion of a ski jumper during the flight phase. Since then, several authors have applied these equations or modified versions of them to analyze ski jumping (Müller, 2009). Transformed to a first order system the differential equations of motion read 3 2 ρw 2_ 3 ½ vx 2m Dðvwx vx Þþ Lðvwz vz Þ 7 6 ρw 6 v_ z 7 6 2m ½Dðvwz vz Þ Lðvwx vx Þ g 7 7 6 7 6 s_ ¼ 6 x_ 7 ¼ 6 7; vx 7 4 5 6 5 4 vz z_ with s(t) being the state matrix of the system and w¼((vwx vx)2 þ(vwz vz)2)1/2. The simulations done here use the modeling approach developed by Müller et al. (1996) which enables consideration of aerodynamic effects on the trajectory due to changing flight position angles. Initial conditions are s(0) ¼(vx(0),vz(0),x(0),z (0))T with v ¼v0 þ vp0. Inrun velocities used for Esto-Sadok HS 106 m, HS 140 m, and Harrachov HS 205 m are v0 ¼ 25.0 ms 1, 26.5 ms 1, 28.5 ms 1, respectively (according to FIS Certificates of jumping hills). A take-off velocity perpendicular to the ramp of vp0 ¼ 2:5 ms 1 was used (Virmavirta et al., 2001; Müller, 2008). In all simulations vw was set to zero (calm wind). At the flight time T the ski jumper´s flight path intersects the hill profile Ph:z(x) at the landing point so that the height above ground is h(x(T),z(T))¼ 0. 2.2. Optimization
νp0
Dðα; βÞ ¼ a0 þa1 α þ a2 β þ a3 α þ a4 β þa5 α 2
2
þa6 β þ a7 αβ þa8 α2 β þ a9 αβ ; 3
2
3
Besides trying to maximize jump length, the athlete has to balance the pitching moment, which depends on his flight posture and the angle of attack and therefore his individual range of admissible angles is limited. In this study, the controls are u (t)¼ (α(t),β(t))T. Both controls α and β are constrained by αmax and βmin(t), respectively. In order to avoid unrealistic position changes of β, the range of admissible β-values was chosen with respect to angles measured in the field (reference jump A in Schmölzer and Müller, 2002) and the abbreviation Δβref A ¼ βref A ðtÞ βmin ðtÞ is introduced. However the values of βmin(t) must not go below a limit of 01, because the tumbling risk is high using such flight positions (Müller et al., 1996). The maximum value of the angle of attack of the skis αmax does not exceed 451 because larger angles have not been observed in the field (Müller et al., 1996, Schmölzer and Müller, 2002, 2005) and because of sudden lift and pitching moment changes were found in wind tunnel measurements with simplified models in the range around 451 (Reisenberger et al., 2004). Optimization studies were made with varying constraints αmax and βmin(t) as listed in Table 1. The constrained optimization problem with free final state and time has the form: Minimize J ¼ ϕðsðTÞÞ ¼ xðTÞ subject to the system of first order dynamic constraints s_ ðtÞ ¼ fðsðtÞ; uðtÞÞ with initial condition sðt 0 Þ ¼ ðvx ðt 0 Þ; vz ðt 0 Þ; xðt 0 Þ; zðt 0 ÞÞT and terminal condition hðxðTÞ; zðTÞÞ ¼ 0
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Table 1 Optimization studies with various boundaries on the controls α and β. Number
αmax [deg]
Δβref
1 2 3 4 5 6 7 8 9
35.0 35.0 35.0 40.0 40.0 40.0 45.0 45.0 45.0
10.0 15.0 10.0 10.0 15.0 10.0 10.0 15.0 10.0
A
[deg]
and the constraints on the controls n o uðtÞ A ðαðtÞ; β ðtÞÞT jαðtÞ r amax ; βðtÞ Z βmin ðtÞ with (
βmin ðtÞ ¼
βref A ðtÞ þ Δβref A ; βmin Z 0 βmin o 0: 0;
A Hamiltonian is constructed for solving the optimization problem: T ℋ sðtÞ; uðtÞ; λðtÞ ¼ λ ðtÞfðsðtÞ; uðtÞÞ; with the adjoint variables λ. Applying Pontryagin's minimum principle (Pontryagin et al., 1962), as done by Remizov (1984), the necessary conditions for u*(t) to be optimal are: ∂ℋðsn ðtÞ; un ðtÞ; λðtÞÞ n s_ ðtÞ ¼ ¼ fðsn ðtÞ; un ðtÞÞ; ∂λ λ ¼ λn n
λ_ ðtÞ ¼
n ∂ℋðsðtÞ; un ðtÞ; λ ðtÞÞ ∂s
s ¼ sn
and n
n
ℋðsn ðtÞ; un ðtÞ; λ ðtÞÞ ¼ minu ℋðsn ðtÞ; uðtÞ; λ ðtÞÞ
The differences between the jump lengths with reference jump A (number 0) and the optimized jumps number 1–9 increase with increasing hill size (Fig. 4). When increasing the range of admissible values of one of the controls α or β, jump lengths become larger, independently of hill size. Fig. 4 shows that jump length increases resulting from all optimization study are more pronounced at larger hills. For example, at the normal hill (HS 106 m) number 9 resulted in an increased jump length of Δl ¼ 6:3 m when compared to the reference jump A, Δl ¼ 12:1 m at the large hill (HS 140 m), and Δl ¼ 40:3 m at the ski flying hill (HS 205 m). A comparison of the 9 optimization studies indicates that several sets of α- and β-time courses result in quite small jump length differences. For example, optimization numbers 3, 5, and 8 for the large hill (broken lines in Fig. 4) resulted in 122.0m, 121.4 m, 122.5 m, and these values are only slightly below the jump length resulting from the best optimization result (number 9) of 125.9 m. This also holds true for optimization studies at the normal and at the ski flying hill. The optimized time courses of the controls α and β for all optimization studies on all 3 hill sizes are shown in Fig. 5a–c. In all 27 optimization studies, the optimal value αn increases up to αmax during the flight. The angle of attack at the beginning of the optimization (t0 ¼0.7 s), termed αn0 , increases when βmin(t) is lowered, and also when the hill size is decreased (compare to Fig. 5a–c and Table 2). The flight times at which αmax has to be reached depend on both the constraints and the hill size. This can be seen in Fig. 5a–c and in Table 3. Fig. 5a–c also shows that the flight time until β has to be kept as low as possible is independent of hill size. In all optimization studies βn decreased as fast as possible towards βn ¼ βmin, and therefore the lowest βn-value was always reached at same flight time (t¼ 1.5 s), independently of hill size. In
for all admissible controls and for all t A ðt 0 ;T n Þ and the transversality condition ∂ϕ ∂h þ υ ; λn ðT n Þ ¼ ∂s ∂s Tn with υ can be obtained by the condition at the final time T n ℋðsn ðT n Þ; un ðtÞ; λ ðT n ÞÞ ¼ 0:
n
The constraints can be enforced by a penalty function (Chong and Żak, 2008): PðuðtÞÞ ¼ ðmax 0; αðtÞ amax Þ2 þ ðmax 0; β min ðtÞ βðtÞ Þ2 : Hereby the optimal controls un(t) can be found from an unconstrained problem by minimizing the convex function n
n
qðc; sn ðtÞ; uðtÞ; λ ðtÞÞ ¼ ℋðsn ðtÞ; uðtÞ; λ ðtÞÞ þ cPðuðtÞÞ with the penalty parameter c ¼ 1010 . The optimal controls un(t) are global maxima as well as the associated jump lengths l. The optimization algorithm starts at t0 ¼ 0.7 s. With an initial guess u0(t) for i¼ 0 it has the following structure:
Fig. 3. Comparison of the flight path obtained with the reference jump A and with the optimized jump number 9. P-Point: 106 m, K-Point: 125 m, and L-Point: 139 m.
1. Solve s_ ðtÞ ¼ fðsi ðtÞ; ui ðtÞÞ forward in time by Heun's method until h(xi(Ti), zi(Ti))¼ 0 and sn(t0) obtained by solving the equations of motions with s (0) up to t0 using reference jump A (Schmölzer and Müller, 2002). _ i ðtÞ ¼ ð∂ℋ=∂sÞðsi ðtÞ; ui ðtÞ; λi ðtÞÞ backward in time by Heun's method 2. Solve λ and λi(Ti). i 3. minu qðc; si ðtÞ; ui ðtÞ; λ ðtÞÞ for all tA (t0,Ti) with Newton's method. 4. Set i¼i þ1 and go to step 1 until J has converged to the minimum value. i
3. Results Fig. 3 shows flight path simulations using the profile of the large hill in Esto-Sadok (HS 140 m). The solid line represents the flight path of the reference jump A (Schmölzer and Müller, 2002) and the broken line the flight path of the optimized jump number 9 with the widest range of admissible values (Table 1). The jump length obtained with reference jump A was 113.8 m, and 125.9 m resulted for the optimized jump, i.e. an increase of 12.1 m.
Fig. 4. Jump lengths resulting from the optimization studies listed in Table 1 for all 3 hill sizes. Optimization study number 0 denotes reference jump A.
A. Jung et al. / Journal of Biomechanics 47 (2014) 716–722
Fig. 5. Optimized time courses for α and β. (a) Normal hill (HS 106 m), (b) Large hill (HS 140 m), (c) Ski flying hill (HS 205 m). Numbers of optimization studies according to Table 1.
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Table 2 The optimized angle of attack αn0 at t0 ¼ 0.7 s as a function of the optimization study number and at all 3 hill sizes. Number
HS 106 m αn0 [deg]
HS 140 m αn0 [deg]
HS 205 m αn0 [deg]
1 2 3 4 5 6 7 8 9
22.2 23.5 25.1 22.1 23.3 24.8 22.0 23.2 24.6
19.5 20.8 22.4 19.4 20.6 22.2 19.4 20.5 21.9
15.9 16.9 18.6 15.6 16.6 18.2 15.6 16.4 17.9
Fig. 6. Optimized jump 9 compared to a modified version with 10% reduced drag and lift areas throughout the whole flight. To obtain a jump length of 125.9 m, an in-run velocity of 27.0 ms 1 is used here (HS 140 m).
Table 3 Occurrence times t αmax and t βn 4 βmin in [s] for all 9 optimization studies at all 3 hill sizes. Number
1 2 3 4 5 6 7 8 9
HS 106 m
HS 140 m
HS 205 m
t αmax
t βn 4 βmin
t αmax
t βn 4 βmin
t αmax
t βn 4 βmin
1.76 1.47 1.27 2.39 1.99 1.66 2.80 2.38 2.03
2.11 1.70 1.43 3.05 2.48 2.12 / / 3.22
2.30 1.93 1.58 3.05 2.61 2.21 2.53 3.08 2.69
2.73 2.24 1.86 3.10 3.19 2.78 / 4.46 4.06
3.11 2.75 2.34 4.03 3.66 3.24 4.51 4.32 3.91
3.64 3.15 2.73 4.73 4.32 3.96 / 5.60 5.39
the flight phase from t βn 4 β , βn increases up to the landing point min of the optimized jump. The time t βn 4 β increases with both min higher αmax and at larger hills, but it decreases with lower βmin(t) (Table 3). The lower αmax the higher the increase of βn, however βn(Tn) is independent from the hill size. Increasing the body-ski angle β in this flight phase results in a jump length increase of 2.4 m, 0.5 m, and 0.0 m for optimization studies 3, 6, and 9 at the normal hill (large hill: 3.5 m, 0.6 m, 0.0 m; ski flying hill: 8.0 m, 1.4 m, 0.0 m). Applying a jump which has been optimized for a given hill to another hill size results in a decreased jump length when compared to the optimal jump obtained for this hill. For example, jump 9 optimized for the large hill (HS 140 m) results in a 1.6 m jump length decrease when applied to the normal hill (HS 106 m) and a decrease of 9.6 m when applied to the ski flying hill. Fig. 6 illustrates that reduced drag and lift areas, e.g. due to the equipment changes, has little effect on the optimal time courses of α and β. A 10% reduction of both drag and lift areas reduced the jump by 9.1 m on the large hill (HS 140 m). In order to reach the same jump length, the in-run velocity was increased to 27.0 ms 1, but αn(t) and βn(t) changed only slightly. Fig. 7a and b compares the drag and lift forces of the optimized jump 9 to the aerodynamic forces occurring with the reference jump A. When compared to reference jump, jump length increases when aerodynamic forces are kept low (due to low α- and βvalues) in the first part of the flight phase and when they increase in a more pronounced way in the second part until this is limited by reaching αmax.
4. Discussion During the transition from take-off to flight, the posture angles and also the angle of attack of the skis with respect to the
Fig. 7. (a) Lift forces and (b) drag forces for reference jump A and optimized jump 9 at the large hill (HS 140 m).
airstream change rapidly (Virmavirta et al., 2005). Adequate series of wind tunnel measurements corresponding to these fast positional changes from take-off until t¼0.7 s are missing. Therefore, this optimization study of the flight phase in ski jumping starts at t¼0.7 s. The reference jump A (Schmölzer and Müller, 2002) which had been developed on the basis of comprehensive field and wind tunnel studies and which was used for various computer simulation studies of ski jumping before (Schmölzer and Müller, 2002; Schmölzer and Müller, 2005; Müller, 2009) was used as the starting basis for these jump length optimization studies by means of Pontryagin's minimum principle (Pontryagin et al., 1962). This approach has already been used before (Remizov, 1984). However, Remizov applied a simplified rigid body model of a ski jumper and only the optimal angle of attack of the skis αn(t) was determined. Here, the optimized time courses of both controls are of predominant importance for the flight phase in ski jumping: the angle of attack αn(t) and the body-ski angle βn(t) were determined. The hip angle and V-angle were kept constant at the value of 1601 and 351, respectively, at which lift is maximal and drag is low (Schmölzer and Müller, 2002). Since the equations of motion can numerically be determined with any desired accuracy, it is the wind tunnel data input which determines the simulation accuracy. Therefore, the optimization study performed here uses the detailed data sets obtained from a reduced scale model (1:(2)1/2) (Meile et al., 2006) which was calibrated by means of highly accurate measurements
A. Jung et al. / Journal of Biomechanics 47 (2014) 716–722
obtained in a large scale wind tunnel using a 1:1 model of a ski jumper with skis (Schmölzer and Müller, 2002). Here, optimization studies were performed without external wind, although this 2D modeling approach is designed to investigate the effects of all initial value and parameter variations which may have an impact on the optimization of the flight. The reference jump (Schmölzer and Müller, 2002) which is used here as the starting point for the optimization represents the time functions of drag and lift area values corresponding to the mean flight position angles of the best 10 athletes measured during several world cup competitions. A detailed representation of the last part of the flight is not included. Therefore, these mean flight positions do not represent the best jump and optimized jumps were substantially larger (Fig. 4). As is to be expected, jump length increases at all 3 hills were largest for the widest admissible control values. The length increases at the normal, large, and ski flying hill with optimized jump number 9 are 6.3 m, 12.1 m, and 40.3m, respectively. This shows that the aerodynamic strategy of the athlete is of predominant importance in ski flying and also plays a leading role on the large hill. The computer model optimization is only limited by the chosen range of admissible values of the controls α and β, whereas – in the real world – these chosen ranges may not be achievable by the athlete for biomechanical and aerodynamic stability reasons. It is of high importance for ski jumping practice that various aerodynamic strategies (combinations of α- and β-time courses) can lead to similar jump lengths (Fig. 4). This enables athletes to win competitions using individual aerodynamic strategies which are suited best for their personal features and abilities. This corresponds to results found in field studies during Olympic Games competitions (Schmölzer and Müller, 2005, Virmavirta et al., 2005). A small decrease of jump length due to a different aerodynamic behavior of the ski jumper can easily be compensated by one of the other performance factors. Even small changes in external wind can easily mask the effects of two different aerodynamic strategies which both would result in comparable jump lengths without wind influence. Consideration of the various parameters which influence jump length substantially (Schwameder and Müller, 1995; Müller et al., 1995; 1996; Schmölzer and Müller, 2002; 2005; Virmavirta et al., 2005; Müller, 2009, Virmavirta et al., 2009) and optimization results found here makes clear that various individual flight styles can have the potential for winning a competition. For example, optimization studies 3, 5, and 8 at the large hill (Fig. 4) resulted in jump lengths which do not differ by more than 1.1 m, although the admissible controls differed substantially in these cases. Athlete's anticipation of the range of control angles possible without taking too high risks (tumbling accidents can easily occur) largely depends on personal experience and on the sensitivity of athlete's sensory-motor system. All 9 optimization studies at all 3 hills showed that the optimized angle of attack αn of the skis increases continuously until αmax is obtained. A continuous increase of αn was also found in the simplified optimization studies of Remizov (1984): αn increased continuously up to 451. The optimization studies result in the advice to reduce the βangles as fast as possible in order to reduce drag rapidly in the first part of the flight. The α-angles should increase continuously (Fig. 5a–c), but not too fast, as this would increase drag. It is interesting to compare field study data which show that this is not the case in real world ski jumping (Müller et al., 1996; Schmölzer and Müller, 2002, 2005; Virmavirta et al., 2005): athletes increase the ski angle of attack rapidly. This appears to be necessary for obtaining a substantial backward rotating torque to stop the angular momentum resulting from the take-off jump.
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In the last part of the flight (Fig. 5a–c), β should increase in order to increase aerodynamic forces when α is limited to a value below 401. This results in a length increase of up to 8.0 m on the ski flying hill. Time courses αn(t) and βn(t) vary on different sized hills because the in-run velocities and therefore the aerodynamic forces differ to each other. This is also important for ski jumping practice. Meanwhile, equipment has changed, in particular the jumping suits are cut tighter and made of thinner material which reduces aerodynamic forces (Müller et al., 1996; Meile et al., 2006). Therefore, results were also compared to optimizations when using reduced drag and lift area values. It can be seen in Fig. 6 that the optimized time courses of α and β using 10% reduced drag and lift areas throughout the flight differ only in a negligible way when jump length is unaltered (due to increase of v0). However, reduced aerodynamic forces may effect flight stability regulation and this may cause α and β limitations. In this anticipatory optimization study the reduction of both drag and lift areas is attributed to the whole system consisting of the jumper and the skis. Further detailed wind tunnel measurements using latest gear are necessary to study possible shortcomings of this simplification. It has been discussed before (Müller et al., 1996) that extremely low β-angles can easily result in tumbling accidents. On the other hand, aerodynamic measurements using flat plates and prismatic bodies as simplified models of a ski jumper indicate that an immediate reduction of the lift coefficient is to be expected when the angle of attack exceeds about 40–451 and this reduces the coefficient of moment rapidly (Reisenberger et al., 2004). This can also lead to tumbling accidents. Reaching high performance is a tightrope walk because optimizing drag and lift area time functions cannot be done without making sure that the flight remains stable: this is the most difficult problem the athlete has to solve. Any mistake or a too daring behavior would lead to severe accidents. It is the pitching moment control which enables the athlete to perform stable flights and to obtain aerodynamically optimized positions throughout the flight (Müller, 2009; Meile et al., 2006; Reisenberger et al., 2004; Müller et al., 1996). Already during the take-off, the control of angular momentum is of highest importance and this effects performance at least as much as a high linear momentum perpendicular to the ramp. Wind tunnel measurements with athletes in latest gear including take-off and transition to flight (up to 0.7 s), the flight phase, and the landing behavior would enable a more detailed understanding of individual flight style optimization strategies and would lead to improved arguments for limitations of regions of admissible controls in the optimization studies.
Conflict of interest statement None declared.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2013.11. 021. References Arndt, A., Brüggmann, G.-R., Virmavirrta, M., Komi, P., 1995. Techniques used by Olympic ski jumpers in the transition from takeoff to early flight. J Appl Biomech. 11, 224–237. Chong, E.K.P., Żak, S.H., 2008. An Introduction to Optimization, 3rd edition WileyInterscience, New York.
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Denoth, J., Luethi, S.M., Gasser, H.H., 1987. Methodological problems in optimization of the flight phase in ski jumping. Int. J. Sport Biomech. 3, 404–418. König, H., 1952. Theorie des Skispringens angewandt auf die Flugschanze in Oberstdorf, Uhrentechnische Forschung – Vom Skisprung zum Skiflug. J.F. Steinkopf Verlag, Stuttgart, pp. 235–253. Meile, W., Reisenberger, E., Mayer, M., Schmölzer, B., Müller, W., Brenn, G., 2006. Aerodynamics of ski jumping: experiments and CFD simulations. Exp. Fluids 41, 949–964. Müller, W., Platzer, D., Schmölzer, B., 1995. Scientific approach to ski safety. Nature 375, 455. Müller, W., Platzer, D., Schmölzer, B., 1996. Dynamics of human flight on skis: improvements on safety and fairness in ski jumping. J. Biomech. 29 (8), 1061–1068. Müller, W., 2008. Performance factors in ski jumping. In: Nørstrud, H. (Ed.), Sport aerodynamics. Centro Internazionale di Scienze Meccaniche (Udine), CISM series, vol. 506. Springer, Wien, New York, pp. 139–160. Müller, W., 2009. Determinants of Ski-Jump performance and implications for health, safety and fairness. Sports Med. 39 (2), 85–106. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F., 1962. The mathematical theory of optimal processes. Wiley Interscience, New York, London. Reisenberger, E., Meile, W., Brenn, G., Müller, W., 2004. Aerodynamic behaviour of prismatic bodies with sharp. Exp. Fluids 37, 547–558.
Remizov, L.P., 1984. Biomechanics of optimal flight in ski-jumping. J. Biomech. 17, 167–171. Schmölzer, B., Müller, W., 2002. The importance of being light: aerodynamic forces and weight in ski jumping. J. Biomech. 35, 1059–1069. Schmölzer, B., Müller, W., 2005. Individual flight styles in ski jumping: results obtained during Olympic Games competitions. J. Biomech. 38, 1055–1065. Schwameder, H., Müller, E., 1995. Biomechanische Beschreibung und Analyse der V-Technik im Skispringen. Biomechanical description and analysis of the V-technique in ski-jumping. Spectr. Sportwiss. 7 (1), 5–36. Straumann, R., 1927. Vom Skiweitsprung und seiner Mechanik, Jahrbuch des Schweizerischen Ski Verbandes. Selbstverlag des SSV, Bern, pp. 34–64. Vaverka, F., Janura, M., Elfmark, M., Salinger, J., 1997. Inter- and intra-individual variability of the ski-jumper′s take-off. In: Müller, E., Schwameder, H., Kornexl, E., Raschner, C. (Eds.), Science and Skiing. E&FN Spon, London, pp. 36–48. Virmavirta, M., Kivekäs, J., Komi, P., 2001. Take-off aerodynamics in ski jumping. J. Biomech. 34, 465–470. Virmavirta, M., Isolehto, J., Komi, P., 2005. Characteristics of the early flight phase in the Olympic ski jumping competition. J. Biomech. 38, 2157–2163. Virmavirta, M., Isolehto, J., Komi, P., Schwameder, H., Pigozzi, F., Massaza, G., 2009. Take-off analysis of the Olympic ski jumping competition (HS-106 m). J. Biomech. 42, 1095–1101.
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Flight style optimization in ski jumping on normal, large, and ski flying hills Alexander Jung a, Manfred Staat a, Wolfram Müller b,n a b
Institute of Bioengineering, Aachen University of Applied Sciences, Heinrich-Mußmann-Straße 1, 52428 Jülich, Germany Medical University of Graz, Institute of Biophysics, Harrachgasse 21/4, 8010 Graz, Austria
art ic l e i nf o
a b s t r a c t
Article history: Accepted 18 November 2013
In V-style ski jumping, aerodynamic forces are predominant performance factors and athletes have to solve difficult optimization problems in parts of a second in order to obtain their jump length maximum and to keep the flight stable. Here, a comprehensive set of wind tunnel data was used for optimization studies based on Pontryagin′s minimum principle with both the angle of attack α and the body-ski angle β as controls. Various combinations of the constraints αmax and βmin(t) were analyzed in order to compare different optimization strategies. For the computer simulation studies, the Olympic hill profiles in Esto-Sadok, Russia (HS 106 m, HS 140 m), and in Harrachov, Czech Republic, host of the Ski Flying World Championships 2014 (HS 205 m) were used. It is of high importance for ski jumping practice that various aerodynamic strategies, i.e. combinations of α- and β-time courses, can lead to similar jump lengths which enables athletes to win competitions using individual aerodynamic strategies. Optimization results also show that aerodynamic behavior has to be different at different hill sizes (HS). Optimized time courses of α and β using reduced drag and lift areas in order to mimic recent equipment regulations differed only in a negligible way. This indicates that optimization results presented here are not very sensitive to minor changes of the aerodynamic equipment features when similar jump length are obtained by using adequately higher in-run velocities. However, wind tunnel measurements with athletes including take-off and transition to stabilized flight, flight, and landing behavior would enable a more detailed understanding of individual flight style optimization. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Ski jumping Optimal control Computer simulation Aerodynamics
1. Introduction The aerodynamic strategy of the athlete has become a predominant performance factor in ski jumping since the “V-style” was introduced by Jan Boklöv in 1985 (Müller, 2009). In ski flying, jump lengths far beyond 200 m have been exceeded many times, the present world record being 246.5 m (J.R. Evensen, Norway, 2011). At a given hill, jump length determinants are the in-run velocity, the take-off velocity perpendicular to the ramp, the weight, and the aerodynamic forces which depend on the equipment and on the flight style (Straumann, 1927; König, 1952; Denoth et al. 1987; Schwameder and Müller, 1995; Müller et al., 1995, 1996). The take-off phase is crucial (Virmavirta et al., 2001; Vaverka et al., 1997; Müller, 2009), but not only in terms of linear momentum: it is the angular momentum which enables the athlete to obtain an advantageous flight position as soon as possible (Arndt et al., 1995; Schwameder and Müller, 1995; Vaverka et al., 1997; Virmavirta et al., 2005, 2009). The body and
n
Corresponding author. Tel.: þ 43 316 380 3913. E-mail address: [email protected] (W. Müller).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.021
equipment configuration throughout the flight determines the drag and lift areas and thereby substantially the jump length (Müller et al., 1995, 1996; Müller, 2009; Schmölzer and Müller, 2002, 2005). The time courses of lift and drag areas can largely be influenced by the athlete by altering the angle of attack α and the body-ski angle β (Fig. 1). Remizov (1984) was the first to publish an optimization study. He applied Pontryagin's minimum principle (Pontryagin et al., 1962) to a simplified model of a ski jumper, using α(t) as the control function. The body-ski angle β(t) is at least as important as α(t) for obtaining a large jump length (Müller, 2009). Both time functions are also essential for stabilizing the flight as they determine the pitching moment together with the angle of the skis to each other (V-angle), which varies during the stabilized flight phase. Individual mean values ranged between about 251 and 401 (Müller et al. 1996; Schmölzer and Müller, 2002, 2005; Müller, 2009). The hip angle γ should be about 1601 as drag is low and lift is at its maximum in this position (Schmölzer and Müller, 2005). During the Olympic Games 2002, Schmölzer and Müller (2005) investigated individual flight styles. They found distinct jump length maximization strategies between the Olympic medalists,
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717
Fig. 1. Flight position of a ski jumper: angle of attack (α), body-ski angle(β), hip angle (γ), and angle of the skis to each other (V).
although jump lengths were close to each other which implies that there exist several optima which may result in similar jump lengths. Besides the choice of D- and L-functions by means of respective flight positions to maximize jump length, athletes also have to consider their individual aerodynamic features and biomechanical, sensory-motor and mental abilities, and, most importantly: flights have to be kept stable (Müller et al., 1996, Schmölzer and Müller, 2005). Athletes need to solve these extremely difficult optimization problems in “real time”. Here, a comprehensive set of wind tunnel data obtained with a reduced scale model (Meile et al., 2006) which was calibrated with measurement results performed in a large scale wind tunnel (Schmölzer and Müller, 2002) was used for optimization studies based on Pontryagin′s minimum principle with both α and β as controls. Various combinations of the constraints αmax and βmin(t) were analyzed for comparison of different optimization strategies. The Olympic hill profiles in Esto-Sadok, Russia (HS 106 m and HS 140 m), and Harrachov, Czech Republic, host of the Ski Flying World Championships 2014 (HS 205 m), were used. In order to anticipate reduced lift (L) and drag areas (D) caused by tighter and thinner jumping suits which are compulsory due to recent FIS ski jumping equipment regulations, optimization studies were also made using reduced L and D values.
2. Methods 2.1. Computer model of a ski jumper´s flight phase The inrun velocity ν0 as well as the take-off velocity perpendicular to the ramp are initial conditions for the flight phase. During the flight the gravitational force Fg ¼ mg (m¼ 72 kg, g ¼ 9.81 ms 2), the aerodynamic forces drag F d ¼ ρ=2 Dw2 and lift F l ¼ ρ=2 Lw2 determine the flight path of a ski jumper′s center of gravity (Fig. 2). The relative wind vector w is the sum of the external wind vw and the velocity v of the ski jumper: w¼vw v. We assume a plane problem with no crosswind so that w¼ (wx 0 wz)T. The air density is a function of the air pressure with ρ ¼ (p)/(RT), where T is the 1 absolute temperature and R ¼ 288:3 JK 1 kg the gas constant. The air density is ρ ¼ 1:15 kgm 3 (ICAO norm-atmosphere) at 650 m above sea level (Esto-Sadok: 600 m, Harrachov: 700 m). D ¼cdA and L ¼ clA are the drag and lift areas which can be measured in a wind tunnel. As introduced by Müller et al. (1995, 1996) tabulated functions D ¼D(t) and L ¼ L(t) based on field analyses and on wind tunnel measurements were used to consider the changing postures and angles of attack during the flight. Meile et al. (2006) performed comprehensive series of wind tunnel measurements (crosssection area: 2.00 1.46 m2) with a reduced scale model (1:(2)1/2) of a ski jumper. Range of measurements for α:20–451, for β:0–201. The hip angle γ and the V-angle were held constant at the advantageous angles 1601 and 351, respectively (Schmölzer and Müller, 2002). Data were calibrated by comparisons to measurements obtained with a 1:1 ski jumper model (height: 1.78 m, ski length: 2.60 m) in a large scale wind tunnel (cross-section area: 5 5 m2) (Schmölzer and Müller, 2002). Calibration factors fd for drag and fl for lift areas are 2.219 (SD ¼ 0.046) and 2.244 (SD ¼0.050), respectively. Both values are higher than 2.000 because in the large scale wind tunnel a highly realistic model of a ski jumping in full gear was used, whereas the reduced scale model was composed of simplified geometries. The data for D(α,β) and L(α,β) were fitted to bicubic polynomials:
Fig. 2. Flight path of a ski jumper and hill profile. Hill parameters for Esto-Sadok HS 106 m (No. 383/RUS 4), HS 140 m (No. 384/RUS 5), and Harrachov HS 205 m (FLY5/ CZE) can be found in the FIS Certificates of Jumping Hills. Hill profiles were modeled piecewise. v0 is the in-run velocity and vp0 the velocity perpendicular to the ramp. θ is the angle of the relative wind vector w and φ is the flight path angle. Lðα; βÞ ¼ b0 þ b1 α þ b2 β þ b3 α2 þ b4 β þ b5 α3 2
þ b6 β þ b7 αβ þ b8 α2 β þ b9 αβ : 3
2
This resulted in maximum errors of 1.64 10 2 m2 for D(α,β) and 1.27 10 2 m2 for L(α,β), respectively. The coefficients ak and bk are given in Appendix A. Straumann (1927) was the first to publish the equations of motion of a ski jumper during the flight phase. Since then, several authors have applied these equations or modified versions of them to analyze ski jumping (Müller, 2009). Transformed to a first order system the differential equations of motion read 3 2 ρw 2_ 3 ½ vx 2m Dðvwx vx Þþ Lðvwz vz Þ 7 6 ρw 6 v_ z 7 6 2m ½Dðvwz vz Þ Lðvwx vx Þ g 7 7 6 7 6 s_ ¼ 6 x_ 7 ¼ 6 7; vx 7 4 5 6 5 4 vz z_ with s(t) being the state matrix of the system and w¼((vwx vx)2 þ(vwz vz)2)1/2. The simulations done here use the modeling approach developed by Müller et al. (1996) which enables consideration of aerodynamic effects on the trajectory due to changing flight position angles. Initial conditions are s(0) ¼(vx(0),vz(0),x(0),z (0))T with v ¼v0 þ vp0. Inrun velocities used for Esto-Sadok HS 106 m, HS 140 m, and Harrachov HS 205 m are v0 ¼ 25.0 ms 1, 26.5 ms 1, 28.5 ms 1, respectively (according to FIS Certificates of jumping hills). A take-off velocity perpendicular to the ramp of vp0 ¼ 2:5 ms 1 was used (Virmavirta et al., 2001; Müller, 2008). In all simulations vw was set to zero (calm wind). At the flight time T the ski jumper´s flight path intersects the hill profile Ph:z(x) at the landing point so that the height above ground is h(x(T),z(T))¼ 0. 2.2. Optimization
νp0
Dðα; βÞ ¼ a0 þa1 α þ a2 β þ a3 α þ a4 β þa5 α 2
2
þa6 β þ a7 αβ þa8 α2 β þ a9 αβ ; 3
2
3
Besides trying to maximize jump length, the athlete has to balance the pitching moment, which depends on his flight posture and the angle of attack and therefore his individual range of admissible angles is limited. In this study, the controls are u (t)¼ (α(t),β(t))T. Both controls α and β are constrained by αmax and βmin(t), respectively. In order to avoid unrealistic position changes of β, the range of admissible β-values was chosen with respect to angles measured in the field (reference jump A in Schmölzer and Müller, 2002) and the abbreviation Δβref A ¼ βref A ðtÞ βmin ðtÞ is introduced. However the values of βmin(t) must not go below a limit of 01, because the tumbling risk is high using such flight positions (Müller et al., 1996). The maximum value of the angle of attack of the skis αmax does not exceed 451 because larger angles have not been observed in the field (Müller et al., 1996, Schmölzer and Müller, 2002, 2005) and because of sudden lift and pitching moment changes were found in wind tunnel measurements with simplified models in the range around 451 (Reisenberger et al., 2004). Optimization studies were made with varying constraints αmax and βmin(t) as listed in Table 1. The constrained optimization problem with free final state and time has the form: Minimize J ¼ ϕðsðTÞÞ ¼ xðTÞ subject to the system of first order dynamic constraints s_ ðtÞ ¼ fðsðtÞ; uðtÞÞ with initial condition sðt 0 Þ ¼ ðvx ðt 0 Þ; vz ðt 0 Þ; xðt 0 Þ; zðt 0 ÞÞT and terminal condition hðxðTÞ; zðTÞÞ ¼ 0
718
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Table 1 Optimization studies with various boundaries on the controls α and β. Number
αmax [deg]
Δβref
1 2 3 4 5 6 7 8 9
35.0 35.0 35.0 40.0 40.0 40.0 45.0 45.0 45.0
10.0 15.0 10.0 10.0 15.0 10.0 10.0 15.0 10.0
A
[deg]
and the constraints on the controls n o uðtÞ A ðαðtÞ; β ðtÞÞT jαðtÞ r amax ; βðtÞ Z βmin ðtÞ with (
βmin ðtÞ ¼
βref A ðtÞ þ Δβref A ; βmin Z 0 βmin o 0: 0;
A Hamiltonian is constructed for solving the optimization problem: T ℋ sðtÞ; uðtÞ; λðtÞ ¼ λ ðtÞfðsðtÞ; uðtÞÞ; with the adjoint variables λ. Applying Pontryagin's minimum principle (Pontryagin et al., 1962), as done by Remizov (1984), the necessary conditions for u*(t) to be optimal are: ∂ℋðsn ðtÞ; un ðtÞ; λðtÞÞ n s_ ðtÞ ¼ ¼ fðsn ðtÞ; un ðtÞÞ; ∂λ λ ¼ λn n
λ_ ðtÞ ¼
n ∂ℋðsðtÞ; un ðtÞ; λ ðtÞÞ ∂s
s ¼ sn
and n
n
ℋðsn ðtÞ; un ðtÞ; λ ðtÞÞ ¼ minu ℋðsn ðtÞ; uðtÞ; λ ðtÞÞ
The differences between the jump lengths with reference jump A (number 0) and the optimized jumps number 1–9 increase with increasing hill size (Fig. 4). When increasing the range of admissible values of one of the controls α or β, jump lengths become larger, independently of hill size. Fig. 4 shows that jump length increases resulting from all optimization study are more pronounced at larger hills. For example, at the normal hill (HS 106 m) number 9 resulted in an increased jump length of Δl ¼ 6:3 m when compared to the reference jump A, Δl ¼ 12:1 m at the large hill (HS 140 m), and Δl ¼ 40:3 m at the ski flying hill (HS 205 m). A comparison of the 9 optimization studies indicates that several sets of α- and β-time courses result in quite small jump length differences. For example, optimization numbers 3, 5, and 8 for the large hill (broken lines in Fig. 4) resulted in 122.0m, 121.4 m, 122.5 m, and these values are only slightly below the jump length resulting from the best optimization result (number 9) of 125.9 m. This also holds true for optimization studies at the normal and at the ski flying hill. The optimized time courses of the controls α and β for all optimization studies on all 3 hill sizes are shown in Fig. 5a–c. In all 27 optimization studies, the optimal value αn increases up to αmax during the flight. The angle of attack at the beginning of the optimization (t0 ¼0.7 s), termed αn0 , increases when βmin(t) is lowered, and also when the hill size is decreased (compare to Fig. 5a–c and Table 2). The flight times at which αmax has to be reached depend on both the constraints and the hill size. This can be seen in Fig. 5a–c and in Table 3. Fig. 5a–c also shows that the flight time until β has to be kept as low as possible is independent of hill size. In all optimization studies βn decreased as fast as possible towards βn ¼ βmin, and therefore the lowest βn-value was always reached at same flight time (t¼ 1.5 s), independently of hill size. In
for all admissible controls and for all t A ðt 0 ;T n Þ and the transversality condition ∂ϕ ∂h þ υ ; λn ðT n Þ ¼ ∂s ∂s Tn with υ can be obtained by the condition at the final time T n ℋðsn ðT n Þ; un ðtÞ; λ ðT n ÞÞ ¼ 0:
n
The constraints can be enforced by a penalty function (Chong and Żak, 2008): PðuðtÞÞ ¼ ðmax 0; αðtÞ amax Þ2 þ ðmax 0; β min ðtÞ βðtÞ Þ2 : Hereby the optimal controls un(t) can be found from an unconstrained problem by minimizing the convex function n
n
qðc; sn ðtÞ; uðtÞ; λ ðtÞÞ ¼ ℋðsn ðtÞ; uðtÞ; λ ðtÞÞ þ cPðuðtÞÞ with the penalty parameter c ¼ 1010 . The optimal controls un(t) are global maxima as well as the associated jump lengths l. The optimization algorithm starts at t0 ¼ 0.7 s. With an initial guess u0(t) for i¼ 0 it has the following structure:
Fig. 3. Comparison of the flight path obtained with the reference jump A and with the optimized jump number 9. P-Point: 106 m, K-Point: 125 m, and L-Point: 139 m.
1. Solve s_ ðtÞ ¼ fðsi ðtÞ; ui ðtÞÞ forward in time by Heun's method until h(xi(Ti), zi(Ti))¼ 0 and sn(t0) obtained by solving the equations of motions with s (0) up to t0 using reference jump A (Schmölzer and Müller, 2002). _ i ðtÞ ¼ ð∂ℋ=∂sÞðsi ðtÞ; ui ðtÞ; λi ðtÞÞ backward in time by Heun's method 2. Solve λ and λi(Ti). i 3. minu qðc; si ðtÞ; ui ðtÞ; λ ðtÞÞ for all tA (t0,Ti) with Newton's method. 4. Set i¼i þ1 and go to step 1 until J has converged to the minimum value. i
3. Results Fig. 3 shows flight path simulations using the profile of the large hill in Esto-Sadok (HS 140 m). The solid line represents the flight path of the reference jump A (Schmölzer and Müller, 2002) and the broken line the flight path of the optimized jump number 9 with the widest range of admissible values (Table 1). The jump length obtained with reference jump A was 113.8 m, and 125.9 m resulted for the optimized jump, i.e. an increase of 12.1 m.
Fig. 4. Jump lengths resulting from the optimization studies listed in Table 1 for all 3 hill sizes. Optimization study number 0 denotes reference jump A.
A. Jung et al. / Journal of Biomechanics 47 (2014) 716–722
Fig. 5. Optimized time courses for α and β. (a) Normal hill (HS 106 m), (b) Large hill (HS 140 m), (c) Ski flying hill (HS 205 m). Numbers of optimization studies according to Table 1.
719
720
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Table 2 The optimized angle of attack αn0 at t0 ¼ 0.7 s as a function of the optimization study number and at all 3 hill sizes. Number
HS 106 m αn0 [deg]
HS 140 m αn0 [deg]
HS 205 m αn0 [deg]
1 2 3 4 5 6 7 8 9
22.2 23.5 25.1 22.1 23.3 24.8 22.0 23.2 24.6
19.5 20.8 22.4 19.4 20.6 22.2 19.4 20.5 21.9
15.9 16.9 18.6 15.6 16.6 18.2 15.6 16.4 17.9
Fig. 6. Optimized jump 9 compared to a modified version with 10% reduced drag and lift areas throughout the whole flight. To obtain a jump length of 125.9 m, an in-run velocity of 27.0 ms 1 is used here (HS 140 m).
Table 3 Occurrence times t αmax and t βn 4 βmin in [s] for all 9 optimization studies at all 3 hill sizes. Number
1 2 3 4 5 6 7 8 9
HS 106 m
HS 140 m
HS 205 m
t αmax
t βn 4 βmin
t αmax
t βn 4 βmin
t αmax
t βn 4 βmin
1.76 1.47 1.27 2.39 1.99 1.66 2.80 2.38 2.03
2.11 1.70 1.43 3.05 2.48 2.12 / / 3.22
2.30 1.93 1.58 3.05 2.61 2.21 2.53 3.08 2.69
2.73 2.24 1.86 3.10 3.19 2.78 / 4.46 4.06
3.11 2.75 2.34 4.03 3.66 3.24 4.51 4.32 3.91
3.64 3.15 2.73 4.73 4.32 3.96 / 5.60 5.39
the flight phase from t βn 4 β , βn increases up to the landing point min of the optimized jump. The time t βn 4 β increases with both min higher αmax and at larger hills, but it decreases with lower βmin(t) (Table 3). The lower αmax the higher the increase of βn, however βn(Tn) is independent from the hill size. Increasing the body-ski angle β in this flight phase results in a jump length increase of 2.4 m, 0.5 m, and 0.0 m for optimization studies 3, 6, and 9 at the normal hill (large hill: 3.5 m, 0.6 m, 0.0 m; ski flying hill: 8.0 m, 1.4 m, 0.0 m). Applying a jump which has been optimized for a given hill to another hill size results in a decreased jump length when compared to the optimal jump obtained for this hill. For example, jump 9 optimized for the large hill (HS 140 m) results in a 1.6 m jump length decrease when applied to the normal hill (HS 106 m) and a decrease of 9.6 m when applied to the ski flying hill. Fig. 6 illustrates that reduced drag and lift areas, e.g. due to the equipment changes, has little effect on the optimal time courses of α and β. A 10% reduction of both drag and lift areas reduced the jump by 9.1 m on the large hill (HS 140 m). In order to reach the same jump length, the in-run velocity was increased to 27.0 ms 1, but αn(t) and βn(t) changed only slightly. Fig. 7a and b compares the drag and lift forces of the optimized jump 9 to the aerodynamic forces occurring with the reference jump A. When compared to reference jump, jump length increases when aerodynamic forces are kept low (due to low α- and βvalues) in the first part of the flight phase and when they increase in a more pronounced way in the second part until this is limited by reaching αmax.
4. Discussion During the transition from take-off to flight, the posture angles and also the angle of attack of the skis with respect to the
Fig. 7. (a) Lift forces and (b) drag forces for reference jump A and optimized jump 9 at the large hill (HS 140 m).
airstream change rapidly (Virmavirta et al., 2005). Adequate series of wind tunnel measurements corresponding to these fast positional changes from take-off until t¼0.7 s are missing. Therefore, this optimization study of the flight phase in ski jumping starts at t¼0.7 s. The reference jump A (Schmölzer and Müller, 2002) which had been developed on the basis of comprehensive field and wind tunnel studies and which was used for various computer simulation studies of ski jumping before (Schmölzer and Müller, 2002; Schmölzer and Müller, 2005; Müller, 2009) was used as the starting basis for these jump length optimization studies by means of Pontryagin's minimum principle (Pontryagin et al., 1962). This approach has already been used before (Remizov, 1984). However, Remizov applied a simplified rigid body model of a ski jumper and only the optimal angle of attack of the skis αn(t) was determined. Here, the optimized time courses of both controls are of predominant importance for the flight phase in ski jumping: the angle of attack αn(t) and the body-ski angle βn(t) were determined. The hip angle and V-angle were kept constant at the value of 1601 and 351, respectively, at which lift is maximal and drag is low (Schmölzer and Müller, 2002). Since the equations of motion can numerically be determined with any desired accuracy, it is the wind tunnel data input which determines the simulation accuracy. Therefore, the optimization study performed here uses the detailed data sets obtained from a reduced scale model (1:(2)1/2) (Meile et al., 2006) which was calibrated by means of highly accurate measurements
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obtained in a large scale wind tunnel using a 1:1 model of a ski jumper with skis (Schmölzer and Müller, 2002). Here, optimization studies were performed without external wind, although this 2D modeling approach is designed to investigate the effects of all initial value and parameter variations which may have an impact on the optimization of the flight. The reference jump (Schmölzer and Müller, 2002) which is used here as the starting point for the optimization represents the time functions of drag and lift area values corresponding to the mean flight position angles of the best 10 athletes measured during several world cup competitions. A detailed representation of the last part of the flight is not included. Therefore, these mean flight positions do not represent the best jump and optimized jumps were substantially larger (Fig. 4). As is to be expected, jump length increases at all 3 hills were largest for the widest admissible control values. The length increases at the normal, large, and ski flying hill with optimized jump number 9 are 6.3 m, 12.1 m, and 40.3m, respectively. This shows that the aerodynamic strategy of the athlete is of predominant importance in ski flying and also plays a leading role on the large hill. The computer model optimization is only limited by the chosen range of admissible values of the controls α and β, whereas – in the real world – these chosen ranges may not be achievable by the athlete for biomechanical and aerodynamic stability reasons. It is of high importance for ski jumping practice that various aerodynamic strategies (combinations of α- and β-time courses) can lead to similar jump lengths (Fig. 4). This enables athletes to win competitions using individual aerodynamic strategies which are suited best for their personal features and abilities. This corresponds to results found in field studies during Olympic Games competitions (Schmölzer and Müller, 2005, Virmavirta et al., 2005). A small decrease of jump length due to a different aerodynamic behavior of the ski jumper can easily be compensated by one of the other performance factors. Even small changes in external wind can easily mask the effects of two different aerodynamic strategies which both would result in comparable jump lengths without wind influence. Consideration of the various parameters which influence jump length substantially (Schwameder and Müller, 1995; Müller et al., 1995; 1996; Schmölzer and Müller, 2002; 2005; Virmavirta et al., 2005; Müller, 2009, Virmavirta et al., 2009) and optimization results found here makes clear that various individual flight styles can have the potential for winning a competition. For example, optimization studies 3, 5, and 8 at the large hill (Fig. 4) resulted in jump lengths which do not differ by more than 1.1 m, although the admissible controls differed substantially in these cases. Athlete's anticipation of the range of control angles possible without taking too high risks (tumbling accidents can easily occur) largely depends on personal experience and on the sensitivity of athlete's sensory-motor system. All 9 optimization studies at all 3 hills showed that the optimized angle of attack αn of the skis increases continuously until αmax is obtained. A continuous increase of αn was also found in the simplified optimization studies of Remizov (1984): αn increased continuously up to 451. The optimization studies result in the advice to reduce the βangles as fast as possible in order to reduce drag rapidly in the first part of the flight. The α-angles should increase continuously (Fig. 5a–c), but not too fast, as this would increase drag. It is interesting to compare field study data which show that this is not the case in real world ski jumping (Müller et al., 1996; Schmölzer and Müller, 2002, 2005; Virmavirta et al., 2005): athletes increase the ski angle of attack rapidly. This appears to be necessary for obtaining a substantial backward rotating torque to stop the angular momentum resulting from the take-off jump.
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In the last part of the flight (Fig. 5a–c), β should increase in order to increase aerodynamic forces when α is limited to a value below 401. This results in a length increase of up to 8.0 m on the ski flying hill. Time courses αn(t) and βn(t) vary on different sized hills because the in-run velocities and therefore the aerodynamic forces differ to each other. This is also important for ski jumping practice. Meanwhile, equipment has changed, in particular the jumping suits are cut tighter and made of thinner material which reduces aerodynamic forces (Müller et al., 1996; Meile et al., 2006). Therefore, results were also compared to optimizations when using reduced drag and lift area values. It can be seen in Fig. 6 that the optimized time courses of α and β using 10% reduced drag and lift areas throughout the flight differ only in a negligible way when jump length is unaltered (due to increase of v0). However, reduced aerodynamic forces may effect flight stability regulation and this may cause α and β limitations. In this anticipatory optimization study the reduction of both drag and lift areas is attributed to the whole system consisting of the jumper and the skis. Further detailed wind tunnel measurements using latest gear are necessary to study possible shortcomings of this simplification. It has been discussed before (Müller et al., 1996) that extremely low β-angles can easily result in tumbling accidents. On the other hand, aerodynamic measurements using flat plates and prismatic bodies as simplified models of a ski jumper indicate that an immediate reduction of the lift coefficient is to be expected when the angle of attack exceeds about 40–451 and this reduces the coefficient of moment rapidly (Reisenberger et al., 2004). This can also lead to tumbling accidents. Reaching high performance is a tightrope walk because optimizing drag and lift area time functions cannot be done without making sure that the flight remains stable: this is the most difficult problem the athlete has to solve. Any mistake or a too daring behavior would lead to severe accidents. It is the pitching moment control which enables the athlete to perform stable flights and to obtain aerodynamically optimized positions throughout the flight (Müller, 2009; Meile et al., 2006; Reisenberger et al., 2004; Müller et al., 1996). Already during the take-off, the control of angular momentum is of highest importance and this effects performance at least as much as a high linear momentum perpendicular to the ramp. Wind tunnel measurements with athletes in latest gear including take-off and transition to flight (up to 0.7 s), the flight phase, and the landing behavior would enable a more detailed understanding of individual flight style optimization strategies and would lead to improved arguments for limitations of regions of admissible controls in the optimization studies.
Conflict of interest statement None declared.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jbiomech.2013.11. 021. References Arndt, A., Brüggmann, G.-R., Virmavirrta, M., Komi, P., 1995. Techniques used by Olympic ski jumpers in the transition from takeoff to early flight. J Appl Biomech. 11, 224–237. Chong, E.K.P., Żak, S.H., 2008. An Introduction to Optimization, 3rd edition WileyInterscience, New York.
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Denoth, J., Luethi, S.M., Gasser, H.H., 1987. Methodological problems in optimization of the flight phase in ski jumping. Int. J. Sport Biomech. 3, 404–418. König, H., 1952. Theorie des Skispringens angewandt auf die Flugschanze in Oberstdorf, Uhrentechnische Forschung – Vom Skisprung zum Skiflug. J.F. Steinkopf Verlag, Stuttgart, pp. 235–253. Meile, W., Reisenberger, E., Mayer, M., Schmölzer, B., Müller, W., Brenn, G., 2006. Aerodynamics of ski jumping: experiments and CFD simulations. Exp. Fluids 41, 949–964. Müller, W., Platzer, D., Schmölzer, B., 1995. Scientific approach to ski safety. Nature 375, 455. Müller, W., Platzer, D., Schmölzer, B., 1996. Dynamics of human flight on skis: improvements on safety and fairness in ski jumping. J. Biomech. 29 (8), 1061–1068. Müller, W., 2008. Performance factors in ski jumping. In: Nørstrud, H. (Ed.), Sport aerodynamics. Centro Internazionale di Scienze Meccaniche (Udine), CISM series, vol. 506. Springer, Wien, New York, pp. 139–160. Müller, W., 2009. Determinants of Ski-Jump performance and implications for health, safety and fairness. Sports Med. 39 (2), 85–106. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F., 1962. The mathematical theory of optimal processes. Wiley Interscience, New York, London. Reisenberger, E., Meile, W., Brenn, G., Müller, W., 2004. Aerodynamic behaviour of prismatic bodies with sharp. Exp. Fluids 37, 547–558.
Remizov, L.P., 1984. Biomechanics of optimal flight in ski-jumping. J. Biomech. 17, 167–171. Schmölzer, B., Müller, W., 2002. The importance of being light: aerodynamic forces and weight in ski jumping. J. Biomech. 35, 1059–1069. Schmölzer, B., Müller, W., 2005. Individual flight styles in ski jumping: results obtained during Olympic Games competitions. J. Biomech. 38, 1055–1065. Schwameder, H., Müller, E., 1995. Biomechanische Beschreibung und Analyse der V-Technik im Skispringen. Biomechanical description and analysis of the V-technique in ski-jumping. Spectr. Sportwiss. 7 (1), 5–36. Straumann, R., 1927. Vom Skiweitsprung und seiner Mechanik, Jahrbuch des Schweizerischen Ski Verbandes. Selbstverlag des SSV, Bern, pp. 34–64. Vaverka, F., Janura, M., Elfmark, M., Salinger, J., 1997. Inter- and intra-individual variability of the ski-jumper′s take-off. In: Müller, E., Schwameder, H., Kornexl, E., Raschner, C. (Eds.), Science and Skiing. E&FN Spon, London, pp. 36–48. Virmavirta, M., Kivekäs, J., Komi, P., 2001. Take-off aerodynamics in ski jumping. J. Biomech. 34, 465–470. Virmavirta, M., Isolehto, J., Komi, P., 2005. Characteristics of the early flight phase in the Olympic ski jumping competition. J. Biomech. 38, 2157–2163. Virmavirta, M., Isolehto, J., Komi, P., Schwameder, H., Pigozzi, F., Massaza, G., 2009. Take-off analysis of the Olympic ski jumping competition (HS-106 m). J. Biomech. 42, 1095–1101.