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The Features of the Landing Slope of a Ski Jumping Hill That Need to be Considered
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 112 (2015) 373 – 378
7th Asia-Pacific Congress on Sports Technology, APCST 2015
The features of the landing slope of a ski jumping hill that need to be considered Kazuya Seoa*, Yuji Niheib, Ryutaro Watanabeb, Toshiyuki Shimanoc, Takayuki Sakaguchia a
Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan Yamagata City Office, 2-3-25 Hatagomachi, Yamagata 990-8540, Japan c Access Corporation, 2-3-4, Minami-1-jo Higashi, Chuo-ku, Sapporo 060-0051, Japan b
Abstract This paper describes the features of the landing slope of a ski jumping hill that need to be considered. A ski jumping hill is composed of the in-run, the take-off table, the landing slope and the out-run. The features of the landing slope that we have considered are the construction fee, the safety of the landing slope, the long flight distance for the interesting game and the difficulty for unskilled jumpers. The construction fee was estimated on the basis of the amounts of material that need to be removed from and brought in to the existing Zao jumping hill in Yamagata city. The safety on landing was estimated on the basis of the landing velocity. The landing velocity is the velocity component perpendicular to the landing slope at the instance of landing, and this needs to be small to reduce the impact and make the landing safer. In order to estimate the landing velocity and the flight distance, it is required to simulate the flight trajectory. The difficulty for unskilled jumpers is estimated on the basis of the variance of the flight distance. It is considered that the flight distances for unskilled jumpers are less than for skilled jumpers because they are unable to satisfy the optimal conditions from take-off through to landing. Therefore, a landing slope for which the variance in the flight distance is large is defined as a difficult slope for unskilled jumpers. © 2015The The Authors. Published by Elsevier © 2015 Authors. Published by Elsevier Ltd. This isLtd. an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University. Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
Keywords: Landing slope; Ski jumping; Safety landing; Construction fee; Variety
* Corresponding author. Tel.: +81-23-628-4350; fax: +81-23-628-4454. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
doi:10.1016/j.proeng.2015.07.209
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
Nomenclature b FDi FDL g D Iyy h L M m Q (U, W) v┴ (Xa, Za) (xb, zb) α βH γ Θ
Width of the landing slope Flight distance around FDL Longest flight distance Gravitational acceleration Drag Moment of inertia of the body–ski combination on its yb –axis Height difference between the old Zao and the new Zao Lift Pitching moment Mass of the body–ski combination yb component of the angular velocity vector (xb, zb) components of the velocity vector Velocity component perpendicular to the landing slope at the instance of landing (xb, zb) components of the aerodynamic force Body-fixed coordinate system Angle of attack Slope of the landing hill at the landing point Flight path angle Pitch angle
1. Introduction Since 2012 the Zao jumping hill in Yamagata city has been host to the annual ladies world cup. A ski jumping hill is composed of the in-run, the take-off table, the landing slope and the out-run. The Zao track was renovated to resemble the ski jump at the Sochi Games in 2013, with a take-off table with an angle of 11 degrees downhill. A further renovation related to the landing slope is being planned for 2015. This is the subject of this study. The features of the landing slope that we have considered are the construction fee, the safety of the landing slope, the long flight distance and the difficulty for unskilled jumpers. In this paper, it will be discussed how to estimate four features of the landing slope, which are the construction fee, the safety of the landing slope, the long flight distance and the difficulty for unskilled jumpers. 2. Construction fee The construction fee was estimated on the basis of the amounts of material that need to be removed from and brought in to the existing Zao jumping hill. Lower cost is, of course, better. The inertial coordinate system is shown in Fig.1. The origin is defined as being at the edge of the take-off table, while the XE-axis is in the horizontal forward direction and the ZE-axis is vertically downward. The height difference between the old Zao and the new Zao at XE is denoted by h(XE) as shown in Fig.1. The width at XE is denoted by b(XE). The amounts of material that need to be removed from and brought in to the existing Zao jumping hill are derived by equation (1).
Amount of materials
³
X E ( tf )
0
h X E b X E dX E
(1)
Here, the flight time is denoted by tf. The construction fee depends on the height at which material needs to be removed from and brought into the existing Zao jumping hill. The greater the height, the more expensive the construction fee. Here, the lowest cost is at ZU (the lowest height) and this is assumed to be 200 Japanese yen per 1 m3, while the highest cost is at ZE=0 (the highest height), which is assumed to be 10,000 yen per 1 m3 on the basis of
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
experience. The cost between ZU and ZE=0 is derived using a linear relationship between cost and height. Therefore, the construction fee, F1, can be estimated using equation (2).
F1
³
X E ( tf )
0
9800 ½ Z E X E ZU 200¾ dX E h X E b X E ® ¯ ZU ¿
Fig. 1. Inertial coordinate system.
(2)
Fig. 2. Height difference between the old Zao and the new Zao.
3. Landing velocity The safety on landing was estimated on the basis of the landing velocity [1]. In order to estimate the landing velocity, the flight trajectory needs to be simulated. This is discussed in section 6. The landing velocity is the velocity component perpendicular to the landing slope at the instance of landing, and this needs to be small to reduce the impact and make the landing safer. The landing velocity, F2, is shown in equation (3), where the flight path angle and the slope of the landing hill at the landing point are denoted by γ and βH.
F 2 vA V sin J E H
Fig. 3. Landing velocity, v┴.
(3)
Fig. 4. Contour map of flight distance. FDL: The longest flight distance denoted by ×, FDi: The flight distance around FDL denoted by ●
4. Flight distance A ski jumping hill should be designed so that it contributes to the creation of an exciting spectacle, which means that the jumpers have longer flight distances. The flight distance is defined by the distance along the profile of the
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
landing slope. The flight distance, F3, is derived from equation (4). The flight trajectory needs to be simulated in order to determine the landing point, XE(tf). This is discussed in section 6.
F3
³
X E tf
0
X
2 E
2
Z E , new dX E
(4)
5. Difficulty for unskilled jumpers In Japan, jumping hills have been designed to produce long flight distances for all jumpers. In other words, jumping hills have been constructed such that they face into a head wind. However, the landing slope in Zao is designed to be a difficult slope for unskilled jumpers. The design concept of Zao is that the flight distances of skilled jumpers are comparable with the hill size, while the flight distances of unskilled jumpers are usually less than this. The flight distances of unskilled jumpers are less than those of skilled jumpers because they are unable to satisfy the optimal conditions from take-off through to landing. Therefore, a landing slope for which the variance in the flight distance is large is defined as a difficult slope for unskilled jumpers. The variance in the flight distance is defined by equation (5), where FDL is the local longest flight distance denoted by × in Fig.4, FDi are simulated flight distances around FDL denoted by ●, and n is the flight simulation number. The abscissa and the ordinate in Fig.4 are design variables, which are the time variation of the ski-opening angle, the hip-bending angle, etc. The ellipse in Fig.4 corresponds to the human error. The jumper is human, not an automaton, so there will be some human error, which makes the flight distance shorter. n
¦ FD FD
2
i
F4
L
i 1
n 1
(5)
6. Flight simulation In order to estimate F2, F3 and F4, the flight trajectory needs to be simulated. It is assumed that the motion of the body–ski combination occurs in a fixed vertical plane. The body-fixed coordinate system is shown in Fig. 5. The origin is defined as the center of gravity of the body–ski combination. In terms of coordinate transformations [2] we then have
X E
U cos 4 W sin 4
(6)
Z E
U sin 4 W cos 4
(7)
Here, (U, W) are the (xb, zb) components of the velocity vector. The equations of motion and the moment equation are
U
1 >X a mg sin 4@ QW m
(8)
W
1 >Z a mg cos 4@ QU m
(9)
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
Q
Ma I yy
(10)
Q 4
(11)
Here, (Xa, Za) are the (xb, zb) components of the aerodynamic force, Q is the yb component of the angular velocity vector, m is the mass of the body–ski combination, g is the gravitational acceleration, Ma is the yb component of the aerodynamic moment, and Iyy is the moment of inertia of the body–ski combination on its yb -axis, respectively. The flight trajectory (XE(t), YE(t), ZE(t)) can be obtained by integrating Equations (6) through (11) numerically. The aerodynamic forces Xa and Za in Equations (8) and (9) are derived from D and L as shown in Eqns. (12) and (13).
Xa
L sin D D cos D
(12)
Za
L cos D D sin D
(13)
The aerodynamic drag, lift and moment in Eqns. (12) through (13) were all obtained during wind tunnel tests as functions of α, β and λ [2]. The wind tunnel data were acquired for α at 5° intervals between 0° and 40°, and for β 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.
Fig. 5. Body-fixed coordinate system and definition of characteristic parameters.
Fig. 6. Comparison between the old Zao landing slope and two examples.
7. Examples of a landing slope Examples of landing slopes are shown in Fig.6. The broken line shows the profile of the old Zao landing slope, the solid line shows the profile of a relatively low cost landing slope and the chain line shows a profile which can produce relatively long flight distances. It is possible to control the construction fee, the flight distance and so on, by changing the profile of the landing slope. The profile of the low cost slope coincides with the old profile especially at greater height (around ZE=10), though the profile is different at lower levels (around ZE=60). On the other hand, the profile which produces long flight distances is higher at XE =40, while it is almost the same height at XE=120. Since the flight distance is defined by the distance along the profile, as given by equation (4), the profile of the chain line produces the longer flight distances.
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
8. Summary Various features of the landing slope for the Zao ski jump, i.e. the construction fee, its safety, longer flight distances to produce more spectacular jumps and the difficulty for unskilled jumpers, are considered. In order to estimate or analyze these features, the flight trajectory and the profile of the landing slope need to be simulated. As a result, it is clear that the profile of the landing slope determines the construction fee, its safety, the flight distance and the difficulty for unskilled jumpers. Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research (A), No. 15H01824, Japan Society for the Promotion of Science. References [1] McNeil A.J., Hubbard M. and Swedberg, A.D., Designing tomorrow’s snow park jump, Sports Engineering, 15 (2012) 1-20. [2] Stevens B. and Lewis F., Aircraft control and simulation, 2nd Edition, Wiley, Hoboken, New Jersey, 2003. [3] Seo K., Watanabe I. and Murakami M. Aerodynamic force data for a V-style ski jumping flight, Sports Engineering, 7 (2004) 31-39.
ScienceDirect Procedia Engineering 112 (2015) 373 – 378
7th Asia-Pacific Congress on Sports Technology, APCST 2015
The features of the landing slope of a ski jumping hill that need to be considered Kazuya Seoa*, Yuji Niheib, Ryutaro Watanabeb, Toshiyuki Shimanoc, Takayuki Sakaguchia a
Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan Yamagata City Office, 2-3-25 Hatagomachi, Yamagata 990-8540, Japan c Access Corporation, 2-3-4, Minami-1-jo Higashi, Chuo-ku, Sapporo 060-0051, Japan b
Abstract This paper describes the features of the landing slope of a ski jumping hill that need to be considered. A ski jumping hill is composed of the in-run, the take-off table, the landing slope and the out-run. The features of the landing slope that we have considered are the construction fee, the safety of the landing slope, the long flight distance for the interesting game and the difficulty for unskilled jumpers. The construction fee was estimated on the basis of the amounts of material that need to be removed from and brought in to the existing Zao jumping hill in Yamagata city. The safety on landing was estimated on the basis of the landing velocity. The landing velocity is the velocity component perpendicular to the landing slope at the instance of landing, and this needs to be small to reduce the impact and make the landing safer. In order to estimate the landing velocity and the flight distance, it is required to simulate the flight trajectory. The difficulty for unskilled jumpers is estimated on the basis of the variance of the flight distance. It is considered that the flight distances for unskilled jumpers are less than for skilled jumpers because they are unable to satisfy the optimal conditions from take-off through to landing. Therefore, a landing slope for which the variance in the flight distance is large is defined as a difficult slope for unskilled jumpers. © 2015The The Authors. Published by Elsevier © 2015 Authors. Published by Elsevier Ltd. This isLtd. an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University. Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
Keywords: Landing slope; Ski jumping; Safety landing; Construction fee; Variety
* Corresponding author. Tel.: +81-23-628-4350; fax: +81-23-628-4454. E-mail address: [email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University
doi:10.1016/j.proeng.2015.07.209
374
Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
Nomenclature b FDi FDL g D Iyy h L M m Q (U, W) v┴ (Xa, Za) (xb, zb) α βH γ Θ
Width of the landing slope Flight distance around FDL Longest flight distance Gravitational acceleration Drag Moment of inertia of the body–ski combination on its yb –axis Height difference between the old Zao and the new Zao Lift Pitching moment Mass of the body–ski combination yb component of the angular velocity vector (xb, zb) components of the velocity vector Velocity component perpendicular to the landing slope at the instance of landing (xb, zb) components of the aerodynamic force Body-fixed coordinate system Angle of attack Slope of the landing hill at the landing point Flight path angle Pitch angle
1. Introduction Since 2012 the Zao jumping hill in Yamagata city has been host to the annual ladies world cup. A ski jumping hill is composed of the in-run, the take-off table, the landing slope and the out-run. The Zao track was renovated to resemble the ski jump at the Sochi Games in 2013, with a take-off table with an angle of 11 degrees downhill. A further renovation related to the landing slope is being planned for 2015. This is the subject of this study. The features of the landing slope that we have considered are the construction fee, the safety of the landing slope, the long flight distance and the difficulty for unskilled jumpers. In this paper, it will be discussed how to estimate four features of the landing slope, which are the construction fee, the safety of the landing slope, the long flight distance and the difficulty for unskilled jumpers. 2. Construction fee The construction fee was estimated on the basis of the amounts of material that need to be removed from and brought in to the existing Zao jumping hill. Lower cost is, of course, better. The inertial coordinate system is shown in Fig.1. The origin is defined as being at the edge of the take-off table, while the XE-axis is in the horizontal forward direction and the ZE-axis is vertically downward. The height difference between the old Zao and the new Zao at XE is denoted by h(XE) as shown in Fig.1. The width at XE is denoted by b(XE). The amounts of material that need to be removed from and brought in to the existing Zao jumping hill are derived by equation (1).
Amount of materials
³
X E ( tf )
0
h X E b X E dX E
(1)
Here, the flight time is denoted by tf. The construction fee depends on the height at which material needs to be removed from and brought into the existing Zao jumping hill. The greater the height, the more expensive the construction fee. Here, the lowest cost is at ZU (the lowest height) and this is assumed to be 200 Japanese yen per 1 m3, while the highest cost is at ZE=0 (the highest height), which is assumed to be 10,000 yen per 1 m3 on the basis of
375
Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
experience. The cost between ZU and ZE=0 is derived using a linear relationship between cost and height. Therefore, the construction fee, F1, can be estimated using equation (2).
F1
³
X E ( tf )
0
9800 ½ Z E X E ZU 200¾ dX E h X E b X E ® ¯ ZU ¿
Fig. 1. Inertial coordinate system.
(2)
Fig. 2. Height difference between the old Zao and the new Zao.
3. Landing velocity The safety on landing was estimated on the basis of the landing velocity [1]. In order to estimate the landing velocity, the flight trajectory needs to be simulated. This is discussed in section 6. The landing velocity is the velocity component perpendicular to the landing slope at the instance of landing, and this needs to be small to reduce the impact and make the landing safer. The landing velocity, F2, is shown in equation (3), where the flight path angle and the slope of the landing hill at the landing point are denoted by γ and βH.
F 2 vA V sin J E H
Fig. 3. Landing velocity, v┴.
(3)
Fig. 4. Contour map of flight distance. FDL: The longest flight distance denoted by ×, FDi: The flight distance around FDL denoted by ●
4. Flight distance A ski jumping hill should be designed so that it contributes to the creation of an exciting spectacle, which means that the jumpers have longer flight distances. The flight distance is defined by the distance along the profile of the
376
Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
landing slope. The flight distance, F3, is derived from equation (4). The flight trajectory needs to be simulated in order to determine the landing point, XE(tf). This is discussed in section 6.
F3
³
X E tf
0
X
2 E
2
Z E , new dX E
(4)
5. Difficulty for unskilled jumpers In Japan, jumping hills have been designed to produce long flight distances for all jumpers. In other words, jumping hills have been constructed such that they face into a head wind. However, the landing slope in Zao is designed to be a difficult slope for unskilled jumpers. The design concept of Zao is that the flight distances of skilled jumpers are comparable with the hill size, while the flight distances of unskilled jumpers are usually less than this. The flight distances of unskilled jumpers are less than those of skilled jumpers because they are unable to satisfy the optimal conditions from take-off through to landing. Therefore, a landing slope for which the variance in the flight distance is large is defined as a difficult slope for unskilled jumpers. The variance in the flight distance is defined by equation (5), where FDL is the local longest flight distance denoted by × in Fig.4, FDi are simulated flight distances around FDL denoted by ●, and n is the flight simulation number. The abscissa and the ordinate in Fig.4 are design variables, which are the time variation of the ski-opening angle, the hip-bending angle, etc. The ellipse in Fig.4 corresponds to the human error. The jumper is human, not an automaton, so there will be some human error, which makes the flight distance shorter. n
¦ FD FD
2
i
F4
L
i 1
n 1
(5)
6. Flight simulation In order to estimate F2, F3 and F4, the flight trajectory needs to be simulated. It is assumed that the motion of the body–ski combination occurs in a fixed vertical plane. The body-fixed coordinate system is shown in Fig. 5. The origin is defined as the center of gravity of the body–ski combination. In terms of coordinate transformations [2] we then have
X E
U cos 4 W sin 4
(6)
Z E
U sin 4 W cos 4
(7)
Here, (U, W) are the (xb, zb) components of the velocity vector. The equations of motion and the moment equation are
U
1 >X a mg sin 4@ QW m
(8)
W
1 >Z a mg cos 4@ QU m
(9)
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Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
Q
Ma I yy
(10)
Q 4
(11)
Here, (Xa, Za) are the (xb, zb) components of the aerodynamic force, Q is the yb component of the angular velocity vector, m is the mass of the body–ski combination, g is the gravitational acceleration, Ma is the yb component of the aerodynamic moment, and Iyy is the moment of inertia of the body–ski combination on its yb -axis, respectively. The flight trajectory (XE(t), YE(t), ZE(t)) can be obtained by integrating Equations (6) through (11) numerically. The aerodynamic forces Xa and Za in Equations (8) and (9) are derived from D and L as shown in Eqns. (12) and (13).
Xa
L sin D D cos D
(12)
Za
L cos D D sin D
(13)
The aerodynamic drag, lift and moment in Eqns. (12) through (13) were all obtained during wind tunnel tests as functions of α, β and λ [2]. The wind tunnel data were acquired for α at 5° intervals between 0° and 40°, and for β 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.
Fig. 5. Body-fixed coordinate system and definition of characteristic parameters.
Fig. 6. Comparison between the old Zao landing slope and two examples.
7. Examples of a landing slope Examples of landing slopes are shown in Fig.6. The broken line shows the profile of the old Zao landing slope, the solid line shows the profile of a relatively low cost landing slope and the chain line shows a profile which can produce relatively long flight distances. It is possible to control the construction fee, the flight distance and so on, by changing the profile of the landing slope. The profile of the low cost slope coincides with the old profile especially at greater height (around ZE=10), though the profile is different at lower levels (around ZE=60). On the other hand, the profile which produces long flight distances is higher at XE =40, while it is almost the same height at XE=120. Since the flight distance is defined by the distance along the profile, as given by equation (4), the profile of the chain line produces the longer flight distances.
378
Kazuya Seo et al. / Procedia Engineering 112 (2015) 373 – 378
8. Summary Various features of the landing slope for the Zao ski jump, i.e. the construction fee, its safety, longer flight distances to produce more spectacular jumps and the difficulty for unskilled jumpers, are considered. In order to estimate or analyze these features, the flight trajectory and the profile of the landing slope need to be simulated. As a result, it is clear that the profile of the landing slope determines the construction fee, its safety, the flight distance and the difficulty for unskilled jumpers. Acknowledgements This work is supported by a Grant-in-Aid for Scientific Research (A), No. 15H01824, Japan Society for the Promotion of Science. References [1] McNeil A.J., Hubbard M. and Swedberg, A.D., Designing tomorrow’s snow park jump, Sports Engineering, 15 (2012) 1-20. [2] Stevens B. and Lewis F., Aircraft control and simulation, 2nd Edition, Wiley, Hoboken, New Jersey, 2003. [3] Seo K., Watanabe I. and Murakami M. Aerodynamic force data for a V-style ski jumping flight, Sports Engineering, 7 (2004) 31-39.