Olympic Bobsled, Luge Skeleton Tracks Olympic and World Cup competitions FIL and FIBT closely regulate tracks & competitions All track designs require certification Precise specifications and rules Committees for design, running competitions, etc. Natural tracks exist
Luge event Luge-rider is feet first and supine Sled surface shape conforms to body Minimum time performance
Bridge connects runners to sled
Luge sled steering Blade-ice contact region L~ 12 cm Blade planes tilted inward from vertical Sled and blade planes deformable about lateral horizontal axis so steerable with opposite shoulder/leg pressure Sleds go where they are steered Blades very thin and nearly flat cross-sectional radius r C ~ m longitudinal radius: r L ~14 m
Track design Every 4 years like clockwork Highly banked curves linked by straight flat sloping sections Udo Gurgel (Leipszig) has designed last 8 tracks Designs “look” same
Straight section design Both walls roughly vertical Flat bottom contains rounded inner corners Allows gentle alignment and return to center
Banked curve cross-section design Outside surface smoothly curved (5 g ’ s possible) Inside vertical wall roughly cylindrical Bottom flat but also has rounded inner corner !
Looking up track
Conclusion of FIL Official Report The sled “appears to have hit the wall at an exceptional angle that caused the sled to compress … result[ing] in the sled serving as a catapult when it decompressed launching … the sled into the air”
FIL Official Report Suggested that circumstances of accident were so complex and exceptional as to make it “unknown and unpredictable” 1. 1.International Luge Federation, Official Report to the IOC on accident of Georgian athlete Nodar Kumaritashvili, at the Whistler Sliding Center, Canada on February 12, 2010 during official luge training for the XXI Olympic Winter Games, accessed December 2011.
1. Higher walls added 2. Fillet was removed from ice track corners even in sections where it is specified by the rules (curious) Track modifications
FIL report mentioned “Squaring off the curve of the ice between the base of the track and the sidewalls of the outrun.” Track modifications
Can derive and integrate ode’s for sled motion Assume particle model for sled. If FIL cause of accident sounds like “black magic” what could offer a more cogent explanation? How about Newton’s laws?
Equations of motion on ice surface Fillet surface adequately approximated by torus
u = “longitude” v = “latitude” Fillet (torus) surface shape Two parameters (angles) characterize location on ice surface
First and second fundamental matrices of the toroidal surface
Normal curvature – instantaneous curvature of 3-D path on ice surface
Gaussian curvature K is product of maximum and minimum (principal) curvatures at a point Sign shows principal curvatures on fillet are of opposite sign!
Equations of motion 2 second order ode’s for u and v Neglects aerodynamic and friction forces but includes gravity and large track normal force
Hold speed v o constant, vary angle o between velocity and tangent to fillet toe. Qualitative character of results depends strongly on entry angle o
(Note: x axis not to scale)
Track design flaw is presence of inner fillet. A fillet at the base of an inside wall can launch a slider into flight across the track. Importance of negative Gaussian curvature 1. Positive lateral curvature (v) turns lateral velocity to vertical 2. Negative longitudinal curvature (u) means contact is eventually lost 3. Thereafter flight path lies in vertical tangent plane to inner wall 4. Ejection occurs if vertical velocity sufficient to clear exterior wall.
Speed exacerbates ejection Top speeds 13% larger than design values 13% larger v o => 13% larger lateral speed 13% larger v o => 27% larger zenith height
Quotes Georgian President Mikahail Saakshvili (2010) “ No sports mistake is supposed to lead to a human death.” Richard Feynman (1986 during space shuttle Challenger investigation) “ For a successful technology, reality must take precedence over public relations for nature cannot be fooled.”
Design and review process Much potential for financial conflicts of interest Cost = C$105M I year effort to get 4 numbers for this study failed
Design and review process Track cost = C$105M Much potential for financial conflict of interest My one-year effort ( s and telephone calls to Udo Gurgel and Whistler Sports Legacy Society ) to get 4(!) numbers for this study failed No independent review process e xists
Conclusions Ice track ejection can be explained with a simple analytic model of fillet surface shape and Newton’s laws. Interaction of the right runner with the fillet resulted in vertical velocity necessary for, and was the cause of, ejection in the Whistler accident. Bending of the bridge was caused by the normal force but was not the cause of ejection. A more open review and investigation process is desirable and could only increase resulting safety of athletes using the tracks.