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IV40 Perpvmm
Prnr pk
ON OPTIMAL RUNNING DOWNHILL ON SKIS RYSZARD MAROASKI Institute
for
Aircraft Engineering and Applied Mechanics. ul. Nowowicjska 24.00-665 Warsaw. Poland
Abstract-In the paper the problem of the minimum time schuss is discussed.The skier is modelled as a point mass moving on the slope. The profile of the slope may be represented by any given function. Initial and final conditions are given. The control function is the aerodynamic drag of the skier’s body. Miele’s method is applied for the optimum control problem stated in such a manner. The nature of the optimal solution is independent of the slope shape.
NOMESCLATURE
initial point final point aerodynamic coefficient of air resistance aerodynamic coefficient of lifting force aerodynamic drag frictional force gravitational acceleration lifling force mass of skier-skis system normal reaction of the ground radius of curvature area of skier’s front view projection coronal plane time of schuss (performance index) time control variable wind velocity skier’s velocity path along the slope slope angle profile of slope dynamic friction cocllicient independent variable density of the air fundamental function derivative second derivative maximum value minimum value
onto
the
PRORLEM FORMULATION
It is a well known fact that record results may be calculated beforehand and, therefore, planned. This paper deals with the gliding portion of a ski race. which may make up 50% of World Cup events and almost 100% of competitions to gain the record downhill speed. Downhill slopes have many difficult and dangerous segments where racers must reduce their speed. The value of the critical speed depends on the skier’s skill, and when the racer surpasses this speed limitation he loses the rate of running or simply falls. Since the World Cup events are determined by fractions of a second, the question is how the racer can control his speed to minimize the time taken to cover such a segment. The speed limitation at the end of the
Received in finul Jiirm 8 Scptrmher 1989. 435
gliding section should be satisfied. For high velocities the most effective way of braking is an extension of the skier’s position to increase the air resistance. For low velocities racers usually control their speed by controlling the motion of skis. Both methods of reducing the speed may be taken into account. In such a statement we meet the typical problem of the optimal control theory. The problem of the minimum time schuss(with the drag as a controller) was put forward and solved by Remizov (1980) by applying Pontryagin’s maximum principle. The main weakness of this method is that it only gives necessaryconditions of optimality. In the standard formulation it completely fails when singular arcs occur. Such faults do not exist in Miclc’s method of linear integral cxtremization by Green’s theorem. In this paper the slope angle may not have a constant value as in the previous one. but it may be a given function of an independent variable. It can be shown that the solution of Remizov’s problem is a special case of the problem considered here. The main aussumptions of the skier model are specified below (most of them arc the same as in Remizov’s paper). 1. The skier is reduced to his centre of gravity and is treated as a point massm.This assumption seemsto be reasonable becausethe distance the racer should cover is much greater than the linear dimensions of his body. In such a model. the displacements of the skier’s ccntre of gravity connected with the extension of his position during braking are neglected. as are displacements caused by amortization flexions of legs during the passage of the gully. 2. The skis are in continuous contact with the slope (no jumps). During the jump the skier’s position (and hence his aerodynamic drag) is not an independent variable which can be chosen arbitrarily. but results from equilibrium of the aerodynamical and gravitational moments. 3. The skier’s velocity u is greater than zero everywhere. This assumption enables us to change the independent variable in the equation of motion which, therefore, may be integrated in the known interval (see later). The question may arise if this assumption is satisfiedat the start, It seemsthat in World Cup events
R. MARO~SKI
436
during the start the skier’s centre of gravity has a velocity greater than zero-it is a kind of flying start where only the ski-sticks are locked. In the present model the value of the velocity may be very low but not equal to zero. 4. The friction between the skis and the slope is described by Coulomb’s law, which means that the value of the frictional force is directly proportional to the normal reaction of the ground. It increases on the concave part of the slope and decreaseson the convex section. 5. The racer is able to reduce the speed of schussby means of aerodynamic drag. This assumption is the same as in Remizov’s paper and it works for high velocities typical of competitions to gain the record speed. The other way of controlling the skier’s speed, by controlling the motion of the skis, is also possible in the considered model as it causeschanges in the value of the friction coefficient 11. Modifications of the method for such a case will be discussed in the conclusions section. 6. The given function q(C). describing the slope profile. has at least the second derivative. This assumption has mathematical meaning and is necessary for further considerations (see the equation of motion). It coincides with assumption 2 and warrants the smoothnessof the slope excluding the sharp edges. 7. The slope angle a (see Fig. I) belongs to the interval O,
P_owsJcn+aJ
IV40 Perpvmm
Prnr pk
ON OPTIMAL RUNNING DOWNHILL ON SKIS RYSZARD MAROASKI Institute
for
Aircraft Engineering and Applied Mechanics. ul. Nowowicjska 24.00-665 Warsaw. Poland
Abstract-In the paper the problem of the minimum time schuss is discussed.The skier is modelled as a point mass moving on the slope. The profile of the slope may be represented by any given function. Initial and final conditions are given. The control function is the aerodynamic drag of the skier’s body. Miele’s method is applied for the optimum control problem stated in such a manner. The nature of the optimal solution is independent of the slope shape.
NOMESCLATURE
initial point final point aerodynamic coefficient of air resistance aerodynamic coefficient of lifting force aerodynamic drag frictional force gravitational acceleration lifling force mass of skier-skis system normal reaction of the ground radius of curvature area of skier’s front view projection coronal plane time of schuss (performance index) time control variable wind velocity skier’s velocity path along the slope slope angle profile of slope dynamic friction cocllicient independent variable density of the air fundamental function derivative second derivative maximum value minimum value
onto
the
PRORLEM FORMULATION
It is a well known fact that record results may be calculated beforehand and, therefore, planned. This paper deals with the gliding portion of a ski race. which may make up 50% of World Cup events and almost 100% of competitions to gain the record downhill speed. Downhill slopes have many difficult and dangerous segments where racers must reduce their speed. The value of the critical speed depends on the skier’s skill, and when the racer surpasses this speed limitation he loses the rate of running or simply falls. Since the World Cup events are determined by fractions of a second, the question is how the racer can control his speed to minimize the time taken to cover such a segment. The speed limitation at the end of the
Received in finul Jiirm 8 Scptrmher 1989. 435
gliding section should be satisfied. For high velocities the most effective way of braking is an extension of the skier’s position to increase the air resistance. For low velocities racers usually control their speed by controlling the motion of skis. Both methods of reducing the speed may be taken into account. In such a statement we meet the typical problem of the optimal control theory. The problem of the minimum time schuss(with the drag as a controller) was put forward and solved by Remizov (1980) by applying Pontryagin’s maximum principle. The main weakness of this method is that it only gives necessaryconditions of optimality. In the standard formulation it completely fails when singular arcs occur. Such faults do not exist in Miclc’s method of linear integral cxtremization by Green’s theorem. In this paper the slope angle may not have a constant value as in the previous one. but it may be a given function of an independent variable. It can be shown that the solution of Remizov’s problem is a special case of the problem considered here. The main aussumptions of the skier model are specified below (most of them arc the same as in Remizov’s paper). 1. The skier is reduced to his centre of gravity and is treated as a point massm.This assumption seemsto be reasonable becausethe distance the racer should cover is much greater than the linear dimensions of his body. In such a model. the displacements of the skier’s ccntre of gravity connected with the extension of his position during braking are neglected. as are displacements caused by amortization flexions of legs during the passage of the gully. 2. The skis are in continuous contact with the slope (no jumps). During the jump the skier’s position (and hence his aerodynamic drag) is not an independent variable which can be chosen arbitrarily. but results from equilibrium of the aerodynamical and gravitational moments. 3. The skier’s velocity u is greater than zero everywhere. This assumption enables us to change the independent variable in the equation of motion which, therefore, may be integrated in the known interval (see later). The question may arise if this assumption is satisfiedat the start, It seemsthat in World Cup events
R. MARO~SKI
436
during the start the skier’s centre of gravity has a velocity greater than zero-it is a kind of flying start where only the ski-sticks are locked. In the present model the value of the velocity may be very low but not equal to zero. 4. The friction between the skis and the slope is described by Coulomb’s law, which means that the value of the frictional force is directly proportional to the normal reaction of the ground. It increases on the concave part of the slope and decreaseson the convex section. 5. The racer is able to reduce the speed of schussby means of aerodynamic drag. This assumption is the same as in Remizov’s paper and it works for high velocities typical of competitions to gain the record speed. The other way of controlling the skier’s speed, by controlling the motion of the skis, is also possible in the considered model as it causeschanges in the value of the friction coefficient 11. Modifications of the method for such a case will be discussed in the conclusions section. 6. The given function q(C). describing the slope profile. has at least the second derivative. This assumption has mathematical meaning and is necessary for further considerations (see the equation of motion). It coincides with assumption 2 and warrants the smoothnessof the slope excluding the sharp edges. 7. The slope angle a (see Fig. I) belongs to the interval O,
