Most recreational alpine skiers enjoy making skidding turning motions with their skis as they head down a slope, being in contrast to that by alpine ski racers, who try to prevent sideslips and associated braking by carving a trace in the snow. Here, though, we only consider the turning motions of recreational alpine skiers, as racing turns are beyond the scope of our study. Kinosita (2) qualitatively considered the turning motion of skis, while Renshaw and Mote (4) analyzed a turning ski during a constant radius, constant speed turn made on a horizontal surface, where a hypothetical thrust was assumed. Recently, we numerically simulated the alpine ski tracks of ski-skier systems on various ski slopes (1), i.e., the equations of motion were solved using a two-dimensional water jet analogy to represent the forces contained in these equations. This led to the present paper, in which the centripetal force that makes such ski turning motions possible is considered to be a skidding force generated by either impacting soft snow or cutting hard snow; an expansion of scope requiring a three-dimensional analysis. Our main objective is to numerically simulate recreational ski tracks of ski-skier systems when the snow is soft or hard, and also to quantitatively show how the placement of the boot and angle of side-cuts respectively effect such ski tracks.
Resistance Forces Acting on a Ski and Equations of Motion
When skidding occurs as alpine skis are turning on a snow surface, resistance forces act on them. These forces are considered to be the longitudinal friction generated between the ski and snow, the snow resistance force corresponding to the system centripetal force, and gravity. We assume that when snow is soft, the resistance force is a “snow impacting force,” whereas when hard, it is a “snow cutting force.” The impact of snow is treated using a three-dimensional water jet analogy, while cutting of snow uses the application of metal cutting theory. The term water jet is commonly used in fluid mechanics, namely, the forces acting on a solid wall being hit by a water jet can be calculated by determining the difference between the jet's momentum before and after it impacts the wall. Metal cutting theory provides the required mechanics for determining the forces needed by machine tools to cut metals. It should be noted that only a single ski is analyzed here, though this does not adversely affect results, i.e., during the curvilinear motion produced by turning, the outer ski is mainly loaded and the inner one plays a less important role in the control of direction.
Snow Impacting Force
A single ski without a side-cut, i.e., a mid-length taper in the width of the ski (Fig. 1), is considered to slide on a snow surface with velocity V and angular velocity ω, while having an angle γ between the ski axis and ski velocity as shown inFigure 2. The origin of the coordinate axes is taken as the location of the ski boot, with the boot's location along the ski length being denoted as η, which is the ratio of the ski length in front of the boot r F to the total ski length l. The depth that the ski penetrates the snow is d j, while the angle of its bottom surface against the snow surface is α. At point r, which is the coordinate along the longitudinal axis of the ski, the relative velocity ν of the snow perpendicular to the ski axis is ωr +V sin γ. As the ski slides through the snow, it applies impacting forces to the snow over length dr, namely, horizontal component F y′ and vertical componentF z′. The ground also applies the forceG z′ to the snow to change the direction of ν. These forces can be calculated using the change in the momentum of snow during timeΔt, i.e., equation (1)
In equation 1, Δm is the mass of snow impacting length dr during Δt, being expressed as:equation (2)
where ρ is the mass density of snow and ϕ the angle between the side of the ski and relative snow velocity. The forces acting on the ski are equal in magnitude and opposite in direction to F y′ andF z′ according to Newton's third law of motion. Substitution of equation 2 into 1 givesF y′ corresponding to the horizontal force acting on the ski. By assuming that the frictional force between the bottom of the ski and snow is negligible in comparison with F y′ andF z′, then F z′ =F z′/tan α. Integration ofF y′ along the front part of the ski axis gives the total force to that part of the ski, as well as the point where the total force acts. For the rear part, we simply take ν = -ωr + V sin γ and follow the same procedure. The snow impacting force for a ski with a side-cut is calculated as previously described(1).
Snow Cutting Force
When skiing on a hard snow surface, the edge of the ski must cut the snow surface to obtain a resistance (cutting) force corresponding to the centripetal force. Lieu and Mote (3) experimentally obtained the cutting force for ice, but not for snow. In addition, they only considered two-dimensional (orthogonal) cutting, which occurs when the velocity of the ski is perpendicular to its longitudinal axis. Renshaw and Mote (4), however, empirically derived the orthogonal cutting force F i′ (N · m-1) for snow as a function of the edging angle α (radian) and cutting depthd C (m), i.e., equation (3)
where K d is the following smoothing function:Equation
F i′ was experimentally observed by Lieu and Mote to be normal to the ski's bottom surface.
To use equation 3 in our analysis of ski turning motion, we applied the mechanics used in metal cutting theory. However, two main differences exist: 1) in conventional metal cutting processes, the rake angle αn is positive, whereas it is negative in snow cutting; and 2) the inclination angle i (Fig. 3) when oblique metal cutting is usually small, whereas it is quite large in snow cutting by a ski. The orthogonal metal cutting shown inFigure 3 corresponds to the experiment by Lieu and Mote(3), in which αn is negative. Consequently, direct application of this theory to ski snow cutting is impossible, although making a rough approximation is still considered feasible. We therefore applied it to estimate the oblique cutting forces from the orthogonal cutting forces given by equation 3. Namely, based on experimental results by Usui and Hirota (7) and Stabler's rule(5) for oblique metal cutting, it is assumed that the horizontal component of oblique cutting force F H and that of the orthogonal one F H′ have the relationF H ≊ F H′; while the vertical component of oblique cutting force F V and that of the orthogonal one F V′ have the relationF V ≊ F V′ (seeFig. 3). Since the transverse component of oblique cutting force F T (see Fig. 3) is still unknown, we calculate the force corresponding to the centripetal force acting on the system per Usui et al. (6), i.e.,equations (4),(5),(6)
where αn = α - π/2. Using known FH and FV, equations 4 and 5 are first solved with respect to N t and F t, with their resultant values being substituted into equation 6 to give FT. The forces acting on length dr of the ski can subsequently be calculated, after which integration along the length of the ski gives the total forces acting on it.
The frictional force in the opposite direction of coordinatex′ (Fig. 2) is determined by multiplying the frictional coefficient μL by the compressive force applied to the snow by the bottom of the ski. This compressive force acts normal to the ski's bottom surface.
Equations of Motion
Since all the forces within the considered system are now known, we can write in ski-slope x- and y-coordinates (Fig. 4) the following associated equations of motion for a corresponding ski-skier system: equations (7),(8),(9)
where Ψ is the angle of the ski slope, β the angle between ski axis and y-axis, g the gravitational acceleration, m the mass, and I the moment of inertia of the system; whileR L and R T are the total forces acting along the longitudinal axis x′ and transverse axisy′, respectively, being caused by either impacting or cutting of snow. Equation 9 considers the rotational motion around the ski boot location (x′ = y′ = 0), where: Equation
Equations 7-9 were numerically solved using the Runge-Kutta method.
RESULTS AND DISCUSSION
All numerical simulations were performed using the following values:m = 65 kg, l = 2 m, ρ = 500 kg · m-3,μL = 0.01, where m is the assumed mass of the skier and ski (60 and 5 kg). For the considered ski-skier system, I is calculated by representing the skier as a 0.4-m diameter solid cylinder(located at x′ = y′ = 0) and the ski as a 2-m-long bar. The coordinate axes x and y are used to represent a ski slope with angle Ψ (Fig. 4). The time interval of the Runge-Kutta method is taken as 0.05 s. The initial conditions of the system at x = y = 0 are x-direction velocity u o = 6 m · s-1, y-direction velocity νo = 6 m · s-1.
The angle between the ski axis and y-axis β0 = (55/180) π. Other parameters were set as: Ψ = (5/180) π, α = π/6,d J = 0.03 m, d C = 0.001 m, η =(r F/l) = 0.59, 0.57, and 0.55.Figures 5 and 6 present simulation results, where the indicated ski length is shown at four times actual size in 1-s intervals. Ski tracks of a downhill turn induced by the snow impacting force are shown inFigure 5 for three different boot locations, where asη increases, i.e., the boot location moves to the rear, the rotational motion becomes much more pronounced and the radius of the curvilinear motion becomes smaller. Figure 6 shows corresponding results induced by the snow cutting force. Note that η has the same general effects on the turning motion as those induced by the snow impacting force. The practical application of this strong effect relates to locating the attachment of bindings that “fix” the placement of the boot on a ski. The numerical results quantitatively show that the attachment point of bindings to ski has a strong effect on the rotational and curvilinear motion.
All present-day skis have a side-cut, since it is thought to enable easier turning, i.e., it changes the ski's attack angle against the snow at the front and rear part of the ski. We previously quantitatively showed the effect of a side-cut for η = 0.5 (center of the ski) using a two-dimensional water jet analogy that corresponds to the three-dimensional one when α = Ψ/2. Our analysis assumed that the sides of the ski had no curvature. The attack angles of the ski's front and rear part were considered to be γ +θF and γ - θR, respectively, whereθF/θR is the angle between the front/rear side of the ski and the ski's longitudinal axis (Fig. 1).Figure 7 shows results using the presented three-dimensional water jet analogy, where the side-cut is varied with η = 0.55. The system properties and initial conditions were the same as those forFigure 5. Note that as θF andθR increase, i.e., a more distinct side-cut is present, the rotational and curvilinear motions become more pronounced. The effect of a side-cut on ski tracks induced by snow cutting will be investigated in the future.
Our main results are summarized as follows: 1) Both the snow impacting force and snow cutting force make it possible for a ski to turn when downhill skiing, and we confirm quantitatively the following common knowledge. 2) The location of the ski boot has marked effects on the radius of curvature of ski tracks when downhill skiing. 3) A ski's side-cut significantly affects the radius of curvature of ski tracks when downhill skiing.
We are currently conducting experiments using sand instead of snow in order to evaluate the presented theoretical results concerning the effect of the snow impacting phenomenon.