News from the American Sports Builders Association                                                       October 2013

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Measurement of Banked Indoor Tracks - Part 2
by Wayne T. Armbrust, Ph.D., ComputomarxTM

The Banked Track Curve Segments

Figure 2, below, illustrates the curve segments of the banked track. In all that follows it will be assumed that the curves at each end of the track are equal, and that each curve is symmetrical about the long axis of the track. Point 0 at the beginning of the curve is the beginning of the transition to the ascending segment of the curve, point 1 marks the end of the transition and the beginning of the ascending segment, point 2 is the end of the ascending segment and the beginning of the transition to the constant banking segment, and point 3 is the end of that transition and the beginning of the constant banking segment.

Continuing, point 4 is the end of the constant banking segment and the beginning of the transition to the descending segment, point 5 is the end of that transition and the beginning of the descending segment, point 6 is the end of the descending segment and the beginning of the transition to the straightaway, and point 7 is the end of that transition at the end of the curve.






The Differential Element of Arc Length

For a banked track, cylindrical coordinates are the natural coordinate system to use. In Cartesian coordinates in three dimensions, the differential element of arc length, ds, is given by

ds2 = dx2 + dy2 + dz2. (4)

In cylindrical coordinates the differential element of arc length is given by

ds2 = dr2 + (rdθ)2 + dz2 (5)

The differential element of arc length in cylindrical coordinates is illustrated in Figure 3.

We can simplify equation (5) by noting that r and z are both functions of θ. We can then write

dr =(∂r/∂θ)dθ (6a), dz = (∂z/∂θ)dθ (6b)

and

ds = √((∂r/∂θ)2 + r2 + (∂z/∂θ)2)dθ (7)

It is the differential in equation (7) that we seek to integrate in a piecewise fashion around the seven segments. Note that for a flat track = ∂r/∂θ = ∂z/∂θ = 0 and equation (7) becomes simply ds = rdθ, which when integrated around the entire curve becomes the well known result πr.

The Banked Track Cross Section

Figure 4 illustrates a cross section of the banked track for lane i (1 ≤ i ≤ 6 typically) at some angle θ around the turn. r0 is the radius to the track side of the curb. di is the slant distance from the curb to the measure line of lane i. α is the angle of banking at angle θ (assumed constant across the cross section) around the curve (0 ≤ α ≤ α0, where α0 is the maximum angle of banking in the constant banking portion of the curve, typically 8° - 12°).

zi is the vertical distance above the straightaway given by

zi = disinα (8a)

and

ri = r0 + dicosα (8b)

Constant Banking Segment Arc Length

We will first consider the easiest portion of the arc length calculation, the constant banking segment (from points 3 – 4 in Figure 2). Throughout this segment, α = α0 and ∂r/∂θ = ∂z/∂θ = 0 and the problem reduces to one dimension as stated in Part I of this paper. Thus, from (7) and (8b),

dsi = (r0 + dicosα0)dθ (9)

Since r0, di, and α0 are constant, we obtain, upon integration,

Sci = (r0 + dicosα0)(θ4 – θ3) (10)

with Sci the arc length of the constant banking segment for lane i, and θ3 and θ4 the angles of the curve traversed (in radians) between points 3 and 4.



Ascending and Descending Segments Arc Length

Because of symmetry, the arc length of the descending segment is the same as the ascending segment, so we will only consider the ascending portion. Because the builders of banked tracks have not shared precise construction details with the author, certain assumptions must be made. Two likely schemes come to mind with regard to the ascending segment. One where the angle of banking increases linearly with the angle traversed, and one where the z coordinate increases linearly. In practice, because the angle of banking is small, there will be little difference in the two approaches.

In what follows we assume the angle of banking increases linearly with angle traversed.

The angle of banking will therefore be given by

α = α1 + (θ – θ1)(α2 - α1)/(θ2 – θ1) (11)

with α1 and α2 the angles of banking and θ1 and θ2 the angles traversed at points 1 and 2, respectively. By substitution into (8a) and (8b), zi and ri become

zi = disin[α1 + (θ – θ1)( α2 - α1)/(θ2 – θ1)] (12a)

ri = r0 + dicos[α1 + (θ – θ1)( α2 - α1)/(θ2 – θ1)] (12b)

∂zi/∂θ and ∂ri/∂θ are given by

∂zi/∂θ = dicos[α1 + (θ – θ1)(α2 - α1)/(θ2 – θ1)][(α2 - α1)/(θ2 – θ1)] (13a)

∂ri/∂θ = - disin[α1 + (θ – θ1)(α2 - α1)/(θ2 – θ1)][(α2 - α1)/(θ2 – θ1)] (13b)

Substituting into (7) and integrating (after simplification), we obtain
Sai = ∫√(r02 + 2r0dicos[α1 + (θ – θ1)(α2 - α1)/(θ2 – θ1)] + di2cos2[α1 + (θ – θ1)(α2 - α1)/(θ2 – θ1)] + di2[(α2 - α1)/(θ2 – θ1)]2)dθ (14)

Where Sai is the arc length of the ascending (and descending) segment of the curve in lane i.

Equation (14) is not integrable in closed form, but since the integrand does not change rapidly with θ it can be readily integrated with high precision using a standard numerical method such as Simpson’s Rule.

In Part III of this paper we will calculate arc length for the most difficult segments of the curve, the transition segments, and summarize the three parts of the paper.


© 2013 American Sports Builders Association 

8480 Baltimore National Pike #307 • Ellicott City, MD 21043 • 410-730-9595 • info@sportsbuilders.org

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