News from the American Sports Builders Association                                                       April 2014

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Measurement of Banked Indoor Tracks - Part 3
by Wayne T. Armbrust, Ph.D., ComputomarxTM

This article is the third in a series and is intended to be read in conjunction with the first and second articles (published in April 2013 and October 2013 issues of Newsline respectively).

Transition Segments Arc Lengths

As with the ascending and descending segments of the curve, certain assumptions must be made due to lack of precise information from manufactures.  Despite this, two requirements for these segments emerge:  1)  The transition segments must join smoothly with the segments on either end, and 2)  Acceleration in the z direction must never be greater than a small fraction of the acceleration due to gravity, either positive or negative, even at the highest sprint speeds.

By joining smoothly with the segments at either end, we mean that not only do the segments join, but that the first derivatives of α with respect to θ are continuous at the junctions.  The limits on acceleration in the z direction will put limits on the lengths of the transition zones depending on the maximum angle of banking α0, and to a lesser extent, the radius of the curve.

The simplest mathematical form to produce the smooth transition zones is a quadratic equation of the form

α = a(θ θs)2+d  (15)

with α the angle of banking at an angle traversed of θ, θs the angle traversed at the beginning of the transition zone, and a and d constants to be determined.

For the transition zone at the beginning of the curve, α equals zero when θ  equals zero and since at point 0 θs equals zero, d equals zero for the transition segment from point 0 to point 1.  The constant a must be chosen such that the transition joins the ascending portion of the curve at point 1 smoothly.

We can rewrite Equation 11 as

α = b(θ θ)  (16)

with

b = (α2 - α1)/(θ2 θ1)  (17)

and θ the value of θ at which α equals zero in Equation 11.  The smoothness requirement determines a; taking the derivative of Equation 17 and setting it equal to the derivative of Equation 15 at θ = θ1 we obtain  

 a = b/2θ1 (18).

Thus, for the transition from point 0 to point 1, the equation for α as a function θ of becomes

α = (b/2θ1 2 = [((α2 - α1)/(θ2 θ1))/2θ1 2.   (19)

Using similar arguments, we can determine the equation for α in the transition region from point 2 to point 3.  In this transition zone we obtain

α = α0 - (b/2(θ3 θ2))(θ - θ3)2  = α0 - [((α2 - α1)/(θ2 θ1))/2(θ3 θ2)](θ - θ3)2.  (20)

Since we assume that the angle of banking is constant across the cross section of the track, Equations 19 and 20 apply for all lanes.  Substituting into (8a) and (8b), zi and ri become

zi = disin[(b/2θ12]  (19a)

ri = r0 + dicos[(b/2θ12]  (19b)

for the transition segment from point 0 to point 1, and

zi = disin[α0 - (b/2(θ3 θ2))(θ - θ3)2]  (20a)

ri = r0 + dicos[α0 - (b/2(θ3 θ2))(θ - θ3)2]  (20b)

for the segment from point 2 to point 3.

∂zi/∂θ and ∂ri/∂θ are given by

∂zi/∂θ = di[cos((b/2θ12)](bθ/θ1) (21a)

∂ri/∂θ = - di[sin((b/2θ12)](bθ/θ1) (21b)

for the first transition segment and

∂zi/∂θ = -di[cos(α0 - (b/2(θ3 θ2))(θ - θ3)2)](bθ/(θ3 θ2))  (22a)

∂ri/∂θ = di[sin(α0 - (b/2(θ3 θ2))(θ - θ3)2)](bθ/(θ3 θ2))  (22b)

for the second.  Substituting into (7) and integrating (after simplification), we obtain

Text Box: θ1

 

St1i = √(r02 + 2r0dicos[(b/2θ12] + di2cos2[(b/2θ12] + di2[bθ/θ1]2)dθ  (23)

 

where St1i is the arc length of the transition segment from point 0 to point 1 in lane i and

Text Box: Θ2

Text Box: Θ3

 

St2i = √(r02 + 2r0dicos[(α0 - b/2(θ3 θ2))(θ - θ3)2] + di2cos2[(α0 - b/2(θ3 θ2)) (θ - θ3)2]

+ di2[bθ/(θ3 θ2)]2)dθ  (24)

where St2i is the arc length of the transition segment from point 2 to point 3 in lane i.  As in the case of the ascending and descending segments, Equations 23 and 24 are not integrable in closed form, but again the integrals can be accurately evaluated using standard numerical methods.

From the symmetry assumed in Part II of this paper, we obtain the arc length of lane i for the entire curve as

Si = Sci + 2Sai + 2St1i + 2St2i.  (25)

Vertical Acceleration Considerations

We now consider accelerations in the z direction.  If we differentiate Equation 19a with respect to θ twice and apply the chain rule, and evaluate the resultant expression at θ equals 0, we obtain an expression for the vertical acceleration ∂2zi/∂t2 at that point.

2zi/∂t2 = (dib/ θ1)(dθ/dt)2  (26)

where dθ/dt is the angular velocity of the runner equal to v/r, where v is the linear velocity and r is the effective radius of the curve at that point (reduced by the banking when applicable).  Performing similar operations on Equation 20a and evaluating the resultant expression at θ equals θ3, the beginning of the constant banking segment, yields

 

2zi/∂t2 = -[dib/(θ3 θ2)](dθ/dt)2cosα0. (27)

Table 1 shows vertical accelerations at the beginning and end of the curve for each lane for a track similar to the IAAF Standard Indoor Track having an angle of banking of 10 in the constant banking segment.  The constant banking segment is 60 of arc with 5 arc transitions joining the 50 of arc ascending and descending segments to the straightaways and constant banking segment.  A running speed of 10 m/s is used for the calculations, which is about the maximum ever expected.  Positive vertical accelerations (making the runner feel heavier) increase going out the lanes, but never exceed 0.21 the acceleration due to gravity.  This is acceptable for the short distance over which this acceleration occurs.

Table 2 shows vertical accelerations at the beginning and end of the constant banking segment for a slightly different configuration.  Here the constant banking segment is 70 with 6 transitions and 43 ascending and descending segments.  Here the negative vertical accelerations (making the runner fell lighter) are never greater than 0.19 g.

In general maximum vertical accelerations increase with the running speed and angle of banking.  They decrease as the transition segments are lengthened and the constant banking segment is shortened.  However, as transition segments are lengthened, the slope encountered by the runners as they ascend and descend the banking increases.  The parameter b is a measure of this slope.   Reducing the length of the constant banking segment negates some of the purpose of having a banked track.  Parameters may be adjusted to suit individual preferences and to match anticipated competitions.  Since vertical accelerations vary with the square of speed and are significant only in the outer lanes, they really are only a consideration for shorter races run all or partially in lanes.

Alternate Approaches

This paper assumes a track where the angle of banking increases linearly with angle traversed.  Other banking schemes are feasible and can be similarly dealt with.  In cases where the angular dependence of the banking is unknown and cannot be readily determined, an alternate procedure may be employed.  Numerous points on the measure line of each lane could be shot from the radius point of the turn using a total station instrument, recording radius, elevation, and azimuth.  This data could then be input to a CAD program which could then produce a smooth arc from the points and calculate the arc length of the curve in each lane.  An iterative process could then be used to determine the angle of azimuth producing a given arc length on a portion of the curve, thus allowing staggers, exchange zones, etc., to be located.  This process would essentially be an automated version of the method of measuring around pins outlined in the first part of this paper, but more accurate.  The disadvantage of this procedure is that it is, as in the case of the pins method, very labor intensive, and since it will involve the services of a surveyor, very costly.


2014 American Sports Builders Association 

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

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