Proyecto de Investigación

Aerodynamic drag is by far the most significant variable for track cycling, accounting for 90% of total resistance over 50kph, and is dependent on the air density, ρ, drag coefficient, Cd, frontal area, A, and

velocity,V (D= 1₂ρCdAV2).

Air density is dependent on the temperature, pressure and relative humidity, all of which should be accounted for and be as close as possible to actual race conditions in order for the highest accuracy to be achieved. Di Prampero et al. [1979] and Olds et al. [1993] did not take into account the relative humidity when creating an expresion for the air density for simplicity, and lumpedCdAρas a single parameter (CdAρ=0.404). These

models would therefore only be useful for analysing athletes at the same atmospheric conditions as were used when generating the models. Olds et al. [1995] did update the model to include a correction term for the relative humidity,CFH, as shown in Equation 3.1, where 1.225 is the density of air at sea level at an ambient

temperature of 288.15K, PB is the absolute pressure (Pa) and T is the temperature (K) at race conditions.

ρ= 1.225PB 760

288.15

T CFH (3.1)

Another common method of calculating air density is by using partial pressures of wet/dry air and the saturation vapour pressure of water, as shown in Equation 3.2, where Pd is the partial pressure of dry air

(Pa),Pv is the partial pressure of wet air (Pa),Rdis the specific gas constant for dry air (JK−1kg−1),Rv is

the specific gas constant for wet air (JK−1_kg−1_{), T is the temperature (K),φis the relative humidity, and} Psatis the saturation vapour pressure of water (Pa).

ρ= Pd RdT + Pv RvT (3.2) Pd=P−Pv Pv =φPsat Psat= 610.78×10(7.5T−2048.625/T−35.85)

Although this empirical correlation is good for estimating the air density, it does not quite agree with measured data. An alternative method for calculating the air density for moist air is described by Davis [1992], shown in Equation 3.3, where P is the pressure (Pa), Ma is the molar mass of dry air,xv is the mole

fraction of water vapour, T is the temperature (K), R is the molar gas constant, Mv is the molar mass of

water, and Z is the compressibility factor. This method calculates the saturation vapour pressure at ambient temperature, an enhancement factor at ambient temperature and pressure, the mole fraction of water vapour, the compressibility factor of air, and finally the air density, using known temperature, pressure and relative humidity data. ρ= P Ma ZRT ! 1₋xv " 1₋Mv Ma #$ (3.3) Temperature also affects air density, which in turn affects the aerodynamic drag. Di Prampero et al. [1979] suggests that a change in temperature by 3°C will cause a change in drag by 1%. Atkinson et al. [2003] suggests that an increase in temperature by 3°C and a decrease in pressure by 2KPa will cause a reduction in air density, and therefore drag, by 4%. These results highlight the importance of ensuring the temperature and pressure used for modelling are as close as possible to race conditions in order for the results to be as accurate as possible.

Air density changes with altitude, and at a lower air density (higher altitude) the aerodynamic drag decreases, enabling athletes to pedal faster for the same amount of power output. However, there is also a decline in maximal oxygen uptake, VO2max, when the air density is lower, and although athletes can acclimatise to

such conditions, a decrease in VO2max will lead to a decrease in power output. The decline in power output

can be assumed to be equivalent to the decline in VO2max [Di Prampero et al., 1979, Olds et al., 1995,

Bassett Jr et al., 1999] and for a trained, acclimatised subjects this can be described by Equation 3.4 [Davis, 1992, Bassett Jr et al., 1999], wherezis the altitude (m) (r2= 0.9729) . Equation 3.4 is useful for estimating

power profiles and finishing times for athletes competing on different indoor velodromes around the world, even when only one set of power or velocity data is available at one known altitude.

P = (−1.122)z2−(1.8991)z+ 99.921 (3.4)

An alternative method for comparing race data on similar terms is described by Bassett Jr et al. [1999], who used Equation 3.5 to adjust all hour records to sea level in order to compare past outcomes on equal terms, where ρis the air density (kgm−3_{), g is the acceleration due to gravity (ms}−2_{), n is a polytropic gas}

coefficient (1.235), z is the altitude above sea level (m), R is the specific gas constant for air (287.1Nm) and T is the temperature (K). Bassett Jr et al. [1999] used this equation to generate a correction factor to account for changes in altitude, where sea level = 1.0.

ρ2 ρ1

= [1₋g(n₋1)(z2−z1)/(nRT1)](1/(n−1)) (3.5)

Olds et al. [1995] corrects for altitude by using default values for temperature and pressure as described by an iterative appoximation method; using an initial pressure, PB, of 101325Pa the pressure at a specific altitude,

z, is calculated by firstly finding an approximation for the acceleration due to gravity, g, then finding the temperature at this altitude, T, and finally finding the pressure for the ith+1 iteration, PBi+1:

g= 9.80665( 6378 6378+ z 1000) 2 T=288.15-0.0065Z PBi+1=PB_288.15−T_0.0065Z ×_287×g_0.0065

Depending on the available data and aim of the mathematical model, any of the above equations are valid for analysing the effects of altitude on cycling performance. The model by Bassett Jr et al. [1999] is more useful for making comparisons between two results at different elevations on equal terms, where as the model by Olds et al. [1995] is more useful for determining the pressure from a known temperature and elevation. The equation for aerodynamic drag is D = 1₂ρCdAV2. The drag coefficient, Cd, varies significantly with

position and size of riders, ranging from Cd = 0.35 for riders on the drops toCd = 1.1 for riders standing

on the pedals [Martin et al., 2007]. This results in a wide variety of values for drag coefficient reported in the literature. Estimates of the frontal area of the rider, rider and bike, or rider, bike and wheels also vary widely throughout the literature, as shown in Table 3.1. The frontal area can be determined by photographic weighing, manual planimetry, computerised planimetry or using Computer Aided Software (CAD). Debraux et al. [2009] compared the methods of using CAD, photographic weighing and computerised planimetry to calculate the frontal area of a cyclist and found no significant difference between the three methods, as shown in Table 3.2. It should be noted that the drag coefficient and frontal area will vary between riders, due to differences in body size and shape.

n Projected Frontal Area (m2₎

Hoods Drops Aero

Capelli et al. (1993) 2 0.394

Davies (1980) 15 0.5

Faria & Cavanagu (1979) - 0.5 0.35 Gross et al. (1983) - 0.399 0.362 McLean (1994) 10 0.387 10 0.465 Neumann (1992) - 0.6 0.5 0.4 Nonweiler (1956) 3 0.396 0.326 Olive (1996) 17 0.605 0.563 0.493 Pugh (1974) 6 0.47 0.46 0.42 Swain et al. (1987) 5 0.318 5 0.378 Wright et al. (1993) 10 0.332

Table 3.1: Reported values for the projected frontal area of cyclists [Olds and Olive, 1999]

Photographic Weighing CAD Computerised Planimetry

Upright Position 0.432 0.430 0.533

Aero Position 0.341 0.338 0.426

Table 3.2: Comparison of three methods to calculate the frontal area of a cyclist (m2_{)[Debraux et al., 2009]}

Olds and Olive [1999] also carried out a study to compare these three methods and found that all methods showed high precision and reliability, and were similar to within 3.3% of each other. They found that the differences between reported values was most likely due to the perspective and distortion effects of photography; the frontal area changed with displacement of the reference dimension, the distance between the camera and the cyclist, and the focal length of the camera. Therefore it is difficult to compare reported values of frontal area with each other unless these factors are also reported in the literature. Although Olds and Olive [1999] found a valid explanation for the differences in calculated frontal areas for cyclists, it is unlikely that the test subjects in each study adopted exactly the same position as each other, nor had the same equipment or set up. Although two athletes may both adopt a “drops” position it is unlikely that both will be in exactly the same position, resulting in a difference in frontal area. More reliable models are those which allow the frontal area and drag coefficient for the specific athlete, set up and equipment to be input into the model, rather than relying on previously reported values in the literature or even correction factors, such as the model by Bassett Jr et al. [1999].

Olds et al. [1993], Di Prampero et al. [1979] and Bassett Jr et al. [1999] have created expressions to calculate the frontal area from the body surface area (BSA) of the rider and shape of the bike, rather than using photographic weighing, manual planimetry or computerised planimetry. Previous data has shown that there is a linear relationship between body surface area, BSA, and frontal area, A, although the accuracy of this relationship varies from r2_{=0.401 to r}2_{=0.757 [Bassett Jr et al., 1999]. A possible reason for this discrepancy}

is explained by Bassett Jr et al. [1999], who state that the slope of the linear regression curve is similar for all studies, but the ratio of BSA to A favours larger subjects and unless this is taken into account the power required for larger subjects to overcome aerodynamic drag will be overestimated. However, a significant change in rider position will have a huge impact on the frontal area regardless of the height and weight of the athlete. Martin et al. [2006] found a 68% difference in drag area for subjects whose body mass and predicted body surface area differed by only 10% and 4% respectively, showing that a difference in drag area is not simply caused by body mass or surface area. Therefore, the method of calculating the frontal area from body surface area should not be used unless no other data is available, and even then this method should take into account differences in body size and only be used as an approximation. The equation by Bassett Jr et al. [1999] used to calculate the frontal area based on body surface area, which does accounts for body size, is shown in Equation 3.6.

A= 0.0293m0.425H0.725+ 0.0604 (3.6)

Barelle et al. [2010] carried out a study to determine the relationship between the frontal area and the body height and mass, and the frontal area and the inclination and length of the helmet. Using nine professional, male cyclists, five different helmets, and three helmet angles (low, usual and high) Barelle et al. [2010] measured the frontal area by computerised planimetry with a reference board located at 0.15m from the crank set; the optimal location of the reference board was determined as a ratio of proportions and distances between body parts. The results showed that the frontal area was sensitive to both body height (H) and mass (m), as shown by Equation 3.7, and related to the helmet inclination (α) and length (L) by a polynomial

regression when the helmet was in the usual position, as shown by Equation 3.8.

A=Acb+Ach

Acb= 0.045×H1.15×m0.2794 (3.7)

Ach= 0.329×(Lsinα)2−0.137×Lsinα (3.8)

Martin et al. [1998] recommended using drag area,CdA, instead of separating the variables of drag coefficient

and frontal area, as drag area is dominated by the separation associated with rider position, shape, size, and surface roughness; as the frontal area changes, the flow over the rider and bike will also change. However, there are still large variations is the percentage reduction in drag area from changes in position in the literature, most likely due to the difference in stature between individual riders. Table 3.3 shows a summary of reported values of the reduction in drag area from the stated reference position. Comparing these results for athletes going from a dropped position to a time trial position, an athlete may reduce their drag by anything between 2.2% and 31% depending on their stature. Models which allow actual values for drag coefficient, frontal area, or drag area to be changed for individual riders are more realistic than those which use a correction factor for different riding positions, such as the model by Bassett Jr et al. [1999], or those based on a drag coefficient for a specific riding position, such as the model by Di Prampero et al. [1979] which is only applicable to riders on a racing bike in a fully dropped position.

Kyle & Kyle Kyle Richardson & Zdravkovich Grappe Burke (1984) (1991b) (1989) Johnson (1994) (1996) et al. (1997)

Reference Position Dropped Dropped Cowhorn Dropped Dropped Upright

Dropped-crouch 20% 5%-13% 8%

Speed-skating 12%

Time-trial 14%-16% 2.2% 24%-31% 12%

Hill descent 28% 24%-31%

Obree 28%

Table 3.3: Percentage reduction in drag area from stated reference position

Values for drag area can be measured from wind tunnel tests, coast down tests or towing tests. Each of these methods has their own advantages and disadvantages. Martin et al. [2006] used linear regression to calculate the drag area based on field tests using a power meter and compared the results to wind tunnel tests. There was no significant difference between the field derived drag area and the drag area recorded from wind tunnel testing. However, regression analysis precision is limited by the effects of changes in the ambient conditions, especially wind [Candau et al., 1999]. Candau et al. [1999] used coast down tests to determine the drag area of cyclists, the results of which lay within the same range as wind tunnel tests. However, Candau et al. [1999] also suggested that coast down tests have a tendency to overestimate drag because there is no drag interaction between the cyclist and the floor, as there is in wind tunnel testing, and that wind tunnel tests have a higher reproducability even compared to coast down tests carried out in a hallway where there are no effects of wind or changes in temperature or grade.