On modelling bicycle power for velodromes: Part I: Formulation for individual pursuits
Michael A. Slawinski, Raphaël A. Slawinski, Theodore Stanoev
OOn modelling bicycle power for velodromes: Part I
Formulation for individual pursuits
Michael A. Slawinski ∗ , Rapha¨el A. Slawinski † , Theodore Stanoev ‡ August 4, 2020
Abstract
For a moving bicycle, the power meters respond to the propulsion of the centre of massof the bicycle-cyclist system. Hence, an accurate modelling of power measurements — on avelodrome — requires a distinction between the trajectory of the wheels and the trajectory ofthe centre of mass. We formulate and examine an individual-pursuit model that takes intoaccount the aforementioned distinction. In doing so, we provide the details of the invokedphysical principles and mathematical derivations.
In this article, we consider a mathematical model to account for power-meter measurements onvelodromes. This work is a mathematization of certain aspects of studies presented by Martin et al(1998) and Underwood (2012). The discussed model is pertinent to individual pursuits. Herein,we assume a cyclist to follow the black-line, in a constant aerodynamic position, with a constantblack-line speed, which is tantamount to a constant cadence.Using this model, for a given cyclist, we can calculate the power required to achieve a desiredtime or — since the relation between power and speed is one-to-one — the time achievable with aparticular power. Also, for repeated laps, we can estimate model parameters from the power andspeed measurements. The method proposed to estimate these parameters is specific to velodromes,and different from the circuit study presented by Chung (2012).We begin this paper by presenting a model and justifying its mathematical formulation. We illustrateboth forward and inverse applications of the model. We conclude this article by a discussion ofresults. This article contains also two appendices. ∗ Memorial University of Newfoundland, Canada, [email protected] † Mount Royal University, Canada, [email protected] ‡ Memorial University of Newfoundland, Canada, [email protected] a r X i v : . [ phy s i c s . pop - ph ] A ug Formulation
A mathematical model to account for the power required to propel a bicycle — along a straightcourse — with speed V → is (e.g., Danek et al, 2020a) P = F ← V → (1)= gravity (cid:122) (cid:125)(cid:124) (cid:123) m g sin Θ + change of speed (cid:122) (cid:125)(cid:124) (cid:123) m a + rolling resistance (cid:122) (cid:125)(cid:124) (cid:123) C rr m g cos Θ (cid:124) (cid:123)(cid:122) (cid:125) normal force + air resistance (cid:122) (cid:125)(cid:124) (cid:123) η C d A ρ ( V → + w ← (cid:124) (cid:123)(cid:122) (cid:125) air flow speed ) − λ (cid:124) (cid:123)(cid:122) (cid:125) drivetrain efficiency V → , where F ← stands for the forces opposing the motion and V → for the ground speed. In particular, m is the mass of the cyclist and the bicycle, g is the acceleration due to gravity, Θ is the slope of a hill, a is the change of speed, C rr is the rolling-resistance coefficient, C d A is the air-resistance coefficient, ρ is the air density, w ← is the wind component opposing the motion, λ is the drivetrain-resistancecoefficient, η is a quantity that ensures the proper sign for the tailwind effect, w ← < − V → ⇐⇒ η = − η = 1 .To consider a steady ride, a = 0 , on a flat course, Θ = 0 , in windless conditions, w = 0 , we write P = C rr mg + C d A ρ V → − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V → . (2)In modelling the power along a straight course, there is no distinction between the ground speed ofthe centre of mass and of any other point of the bicycle-cyclist system. The distinction appears ifthe cyclist deviates from a straight course by leaning, which becomes pronounced on a velodrome.Let us consider a velodrome whose black-line distance is C , and the banking angle and radius are θ and r , respectively. Also, let us assume that the centre of mass of the bicycle-cyclist system is h above the ground, without lean. If so, the radius of the centre-of-mass trajectory, r CoM , is shorterthan r by h sin ϑ , where ϑ is the angle, measured from the vertical, at which the cyclist leans. Hence,the distance traveled — in one lap — by the centre of mass is shorter than the black line by2 π r − π r CoM (cid:122) (cid:125)(cid:124) (cid:123) ( r − h sin ϑ ) = 2 πh sin ϑ . (3)Thus — neglecting a progressive leaning and straightening between the straights and the banks — thedistance travelled by the centre of mass, along the straights and along the banks, for a single lap, is C − πr and 2 π ( r − h sin ϑ ) , (4)respectively.Given a laptime, t (cid:9) = CV → , (5) Herein, we assume that the position of a cyclist on a bicycle remains the same, which is a reasonable assumptionfor an individual pursuit, after the initial acceleration. Hence, h is constant, and the change of the height of the centreof mass is due only to the lean angle, ϑ . igure 1: Christiaan Huygens confident of a centripetal force, which he mathematized in 1659 we obtain V → , which is the average black-line speed, per lap. For the centre of mass, if we assumea constant black-line speed, t (cid:9) = C − πrV → + 2 π ( r − h sin ϑ ) V (cid:120) , (6)where V → is the average centre-of-mass speed along the straights and V (cid:120) is the average centre-of-mass speed along the banks. V → is both the black-line speed and the centre-of-mass speed along thestraights, since the lap average of a constant speed is the same as the average for any lap segment,and — along the straights — there is no distinction between the speed of the centre of mass and thespeed of any point of the bicycle-cyclist system.The lean angle of a cyclist — as illustrated in Figures 1 and 2, and entailed by expression (18), below,as well as discussed in Appendix A — is ϑ = arctan F cp F g , (7)where the magnitude of the centripetal force is F cp = m V (cid:120) r CoM , (8)and of the force of gravity is F g = m g ; hence, V (cid:120) = (cid:112) g ( r − h sin ϑ ) tan ϑ . (9)Inserting expression (9) into expression (6), we obtain an equation for ϑ , whose solution entails V (cid:120) .Following expressions (4), the proportion of distance travelled, per lap, by the centre of mass, is1 − πrC and 2 π ( r − h sin ϑ ) C , (cid:104) V (cid:105) = 11 V → − πrC − πh sin ϑC + 1 V (cid:120) π ( r − h sin ϑ ) C − πh sin ϑC (10a)= V → V (cid:120) ( C − πh sin ϑ ) C V (cid:120) + 2 π (cid:0) r (cid:0) V → − V (cid:120) (cid:1) − V → h sin ϑ (cid:1) . (10b)The average power per lap is P = 11 − λ V → V (cid:120) ( C − πh sin ϑ ) C V (cid:120) + 2 π (cid:0) r (cid:0) V → − V (cid:120) (cid:1) − V → h sin ϑ (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) (cid:104) V (cid:105) (11a)C rr mg (cid:18) − πrC (cid:19) (11b)+ C rr m g (sin θ tan ϑ + cos θ ) (cid:124) (cid:123)(cid:122) (cid:125) N cos θ + C sr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m g sin( θ − ϑ )cos ϑ (cid:124) (cid:123)(cid:122) (cid:125) F f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin θ π r CoM (cid:122) (cid:125)(cid:124) (cid:123) ( r − h sin ϑ ) C (11c)+ C d A ρ (cid:18) − πrC (cid:19) V → + 2 π r CoM (cid:122) (cid:125)(cid:124) (cid:123) ( r − h sin ϑ ) C V (cid:120) , (11d)with ϑ , V → and V (cid:120) that result from equations (5), (6) and (9). Herein, C sr is the coefficient of thelateral friction and | | stands for the magnitude. If C → ∞ , expression (11) reduces to expression (2),as expected.The second fraction in factor (11a) is the average centre-of-mass speed, per lap, which combines,proportionally, V → , along the straights, and V (cid:120) , along the banks. Summand (11b) is the rolling re-sistance along the straights. Summand (11d) is the air resistance, which is a function of proportionsbetween V → , along the straights, and V (cid:120) , along the banks. A velodrome, such as V´elodrome de Bordeaux-Lac, has C = 250 , r = 23 , θ = π/ .
23 ,along the straights; the latter is not zero to avoid an excessive change of track inclination between the banks and thestraights. We could refine the model by including, (cid:0) C rr cos Θ + C sr sin Θ (cid:1) m g (cid:18) − πrC (cid:19) , to accounts for the force of gravity along the straights, which corresponds to summand (11c), with ϑ = 0 . However,since Θ is small, cos Θ ≈ ≈ rr m g (cid:18) − πrC (cid:19) , which is summand (11b). We could refine the model by including the effect of air resistance of rotating wheels (Danek et al, 2020a, Ap- igure 2: Force diagram: Inertial frame
To formulate summand (11c), we use the relations among the magnitudes of vectors N , F g F cp and F f , illustrated in Figure 2. In accordance with Newton’s second law, for a cyclist to maintain ahorizontal trajectory, the resultant of all vertical forces must be zero, (cid:88) F y = 0 = N cos θ + F f sin θ − F g . (12)In other words, F g must be balanced by the sum of the vertical components of normal force, N ,and the friction force, F f , which is parallel to the velodrome surface and perpendicular to theinstantaneous velocity. Depending on the centre-of-mass speed and on the radius of curvature forthe centre-of-mass trajectory, if ϑ < θ , F f points upwards, in Figure 2, which corresponds to itspointing outwards, on the velodrome; if ϑ > θ , it points downwards and inwards. If ϑ = θ , F f = .Since we assume no lateral motion, F f accounts for the force that prevents it. Heuristically, it canbe conceptualized as the force exerted in a lateral deformation of the tires.For a cyclist to follow the curved bank, the resultant of the horizontal forces, (cid:88) F x = − N sin θ + F f cos θ = − F cp , (13)is the centripetal force, F cp , whose direction is perpendicular to the direction of motion and pointstowards the centre of the radius of curvature. According to the rotational equilibrium about thecentre of mass, (cid:88) τ z = 0 = F f h cos ( θ − ϑ ) − N h sin( θ − ϑ ) , (14)where τ z is the torque about the axis parallel to the instantaneous velocity, which implies F f = N tan( θ − ϑ ) . (15)Substituting expression (15) in expression (12), we obtain N = m g cos θ − tan( θ − ϑ ) sin θ = m g (sin θ tan ϑ + cos θ ) . (16)Using this result in expression (15), we obtain F f = m g (sin θ tan ϑ + cos θ ) tan( θ − ϑ ) = m g sin( θ − ϑ )cos ϑ . (17) pendix D), which would require introducing another resistance coefficient to summand (11d), if the wheels are thesame, or two coefficients, if they are different. F f , is a dissipative force, it does negative work. Hence, the work doneagainst it — as well as the power — need to be positive. For this reason, in expression (11c), weconsider the magnitude of F f .To relate F cp and ϑ , we use results (16) and (17) in expression (13), to obtain F cp = N sin θ − F f cos θ = m g tan ϑ , (18)which is tantamount to expression (7). Examining expressions (8) and (18), we see that the leanangle is a function of the centre-of-mass speed and of the radius of curvature for the centre-of-masstrajectory; it is independent of mass or the track inclination. Figure 3: F g in terms of vertical components of N and F f In terms of solutions (16) and (17), expression (12) — as a function of ϑ , for a fixed value of θ — isshown in Figure 3. F g is constant, as required. Also, as required, F f = 0 and N cos θ = F g , at ϑ = θ . For ϑ < θ , F f points outwards, hence — in accordance with Figure 2 — it is positive. For ϑ > θ , F f points inwards and, hence, is negative. The crossing of two curves corresponds to ϑ atwhich the vertical components of N and F f are equal to one another. Let us consider the following values. For the bicycle-cyclist system, m = 111 , h = 1 . d A = 0 . rr = 0 .
002 , C sr = 0 .
003 and λ = 0 .
02 . For the velodrome, C = 250 , r = 23 and θ = 0 . g = 9 .
81 and ρ = 1 .
225 .Let the laptime be such that, according to expression (5), the corresponding black-line speed is V → = 12 . In accordance with expressions (6) and (9), ϑ = 0 . V (cid:120) = 11 . P = 229 . V (cid:120) < (cid:104) V (cid:105) = 11 . < V → , as expected.The values of summands (11b), (11c) and (11d), which represent the forces opposing the movement,are, respectively, 0 . . . For consistency with power meters, whose measurements are expressed in watts, which are kg m / s , we use the SI units for all quantities. Mass is given in kilograms, kg , length in meters, m , and time in seconds, s ; hence, speedis in m / s ; angles are in radians. P = 243 . . . N = [ − . , . F g = [0 , − . F cp = [ − . , F f = [181 . , . F cp . Also, the orientation of F f is θ = 0 . For this numerical example, expression (11) can be written as229 . . d A1 − λ (cid:124) (cid:123)(cid:122) (cid:125) X +11530 . rr − λ (cid:124) (cid:123)(cid:122) (cid:125) Y +1124 . sr − λ (cid:124) (cid:123)(cid:122) (cid:125) Z . (19)In contrast to expression (1), for expression (2), and for its extension, given by expression (11),the resistance coefficients appear only as ratios. Hence, even with many independent equations, wecannot obtain — as an inverse solution — the values of C d A , C rr , C sr and λ , but only the ratios, X , Y and Z .To obtain the values of X , Y and Z , we perform the least-squares fit of ten equations analogous toequation (19), with V → ∈ (11 . , .
5) , whose matrix representation is . . . . . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XYZ . (20)The least-squares solution of system (20) is X = 0 . Y = 0 . Z = 0 . λ = 0 .
02 , we obtain C d A = 0 . rr = 0 .
002 , C sr = 0 .
003 , as expected.For the measured values of P — as opposed to the modelled ones, for which three equations sufficeto solve for X , Y and Z — a redundancy of the laptime information allows us to estimate them,and to obtain statistical information about the empirical adequacy of a model, which, however7ophisticated, remains only a mathematical analogy for a physical realm. This redundancy couldcorrespond to different laptimes during a single ride.An insight into the consistency of information can be gained by writing each equation of system (20)as 1 = aP C d A1 − λ (cid:124) (cid:123)(cid:122) (cid:125) X + bP C rr − λ (cid:124) (cid:123)(cid:122) (cid:125) Y + cP C sr − λ (cid:124) (cid:123)(cid:122) (cid:125) Z , and plotting a/P , b/P and c/P . For system (20), they are collinear. For measurements, thedeparture from the collinearity is indicative of the quality of the model and of the measurement errors.Within a model, such a plot can be used to study the sensitivity of X , Y and Z to perturbations.To estimate C d A , C rr and C sr , the value of λ needs to be given independently or be assumed. It iscommonly accepted that, for high-quality track bicycles, λ ∈ (0 . , .
03) . Also, if the power meteris in the rear hub, as opposed to being in the pedals or the bottom bracket, λ ≈ Expression (11), together with expressions (5), (6) and (9), allows us to calculate the power requiredto achieve a desired individual-pursuit time or the time achievable with a particular power. Italso allows us to quantify the effects of the bicycle-cyclist weight, air resistance, rolling resistance,drivetrain resistance and lateral friction, as well as of the velodrome size, steepness of its banksand tightness of its curves. Furthermore, a quantification of these effects lends itself to a study ofoptimization of a cyclist’s effort (Danek et al, 2020b, Appendix A).Our formulation and examination of an inverse solution for expression (11) shows that we can inferonly the ratios, C d A / (1 − λ ) , C rr / (1 − λ ) and C sr / (1 − λ ) . Nevertheless, the proposed solutionallows us to gain an insight into the consistency between the measurements and the model.In view of presented results, we conclude that accurate inferences based on the power-meter mea-surements on a velodrome require a distinction between the trajectory of wheels, which herein weassume to coincide with the black line, and the trajectory of the centre of mass. The forces involvedand relations among them, as well as other entailed quantities, are functions of the latter.If necessary, to model highly accurate measurements or for a specific scope of investigation, themathematical model stated in expression (11) can be refined in a manner suggested in footnotes 2and 3. In contrast to these refinements, a progressive leaning and straightening — neglected toformulate distances in expression (4), and resulting in two constant centre-of-mass speeds, V → and V (cid:120) , in expression (6)— cannot be achieved within this model, as discussed in Appendix C.Also, the assumption of a constant black-line speed cannot be relaxed within this model. It isnecessary to formulate expressions (6) and (10).The results presented in this article might be an a posteriori reassurance and comfort for Michael Hutchin-son (2006, p. 251), in his attempt to achieve immortality the hard way,Ride fast—in the end that’s all it ever comes down to. The pressure of another curve,the relief of the simple straight. But the straight’s short respite is never enough. Myshoulders are awful. My arms hurt. And every few seconds I have to manage anotherbanked curve. Each one demands a little more effort, a little more concentration. As the8hysical toll mounts, the balance and rhythm aren’t offering the protection from realitythat they did. I need some sort of reassurance. Some comfort.The power required to maintain the same speed — with respect to the black line — is less on a curvethan on a straight. In a certain manner, the curves provide a short respite. Also, thanks to thecurves, the distance travelled by a centre of mass, within a given time, is shorter than with respectto the black line. A Lean angle
Expressions (7) and (18) imply that the lean angle, ϑ , depends only on the centripetal accelerationof the cyclist’s centre of mass and the acceleration of gravity. In other words, the lean angle dependsonly on the centre-of-mass speed and the radius of curvature of the centre-of-mass trajectory, noton the track inclination, θ , even though both the normal force, stated in expression (16), andthe frictional force, in expression (17), do depend on track inclination. However, the θ -dependencecancels out of the centripetal force, stated in expression (18), by a seldom used trigonometric identity.Given the generality of this result, it would be satisfying to obtain it in a manner that explains it inthe context of physics. To this end, let us analyze the situation from inside the noninertial frame ofthe cyclist. Specifically, we consider the frame comoving with the cyclist around a curve, with thecyclist considered as a point mass. We neglect the additional accelerated motion resulting from therotation of the cyclist about an axis through the centre of mass.As illustrated in Figure A1 and in contrast to Figure 2, in this noninertial frame, instead of theforces in the horizontal direction summing to the centripetal force, F cp , the forces in this direction,including the fictitious centrifugal force, F cf = − F cp , sum to zero. The centrifugal force must betaken to act at the centre of mass, since otherwise the torque about the centre of mass, stated inexpression (14), would be affected. Figure A1:
Force diagram: Noninertial frame
To proceed, we invoke the vector identity, (cid:88) τ = R CoM × (cid:88) F + (cid:88) τ CoM , where (cid:80) τ is the net torque about an arbitrary point, (cid:80) τ CoM is the net torque about the centre ofmass, R CoM is the position vector of the centre of mass, and (cid:80) F is the net force. From this identitythere follows the well-known result that if the net force is zero and the net torque about the centreof mass is zero, the net torque about any other point is also zero. In particular, let us consider thetorque about the point of contact of the tires with the surface, (cid:88) τ z = h F g sin ϑ − h F cf sin (cid:16) π − ϑ (cid:17) = 0 , ϑ = F cf F g = F cp F g , (A.1)where expression (A.1) is equivalent to expressions (7) and (18). Expression (A.1) manifestly holdswhether the curve is banked or unbanked.Physically, from inside the noninertial frame of the cyclist, the value of the lean angle that obtainsfor specific values of speed and radius is the one that makes the gravitational torque balance thecentrifugal torque. In particular, this condition makes no reference to track inclination. B Harmonic mean
To calculate the average power over a lap, we require total work and time, W = (cid:73) F ( s ) d s and T = (cid:73) d t = (cid:73) d sV ( s ) , to write P = WT = (cid:73) F ( s ) d s (cid:73) d sV ( s ) . If we consider n segments along which F and V are constant, we write P = n (cid:88) i =1 F i ∆ s in (cid:88) i =1 ∆ s i V i s s = n (cid:88) i =1 F i ∆ s i ∆ s n (cid:88) i =1 V i ∆ s i ∆ s . Since in our case, there are only two segments, we write P = F → ∆ s → ∆ s + F (cid:120) ∆ s (cid:120) ∆ s V → ∆ s → ∆ s + 1 V (cid:120) ∆ s (cid:120) ∆ s , (B.1)where → denotes straights and (cid:120) denotes banks. The numerator is the distance-weighted arithmeticmean, F . The denominator is reciprocal of the distance-weighted harmonic mean, (cid:104) V (cid:105) = (cid:88) i =1 ∆ s i ∆ s (cid:88) i =1 V i ∆ s i ∆ s = 11 V → ∆ s → ∆ s + 1 V (cid:120) ∆ s (cid:120) ∆ s , which — for the average centre-of-mass speed — results in expression (10a).10n contrast to (cid:104) V (cid:105) , the distance-weighted arithmetic mean is V = (cid:88) i =1 V i ∆ s i ∆ s = V → ∆ s → ∆ s + V (cid:120) ∆ s (cid:120) ∆ s , (B.2)which, in view of expression (B.1), confirms that P (cid:54) = F V . The harmonic mean is less than thearithmetic mean; it is skewed toward slower speeds. Thus,
F V would overestimate the averagepower.For our numerical example, following expression (10), (cid:104) V (cid:105) = 11 . F (cid:104) V (cid:105) = 229 . V = 11 . F V = 229 . V → and V (cid:120) . It wouldnot be so, for an average of an upwind and downwind segments, discussed by Danek et al (2020b,Appendix A), where also the harmonic mean is used. C Transition between curves and straights
In this article, to consider the velodrome in question, we assume that, along the banks, the radiusof curvature is constant. Hence, the track is composed of two semicircles and two straights; this isthe case of the light grey oval in Figure C1. The dark grey and black ovals also represent a trackwhose C = 250 , but their radii of curvature are not constant; they are r = 23 , at the interceptswith the horizontal axis, and r → ∞ , at the intercepts with the vertical axis. The dark grey oval iscomposed of an ellipse (Benham et al, 2020), whose semiaxes are 23 and 30 , and of two straights.The black oval, in polar coordinates, is r ( φ ) = 28 . (cid:0) . φ cos φ + 0 . φ (cid:1) , φ = [ 0 , π ) , (C.1)where the coefficients are found numerically by invoking the concept of the arclength and curvatureto ensure that C = 250 , r min = 23 , r max → ∞ . Figure C1:
Velodrome tracks
These ovals share important geometrical properties, namely, their circumference and their radii ofcurvature, at the horizontal-axis and vertical-axis intercepts. However, the model presented in this11rticle applies explicitly to the light grey oval. Its application to other similar ovals entails a decreasein accuracy.For the light-grey oval, a model requires two constant centre-of-mass speeds, for the straights andfor the circular banks. For the dark-grey oval, an explicit model requires the centre-of-mass speedto be represented by two functions, where the speed along the straight is constant, but along thebanks is not, due to the changing radius of curvature along the elliptical bank. For the black oval,an explicit model requires the centre-of-mass speed to be represented by a single function, whichdepends on the continuously changing radius of curvature.Let us examine the black-oval model. Its average curvature, for the length of a lap, is κ = C (cid:90) κ ( s ) d s C (cid:90) d s = π (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( φ ) + 2 (cid:18) ∂r ( φ ) ∂φ (cid:19) − r ( φ ) ∂ r ( φ ) ∂φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:115) r ( φ ) + (cid:18) ∂r ( φ ) ∂φ (cid:19) (cid:115) r ( φ ) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φ π (cid:90) (cid:115) r ( φ ) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φ = π (cid:90) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( φ ) + 2 (cid:18) ∂r ( φ ) ∂φ (cid:19) − r ( φ ) ∂ r ( φ ) ∂φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( φ ) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φ π (cid:90) (cid:115) r ( φ ) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φ ;a numerical integration, with r given in expression (C.1), results in κ = 0 . r := 1 / κ = 39 . r = 23 , along the banks, and infinity along thestraights. For the black-oval model, in a manner analogous to expressions (3) and (4), the distancetraveled — in one lap — by the centre of mass is π (cid:90) (cid:118)(cid:117)(cid:117)(cid:117)(cid:116)(cid:16) r ( φ ) − h sin ϑ (cid:17) + ∂ (cid:16) r ( φ ) − h sin ϑ (cid:17) ∂φ d φ = π (cid:90) (cid:115)(cid:16) r ( φ ) − h sin ϑ (cid:17) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φ , where ϑ is the average lean angle. Thus, given a laptime, in a manner analogous to expression (6),we write t (cid:9) = π (cid:90) (cid:115)(cid:16) r ( φ ) − h sin ϑ (cid:17) + (cid:18) ∂r ( φ ) ∂φ (cid:19) d φV , (C.2)12here V is the average speed, which — following expression (9) — we write as V = (cid:112) g r tan ϑ . (C.3)To compare the resulting power with the numerical example in Section 3, we let the laptime be such Figure C2:
Left-hand and right-hand sides of equation (C.2) that, according to expression (5), the corresponding black-line speed is V → = 12 . In accordancewith expressions (C.2) and (C.3), and as shown in Figure C2, we obtain ϑ = 0 . V = 11 . P = 236 .
415 . If weconsider the average centre-of-mass speed, per lap, for the light-grey oval, which — in accordancewith expression (10) — is V = 11 . P = 232 .
223 . Another comparison is the distancetravelled by the centre of mass. For the black oval, it is 247 .
662 ; for the light-grey oval, it is 246 .
024 .
Figure C3:
Track inclination
The model based on the light-gray oval requires fewer approximations, within the realm of math-ematics. This is a consequence of the idealization of a velodrome track, which is greater for thelight-gray oval than for the black oval, due to the assumption of a constant curvature and no transi-tion between the curves and the straights. Thus, in spite of more mathematical approximations, thelatter might exhibit a superior empirical adequacy. A further examination of this question, whichis essential to the concept of modelling, requires experimental results. The conclusiveness of suchresults, however, might also be questionable, in view of the similarity of P = 236 .
35 , for the blackoval, and P = 229 . We could refine the black-oval model by including the effect of the track inclination, which — given a minimumand maximum values of inclination along the oval, stated in expression (C.1), as well as an interpolation formulabetween them — is illustrated in Figure C3. Hence, this effect could be expressed as a continuous function of distance. cknowledgements We wish to acknowledge Len Bos for his mathematical insights, Roger Mason for his perceptivecomments, David Dalton, for his scientific editing and proofreading, Elena Patarini, for her graphicsupport, and Roberto Lauciello, for his artistic contribution. Furthermore, we wish to acknowledgeFavero Electronics for inspiring this study by their technological advances and for supporting thiswork by providing us with their latest model of Assioma Duo power meters.
Conflict of Interest
The authors declare that they have no conflict of interest.
References
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