On modelling bicycle power-meter measurements: Part I. Estimating effects of air, rolling and drivetrain resistance
OOn modelling bicycle power-meter measurements: Part I
Estimating effects of air, rolling and drivetrain resistance
Tomasz Danek ∗ , Michael A. Slawinski † , Theodore Stanoev ‡ May 9, 2020
Abstract
Power-meter measurements together with GPS measurements are used to study the modelthat accounts for the use of power by a cyclist. The focus is on estimating the coefficients ofthe air, rolling and drivetrain resistance, uncertainties of these estimates, as well as relationsbetween them. Expressions used in the main text are derived in the appendices.
Using power meters to study cycling performance combines two distinct realms that do not have anexplicit relation between them: mechanical measurements and physical conditions. A hypotheticalrelation between them is offered by mathematical modelling. For instance, the answer to a ques-tion — is a high power output that results in a low ground speed a consequence of a strong headwind,steep climb, their combination or a completely different factor, such as an unpaved road? — cannotbe obtained from power meters alone; it can only be postulated, under various assumptions, as amodel, without a claim to uniqueness of solution.Many studies examine the physics of cycling. For instance, there are wind-tunnel studies to measurethe aerodynamics of bicycle wheels (e.g., Greenwell et al, 1995), and the studies to estimate theaccuracy of power measurements based on the frequency of the pedal-speed measurements (e.g.,Favero, 2018). There are studies to examine power required to overcome the effect of winds, takinginto account tire pressure, wheel radius, altitude and relative humidity (e.g., Olds et al, 1995), aswell as the aerodynamic drag, rolling resistance, and friction in the drivetrain (e.g., Martin et al,1998). There are studies to estimate model parameters from measurements on the road (e.g., Chung,2012) and to devise optimal speeds for time trials on closed circuits in the presence of wind (e.g.,Anton, 2013). There are studies to investigate the aerodynamics of track cycling to predict theindividual-pursuits times (e.g., Underwood and Jermy, 2014) and to simulate cyclist slip and steerangles necessary to navigate turns on a banked track (e.g., Fitton and Symons, 2018). There aregraduate theses in mechanical engineering (e.g., Moore, 2008; Underwood, 2012).Be that as it may, the science of cycling is a rich field that combines theoretical, computational andexperimental aspects of such disciplines as mathematical physics, continuum mechanics, as well asthe optimization and approximation theories.In this article, we consider power-meter measurements and a mathematical model to examine theconversion of power generated by a cyclist into motion of a bicycle. In particular, we infer the values ∗ AGH–University of Science and Technology, Krak´ow, Poland, [email protected] † Memorial University of Newfoundland, Canada, [email protected] ‡ Memorial University of Newfoundland, Canada, [email protected] a r X i v : . [ phy s i c s . pop - ph ] M a y igure 1: Left-hand plot: force applied to pedals, ¯ f (cid:8) = 193 . ± .
05 ; right-hand plot: circumferentialspeed of pedals, ¯ v (cid:8) = 1 . ± . of the air, rolling and drivetrain resistance by seeking an acceptable agreement between obtainedmeasurements and model retrodictions. To do so, we combine classic formulations of fluid mechanicswith innovative optimization methods, which extend the work of Cavazzuti (2012). We also invokeaspects of approximation theory to comment on the empirical adequacy of estimated values.We begin this article by presenting the power-meter and GPS measurements. Subsequently, weformulate and discuss a mathematical model to connect these two types of measurements. Usingthis model and data from a flat segment of several kilometres — in Northwestern Italy, betweenRivalta Bormida and Pontechino — that did not require any braking, we estimate the effects of theair, rolling and drivetrain resistance, as well as examine their uncertainties. We conclude by adiscussion of results.This article contains also several appendices, where we present the derivations of expressions used inthe main text to emphasize their assumptions and, hence, limitations. These derivations are familiarto mathematical physicists, but might be less so to a broad range of sport scientists; as such, theseappendices might be viewed as a brief auxiliary tutorial to facilitate the understanding of the maintext. Power is a rate at which work is done; hence, it is equal to the amount of work divided by the timeit takes to do it, which is tantamount to the product of force and speed, P = f (cid:8) v (cid:8) . (1)In the context of cycling, f (cid:8) is the force applied to pedals and v (cid:8) is the speed with which therotating pedals cover the distance along the circumference of their rotation, which means that v (cid:8) isproportional to the length of the crank.In this study — with the method used by Favero (2018) — v (cid:8) is an instantaneous speed, not anaverage per revolution; so is f (cid:8) . The importance of such an approach is illustrated in Figures 1 and2, which present the averages over pedal revolutions for the entire course, not the averages over asingle pedal revolution or over a specific period; these averages discussed in Appendix A.For measurements presented in Figure 1, the covariance between the two quantities is cov( v (cid:8) , f (cid:8) ) = − . v (cid:8) f (cid:8) = 258 .
811 , which is consistent with P = 258 . 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 , change in speed in m / s , and force in newtons, kg m / s ; angle is in radians. igure 2: Left-hand plot: force applied to pedals, ¯ f (cid:8) = 193 . ± .
75 ; right-hand plot: circumferentialspeed of pedals, ¯ v (cid:8) = 1 . ± . Figure 3:
Power: P = 258 . ± . in the caption of Figure 3, and based on power-meter measurements. This is distinct from the productof the averages, v (cid:8) f (cid:8) = 262 . v (cid:8) f (cid:8) + cov( v (cid:8) , f (cid:8) ) =262 . − . . v (cid:8) and f (cid:8) — is illustrated in Figure 3. At the same time as the power-meter information is collected, the GPS collects information aboutspeed, whose average values over the entire segment are illustrated in Figure 4, and altitude, il-lustrated in Figure 5, where its values are grouped within speed intervals and the correspondingstandard deviations are illustrated by error bars. The average altitude over the entire segmentis h = 145 . ± .
355 , and the median is 146 . ≈ h , which is indicative of little change of altitude.To relate the measured power to the surrounding conditions, we mediate between the two sourcesof information by a mathematical model. Herein, the power meter provides information regardlessof the conditions, GPS provides information about the surroundings independently of the poweroutput, and a mathematical model — based on physical principles — serves as a hypothetical relationbetween them. It must remain hypothetical since there is no explicit relation between the two sourcesof information. 3 igure 4: Ground speed, V → = 10 . ± . (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) Figure 5:
Altitude
Let us consider the following mathematical model. P = F ← V → (2)= 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 motion, and V → is the ground speed of the bicycle. F ← consists of the following quantities. • m : mass of a cyclist and a bicycle • g : acceleration due to gravity, whose effects are illustrated in Figure 6 • θ : slope • a : change of ground speed • ρ : air density, ρ = 1 .
225 exp[ − . h ] , (3)where h is the altitude above the sea level • w ← : wind component opposing the motion In this study, only translation — in contrast to rotation — of a mass is considered explicitly. The effects ofrotation of wheels and cranks are included implicitly in coefficients C d A , C rr and λ . In Appendix B, we discuss howto accommodate explicitly the wheel rotation. In this study, we do not consider air density as a function of humidity or changeable atmospheric pressure. C rr : unitless rolling-resistance coefficient; in a manner analogous to the friction on the planeinclined by θ , C rr is a proportionality constant between the maximum force, mg , and the forcenormal to the surface, mg cos θ • C d A : air-resistance coefficient; a product of a unitless drag coefficient, C d , and a frontalsurface area, whose units are m • η : a unitless quantity whose absolute value is equal to one and that ensures the proper signfor the tailwind effect, η = sgn( V → + w ← ) = V → + w ← | V → + w ← |• λ : unitless drivetrain-resistance coefficient to account for the loss of power between the powermeters and the propelling rear wheel; if power meters are in the pedals, λ includes the resistanceof bottom bracket, chain, rear sprocket and rear-wheel hub; it also includes losses due to theflexing of the frame; if power meters are in the rear-wheel hub, λ ≈ • change in elevation: increases the required power if θ > θ < θ = 0 ; it is associated with the change in potential energy, • change in speed: increases the required power if a > a < a = 0 ; it is associated with the change in kinetic energy, which is not lost unless the riderbrakes, • rolling resistance: increases the required power, • air resistance: increases the required power if the speed of the air flow relative to the cyclistis positive, ( V → + w ← ) > ⇒ η = 1 , decreases if ( V → + w ← ) < ⇒ η = − V → + w ← ) = 0 .Similar models — exhibiting a satisfactory empirical adequacy — are used in other studies (e.g., Mar-tin et al, 1998). In general, obtaining a unique result for the values of the resistance coefficients by minimizing themisfit between the right-hand side of expression (1), which represents measurements, and the right-hand side of expression (2), which represents a model, is impossible. Different combinations ofvalues give the same result. Also, the misfit function might have several minima, with the globalone not necessarily localized in the region for which the values have any physical interpretations.For instance, as stated by Chung (2012),Remember, C d A is an area. You can’t have negative area.Moreover, the optimization relies on measurements, which are subject to experimental errors, in-cluding limitations of the GPS accuracy.To accommodate these issues in our search for the values of C d A , C rr and λ , we group the valuesof the ground speed, V → , illustrated in Figure 4, in intervals of 0 . . . A formulation of C d A is presented in Appendix C. In this study, we consider only the effect of the translation speed, V → , upon the air resistance. Effects due torotation of wheels are discussed in Appendix D. igure 6: Isaac Newton subject to the effects of gravity in expression (2). (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31)
Figure 7:
Power
To avoid spurious results, groups are restricted to those that contain at least five values. The modeis 9 . a , θ and ρ , for each group. They are obtained from the GPS measurements: a and θ as the temporal andspatial derivatives of the measured speed and altitude, respectively, and ρ by using expression (3);these values are illustrated in Figures 8–10, with standard deviations illustrated by error bars.The average change of speed — over the entire segment — is a = 0 . ± . θ = 0 . ± . ρ = 1 . ± . (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) Figure 8:
Change of speed (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) Figure 9:
Slope (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31)
Figure 10:
Air density
To estimate the values of C d A , C rr and λ , we write expression (2) as f = P − mg sin θ V → + m a V → + C rr mg cos θ V → + η C d A ρ ( V → + w ← ) V → − λ (cid:124) (cid:123)(cid:122) (cid:125) F ← V → , (4)and minimize the misfit, min f . The grouped values, with their standard deviations, are used asinputs for a local optimization. Each group is treated separately and, hence, the statistics of itsinput parameters is different than for the entire set. In view of the expected values, a starting pointfor a local optimization is C d A = 0 . rr = 0 .
005 and λ = 0 .
035 . Also, m = 111 , g = 9 .
81 and w ← is set to zero.The process is repeated ten thousand times. The input values are perturbed in accordance with theirGaussian distributions, since — according to the central limit theorem — measurements affected bymany independent processes tend to approximate such a distribution. We obtain optimal valueswith their standard deviations,C d A = 0 . ± . , C rr = 0 . ± . , λ = 0 . ± . , shown in Figure 11. As illustrated in Figure 12, these values result in a satisfactory minimizationof misfit for expression (4). Using these values, together with the average values, over the entiresegment, assuming a = θ = w ← = 0 , letting m = 111 , g = 9 .
81 , we obtain, in accordance withexpression (2), P = 255 .
315 , which is consistent with P = 258 . Figure 11:
Optimal values; left-hand plot: C d A = 0 . ± . rr = 0 . ± . λ = 0 . ± . igure 12: Misfit of equation (4): f = 0 . ± . As stated in the subtitle, the main purpose of this article is a study of estimating effects of air,rolling and drivetrain resistance. Herein, restricting the study to a range of speeds, as illustratedin Figure 4, is consistent with time trialing on a flat course. Also, it excludes sporadic events, suchas tight corners and brief mechanical failures, for which braking or stopping diminishes the qualityof estimates of C rr and λ , and during which an aerodynamic position is not maintained, thusdiminishing the quality of estimating C d A . Furthermore, a restricted range enhances the validity oftreating C d A , C rr and λ as constants, even though, in general, they are functions of speed.For flat courses, the force of the air resistance, which is expressed by the fourth summand of expres-sion (2), is proportional to the square of speed and, hence, becomes dominant. Hence, C d A playsa significant role, in contrast to steep climbs, for which the numerator is dominated by the firstsummand, which accounts for the force against gravity.The sensitivity to the force of gravity can be quantified by studying relations among m , θ and V → .For instance, for any cyclist — with a given power output — the dependence of speed on mass isgreater on a climb than on a flat course, and this dependence can be quantified, using ∂V → /∂m andthe implicit function theorem, discussed in Appendix E.In estimating the effects of the air, rolling and drivetrain resistance, we recognize that the right-handside of expression (2), which is a forward problem, invokes C d A , C rr and λ with their independentphysical meanings. The misfit minimization of equation (4), min f , however — for which the left-hand side is the measured value and the right hand side is the retrodiction of a model — treats C d A ,C rr and λ as adjustable parameters; for instance, C d A and C rr , which — as physical quantities — areindependent of one another, become inversely proportional to one another, for η = 1 . In general,following the implicit function theorem, this inverse relation is ∂ C d A ∂ C rr = − ∂f∂ C rr ∂f∂ C d A = − m g cos θ η ρ ( V → + w ← ) , where f is given in expression (4); in the present study, ∂ C d A ∂ C rr = − m gρ V → = − . < . Relations between the rates of change of any two quantities in expression (4) can be insightful inexamining the behaviour of a model. To study the performance of a cyclist, as opposed to behaviours8f a model, only a few among them are pertinent; others — such as ∂ C d A /∂ C rr — are not endowedwith a physical meaning.Maintaining the physical meaning of C d A , C rr and λ remains a challenge. To extract λ , one mightconsider a statement of Chung (2012).Many models include a term for overall drivetrain efficiency, η , but all of the datafiles I’m looking at come from Power Taps which, in theory, should be downstream ofdrivetrain losses, i.e., η = 1 . If you have an SRM, which measures power at the crank(i.e., upstream of drivetrain losses), you will want to decide how to model drivetrainlosses. Martin et al. presumed a fixed percentage loss of 2.3% of power (i.e., η = 0 . λ = 0 , used by Chung (2012), increases the stability of extracting the remaining twocoefficients thanks to the disappearance of the denominator in expression (2). Otherwise, a divisionof the entire expression is a scaling that contributes to nonuniqueness.However, even if λ = 0 , “prying apart C rr and C d A” , in words of Chung (2012), remains a challenge,for which he suggests a scenario wherein[i]f we test using the same tires and tubes on the same roads on the same day at thesame pressure then C rr is a constant and we can concentrate on estimating changes inC d A .An estimate of individual values of C d A , C rr and λ , as physical quantities or, at least, their relativechanges is an important aspect of our work. We wish to ensure a sufficient accuracy to examineeffects — on C d A — of aerodynamic equipment and riding position, as well as the efficiency of draftingfor a team time trial. Also, we wish to examine the effects — on C rr — of tire width and pressure.Furthermore, we wish to examine the effects — on λ — of different sizes of the chainring and cog thatresult in the same ratio, as well as a the efficiency of a fixed gear versus freewheel. Acknowledgements
We wish to acknowledge Len Bos and Yves Rogister, for our fruitful discussions, David Dalton, for hisscientific editing and proofreading, Elena Patarini, for her graphic support, and Roberto Lauciello,for his artistic contribution. Furthermore, we wish to acknowledge Favero Electronics, includingtheir Project Manager, Francesco Sirio Basilico, and R&D engineer, Renzo Pozzobon, for inspiringthis study by their technological advances and for supporting this work by insightful discussions andproviding us with their latest model of Assioma Duo power meters.
References
Anton AB (2013) Optimal time-trial bicycle racing with headwinds and tailwinds arXiv:1309.1741[physics.pop-ph]Birkhoff G (1950) Hydrodynamics: A study in logic, fact, and similitude. Princeton University PressCavazzuti M (2012) Optimization methods: From theory to design scientific and technological as-pects in mechanics. Springer herein, η ≡ − λ A power meter in the rear hub http://anonymous.coward.free.fr/wattage/cda/indirect-cda.pdf
Favero (2018) Influence of angular velocity of pedaling on the accuracy of the measurement of cyclistpower. URL https://drive.google.com/open?id=1zAb1KFh35RRQOo632FNz8nHbhF-PtJDz
Fitton B, Symons D (2018) A mathematical model for simulating cycling: Applied to track cycling.Sports Engineering 21:409–418Greenwell D, Wood N, Bridge E, Addy R (1995) Aerodynamic characteristics of low-drag bicyclewheels. The Aeronautical Journal 99(983):109–120Martin JC, Milliken DL, Cobb JE, McFadden KL, Coggan AR (1998) Validation of a mathematicalmodel for road cycling power. Journal of Applied Biomechanics 14:276–291Moore JK (2008) Aerodynamics of high performance bicycle wheels. Master’s thesis, University ofCanterburyOlds TS, Norton KI, Lowe ELA, Olive S, Reay F, Ly S (1995) Modeling road-cycling performance.Journal of Applied Physiology 78(4):1596–1611Underwood L (2012) Aerodynamics of track cycling. PhD thesis, University of CanterburyUnderwood L, Jermy M (2014) Determining optimal pacing strategy for the track cycling individualpursuit event with a fixed energy mathematical model. Sports Engineering 17:183–196
A On effects of averaging pedal speed per revolution forpower calculations
A.1 Preliminary remarks
Since, in expression (1), v (cid:8) is proportional to the cadence, it is common to simplify circumferentialspeed measurements by considering only the cadence. This means that—instead of measuring thespeed instantaneously along the revolution—the measurement is performed only once per revolutionand the resulting average is used in subsequent calculations. In this appendix, we examine the effectsof such a simplification.Referring to an expression equivalent to expression (1)—that invokes torque and angular velocityinstead of the force and circumferential speed—Favero (2018) state that[t]he torque/force value is usually measured many times during each rotation, while theangular velocity variation is commonly neglected, considering only its average value foreach revolution. [ . . . ] Favero Electronics, to ensure the maximum accuracy of its powermeters in all pedaling conditions, decided to research to what extent the variation ofangular velocity during a rotation affects the power calculation.To examine the effect of including speed variation during a revolution, let us consider the followingformulation to gain analytical insights into the empirical results obtained by Favero (2018). A.2 Formulation
Consider a pedal whose revolution takes one second; hence, its circumferential speed is v (cid:8) ( θ ) = v (1 + a cos(2 θ )) , θ ∈ (0 , π ] , (A.1)10 igure A1: Circumferential speeds corresponding to expression (A.1): a = 0 .
25 , r = 0 .
175 and a = 0 , r = 0 .
175 ; the former shown in black and the latter in grey
Figure A2:
Applied force corresponding to expression (A.2): b = 0 . c = π/ f = 200 where v = 2 πr/ r is the crank length and θ is the angle. Expression (A.1) is illustrated inFigure A1.Assume the magnitude of the tangential component of force applied to both pedals during thisrevolution to be f (cid:8) ( θ ) = f (1 + b cos(2( θ + c ))) , θ ∈ (0 , π ] , (A.2)where f is a constant and c is an angular shift between v (cid:8) and f (cid:8) , which is a constant whose unitsare radians. Expression (A.2) is illustrated in Figure A2.In accordance with expression (1), the instantaneous power, at θ , is P (cid:8) ( θ ) = f (cid:8) ( θ ) v (cid:8) ( θ ) , and theaverage power over the revolution is P (cid:8) = 12 π π (cid:90) P (cid:8) ( θ ) d θ = (2 + a b cos(2 c )) πrf . (A.3)If we consider the average value of speed, v (cid:8) = 12 π π (cid:90) v (cid:8) ( θ ) d θ = 2 πr , (A.4)then, P (cid:8) = 12 π π (cid:90) f (cid:8) ( θ ) v (cid:8) d θ = 2 πrf , (A.5)11 igure A3: Instantaneous power corresponding to expressions (A.3) and (A.5); the former shown in blackand the latter in grey, with averages of 227 and 220 , respectively over one revolution; a , b and c have no effect on P (cid:8) . Examining expressions (A.3) and (A.5), wesee that the former reduces to the latter if a b = 0 or if c = π/ c = 3 π/ f (cid:8) = 12 π π (cid:90) f (1 + b cos(2( θ + c )) d θ = f , (A.6)which is the average force per revolution that results from expression (A.2) .The integrands of expressions (A.3), with c = 0 , and (A.5) are illustrated in Figure A3. Therein,following expressions (A.1) and (A.2), the integrand of expression (A.3) is P (cid:8) ( θ ) = f (cid:8) ( θ ) v (cid:8) ( θ ) = f (1 + b cos(2( θ + c ))) v (1 + a cos(2 θ )) . Invoking trigonometric identities and rearranging, we write it as P (cid:8) ( θ ) = f v ab c ) (cid:124) (cid:123)(cid:122) (cid:125) constant + a cos(2 θ ) + b cos(2( θ + c )) (cid:124) (cid:123)(cid:122) (cid:125) double frequency + ab θ + 2 c ) (cid:124) (cid:123)(cid:122) (cid:125) quadruple frequency . However, the effect of the third summand is small enough not to appear in Figure A3. For instance,if we let c = 0 , the double-frequency term becomes ( a + b ) cos(2 θ ) and the quadruple frequency termbecomes ab cos(4 θ ) . If a < b < A.3 Numerical examples
If we let a = 0 .
25 , b = 0 . c = 0 , f = 200 , v = 2 πr and r = 0 .
175 , expression (A.3) results in234 watts, as the average power per revolution, and expression (A.5) in 220 watts. The approachthat neglects speed variations during the revolution can also overestimate the average power. If welet c = π/ P = 206 and expression (A.5) remains unchanged. Theseresults are illustrated in Figure A4.Expressions (A.1)–(A.6) refer to a single revolution. Hence, the values resulting from expres-sions (A.3) and (A.5) remain the same, regardless of cadence; they are averages over one rotation.12 igure A4: Average power corresponding to expressions (A.3) and (A.5); the former shown in black andthe latter in grey; the former depends on ab , the latter does not; the former depends on c , the latter doesnot; c = 0 , for the increasing line, c = π/ c = 3 π/ Figure A5:
Circumferential speeds corresponding to expression (A.7): a = 0 .
25 , r = 0 .
175 and a = 0 , r = 0 .
175 ; the former shown in black and the latter in grey
If the pedaling is smoother, as one might expect for higher cadences, the values of a and b be-come smaller. Since these values are smaller than unity and appear as a product, expression (A.3)might approach expression (A.5). If we let a = 0 . b = 0 . c = 0 , f = 200 and r = 0 .
175 ,expression (A.3) results in P = 223 ; the value of expression (A.5) remains unchanged.Furthermore, for a single revolution, there is a unique pair of force and speed that results in a powergiven by expressions (A.3) and (A.5). However—for a given time interval and various cadences—there are many pairs of force and speed that result in the same value of power.For expression (A.1) to correspond to two revolutions per second, we modify it to be v (cid:8) ( t ) = 4 πr (1 + a cos(8 πt )) , t ∈ (0 , , (A.7)where t stands for time; expression (A.7) is illustrated in Figure A5. Accordingly, we modifyexpression (A.2) to be f (cid:8) ( t ) = f (1 + b cos(8 π ( t + c ))) , t ∈ (0 , , (A.8)where c is a time shift between v (cid:8) and f (cid:8) , which is a constant whose units are seconds; expres-13 igure A6: Applied force corresponding to expression (A.2): b = 0 . c = π/ f = 200 sion (A.8) is illustrated in Figure A6. Hence, expression (A.3) becomes P (cid:8) = (cid:90) v (cid:8) ( t ) f (cid:8) ( t ) d t = (cid:90) v (cid:8) ( t ) (cid:122) (cid:125)(cid:124) (cid:123) πr (1 + a cos(8 πt )) f (cid:8) ( t ) (cid:122) (cid:125)(cid:124) (cid:123) f (1 + b cos(8 π ( t + c ))) d t = 2(2 + ab cos(8 πc )) πrf . (A.9)Examining expressions (A.3) and (A.9), we see that to keep the same average power per second —with ab = 0 or c = 0 — we need to halve the value of f . Otherwise, the ratio is2 + ab cos(2 πc )2 (2 + ab cos(8 πc )) . In accordance with expression (A.3), which is tantamount to the power averaged over one second — ifthe cadence is one revolution a second — and given a = 0 .
25 , b = 0 . c = 0 , f = 200 , r = 0 .
175 ,we have P (cid:8) = 234 . With a cadence of two revolutions a second, in accordance with expression (A.9),the same average power is obtained with f = 100 . Thus, among many possible pairs that result in P (cid:8) = 234 , we have (60 rpm, f = 200) and (120 rpm, f = 100) .If a = 0 , in accordance with expression (A.3) and (A.5), P (cid:8) = 2 πrf , per second, and, in accordancewith expression (A.9), P (cid:8) = 4 πrf , per second. Thus, to keep the same average power, we againhalve the value of f . If the original value, at 60 rpm, is P (cid:8) = 234 , the corresponding value iscalculated to be f = 213 , instead of 200 . This results in a different — and less accurate — pair, dueto neglecting speed variation during a revolution. A.4 Closing remarks
As illustrated in this appendix, there is a discrepancy between the power-meter calculations resultingfrom the use of the instantaneous-speed and average-speed information. Removing this discrepancyis crucial for a variety of information that rely on power measurements, as is the case of this paper.Let us conclude by addressing the issue of sampling with regards to the discrepancy in powercalculation resulting from v (cid:8) as opposed to v (cid:8) ( θ ) . Let us consider the gear of 54 ×
17 . Fora road bicycle, one revolution results in a development of 6 .
67 metres. Hence, for the speed of48 . igure B1: Rolling without slipping: angular speed, ω , circumferential speed, v = ω r , where r is radius,and bicycle speed, V → = v B Rotation effects: moment of inertia
To include the effect of rotation upon change of speed in the model stated in expression (2), weconsider the moment of inertia, which is mr , for a thin circular loop, and mr / r stands for their radius. Relating the angular change in speed to the circumferential one — bya temporal derivative of v = ωr , where v is the circumferential speed and ω is the angular speed —the magnitudes of the corresponding rotational force are F r = mra and F r = mra/ F = ma and F = ma/ v = ωr is the circumferential speed. To show that it is equal to theground speed of the bicycle, V → , let us consider the point of contact of the wheel and the road. Theground speed of that point is the sum of the circumferential speed of the wheel, at that point, andthe speed of the bicycle. Since — under assumption of no slipping — the ground speed of that pointis zero and the other two speeds refer to velocities in the opposite directions, we have v = V → . Thus,as illustrated in Figure B1, the circumferential speed of the wheel is the same as the ground speedof the bicycle; the same is true for the change of speed, as required.To consider a bicycle with one standard wheel and one disk wheel, we denote the mass of theformer by m w , and the mass of the latter by m d . Thus, the second summand in the numerator ofexpression (2), which is a linear force, becomes ( m + m w + m d / a .Note that m contains both m w and m d to account for the translational and rotational effects; theformer depends on the total mass and the latter on the mass of the wheels only. For a standardwheel, m w ≈ . m d ≈ . r = 0 .
31 .
C Air-resistance coefficient
To formulate the air-resistance force in expression (2), we assume that it is proportional to thefrontal area, A , and to the pressure, p , exerted by air on this area, F a ∝ pA , where p = ρV hasa form of kinetic energy and V = V → + w ← is the relative speed of a cyclist with respect to the air; p is the energy density per unit volume. We can write this proportionality as F a = C d AρV , where C d is a proportionality constant, which is referred to as the drag coefficient.A more involved justification for the form of the air-resistance force in expression (2) is based ondimensional analysis (e.g., Birkhoff, 1950, Chapter 3). We consider the air-resistance force, which15s a dependent variable, as an argument of a function, together with the independent variables, towrite f ( F a , V, ρ, A, ν ) = 0 ;herein, ν is the viscosity coefficient. Since this function is zero in any system of units, it is possibleto express it in terms of dimensionless groups, only.According to the Buckingham theorem (e.g., Birkhoff, 1950, Chapter 3, Section 4) — since there arefive variables and three physical dimensions, namely, mass, time and length— we can express thearguments of f in terms of two dimensionless groups. There are many possibilities of such groups,all of which lead to equivalent results. A common choice for the two groups is F a ρ A V , which is referred to as the drag coefficient, and V √ Aν , which is referred to as the Reynolds number. Thus, treating physical dimensions as algebraic objects,we can reduce a function of five variables into a function of two variables, g (cid:32) F a ρ A V , V √ Aν (cid:33) = 0 , which we write as F a ρ A V = h (cid:32) V √ Aν (cid:33) , where the only unknown is F a , and where h is a function of the Reynolds number. Denoting theright-hand side by C d , we write F a = C d A ρ V , as expected. In view of this derivation, C d is not a constant; it is a function of the Reynolds number.In our study, however — within a limited range of speeds — C d is treated as a constant. Furthermore,since A is difficult to estimate, we include it within this constant, and consider C d A .
D Rotation effects: air resistance
To include the effect of air resistance of rotating wheels in the model stated in expression (2), anothersummand is to be introduced to the numerator, namely, C w πr ρ ( V → + w ← ) , where r is the wheel radius. Such a summand is formulated by invoking dimensional analysis in amanner analogous to the one presented in Appendix C.To combine rotational air resistance with the translational one, we use v = ωr , where v is thecircumferential speed and ω is the angular speed, and the fact that — as discussed in Appendix Band illustrated in Figure B1 — the circumferential speed, under the assumption of rolling withoutslipping, is the same as the ground speed of the bicycle, V → .16onsidering two standard wheels, we write P = mg sin θ + ( m + 2 m w ) a + C rr mg cos θ + η ρ (2 C w A ◦ (cid:122)(cid:125)(cid:124)(cid:123) πr +C d A f ) ( V → + w ← ) − λ V → ; (D.1)herein, in contrast to expression (2) and as discussed in Appendix B, the change of speed, expressedby the second summand, contains effects of the moment of inertia due to rolling wheels. Theair resistance, expressed by the fourth summand, distinguishes between the air resistance due totranslation of a bicycle from the air resistance due to its rolling wheels. A f is the entire frontal areaand A ◦ is the wheel side area. An examination of the effect of two different wheels requires theintroduction of two coefficients, one for each wheel.In this study, the quality of available information renders the extraction of values for the resistancecoefficients difficult. Even though the data obtained from the power meter has high accuracy, thedata based on the GPS measurements introduces the uncertainty that renders an accurate extractionof even three parameters a numerical challenge. Extraction of four or five parameters requires furtherstudies and, above all, more reliable data.In the meantime, we can consider forward estimates, such as gaining an insight into the effect of adisk wheel. Following expression (D.1) — under windless conditions, w ← = 0 , on a flat course, θ = 0 ,and with a steady tempo, a = 0 — we write the required powers as P n = C rr mg + ρ (2 C w n A w + C d A f ) V → − λ V → and P d = C rr mg + ρ ((C w n + C w d )A w + C d A f ) V → − λ V → , where we distinguish between the drag coefficients of a normal wheel and a disk wheel. The differencein required power is ∆ P = C w n − C w d − λ ) A w ρ V → . Letting C w n ≈ .
05 and C w d ≈ .
035 (Greenwell et al, 1995) means that, for a standard wheel,C w A ◦ ≈ .
015 , and for a disk wheel, C w A ◦ ≈ .
01 . Both values are significantly smaller thanC d A f , as expected. Letting r = 0 .
31 , ρ = 1 .
204 , λ = 0 . P ≈ . V → . For V → = 10 .
51 , we have ∆ P ≈ . P = 258 . P/P ≈ .
3% , to maintain the same speed. The disadvantages of the weight of a disk wheel almostdisappear for a flat course, θ = 0 , and a steady tempo, a = 0 ; m remains in the third summand,only. E Rates of change
It is insightful to examine the relations between the rates of change of quantities that appear inexpression (2). For instance — for the case examined herein — what increase of speed would resultfrom an increase of power by 1 watt? Using expression (D.1) and the implicit function theorem, ∂ C d A f /∂ C w A ◦ = − d A f and C w A ◦ . We expect, C w A o (cid:28) C d A f ; however,the optimization programs treat them as two adjustable parameters of equal importance.
17o answer this question, we need to find ∂V → /∂P . To do so — without solving equation (2) for V → as a function of P , which is a cubic equation — we invoke the implicit function theorem. Let usconsider expression (4). As required by the theorem, f possesses continuous partial derivatives inall its variables at all points, except at λ = 1 , which is excluded by mechanical considerations. Also,as required by the theorem, f = 0 , in the neighbourhood of interest, which is true as a consequenceof equation (4) and is illustrated in Figure 12. Hence, in accordance with the theorem, among manyrelations between quantities in this expression, we can consider, for instance, ∂V → ∂P = − ∂f∂P∂f∂V → (E.1a)= 2 (1 − λ )2 m a + η ρ C d A ( V → + w ← ) (3 V → + w ← ) + 2 m g (C rr cos θ + sin θ ) . (E.1b)Expression (E.1b) is valid only in the neighbourhood of a point for which a combination of values — P , λ , m , g , θ , a , C rr , η , C d A , ρ , V → , w ← — results in f = 0 . It is not valid for arbitrary valuesof these quantities. However, this restriction is not a significant limitation for a study of cyclingperformance, since f = 0 is a criterion for an empirical adequacy of a power-meter model.In accordance with Section 3, inserting the fixed values, m = 111 , g = 9 .
81 , w = 0 , which entails η = 1 , the averages of values obtained from measurements, P = 258 . V → = 10 .
51 , andmodelling, C d A = 0 . rr = 0 . λ = 0 . θ = 0 , as an average overthe entire segment, which is consistent with its flat topography, and a = 0 , which is consistent witha steady tempo, we obtain the sought answer, ∂V → ∂P = 0 . , which means that an increase of power by 1 watt results in an increase of speed of about 0 .
018 metresper second. Conversely, in accordance with expression (E.1a), within this neighbourhood, ∂P∂V → = 56 . , which means that an increase of speed by 1 metre per second requires an increase of power of about57 watts. Thus — within the neighbourhood of f = 0 — a 9 ..