On maximizing VAM for a given power: Slope, cadence, force and gear-ratio considerations
OOn maximizing VAM for a given power
Slope, cadence, force and gear-ratio considerations
Len Bos ∗ , Michael A. Slawinski † , Theodore Stanoev ‡ August 25, 2020
Abstract
For a given power, the mean ascent speed (VAM) increases monotonically with the slope.Also, to maximize the ascent speed, the slope needs to be constant. These properties — which aretheorems stemming from a standard mathematical model to account for the power required topropel a bicycle — constitute a mathematical-physics background upon which various strategiesfor the VAM maximization can be examined in the context of the maximum sustainable power asa function of both the gear ratio and cadence. Recently established records provide an empiricalsupport for these analytical results, which are based on theoretical considerations.
In this paper, we consider a mathematical model that accounts for power-meter measurements. Inaccordance with this model, VAM ( velocit`a ascensionale media ), which is the mean ascent speed,increases monotonically with the slope. We show that VAM is maximized along a slope that isconstant.We begin this paper by presenting a model to account for the power required to maintain a givenground speed. Subsequently, we express VAM in terms of the ground speed and the slope. Weproceed to prove that — for a given power — VAM increases monotonically with the slope, and thatthe maximization of VAM requires a constant slope. To gain an insight into these results, weexamine a numerical example wherein we vary the values of cadence and gear ratio. We concludeby suggesting possible consequences of our results for VAM-maximization strategies. ∗ Universit´a di Verona, Italy, [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 ] A ug Power
A standard mathematical model to account for the power required to propel a bicycle with speed V is (e.g., Danek et al., 2020) 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. Herein, m is themass of the cyclist and the bicycle, g is the acceleration due to gravity, θ is the slope of a hill, a is thechange of speed, C rr is the rolling-resistance coefficient, C d A is the air-resistance coefficient, ρ is theair density, w ← is the wind component opposing the motion, λ is the drivetrain-resistance coefficient, 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 , in windless conditions, w = 0 , we write expression (1) as P = mg sin θ + C rr mg cos θ + C d A ρ V − λ V . (2) Commonly, VAM is stated as a mean ascent speed in metres per hour. Since expressions (1) and(2) are commonly expressed in the SI units, let us defineVAM := 3600 V ( θ ) sin θ . (3)As we proceed to prove, for a given value of P , VAM is a monotonic function of θ ; hence, there isno maximum.Setting the power to P = P , and using expression (2), we write V + A ( θ ) V + B = 0 , (4)where A ( θ ) = mg C d A ρ (sin θ + C rr cos θ ) , (5a) B = − (1 − λ ) P C d A ρ , (5b)to state the following theorem. Theorem 1.
For a given power, P = P , V ( θ ) sin θ increases monotonically for (cid:54) θ (cid:54) π/ .Proof. Expressions (4), (5a) and (5b) determine V = V ( θ ) . Then,dd θ ( V ( θ ) sin θ ) = V (cid:48) ( θ ) sin θ + V ( θ ) cos θ , (6)2here (cid:48) is a derivative with respect to θ . The derivative of expression (4) is3 V ( θ ) V (cid:48) ( θ ) + A (cid:48) ( θ ) V ( θ ) + A ( θ ) V (cid:48) ( θ ) = 0 , from which it follows that V (cid:48) ( θ ) = − A (cid:48) ( θ ) V ( θ )3 V ( θ ) + A ( θ ) . (7)Using expression (7) in expression (6), and simplifying, we obtaindd θ ( V ( θ ) sin θ ) = V ( θ ) 3 V ( θ ) cos θ − A (cid:48) ( θ ) sin θ + A ( θ ) cos θ V ( θ ) + A ( θ ) . (8)Upon using expression (5a), the latter two terms in the numerator of expression (8) simplify to − A (cid:48) ( θ ) sin θ + A ( θ ) cos θ = mg C d A ρ (cid:16) − (cos θ − C rr sin θ ) sin θ + (sin θ + C rr cos θ ) cos θ (cid:17) = mg C rr12 C d A ρ . Hence, dd θ ( V ( θ ) sin θ ) = V ( θ ) 3 V ( θ ) cos θ + mg C rr12 C d A ρ V ( θ ) + A ( θ ) > , (cid:54) θ (cid:54) π , (9)as required. Since expression (9) is positive, VAM is — within the range of interest — a monotonicallyincreasing function of θ .From this theorem, it follows that the steeper the incline that a rider can climb with a given power,the greater the value of the corresponding VAM.Recently, Lachlan Morton (VeloViewer, 2020) established an Everesting record, which consists ofriding an uphill repeatedly in a single activity until one climbs the height of Mount Everest (8,848metres). To do so — in a manner consistent with Theorem 1 — he chose a steep slope, whose averagegradient is 11 .
1% .Since there is no local maximum, there is no specific value of θ that maximizes expression (3), fora given power. Hence — based on expression (2), alone — we cannot specify the optimal slope tomaximize VAM. To examine such a question, one needs to include other considerations.For instance, we can consider P = f v , (10)where f is the force applied to the pedals and v is the circumferential speed of the pedals. These aremeasurable quantities; they need not be modelled in terms of the surrounding conditions, such asthe mass of the rider, strength and direction of the wind or the steepness of an uphill. To proceed,let us specify (cid:96) c , which is the crank length, g r , which is the gear ratio, r w , which is the wheel radius,and c , which is the cadence, with all quantities expressed in the SI units. Hence, the circumferentialspeed is v = 2 π (cid:96) c c , (11)with the corresponding ground speed of V = 2 π r w c g r , (12)which allows us to examine VAM as a function of the cadence and gear ratio.3 Brachistochrone
Prior to examining VAM as a function of the cadence and gear ratio, let us return to expressions (4),(5a) and (5b) to consider the brachistochrone problem, which herein consists of finding the curvealong which the ascent between two points — under the assumption of fixed power — takes the leastamount of time.
Theorem 2.
The path of quickest ascent from (0 , to ( R, H ) , which is an elevation gain of H over a horizontal distance of R , is the straight line, y = ( H/R ) x , where (cid:54) x (cid:54) R and y ( R ) = H .Proof. The relation between speed, slope and power is stated by expressions (4), (5a) and (5b),which determine V = V ( θ ) , where θ is the slope, namely,tan θ = d y d x , θ = arctan (cid:18) d y d x (cid:19) . The traveltime over an infinitesimal distance, d s , is d t = d s/V , where V is the ground speed. Sinced s = d x + d y , we write d t = (cid:115) (cid:18) d y d x (cid:19) d xV ( θ ) = (cid:113) y (cid:48) ) V (arctan( y (cid:48) )) d x , where y (cid:48) := d y/ d x . Hence, the ascent time is T = R (cid:90) (cid:113) y (cid:48) ) V (arctan( y (cid:48) )) d x =: R (cid:90) (cid:112) p V (arctan( p )) (cid:124) (cid:123)(cid:122) (cid:125) F d x , where p := y (cid:48) ( x ) . Thus, we write the integrand as a function of three independent variables, T =: R (cid:90) F ( x, y, p ) d x . To minimize the traveltime, we invoke the Euler-Lagrange equation, ∂∂y F ( x, y, p ) = dd x ∂∂p F ( x, y, p ) . Since F is not an explicit function of y , the left-hand side is zero,0 = dd x (cid:40) ∂∂p (cid:112) p V (arctan( p )) (cid:41) , (13)which means that the term in braces is a constant. To proceed, we use the fact that p = tan θ = ⇒ ∂∂p = d θ d p ∂∂θ = cos θ ∂∂θ and ∂∂p (cid:112) p V (arctan( p )) = cos θ ∂∂θ (cid:112) p V (arctan( p )) = cos θ ∂∂θ √ θV ( θ ) = cos θ ∂∂θ √ sec θV ( θ )= cos θ dd θ θ V ( θ ) , (cid:54) θ (cid:54) π/ x (cid:18) cos θ dd θ θ V ( θ ) (cid:19) ≡ , which, using the fact that θ = arctan( y (cid:48) ( x )) = ⇒ d θ d x = 11 + ( y (cid:48) ( x )) y (cid:48)(cid:48) ( x ) and dd x = d θ d x dd θ , we write as y (cid:48)(cid:48) ( x )1 + ( y (cid:48) ( x )) dd θ (cid:18) cos θ dd θ θ V ( θ ) (cid:19) ≡ . Hence, there are two possibilities. Either y (cid:48)(cid:48) ( x ) ≡ (cid:124) (cid:123)(cid:122) (cid:125) (a) or dd θ (cid:18) cos θ dd θ θ V ( θ ) (cid:19) ≡ (cid:124) (cid:123)(cid:122) (cid:125) (b) . We claim that (b) is impossible. Indeed, (b) holds if and only ifcos θ dd θ θ V ( θ ) ≡ a , where a is a constant. It follows thatdd θ θ V ( θ ) = a sec θ ⇐⇒ θ V ( θ ) = a tan θ + b , where a and b are constants, andcos θ V ( θ ) = 1 a tan θ + b ⇐⇒ V ( θ ) = 1 a sin θ + b cos θ . This cannot be the case, since it implies that there exists a value of θ ∗ for which a sin θ ∗ + b cos θ ∗ = 0and V ( θ ∗ ) → ∞ . But V + A ( θ ) V + B = 0 , so any root of this cubic equation is such that | V | (cid:54) max { , | A ( θ ) | + | B |} . Thus, it follows that (a), namely, y (cid:48)(cid:48) ( x ) ≡ y ( x ) is a straight line.For the ascent from (0 ,
0) to (
R, H ) , the line is y ( x ) = ( H/R ) x , as required.In the context of VAM, there is a corollary of Theorem 2. Corollary 1.
The path of the quickest gain of altitude between two points, (0 , and ( R, H ) , is thestraight line, y = ( H/R ) x , where (cid:54) x (cid:54) R and y ( R ) = H . The aforementioned record of Lachlan Morton (VeloViewer, 2020) was established — in a mannerconsistent with Theorem 2 and its corollary — on a hill whose average and maximum gradients are11 .
1% and 13 .
2% , respectively.Let us emphasize that Theorem 2 and its corollary are statements of mathematical physics, whoseconsequences might be adjusted by other factors to propose an actual strategy. For instance, itmight be preferable — from a physiological viewpoint — to vary the slope in order to allow a riderperiods of respite. 5
Numerical example
To examining VAM as a function of the cadence and gear ratio, let us consider the following values.For the bicycle-cyclist system, m = 111 kg, C d A = 0 . , C rr = 0 . λ = 0 . (cid:96) c = 0 .
175 m, g r = 1 . r w = 0 .
335 m. For the external conditions, g = 9 .
81 m/s and ρ = 1 . .Let us suppose that the power that the rider keeps during a climb is P = 375 W and that the cadenceis c = 1 , which is 60 rpm, and — in accordance with expression (11) — results in v = 1 . f = 341 . V = 3 . V into expression (4), we obtain45 . θ + 3543 . θ − . , whose solution is θ = 0 . . ◦ , which results in VAM = 1039 . θ corre-sponds to a grade of 9.1840 % . Let us consider the effect on the value of VAM due to varying the cadence and gear ratio. To do so,we use expression (12) in expressions (3) and (4) to obtainVAM = 7577 . c g r sin θ (14)and 1 . c g r ) + 30 . c g r cos θ + 2362 . c g r sin θ −
375 = 0 , (15)respectively. In both expressions (14) and (15), c and g r appear as a product.For each c g r product, we solve equation (15) to obtain the corresponding value of θ , which we usein expression (14) to calculate the VAM. We consider c g r ∈ [0 . , .
75] ; the lower limit can represent c = 0 . g r = 1 ; the upper limit can represent c = 1 . g r = 2 . v = 0 .
55 m/s and v = 1 .
65 m/s . Since P = 375 W , in accordance with expression (10), the forces applied to thepedals are f = 682 N and f = 227 N , respectively.Examining Figure 1, we conclude that lowering c g r increases the slope of the climbable incline and,hence — for a given value of power — increases the VAM, as expected in view of Theorem 1.The VAM is determined not only by the power sustainable during a climb, but also the steepness ofthat climb. As stated in Theorem 1, the steeper the slope that can be climbed with a given power,the greater the VAM. Hence, as illustrated in Figure 1, the highest value corresponds to the lowestcadence-gear product. A maximization of VAM requires an optimization of the force applied to thepedals and their circumferential speed, along a slope, in order to maintain a high power. The presented results constitute a mathematical-physics background upon which various strategiesfor the VAM maximization can be examined. In particular, in the context of physiological consider-ations, one could examine the maximum sustainable power as a function of both the force applied6 igure 1:
VAM as function of c g r , with P = 375 W to the pedals and their circumferential speed. For instance, one might choose a less steep slopeto allow a higher c g r product, whose value allows to generate and sustain a higher power. Onemight also consider a very short and an exceptionally steep slope, while keeping in mind the issueof measurement errors due to the shortness of the segment itself. In all cases, however, the slopeneeds to be constant. Figure 2:
Slope, in % , as function of m , with P = 375 W and c g r = 2 . Let us revisit expression (14), whose general form, in terms of c g r , is8 π r w ( c g r ) + 2 π r w m g (sin θ + C rr cos θ ) C d A ρ c g r − P (1 − λ ) C d A ρ = 0 . Using common values for most quantities, namely,C d A = 0 .
27 m , C rr = 0 . , λ = 0 . , ρ = 1 . / m , r w = 0 .
335 m , and, for convenience of a concise expression, below, we invoke the small-angle approximation, whichresults — in radians — in sin θ ≈ θ and cos θ ≈ − θ / θ = 50 − . (cid:115) . . c g r − . P m c g r , igure 3: VAM as function of m , with P = 375 W and c g r = 2 . where m is the mass of the bicycle-cyclist system, and the product, c g r , might be chosen to corre-spond to the maximum sustainable power, P . This product is related to power by expression (10),since c is proportional to v , and g r to f . For any value of c g r , there is a unique value of P , whichdepends only on the power of a rider.As expected, and as illustrated in Figure 2, the slope decreases with m , ceteris paribus . Also, asexpected, and as illustrated in Figure 3, so does VAM. For both figures, the abscissa, expressed inkilograms, can be viewed as the power-to-weight ratio from 6 .
25 to 3 .
41 , whose limits represent thevalues sustainable — for about an hour — by an elite and a moderate rider, respectively.
Acknowledgements
We wish to acknowledge Elena Patarini, for her graphic support, and Favero Electronics for inspiringthis study by their technological advances and for supporting this work by providing us with theirlatest model of Assioma Duo power meters.
Conflict of Interest
The authors declare that they have no conflict of interest.
References
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