The Weight vs. Wind Resistance Tradeoff - metric version - by Nick Hein

The Weight vs. Wind Resistance Tradeoff

Making Sense of it - Metric update

By Nickolas Hein 1999

ABSTRACT

It is generally acknowledged that a faired bike will be heavier than an unfaired bike. There have been many general debates about whether the extra speed of a faired bike on level-ground and downhill speed makes up for the reduced speed when climbing hills. This paper describes the general terms involved in the tradeoff and concludes with a systematic method that can be used to determine the tradeoff in terms of overall trip time. It also includes simplified formulas for level ground speed, hill-climbing speed and for determining the speed at which a heavier/faired bike is faster than a lighter/unfaired bike. The method consists of developing a plot that characterizes the bike, showing speed vs. slope then creating a plot that characterizes the terrain, showing slope vs. probability. The two are combined to present "time at speed" vs. speed. By integrating this curve total trip time over the trip distance is derived.

Definitions of Terms and Units

Symbol Metric English Description
P Watts ft*lb/sec Power provided by the rider. For simplicity it is assumed that power losses from mechanical inefficiency are already removed.
v m/s ft/sec, mph Velocity of the vehicle at current position
W kgf, N lb Total weight of vehicle and rider
M kgm slug Total Mass of vehicle and rider
a m/s2 ft/s2 Acceleration
nx .none   Non-dimensional acceleration (fraction of gravity).
g .none   Gradient of terrain at current position
(slope is 100 times g, measured in %)
m .none   Rolling friction coefficient of tires
r .kg/ft3 slug/ft3 Atmospheric air density
g .m/ s2 ft/ s2 Acceleration of gravity
Cd   none Drag Coefficient based on frontal area
A .m2 ft2 Frontal Area
CdA .m2 ft2 The 2 terms above combined as "effective" frontal area
(.5rV2) or q N/m2 or

Pascals

lb/ft2 Dynamic air pressure (also known as impact pressure)

 

DERIVATION OF EQUATIONS

In order for a vehicle to move forward its rider provides an amount of power to overcome certain resisting forces. The total force resisting forward motion of a vehicle comes primarily from the effects of slope, rolling and wind resistance. The forces due to each of these terms are as follows:

Rider Force: Frider = Prider * h/ V
Slope: Fslope = W sin g (or Wg for small slopes)
Rolling: Frolling = W m
Wind: Fwind = (.5rV2) * CdA

The total force equation (which equals mass times acceleration or M a ) becomes:

SF = M a = (Prider * h )/ V - Wg - Wm - (.5rV2)CdA

In order to simplify the presentation slightly Prider * h will be abbreviated as P, the power available at the road surface. Mass can also be expressed as W / g so this becomes:

W/g * a = P / V - Wg - Wm - (.5rV2)CdA
Or:
W* a/g = P / V - Wg - Wm - (.5rV2)CdA

Observing that a/g is the same as longitudinal acceleration in g’s, Nx , and dividing both sides by weight results in:

Nx= P - g - m - (.5rV2)CdA
WV W

In general terms, a rider/vehicle is characterized by its Power/Weight ration (important for acceleration and climbing) and Weight/Drag ratio (important for coasting). Power, weight and drag will be constant for a given vehicle with a given rider and payload. Therefore the parameters (P/W) and (W/ CdA) will be introduced. These will allow us to generalize the performance analysis for all vehicles of different sizes that are similar in streamlining and carry riders that are similar in power output for their weight (instead of having an analysis that only applies to one combination of W, P and CdA). Rearranging the equation with these parameters gives:

Nx = (P/W) - (.5rV2) - m - g             (1)
V (W/CdA)

It may improve understanding of this equation if you look at the wind resistance terms as being 1 / (W/D) and recognizing that W/D is exactly analogous to L/D for aircraft – it is the slope (or glide angle) at which a given speed can be maintained in the absence of any power input. More will be presented on this in the next section.

In physical terms this equation obviously shows that you can find out your expected acceleration if you know all of the input terms. Beyond that, however, because one of the terms is slope, all of the terms can be interpreted as having the same magnitude and units of slope – they will be defined here as "equivalent slope" terms. In other words, if Nx turns out to be positive, then its magnitude is not only the acceleration you will see at the current slope, it is also the amount of additional slope you can climb. Similarly (P/W)/V is the maximum slope capability for a given power input and weight at a known speed. Rolling resistance will always reduce your hill-climbing capability by a constant (but small) amount. Wind resistance reduces hill-climbing capability by an amount proportional to V2.

 

SIMPLIFICATIONS

This derivation already takes the assumption that slopes are small so that sin g = g. It is also instructive to look at some special cases based on some more rash assumptions. Our most memorable and lasting moments on HPVs tend to be long slow climbs and high speed cruising on level ground. There is a simplification for each of these cases. First, in either case a steady speed will be assumed so Nx = 0. Now 2 test bikes will be defined that will be used for examples throughout the rest of the paper:

Vehicle 1:
Unfaired bike
Weight 25 lb (11.4 kg)
CdA 4 sq ft (.372 sq m)

Vehicle 2:
Hypothetical Recumbent Streamliner
Weight 50 lb (22.8 kg)
CdA 1 sq ft (.093 sq m)

Both are assumed to have the same drivetrain efficiency and a rolling friction coefficient (m) of .002. The rider of both bikes weighs 150 lb. (68.2 kg) and puts out .25 HP (138 ft*lb/sec) or 189 Watts – a typical output for a healthy young male. A mechanical efficiency has already been presumed so that this is the output at the wheel. The rider is putting out slightly more to overcome the drivetrain friction. Note that for performance analysis this is all that needs to be known about the bike. Other details of the design can remain anonymous.

Low Speed:
At low speeds wind resistance can be neglected and the power due to rolling resistance is also small. Most of your work is done in climbing. Eliminating terms and rearranging the acceleration equation to solve for slope gives:

V = (P/W)
g

In other words, the your speed is directly proportional to how much power you can put out per pound of total weight. The the steeper the climb the more you slow down. In the extreme case of climbing straight up, your speed will be equal to your Power-to-weight ratio. In fact, the vertical speed component is always equal to P/W under this assumption. So all you need to know to determine how long it will take to climb a hill is its height. As an example, here is how our two example bikes will perform.

Bike Unfaired Streamliner
Climb Rate (P/W) 16 m/min 47 ft/min 13.7 m/min  41 ft/min
Time to climb 300 ft. (100 m) hill 6 min 7.3 min

Table 1. Hill-climbing Performance Comparison

It takes 1.3 minutes longer to climb the same hill on the faired bike because the speed isn’t high enough for the fairing to give a benefit. Judging only from this case many prospective streamliner customers would opt to leave off the fairing. However, this isn’t the entire story.

High Speed:
On level ground you would like to know how fast you can go for a given amount of power. You can still neglect power lost due to rolling resistance. The acceleration equation becomes:

(P/W) = (.5rV2)
V (W/CdA)

Solving the acceleration equation for velocity yields:

V =    2P   1/3
rCdA

If you happen to carry around a calculator that does cube roots you can predict your level-ground speed knowing only your power output and the effective frontal area of the bike. For example, the rider on Bike 1 (unfaired) could expect to cruise along on level ground at 31 ft/s or 21 mph (34 km/hr). By instead riding a streamlined bike (Bike 2) with the effective frontal area his speed would be 33 mph (53 km/hr). Notice that weight is completely absent from this equation. In reality the effect of weight on rolling resistance would cause actual speeds to be 3-5% slower.

An interesting exercise is to evaluate the wind resistance term at 32 km/hr (20 mph). It just so happens that in the metric system of units, .5rV2 (commonly known as dynamic pressure), is equal to about 5 kgf/m^2 (1 lb/ft^2 in English units). If you again ignore rolling resistance and assume coasting, you can find out the gradient required to coast at this constant speed. In the author’s experience when riding around town a lot of time is spent at this speed, and this result will characterize how much effort is required to do that. Assuming a known value of CdA, the ratio of W/ CdA is the grade that will permit you to coast at 32 kph (20 mph). In equation form g = CdA/ W. The inverse of this term is W/D. As mentioned before this term is exactly analogous to L/D commonly used to express aerodynamic performance of aircraft. For the examples used previously:

 

Bike Unfaired Streamliner
CdA/ W
Metric Units
5 x . 376 / 79.5 5 x .093 / 90.7
English Units 4 / 175 1/200
Grade .023 or a 2.3% grade .005 or a .5% grade
W/D 44 200

Table 2. W/D (~L/D) Comparison of Example Bikes

 

Since dynamic pressure is quadrupled if speed is doubled to 64 kph (40 mph) the streamliner would have a W/D of 50 while the unfaired bike would be down to 11. Consequently the grades required for these bikes to maintain that speed while coasting increase to 9.1% and 2% respectively. Typical aircraft cruise L/D can range from 40-60 for low-speed high-performance gliders (typically cruising at about 60 mph). By taking advantage of streamlining, bicycles can achieve the kind of cruising efficiency that has only been possible with high-performance aircraft, at a much lower weight and cost.

This analysis has given us the ability to prove that a lighter bike is faster up hills and a streamlined bike, usually heavier, is faster on the flats. What is needed is a way to tell which bike is faster all-around in terrain with arbitrary combinations of hills and level roads.

 

THE TRADEOFFS

So now we know the important parameters at the extreme ends of the speed scale. At low speeds Power/Weight is most important while at high speeds Weight/Drag dominates. However, the real question all along has been:

"Given the P/W and W/D for two bicycles which one will be superior for getting to the end of the same ride in less time?"

Knowing this will permit you to find out, given the type of terrain that you ride in, when either vehicle is better for getting around faster. The acceleration equation has given us a way of characterizing the vehicle based on its P/W and W/D ratios. By plotting the speed capability vs. gradient it will be apparent that the curves for two different vehicles will cross at a certain speed. Above this "break-even speed", the more streamlined vehicle will be at a speed advantage in spite of a greater weight. Below break-even speed, the lighter and less-streamlined vehicle will be faster in spite of greater drag.

Solving the steady-state (Nx = 0) acceleration equation for gradient yields the equation:

g = (P/W) - m - (.5rV2) (2)
V (W/CdA)

This will be referred to as the slope equation. Substituting a range of values for velocity yields a plot of the gradient capability for the two example vehicles as shown in Figure 1. This can be considered to be the speed you can go at any given slope. The point where these curves cross is the break-even speed above which the streamlined bike takes less effort and below which the lighter bike takes less.


Figure 1. Speed vs. Gradient Capability for the Two Example Vehicles
(10 mph = 16 km/hr)

Alternatively, you can set the slopes equal for the two bikes and can solve for the velocity at which they will go the same speed. Doing so yields the equation:

Vbe =           2 [(P/W)1 – (P/W2)]       1/3
r [1/(W/CdA)1 – 1/(W/CdA)2]           (3)

For the example bikes the breakeven speed is approximately 6 mph (10 kph). If you ride a given bike with a cycling computer you probably have an idea how much time you spend above or below a given speed and you can make a pretty good guess at this speed on your current bike. Knowing this will help you decide whether a heavier streamlined bike will save you time. For most riders this is about all that is needed since you may only do this when you decide whether to get a new bike. However, if you’d like to compare bikes without spending all that time riding them this equation comes in handy. The breakeven speed is handy to know even if you live in completely flat terrain. Remember that the acceleration equation treats acceleration the same as slope. If you’re going slow because you’re accelerating it’s the same as going slow because you’re climbing.

Although the speed vs. gradient curve and acceleration equation completely summarize the performance of each vehicle over the entire conceivable range of slopes anyone might encounter, what is still needed is a way to determine how much of the terrain is steeper than the break-even slope.

To characterize the terrain, in theory, we could each ride around our own home range with an altimeter and odometer and get terrain profiles for each route we ride. Using a time-history simulation program you could then find out whether it takes longer overall to ride one vehicle instead of the other. Although this is exact and rigorous, it gives the unsatisfying result that it needs to be repeated for every new route of interest. A general approach is to assume a standard probability distribution curve (bell curve) adjusted for the local terrain characteristics. An example of such a curve is shown in Figure 2.


Figure 2. Standard Slope Distribution for Terrain With 10% Maximum Grade

A standard distribution is a bell curve where 98% of all occurrences are within 3 standard deviations of the mean. The difference between flat and hilly terrain is characterized by the magnitude of slope within 3 standard deviation to either side of flat (0%). A distribution for flat terrain would fit between 1-2% on either side. Hilly terrain would have a range on either side of zero corresponding to the steepest slope. Obviously it is unlikely that any real terrain fits this nice neat model. The author’s own home range is characterized by large flat river valleys, long steep climbs up the ridges and rolling hills on the ridgetops, yielding a distribution curve that is bimodal. So the curve for any area may have some lumps in it that make it deviate from the standard distribution. However, it is useful to have this standard bell curve even if it is not exact.

It would be nice if we could look at the speed vs. gradient plot to figure out the breakeven grade, then take that to the gradient distribution plot and determine which bike is better, based on how much of the route was steeper than the breakeven slope. It would be nice but it might be wrong. The distribution tells you how much of the distance you are going faster. What you really want to know is how much time you spent going faster. Then you can figure total trip time. Getting this answer requires an additional step.

Because of the way the slope equation is written (re-writing it is messy), we put in even values of speed to get non-integer values of slope. Since all of this analysis was done in an Excell spreadsheet, the slopes were interpolated to get integer values of slope and their corresponding speeds. Probability of slope was then turned into distance of slope from which a time could be calculated. Summing the times gives the total trip time for the standard normal route. The results are given for the two example vehicles in Figures 3 and 4. An assumption was made that 40 mph was the safe speed limit and therefore in situations where the vehicle exceeds 40 mph brakes are used to hold speed. This is the cause of the odd-looking wiggles in Figure 4 for Bike 2. This curve is then integrated to get a total trip time for a given distance.


Figure 3. Distance at Speed
(3000 ft = 1000 m)


Figure 4. Time at Speed 
(10 mph = 16 km/hr)

The following table presents the results along with a comparison of the time it takes to ride the actual route. The example situation I’ve used is based on my actual commute, which is a 25 mi. round trip with a maximum 10% grade. The analysis has shown that it’s reasonable to expect that the more streamlined bike will complete a hilly route in less time even though it has a greater weight. In the real world this result holds, although the total trip time is longer for both bikes

Vehicle

Calculated time (min)

Percent Difference

Actual (Approx.)

Percent Difference

Unfaired

90:00

 

110

 
Faired

78:30

-13%

100

-9%

The difference between the calculated and actual times is likely due to the following. This entire solution process has assumed quasi-steady conditions. That is, at any given slope the vehicle will have a steady speed. In reality we know that you zoom down a slope and might have much higher speeds at zero grade than this shows, or when cresting a hill you will have a slower speed at zero grade than this shows (until you accelerate). In addition, the route has quite a few stoplights and some heavy traffic to slow down for. However, this method provides a way of making gross comparisons quickly, after which you can run a more detailed simulation using actual terrain data if necessary.

 

CONCLUSION

Here is a summary of the concepts and equations presented:

  1. The faster of 2 bikes depends on their weight and streamlining. These are characterized by the ratios of Power/Weight and Weight/Drag. Weight/Drag is exactly analogous to the Lift/Drag ratio commonly shown for aircraft performance.
  2. The acceleration of a bike (as a fraction of gravity) is due to 4 components as shown by the following equation in terms of the Power/Weight and Weight/Drag:
    Nx = (P/W) - (.5rV2) - m - g       (1)
    V (W/CdA)
  1. Since slope is one of the terms in this equation, all of the terms can be interpreted as equivalent slopes.
  2. The following equations permit simplified analysis for the low-speed (steep climb) and high-speed (flat-terrain) cases:
    Low-Speed V = (P/W)                High Speed V =    2P   1/3
    g rCdA
  1. The slope a bike can climb vs. speed can be calculated from the equation:
    g = (P/W) - m - (.5rV2)        (2)
    V (W/CdA)
  1. There is a break-even speed (Vbe) below which the lighter, less streamlined bike will be faster and above which the heaver, more streamlined bike will be faster. This speed can be found from the equation:
    Vbe =          2 [(P/W)1 – (P/W2)]       1/3
    r [1/(W/CdA)1 – 1/(W/CdA)2]      (3)
  1. Terrain can be characterized by a standard deviation with the bounds set by the maximum slope in the riding range. Using the slope vs. speed characterization of the bike and the Probability vs. Slope characterization of the terrain an estimate can be made of the difference in time it takes 2 bikes to complete any distance in that terrain.

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