From: andrew cooke <andrew@...>
Date: Sun, 20 Dec 2020 16:58:17 -0300
On the Internet, when discussing lower gearing for a bike, you can expect
someone to comment "just learn to climb". Which drives me crazy, because the
problem is not the cyclist, but the gears.
Gearing on most bikes sold to new cyclists is not that different to the
gearing that professional cyclists use. Yet professional cyclists are hugely
more powerful than new cyclists (or even experienced amateurs). A reasonably
fit, new cyclist can expect to produce around 100W sustained power. A pro can
maintain 4x that, or more.
The physics of cycling uphill - at least at speeds slow enough for air
resistance to be negligible - is simple. It's so simple that it's basically
linear - you can adjust things by scaling relative numbers. In other words,
if a newbie cyclist has 1/4 the power of a pro they should have gears that are
4x as easy. This is not the case.
To illustrate this I've written a small program that calculates the gearing
you would need (expressed as front x back number of teeth on the gears) for
different power output, gradients, and cadences.
The power levels cover the range from newbie (100W) to pro (400W+), the
gradients from "normal" hills (6%) to the steepest the Alps can offer (14%),
and the cadences reflect climbing standing (30rpm), grinding slowly sitting
(60rpm), and spinning (90rpm) while sitting.
Note that I have assumed a rider of about my weight (65kg, 143lb). If you
(plus bike) weigh twice what I do then you need gears twice as low. Again,
it's simple scaling.
Full results (and the program) are below. Here I'll pick a few interesting
numbers:
* A 400W pro can spin (90rpm) up a 12% gradient using 39x27. This is
typical of the lowest gearing on a professional bike (which makes sense).
* A 100W newbie, to do the same, would need 26x72. That's a 26 tooth gear
at the front and an 72 tooth rear - so extreme it's not even available on
mountain bikes (a 200W rider would need 26x36, which is a MTB gear).
* A 200W amateur rider, with 34x28 gears (about the lowest most new road
bikes go) can spin (90rpm) up a gradient of around 7%, but can manage over
14% if they learn to climb standing at a low cadence (30rpm).
I think the last point is the source of "learn to climb" - learn to ride
standing at low cadence. But note that this is only an option if you're
a similar weight to me (and many riders seem to weigh much more).
Finally, remember that these are broad generalisations, based only on work
against gravity (ignoring rolling resistance and wind resistance). But when
we're dealing in differences of a factor of 4, a rough categorization within
10% or so is fine.
Andrew
Cyclist generating 100W
-----------------------
Climb of 6%
30rpm 26x12 30x14 34x16 39x18
60rpm 26x24 30x28 34x31 39x36
90rpm 26x36 30x41 34x47
Climb of 8%
30rpm 26x16 30x18 34x21 39x24
60rpm 26x32 30x37 34x42 39x48
90rpm 26x48
Climb of 10%
30rpm 26x20 30x23 34x26 39x30
60rpm 26x40 30x46
90rpm 26x60
Climb of 12%
30rpm 26x24 30x28 34x31 39x36
60rpm 26x48
90rpm 26x72
Climb of 14%
30rpm 26x28 30x32 34x36 39x42
60rpm 26x56
90rpm 26x84
Cyclist generating 200W
-----------------------
Climb of 6%
60rpm 26x12 30x14 34x16 39x18
90rpm 26x18 30x21 34x23 39x27
Climb of 8%
30rpm 34x10 39x12
60rpm 26x16 30x18 34x21 39x24
90rpm 26x24 30x28 34x31 39x36
Climb of 10%
30rpm 26x10 30x11 34x13 39x15
60rpm 26x20 30x23 34x26 39x30
90rpm 26x30 30x34 34x39 39x45
Climb of 12%
30rpm 26x12 30x14 34x16 39x18
60rpm 26x24 30x28 34x31 39x36
90rpm 26x36 30x41 34x47
Climb of 14%
30rpm 26x14 30x16 34x18 39x21
60rpm 26x28 30x32 34x36 39x42
90rpm 26x42 30x48
Cyclist generating 300W
-----------------------
Climb of 6%
Note: speed > 20kmh, air resistance significant
60rpm 34x10 39x12
90rpm 26x12 30x14 34x16 39x18
Climb of 8%
60rpm 26x11 30x12 34x14 39x16
90rpm 26x16 30x18 34x21 39x24
Climb of 10%
30rpm 39x10
60rpm 26x13 30x15 34x17 39x20
90rpm 26x20 30x23 34x26 39x30
Climb of 12%
30rpm 34x10 39x12
60rpm 26x16 30x18 34x21 39x24
90rpm 26x24 30x28 34x31 39x36
Climb of 14%
30rpm 30x11 34x12 39x14
60rpm 26x19 30x21 34x24 39x28
90rpm 26x28 30x32 34x36 39x42
Cyclist generating 400W
-----------------------
Climb of 6%
Note: speed > 20kmh, air resistance significant
90rpm 30x10 34x12 39x13
Climb of 8%
Note: speed > 20kmh, air resistance significant
60rpm 34x10 39x12
90rpm 26x12 30x14 34x16 39x18
Climb of 10%
Note: speed > 20kmh, air resistance significant
60rpm 26x10 30x11 34x13 39x15
90rpm 26x15 30x17 34x20 39x22
Climb of 12%
60rpm 26x12 30x14 34x16 39x18
90rpm 26x18 30x21 34x23 39x27
Climb of 14%
30rpm 39x10
60rpm 26x14 30x16 34x18 39x21
90rpm 26x21 30x24 34x27 39x31
Python 3 code:
circumference = 2.14 # m (measured from rolling road bike)
g = 9.8 # m/s2 (gravitational acceleration)
mass = 8 + 65 # kg (bike + me)
front_gears = [26, 30, 34, 39] # teeth
gradients = [6, 8, 10, 12, 14] # %
powers = [100, 200, 300, 400] # W
cadences = [30, 60, 90] # standing, grinding, spinning
for power in powers:
print(f'\nCyclist generating {power}W')
print('-----------------------\n')
vertical_speed = power / (mass * g) # m/s
for gradient in gradients:
print(f' Climb of {gradient}%')
horizontal_speed = vertical_speed * 100 / gradient # m/s
if horizontal_speed * 3.6 > 20:
print(' Note: speed > 20kmh, air resistance significant')
wheel_rpm = 60 * horizontal_speed / circumference
for cadence in cadences:
subtitle = False
for front_gear in front_gears:
gear_ratio = cadence / wheel_rpm
rear_gear = int(front_gear * gear_ratio + 0.5) # round
if rear_gear > 9 and (rear_gear <= 50 or not subtitle):
if not subtitle:
print(f' {cadence}rpm ', end='')
subtitle = True
print(f'{front_gear}x{rear_gear} ', end='')
if subtitle:
print()
print()
Cadence v Speed
From: andrew cooke <andrew@...>
Date: Sat, 26 Dec 2020 10:24:43 -0300
The above isn't clear enough on how important cadence is for climbing. Choosing gears for a bike has to meet three criteria: * High cadence on steep climbs (pressure on lowest gears) * Not spinning out on steep descents (pressure on highest gears) * Reasinable spacing (pressure on number of gears) Current solutions are largely failing new riders for the first of these. Andrew