## [Bike] Gearing For Real Cyclists

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

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
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
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