Maximal Braking on a Standard Bicycle
Version 1.0
Joseph S. Riel
joer@k-online.com
5 July 2005
Introduction Solve for amax gives
The maximum steady-state braking deceleration amax = g hcm /`f . (3)
achievable by a cyclist riding a standard bicy-
cle is limited by pitch-over [1]. In this article This shows that the maximum deceleration
an expression is derived for the maximum de- depends on the location of the CM of the BR.
celeration on a level road. It is then extended
to a road with a slope. A simple method is de-
scribed for measuring the position of the center
Slopes
of mass (CM) of a cyclist.
Now consider the case where the road is not
level, that is has a slope s.
Level Ground Fx : m amax = FB, (4)
Consider a bicycle and rider (BR) braking on
M|P1 : 0 = FB hcm mg(`f cos + hcm sin ).
(5)
level ground. If the coefficient of friction be-
tween tire and road is large enough (and it gen- Solving for amax gives
erally is), the bicycle rotates about the ground
contact point of the front wheel when the accel- `f
amax ( ) = cos + sin g. (6)
eration exceeds a maximum value, amax . This hcm
value is computed by assuming that the
For slopes in the range (20%, 20%), cos 1,
Fx : m amax = FB , (1) sin , so
M|P1 : 0 = FB hcm mg`f . (2) amax (s) amax (0) + s g. (7)
1
For example, if the maximum braking ac- The parameters kf and kh correspond to the
celeration to avoid pitchover on level ground value and (negative) slope of kwr at ks = 0, re-
is 0.7 g, then when descending a 10% slope spectively. The location of the CM is given by
(s = 0.1) the maximum braking acceleration
is 0.7 g 0.1 g = 0.6 g. `f = `wb kf , (11)
hcm = `wb kh + Rw . (12)
The wheelbase of the bicycle, `wb , is 100 cm,
Locating the Center of Mass the wheel radius, Rw , is 34 cm. Using values of
k and kh extracted from the graphs in figure 2
As shown in the preceding analyses, the loca- f
gives
tion of the CM of the BR significantly affects the
available braking deceleration. This section de- `f(on) = (100 cm)(0.58) = 58 cm,
scribes a simple measurement technique for lo-
cating the CM of the BR. hcm(on) = (100 cm)(0.58) + 34 cm = 92 cm,
The fractional weight, kwr , on the rear tire is `f(behind) = (100 cm)(0.71) = 71 cm,
hcm(behind) = (100 cm)(0.52) + 34 cm = 86 cm.
Wf
kwr = 1 , (8)
Wtot The maximum braking acceleration on level
`f `h ` /` ground is then
= p s wb , (9)
`wb `wb 1 (`s /`wb )2
amax(onsaddle) = 58/92 g = 0.63 g,
ks
= kf kh p , (10) amax(offsaddle) = 71/86 g = 0.83 g.
1 ks2
where Wtot is the total weight of the BR, Wf is
the weight on the front wheel, `f , `wb , and `s are References
shown in figure 1, and kf , kh , and ks are the re-
spective lengths normalized to the wheel base, [1] David Gordon Wilson with Jim Papadopou-
that is, `f /`wb , `h /`wb , and `s /`wb , respectively. los. Bicycling Science. Massachusetts Insti-
To determine kf and kh , measure kwr while vary- tute of Technology, third edition, 2004.
ing the height of the rear axle above the front
axle. This can be readily done with a bathroom
scale and an assortment of bricks or blocks of
wood to raise the rear or front of the bicycle.
Figure 2 plots the data for a rider on a stan-
dard (lightweight) road bike in two positions:
sitting on the saddle and hanging off (behind)
the saddle.
2
CM
Wtot
`h
hcm
`f
`wb
Rw
Wr
`s
Wf
Figure 1: Locating the CM
0.8 +
+ +
+
+ +
0.7 +
+ +
+
+
+ Behind the saddle
+ +
+ + +
0.6 + +
kwr
+ +
On the saddle +
+
+ +
0.5 +
+
+
0.4 +
0.3 0.2 0.1 0 0.1 0.2 0.3
ks
Figure 2: kwr versus ks . The lower curve is for
the cyclist sitting on the saddle; the upper curve
is for the cyclist behind the saddle. For both po-
sitions hands are in the drops, gripping the brake
levers. Both data and fitted curves are shown for
each position.
