Thursday, October 15, 2009

Analysis Of The Bicycle Endo

The endo, short for end-over or end-over-end, is a type of pitch over crash where the cyclist goes over the handlebars, the weight offset of which causes an inertial moment to act about the front wheel resulting in rear portion of the bicycle to flip in the air above and behind him.

Usually, the cyclist, as a sudden reflex action, yanks out their hands or legs at some point to cushion the impending fall and ends up letting go of handlebar control. Meanwhile, the bicycle is bound to fall either sideways, due to its motion about the steering axis or right on top of the cyclist. In the latter scenario, the saddle or even the rear wheel itself could land on the cyclist's body.

Again, an endo is a crash that could cause injury. It is not a bicycle trick. That one has another name. Its called a 'stoppie' or a 'wheelie'. An endo is caused due to strong front wheel braking or when the bicycle hits a curb, a structure more rigid than the wheel itself. Endos may also occur if the front wheel is loose, i.e, if it is not secured properly to the fork dropouts by the quick release skewers.

In this post, we will cover the "endo parameter", study the relationship between braking force and endo parameter on level ground, outline some common reasons for endos, check out a video analysis of an endo and finally study the relationship between gradient of the road and endo parameter through a literature source.


Braking a bicycle naturally upsets equilibrium and transfers weight to the front wheel. With a stark increase in the overall braking force, the load on the rear wheel approaches zero, after which the rear wheel will start to lift off the ground. Hard braking may stop the bicycle but Newton's first law reigns supreme as the cyclist's body continues in motion in the headed direction. This rider motion has some momentum. If not self-controlled, the rider will flip over the handlebars and the bicycle will pitch-over as well. What results is the endo.

It turns out that while outrageous situations cannot be helped, some factor of safety from bicycle design and rider positioning skill can provide for a cushion against pitch-over tendency in the above mentioned situations.

I'll call my main parameter of interest the pitch-over parameter (or endo parameter for lack of a better word) - A/H - as can be seen in the diagram below :

Fig 1 : Free Body diagram of a bicycle-rider system just at pitch-over. Now you may be able to appreciate from geometry and c.o.g as to why recumbents and tandems are stable in pitch-over. O is the point signifying the contact point of the front wheel with the ground.


W = combined rider-bicycle weight
Wf = normal load on front wheel
Wr = normal load on rear wheel
Ff = braking force at front wheel
Fr = braking force at rear wheel
Fb = braking reaction (mass times deceleration)
L = wheelbase
A = location of center of gravity (c.o.g) aft of front wheel
B = location of c.o.g forward of rear wheel
H = height of c.o.g

The endo parameter, A/H (a ratio), in the combined bicycle-rider system should be large enough to avoid front pitch-over. Obviously the vertical height, H, of the center of gravity (c.o.g) and the location of the c.o.g aft of the front wheel, A, are going to vary with variation in rider's weight, height and sitting position.

This highlights why its important to get a proper bike fit for the type of bicycle you wish to ride. Its not just a question about comfort. Its also a question about safety. Enlarged riders who overwhelm miniature bikes not made for their size will quickly find out what they're doing wrong. All they have to do is hit the front brakes hard and they're right on target to be turned into human projectiles.

Looking at the free body diagram above, we can deduce that the system is in static equilibrium about the front wheel contact point O if the sum of the moments due to all forces about that point is zero. In other words, rotation is just initiated at :

Fig 2 : Endo parameter relationship to braking force. This also gives us an expression for the braking force at the point of the pitch-over.

From this simple relation for level ground, we see that endo parameter is equal to the braking force as a percentage of total weight at just about the initiation of the endo. Braking force is a function of the co-efficient of friction at the tire-road interface.

Before the initiation of pitch-over, the braking force-weight ratio is lesser than the endo parameter. Well after the pitch-over has been initiated, the endo parameter falls lesser than the braking force-weight ratio.

We can now infer that making A/H larger is better for safety. Otherwise, a lesser braking force relative to total weight will be sufficient to initiate pitch-over. How? Simply because the braking force-weight ratio catches up with the endo parameter sooner. Oops.

A/H can be fixed to be greater with good bicycle design and proper fit. It can also be superficially made larger by the cyclist while riding by positioning his body rearward (relative to bottom bracket) as the following picture shows :

Fig 3 : A cyclist ducks and shifts his c.o.g rearward to increase his endo parameter

People succumb to pitch-overs because of other factors too. They may not be skilled enough to increase the endo factor, A/H. They also may not be skilled enough to modulate and may tend to hitting the front brakes really hard without realizing that a front brake can cause more deceleration than a rear brake. For comparison, front brakes generate upto 0.5g's of retarding force whereas rear brakes produce a max of 0.1 or 0.2g's. Note that maximum deceleration is limited by the co-efficient of friction between tire and road and the normal load.

You can notice front braking power for some mountain bikes through a speed vs time graph.

Fig 4 : Speed vs Time graphs of MTB's (Courtesy : Beck Forensics)

Obviously, the graphs show that you can bring a bike to a stop faster using the front brakes alone than the rear brakes. Using both brakes is even better for reducing stopping distance even further.


Beck Forensics did an interesting little video analysis of an endo. The following image as well as the snippet below it is taken from a web sampler of their book Bicycle Collision Investigation. It shows the steps involved in an endo before the crash.

A mountain bicyclist traveling at about 22.5 mph (36.2 kph) applies only the front brake. Once the front wheel is nearly locked, the rear wheel starts to lift up. At about 0.20 seconds, the rider really has no chance to recover. At about 0.33 seconds, he releases the brake and prepares his right hand, and then his left hand for landing. The cones are shaped in 25 foot (7.6 m) intervals and the grade is about -2% (descent). (Courtesy : Beck Forensics)


This section is a little more involved. It uses the same analysis techniques shown above to derive a relationship between the "endo parameter" and braking force-weight ratio with the percentage gradient of the ground. You will see that the chances of an endo are more likely on a descent.

The following literature is from one digest of IHPVA (2001), written by a retired engineer named Frederick Matteson. Click on the series of images to zoom the text. Alternatively, you can also read the paper here.


Page 1 : Click to zoom

Page 2 : Click to zoom

Page 3 : Click to zoom


Budbrake : Proportional Brake Control For Safer Bike Stops
Dynamic Stability Of Bicycle Design : Part 1
Dynamic Stability Of Bicycle Design : Part 2
Dynamic Stability Of Bicycle Design : Part 3
Dynamic Stability Of Bicycle Design : Part 4

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Phil said...

Thanks Ron. I love your articles and really appreciate the time you put into this. As usual, I'll have to go home and take a good look at this :)

Anonymous said...

It would be interesting to do a comparison of a 26er" vs 29er" mountain bike to see if which is less likely to endo.

Will said...

just curious. what is the cog for normal bikes? i have been trying to hunt this down. how can one assess with simple tools where the cog lies if they're sitting on a bicycle? any ideas?

Bluenoser said...

As always Ron I love your posts. Downhill has always been my nemesis.

But 22miles per hour when locking up the front brake? Come on... that's just looking for disaster.

There has to be the let go factor factored in. Where if you feel you're going over the bars you let go the brake lever.