We are told there's nothing easier than riding a bike. The reality is when it comes to staying upright, there is nothing more complicated. The mathematical formula which explains the motion of a bicycle looks like it could be used to split the atom and took scientists from three different countries years to devise.

It also explains the frustration and angst suffered by so many dads watching their struggling youngsters totter, wobble and crash as they attempt to master two wheels for the first time.

The complex equation, which uses 31 numbers and symbols, plus nine pairs of brackets, looks like this:

What this actually boils down to is:

Inertia forces + gyroscopic forces + the effects of gravity and centrifugal forces = the leaning of the body and the torque applied to the handlebars of a bike.

Or put more simply if you don’t peddle fast enough to keep moving while keeping the bike straight, you fall over.

The equation produced by boffins from universities in Holland, the USA and Nottingham has come to light during research by Halfords to compile tips for parents teaching their children to ride a bike.

With National Cycle Week starting Mon June 20, more people than ever are being encourage to get on a bike and Halfords was hoping to help cut down on the agony suffered by youngsters when going though that rite of passage when first taking to two wheels.

Commercial Director, Paul McClenaghan said “It turns out that getting off on the right foot on a bicycle, ditching the stabilisers and speeding away from your anxious parents is actually much more complex that people realised.

“Once you master the technique as the saying goes it’s something you never forgot, but there is a great deal of science behind the skill. We make all our own bikes and are hoping that this work will help make bike riding as easy as possible.”

Dr Arend Schwab of Delft University of Technology in the Netherlands who helped develop the equation explains that ever since the inventions of the bicycle in the 1860’s mathematicians have been trying to use Newton’s laws of motion to explain its unique movement and ability to balance.

“People more than a hundred years ago were trying to figure out why a two wheeled bicycle, given forward momentum, like a push, would seem to balance by itself,” said Dr Schwab.

The meticulous mathematical account of bike riding and their continued research may eventually lead to better bike design with improved stability and safety, something that has also attracted the attention of British bike retailing giant Halfords.

Dr Schwab explained: “Using our equation we can simulate the motion of a bike and predict whether it will remain stable or not, under certain conditions, such as if it goes over a bump, or is hit by a gust of wind.

“This equation is aimed at enabling a bike designer to change certain features and to see the overall finished effect on the bike, without having to actually manufacture it first.

“For instance if you are designing a folding bike with smaller wheels or one with a shorter wheel base this equation allows you to interpret how design changes will affect the stability and behaviour of the bike,” Dar Schwab added.

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**More information:**
-- For further information on Dr Schwab and his colleagues’ research, visit: audiophile.tam.cornell.edu/~als93/Bicycle/index.htm

-- Research paper: J. P. Meijaard, Jim M. Papadopoulos, Andy Ruina, A. L. Schwab, 2007 ``Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,'' Proceedings of the Royal Society A 463:1955-1982. doi:10.1098/rspa.2007.1857

## hush1

Such a bike never "tips over". It remains upright, whether stationary or in motion. A force applied to tilt the bike, if removed again, results in the bike erecting itself to the upright position again.

Of course, gyroscopes have been tried, to achieve this effect.

To imagine this, it helps to picture a tight rope walker, who's center of gravity is so far below the rope's surface, that the tight rope walker needs to apply a additional force to his feet to remove his feet from the rope.

Of course, the bike, the tight rope walker and their centers of gravity all have to lie along the same vertical plane.

## hush1

should read..."...an additional force..."

ugh, grammer! :)

## MikeMike

## trekgeek1

What the hell are you talking about? There is no way you can put the center of gravity below the object. The center of gravity must lie within the bounds of the object. Not necessarily within the material, but within it's dimensions( a boomerang shaped object will have its center in empty air, but within the dimensions of the object). I've seen these bikes at science museums where a large counter weight hangs below the cable on which the bike rolls along, but good luck doing that on concrete sidewalks. Nonsense, absolute nonsense.

And of course, if I've made an egregious error, please correct me Phyorg community. This isn't my field of expertise.

## hush1

Well, your museum bike experience approaches the practical solution I have in mind. And yes, you are absolutely correct. The center of gravity does lie within the bike's (slightly modified) dimensions.

You have made no egregious error. I see nothing to correct in your reasoning. Your labeling, "Nonsense, absolute nonsense" might change later though.

And yes, I will need to test that on any surface, concrete sidewalks included. And with luck, I'll patented it, if it works.

## frajo

## RobertKarlStonjek

Instead, a mass that can lean and actuators that can control the handlebars and peddles would be able to ride a bike using this formula, but the bike would fall over below a certain speed (but experienced humans do not).

The equation is accurate for the scenario they modelled (centrifugal force etc) but does not apply to actual bike riding, especially at low speed.

## Doug_Huffman

Ride smart, not hard. Ride recumbent. Ride tricycle. Ride Greenspeed.com.au

## RobertKarlStonjek

This has several applications ~ balancing at low speed is greatly enhanced; fast acceleration can be achieved by angling the bike as the peddles are depressed (see road racers sprint to the finish line, as in the 'Tour de France'); and mountain bike riders use the technique to gain more traction on uneven terrain, balance at low speed, and to allow the legs to act as shock absorbers over bumps.

So it the mathematical problem is far from insurmountable and most probably could be appended to the current solution. There are two dynamics to consider ~ the lateral acceleration caused by suddenly angling the bike (as when countering pressure to the peddles) and the change in balance point, as when riding at low speed. Obviously the scientists who formulated the above equation never leave the saddle when riding...

## hush1

## PTK

## jimbo92107

## Ronan

## Kenfreeman

The Schwab/Ruina/Papadopoulos paper cited gives eigenvalues for a sample bike design and shows how stability/instability zones emerge as speed is increased from zero. In this respect it does address very low speed riding, though did not specifically discuss the track stand. It is progress to improve our understanding of a piece of a problem, en route to understanding the whole.