Elasticity is usually something to avoid in a bicycle, and much effort is invested into making frames as rigid as possible within an given weight. Flexibility in the frame can lead to instability at high speeds and it is also believed to cause loss of energy.

Springs (green) added between the crank and the frame of a bicycle. When turning the crank, the springs get stretched and relaxed alternately. Can this be an advantage?

So the idea of adding springs to the crank mechanism appears at first sight not to be a very good one. So why even bother?

Well, the crank moment over a cycle varies much like a sine function, and this causes speed variations - more at low speeds than at high speeds. These variations in speed are believed to be undesirable, so coaches are trying to teach young competitive bicycle riders to apply the torque as evenly as possible over a cycle. This is difficult to do because the human leg is not well equipped to provide torque near the top and bottom dead centers of the pedal cycle. In fact, if we require the body to provide a constant crank torgue over a cycle, then we can compute a muscle activity profile like this:

The envelope of muscle activity over a cycle with constant crank torque. The two sharp tips of the graph are the dead centers - much muscle activity is necessary to drive the crank

Here's another skeptical thought: Obviously there should be two springs because of the symmetry of the pedals and the two legs. But wouldn't this just make the the effect of the springs cancel each other out?

The questions are:

1. Can springs on the crank mechanism make the torque generation more homogeneous so that the rider does not perceive the dead centers?
2. If springs are beneficial, which stiffness should they have, and how should the be mounted?.

We can define the following parameters of the problem:

	L0: The slack length of the spring.
	F1: The spring force at 50% strain
	SpringRad: The radius of the fixation of the spring on the crank
	SpringAngle: The angle of the fixation of the spring on the crank.
	SpringY: The vertical coordinate of the fixation of the springs on the bicycle frame measured from the crank center.

We used the console version of the AnyBody Modeling System, and we constructed a very simple optimization routine inside Matlab to call AnyBody as a subroutine. The optimizer vas based on golden section search optimization along the parameter axes in the deign space. In other words, the algorithm sequentially optimized with respect to each parameter while keeping the others constant.

With optimization it is always good to know a little bit about the kind of design space you are operating in. So we initially made a parameter study of two parameters and made a surface plot  of the result. The parameters were F1 and SpringAngle.

The variation of maximum muscles activity as a function of the angle of the mounting point of the spring on the crank.

The figure shows clearly that the design space is non-convex. This means that it is likely to have several local optima.

A subsequent optimization of all the parameters succeeds to reduce the maximum muscle activity over a cycle to 18.1% where the value without springs is 39%. The profile is very even with only a drop in perceived effort that accidentally occurs just at the dead centers.

Muscle activity envelope over a crank cycle with optimized spring parameters.

The optimized values of the design parameters are these:

L0 = 0.2897 m
F1 = 114.5789 N
SpringRad = 0.1673 m
SpringAngle = 49.2547 degrees
AnyVar SpringY = 0.4319 m
 

Configuration of optimized springs.

The force variation in the two springs over the cycle reveals that they become slack when they are close to the top of the crank.

Force variation in the two springs over the cycle. Notice the flat piece at the top of the curves where the springs become slack.

The springs are rather long, and it might be necessary to limit the length or the radius of fixation on the crank. The optimization capability makes it very easy to impose that type of constraints and repeat the optimization within the given limits. The entire optimization of this very simple model takes only a few minutes on a laptop PC.