Does there exist optimal damping setting for all terrains? When ζ = 1, there is a double root γ (defined above), which is real. The system is said to be critically damped. A critically damped system converges to zero as fast as possible without oscillating (although overshoot can occur).


Wikipedia

Does this mean that for constant rider+bike weight, there is optimal setting for the fron and rear shock? I had an impression that the terrain and style of riding affect the decision of settings.

Furthermore, if there is optimal damping, does it mean the rebound and compresison damping should be equal (as in the simple model, discussed in wikipedia)? If not, is this because the linear time-invariant model of second order is unable to represent the cyclist as a system adequately? Why?

EDIT (clarifications):


Hmmm, so let's ignore pedaling efficiency. The question still remains relevant.    
Optimal as in absorbs terrain irregularities best, resulting in smoothest (closest to straight line) trajectory of the rider.
Without digging too deep into the physics of damped motion now, I say "there isn't an optimal damping setting for all riders and terrains".

Let's consider first the critical damping case as defined in the wikipedia article. This is based on the case that you "excite" your system (i.e. set it into motion) once and then leave it alone until it has calmed down again due to damping. The best approximation in real life cycling would an isolated pothole or curb on an otherwise smooth road: once you've hit that obstacle, your suspension compresses and then returns back to its initial state, if there is no second bump during the time it takes the suspension to relax again. As the suspension is only uncompressing here, it's also only rebound damping that matters.

So what happens now as we go to a rougher surface where the time between bumps is shorter than the time constant of the damping, i.e. the time it takes the suspension to return into its initial state? Lets say the first bump you hit fully compresses the fork and as it has relaxed halfway back you hit another bump of the same strength which will set your fork again to full compression (if it could it would maybe even compress further). On the second bump your fork will only give you half of its travel as it is still halfway compressed from the first one. now consider an even rougher surface where the suspension can only uncompress 10% of its travel between bumps and you (hopefully) can easily see that sooner or later you will ride with your suspension nearly fully compressed, leaving you with no suspension at all.

To overcome this, you can reduce rebound damping which enables your suspension to uncompress faster at the cost of overshooting. Therefore your suspension gets more wobbly but it can react better to frequent bumps.

Compression damping comes into play when the suspension compresses (surprise?). A stronger compression damping reduces the speed the suspension compresses which is also the reason why most higher priced forks and shocks realize their lockout mechanism through high compression damping – this way you still have some suspension left for the really hard bumps. A stronger compression damping on your rough surface also means that your suspension is using less travel for the same sizes bumps, however, this time it is not converging towards the maximum compression level but tends to stay at sag level, i.e. the compression it has under rider weight. As a consequence, you can "save" a bit of travel for the really hard bumps if you raise your compression damping at the cost of a harder riding experience and a slower reaction of your suspension.