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I'm no expert at parameters like optimal frame height or handlebar to saddle distance. This article focuses on more general geometric aspects affecting stability and handling of bikes, both upright and recumbent.
Let's say the dimensions are positive in the direction of the arrows. Letter marking was chosen arbitrarily.
A crucial parameter determining dynamic stability of a vehicle. Road forces act on the wheel at point B, steering axis intersects the road at point A. If we divert the wheel from straight course, road force Ft together with inertial force of the vehicle Fv at distance r create a torque which centers the wheel again (negative feedback).
General experience says the optimal trail for a bicycle is somewhere around 50 mm. Shorter one makes the steering quicker and more responsive, but the self-centering effect (necessary for hands-free riding, for example) diminishes. Longer trail makes the bike too stable and it's difficult to force it into a turn.
Theoretically, trail can go all the way to the negative direction, which means the wheel is pushed before the steering axis rather than pulled behind it. The torque from road and inertial forces then works the other way round: it pushes the wheel further out of straight direction rather than centering it (positive feedback). A bike with classic fork and handlebars is completely unridable in such configuration, but if we overcome the destabilizing dynamic torque with strong static centering torque (see further) and involve legs in the steering, it can work. A living example is Python lowracer:
Trikes use positive trail too. Probably the only occasion where neutral trailless geometry is better is rickshaws with just one of the rear wheels driven - a trail would cause uncomfortable torques on the handlebars.
Another important element of dynamic stabilization. If we tilt a spinning flywheel with a torque Mx, torque My appears and, coincidentally, steers the wheel exactly where it needs to be to cancel the tilt:
The steering torque derives from a vector product of rotation and Mx. To keep things simple, all we need to know is the stabilizing effect grows with rotation speed and moment of inertia (i.e. diameter and mass) of the wheel. Gyro effect is considerable with standard-sized bicycle wheels (26" and larger), you pretty much don't need to hold your handlebars to stay upright in the whole range of usual cruising speeds. With smaller wheels (folding bikes etc.), it kicks in at higher speeds only, and sometimes is not strong enough for self-stabilization.
A static torque which pushes the steered wheel out of centre position, regardless of speed and direction of ride. This is what makes standard bike's handlebars turn sideways when the bike stops moving. It's because the front end stands on a "hill" (point A) when centered, and falls down to point B when turned:
This is generally no problem while riding since the static destabilizing torque is negligible when compared to dynamic stabilizing torques.
The abovementioned Python has negative flop because the whole frame lifts when the front end steers out of centre:
It is possible to combine advantages of both approaches. With steering angles greater than 90°, we get both positive trail and negative flop:
The design handles very well (tested by Balor here). Disadvantages are purely mechanical: more stress on the fork (torques from static load and braking add rather than subtract), headset too far forward for normal direct handlebars, and therefore more complexity and weight.
In case we can't live without static centering and we can't solve it by geometry, a spring can do the trick.
Quite important thing for self-stability. As Delft university researchers found (see here), it is possible to build a stable bike without any trail or gyroscope at all, just with correctly distributed mass around front fork. The point is to make the handlebars turn into the lean by their inertia, straightening the bike up again. It seems to work best when handlebars' centre of mass is a little bit (few centimetres) in front of the steering axis, which is where most standard bikes have it. If we put it behind the axis (typical for recumbents with direct tiller steering), inertia causes turning out of the lean, resulting in a fall. If we put it too far forward, the inertial negative feedback would be too strong, making the bike oscillate or fall over to the other side. But with a big weight in front of steering axis (typically a heavy basket on handlebars or MBB front drive of a recumbent) and steering angle under 90°, static wheel flop is much stronger than any inertial effects, so the wheel swings all the way to one side and stays there.
Rider's hands also count in the weight distribution around handlebars. If they push against standard handlebars with a stem or hang on a recumbent's overseat tiller, they add some static centering torque.
Height of centre of mass above the ground affects the stability too. Bodies in stable equilibrium (i.e. multitrack vehicles) are generally the more stable the lower their CoG is and the further apart their wheels are. So on a trike, it's always good to put seat and cargo as low as possible. Bodies in unstable equilibrium (singletrack vehicles) are the more stable the higher their CoG is (an old analogy is balancing a broomstick vertically on your palm vs. balancing a pencil). That's why a standard upright bike is easier to balance than a 1 m tall lowracer. Pannier manufacturers say putting cargo low down is good for a bike's stability, but that only applies when stopped: holding the loaded bike upright needs less force then.
Last kind of stability is the front-rear one, which determines how hard you can brake before getting thrown over the handlebars, or how steep hill you can climb without flipping backwards. Here the rule is simple: the lower your centre of gravity and the longer your wheelbase, the better. A low recumbent can brake all the way to the tyre adhesion limit. On taller machines, you either shift your weight to the rear somehow, or avoid braking that hard.
Driven front wheel and steered rear is sort of holy grail of recumbent designers: it would allow short chain without any leg steering issues. The only problem is it doesn't work:
Horizontal axes of the figures show time, vertical show lateral position. Black line is CoG position which we knock to one side at the beginning. Blue line is front wheel's contact patch position, red line is rear wheel. The dashed vertical line shows the critical moment where we either have the wheels back under our CoG, or we fall. First figure plots the corrective action for front steered wheel, second for the rear. With FWS, stabilization occurs immediately after the handlebars are turned: one wheel quickly moves under the new CoG position, the other one slowly follows. With RWS, one wheel quickly moves further away from the CoG, which is bad; the other slowly moves in the correct direction, but can't catch up. We can work around this by moving the centre of gravity as far forward over the front wheel as possible; the light rear end can then flap around without affecting the lean directly, like an aircraft rudder. But it means we get an equivalent of an old penny-farthing with all of its drawbacks (forward flip under braking and no useful cargo space), not a practical machine. The rearmost place where we can put steering axis on a singletrack vehicle is about halfway between the wheels - something like Python, but with the seat on the front half and reversed handlebars controlling the rear - like this lowracer. When the rear half steers, the front half steers too (in the "correct" opposite direction). Probably not self-balancing (gyroscopic torques of the wheels cancel out and the trail is far below zero), but perfectly ridable.
Rear wheel steering is possible on multitrack vehicles. Slow machines (forklifts, cargo trikes) use it extensively, fast ones don't - the problem is dynamic unstability (oversteer). I tested it on a model:
The result was an appropriate negative trail and some damping do the trick, the little trike returned to straight line after being pushed sideways. I don't know if it would work in the whole range of practical speeds, but any deviations should be correctable by stronger static centering. Anyway, it was already successfully tested in full size by Highlander:
If the front wheel is driven and the bottom bracket with pedals is attached to the front (steerable) fork, pedaling forces interfere with the steering. Turning torque is equal to pedal force times its perpendicular distance from steering axis:
There are multiple possible solutions. Python (image below left) has the pedals so far from steering axis that you just need to push slightly outward and the r arm (and therefore the torque) disappears. Cruzbike (below right) counteracts the torque by pulling on the handlebars.
Nothing is perfectly rigid and each body has some resonant frequency where it can oscillate easily. Bicycles and motorcycles oscillate torsionally: front wheel wobbles side to side and the frame twists behind it (see this video for an example). The periodic impulses powering the oscillations probably come from imperfectly balanced wheels. At a certain (rather high) speed, frequency of wheel vibration can align with resonant frequency of the frame, undamped oscillation grows quickly and if you don't stop it somehow, you crash. How to stop it? Logically we need to get out of the resonant frequency, so we must either slow down or speed up (I'd prefer the former). Or maybe we could alter the frequency by shifting our weight, changing our grip on the handlebars, position on the seat etc., but I have no idea if it would possibly work. That much for theory. In practice, you have about half a second to figure it out, so make your decision quickly.
I have experienced this phenomenon only on my road bike when it has some cargo on its rack, rolls at over 20 km/h and when I put my hands off the handlebars. But as long as I hold them, nothing happens. None of my other bikes allows hands-free riding, so there are no more observation data.
Another type of oscillation is vertical "pogo" caused by resonance of legs on pedals with something springy, typically suspension forks. This is not dangerous, it just eats energy. A quick solution is to pedal more smoothly or at a different frequency. A "proper" solution is to change the relative position of chain and the swingarm axis, but it's never going to be perfect because there's a different combination of chain pull and inertial forces for each gear. I have no practical experience with suspension, so all I can do is point you to someone who has - see for example this analysis by Charlie Ollinger.
I don't know if this brief overview helps you somehow. I hope it at least doesn't do any harm :-).