The usual description given for bow tiller is the difference in the orthogonal distance from the string to the top and bottom ends of the riser
i.e the distance (a) minus (b) in the diagram. If (a) is greater than (b) this is called a positive tiller, if (b) is greater than (a) this is
called a negative tiller and if (a) and (b) have the same value this is referred to as a zero tiller.
Another way to look at tiller is that as the modern Olympic bow is symmetric with respect to the grip, with zero tiller the riser is parallel to the bow string. If we take the string as vertical, with a positive tiller then the top of the riser leans forward and with a negative tiller the top of the riser leans backwards.
When it comes to the functional purpose of tiller then the usual answer is that tiller is used so that the upper and lower limbs "close" simultaneously which unfortunately is completely wrong.
Tiller like most topics in archery is not understood in any great detail. The aim of this section is to try to indicate where tiller came from historically and what meaning it has for the modern Olympic recurve bow.
Tiller is the result of two geometrical effects caused by the sensible decision to shoot the arrow over and not through the bow hand.
a) Arrow generated tiller
For every action there is an equal and opposite reaction
Because the string is pushing the arrow forward (action) the arrow is pushing the string backwards. As the arrow is pushing the string backwards above the bow centre
this generates a torque on the riser via the limbs acting to tip the top of the riser backwards towards the archer pivoting around the grip.
b) Limb Energy generated tiller
Because the string is pulled back above the mid point, the resulting leverage difference between the top and bottom limbs results in the top being drawn (bent) more than the bottom limb. As a result more energy is stored in the top limb at full draw than the bottom limb. During the power stroke each limb generates a force (action/reaction again) on its associated limb pocket. Because more energy is stored and hence released by the top limb in the same time than in the bottom limb, the (reaction) force on the top limb pocket is higher than the lower pocket. The result is a net torque acting on the riser to tip the top of the riser backwards towards the archer pivoting around the grip.
So both the arrow and limb energy effects add to tip the top of the riser backwards during the shot.
As mentioned both limbs "close" at the same moment as closure is determined by the string tension which acts effectively simultaneously on both limbs. During the power stroke the bow limbs are coupled together at the butt end via the riser and at the limb tip end via the string so the limbs do not act independantly. The overall effect of limbs with different starting energies coupled together is a high frequency limb vibration during the power stroke, something like 400Hz.(see Recoil and vibration in an archery bow equipped with a multi-rod stabilizer ).
The Beiter video of a string walked arrow release provides some illustration of how bow limb movement is coupled. At the moment of release there is no real tip to tip connection between the limb tips as the string has slack in it. This slack rapidly disappears as in this case the limbs can move at different speeds. Once the slack is removed and both limb tips are connected at the arrow nock then joint movement of the limbs (with oscillation) establishes itself.
As it's not a good idea to shoot an arrow through the archers bow hand it follows that where the bow is pushed and the bowstring is pulled are at different vertical positions. Historical bow designs vary but lets assume the bow at bracing height is vertically symmetrical with bow hand push being at the geometrical center of the bow and the point where the string is pulled (the nocking point) being above the geometrical center of the bow. The top and bottom limbs are assumed to be identical in geometry and strength. When the bow is drawn it is assumed that the center section (riser) remains vertical. The verticality is maintained by the archer gripping the handle and/or the frictional force between the tab and the bow string.(to get this concept imagine what would happen if the archer held the handle freely like a modern recurve archer and instead of a tab drew the bow with a pulley; instead of being drawn the bow would just rotate to the horizontal as the pulley ran along the bow string to its top end).
As the distance from the nock point to the top limb tip is shorter than the distance from the nock point to the bottom limb tip it follows from the geometry that at full draw the top limb has been pulled back (drawn) more than the bottom limb. As the energy stored in each limb depends on how much it is drawn (the area under the individual limb force draw curve) it follows that at full draw more energy is stored in the top limb than the bottom limb.
The consequences of shooting a bow with an initial energy imbalance between the top and bottom limbs is a) a lot of limb vibration during the power stroke and b) a torque is generated on the bow handle during the power stroke acting to rotate the top of the handle towards the archer.
In the diagram the blue lines represent the forwards accelerations of the limbs and arrow and the red lines the corresponding horizontal components of the reaction
forces on the riser pockets. The arrow acceleration reaction on the riser acts through the string & limbs so adds to the top and bottom limb reaction forces. As
more energy is stored in the top limb then the bottom limb and the arrow (nock point) is above the handle pivot point it follows that the horizontal reaction force
on the top limb pocket is greater than the horizontal reaction force on the botom limb pocket. The overall horizontal reaction force on the handle can thus be represented
by a single equivalent force acting above the bow pivot point. There is therefore a torque acting on the handle to rotate the bow in the direction for the top
of the bow to rotate towards the archer.
A consequence of of the handle rotating during the power stroke is that when the limbs bottom out (basically when the string starts to decelerate the limbs, which of course must happen to both limbs simultaneously by definition ) the energy remaining in the limbs is not equal. The symptoms of "poor tiller" are the top limb rotating back towards the archer (the consequence of handle torque) and a lot of post shot limb flutter (the consequence of the limb energy difference at limb closure). To illustrate the effect of a limb energy imbalance at closure place a bow string upwards in a bow vice or across your knees and push down on one limb tip. The limbs are "closed" i.e. static but have different amounts of stored energy. if you release the pushed on limb tip you get limb flutter i.e. a coupled oscillation between the limbs.
The historical solution to the tiller problem was for the bowyer to make the bottom limb stronger than the top limb (either stiffer and/or shorter). Now although the bottom limb is drawn less than the top limb because it is stiffer it stores more energy per unit of draw. Tillering the bow was adjusting the bow energy storage between the limbs so that the net horizontal reaction force components on the top and bottom bow sections during the power stroke were equal.I.e. the net resultant recoil force equivalent acted through the bow pivot point. The limbs still vibrated during the power stroke but the torque on the bow handle in theory could be eliminated. In general with traditional bows the bow grip is often gripped firmly so any remaining torque on the handle is opposed by the archer's hand. Preventing any rotation of the riser results in the limbs having pretty much the same residual energy at limb closure so minimising any post shot limb flutter.
A consequence of having the bottom limb stronger then the top limb was that at bracing height the bow was no longer vertically symmetric. The distance (b) was smaller than the distance (a) i.e. the bow has a positive tiller in the conventional sense. The amount of positive tiller is a reflection of the difference in strength between the upper and lower limbs.
There are number of differences between the Olympic recurve and the traditional bow as regards tiller.
The normal set up for an olympic bow with zero tiller limbs is with zero tiller. (Distances (a) and (b) are equal). For the minority of limbs that are not manufactured with equal strength then the tiller set up is the one specified by the limb manufacturer (typically up to 5mm positive). Of course manufactured limbs which are supposed to be identical may not be so and so the actual measured tiller might be slightly off the nominal zero value. In practice the measured tiller should be between zero and the natural limb tiller. In all cases though the limb exit angles from the riser are generally recommended to be the same for the top and bottom limbs. ( In practice when the effect of variations in tiller as regards bow performance have been physically tested the result seems to have been that tiller variations have an insignificant effect on bow performance. So although "zero tiller" is the recommendation small variations appear to be irrelevant). There is sometimes confusion here between tiller and nocking point. Changing the nocking point affects the bow tuning so it will impact on bow performance. Changing the physical tiller will physically move the nocking point so unless the nocking point is reset you can get the impression that changing tiller changes the bow performance.
As for the traditional bow the nock to limb tip string distances will be greater for the lower limb than the upper so at full draw you have more energy stored in the upper limb. You will also have the effect of the accelerating arrow being above the bow pivot point. Overall during the power stroke you will have the same equivalent horizontal reaction (recoil) force acting above the pivot point generating a torque on the handle as for the historical bow and in the sense for the top of the bow to rotate towards the archer.
For the Olympic recurve bow you can use the stabiliser system to counteract the this recoil torque by pushing the bow center of mass (COM) forwards and/or down with respect to the bow pivot point. Put the COM in the correct area with the long rod and end weight and the recoil torque is cancelled out by the gravitational torque. This provides for zero riser rotation while the archer allows the bow to pivot freely against the bow hand. This topic is discussed in more detail in the section on bow stabilisation. The simple test on having the bow COM in the right area is to observe that the long rod does not visibly rotate after the shot i.e. there is little riser angular acceleration during the shot. If the riser does not rotate during the power stroke than the two limbs will have similar amounts of energy at limb closure resulting in the minimum mount of post shot limb flutter. Some example photo analysis of post shot long rod behaviour is located here Rod rotation
The consequence of having limbs of different strengths with traditional bows resulted in the positive tiller value at bracing height. The consequence of having
identical limb strengths and the same limb exit angles with the Olympic recurve bow results in the zero tiller value at bracing height. It is possible however to
have a pseudo-tiller value with an Olympic bow by having different limb exit angles. This is an option favoured by a minority of archers, probably because it reflects
the traditional bow set up rather than there being any known advantage in taking such an approach.
If say you start with zero tiller and screw say the top limb bolt out than you end up with a positive tiller. Screwing a limb bolt out increases the corresponding riser end to string distance. The limb of course has the same strength, what has changed is the limb's bracing height geometry. The limb has a lower pre-tension and it is this that is increasing the limb tiller distance. Using psuedo-tiller will reduce the energy imbalance between the limbs at full draw as the limbs start with a different pre-tension. Changing the limb angle to the riser will also change the magnitude and direction of the reaction force on the riser end so a positive pseudo-tiller will probably reduce the net torque acting to rotate the riser in the usual sense during the power stroke. (The more the limb bolt is unscrewed the more the limb momentum change is in the up-down direction rather than a forward back direction; Compound bows take this to the extreme). The effect of pseudo-tiller on in shot limb vibration and post shot limb flutter is not known.
For further reading on this topic see Tiller references