Hooke's Law:

Hooke's Law says that the restoring force due to a spring is proportional to the length that the spring is stretched, and acts in the opposite direction.

If we imagine that there are no other forces, and let x(t) represent the distance the spring is stretched at time t then the restoring force might be represented as -K²x(t) where K > 0.

The dimensions of an object tend to change when forces are applied to the object. For example, when opposite forces are applied to both ends of a spring, the spring is either stretched or compressed. Unless the spring is damaged, it will return to its original dimension when the forces are removed. Objects that return to their original dimensions after the applied forces are removed are called "elastic" objects. Their study has led to the study of interesting relationships. Many objects react this way: Arrows when shot, golf balls when struck, etc. Those objects are much harder to gather data for analysis than a spring so we will stick with a simple spring.

On the other hand, we accept the relation that the total force on an object is the product of its acceleration and its mass (our old friend I. Newton). Since acceleration, a, is the rate of change of velocity, v, with respect to time, and velocity is the rate of change of position with respect to time, we have a(t) = v'(t) and v(t) = x'(t), so a(t) = (x')'(t), that is, acceleration is the derivative of the derivative of position with respect to time. Usually we drop the parentheses and write x'' in place of (x')', and refer to x'' as the second derivative of x (with respect to whatever the variable is). Thus for our spring, the total force is mx''(t), where m > 0 s the mass of the spring. Thus Hooke's Law tells us that if there are no other forces (no gravity, no air resistance, etc.) then

Understanding a system of coupled springs, such as modeled by Spring Chain, requires a small knowledge of single spring physics and differential equations. The motion of a mass attached to a spring can be derived from Hooks law, which relates the force exerted by the spring to the spring’s displacement from equilibrium. Simply stated, Hooks law says:

where k, known as the spring constant, is a measurable characteristic of the spring. According to this relationship, the acceleration of the mass attached to the spring will be directly proportional to the displacement from equilibrium. From this equation, it can be shown that for an undamped and undriven spring-mass system, the position of the mass as a function of time is given by the equation

where w is equal to the square root of the spring constant divided by the mass, and d is related to the distance the mass is from equilibrium at time t=0.

For a system of coupled oscillators, the system becomes slightly more complicated. The force on any single mass now will depend on that mass’s separation distance from its two neighboring masses on either side. In such a situation, Hooks law becomes:

where xn+1 and xn-1 are the positions of the particle before and after the particle for which the position is being calculated. As you will note, this makes for a slightly more difficult differential equation, because the acceleration of one mass depends not only on its position, but the position of the two particles next to it. Consequently, this is a “coupled” differential equation which will be much easier to solve using numerical methods and a computer than using pencil and paper.

Spring Chain will solve this differential equation for you.

Introduction

Lets face facts if not a little reality. The dynamics of springs mean absolutely nothing without apposing forces in mass and weight.  With this in mind we can also apply the following.

When you shoot an arrow the nock end moves forward, from side to side (Archers Paradox) and up and down. Tiller relates to the up and down movement of the nock. One aim of the bow setup is to get the arrows leaving the bow without any rotation in the vertical plane.

If the arrow leaves the bow with say the arrow rotating in the pile upwards direction then drag forces will push the arrow upwards and vice versa if the pile is rotating downwards. If during the shot the string force runs above or below the arrow center of mass then a torque is generated producing rotation of the arrow shaft in the vertical plane. As the nock of the arrow during the shot moves up and down relative to the arrow center of mass both the magnitude and the direction of the vertical torque on the arrow will vary. The resultant arrow rotation when it leaves the string will be total of all the accumulated rotation acquired during the shot. The aim is to have the total accumulated angular momentum equal to zero when the arrow leaves the string.

Factors which affect the up and down movement of the nock and the net arrow angular momentum are: Limb spring behavior Draw geometry gravity Arrow orientation (nocking point tuning) the archer Arrow FOC

Limb Spring Behavior

The following diagram illustrates how the spring behavior affects the vertical nocking point position. You have two springs of different strength connected at the ends by a string which runs around a pulley. The tension in the string is constant. In order to get a constant tension the weaker spring, at the 'full draw' position has to stretch more.

If the pulley is moved slowly towards the springs the weaker spring reduces in length more than the stiffer spring and so the 'nocking point' on the string moves towards the weaker spring (ref). In the case of a bow the two springs are the upper and lower limbs and the nocking point movement is up or down. The string tension is defined by both the limb spring force and the string/limb geometry (see section on draw force). The limb spring force and the geometry both vary in a complex way as the arrow moves forward as the shape of the limbs themselves vary. The nocking point may move up and down as the arrow moves forward as the relative spring strengths' vary.

The standard definition of 'Tiller' is the relative distance at a right angle to the string to the bottom of each limb respectively. I am not sure this is a useful measurement to have as you are only measuring the relative 'spring strength' at one point and as it is at the bracing height the least important point at that. The most sensible bow tiller to have is zero i.e. the distances measured at bottom and top limbs being the same. This is easy to check and if say you change the bracing height it as a simple setup to re-establish.

Draw Geometry

When the bow is drawn the draw force in most cases acts between the nocking point and the bow grip (The exception is the bare bow archer who string walks).  When the arrow is shot the string force is acting on a line between the nocking point and around the arrow rest position. As the arrow rest is a couple of inches above the throat of the grip this geometry change may result in a vertical nock movement effect. In effect, at full draw prior to release (assuming the springs are equal)  the bottom limb is drawn more than the top limb. On release the limb springs are not balanced with respect to the arrow/string force and the nocking point moves downwards as the lower limb moves forward more than the top limb. You can offset this effect by making the lower limb stiffer than the top limb so it doesn't move as much. Compensation for this effect is usually built into the bow design.

Gravity

When the arrow is moving forward on the bowstring the nocking point is vertically fixed (to the string) so as the arrow falls under gravity the arrow rotates and its center of mass drops. Torque wise this is equivalent to the nocking point moving upwards. The time duration of the arrow on the string is very small so the effect is negligible.

Arrow Orientation (Nocking Point Tuning)

The objective is to get the arrow leaving the string with zero net vertical rotation. Once the bow limb setup/bracing height is established the archer can vary the net vertical rotation out of the bow by changing the initial vertical torque the arrow gets from the string. This is done by adjusting the initial vertical height difference between the arrow's nocking point and its center of mass (otherwise known as nocking point tuning). This can be done by adjusting the vertical position of the arrow rest or, as happens in practice, the nocking point. If the nocking point is above the arrow center of mass then the initial vertical torque will be in the direction to rotate the pile end of the arrow downwards (and vice versa). The bigger the vertical gap between the nocking point and the arrow center of mass then the higher the initial torque is. Thus by moving the nocking point up and down the string both the direction and magnitude of the initial vertical torque can be adjusted. This adjustment is made on a trial an error basis, using paper tear or bare shaft tuning etc. methods, until the arrow leaves the bow with zero vertical rotation.

The bow setup is always made so that the 'tuned' nocking point position has the arrow slanting down to the arrow rest. If the arrow was slanting up to the rest then on the shot the arrow shaft would be dragging over the rest and there is the possibility that you could get a vertical torque input to the arrow from its reaction against the rest.

The Archer

Nocking point tuning includes the effects of how the archer holds and releases the bow string whatever the finger pressure balance the archer uses. Variations in the loose will introduce variations in vertical nock movement, particularly if the 'final' pressure point varies between being below and above the nock. The action of the bow hand can also effect the tiller (one of the effects that comes under the catch all of 'bow hand torque'). If say the archer 'heels' the bow hand then one effect is to draw the lower limb relatively more than the upper limb affecting the bow tiller. The arrow leaves the bow rotating pile upwards and the arrow ends up hitting high.

Arrow FOC

When the nocking point moves say upwards the effect on the arrow is to rotate it. The point about which the arrow rotates is determined by the FOC (position of the center of gravity). The higher the FOC then the further forward will be the rotation point. The change in the vertical gap between the nock and the arrow center of mass for any nock movement is higher (hence increased torque) the higher the FOC. In addition the acquired arrow angular acceleration and hence rotation (as discussed in the section on FOC with respect to fletching torque) depends on the arrow FOC.