Draw Force Curve and Energy Storage of a Bow
The draw force curve of a bow describes the bow’s draw force as a function of draw length. It can be measured by recording the draw force from brace height to full draw, for example in one-inch increments. At brace height, the draw weight is zero.
The draw force curve shows how well a bow stores energy. The stored energy of a bow can be calculated directly from this curve. It is a well-known fact that recurve bows are fast, and one reason for this is that they store more energy than straight bows. This is due to their higher initial pre-tension, which results in higher string tension at brace height compared to straight bows.
A bow stores energy when it is bent, but the energy stored in the bow when it is bent from its resting state (without a string) to brace height is impossible to determine. Fortunately, this is not necessary, because that energy is not usable for propelling an arrow. However, this initial bending defines the pre-tension of the bow, which can be seen in the draw force curve: the greater the pre-tension, the more rapidly the draw weight rises with draw length.
There is also a geometrical factor that affects the energy storage of a bow. As a result, at some point the increase in draw weight begins to level out, and at long draw lengths the draw weight increase accelerates again.
The most important information provided by the draw force curve is the bow’s energy storage. This can be calculated by determining the area under the draw force curve, either numerically, by integration, or using suitable software. However, the absolute energy storage alone is not very informative, because it does not directly indicate how “good” the bow is. The important parameters emerge when the stored energy is compared to a reference value. There are two commonly used options:
1. Energy Storage Factor (ESF)
The energy storage factor (ESF) describes how much energy a bow stores compared to a perfectly straight draw force curve. The unit is percentage (%). For example, consider a reflexed bow that stores a certain amount of energy. The reference line is a straight line from brace height to the full draw weight of a corresponding (theoretical) bow. By comparing the area under the actual draw force curve to the area under this straight reference line, we obtain the ESF.
Typical values are:
• Reflexed bows: approximately 105–115%
• Straight bows: close to 100%
• Deflexed bows: below 100%
It is often assumed that the draw force curve of a regular straight bow is very close to a straight line, although in reality it is never perfectly straight. In short, the energy storage factor (ESF) describes the normalized energy storage of a bow.
2. Stored Energy per Peak Draw Force (SE/PDF)
Stored energy per peak draw force (SE/PDF) describes how much energy the bow stores relative to its full draw weight. This is a “New World” parameter based purely on British units. Personally, I do not like this parameter. For example, if a straight 40# bow stores 40 ft·lbf of energy, its SE/PDF is 1.00. A reflexed bow stores more energy, so its SE/PDF is higher, typically around 1.10–1.20. This parameter is problematic because it compares two different physical quantities that should not be directly compared—essentially, apples to oranges.
Comparison of ESF and SE/PDF
There are some interesting differences between these two parameters, as shown in the Table. As can be seen, the ESF decreases as draw length increases. Although a longer draw length stores more total energy, the bow begins to stack, meaning that it stores less energy per unit of draw length as the draw length increases. Therefore, the relative energy storage decreases.
In contrast, SE/PDF increases with increasing draw length (with the deflexed bow being an exception). The reason is that at longer draw lengths the bow stores more energy overall. For example, the energy stored between 30″ and 31″ is much greater than the energy stored between 11″ and 12″. Thus, the stored energy increases proportionally more at longer draw lengths.
Because of this fundamental difference between the two parameters, the energy storage factor (ESF) is the only parameter that properly describes how well a bow stores energy. For example, when comparing the SE/PDF values of a reflexed bow (28″ draw) and a deflexed bow (28″ draw), they may appear to be equally good—but the ESF reveals the true difference. Hopefully, this explains why SE/PDF is not a useful parameter.
Notes on the Table and Images
Please refer to the Table and the images. The values in the Table and the curves in the images are generated using the VirtualBow program. The absolute draw weight values are not important. The modeled bow is a very simple design: 1700 mm (67″) long, with a width taper from 35 mm to 10 mm and a thickness taper of 0.006. The reflexed bow is reflexed 4" and the deflexed bow is deflexed 4". Brace height is 6" from the belly. In the Table, “dw (ft·lbf)” represents stored energy.
The images show:
• The draw force curve (bold blue line)
• A reference line from zero (brace height) to full draw (32″)
• A stiffness line (purple line)
The stiffness line is particularly interesting. Mathematically, it is the derivative of the draw force curve. It describes how the draw weight changes per inch of draw length. For example, in the image Reflex_bow_32_inch_draw.png, the draw weight increases by about 2.5# between 12″ and 13″, and by over 3# per inch between 31″ and 32″. Thus, while the draw weight always increases, the stiffness indicates how rapidly it increases. If the stiffness were negative, the draw weight would decrease. The minimum point of the stiffness curve (around 17″–18″ for this specific bow) marks the point at which the draw weight increase is at its minimum. In my opinion, this point could define the stacking point of the bow. From that point onward, the draw weight increase begins to accelerate. The more rapidly the draw weight increases, the more the bow feels like it is stacking.
Conclusion
The main purpose of this discussion is to explain bow energy storage, what it tells us, and how it should be used. In my opinion, the energy storage factor (ESF) is the only suitable parameter for evaluating how efficiently a bow stores energy.
