Technical Discussions

This page is the launching point for a number of technical discussions on bow performance.   The discussions are designed to by read in order because the material builds on the previous post.   I have attached an Excel spreadsheet at the end of each page that implements the calculations.   The files are also collected at the bottom of this page.

1) Understanding the Bow Draw Curve - It all starts here!

In the world of archery energy is conserved, and the energy to launch the arrow comes from the archer.  As you draw the bow string you add potential energy to the bow.  No mater what type of bow we are shooting you must put energy in to get energy out.   It is not less work to draw a compound bow, it is simply easier to hold the bow at full draw because the cams reduce the draw weight at full draw.  A compound bow adds potential energy in the middle of the draw cycle while a conventional bow adds most of the potential energy at the end of the draw cycle.  In this discussion we will compute the energy stored by a compound and conventional bow and calculate maximum arrow velocities.



In the last post I described the draw force curves of compound and traditional bows.   The compound bow was able to store more potential energy because the archer pulled the bow string for a longer distance at full draw weight.   More stored energy resulted in more kinetic energy for the arrow and faster arrow velocity at the point of nock release.  But what is happening to the arrow during the time from string release to nock release?  Clearly the arrow starts with a velocity of zero and reaches full velocity as the arrow leaves the bow.   The velocity profile during this cycle will determine the forces on the arrow and arrow rest, launch time, and ultimately the size of our arrow groups.

WOW!  How does this bow really work?   This was my first response to pulling a compound bow for the first time.  After tracing the path of the bow string and power cables over the pulleys and cams I wanted a mechanical model that described how the bow worked.  I tried to watch the cams move as I pulled the bow and determined that this can be a bad idea.   Besides the intellectual curiosity of producing a bow force-draw model, this type of information helps in the fine tuning of the bow.   In my search of the internet and published literature I have found a number of qualitative descriptions of compound bow function, but no specific models of cam function.   I wanted a compound bow model that would let me play with cam geometry and see the effect on a modeled force-draw curve.   A complete engineering model of the mechanics of a compound bow is fairly complex.  However,  as you will see from this discussion, it is possible to produce a very reasonable model for a compound bow force-draw curve using a simple excel spreadsheet. 
4) Arrow Trajectories: Arrow Flight in the Real World
One of my biggest motivations to understand the physics of arrow flight was to model the trajectory of an arrow under real shooting conditions.   This means calculating the motion of an arrow under the influence of aerodynamic drag, at any shooting height, and at any launch angle.  In this post I develop a full physical model for arrow flight including aerodynamic drag.  The calculations are easily implemented in Excel.

In the last post I detailed the physics of arrow flight including the real effects of aerodynamic drag on the trajectory of an arrow.   These calculations will not make you a better archer, but they do provide significant insight into the techniques and equipment that can improve archery success.  While turkey hunting the other day with Mike Guarino of Maine Wilderness Tours we got into a discussion on the appropriate correction for shooting at a target above or below the archer.   For the bow hunter, I argue that estimating range is the single biggest factor in hitting your target and that shooting elevation has almost no effect on shot placement.  This blog details the calculations used to support this claim.   

6) Virtual Mass - The key to understanding arrow speed
In my last post I showed that accurate range estimation is the single biggest factor in hitting your target in a hunting situation.  However, shooting errors due to poor range estimates can be reduced by shooting faster arrows resulting in a flatter arrow trajectories and decreased down-range arrow drop.   Lots of archery products claim to increase arrow speed, but in reality only the potential energy of the bow (draw weight and draw length) and arrow mass will change arrow speed to any significant extent.   Purchasing a new, more powerful bow is a big investment.  This post explores how arrow speed is influenced by the mass and design of the arrow.

In my previous posts I detailed the physics behind calculating the trajectory of an arrow shot from a bow under real conditions experienced by an archer.   The most important detail in calculating correct trajectory calculations is knowing the correct drag coefficient for the arrow and including the effect of drag on arrow trajectories. In this post I expand these calculations to include typical hunting rifles, muzzle loaders, and bows with examples for each. I have tried to make this post a bit more theory light and application heavy. By comparing the trajectories of three different projectiles (arrows, 30 caliber bullets, and 50 caliber muzzle loader rounds) if becomes easier to understand the similarities and differences of these three weapons.

In my past posts I described trajectory calculations that show that only the horizontal distance matters when aiming a bow or riffle.  The horizontal distance is referred to as the ballistic distance by some range finder manufacturers. You should aim your weapon based on the horizontal distance even though the true distance to the target is longer. The calculations are confirmed by our experience shooting from a tree stand - aim based on the horizontal distance. However our calculations and experience don't answer the question - Why does this work?  This post tries to answer this question for real-world hunting conditions.

To calculate the trajectory of an arrow we only need to define the three forces that accelerate or decelerate the arrow: 1) force of acceleration from the bow toward the target, 2) force of acceleration toward the earth due to gravity, and 3) force of deceleration due to aerodynamic drag on the arrow.   The acceleration forces of the bow and gravity are well understood and covered in full detail in past posts.  This post addresses the deceleration forces on the arrow due to aerodynamic drag.  Computing the drag is not simple, but it is critical in defining the trajectory of an arrow (and all ballistic projectiles).


Archery shooting machines are a fantastic tool for bow tuning and evaluating arrow flight.   Several commercial shooting machines such as the Hooter Shooter and the Kwik-Shooter are capable of shooting arrows through the same hole at a target 20 yards away.  Unfortunately, these devices cost $500 to $1200 making them more appropriate for professional archery shops.   Looking at the drawings of these devices and watching a few YouTube videos of home-built shooting machines motivated me to try and build my own shooting machine.   My goal was to build an effective shooting machine for under $100 that could shoot arrows as well as the commercial machines.   This post details my design, materials, construction, and shooting machine performance.


In my last post I described the design and construction of a shooting machine to evaluate bow and arrow performance.   My results with the shooting machine identifies some arrows shooting consistently high and to the left compared to the rest of the arrow group.  What is causing this difference in arrow flight?   Would I get similar results with brand new arrows?   Does a particular brand of arrows fly better than others?   A shooting machine is one way to answer these questions, but for many archers this is overkill.  The archery discussion boards are full of debate over the benefits of paper tuning to optimize bow and arrow performance, and paper tuning is simple and inexpensive.  I decided to test the effectiveness of paper tuning using the shooting machine to remove the human element from the arrow flight.   WOW did this open up a can of worms.   This post details my results paper tuning three different types of arrows all shot from the same bow with a shooting machine.

What is the expected arrow shot pattern for an average, good, and exceptional archer?   What does the grouping of five, ten, and one hundred arrows tell us about our ability to reproducibly shoot a bow?   I will argue that even the best archer shoots fewer arrows in the center of the target than they should if errors in shooting were truly random - IN ONE DIMENSION.  Analysis of archery shot patterns is consistent with a two dimensional error model where the archer's shots have normally distributed errors in both the horizontal and the vertical.  These errors are independent of each other and follow a typical Gaussian pattern centered on the target center.  The combination of the errors results in a shot pattern in the shape of a donut where maximum shot density occurs away from the target center leaving a hole in shot density at the target center.  The size and depth of the hole is related to the standard deviation of shots in the horizontal and vertical dimensions.

President Obama's resounding reelection success resulted, in no small part, to a campaign operation that used demographic information to understand and organize the american electorate in favor of the Democrats. While sitting in a tree stand during the fall hunting season I had plenty of time to reflect on the sophisticated use of data used to plan th presidential campaign and wondered if similar data was available on the movement patterns of deer. Deer density in Maine is generally less than 20 deer per square mile which equates to an average of one deer in a 32 acre parcel of land.   A successful archery hunt will require being in the right location at the right time to intercept the movement of the deer, and yet even a prime hunting location can result in zero deer sightings during the afternoon hunt. This post explores what we know about deer movement in terms of absolute distance traveled and the expected path of the deer. Using this data it is possible to develop a statistical picture of deer movement to compare with our hunting experience. 

During the winter of 2011/12 I built a shooting machine to help answer fundamental questions about arrow trajectory without the complications of human error in shot placement.  In this post I use a shooting machine to compare the target groups of six different arrows made from carbon shafts with different fletchings and tips.  As a bit of background, the reader is directed to the excellent post by Michael Larson,  Fletching Review: speed, drop, ease of use and more comparing the flight speed and drop of arrows made with an array of fletching materials.   Larson's results show the predicted effect of arrow mass on arrow speed - heavier arrows fly slower from the same bow.   His data also shows that differences in fletching design does effect the drag on the arrow.  These effects may be quantified through the trajectory calculators detailed in my previous post.  However, Larson was not able to evaluate the target groups of his test arrows because he did not have a shooting machine.   This post is a start at correcting that problem.

Most bow hunters set the sight pins on their bow by simple trial and error.  The top pin is often assigned to a twenty yard range and the entire sight is moved up or down until the shot angle is correct.   Next, the maximum range pin is moved up or down within the sight until the long range shot angle is correct.   Finally, all other pins are set by splitting the difference between the short and long range pins.   This approach works remarkably well except for very short or long shots.  The challenge is that empirical pin adjustment is time consuming and inconvenient if you change any aspect of your shooting setup (arrow weight, drag, or bow draw weight).  In this post I work out the specific geometry for setting up sight pins and provide a free Excel spreadsheet to perform the calculations.  These tools allow the archer to modify their bow properties and predict the effect of setup changes on pin placement with a high degree of confidence.


In the last post I provided the equations and Excel tools to calculate sight pin geometry.   With this capability it is now possible to compute the errors created from bow torque.  In this post, I will define torque as the rotation of the bow around the axis of the bow arm.   Two types of torque are possible.  Torque during release assuming the sight pin was on the target just before release, and bow torque during the sighting process.   The effect of bow torque on arrow placement can be significant.  

17. A Free Application to Measure Arrow Speed

This post compares three modern methods for measuring arrow velocity; optical chronographs, doppler chronographs, and a new laptop-based audio application that measures arrow speed by recording the time between the sounds of arrow launch and target contact.  This post includes a link to the free Technical Archery Arrow Speed application.  I will begin with a bit of theory on each method and then provide more details on the new audio speed application.

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