A Gear Head's Analysis of Compound Bow Function

    WOW!  How does this bow really work?   This was my 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 was
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 the discussion below, it is possible to produce a very reasonable model for a compound bow force-draw curve using a simple excel spreadsheet.  

    I started by building a general model of my Stinger PSE, single cam bow.   All of the mechanical advantage comes from the lower cam.  This cam reels out the bow string from the blue string cam and reels in the limbs of the bow using the red power cam attached to the green power cable.   It is the compression of the bow limbs that provide the force to drive the cam and bow string.   The pulley at the top of the bow is a simple wheel that transfers string from the lower cam to the upper half of the drawn string.   For the ideal bow, with linear nock travel, the top and bottom travel of the bow string must be identical.   Therefore, it is only necessary to model the travel of the lower half of the bow string because the upper half of the string will be the same.

    The attached Excel spreadsheet details all of the calculations.  Feel free to download the spreadsheet and modify the cam shapes for your particular bow.  I started by measuring the distance from the axis of rotation of each cam to outside of the cam at 11.75 degree intervals around the circumference of each cam.  These measurements were then converted into x,y coordinates that define the circumference of each cam (blue lines in the two figures below).

    The string feed cam turns the bottom cam assembly as the archer pulls on the bow string (green line is the bow string in the figure above).  The greater the diameter of the cam the less cam rotation required to deliver a length of bow string.  It is the changing diameter of the cam that is critical to compound bow function.  In practice I didn't calculate cam rotation with increasing draw length.  Instead, I computed the length of bow string that unwinds from the cam for each 11.75 degree rotation of the cam, a simple numerical trick that makes the calculations easier.

    The power cam is attached to the string feed cam and, therefore, must rotate around the same axis of rotation as the bow string is pulled.  Initially the diameter of this cam (measured from the axis of rotation) is at a maximum distance.  As the power cam rotates it pulls in on the green force cable compressing the bow limbs.   This is the power source for the bow.  According to Hooke's law, the force on the limbs will be;

where k is the spring force constant of the limbs and Delta Y is the distance the limbs are compressed.  An important point is that the force on the limbs increases continuously as you draw the bow.  This force will reach well over 200 pounds and would exert close to 200 pounds of force on the archer if the string cam and power cam had the same shape which, of course, they don't.   As a cautionary note, if you use a bow press to service your compound bow it must withstand much more force than the draw weight of the bow!  

    The compound bow draw force curve experienced by the archer is defined by the mechanical advantage generated by the different shapes of the two cams.  Think of these cams as gears on a bicycle.  Big gears in the back and small gears in the front of a bike make it easy to peddle up hill.  This hill incline is a good analogy for the limb force. The mechanical advantage on a bike is the ratio of the gear diameters.   When we ride up hill we shift to a lower gear and turn the peddles more times at a reduced force.  Similarly, the mechanical advantage of the bow is the ratio of string distances traveled as the cam rotates over some small interval (for example, the blue wedge in the figures above).  In the beginning of the draw cycle both cams have the same diameter and same change in circumference over a small interval of rotation, but this changes rapidly as the cams rotate.   The mechanical advantage of the bow at any angle is the ratio of these string distances:

The figure below shows the mechanical advantage computed as a function of cam rotation.

The large increase in mechanical advantage at 200 degrees rotation is what causes the let off of the bow.  At 200 degrees about 14 times more bow sting is being released compared to the length of power cable being reeled in.  The force on the limbs at this point, 200 pounds, divided by 14 is about 14.3 pounds.  This is close to the actual draw weight at the let off point of the bow.  The final factor in the draw curve is the changing angle of the bow string relative to the axis of the bow.  The force the archer experiences is always less than the bow string force at the pulleys because you only need to pull with the horizontal component of the force vector.   (See the first figure for this geometry correction.)  Combining all these factors provides an equation for Fstring.   This is the force on the archer and the arrow.

    Calculating F at each cam angle and then plotting F against the sum of Dstring results in he computed force-draw curve is shown below (we plot against the sum of Dstring because Dstring is the distance over a small change in cam angle).   This curve is a function of three things, increasing limb force as the limbs are compressed (Flimb), changing mechanical advantage (M), and the angle of the bow string (sin(theta)).

    Comparing the computed curve to my actual force-draw curve you see that the model gets all the general characteristics of the force-draw curve correct, a 60 pound maximum draw weight, the correct draw let off, and the force wall at full draw.   My simple computed curve underestimates the draw distance because I don't correct for motion of the cams and pulley along the axis of the arrow (I leave this model modification to an ambitious high school physics student interested in building their own bow.).   With this simple model it is now possible to play with the shape of the cams to create hard, medium, and soft shooting bows.  The attached Excel spreadsheet provides examples of the calculations for a medium cam bow as a starting point for other bow designs.

Whitney King,
Jun 17, 2011, 6:09 PM