## I have a question about the physics an arrow

 Quote by southshorerat We can use any terminology that will work. If we need to call the Centre of Drag and the Centre of Mass the two nodal points of the first transverse bending mode of the arrow I am fine with that. I thought there may be a difference in terminology when I began the thread. Is there a formula that a guy with no physics background can use to identify the front nodal point of the first transverse bending mode.
To me, the "front nodal point of the first transverse bending mode" of an arrow would refer to the following:

An arrow can bend in the middle. If you allow it to do so, it will resonate. This is a sort of standing wave. In a standing wave there are "nodes" where the amplitude of the wave is zero and "anti-nodes" where the amplitude of the wave is maximized.

An arrow that is bending resonantly around its middle, free from external supports, viewed from a frame of reference where its center of mass is at rest will have two nodes and three anti-nodes. One anti-node will be roughly centered, two more anti-nodes will be at the ends of the arrow. The two nodes will be between the anti-nodes.

The "front nodal point of the first transverse bending mode" would refer to the node that is nearer the arrow's tip.

In bending modes that correspond to higher harmonics there would be more nodal points, of course.

A quick trip to Google finds references in which similar terminology is used (in reference to glockenspiels and violins).

http://www.sarahtulga.com/Glock.htm
http://www.slac.stanford.edu/pubs/be...8-2-atwood.pdf

You can see similar behavior if you take a dangling rope and spin it. It is fairly easy to excite the first mode -- one node near your hand and one below. With practice you can get excite modes with additional nodes.
 Just joined but I'm hoping I can help here. I am a competitive traditional longbowman and P/T physics hack. First, the FOC, in archery terms, isn't the balance point itself but the ratio between the distance between the arrow's centrepoint and the c of G to the overall length of the shaft. If it is zero then the the shaft balances in the middle. Typicially shafts are tip weighted to produce 10-25% FOC where 25% would mean the balance point would be 1/4 of the distance down the shaft. Second, it sounds like the original post was about trying to tune the centres of mass and drag to the node points when the arrow begins flight and is oscilating. This oscilation is known as archer's paradox. When an arrow is fired from a traditional bow, the shaft bends slightly as it goes around the bow due to slightly offset forces. This creates the oscilation for the first 15-20 metres. The node points are NOT the same as the c of G or D points. If the balance point was dead centre of the shaft, then the oscilation would be quite clean and shouldn't deflect the arrow at all but as the point moves forward with a heavier tip, the oscilation becomes unbalanced and the arrow will start to "kick". This is balanced by the type and size of fletching. However it is necessary for the c of G to be ahead of the c of D for stable flight. A final kicker is that most arrows use helical fletching that imparts spin to the arrow which also mitigates flexion of the shaft. I think the poster is suggesting that if the c of G and c of D could be moved onto the front and rear resonant node points, that this would help the flight of the arrow. Probably at a FOC of about 18% and c of D back a little more, depending on shaft material, etc. An interesting thought though I cannot help with the math. The hunting bows that I've seen use brush aperature rests so there should be virtually no archer's paradox and the shaft's are carbon fibre and are extremely light and stiff so I don't think internal resonance should enter into the flight characteristics. Internal vibrations of the bow probably have more impact. As for crossbows the issue is shaft speed vs. length for stability. Whole 'nother ball of wax. Lastly, you mention placing the c of G point on the rest? I shoot with about a 2" overhang on a 31" shaft, so the c of G is well back of the rest and I can't imagine increasing the length of the arrow to move 25% or more of the shaft in front of the bow. That said on hunting shafts the story may be different although, again, there shouldn't be much flex in them. But part of the stability of the shot is for the shaft to be on the rest as long as possible and for the nock point to move exactly along the shaft's long axis so putting the c of G on the rest would be counter-productive. Hope this helps.
 Tom, some archers feel that if you can locate the front node which we call the Centre of Mass also called the front nodal point of the first transverse bending mode and set it on an arrow rest that drops away at the beginning of the shot you can improve the shooter/bow consistency. The method these archers are using is to take a finished arrow and place it on the corner of a table at an estimated point of where they think the Centre of Mass or front nodal point of the first transverse bending mode is located. They then pull down on the shaft to bow it and let it go then watch the results. Then they move it and repeat the process until the arrow responds with little or not bounce. They feel this is where the Centre Of Mass or front nodal point of the first transverse bending mode is located. My reason for coming here is to see if a mathematical formula exists that I could use to identify the Centre of Mass also referred to by you folks as the front nodal point of the first transverse bending mode. There are two types of arrow rests used in archery, a fixed type where the arrow is in contact with it though out the entire length of the arrow. The second is a drop away that falls as soon as the bow is shot. If we can apply a formula that will identify the front nodal point of the first transverse bending mode it is believed it will improve consistency of shooters using a drop away arrow rest. One more point, the front nodal point of the first transverse bending mode will not be in the same location for each archer. We custom fit shafts to each archers setup. Draw length, draw weight, choice of component weights and arrow length are not always the same from archer to archer. Because of this I am assuming the front nodal point of the first transverse bending mode will not be in the same location from shooter to shooter, is this a correct assumption?
 OK, I've looked at video footage of the rest that you are talking about and my general impression is that the location of the rest vs. CofM is of little importance. That the rest falls away cleanly is much more important. The arrow has barely moved when contact with the rest is removed so there is no effect from that standpoint. Second, I reiterate that your idea that the CofM and the front nodal point being co-located is false. There is a relationship between the to points and it is possible to locate the CofM at the nodal point but it is neither necessary nor particularly desirable. It appears the key to tuning this type of bow is to match the CofM distance from the nock with the thrust vector angle(the path that the nock itself travels). This will vary from bow to bow but not from archer to archer (assuming a consistant draw). It's a question of time; the rate that the CofM "falls" after the rest drops away compared with the rate of acceleration from the string to ensure that at the point the nock releases from the string (i.e. when thrust becomes zero) that the shaft is exactly parallel with the launch trajectory. So the CofM must be initially slightly above the thrust vector so that it can "fall" to it. There will be one "ideal" angle for every CofM location on the shaft but you are probably looking for the best CofM location for stable flight. That would have to be determined for the shaft itself independent of the bow setup. In physical terms stability usually comes when the CofM and CofDrag are furthest appart but its a question of balance: shaft speed, impact energy, bow weight, fletch size(drag force), cam force delivery (snap), etc. I would focus on the nock/rest angle to tune the bow for a given shaft CofM. I'm assuming there is a micrometer height adjustment for the rest. the video that I saw was of a particularly well tuned bow and the arrow appeared to move particularly straight along the thrust line with no perceptible oscilation. My two cents. Tom.