Loop Dynamics

[Walter]

Thanks to Gordy we have a consensus that angular momentum around a semicircular loop with a tethered cast and level line is the same in the loop centric frame and the fixed or earth centric frame. For those of you who aren't sure what I mean by the different frames, the fixed or earth centric frame means that the observer is watching a fly cast and is not moving relative to the fly caster. The only thing that appears to be moving to the observer is the line (and the rod during the cast of course). For this observer the fly leg is moving at some speed X, the rod leg speed is zero and the leading edge of the loop is moving at speed X/2. In the loop centric frame, the observer is moving at the same velocity as the leading edge of the loop of the line. They could be anywhere - in the middle of rotation of the loop or possibly on a moving cart travelling parallel to the fly line. To this observer the fly leg seems to be moving at speed X/2, the rod leg is moving in the opposite direction at speed X/2 and the leading edge of the loop is not moving.

In the following charts I am using a level line, with linear density 0.001 kg/m, travelling left to right and the loop is semicircular with a diameter of 1 meter. The speed of the fly leg (top of the loop) is 20 m/s in the fixed frame. The X axis is in degrees from 0 to 180 with 0 being the top of the loop and 180 being the bottom of the loop.

The first chart is the magnitude of the velocity (speed) of a point around the loop. In the fixed frame the line enters the loop at speed 20 m/s and leaves the bottom of the loop at 0 m/s. In the loop centric frame the speed is constant:

The next chart is the angular momentum around the loop:

And the final chart is the accumulated angular momentum around the loop in both frames. You can see that the total amount of AM is the same in both frames.

I have some other results that may be interesting, but I want to verify them before posting.

I know Gordy had some comments/questions from the topic that started this. I'll try to capture those and respond as I post additional results.

[VGB]

Thank you for that information Walter, it is very interesting.

Ever since we had the discussions about drag around the loop back in the day, I’ve been intrigued by the velocity field modelled by Dr Perkins and the resultant distribution of AM around the loop. I think that this distribution is why we see loops produced by good casts that tend to be pointed, as opposed to semi circular. I tend to see semi circular loops when V0 is low or the loop is opening.

I also suspect that Dr Gatti-Bono reused the same velocity field when she predicted that

Fly lines are intentionally tapered to improve casting performance; and bending rigidity, while rather small, is essential in the regions of low tension and compression that develop
on the lower part of the loop.

Regards

Vince

[Walter]

Vince,

I agree with you. I think when we start to explore the am and velocity around the loop we are going to see at least part of the reason why semicircular loops turn into pointy ones as they unroll. What I've modeled so far is just the loop only and does not take into account any other forces such as drag and gravity.

[Walter]

Gordy,

Sorry I can't respond to your questions/comments yet but I'm working my way towards it. In the meantime I've been working on comparing the angular momentum when shooting line in the fixed and loop centric frames. Here are some more results based on the previous values (fly leg speed 20 m/s, line linear density .001 kg/m, loop diameter 1 m) with the addition of a constant rod leg velocity of 5 m/s in the same direction as the fly leg.

The first chart is the distribution of the magnitude of velocity in the two frames:

Shooting Line Magnitude of Velocity.jpg (32.84 KiB) Viewed 615 times

The next is the angular momentum around the loop in the two frames:

Shooting Line AM around loop.jpg (38.95 KiB) Viewed 615 times

And the last is accumulated angular moment around the loop:

[VGB]

Hi Walter

I agree with you. I think when we start to explore the am and velocity around the loop we are going to see at least part of the reason why semicircular loops turn into pointy ones as they unroll. What I've modeled so far is just the loop only and does not take into account any other forces such as drag and gravity.

I’m fairly sanguine about drag around a semi circular loop and gravity until you add the legs. I’d be more inclined to think that fly leg momentum and rod leg tension will have a greater effect at the boundaries of the loop face. I did pose that as a question on the COAM thread and only Graeme responded by agreeing with that premise.

Very interesting results above. I have some questions but will give Gordy first dibs, it was his questions that generated the thread.

Regards

Vince

[gordonjudd]

Here are some more results based on the previous values (fly leg speed 20 m/s, line linear density .001 kg/m, loop diameter 1 m) with the addition of a constant rod leg velocity of 5 m/s in the same direction as the fly leg.

Walter,

Your results seem consistent since shooting line reduces the tangential velocity going around the loop while increasing the loop velocity over the ground (assuming the fly leg velocity stays the same). I would expect that pull back would have the opposite effect and cause the AM in the moving frame to be bigger than the AM in the earth frame.

Gordy

[gordonjudd]

I think that this distribution is why we see loops produced by good casts that tend to be pointed, as opposed to semi circular.

Vince,
Why would the distribution of AM around the loop cause it to morph? Momentum is not a force so I don't see how it could produce a change in the shape of the loop.

Rather I would agree with Torsten that aerodynamic drag is responsible for most of the loop morph we see.

The wedge shape is usually a result of loop morphing, such loop shapes begin with a roundish loop and morph into a pointy loop shape.
Major factors that promote in my opinion the change of shape are the trajectory of the line, tension and aerodynamic forces (especially the drag force). If the line speed is high enough even large rounder loops morph into this shape.

Gordy

[VGB]

Rather I would agree with Torsten that aerodynamic drag is responsible for most of the loop morph we see.

The wedge shape is usually a result of loop morphing, such loop shapes begin with a roundish loop and morph into a pointy loop shape.
Major factors that promote in my opinion the change of shape are the trajectory of the line, tension and aerodynamic forces (especially the drag force). If the line speed is high enough even large rounder loops morph into this shape.

Why do you think that the cause of morphing is aerodynamic drag Gordy, when Torsten is clearly stating that high line speed is sufficient to cause it? Drag after all is just a product of relative velocities between the line and the atmosphere.

Moreover, wasn’t one of the central tenets of your COAM theory that drag around the loop can be considered negligible

Vince

[Walter]

Why would the distribution of AM around the loop cause it to morph? Momentum is not a force so I don't see how it could produce a change in the shape of the loop.

Gordy -
Just throwing this out there,

AM itself is not a force but it is the product centripetal and tangential forces. If the forces are not constant but the mass is evenly distributed, then I think the semicircular loop shape is not going to remain stable. I don't have any model to support this, and I don't know how it will change in response to those unbalanced forces that are calculated is something that could be explored as part of this topic.

[Walter]

Gordy,

I think I have a partial answer to one of your questions about why a tilt would change the am calculations. The issue is how do we define tilt and does it actually affect the am. In the loop centric frame the frame of reference is based on the orientation of the legs. Change the angle of the loop and you also change the orientation of your cartesian co-ordinate system. In the fixed frame the cartesian co-ordinates are always the same. Also remember that we've been looking at this from the point of view of the center of the loop and the direction of the linear momentum that the angular momentum is based on is not constant from that point of view. If you want to look at a true tangent in the fixed frame you would need to consider using the bottom of the loop. With that reference point the velocity of the linear momentum vector is tangential at the top of the loop. Unfortunately, the model I've generated is based on motion relative to the center of the loop and it would take some work to make that sort of a change. I have worked through some of what the velocity and momentum would look like with a 10 degree tilt counterclockwise.

velocity with tilt.jpg (24.94 KiB) Viewed 537 times

angular velocity with tilt.jpg (35.78 KiB) Viewed 537 times

Accumulated angular velocity with tilt.jpg (31.67 KiB) Viewed 537 times