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Vodkaman

Death roll

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Many people have commented by post and PM that what I have written was interesting but difficult to understand. Explaining these theories is definitely not my strong point and I hold my hands up and plead guilty. One contributing factor to this sad admission is that all this theory is new to me also and I haven

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Tow Line inline with the ballast? So by tweaking the towline we can find the spot where the ballast and towline align during bait operation? By putting the towline inline with the ballast are we talking about center weighting on a typical bait design? As opposed to placing the ballast as low as possible? Are we creating overstability like a center of pressure thing? Why have I not tried the metal lip? Being a "x" tinknocker I am ashamed! Gotta go...my brain is thinking and nothing is happening! I am not grasping the towline thing!

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Vman, you'll need to dumb it down for me as well. I think by "in line" he means when looking at the bait from the side. Is the weight on a horizontal plane with the tow eye?

If the ballast is too low, you will have the most stable bait, but it will likely have the least active movement if all other variables are the same.

If you move the ballast toward the center of the bait (vertically if you are looking at the bait from the side, or depthwise if you drill for ballast depth), you will begin to create instability.

Deathroll occurs when you go to far toward or beyond center depending on the bait shape, lip, hooks, etc.

The best example I can describe in words is that of a boat: If you lay down in the bottom of the boat, your weight will cause the boat to sway a little less in choppy water. If you stand up on the seat, you will take a spill.

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Love your efforts "Vodkaman", have one for me. As you say, most of us have toiled away on a "dream lure" only to find no matter where you bend the eye, weight the back hooks, we get the death spiral. I have read your post twice now and much of it has sunk in and confirmed a lot of what I have thought, but was too P'd off to try. I admire your tenacity and lateral thinking, but wait until the mathmaticians get hold of it and minutley anaylise it- there are so many grey areas in this "Science" and most of us have not the brainpower or time to study the pure hydro dynamics of lures or submarines. If we can tweek a lip, drill a weight or move a hook eye and it works consistantly (and catches fish), that's enough proof for me, but if you get some reasoning behind doing all these things, so much the better. Look forward to seeing your drawings, "a picture paints a thousand words". Thanks again, have another one on me and pardon the spelling. Pete

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Vodkaman,

For the last few years I have had an old fluid dynamics book I'm still pondering after...lol

Even the stuff I think I begin to understand then confuses me more. :)

Water is an incompressible fluid, yet an airfoil in water still behaves like a wing as air is a fluid. Because of Bornoulli's law, water must travel (in effect) faster over the top, thus lowering the relative pressure and creating lift. So why do almost all of our lures look like wings, and create lift, when we are trying to get them to dive? lol

Don't even get me started on lamillar flow, Reynolds, turbulence, etc.

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Clarification. Re-examining these theories with drawings has been enlightening and educational to me and I hope at least some of you get something out of the brain strain (yours and mine).

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Diagram 1 shows the lure in its normal swimming orientation. In this example, a deep diving shallow lip and a suitable line angle of about 40 deg to the water surface. While the lure is moving, this angle remains constant and represents how deep the lure will swim (a discussion for another day).

The diagram shows that when the lip forces below the tow line are equal to the body forces above the tow line, the lure is balanced. If the lure attitude is disturbed by a rock etc, the out of balance forces will quickly rotate the body back to its balanced attitude. This balance is pivoted about the line tie.

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As a crude example to describe this balance or equilibrium, diagram 2 shows a long stick with a brick tied at one end and an egg at the other. A length of string is loosely tied in a loop around the stick. The stick is threaded through the loop until it balances. The balance point is found to be closer to the brick than the egg, but still, a balance point is found. The brick represents the lip forces, the egg represents the body forces and the string represents the tow line (fishing line).

If a side force (push) is applied to the brick, the stick will rotate smoothly around the string. This is how I believe the lure moves, rotating around the tow line. This is of course, a gross over simplification of what is actually happening, but the principle is close enough to reality and easy enough to understand, to make the theory useful.

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Next we add gravity to the theory. Diagram (3) is identical to diagram (2) only rotated through 90 deg. It describes the same stick loosely nailed to a wall so that it can rotate. The position of the nail is the same as the string loop. The experiment represents a view of the lure looking along the tow line (fishing line). The nail represents the line tie, the brick represents the lip forces and the egg represents the ballast. In true life, the egg represents a combination of various forces, including the buoyancy, ballast, hook, eyelet, hangar wire and lip weights. The ballast position is close enough to this centre of forces so that they can be ignored for the purposes of this exercise.

If a side force (push) is applied to the brick, it will be noticed that the stick will stop in a random position, not necessarily vertical. It will also be noticed that virtually no effort is required to move the brick, this is because the brick is nailed at the balance position. Referring back to the original article, this represents the start of the tow line/ballast offset instability. At this point erratic movement of the lure can occur. For convenience I have called this type 2 death roll. This type is generally encountered on deep divers, but can be encountered when experimenting with rear mounted ballast, this is how I identified the instability.

Erratic behaviour of the lure will occur with this geometry setup, as there is no force trying to keep the lure vertical. The lure could randomly flip over into death roll.

If the egg is moved further away from the pivot nail (diagram (4)), the stick will swing around to vertical, with the egg at the bottom. This represents death roll, the lure swings around the tow line in a wide arc.

If the egg is moved towards the pivot nail (diagram (5)), the stick will swing around to vertical, with the brick at the bottom. If the brick is pushed, it will return to the vertical, this represents stable lure action.

Remove the egg and nail the stick at the new balance point. Re-attach the egg to the stick at the nail pivot. Once again, if the brick is pushed, the stick rotates freely, stopping randomly, not necessarily vertical. It will also be noticed that, as before, virtually no effort is required to move the brick, this is because the brick is nailed at the balance position. Referring back to the original article, this represents the start of the tow line/ballast instability. At this point erratic movement of the lure can occur. For convenience I have called this type 1 death roll. This type is generally encountered on shallow swimming lures, but can be encountered by experimenting with extremely low mounted ballast, an unrealistic situation.

Erratic behaviour of the lure will occur with this geometry setup, as there is no force trying to keep the lure vertical. The lure could randomly flip over into death roll.

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Diagram 7 shows type 1 death roll, the ballast sitting roughly on the tow line. A shallow lure with a 70 deg lip is represented. The two lines of instability are drawn in. positioning the ballast anywhere between the lines will achieve normal action.

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Diagram 8 shows type 2 death roll, the ballast sitting on the type 2 instability line. A deep diving lure with a 10 deg lip is represented.

Conclusions.

If a new lure goes into death roll, the above theory will identify what type of instability you are dealing with and will offer up a solution. Method, while swimming the lure, make a mental note of the angle that the line makes to the water, while the lure swims roughly horizontal.

Hold the lure horizontal in your hand and hold the line at its swim angle (as noted previously). If the line points directly (or just slightly above) the centre of the ballast, then type 1 death roll is the cause. Solutions to the problem include raise (or lower) the ballast location, raise (or lower) the line tie eye position, increase (or decrease) the lip size.

If the ballast location is some distance from the tow line, this would indicate type 2 death roll. Solutions include; lower the ballast or move the ballast forward, raising the ballast would make the condition worse. Raise the line tie location or reduce the lip size.

In both type 1 & 2 cases, the ideal solution is to move the ballast, as moving the eye or changing the lip will change the tow line angle and affect the swim depth of the lure. For prototyping, I would always give the first prototype an extended line tie eye for easy testing of the geometry.

If all the above is true, then lowering the ballast on a deep diver should increase the action of the lure by reducing the inertia of the ballast, effectively making the lip forces stronger. This can be visualised in diagram 5, as the egg (ballast) is moved closer to the pivot nail, the effect of the brick (lip forces) becomes stronger, swinging back to the vertical faster. This is contrary to what has been discussed on TU before, including me. I firmly believed that as the ballast was lowered, the action would be damped out. For deep divers, the opposite is true, as the ballast is lowered, the action becomes wider.

For shallow lures with steep angled lips, referring to diagram 6, if the egg (ballast) is lowered from this location, the inertia opposes the lip and the action is damped.

The above theories fully explain everything that I have experienced while prototyping. But I have only worked on a limited number of lure setups, I have not done any work with deep divers. I invite you to test out the theories with any lures that roll out of control and report back. If you disagree with the above, say so. By arguing the different cases we will (hopefully) get to the truth.

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Clemmy. The fact that water is an incompressible fluid has bothered me too, in my pursuit of explanations of the behaviour of the hard bait. So, here are a couple of real life examples of the forces in motion.

A sauce pan filled to one inch of the rim and stirred. The water at the edge rises and spills over, the water at the centre sinks down, but the volume is unchanged. The shape of the surface reflects the pressure differences through the vortex. If you throw some sand into the pan, it will gather at the centre of the pan. We have all witnessed debris being drawn into the plug hole vortex after washing the dishes, haven’t we?

As for flight, there are two theories of flight. The first being the traditional aerofoil theory that you are referring to. It has applications in lure design, mostly body shapes and has never really been addressed. Another one on the list of articles that need writing.

The second theory is that employed by all flying insects. Normal flight is only effective down to about 6 inch wing span, any smaller and the aerodynamic forces are not strong enough to sustain flight. Insects use vortex theory, the same theory that the lip on the lure employs to impart the side movements.

The fact is that the pressure on the front of the lip is not what makes the lure swing from side to side, it is the low pressure vortex that forms behind the lip. The vortex forms down one edge of the lip at a time, alternating from side to side. This ‘sucks’ the lip from behind, from side to side. A web search on vortex shedding will explain all.

I researched Reynolds numbers when I researched vortices, the only information to be gained from the Reynolds number is the minimum (and maximum) speed that the lure will swim according to the rules. This pans out at about half a crank per second. Experiments proved this fact, the lure kicks in at a certain speed.

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Vodkaman, How did you determine that tow line angle (in diagram B) was in line with with the ballast. You have a line marked line of instability going through the ballast that parrales the tow line. So if I move the eye( tow line) where does the line of instability move. A parralel line can be drawn to the tow line angle reguardless of where I place the ballast.

Diagram7 you said the two line of insability are drawn, they are not marked so are you refering to the verticle lin through the ballast and the tow line and ballast line

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Vodka I think you diagrams are confusing not only us but yourself too ;).emoticone_vodoo.gif

Although hydrodynamics explains a lot about how the action in a lure is produced; to get to the point where we have conclusive parameters that affects a lure we need to look at the various forces in whole and combine the various forces with force vector diagrams (high school math ;)). Unfortunately calculating and breaking down all the component force vectors might require finite analysis and a lot of lab equipment for measuring them directly or indirectly :(

Go along the line of force vectors then you'll see where the resultant force is acting (direction, magnitude and where). Which would explain why the lure does certain things. We will still need to do force vectors for all 3D axis to determine the combined action in a lure.

That said, lure making is all about making lures that are symetrical as much as you are able.

A good lure for me is a lure with absolute symmetry and body shape that enhances the action instead of dwelling on a lip that makes the action. For example, take a Rapala FatRap, remove the lip and just put a nose tow eye and observe that it still wobble, check the frequency and design a lip that enhances and compliment that action you get a nice lively lure. Fit a lip that fights the natural wobble of the body; you get a lure that doesn't perform as well.

Theory is good but that won't help if your building is lousy. And for empirical analysis, how many of us doing this as a hobby can afford a software like SolidWorks and the embeded COSMOSFloWorks for simulation of the lure u design prior to building. Even if you can, how many have the CAD/CAM CNC facility to translate a 3D model to prototype. Like I always emphasize, it's still the fishes who will be our final judge and we just do not have enough studies on the target fish to say for sure what they want.

Actually if you have a lure that will guarantee 100% success in catching fish every single cast if the fishes are there maybe fishing won't be as fun anymore. Trust me, I got an opportunity to fish a barramundi pond with 6kg fishes and it was 1 cast on fish no matter what u throw, I got bored after the 10th fish, feels more like work than fishing then LOL.

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Water is an incompressible fluid, yet an airfoil in water still behaves like a wing as air is a fluid. Because of Bornoulli's law, water must travel (in effect) faster over the top, thus lowering the relative pressure and creating lift. So why do almost all of our lures look like wings, and create lift, when we are trying to get them to dive? lol

Actually Clemmy, look at some deep diving lures, some of them are actually a reversed airfoil, designed to dive. I usually check tow eye position and body shape of lures I design without a lip. A diving lure will actually dive a bit without lip if you design it right. As we understand these forces, we should take advantage of them and use it to help with the action of the lures we design.

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LaPala,

As I stated in the Lure Action thread, A lipless lure may well dive, but I believe a reverse airfoil design is more likely an attempt to increase action.

As far as looking at all the factors, moments, forces and vectors, yes it can be VERY confusing. But IMHO simply because it's hard to put it all together is no reason not to try. You used the example of a lipless Fat Rap. Why would you put in a certain shape of lip? Why a given length or size? Why a certain material or thickness? Angle? Line tie position?

You see, you are, in attempting to build lures, doing these calculations anyway, just using heuristics and experience rather than trying to understand the reason why.

Otherwise you are left with random experimentation, which can be fun too of course...

Clemmy

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Clemmy. The fact that water is an incompressible fluid has bothered me too, in my pursuit of explanations of the behaviour of the hard bait.

Vodkaman,

Oh dear me... this is a freshman-level mistake. Water is indeed compressible. This fact is well-recognized by physicists, engineers, and analytical biologists, among others, and certainly does affect computational dynamic fluid analysis.

The numerical value for the compressibility coefficient for water is 4.59 x 10-4 /MPa. This factor is best not ignored when constructing theoretical hydrodynamic models, considering the accuracy we usually expect from theoretical modelling. Results will not reflect real-world conditions. Sort of like averaging averages. Since the density gradient in water can exceed 1%, the density change for nitrocellulose (wood) across the gradient might exceed 10%, quite significant! For more information, see Svedberg (1940), Mossiman and Signer (1944), and Fujita (1956).

On a more familiar level, perhaps, we are all aware that lakes stratify, and that warm water floats upon cold water. This, of course, is because cold water is denser than warm water, so even the non-physicists among us will have a basic understanding that water can change it's density (compressiblity). Many folks also recognize that their lures will float or sink or swim differently, depending on water temp. Additionally, the water density at the bottom of a lake will always be higher than at the meniscus, regardless of water temperature, simply due to the weight of the water above. There's your 4.59 x 10-4 /MPa density change at work. Dynamic density gradients are also deterministic in the analysis of vortex propagation, and would likely go a long ways to furthering (and possibly clarifying?) your modelling of the behavior of the hard bait. On the other hand, I suspect LaPala is on to something.

Now, normally I allow a wide latitude between theory and practice-- the real world being what it is-- but I must take you to task on this oversight. If water wasn't compressible, the oceans would be over 100 meters higher than they actually are-- and that would spoil whole lot of really nice largemouth water!

I'll return you to your thread now. Cheers and three beers! beers.gif

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Skeeter jones. In diagram 7, the point was that if the lure is built in such a way that the tow line direction was in line with the ballast then instability would or could occur. Because the ballast no longer has any momentum (force x distance), it no longer has any stabilizing effect. The minutest asymmetry of the lip would cause it to pull one side more than the other and with no damping, would very quickly spiral out of control.

In diagram 8, the line of instability represents the perpendicular distance of the ballast from the tow line, as moments are actually force x perpendicular distance. When the ballast sits on this line, its moment equals the forces on the lip, a bit like opposite balance weights on a wheel. At this point, the lure is at the point of instability. The slightest fault on the lip would tip it over. If the weight was moved further rearwards, thus moving the ballast beyond this line, the lure spirals and death roll occurs.

I arrived at these conclusions by varying the position of the ballast from the front to the rear of the body on the same lure. A ballast position is reached were the instability starts. This is my interpretation of what is happening.

Regarding the two lines question. You are correct, the line has been missed out. The line should be parallel to the tow line, the same as in diagram 8. Apologies for the confusion.

Sagacious, you are indeed correct. You have corrected a huge hole in my assumptions. In fact it has been a point that has severely hampered me in all the time that I have been doing this study into how the lure works. Vortices and any other flow effects are reliant on the fluid being compressible, so I thank you, my job now becomes that much easier.

For the last hour I have been trying to think where did that incorrect assumption come from as I studied fluid dynamics for five years, while completing my HND in aerodynamics. The only thing that I can come up with is that every question on fluids on the course started, ‘an incompressible fluid, of density blah blah, traveling at blah blah’. I guess the incompressible part stuck in my head. You got me, your not the first and I’m SURE that you won’t be the last, thanks again.

Lapala. I would have to agree with you except I know exactly what I wanted to say, but on re-reading the article, I realize that I probably caused more confusion than clarification. This subject is probably one of the most difficult to understand. If I were to consider every single component and take into account their effect on the death roll, the problem would be too great and would likely give it up. The idea is to establish the major factors and analyze the effect of those factors in isolation, in establishing the cause of death roll. I have done the same with each of the aspects of a lures operation, in the hope that all these individual snippets of theories will eventually pull together and allow me to design the lure that I want and hopefully help a few others on the way.

I feel that I am closer to getting the big picture, but how close, I cannot answer. I would certainly accept alternative ideas on the death roll question. Although I think that the theory is plausible, I have to admit that I am not 100% confident. I will certainly be re-visiting the subject in the future (groan!!), when I think of a way of explaining it more clearly. I will take on board the vector idea and give it some thought.

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Good discussion all the way around. V'man I like the way you're working, changing one parameter at a time on your working model and observing and interpreting the consequences. This is basically the same methodology that Skeeter has used to train many beginning crankbait builders on this site BTW, teaching people to learn the basics of ballast, tow-point, lips, and their interrelationships; i.e., not just telling someone how to build a particular crankbait, but giving them the tools to understand and further expand the process. I can appreciate your quest to nail down as many absolute truths as possible in order to develop a written language, where only a spoken language previously existed, one that could only be learned through actual field experience (as I learned it, through a zillion days on the water) as opposed to opening a book. To accomplish this task will indeed greatly enhance our knowledge base, giving builders another way to learn and tweak into the world of crankbaits. I would like to have been able to read all this stuff when I was twelve years old...just imagine having had this knowledge in 1965!:drool::lol:

Dean

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Vodkaman, Sorry I don't understand the perpendicular distance from the tow line to the ballast. Do I measure the distance from the eye to the ballast?. I understand that the tow angle will be different on every bait based on lip length. Are you by chance extending the tow line angle below the lure and measuring the distance from the ballast to where that line would intersect.???

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Dean, thanks, that’s the basic idea.

Hazmail, you also have the basic idea, except the forces on the lip are higher because it is a flat plate. example, imagine you want to move a 1 inch plate through the water as easily as possible. You would add a fairing around it shaped like a wing section or torpedo. The shape would slip through the water with reduced resistance. That is what is happening as the water is passing over the body.

So for the body to balance the lip, the body area will be greater. The ratio between the two could be calculated, but is way beyond my capabilities. An eyeballed ratio of about 2:1, but don't quote me. It would be nice if it was a constant ratio, but it will vary with changes in body shape.

Skeeter Jones. There is nothing to measure, the line of instability is an imaginary line. If the ballast is moved beyond this line, the bait becomes unstable. The line will be a different distance for each bait geometry. The important thing is to understand that it is there.

So what was the point of the thread? How can the theory be used? The document tries to explain what causes the death roll instability, were the lure rolls too far and cannot recover and spirals out of control.

In the case of deep divers, as depicted in diagram 8. to correct the instability, the ballast must be mover forward of the line of instability. This can be achieved in a number of ways. A) physically move the ballast forward. B) move the tow eye forward. C) trim the lip shorter. D) shorten the body from the tail. Selections B, C and D all alter the lip/body balance and cause the dive angle and the tow line angle to be reduced. This in turn moves the line of instability further aft as it is parallel to the tow line.

On a steep lipped, shallow diver, see diagram 7. the weight is shown sitting on the tow line. This unstable condition will give the most action, it may or may not lose control. If it does, a slight adjustment of the ballast fore or aft would solve the problem, also B,C and D would achieve the same.

So, is that the full death roll story? I doubt it.

Does the theory work? Yes, in my experiments so far.

Is the theory correct? I can’t say for sure.

Should I have released the document? If I didn’t release it, I wouldn’t get feed back. If I am definitely wrong, someone out there will correct me and because it seems to work, it cannot do any real harm.

Any experiences or observations around this subject will be gratefully received, if you make a lure that death rolls, DON’T throw it away. Tell me about it, photo’s would help, with a description of ballast location etc, either by PM or post.

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Ok Im not sure if the attachment is going to make it here. Dave if it does see if I have the right location where the ballast would influce the lever arm on the lure. This made sense to me because you said if the ballast was moved forward it would stabilize the lure . It would move the ballast influence point closer to the line tie ,which is the pivot or fulcrum of the lever arm.

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I am going to ask another question here as well. I like this thread and I hope yall don't get tired of me asking the questions. Vodkman you said one of the solutions was to cut the back end of the lure down. This made me think about the amount of dive angle or pitch on a lure. Does more pitch/dive would cause more of the body to be exposed to the water forces, creating more pressure ,more weight on the end of the lever. Reducing the length of the body would reduce the weight/force.

When the lure body is piched down at a steep angle, does the vorticies produced by the lip have less effect on the lure. I am picturing the tail of the lure being above the horozonal line and the vorticies have less surface area to hit.

By the way in my attachment F1 is the force on the lip before the line tie . F2 is the lip force behind the line tie and the combine weight of the lure plus water pressure..

I have checked several deep diving lures and they seem to have about a 10 degree lip angle. Is this about right?

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A lip is essentially a flat plate. It has little drag at level or zero degrees. As dive angle increases, so does the angle of the flat plate, so drag (and thus vortex shedding) is increased until the max of 90 degrees. I believe the attitude of the bait body may or may not take advantage of the von Karman street of vortex shedding based on design. But do not forget that the body (and everything else connected, including the line) imparts it's own effects in terms of drag and vortices.

Did you no that a tuna cannot physically swim as fast as it does? It was a mystery until researchers showed that they are using the vortices created by their bodies, detected by their lateral lines, as springboards for their tail to push off of....

Clemmy

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Skeeter Jones. No wonder this thread is confusing, I keep making mistakes. I should, of course, said that the body should be lengthened and NOT shortened, well spotted.

First off, I welcome any and all questions. Sometimes the query makes me re-examine what is in my mind and in some cases improve the theory.

Regarding your diagram, it is more or less correct, but I cannot see any point in getting picky as it would be off the subject. I feel the need to discuss the line of instability more, as a result of your PM to me. It is a question of static and dynamic (moving) forces. The dynamic forces at the lip and the body override the effect of the ballast on the angle that the bait swims as the forces are much greater than the ballast.

However, when a deep diver at the surface is set in motion, the lure quickly reaches a balance and adopts its dive angle, of say 40 degrees. At this angle, the ballast is raised up and becomes top heavy. There comes a point at which this top heavy weight becomes too much and the lure will become unstable and roll over. This is where I draw the line of instability. The line is drawn parallel to the tow line because it is the perpendicular distance that is important.

When the ballast is sitting on this line, the bait is on the edge of instability and erratic swim actions are more likely to occur. I think that if the motion started at depth, the problem would probably not occur, as the bait would be swimming more horizontally and the raised ballast problem would go away. Unfortunately, the movement starts at the surface.

We cannot at this time calculate the position of this line, but as long as we know that it is there, we can explain what has gone wrong when the bait corkscrews and scares all the fish. Trim the lip or move the ballast forward.

Regarding your question about the lip forces having less effect as the lure swims deep and horizontal. I struggled with this one and still do. The lure is being towed in the line direction. It is this direction that the forces are acting, not the direction that the lure is traveling. The lure takes all that is given to it, ie lip forces, vortices, turbulence, ballast, buoyancy etc and then takes the path of least resistance, a basic law of nature. So the forces on the lip remain constant regardless of depth.

Clemmy. Only now are people starting to understand the true purpose of the lateral line. It had nothing to do with sensing the presence of other fish by pressure differences. I think that scales evolved for the same purpose, to re-capture the vortex energy that the fish expends and get a free rive when lurking behind a rock in moving water.

Experienced lure designers instinctively know what is wrong and know what to do to fix it, in fact they probably haven’t made a lure that suffers this problem for years. To them, this post may well represent a waste of time and effort. I am new to this game and getting on in years, I simply haven’t got the time to gain the years of experience required to make a perfect lure every time. I feel that an understanding of how it works could shorten this learning curve, it certainly has worked for me. Hopefully as I learn more, I will find a way of explaining this stuff more clearly.

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You guys are great in my book. I have learned quite a bit about the science involved in making a lure. In my line of work its hard to fix something if I don't understand why it work and what each part does to make it work. I like to tournament fish and I want the lure that will knock em out and bring them to the boat . I want some erattic action. Why are there different ratios of reels? Because the slower retrive allows the lure to wiggle ,waggle,wobble more. A lure may death roll with a 7:1 ratio but be stable with a 5:1 based on the velocity of travel. I'm taking notes and have a lot of searching to do to try and find the answers I need. Thanks a bunch.

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