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23 November 2024 17:42
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Welcome to the gyroscope forum. If you have a question about gyroscopes in general,
want to know how they work, or what they can be used for then you can leave your question here for others to answer.
You may also be able to help others by answering some of the questions on the site.
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Question |
| Asked by: |
Luis Gonzalez |
| Subject: |
The Root Cause of Precession (your comments are welcome) |
| Question: |
Precession results from how the laws of physics determine stable spin motion and how stability remains in the face of external forces interacting in non-intuitive ways. The scarcity of adequate information in this area of physics begs for a sense of organization starting with the easy basics and going on to the more complex. For that reason I have compiled a list of the things we know intuitively about circular motion and from that build up gradually until we encounter precession in a logical manner. This may or may not be the first time these points have been organized and developed in this manner, but than we all do things in different ways.
Starting with the simple basics, we will look at the properties of spinning objects, purposefully avoiding the configuration of a gyro on a tower under gravity (which we will cover later).
Rules of Stable Spin:
#1 - The center core of the spin-axis is generally the least dynamic.
#2 - Objects (including a disk) can spin around ANY axis of symmetry (i.e. any axis that distributes mass symmetrically).
#3 - Masses outside the center of spin tend to continue in a straight line motion (per Newton) and are kept in check by the centripetal force imbued by the cohesive forces of the material. The centrifugal effect causes larger mass to preempt the outer perimeter of spin, depending on the geometric shape of the object.
#4 - The most stable axis of spin has the highest symmetry, and/or provides the longest spin radius to the largest proportion of mass, symmetrically (consider a disk).
#5 - Depending on its geometrical shape, an object spun around multiple different axes will settle into a spin-axis that is either, a) the average of the different axes of spin, or b) the sum of the spin-vectors, or c) an axis that conforms to the most suitable axis of spin as stated in rule #4 above (consider a disk).
#6 - An object in a stable spin-axis tends to stay in that configuration even if other spin-axes configurations of greater stability exist. The reason is that to change its spin-configuration from one stable spin-axis to another stable spin-axis, the object needs to cross fields of configurations that have highly unstable spin-axes.
#7 - To induce a switch among stable axis configurations requires an external force of sufficient magnitude applied in the appropriate direction.
#8 - As long as the torque-axis “precedes” i.e. continues to exist ahead of the spin-axis, precession continues to occur (as soon as the 2 axes are lined-up parallel, precession stops).
These (maybe incomplete) set of rules explain how objects can spin and can settle into one of many stable spin axes with different levels of stability. A coin twirling on a smooth surface illustrates how a disk is able to spin on an axis of symmetry that is not optimal for a disk (as compared to the spin of a gyro, which spins on the main axis and is in an optimal spin configuration for a disk.)
Gyro flywheels happen to spin in the most stable axis-configuration (for a disk) because they spin on the axis of highest symmetry that also provides the longest spin-radius to the largest proportion of the mass of the disk (per rule #4 above). These 2 factors provide a strong deterrent to change from a highly-stable spin configuration. Well balanced gyros in a stable spin can self-correct most induced wobbles.
Note that change to a stable configuration can be induced by a sufficient external force applied in the appropriate direction. For example, a spinning disk may be made to wobble by introducing a specific type of interference. The cause of a wobble can be of sufficient magnitude and/or can have a harmonic resonance capable of changing spin-axis configurations (e.g. frictions or imbalances can produce destructive harmonic frequencies).
Some of us may perceive that the existence of multiple spin-axes, each with a different level of stability, represent different energy levels (food for thought).
Here comes what I believe to be an innovative view of precession’s root cause (the why & how). The tendency to maintain stable configurations of spin is (probably) the most significant contributing factor to what we know as “precession.” The dynamic self-correcting state of spinning disks make gyros under torque, change orientation in order to maintain their configuration of high order spin-stability (is this statement new?).
Consider that an external torque attempts to change the spin-axis configuration of a disk to a spin-axis of lesser stability. To visualize how this is so, think what the same torque does to a non-spinning gyro; it makes the disk twirl like a coin twirling on a smooth surface. However, in a spinning gyro, the spin-axis would have to cross fields of high-instability in order to reach the relatively less stable “twirling-type” of spin.
When an applied torque attempts to introduce this (coin-like) twirling-type of spin to an already spinning disk, it attempts to both, twirl (on a lesser-stable-axis) and at the same time spin about the main axis of the disk. This extremely chaotic motion is highly-unstable and would violate many rules of spin. (Note - The chaotic motion is not necessarily impossible, but would require the disk to exist in an axis of highly complex symmetry and dubious stability, which theoretically needs a different level and sort of energy).
Therefore, in absence of unusual, strong types of energy, the torque applied to a spinning gyro will cause the disc to respond by repositioning itself into an orientation that allows it to maintain the highest symmetry of its original spin-axis configuration.
The only way the gyro can maintain this original stable spin-axis is by “following” the spin-pattern of the torque, and making its spin-axis parallel to (or the same as) the spin-axis of the torque (through changing axis-orientation rather than changing its axis-configuration). In other words, the spinning disk will seek the position where the direction of the torque is congruent with the direction of stable spin (i.e. where their spins are in the same plane and their axes are parallel or the same). Most important, a gyro’s degrees of freedom allow for a change in orientation for the spin-axis, and THIS BEHAVIOR IS AT THE CORE OF PRECESSION!
In case you missed the meaning of my explanation, symmetries of mass dictate an object’s stable-spin configurations, and gyros have sufficient degrees of freedom to adjust their orientation so that they can maintain a stable spin-configuration when an external force/torque attempts to disrupt the stability. These concepts appear initially complex but become simple with time and familiarization.
This rational development of ideas, starting with commonsense reasons for how spin occurs, followed by spin’s interactions when coupled with external forces within a gyro’s degrees of freedom, is the basis upon which my definition of precession is built; the resulting theory model is consistent and based on premises that can be proven.
With this basic understanding of rules surrounding spin, we may take a fresh look at my basic explanation of precession, and to how it extends to the behavior of the familiar toy-gyro (which has an asymmetric axle, gyro-on-a-tower under gravity configuration). So let’s start by exploring the meaning of the word “precession.”
Roget’s Thesaurus associates the word, “precession” with “precedence” meaning previous-ness; and with “preceding” meaning to lead or to go in front of something. It appears that ‘precession’ describes how the “torque-axis” relates to “the spin-axis” in the dynamics of gyros; i.e. the torque-axis always precedes the spin-axis during precession.
I have explained elsewhere that the toy gyro configuration (on a tower under gravity) derives its ability to continue rotating (beyond 180o) from the inability of the spin-axis to ever catch-up with the torque-axis (derived from gravity). As precession moves the toy-gyro around horizontally, the pivot-axis (which is derived from the axle resting upon the tower plus gravity) also moves around horizontally. Thus the torque which results from the pivot (gravity pulling down on the gyro side of the axle while the tower maintains height on the other end of the axle), also moves horizontally. Since both axes move horizontally, at the same rate, and in the same direction, the spin-axis never catches-up with the torque-axis; hence they remain orthogonal (clear?).
This “eternal” chase is proper only to the toy-gyro (under gravity). In contrast, the spin-axis of gyros with mechanically induced torque (e.g. in gimbals) manages to catch-up with the torque-axis (always) within 180o span of angular precession (see Victor’s drawing).
A question has been raised whether we should call the 90o motion induced mechanically on a gyro (e.g. on gimbals), by a different name rather than precession! Though this motion is independent from gravity, it also occurs at 90o to the torque applied and relies on the exact same basic mechanics. The main difference is that the device with mechanical torque (e.g. a gyro in gimbals receiving a twist to the outer cage) produces a motion at 90o in a way that the spin-axis can catch-up to the torque-axis! In other words the gyro re-aligns the orientation of its spin-axis (NOT the configuration) until it gets to a stable position (as illustrated by Victor Geere’s drawing, when both axes lineup). Once both axes lineup the torque-axis ceases to precede or lead (as opposed to the toy gyro), and the 90o precession motion also STOPS. Rule #8 says that as long as the torque-axis “precedes” the spin-axis precession continues to occur (as soon as the 2 axes are lined-up parallel, precession STOPS).
In a strict sense, the 90o motion caused mechanically (e.g. gyro in gimbals) is indeed as much a form of precession as the one created by gravity for the following reasons: 1) Both motions result from the same type of interactions between torque and spin. 2) Both motions follow the same rules of physics. 3) Both motions occur at 90o to the applied torque. 4) Later we may see that both motions also occur at 90o to the spin of the gyro flywheel.
These 4 points apply to the 90o motions of precession in toy gyros as well as to the 90o motion of gyros in gimbals; both come from, and follow the same rules.
In my opinion both 90o motions are the same with a difference in the mounting configuration due to the use of gravity as the source of torque, versus the use of mechanical torque.
Therefore, the word “precession” is appropriate to describe all 90o motions that result from interactions of spin with torque (what else would you call it and why?).
In the same way, since it is the “acceleration” of gravity which creates the force for the torque, I don’t agree to use the term “accelerated” to differentiate mechanically driven torque, because all torque is always derived from “acceleration” whether it is mechanically induced or from gravity.
Other individuals argue that the word “precession” should only apply to gyros under gravity. This creates an artificial break in the language used to describe precession. The resulting, discontinuity to the flow of thoughts, creates a block to understanding and prevent explaining the root causes of precession.
If we want to eventually bring some clarity to the subject, I say it is wiser to stick to unequivocal terms. The subject is complex enough without adding unnecessary ambiguity. We can use existing, well known, well defined terms and differentiate them with appropriate qualifiers if we need to. I bring this up because complexities are difficult by definition, and creating ambiguous terms can only add obscurity and perpetuate widespread ignorance of the subject.
Admittedly, the facts contain numerous interrelated details and this makes the subject difficult to address in a quick easy manner. Sticking to correct appropriate terms allows easier connection of the many facets, and permits at least a few bright and dedicated individuals to eventually break through and understand.
(Excuse me before I fall off my soap-box as I get on my high horse.)
I believe that this posting (in conjunction with others) completes a comprehensive (though not concise) explanation of precession’s behavior including causes and effects, from a perspective of classical physics.
Good science does not need to state the “what is” of a phenomenon; good science needs only explain the “how” in a way that permits quantification and/or establishes acceptable connections to previously proven, accepted rules.
Any fuzziness in making these connections may bring into question the style of writing, the person writing it, and the choice of forum for that matter. However, dedicated bright individuals with a fair understanding of real physics will eventually decipher that the facts are correct and stated in consistent language. Brighter individuals will infer how knowing these facts allow one to prescribe measurements and verify the coherence of the model theory. It may take some time (all good things take time), but time will tell.
I have yet to find as complete a layman’s explanation for precession, as is stated here. In contrast, one of my college physics books explains precession using a rule-of-thumb (literally) interpretation of angular accelerations. Though the explanation in that book is a bit ridiculous, it does make use of the same rule of thumb used by electric engineers and, right or wrong, appears to create a faint connection between the two fields.
I feel that I have conveyed a perspective of how I see precession occurring as a result of known intuitive interactions. These explanations are better understood from the perspective of a gyro in gimbals, where the acceleration for torque is produced mechanically, and not by gravity (this configuration provides a simpler model with fewer variables).
I have, elsewhere explained how these interactions that occur on the gyro-disk, can project their effects into a motion of translation that causes toy gyros to revolve (like a planet). Therefore I will leave that explanation for the readers to find among my other writings, if they wish to complete those pieces of the puzzle.
Basically, precession in a gyro-on-a-tower configuration results from an interaction of forces that occur on the gyro disk, not at the center of rotation. In the tower-configuration, planetary-style precession is driven from the gyro mass at the perimeter. The center of rotation of the system does not provide direct drive force to the planetary-style motion of precession; the center of the system only provides a pivot point.
I welcome comments relevant to the posting.
Thank you, Luis |
| Date: |
12 September 2006
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Answers (Ordered by Date)
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| Answer: |
nitro macmad - 09/11/2007 18:10:09
| | | Dear luis
I am ashamed that it has taken so long for me to reply to your excellent observations - nice one!
Kind regards
NM
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| Answer: |
Luis Gonzalez - 10/11/2007 19:57:44
| | | Thank you Dear Nitro,
Please think nothing of it.
I am glad that someone with sufficient experience provided such positive remarks.
I consider the content of this thread a primer to spin-basics, some of which we intuitively consider when dealing with gyros.
I thought it a good idea to explicitly state these implicit concepts to eventually have a compendium on Rules of Spin. There is nothing like having a comprehensive understanding of cause-and-effect rules to break apart complex issues.
I am still interested in expanding the list and or the items stated in the list. Currently I am working on explicitly describing connection between the “rules of spin” and the laws of motion.
Another ambitious endeavor, in which I am frankly having a bit of a thinking block, is in trying to quantify gravity induced and mechanically induced precession/deflection through strictly angular means (i.e. without recourse from linear vectors).
Best Regards,
Luis
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| Answer: |
Sandy Kidd - 12/11/2007 07:43:35
| | | Luis,
You said:
Other individuals argue that the word “precession” should only apply to gyros under gravity. This creates an artificial break in the language used to describe precession. The resulting, discontinuity to the flow of thoughts, creates a block to understanding and prevent explaining the root causes of precession.
If we want to eventually bring some clarity to the subject, I say it is wiser to stick to unequivocal terms. The subject is complex enough without adding unnecessary ambiguity. We can use existing, well known, well defined terms and differentiate them with appropriate qualifiers if we need to. I bring this up because complexities are difficult by definition, and creating ambiguous terms can only add obscurity and perpetuate widespread ignorance of the subject.
Admittedly, the facts contain numerous interrelated details and this makes the subject difficult to address in a quick easy manner. Sticking to correct appropriate terms allows easier connection of the many facets, and permits at least a few bright and dedicated individuals to eventually break through and understand.
“Precession” in my physics book says “following”
Luis you are obviously not aware of the significance of separating gyroscopes from flywheels.
I thought I had made the difference abundantly clear.
Precession should only be applied to gravity accelerated systems.
They are a unique and special case.
Flywheels or spinning discs not specifically subjected to gravitational acceleration are totally different animals.
I have noticed that not one of the contributors to this site, yourself included, has explained or ventured to explain, why a disc must spin, to produce the affects you have just gone at great lengths to attempt to explain away.
I would have thought that this was a (the) fundamental issue, and worthy of some explanation from yourself.
Then you would not have to present a case for following outdated thinking.
You said:
“Sticking to correct appropriate terms allows easier connection of the many facets, and permits at least a few bright and dedicated individuals to eventually break through and understand”.
Luis the subject is not at all complex. You are making it complex.
You have been attempting to explain precession away for a few years now Luis, have you not? With not a lot of success methinks.
I have just questioned your lack of explanation relating to spinning discs.
How about the “correct appropriate term” gyroscopic torque couple?
I would like to hear or see your explanation for this “correct appropriate term”.
Maybe its just there because somebody said it was, and that is good enough for Luis.
If you wish to invoke the “correct appropriate terms” as you suggest, you could be dabbling with this thing for a very long time, and it is probably one of the main reasons why very few of you will ever get anywhere with it.
Sandy Kidd
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| Answer: |
Luis Gonzalez - 15/11/2007 00:47:49
| | | Sandy,
I thought we were past the discussion on semantics (the date of the posting is over one year old).
We must be looking at different dictionaries.
I don’t know if I may or may not be able to answer your questions because I am not sure what you are asking.
If you frame your questions in a way that I may understand them, perhaps I can help (if I know the answer).
If you have a question ask it.
I see you found a badly formed phrase to repeat over and over; I guess everyone gets a bone now and then.
Luis
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| Answer: |
Sandy Kidd - 15/11/2007 07:08:11
| | | Luis,
Your answer was almost word for word what I expected from you.
I am well aware of the date of the original posting. Has anything changed?
I do not think the pair of questions were so badly constructed a man of your intelligence could not understand them, and the significance of either.
The answer to one answers the other.
I don’t think there is any simpler way of asking what I asked.
You are repeatedly on about using accepted terminology, so I am asking a question relating to accepted terminology.
I shall make it simple and refer to one fundamental item only, seeing as you find it so hard to understand.
“What creates the gyroscope torque couple?”
I do not think there is simpler way this question can be asked.
Sandy
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| Answer: |
Harry L. - 15/11/2007 09:11:32
| | | @Sandy Kidd
Q: “What creates the gyroscope torque couple?”
A: The torque couple is created by the alteration of angular momentum of the spinning mass as a result of the angular momentum of the spinning mass and an additional orthogonal momentum applied to the spinning mass through gravity OR any other tilting torque.
Hope you understand my Englisch translation...
I don't know why this fact is so difficult to understand!?
Regards,
Harry K.
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| Answer: |
Luis Gonzalez - 17/11/2007 00:19:45
| | | Sandy,
It looks like you are better at mind reading than at designing gyro-propulsion devices (word for word?).
Harry’s answer to your question above is as good as or better than anyone else can provide (assuming we understood what you intended with your question).
I also agree with Harry that this is not a particularly difficult point for bright people interested in this subject, again assuming we understand the intent of your question.
My own (more elaborate though less expressive) explanation as to “why a disk must be spinning” in order to produce precession/deflection has peen posted more than once in this and other threads of this forum.
In its most basic form, my perspective is that precession/deflection are the result of angular force (torque) changing or modifying the direction of an existing angular motion (spin) in a flywheel. That is why the “disk” must be spinning in order to produce the effect we call precession. If you don’t see the connection by now, I doubt you ever will.
I intended this thread to focus on documenting the basic rules of spin, to eventually clarify why (and up to what point) a spinning object (such as a disk) behaves in the orderly way we observe when a torque (of certain type and magnitude) is applied. These rules of spin (when taken together with the explanation for precession/deflection) clarify the shroud of mystery that hangs around most of the observed gyro “anomalies” (or phenomena).
It appears to me that your question may be intended to clarify your own confusion. You see, I read your posting of 09/10/2007 12:42:57 in thread http://www.gyroscopes.org/forum/questions.asp?id=768. In that posting Sandy says: “I have previously made comments about the fictitious gyroscopic couple but no one appears to be able or willing to enlighten this old man how this couple is created, and what part it plays in the generation of gyroscopic torque. Can someone prove to me that it is not just another “convenient” gyroscopic action which I personally think is just a myth born from misunderstanding.” These were your own words!
I also detect dishonesty in a man who claims in one place that the term “Gyroscopic couple” is fictitious and a myth, and then tries to bait me by telling me (in this thread) that his question “What creates the gyroscope torque couple?” is a perfectly good question.., in a cheep attempt to coerce me to answer it.
Sandy, explanations based on known science are well founded.
Explanations (statements) based solely on self claimed authority are worthy of the term “explain away”, as if in vain attempts.
We have different opinions about what is worth pursuing, to discover whether gyro-propulsion is possible.
What is complex to some is simple and clear to others (and visa versa). There are perfectly good reasons that we don’t understand everything that the other states or explains.
The bottom line is that, despite many claims, no one has successfully produced undisputable gyro propulsion yet. The highest level of success at this point is “the space inch-worm”.
Thats all.
Luis
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Nitro MacMad - 17/11/2007 20:11:12
| | | Dear Luis,
I find the sniping between you and Sandy most distressing. This subject is most difficult and has to be approached from both the intuitive and scientific (nutter’s and pedant’s if you will) view if a reasonable breakthrough is to be achieved.
You may (though without being allowed to poke your fingers in the nail holes you may chose not to) take as read that your (well described) “space inch worm” exists.
While “the space inch worm” is not perhaps the longed for “Flash Gordon” answer, It should be realised that the first internal combustion engine, that my memory says was a less than successful gunpowder fuelled nightmare, held little promise in its day. I am sure that its individual explosions would have made it be regarded as a “terrestrial inch worm” until a flywheel was dreamt up by some, now long forgotten, nutter.
Kind regards
NM
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Luis Gonzalez - 18/11/2007 18:31:51
| | | Dear Nitro,
I also find personal attacks disturbing. We should be able to vigorously argue for and against the merit of ideas without condescending remarks to the person.
People should feel free to contribute at all levels of creativeness that their minds will allow them, knowing that others may find errors in their logic or facts, and that’s OK.
Arguing vigorously about ideas (concepts, models, etc) with intelligence and logic and backing it with facts is great fun; however needing to slap back at someone who offends personally is a complete waste of effort.
Sandy contributes excellent material. However I am irritated by any one who bolster their arguments with personally derisive statements that have nothing to do with the subject matter.
I agree that the “space inch-worm” marks the first step toward technology of great potential. That is why I have pursued attempting to demystify some areas, by exposing their hidden complexity.
All half statements need to be pursued with questions, and if no answer results then their complexity must be unpacked by logic and by supporting experiments. A public forum is a bad place to try to hide things.
I will try to make some time to respond to your posting of earlier this month on “precessional cause”.
Best Regards,
Luis
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| Answer: |
Sandy Kidd - 19/11/2007 07:09:43
| | | Harry K
I can get what you wrote out of any physics book.
You did not explain “where” this couple is created?
Luis,
You at one stage stated that I was mistaken and that I was misunderstanding what I was seeing.
You are quite prepared to make wild statements like that without any knowledge of the subject at all, and you wonder what p***** (rhymes with kisses) people off.
Am I some kind of idiot?
Did you test my claims?
I do not misunderstand what I see, and I also make absolutely certain before I go to print.
To me that was a very personal attack.
What's good for the goose.
Sandy
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| Answer: |
Harry K. - 19/11/2007 14:30:59
| | | @Sandy Kidd
Where? Directly at each spinning mass point depending on the level of influence of the rectangular tilting torque (changing per cycle between 0-max-0-max-0).
The angular momentum vector of the spinning mass plus the rectangular tilting momentum vector force each mass point to change its spinning plane and thus the spinning axis must rotate accordant to allow this. This behaviour at each mass point and corresponding diametric mass point cause the rectangular "deflected" gyroscope torque couple.
Now you may reply what I wrote is stated in any physics book as well... And yes you are right because this may have a reason...
Regards,
Harry K.
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Luis Gonzalez - 20/11/2007 02:00:57
| | | Sandy,
Take your time reading through this thread and note that the word “misunderstanding” was only used in a direct quote from you; I even provided the correct thread, date, and time that you made the statement. I used your quote because it flew in the face of the question that you posed to me in this thread.
Now, if you are referring to something I said in another thread, than please respond to it where it happened so we don’t start using statements out of context. If one responds to others at eye level without talking down, then others will take care not to upset. What’s good for the goose…
I don’t make wild statements Sandy. Knowledge of any subject is determined by the quality of the effort, and the depth and willingness of the instructors, not by length of time spent. It is easier to see farther when you stand in the shoulders of giants. Ignorance always presupposes ignorance in others. It would be better if we judged each other by the quality of the thoughts we express. My guess is that jealousy can sometimes blind us.
I don’t wonder who is upset with what I write; I know who is and who is not, and why. I know that open forums attract a wide variety of personalities, and some can not handle criticism of their ideas very well.
I prefer not to have sacred caws but to judge each individual by the content of their mind, at least the part they let others see. Thinking of others as ignorant and stupid (including scholars) can not produce a lasting satisfaction.
I believe you observed well what you saw; however, judging by the obscurity of your theory, the unlikely resulting statements such as “sheds mass” and the complete abhorrence of math and explanatory equations, your conclusions appear misguided. Top this with being unable to reconcile observation with the laws of motion,… no explanation or extension, and credibility becomes flimsy at best.
I am sorry if my observations are brutally honest. I only say what I and probably others observe in this forum.
Luis
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Graeme - 20/11/2007 08:13:18
| | | Hello Luis,
Maybe the maths or theory for this phenomena has not been developed yet, this might make the current mathematics model defunct, just a thought. As I said in an earlier post "freedom to think" we don't know everything, and I’m sure the future holds many surprises in various areas of science. Many people may feel defensive if the above comment on mathematics were the case, as they might not have a basis to move forward, unless they develop new a mathematical theory, that's how most concepts become the normal way of thinking and are then taught in academic circles. How did they work it all out in the past? By practical experiments, rewriting of many theories until it fitted what was happening physically.
Graeme
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Luis Gonzalez - 21/11/2007 02:08:33
| | | Thank you for your thoughts Graeme,
You are right that some of the math has not yet been developed or made widely public.
The appropriate math will present extensions and only minor adjustments to the existing paradigms.
I suspect man’s knowledge is like a drop of water in an ocean of knowledge yet to be known
Historically science has moved forward largely through experimentation; however, many subtle advances in much of recent science have relied heavily on theory which rests solidly on math.
Use of computer model simulations currently create numerous subtle innovations daily to evolve business, biology, and other applications; all built upon math.
Math is not the only path but it can be a disarming tool when you have a stubborn problem that resist being resolved.
Others in this forum have stated that gyro-propulsion will require a multifaceted effort from different sides of science and engineering.
I want to help define the less obvious concepts that appear to elude many. My goal is to relate the concepts so that they fit well together from the perspective of the known laws of science. The equations will emerge with time and we will see that the established basics do not really need to be thrown away; they only need to be expanded.
Regards,
Luis
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