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Evan
12-19-2005, 12:49 AM
I had a productive weekend working on my mill project. In the course of designing this project I spent a lot of time studying the online machine design courses available from MIT.

One of the less obvious things about machine design is the role that simple mathematics plays. In particular, a value called the Golden Mean. This is an irrational number with the value of the ((square root of 5) + 1) divided by 2. It is approximately 1.618. This value has been known since the time of the ancient greeks. It has special mathematical significance.

The Golden Mean is sometimes called the "most" irrational number. Irrational numbers can be represented by an infinite series of ratios (fractions) summed. Pi for instance is approximately equal to 355/113 which gives 3.14159282...

All irrational numbers can be represented by a fraction and series of fractions summed together to any desired level of accuracy. The "more" irrational the number is the smaller the numerator and denominator of those fractions must be. In the case of the Golden Mean it can only be represented by the sum of fractions that contain the number one.

It is the series gm=1+(1/(1+(1/(1+(1/(1+(1/....

So, what does this have to do with machines? The Golden Mean can be expressed as the Golden Ratio, 1/1.618... This in turn is used to make a Golden Rectangle where the ratio of the height to width is the Golden Ratio.

A rectangular surface which is a Golden Rectangle has no fundamental or harmonic vibrational modes in the long direction that are common with the short direction.

This is particularly applicable to the design of carriage and way systems with sliding or rolling element bearings, such as a lathe carriage or mill table. Linear bearings placed at the corners of a Golden Rectangle will have no common resonant modes in the x-y directions. The system is inherently self damped.

Probably not coincidentally, the carriage on my SB9 has a length to width ratio very close to the Golden Ratio. Also not coincidentally, the linear bearings on my mill are at the corners of a Golden Rectangle. http://bbs.homeshopmachinist.net//biggrin.gif

I highly recommend the MIT courses for anyone wanting to learn more about machine design. They are all online and the lecture notes are free for anyone.

http://ocw.mit.edu/OcwWeb/Mechanical-Engineering/index.htm

12-19-2005, 03:02 AM
Thanks Evan. This is the most informative and interesting OT post I've seen here in a good long while.

sauer38h
12-19-2005, 03:14 AM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
Linear bearings placed at the corners of a Golden Rectangle will have no common resonant modes in the x-y directions. The system is inherently self damped.</font>

Unfortunately this doesn't do you much good. You don't need common resonant modes in order for one mode to drive another. The reason is that the resonant peak in the Bode plot (frequency vs. amplitude curve) has a finite width. That is, vibrational energy exists at frequencies other than the resonant frequency, so modes need not have the same resonances to talk to each other.

Also, the fact that the resonant modes are not easily calculated does not mean that they don't exist. Experiments with plates of various shapes show that vibratory modes are very simple and obvious if the plate shapes are simple (like, say, rectangular, or square, or circular). The modes are more complex if the shape of the plate isn't of simple form. The way these are demonstrated experimentally is to set up a plate of suitable shape with the desired edge supports (clamped or simply supported, maybe even free if the experimenter is sufficiently ingenious). A loudspeaker voice coil is glued to the plate, and a support for the speaker magnet structure is cobbled up. When the voice coil is driven with an electrical signal, usually a sinusoid or white or pink noise, the plate will vibrate. The modes can be shown by dumping table salt on the plate. At the points of maximum amplitude, the salt grains will bounce off, and will tend to come to rest, more or less, at the nodes (the points of minimum vibration).

The other problem is that the system is not "self-damped". Passive damping requires the removal of energy from the vibrating system. That can't be done with masses and springs or elastic plates, which are energy storage devices, not energy dissipating ones. The only way to add damping is to get some dry friction in there (as in a Lanchester damper), or fluid friction, or elasteromic hysteresis, or in some cases magnetic hysteresis, or by radiating energy away as sound waves. Alternatively, an active control system can do it, if it has a velocity feedback term (as in the "D" in a PID controller). Mathematically the controlled system can then be identical to a system with mechanical (generally viscous) damping.

Despite all that, the Golden Ratio does have some peculiar properties. Pick two numbers at random (both positive). Add them together. Add that result to the second of the two original numbers. Add that result to the first result. Continue adding the last number to the immediately previous number. After a few terms, take the ratio of the last two numbers. If the ratio doesn't equal 1.618..., try doing it for a few more terms.

Leigh
12-19-2005, 03:16 AM
Thanks very much, Evan. That's extremely interesting. I suppose I must now pursue the MIT course, like I need another project http://bbs.homeshopmachinist.net//biggrin.gif

------------------
Leigh W3NLB

pgmrdan
12-19-2005, 06:36 AM
To me one of the most interesting things about the golden ratio is its use in the appearance of objects. Its surprising how many objects in nature, including ourselves, have proportions equal to the golden ratio. It's an esthetically pleasing ratio.

I've read about its use in the design of woodworking projects, say a wall mirror. When the ratio of the height to the width,or vice versa, is the golden ratio it has a more pleasing appearance. Table height to lenght and length to width are some other uses. Many pieces of furniture are designed using the golden ratio.

There are many applications for this number.

Dan

lynnl
12-19-2005, 07:05 AM
I saw a recent article in either Fine Woodworking or American Woodworker explaining this, from the aesthetic standpoint.

That was the first I'd ever known of it (at least that I can remember).

Evan
12-19-2005, 08:07 AM
"Unfortunately this doesn't do you much good. "

A golden rectangle still has vibrational modes. However, modes common to one axis are much less able to excite modes in the other axis. That is why it is commonly used in the design of speaker cabinets. It is a form of damping, or at the least a way to eliminate an entire series of resonances.

J Tiers
12-19-2005, 08:15 AM
<font face="Verdana, Arial" size="2">Originally posted by pgmrdan:
It's an esthetically pleasing ratio.

Dan</font>

That is often said, yet items constructed using that ratio can look unpleasing to many folks. It can even look clumsy.

I don't think it is anything "magic",......... or perhaps I do...... I believe it was derived originally as some part of alchemy.....

all it takes is ONE undamped resonance mode..........

That said, it is an interesting ratio.

EDIT
On second thought, I might be thinking of the proportion based on body measurements in the alchemy connection. The golden section does indeed go back to the greeks.

[This message has been edited by J Tiers (edited 12-19-2005).]

12-19-2005, 09:31 AM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
This value has been known since the time of the ancient greeks.[/URL]</font>

Evan, are you absolutly sure that shouldn't read....."This value has been known since the time of the ancient geeks."

Actually I'm not surprised to hear of it's importance in machine design. I feel that most artworks that I enjoy are filled with the numbers 3 and 5's. Every motif I've applied to engraving has been based on these numbers. They hold more interest than boredom. I think it's because people like to divide things up into neat packages to deal with them and 3 and 5 trip them up.

[This message has been edited by Your Old Dog (edited 12-19-2005).]

Evan
12-19-2005, 10:09 AM
"Despite all that, the Golden Ratio does have some peculiar properties. Pick two numbers at random (both positive). Add them together. Add that result to the second of the two original numbers. Add that result to the first result. Continue adding the last number to the immediately previous number. After a few terms, take the ratio of the last two numbers. If the ratio doesn't equal 1.618..., try doing it for a few more terms."

That is because the Golden mean is derived from the Fibonacci Sequence. The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, ...

In the Fibonacci Sequence the next number is derived by adding the previous two. Any sequential pair of numbers in the Fibonacci sequence form a single ratio that converges on the value of the Golden Mean.

Rustybolt
12-19-2005, 10:24 AM
This is why I like this board.Thanks Evan

BillH
12-19-2005, 10:30 AM
Very interesting, Im going to look for this golden ratio in things, I wonder if some parts of the running gear on steam locomotives used the golden ratio? Well if I ever make my own mill or cnc router, I'll know to use it, thanks.

mklotz
12-19-2005, 10:52 AM
An interesting way to arrive at the golden mean is the following.

Cut a bar (length = 1 unit) into two pieces of length x (the longer) and 1-x
(the shorter) such that the ratio of the shorter piece to the longer piece is
the same as the ratio of the longer piece to the original bar.

In mathematical terms this becomes:

(1-x)/x = x/1

or:

x^2 + x - 1 = 0

which has the solution:

x = (sqrt(5)-1)/2 (golden ratio = 0.618...)

and from which, very symmetrically, we have:

1/x = (sqrt(5)+1)/2 (golden mean = 1.618...)

Regards, Marv

E-mail: mklotz@alum.mit.edu
Home Shop Freeware - Tools for People Who Build Things
http://www.myvirtualnetwork.com/mklotz

[This message has been edited by mklotz (edited 12-19-2005).]

Evan
12-19-2005, 11:06 AM
And that is closely related to the fact that a golden rectangle can be infinitely subdivided into progressively smaller golden rectangles the corners of which form an Archimedes spiral seen everywhere in nature. Examples are the pattern of the seeds in a sunflower and the shell of the nautilus.

mklotz
12-19-2005, 11:16 AM
So, Evan, let's turn this into a really philosophical thread.

Does mathematics merely model nature or does nature follow mathematical rules
that are imposed on it somehow?

Put another way, did man create mathematics to model the real world or did he
discover the (mathematical) rules of nature that exist outside of his own
intellect?

Regards, Marv

E-mail: mklotz@alum.mit.edu
Home Shop Freeware - Tools for People Who Build Things
http://www.myvirtualnetwork.com/mklotz

Evan
12-19-2005, 11:38 AM
Mathematics is inherent in nature. We didn't invent it, we discovered it. It's everywhere in everything. It is no coincidence that machines work better if they follow these simple principles. Symmetrical machines tend to work better than asymmetrical machines. Symmetry results in balanced forces.

Machines that look good as a result of form following function (AND that follows correct design) work better than ones that don't.

Some of the most difficult problems are those needed to describe what seem very simple natural phenomena. There are many "intractable" problems that nature solves inherently.

One example is finding the point that gives the shortest possible distance to an array of other points. This is a trivial problem for many restricted cases such as the corners of a cube or points lying on the circumference of a circle. In those cases the center satisfies.

But, if the points are numerous and distributed at random distances and directions the problem rapidly becomes intractable. However, if these points are simply plotted on a grid and equal virtual rubber bands tied together with the ends "attached" to "pins" at those points the knot where the bands are joined automatically finds the point of least energy which is also the point that has the shortest possible distance to all the other points at once.

These sort of least energy solutions appear everywhere in nature.

Lew Hartswick
12-19-2005, 11:41 AM
Marv I think youre getting WAY to philosophical
for a bunch of "chip makers" :-)
...lew...

Evan
12-19-2005, 11:47 AM
Incidentally, the base of my mill is also a golden rectangle...

So is the ratio of the x-y axis travels and therefore the ratio of the length of the x-y axis ways.

[This message has been edited by Evan (edited 12-19-2005).]

mklotz
12-19-2005, 12:34 PM
I have to demur when you say that, "We didn't invent it. We discovered it."

The implication is that there is a single set of mathematical principles that
govern the natural world and somehow, almost magically, we've managed to find
the one mathematics that describes it.

Until we can prove that an entirely different 'mathematics' that describes what
we see doesn't exist, we can't say that we've discovered *the* mathematics that
is an inherent part of nature.

Personally, I think we invented 'our' mathematics in the process of solving
some early problems and then went on to 'impose' it on nature in the process
of describing later, more complex problems. The fact that we were able to do
that in an astoundingly successful way is truly mind-boggling.

For a far more scholarly treatment of this subject than I can muster, refer to
Eugene Wigner, in his treatise, "The Unreasonable Effectiveness of Mathematics
in the Natural Sciences",

I imagine the question will never be answered until we contact an alien
intelligent civilization and get to read their physics texts.

Regards, Marv

E-mail: mklotz@alum.mit.edu
Home Shop Freeware - Tools for People Who Build Things
http://www.myvirtualnetwork.com/mklotz

Evan
12-19-2005, 12:56 PM
Early development of mathematics was based on observation of the real world, not just some isolated intellectual exercise. The foundation of the pythagorean theorem was laid by Thales, a greek who discovered how to calculate the height of a pyramid by the length of it's shadow. The entire area of geometry and trigonometry is intimately related to the natural universe. I don't subscribe to the idea that we have somehow "accidentally" created a logcal system that just happens to fit what we observe. The universe is in general "ruled" by the number 2 and all of the permutations and variations that derive from it. Roots and squares are in everything, everywhere.

The absence of another system is of course not proof of anything since the only thing that no evidence proves is that you have no evidence.

pgmrdan
12-19-2005, 01:29 PM
"Does mathematics merely model nature or does nature follow mathematical rules
that are imposed on it somehow?"

We attempt to model nature using mathematics as the tool. We sometimes come fairly close but usually not close enough. We sometimes miss by a mile, figuratively speaking.

Nature follows rules and we're trying to find the mathematical models for those rules.

How long have we been trying to develop a model of weather, ocean currents, or particle motion?

After studying Euclidian geometry for years it's interesting to study two of the non-Euclidean geometries.

Physics, philosophy, and religion are more alike than many people realize. Do they merge? http://bbs.homeshopmachinist.net//smile.gif

[This message has been edited by pgmrdan (edited 12-19-2005).]

sauer38h
12-19-2005, 01:37 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
That is because the Golden mean is derived from the Fibonacci Sequence. The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, ...

In the Fibonacci Sequence the next number is derived by adding the previous two. Any sequential pair of numbers in the Fibonacci sequence form a single ratio that converges on the value of the Golden Mean.</font>

Except in discussion of this particular ratio, I have never seen the assertion that the Fibonacci sequence will converge from a start of any two random numbers. The Fibonacci sequence starts with 1. And of course all Fibonacci numbers are integers. This is not a requirement of the scenario I mentioned.

sauer38h
12-19-2005, 01:44 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
However, modes common to one axis are much less able to excite modes in the other axis. That is why it is commonly used in the design of speaker cabinets. It is a form of damping, or at the least a way to eliminate an entire series of resonances.</font>

This is like asserting that an oddly-shaped room has no standing waves. Unfortuantely, it does. They're not as easily identified as they are in a rectangular room, but you can still measure them and hear them. You don't want two independent resonances to be at the same frequency or to have the same spatial nodes, but any room ratios which aren't low-integer ratios will eliminate that. I recall a fad for making listening rooms oddly-shaped (as in, non-parallel walls), but that's not really the same thing as odd ratios.

And it still has nothing to do with damping. Moving resonances around from where they hurt to where they don't can be useful. I once designed a resonance-control device which did just that, in a very specific application. But damping it ain't.

Evan
12-19-2005, 01:45 PM
Actually in the general case the formation of a new number by adding any two positive integers and then adding that result to the previous number in the manner of the Fibonacci sequence forms a ratio that always converges on Phi (1.618...) This is a general rule and is not restricted to the Fibonacci sequence although the Fibonacci sequence converges more quickly.

Evan
12-19-2005, 01:49 PM
The difference between damping and lack of resonance is a moot point. A machine with no resonances would not need damping.

sauer38h
12-19-2005, 01:57 PM
<font face="Verdana, Arial" size="2">Originally posted by Your Old Dog:
Actually I'm not surprised to hear of it's importance in machine design.</font>

Well, I am. I've been in professional machine design for longer than I care to admit, and the only time I've heard it mentioned or considered was during a discussion of office space, and why our furniture didn't seem to fit right in the cubicles (a non-trivial concern, to be sure, but not really machine design). The problem is that the ratio, while mathematically interesting, doesn't really do anything useful. Some people think it looks nice .... or rather, some people claim that other people think it looks nice.

sauer38h
12-19-2005, 02:03 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
The difference between damping and lack of resonance is a moot point. A machine with no resonances would not need damping.</font>

Not at all. The two concepts are independent, though related in various ways. To assert that the difference is moot is like claiming that stress and strain are indistinguishable, or that differential and integral calculus are identical.

sauer38h
12-19-2005, 02:05 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
Actually in the general case the formation of a new number by adding any two positive integers and then adding that result to the previous number in the manner of the Fibonacci sequence forms a ratio that always converges on Phi (1.618...) This is a general rule and is not restricted to the Fibonacci sequence although the Fibonacci sequence converges more quickly.</font>

Which was exactly what I pointed out in my first post. It is not derived from the Fibonacci sequence - it is a far more general case.

Evan
12-19-2005, 02:14 PM
A lack of resonance will have the same effect as damping. That doesn't mean the two are the same, only the result is. I frequently use added weight to lower resonances of parts I am machining to a point that no resonance occurs. That isn't damping either but it works just as well. Here is an example. The lead bar clamped to the plate lowers the resonant frequency of the part below the vibration frequencies produced by the operation.

http://vts.bc.ca/pics/flycut2.jpg

If this can be achieved by making the assembly non-resonant in the first place the same effect is achieved.

I suggest you take it up with the professors at MIT. I don't recall exactly which course it was mentioned in but it was clearly pointed out in a section on the design of linear bearings.

Evan
12-19-2005, 02:20 PM
Yes it is derived from the Fibonacci sequence. Other such sequences are general cases of Fibonacci series.

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

[This message has been edited by Evan (edited 12-19-2005).]

dberndt
12-19-2005, 02:31 PM
Just looking around on the MIT open courseware site and though I can find what I assume to be the course Evan is talking about 2.72? I see everything but the actual lectures online? Can someone point out what I'm doing wrong here?

http://ocw.mit.edu/OcwWeb/Mechanical-Engineering/2-72Elements-of-Mechanical-DesignFall2002/CourseHome/index.htm

BillH
12-19-2005, 02:42 PM
Hmm, the class is based on groups, glad I never went to MIT.

Evan
12-19-2005, 02:56 PM

http://pergatory.mit.edu/2.007/lectures/lectures.html

12-19-2005, 04:48 PM
The teenaged neighbor girl is definitely not rectangular but her aspect is golden. She's a good student at school and sensible too. She's even nice to old farts. Maybe golden to too small an adjective to apply to someone whose future seems bright with beauty and accomplishment.

sauer38h
12-19-2005, 09:38 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
I suggest you take it up with the professors at MIT.</font>

I did, and rather a lot of it, when I was there.

sauer38h
12-19-2005, 09:43 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
Yes it is derived from the Fibonacci sequence. Other such sequences are general cases of Fibonacci series.</font>

Ah, there's the problem. You are using an unconventional definition of the word "derivation" in a mathematical context. It does not mean "somehow connected".

The only really interesting thing about the Golden Mean is that it can be derived in several independent ways, and comes out the same no matter what the derivation.

[This message has been edited by sauer38h (edited 12-19-2005).]

sauer38h
12-19-2005, 09:53 PM
<font face="Verdana, Arial" size="2">Originally posted by Evan:
I frequently use added weight to lower resonances of parts I am machining to a point that no resonance occurs.</font>

You are using an unconventional definition of "resonance". To those who do it professionally, a resonance is a property of the structure. It will show up with, say, white noise excitation. A resonant response may or may not be excited by various inputs. Narrowband excitations may miss it entirely. But a lack of a resonant response is not the same as saying that the resonance is gone. That is, "resonance" and "resonant response" are not the same animal. Think of one as cause, and one as effect.

Evan
12-20-2005, 12:19 AM
I didn't say it was gone. I said the resonance (the frequency) was lowered when the weight is added. Add weight to a guitar string and see what happens.

[This message has been edited by Evan (edited 12-20-2005).]

Yankee1
12-20-2005, 08:19 PM
Hello All,
First want to say,A very interesting subject. If the resonant response frequency is known would it not be better to design above or below it. I remember an example where 100 pairs of tuning forks were attached to a single wire a long distance apart forks 1 through 100 were on one end of the wire and their identical mates were on the other end. When for # 35 was struck only fork #35's identical mate on the other end would respond. I know in electrical systems that the inductive reactance must be equal to the capacitive reactance to become resonant at a specific frequency.I understand what Evan said about adding the lead bar which would change the inductive reactance. What interests me about this subject it ability to determine node points
as cound be applied to rifle barrel node points. I found the salt and white noise information highly interesting.
Thanks for the subject .
Regards Chuck

Evan
12-21-2005, 12:38 AM
The usual approach to dealing with resonances is to simply add mass. Hopefully that will lower the resonant frequencies below whatever excitations will be encountered. There is also damping inherent in most materials from internal friction. An exception is glass which is considered a perfectly elastic material. That's why it makes such nice bells.

I was machining (flycutting) a part a couple of nights ago and I ended up clamping a small 5 lb anvil to it to get rid of chatter. Worked great.

On my mill project I have paid special attention to eliminating the potential for chatter caused by resonances. This is especially important as it uses round rod ways for all axes. One way to help achieve this is simply using massive sections. The mill is almost entirely aluminum as I don't have the facility to machine steel or cast iron in the sizes I need. Even though aluminum and only a bench top design it will weigh around 300 lbs.

If it were steel in the same size sections it would be around 900 lbs. That isn't the only method I am using. Where the use of something like the golden ratio isn't applicable I have taken care to design with prime numbers. The X ways are 21 inches long, the Y ways are 17 inches and the Z ways are 23. The center to center distance of the two Z columns is 13 inches and the overall depth is 17" while the overall width is 23".

The point of using prime numbers for the lengths of the ways ensures that any common harmonic frequencies will be of a very high order and therefor negligible.