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Math and machines, the Golden Mean

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  • Math and machines, the Golden Mean

    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.

    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...ring/index.htm
    Free software for calculating bolt circles and similar: Click Here

  • #2
    Thanks Evan. This is the most informative and interesting OT post I've seen here in a good long while.

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    • #3
      <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.

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      • #4
        Thanks very much, Evan. That's extremely interesting. I suppose I must now pursue the MIT course, like I need another project

        ------------------
        Leigh W3NLB
        Leigh
        The entire content of this post is copyright by, and is the sole property of, the author. No assignment
        of title nor right of publication shall ensue from presentation of this material on any computer site.

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        • #5
          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

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          • #6
            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).

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            • #7
              "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.
              Free software for calculating bolt circles and similar: Click Here

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              • #8
                <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).]
                1601

                Keep eye on ball.
                Hashim Khan

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                • #9
                  <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).]
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                  It is true, there is nothing free about freedom, don't be so quick to give it away.

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                  • #10
                    "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.
                    Free software for calculating bolt circles and similar: Click Here

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                    • #11
                      This is why I like this board.Thanks Evan

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                      • #12
                        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.

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                        • #13
                          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: [email protected]
                          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).]
                          Regards, Marv

                          Home Shop Freeware - Tools for People Who Build Things
                          http://www.myvirtualnetwork.com/mklotz

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                          • #14
                            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.
                            Free software for calculating bolt circles and similar: Click Here

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                            • #15
                              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: [email protected]
                              Home Shop Freeware - Tools for People Who Build Things
                              http://www.myvirtualnetwork.com/mklotz
                              Regards, Marv

                              Home Shop Freeware - Tools for People Who Build Things
                              http://www.myvirtualnetwork.com/mklotz

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