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

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