*Author’s Note: I would like to thank Dr. Timber Yuen. The analysis I did below was directly learned in his Machine Dynamics course as part of my degree in Manufacturing. Dr Yuen’s practical problem solving teaching is a refreshing and needed approach where many engineering students are ‘drowning’ in math and not able to solve real world problems.*

The Sieg X2 Mini Mill is know for the wet noodle characteristics of the column. In particular the tilting column variation of the X2 (the most common variation) has extreme chatter and vibration issues when trying to take anything more than very small depths of cut in steel. The reputation is such that Little Machine Shop has removed the Sieg’s tilting option on its mills in order to improve rigidity.

The other day I was single point fly cutting some tall plates with the Sieg X2 (no, I didn’t strip the plastic gears … yet) and noticed the column vibration was very significant.

I decided I should investigate what was going on. Information on additional column support on the X2 is very plentiful around the web and I could have simply manufactured some form of column brace based on the modifications others have done. But I wanted to learn more about the vibration issue before I went directly to a solution. I though, hey that mill column looks a lot like a simple spring – mass – damper system. The spring, well that’s the column, the mass – that’s the spindle housing and motor, and the damping – well there shouldn’t be much.

Firstly, I wanted to figure out what the natural frequency the column vibration. How do you do this? Most times an accelerometer would be mounted to the column. I didn’t have an accelerometer handy. Or did I? I started to think about the smart phone I owned. Most smart phones have accelerometers built in. I downloaded some software that retrieved data from the accelerometer, attached my phone to the column (zip ties work – electrical tape works well but leaves sticky glue on your screen!) and proceeded to strike the column with a dead blow hammer on the spindle housing in the Y direction and plot the response.

I plotted the response in Excel. The output from the accelerometer was in m/s². I used the phone’s Z axis output only.

Now is probably a good time to comment a little about the sample rate from the accelerometer. My cell phone is an inexpensive Alcatel Pixi. The maximum sample rate from the accelerometer I could achieve is 100 Hz. This is why the above chart looks choppy. I would have preferred something higher – say 500 Hz, but the data is good enough to make some general observations.

From the graph I found the period of the vibration to be 0.03941 seconds. The inverse of the period is the frequency, which is 25.374 Hz. 25.374 Hz is 1522 rpm. From this point on some math is involved, you can view it in the spreadsheet posted below. If you want me to detail the math used, send me an email. The mass was approximated using the mass of the spindle head and 0.23 x the mass of the column. Using the data the following are calculated:

Damping Factor | 0.110491 | |

Natural Frequency | 25.53079 | Hz |

Natural Frequency | 160.4147 | rad/s |

Weight | 45 | lbs |

Mass | 20.45455 | kg |

K (spring rate) | 526354 | N/m |

C (damping) | 12.65826 | kg/s |

The low amount of damping is expected. The low K value had me scratching my head a bit so I decided to calculate what the K value should be based on a fixed cantilever beam. I estimated the moment of inertia using a square tube. Again, I’ll spare the detailed math.

Ixx (Moment of Inertia) | 1.68 | in^4 |

Ixx | 699268.795 | mm^4 |

Ixx | 6.99269E-07 | m^4 |

Length | 17 | in |

Length | 431.8 | mm |

Length | 0.4318 | m |

Young’s Modulus | 12000000 | psi |

Young’s Modulus | 82737120000 | N/m^2 |

Calculated K | 2155846.764 | N/m |

Weight | 45 | lbs |

Mass | 20.45454545 | kg |

Calculated Natural Frequency | 324.6489688 | rad/s |

51.66948815 | Hz | |

3100.169289 | rpm |

Whoa! That’s a lot higher than what we measured! What does this mean? Something must be adding to the ‘springiness’ of the system. I concur with most around the web that the large titling interface isn’t very good.

Now before we go into improving the stiffness of the system, we should ask ourselves why we are doing it. When I was single point flycutting, I was fly cutting at an RPM of around 500 – 600 rpm. This is about 10 Hz. Our measured natural frequency of the system is 25 Hz. This condition where we are applying a load and taking it off is type of rotating unbalance problem. The frequency ratio, simply the operating frequency divided by the natural frequency, gives an indication how close you are to resonance, and helps you figure out what the machine response will be. In this case the frequency ratio, or r, is 0.4. What does this mean? Well avoiding all the math, a quick chart for rotational unbalance, (from Dr. Yuen’s spreadsheets – thank you!) gives a more clear picture:

At 10 Hz or r = 0.4 and zeta = 0.1 we are approaching the sharp peak where r = 1. That’s really bad! And the force chart shows the same story:

Since I really can’t do anything about the damping in the system I want to try to increase the stiffness of the system and thus operate at a lower frequency ratio, r. Since the calculated stiffness should be closer to 50 Hz, I decided to fabricate a plate and mount it on the column, as well as add additional support for the base. If you want additional pictures or drawings of the bracket send me an email and I’ll try to get them to you.

The bracket allows the mill to be trammed in the X direction, but removes the titling ability. I never really used it anyway. When I made the bracket I scraped it as flat as I could. After installing I trammed the mill in the X and Y axis to within .0005″ (hence the shims). I remounted my cell phone to the mill and determined the new natural frequency.

As you can see the data is becoming more choppy. This is due to the increased frequency and the 100 Hz limitation by my phone. From the graph I found the period of the vibration to be 0.022 seconds with the inverse or frequency to be 45 Hz. That’s better! The rest of the math shakes down below.

Damping Factor | 0.089214 | |

Natural Frequency | 45.5025 | Hz |

Natural Frequency | 285.9006 | rad/s |

Weight | 45 | lbs |

Mass | 20.45455 | kg |

K (spring rate) | 1671937 | N/m |

C (damping) | 13.64477 | kg/s |

The damping factor stays about the same (small change is due to experimental error!).

What type of improvement will you see with this? Take my fly cutting scenario. The new frequency ratio is 10 Hz / 45 Hz = 0.2. From the rotating unbalance chart above if you move from r = 0.4 (where we were) to r = 0.2 the displacement decreases by a factor of over 5! That is a pretty large reduction in displacement.

In conclusion, as many know already, the standard titling arrangement with 36 mm nut is not the best setup. Adding a bracket or additional support is required. At least now I have a quantifiable reason why.

You can download the spreadsheet if you want to: X2VibrationAnalysis.

Well done and well written. I balanced a small electric motor rotor by trial and error and from that experience I share your interest in vibration, vibration analysis and balancing and would like to see the math.

TIA Carl

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Hey Carl,

I’ve had a lot of interest on the math so I’m working on a detailed math post. I have all my notes, I just have to figure out how to get it ‘presentable’ on the screen. Thanks for reading, and I appreciate your comment!

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For your calculation, why didnt you choose peak to peak values? Your (Time, Acceleration) values seem to be (6.3870311, .766) for the first set and (6.4264097, -0.306) for second set. however for for x1 and x2 you used .919 and .459 to compute your damping value. Why?

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