**Author details**

J. G. Detoni\* , F. Impinna, N. Amati and A. Tonoli

\*Address all correspondence to: joaquimd@gmail.com

Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Turin, Italy

**Chapter 4**

**Free Vibration Analysis of Spinning Spindles:**

Seyed M. Hashemi and Omar Gaber

http://dx.doi.org/10.5772/51174

**1. Introduction**

Additional information is available at the end of the chapter

**A Calibrated Dynamic Stiffness Matrix Method**

The booming aerospace industry and high levels of competition has forced companies to constantly look for ways to optimize their machining processes. Cycle time, which is the time it takes to machine a certain part, has been a major concern at various Industries deal‐ ing with manufacturing of airframe parts and subassemblies. When trying to machine a part as quick as possible, spindle speed or metal removal rates are no longer the limiting factor; it is the chatter that occurs during the machining process. Chatter is defined as self-excited vi‐ brations between the tool and the work piece. A tight surface tolerance is usually required of a machined part. These self-excited vibrations leave wave patterns inscribed on the part and threaten to ruin it, as its surface tolerances are not met. Money lost due to the destructive nature of chatter, ruining the tools, parts and possibly the machine, has driven a lot of re‐ search into determining mathematical equations for the modeling and prediction of chatter. It is well established that chatter is directly linked to the natural frequency of the cutting

The first mention of chatter can be credited to Taylor [18], but it wasn't until 1946 that Ar‐ nold [3] conducted the first comprehensive investigation into it. His experiments were con‐ ducted on the turning process. He theorized that the machine could be modelled as a simple oscillator, and that the force on the tool decreased as the speed of the tool increased with relation to the work piece. In his equations, the proportionality constant of the speed of the tool to the force was subtracted from the damping value of the machine; when the propor‐ tionality constant increases beyond the damping value of the machine, negative damping occurs causing chatter. This was later challenged by Gurney and Tobias who theorized the now widely accepted belief that chatter is caused by wave patterns traced onto the surface

> © 2012 Hashemi and Gaber; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Hashemi and Gaber; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

system, which includes the spindle, shaft, tool and hold combination.
