Using mode shapes to overcome vibration

Understanding how machine tool vibration starts is essential to controlling it when machining. That understanding starts with “stiffness,” which from an engineering point of view is the ratio between a force and the displacement caused by the force. If the force is static (not time-varying), then the displacement is constant, and the relationship between force and displacement is typically written as F = kx, where F is the static force, x is the displacement, and k is the static stiffness. (Many mechanical springs produce a linear relationship between force and displacement, and the k parameter is often called the “spring constant.”)

All physical systems, including machine tools, deflect in response to force, and thus exhibit stiffness. In machine tools, users typically want the cutting edge to exhibit a high stiffness so the cutting tool deflects as little as possible under the influence of the cutting force. To make machine tools as stiff as possible, they are typically constructed using large, solid, metal components.

However, cutting forces are generally not static. For example, as the teeth of a milling tool enter and exit a cut, the cutting force varies. The time-varying cutting forces cause time-varying deflections, or vibrations. The amplitude of the vibration depends not only on the size of the cutting force, but also on its frequency and the machine tool structure. The machine tool has certain frequencies at which it would “like” to vibrate. When the cutting force matches one of those frequencies, the resulting vibration is substantially larger than when the cutting force does not match.

Courtesy of S. Smith

Figure 1. A spindle model and two mode shapes.

The frequency-dependent relationship between time-varying force and vibration is called “dynamic stiffness.” The frequencies at which the machine tool would like to vibrate are called its natural frequencies, and most machine tools have many of them. People often encounter a natural frequency while driving a car with an unbalanced wheel, for example. The unbalanced wheel causes a variable force (once per revolution), which excites the car. At a certain speed, the frequency of excitation matches one of the natural frequencies of the car structure, and the resulting vibration can be quite large.