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.

Each of the natural frequencies of the machine tool has a characteristic deflection pattern associated with it. When a structure vibrates in one of its natural frequencies, it vibrates with the corresponding pattern, which is called a “mode shape.” Each natural frequency has its own mode shape. Figure 1 shows a simple model of a spindle, including bearings, shaft, toolholder and cutting tool. Two of the characteristic mode shapes are shown schematically below the model. The solid black line shows the spindle in its position without deformation, and the red line shows the characteristic deformation. 

Although mode shapes include the entire structure, a mode shape is often dominated by the deflection of a particular element of the structure. The top mode shape in the figure could be characterized as a spindle bending mode, while the bottom mode shape could be characterized as a cutting tool mode. The mode shapes of machine tools can be calculated using a finite element model, as was the case in this example, or they can be experimentally measured using modal analysis techniques.

Knowledge of mode shapes can solve machining vibration problems. The frequency of a problem vibration is often close to a structure’s natural frequency, and it is possible to change the frequency of excitation or modify the structure so the problem mode is stiffer.

For example, if the spindle bending mode in Figure 1 caused a vibration problem, stiffening the bearings would not be much help. In the problem mode shape, there is little deflection in the bearings. A better solution would be to increase the diameter of the spindle shaft—stiffening it—or change the spacing between the bearings. 

If the problem vibration was related to the cutting tool mode in Figure 1, increasing the shaft diameter would not be much help, because the spindle shaft barely bends in this tool mode. The tool does not bend much in this mode either. The reason for the vibration is mostly the flexibility of the connection between the tool and toolholder. In this case, switching to a stiffer connection style would help.

Machine tool structures are complicated assemblies of many different components, including shafts, housings, bearings and springs. When these components are assembled, the resulting machine has many natural frequencies, each with its own characteristic mode shape. Measuring or modeling the natural frequencies and mode shapes is a useful step in avoiding or eliminating vibration problems. CTE

About the Author: Dr. Scott Smith is a professor and chair of the Department of Mechanical Engineering at the William States Lee College of Engineering, University of North Carolina at Charlotte, specializing in machine tool structural dynamics. Contact him via e-mail at kssmith@uncc.edu.