**Author**Cutting Tool Engineering

The selection of machining parameters (axial DOC, radial DOC, chip load, spindle speed and surface speed) is constrained by many factors. These include tool life, cutting edge strength, workpiece surface quality, chatter avoidance and chip evacuation.

Two fundamental limitations on machining performance are the spindle motor’s available torque and power. The specifications of the torque and power limitations are often available in torque and power curves from machine tool builders. Understanding those curves starts with understanding the terms. The words force, torque, work and power are often used loosely, but have specific technical definitions.

Force causes an object with mass to accelerate in a linear direction, and the acceleration is directly proportional to the force (F = ma). Linear motion (x) has units of meters. Velocity (v) is the positional rate of change and has units of meters/second. Acceleration (a) is the rate of change of velocity and has units of meters/second^{2}.

The unit of force in the metric system is the newton. There are approximately 4.45 newtons per pound of force (for those who prefer the English system of measurement). The unit of mass in the metric system is the kilogram. Because F = ma, 1 newton is dimensionally the same as 1 kg-m/sec.^{2}.

If an object rotates around an axis (like a spindle), its angular position (θ), typically measured in radians, is analogous to position in the case of linear motion. A radian is a dimensionless measure of angle, and there are 2π radians per rotation. The rate of change of angular position is angular velocity (ω), measured in radians/second, or just 1/second. The rate of change of angular velocity is angular acceleration (α), measured in radians/second^{2}, or just 1/second.

The rotational counterpart to mass is moment of inertia (I), which has units of newton-meter-second^{2}, or equivalently kilograms-meters^{2}. Similar to it being more difficult to accelerate a massive object, the concept of moment of inertia expresses that it is more difficult to cause rotational acceleration for an object with a large mass or where the mass is positioned far from the axis of rotation.

In the same way that force causes linear acceleration of a mass, torque causes angular acceleration of a mass moment of inertia (T = Iα). Torque has units of newton-meters. You might think of it like a hand on a wrench. Torque can be increased by pushing harder or by increasing the length of the wrench.

In a spindle motor, torque is caused by a current flowing in a loop in a magnetic field. The magnetic field might be created by permanent magnets or in an electromagnet. There is a limit on the available torque caused by the strength of the magnetic field and the maximum current the wire can carry, for example.

In linear motion, the work performed by a force is given by force times distance, and it has units of newton-meters, or joules. In rotational motion, work is given by torque times rotation, or newton-meters times dimensionless radians, so the work in the rotational case also has units of joules.

Power is work per unit of time. In linear motion, power is given by force times velocity, in newton-meters/second, or watts. In rotational motion, power is given by torque times rotational velocity (P = Tω), again in watts.

The torque and power available in a spindle changes as the spindle speed changes. The graphs illustrating those changes are the torque curve and the power curve. Many varieties in design exist, but there are two basic forms of the torque and power curves: constant torque and constant torque followed by constant power.

Figure 1 shows the torque and power curves for a constant-torque motor. The torque is constant throughout the speed range of the spindle, and the power linearly increases as the speed increases. In this case, the spindle motor produces about 1kW per thousand rpm (5kW at 5,000 rpm, 10 kW at 10,000 rpm and so on). This design is common for high-speed spindles.

*Courtesy of S. Smith*

**Figure 1. The torque curve and power curve for a constant-torque spindle.**

Figure 2 shows the torque and power curves for a spindle motor with constant torque followed by constant power. This kind of design is more common for spindles intended for use in the lower speed range, and, typically, the torque is higher. These spindles reach their maximum power at a lower speed, then torque is reduced.

**Figure 2. The torque curve and power curve for a spindle with constant torque followed by constant power.**

In both cases, the average power consumed in the cut must be less than the available power on the spindle motor. The power is approximately proportional to the metal-removal rate, so the power curve places a fundamental limit on the mrr. In both instances, there is little power available at low spindle speeds. While the instantaneous power may exceed the limit (for a short time the spindle acts like a flywheel), a cutting power greater than the available spindle power causes the spindle to stall. NC programmers should compare the programmed mrr and power against the power curve when writing a program. 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.*

### Related Glossary Terms

- chatter
#### chatter

Condition of vibration involving the machine, workpiece and cutting tool. Once this condition arises, it is often self-sustaining until the problem is corrected. Chatter can be identified when lines or grooves appear at regular intervals in the workpiece. These lines or grooves are caused by the teeth of the cutter as they vibrate in and out of the workpiece and their spacing depends on the frequency of vibration.

- metal-removal rate
#### metal-removal rate

Rate at which metal is removed from an unfinished part, measured in cubic inches or cubic centimeters per minute.

- numerical control ( NC)
#### numerical control ( NC)

Any controlled equipment that allows an operator to program its movement by entering a series of coded numbers and symbols. See CNC, computer numerical control; DNC, direct numerical control.