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MACHINE-TOOL CHATTER and STICK-SLIP MODELLING
For an illustration of the effect of machine-tool chatter visit
http://www.youtube.com/watch?v=2KceXIKPWEo
The following example is taken from: G James, Modern Engineering Mathematics (4th edition) p 871.
The system depicted in figure 1 represents a simple model for the phenomenon of machine tool chatter, in which vibrations occur due to a sequence of stick-slip motions between the
machine-tool and the piece being worked.
A block of mass M (kg) is connected by a spring to a rigid support and rests on a belt being
driven at a constant speed v (m/s). The block moves with the belt until the extension x in the
spring is sufficient for the resulting restoring force to overcome the static friction between the
block and belt. The block then slides relative to the belt until the slip velocity between the
two surfaces in contact is zero, when static friction takes effect again.
x
M v
Figure 1: Simple Stick-Slip Model
When the block starts to slip it is moving → with speed v. The magnitude of the restoring force
increases and the block decelerates until it has zero velocity. The direction of sliding motion is
then reversed; the block begins to slide ← and as the extension in the spring decreases so does
the restoring force. When the restoring force balances the sliding friction force the acceleration
becomes zero. The block continues to move ← slowing down under the action of an increasing
positive force (friction force > restoring force) until the block again has zero velocity. The block
then slides → with a velocity that increases until it matches that of the belt. At this moment
the block ‘sticks’ again.