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日期：2024-08-14 08:48

School of Mathematics, Statistics & Physics

ENG2031 - Mathematical Modelling

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.

The aim is to investigate the block oscillations. In particular how the amplitude A and

period T depend on M and v. What other parameters are important?

ENG2031 Chatter March 2024

(1) A Numerical Study

The MATLAB script chatter.m provides a computational tool for this study.

Use this script, with the values allocated from the following table (using the last digit of

your student number ), to investigate how A and T vary with both M and v.

As a starting point consider M (kg) and v (m/s) in the ranges

1 6 M 6 10, 0.1 6 v 6 1.

Take g = 9.81 (m/s2

) for the constant of gravitational acceleration.

last digit µs µd k (kN/m)

0 0.9 0.6 1

1 0.8 0.5 1

2 0.7 0.4 1

3 0.6 0.3 1

4 0.5 0.2 1

5 0.9 0.6 2

6 0.8 0.5 2

7 0.7 0.4 2

8 0.6 0.3 2

9 0.5 0.2 2

Produce graphs showing how A varies with M (for fixed v) and with v (for fixed M).

Similarly for T.

Comment on the qualitative behaviour of the system, as illustrated by your graphs.

Is this behaviour what you expected?

Can you interpret the behaviour in terms of the underlying mechanics?

In presenting your numerical results try and summarize the data as concisely as possible;

avoid too many separate graphs!

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Computational studies like this can provide valuable insights into the behaviour of com plicated systems.

Sometimes it is possible to analyse a model using more theoretical (mathematical) tools.

For this problem we can use some of the mathematical techniques taught in ENG1001.

ENG2031 Chatter March 2024

(2) A Mathematical Study

Usually, in performing a mathematical analysis, it is desirable to keep the working as

general as possible. So, in what follows, do not substitute numerical values for the

parameters M and v (or µs

, µd and k). The aim is to develop a general formula for the

amplitude A.

(2.1) Use Coulomb’s law of static friction to write down a formula for the displacement x0

of the block at the moment it begins to slip. What other law is used in constructing

this formula?

(2.2) Coulomb’s law of dynamic friction provides a simple model for the dry-friction force

experienced by the block when sliding. Use this law, and Newton’s second law, to

formulate a second-order differential equation that models the displacement of the

block during sliding.

(2.3) Using your ENG1001 lecture notes find the general solution of your differential

equation, and from this determine the amplitude A and frequency ω of the oscilla tions during the slipping motion.

Note 1: To calculate the arbitrary constants in your general solution you will need to

specify initial conditions. Set t = 0 at the moment the block begins to slip.

What are the values of x(0) and ˙x(0)?

Note 2: You should find the following formula useful

a cos(ωt) + b sin(ωt) = A cos(ωt − φ) with A =

√

a

2 + b

2

, tan(φ) = b

a

.

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Your report should set out clearly the detailed working for this mathematical analysis.

(3) A Comparative Study

The motion of the block (when slipping) has been modelled using a differential equation.

This equation has been solved in two different ways; numerically (using chatter.m) and

mathematically (using ENG1001.notes).

Do the two methods predict the same results for A?

What can you say about T and ω?

Your report should include a comparison of results obtained from parts (1) and (2).

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ENG2031 Chatter March 2024