Estimate effect on predictions

Brake Squeal: Theory and ExperimentTore Butlin

tb267@cam.ac.uk jw12@eng.cam.ac.ukProf. Jim Woodhousewith

Introduction

Vehicle brake squeal is well known to be a twitchy phenomenon that is still not fully understood. Validating predictive models is hindered by a variety of factors:

=

F

N

GG

GG

v

u

2221

1211

1

1

‘disc’ ‘brake’

NF

F

u1

v1

v2

u2

dis

c ro

tatio

n

r

kn

=

F

N

HH

HH

v

u

2221

1211

2

2

Unstable if any solutions have a negative imaginary part

01

)()()()( 12111211 =++++++ HGωεωµωωµω ik

HHGGn

Model Stability Criteria

Generate initiation

F = [µ0+εVsliding]N

Repeat under same conditions Repeat under varying conditions

stable

unstable

Compare

This research uses a linear model of a single point sliding contact to describe a pin-on-disc test rig.

Transfer function matrices (G & H) relate displacements (u & v) to forces (N & F)

Assume N & F related by linearised velocity dependent friction law:

ConclusionsA first order perturbation analysis provides a useful estimate of the effect of uncertainties on predictions;

A new test method allows some degree of confidence that measurements should be predicted by linear theory;

The tests allow the question of sensitivity to be explored experimentally;

Sensitivity of predictions to measurement uncertainties;

Difficulty in obtaining repeatable results;

Determining whether a given occurrence of squeal would be expected to be predicted by the model.

01

1 22220 =

+++

tkHGNiωε

Gives two characteristic equations:

Determine effect on predictions

Pre

dic

tio

nT

est

Quantify uncertainty

40 consecutive measurements of one peak of a pin mode. Standard deviation as percentage of the mean: natural frequency, 0.01%; modal amplitude, 13%; damping factor, 0.5%.

Vary modal parameters representatively and calculate complex roots of characteristic equation for every combination: 13824 root evaluations required to generate above figure.

Estimate effect of uncertainty using a first order perturbation analysis. Solid lines give estimated deviation from nominal prediction: 1 root evaluation required to generate above figure.

1 2 4 8 1 2 4 8

clockwise anticlockwise

sym

met

ric

disc speed(rpm)

perturbation

0g1g

4g14g

0g1g0g1g

asym

met

ric

Increase N0 until squeal triggered and record initiation. Fit complex pole from frequency and rate of growth if well approximated by exponential curve.

Cycle N0 to obtain as many initiations as possible within 40 second test period. Record range of parameters for post-processing.

Repeat test for a combination of parameters. Repeat sequence for 20 days. Approximately 6,000 useful squeal initiations recorded.

Comparison of measured unstable poles (x) with prediction taking into account uncertainties (|). Good agreement in general but raises some questions.

The tests quantify repeatability over time;

Large dataset has potential to allow statistically significant conclusions to be drawn.

Bac

kg

rou

nd

Res

ult

s

Future WorkCareful analysis of data to begin to answer the following questions:

Do the model predictions agree with measurements?

Can high sensitivity to small perturbations be observed in test data?

Over what time-scales are tests repeatable?

920 922 924 926

42

41

40

39

38

37

36

35

34

33

|H1

1| (

dB)

0 5000 10000 15000

200

150

100

50

0

-50

-100

-150

-200

imag

200

150

100

50

0

-50

-100

-150

-200

imag

0 5000 10000 15000

0 5000 10000 15000real (Hz)

200

150

100

50

0

-50

-100

-150

-200

imag

4.51 4.515 4.52

u 2

0 10 20 30 40

u 2

time (s)

real (Hz) real (Hz)frequency (Hz)

time (s)