Estimate effect on predictions
Brake Squeal: Theory and ExperimentTore Butlin
[email protected] [email protected]. 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?
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