Estimate effect on predictions Brake Squeal: Theory and Experiment Tore Butlin [email protected] [email protected] Prof. Jim Woodhouse with 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 G G G G v u 22 21 12 11 1 1 ‘disc’ ‘brake’ N F F u 1 v 1 v 2 u 2 disc rotation r k n = F N H H H H v u 22 21 12 11 2 2 Unstable if any solutions have a negative imaginary part 0 1 ) ( ) ( ) ( ) ( 12 11 12 11 = + + + + + + H G ϖε ϖ μ ϖ ϖ μ ϖ i k H H G G n Model Stability Criteria Generate initiation F = [μ 0 +εV sliding ]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: Conclusions A 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. 0 1 1 22 22 0 = + + + t k H G N iϖε Gives two characteristic equations: Determine effect on predictions Prediction Test 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 symmetric disc speed (rpm) perturbation 0g 1g 4g 14g 0g 1g 0g 1g asymmetric Increase N 0 until squeal triggered and record initiation. Fit complex pole from frequency and rate of growth if well approximated by exponential curve. Cycle N 0 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. Background Results Future Work Careful 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 |H 11 | (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 15000 real (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)