Vol. 16 ( ), No JASCOME INDIVIDUAL IDENTIFICATION OF CONTACT PRESSURE DISTRIBUTION OF INNER AND OUTER BRAKE PAD USING INVERSE ANALYS

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1 Vol. 16 ( ), No JASCOME INDIVIDUAL IDENTIFICATION OF CONTACT PRESSURE DISTRIBUTION OF INNER AND OUTER BRAKE PAD USING INVERSE ANALYSIS 1) 2) 3), 4), 5) Masanori INOUE, Kenji AMAYA, Yuki ONISHI, Yuki NAKAGAWA and Koichi HATORI 1) ( , m inoue@akebono-brake.com) 2) ( W8-36, kamaya@a.mei.titech.ac.jp) 3) ( W8-36, yonishi@a.mei.titech.ac.jp) 4) ( W8-36, ynakagawa@a.mei.titech.ac.jp) 5) ( , k-hatori@akebono-brake.com) A novel method to identify each inner and outer side of contact pressure distribution between brake pads and the rotor is proposed. The proposed method identifies unknown contact pressure distribution from minute rotor deformation measured with sensors integrated in the rotor. Since the rotor deformation is expected to be minute, a displacement magnifier is developed. In order to identify inner and outer side of contact pressure distribution individually, geometries of the rotor and the displacement magnifier are proposed. Tikhonov regularization is employed in order to overcome ill-posedness. The verification of the proposed method is numerically performed using FEM. Identified contact pressure distribution is well-correlated to exact contact pressure distribution and feasibility of this method is confirmed. Key Words: Inverse Analysis, Contact Pressure Distribution, Automotive Brake System 1. - (1) (2) (3) (4) Kim (5)(6)

Fig.1,,. Fig.1 1 Knuckle Caliper fingers Rotor Caliper Outer pad Rotor Inner pad Piston Hub Cross section of the caliper Fig.

1 3.2 3.3 Rotor deformation Point load response function Observation equation 10 Fig.4 (8) 2 campaniform sensillum (9) Fig.5 Fig.5 1 2.

Sensors measure rotor deformation along z axis. 10 mm 16.3 mm Rotor deformation Strain gauge First lever φ16 mm Second lever Fulcrums Fig.

2 Fig.1,,. Fig.1 1 Knuckle Caliper fingers Rotor Caliper Outer pad Rotor Inner pad Piston Hub Cross section of the caliper Fig. 1 Schematics of the general disc brake system for automotives and the cross section of the brake caliper Fig Rotor deformation Point load response function Observation equation 10 Fig.4 (8) 2 campaniform sensillum (9) Fig.5 Fig S/N Rotation direction Contact areas Sensor hole Sensors θ Optical encoder R z Data logger Fig. 3 Schematic of sensors location. Sensors measure rotor deformation along z axis. 10 mm 16.3 mm Rotor deformation Strain gauge First lever φ16 mm Second lever Fulcrums Fig. 4 Schematics of the displacement magnifier. Tikhonov regularization Unknown contact pressure distribution Fig. 2 Flow diagram of the proposed method. Effort Strain gauge Rotor deformation First lever 3.1. (5)(6) Fig.3 S/N Fulcrums Second lever Rotor deformation Fig. 5 Schematic of the deformed displacement magnifier (5)(6)

3 Fig.6 2 Fig. 6 Hole for inner sensor Holes to reduce stiffness Fin Hole for outer sensor Outer surface Inner surface Schematic of the stiffness manipulation to emphasize sensitivity of the specific surface of the rotor. Fig.8 Fig.9 Fig.8 Fig Fig. 8 Rotor deformation [pm/pa] deg. Inner sensor Outer sensor 8 deg Positon of the point load [deg.] An example of point load response functions. Point load is applied on the inner surface. Fig.7 Fig.7(a) Fig.7(b) Fig.7(c) Rotor deformation [pm/pa] Inner sensor Outer sensor -16 deg. 8 deg Position of the point load [deg.] Contact point Rotor Point load dx dy Outer surface Fig. 9 An example of point load response functions. Point load is applied on the outer surface. Contact points Inner surface Displacement magnifier Undeformed shape (a) (b) (c) Fig. 7 Schematic of the modified displacement magnifier to emphasize sensitivity of the specific surface of the rotor. Fig.7(b) dx Fig.7(c) dy dy dx Fig.8 Fig (1) y = Hx (1) y = [y o,..., y, y ors o ir 1,..., y irsi (2) y orj = [y orj θ 1,..., y orj θ n y irk = [y irk θ 1,..., y irk θ n x = [x o,..., x orpo, x i,..., x irpi (3) x orj = [x orj θ 1,..., x orj θ mo x irk = [x irk θ 1,..., x irk θ mi [ ] hoo h oi H = h io h ii (4)

4 Fig.10 s o s i R y (2) y o y i (y o y i ) (y ors o y ir si ) j k n Fig.10 R p o p i θ m o m i x (3) x o x i (R 1 ) (R po R pi ) j k θ Fig. 10 Calculated areas Discretized contact areas θ R Schematic of discretized identification areas. H (4) 4 h oo h io h oi h ii H (3) x H Fig.8 Fig.9 Toeplitz 3.4. Tikhonov (7) Tikhonov (5) Tikhonov (6) J = Hx y 2 + α x x 2 (5) ˆx = arg min J(x) = (H T H + αi) 1 (H T y + α x) (6) x α I x Ri α L-curve (10) GPa 0.3 Fig.10 Fig.10 R 5 (s o = s i = 5) Fig.10 θ (n = 85) 5 R Fig.10 R 5 (p o = p i = 5) θ (m o = m i = 34) ( RPM) (NASTRAN SOL101) Fig.10 R (10 3 mm) 1 m/pa Fig.11 Fig.11 R R5( ) (5)(6) 1 %

5 Rotor deformation [pm/pa] R2 R3 R4 R Position of the point load [deg.] α m 2 /Pa 2 R5 Fig.16 Fig.20 Fig.21 Fig. 11 Simulated point load response functions of all inner sensors. Point load is applied on inner surface at R Fig.12 Fig.13 Fig.14 m Fig. 15 Result of L-curve method. Fig. 12 (a) Exact contact pressure distribution applied on outer(a) and inner(b) surface. (b) Fig. 16 on. Rotor deformation [μm] Fig. 13 Rotor deformation [μm] Fig Outer - Inner - R2 R3 R4 R Simulated output of inner sensors. Outer Inner R2 R3 R4 R Simulated output of outer sensors. Fig.15 L-curve α m 2 /Pa 2 m, Pa m/pa (5) Fig. 17 on R2. Fig. 18 on R3.

6 Fig. 19 on R4. Fig. 20 on R5. Fig. 21 (a) Contour plot of identified contact pressure distribution on outer(a) and inner(b) surface. 5. Tikhonov (b) (1) Söderberg, A. and Andersson, S. Simulation of wear and contact pressure distribution at the pad-to-rotor interface in a disc brake using general purpose finite element analysis software Wear 267(2009) pp (2) Magnier, V., Brunel, J. F. and Dufrénoy, P. Impact of contact stiffness heterogeneities on friction-induced vibration International Journal of Solids and Structures 51(2014) pp (3) Brimacombe, J. M., Wilson, D. R., Hodgson, A. J., Ho, K. C. T. and Anglin, C. Effect of calibration method on Tekscan sensor accuracy Journal of Biomechanical Engineering 131(2008) pp (4) Kim, J. Y., Kim, J., Kim, Y. M., Jeong, W. and Cho, H. An improvement of brake squeal CAE model considering dynamic contact pressure distribution SAE International Journal of Passenger Cars- Mechanical Systems 8(2015) pp (5) ( ) (6) 29 (2016). (7) Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V. and Yagola, A. G. Numerical methods for the solution of ill-posed problems Mathematics and Its Applications 4(2007) pp (8) Jensen, B. D. and Howell, L. L. Identification of compliant pseudo-rigid-body four-link mechanism configurations resulting in bistable behavior Journal of Mechanical Design 125(2004) pp (9) Vincent, J. F. V., Clift, S. E. and Menon, C. Biomimetics of campaniform sensilla: Measuring Strain from the Deformation of Holes Journal of Bionic Engineering 328(1995) pp (10) Hansen, P. C. Analysis of discrete ill-posed problems by means of the L-curve SIAM Review 34(1992) pp

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