Analysis Of Disc Brake Squeal Using The Finite Element Method

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Page iAnalysis O f Disc Brake Squeal Using TheFinite Element MethodbyZaidi Bin Mohd RipinA thesis submitted in accordance with the requirements for the degree o fDoctor of PhilosophyDepartment o f Mechanical EngineeringUniversity o f LeedsLeeds, United KingdomSeptember 1995The candidate confirms that the w ork submitted is his own and that appropriate credithas been given where reference has been made to the work o f others

Page iiA b s tr a c tThe problem o f disc brake squeal has been examined by developing a finite elementmodel o f the coupled pad-disc system , conducting complex eigenvalue analysis andassociating unstable modes with potential squeal problem areas. A key issue in thisprocess is the representation o f the contact pressure distribution at the frictionalinterface between the disc and the pad. Non-linear contact analysis using the finiteelement model o f the pad revealed that contact is only partial at the pad-disc interfaceand that the contact pressure distribution depends on the friction coefficient, Y oung’smodulus o f the friction material and the way the applied pressure is distributed on thepad backplate. A new method is proposed in which interface contact stiffness is relatedto brake line pressure using a statistical approach based on the measured surfaceproperties o f the interface. Complex eigenvalue analysis o f the coupled pad-discsystem has shown that unstable modes exist within different ranges o f contact stiffnessthereby providing an explanation o f the effect o f varying line pressure on squeal. Thetw o most unstable modes from the analysis show good correlation with experimentalsqueal results. The coupled model is then used for parametric studies the results o fwhich indicate thathigh coefficient o f friction and uniform contactpressuredistribution increase instability whilst a trailing edge biased pressure distribution and ahigh support stiffness at the pad backplate reduce it. Limiting the disc symmetry byintroducing equispaced slots was shown to be effective in reducing instabilitiesinvolving diametral modes o f the disc with the same order o f symmetry only Othermodes were stabilised by increasing the rigidity o f the pad. The overall results suggestthat either the pad or the disc can be mainly responsible for the instability dependingon the mode thus unifying the different approaches to disc brake squeal and enablingthe most appropriate component to be targeted for squeal abatement purposes.

Page iiiT ab le o f con tentsA b s t r a c t. iiT ab le of c o n te n ts . iiiList o f F ig u re s . viiList o f T a b le s .xiiA c k n o w le d g e m e n ts. xiiiD eclaration of O r ig in a lity .xivG en eral L a y o u t a n d P resen tatio nxvC h a p t e r O n e - In tro d u c tio n1.1 Background. 11.2 Present W o rk .31.3 Thesis Organisation.4C h a p t e r Tw o - L ite r a tu r e Review2 1 Introduction. 62.2 Definitions.72.3 Squeal M odels. 82.3 1 Early w o rk . 82.3.2 Variable friction m odel.92.3.3 Sprag-slip model.142.3.4 Pin-on-disc models.162.3.4.1 Friction induced self-excited oscillations. 162.3.4.2 Geometrically constrained instability. 212.3.5 Lumped parameter models. 232.3.6 Analysis o f squeal using the finite element m ethod. 262.3.7 Other theoretical w o rk .312.4 Experimental W o rk .312 .4 .1 General experimentation techniques. 312.4.2 Experiments to determine the effect o f component dynamics on squeal. 322.4.3 Experiments on the effect o f contact stiffness on squeal.332.4.4 Experiments on the effect o f separation o f doublet modes.332.4.5 Experiments using double-pulsed laser holography. 34

Tabic o f contentsiv2.5 Discussion.362.6 Conclusions. 39Chapter Three - M ethodology O f Present Study3.1 Introduction - The Need for A N ew Approach .413 .2 Selection o f Experimental Results for Validation433.3 Operation o f a Fist-Type Sliding Calliper. 443.4 Pressure Representation.473.5 Interface Element. 503.5.1 Variable friction force with variable normal force.503.5.2 Follower friction force effect.573.5.3 Comparison with Hulten friction element. 593.6 Component Interaction. 623.7 Stability Analysis.633.8 Complex Eigenvalues Analysis Using The Finite Element M ethod.653 .9 Method o f Analysis.743 10 Summary. 75Chapter Four - Modal Analysis of the Disc and Pad4.1 Introduction. 764.2 General Method o f Analysis.774.2.1 Limitations in the analysis.774.2.2 General substructuring theory. 784.2.3 Substructuring applied to eigenvalue problem804.2 .4 Selection o f master degrees o f freedom814.3 Eigenvalue Analysis o f the Disc. 824.3.1 Disc model definition. 824.3.2 Results. 924.4 Eigenvalue Analysis o f the Pad944.4.1 Pad model definition. 944.4.2 Results. 994.5 Comparison with Experiments. 1014.5.1 The experimental results for the free-free disc. 1 2

Tabic o f contentsv4.5.2 The experimental results for the free-free pad. 1074.6 Discussion. 1104.7 Summary.113Chapter Five - Contact Pressure Analysis O f The Brake Pad5.1 Introduction. 1145.2 Method o f Analysis.1155.2.3 Finite element models. 1165.4 Results. 1 '75.4.1 Effect o f circumferential friction. 1175.4.2 Effect o f magnitude o f pressure. 1225 .4.3 Effect o f abutment constraint. 1255 .4.4 Effect o f friction material modulus. 1275.5 Discussion. 130Chapter Six - Contact Stiffness Determination6 .1 Introduction.1336.2 Method for Interface Contact Stiffness Determination. 1346.2.1 Stress-strain relationship m ethod.1346.2.2 Experimental m ethod.1356.2.3 Random process approach. 1376.2.3.1 Theory o f random process approach. 1386.3.Comparison o f the Three M ethods.1466.4 Contact Stiffness Determination by The Random Process A pproach. 1476.4.1.Measurement o f surface roughness o f the disc and the pad. 1486.4.2 Calculation o f autocorrelation function.1486.4.3 Results from the autocorrelation function o f the disc and pad sufaces. 1506.4.4 Results o f the contact stiffness calculation.1536.4.4.1 Effect o f friction material properties on contact stiffness. 1566 .4.4.2 Effect o f disc material on contact stiffness.1576 4 4.3 Effect o f disc rms roughness on contact stiffness.1586 .4.4 .4 Effect o f brake pad surface roughness on contact stiffness1596.4.5 Example o f nodal contact stiffness calculation. 160

Tabic o f contentsvi6.5 Discussion. 161C h a p t e r Seven - S tability Analysis o f th e C oupled Pad-D isc System7.1 Introduction.1647.2 M ethodology. 1657.2.1 Interpretation o f complex eigenvalues.1677.3 R e s u lts . 1707.3.1 Effect o f contact stiffness , K c .1707.3.2 Effect o f support stiffness , K s.1767 3 .3 M ode shapes.1817.3 .4 Variation o f phase angle with contact stiffness. 1867 4 Discussion. 1 )()7.4.1 Low pressure range squeal.1927.4.2 High pressure range squeal.197C h a p t e r Eight - P a ra m e tric Studies O f T h e C oupled Pad-Disc System8.1 Introduction. 2028.2 M ethodology. 2038.3 Effect o f Friction Coefficient.2038.4 Effect o f Abutment Stiffness. 2078.5 Effect o f Contact Stiffness Distribution. 2108.6 Effect o f Pad Flexibility.2158.7 Effect o f Different Disc Material.2178 8 Effect o f Disc Symmetry. 2238.9 Discussion. 232C h a p t e r Nine - Conclusions a n d R ecom m endations9 .1 Conclusions.2369.2 Recommendations for Future W o rk . 240R eferences. 241A p p en d ix I - Programme Listing for Contact Stiffness Calculation in C . 250A p p e n d ix II - Input Data For Stability Analysis Using TheANSYS Revision 5.0A Finite Element Analysis Package256

Page viiList of FiguresFigure 2.1Sliding block on plane.9Figure 2.2aBlock on sliding plane. 11Figure 2.2bKinetic friction coefficient as a linear function o f sliding speed. 1 1Figure 2.3Block on sliding plane.13Figure 2.4Sprag-slip model. 15Figure 2.5Slider model for analysis o f dynamic coefficient o f friction. 18Figure 2.6Eight degrees o f freedom lumped parameter model o f a disc brakesystem.24Figure 3.1Exploded view o f the caliper assembly. 45Figure 3.2Schematic diagram o f the operation o f the sliding caliper disc brake.46Figure 3.3The brake pad and the abutment arrangement.47Figure 3 .4Representation o f contact pressure effect using distributed contactstiffness.50Figure 3.5Interaction o f tw o masses with friction effect included.51Figure 3.6A pad model coupled to the disc with friction elements. 51Figure 3 .7The effect o f normal displacement to the variable normal force.53Figure 3 .8Simplified model o f the coupled pad-disc system and the forces actingon the interface with friction. 55Figure 3.9Non-conservative follower forces on deformable surfaces.57Figure 3.10Follower friction force as applied to the two nodes onthe same surface.59Figure 3.11Friction counter-coupled model. 60Figure 3.12Idealised pad model used by Hulten and the associated nodalrelationship with friction effect included. 61Figure 3 .13Discretization o f the domain by 3 nodes triangular elements. 66Figure 3 .14Three dimensional body.67Figure 3.15Schematic diagram o f a body with distributed massshowing velocity components.71Figure 4 .

The problem of disc brake squeal has been examined by developing a finite element model of the coupled pad-disc system , conducting complex eigenvalue analysis and associating unstable modes with potential squeal problem areas. A key issue in this process is the representation of the contact pressure distribution at the frictional .

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