Brake Squeal Analysis in MD NastranPast, Present and Future

Gaowen Ye, Hemant Patel, Joe MaronickMSC Software

2

Agenda

Introduction

Traditional Linear Approach

Present Enhanced Linear Approach

Future Linear/Nonlinear Approach

Concluding Remarks

3

What is Brake Squeal?Friction-induced coupling modes,dynamically unstableCreate noise, commonly known as Brake Squeal

Why Brake Squeal Analysis?To predict the existence of unstable modes or undamped rootsModify and optimize structures and material properties to remove unstable modes & eliminate brake squeal

Introduction

4

Introduction

Complex Eigensolutions – TheoryEquation of motion

where p = α + iω and α = real part of solution ω = imaginary part of solution

Stable/unstable modesα < 0 Stable mode

α > 0 Unstable mode

Damping coefficient

g −2α / |ω| = 2ξ≈

Imaginary

Real

Hessenberg Method

Lanczos Method

Inverse Power Method[ ]{ } )1(02 =++ uKBpMp

5

IntroductionAdvancements in Brake Squeal Analysis

Past: linear approachTraditional MSC Nastran approach Many successful applications over number of yearsComplex eigenvalue solution sequences

Direct Method: small problemModal Method: recommended

Present: enhanced linear approachLeverage contact approach + complex eigensolutionMultiple runs

MD Nastran R3: linear/nonlinear approachAnalysis chainingSingle runNonlinear effects

6

Past: Traditional Linear ApproachA Simple Friction Mechanism

Fpy

N

Fry=-FpyK

Rotor

Pad

Y

Z

Assumptions:The speed of the sliding surface is assumedto be much less than the speed of the traveling vibrational waves. Therefore, the elements representing the surface may be limited to small motions and the traveling wave effects are ignored.Pure sliding friction is assumed. The magnitude of the pad vibration may be very small for the onset of the unstable mode. The analysis will be invalid when the vibrationalvelocities exceed the surface velocity.A static preload is assumed to be large enough to maintain full contact on the pad surface. The frictional coefficient is assumed to be constant. (However, it could be varied over the contact region.)

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Past: Traditional Linear Approach

A large spring K is used to calculate the normal compression force N and frictional forces F between pad and rotor

Where : a large spring: normal compression force: normal displacements of pad and rotor: frictional forces on the pad and rotor: friction coefficient

N

)()()(

3NFF2uuKN

rypy

rzpz

μ=−=−−=

rypy FF ,μ

K

Fpy

N

Fry K Rotor

Pad

rzpz uu ,

8

Past: Traditional Linear ApproachMatrix form of frictional forces

The matrix terms in Eq.(4) are input to the model for each contact point directly as DMIG data (K2PP)resulting in an unsymmetricfrictional matrix

Fpy

N

Fry K Rotor

Pad

)4(11

11

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

−=

⎭⎬⎫

⎩⎨⎧

rz

pz

ry

py

uu

KFF

μ

9

Past: Traditional Linear ApproachSpecial modeling details & Nastran inputs for brake squeal analysis

(refer to sec. 5.2 of advanced dynamic analysis user’s guide for details)

Congruent meshes are needed to calculate the contact forces between pads and rotors using dummy scalar spring elements “ELASi” at all contact pointsK2PP case control and many DMIG entries are needed to incorporate unsymmetric frictional stiffness matrix

VERY TEDIOUS and TIME CONSUMING MODELINGDays or weeks are not uncommon

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Present: Enhanced Linear ApproachGreatly reduces pains & cost of model preparation from weeks/days to hours/minutes:

No need to define spring elements between pads & disksNo need to input DMIG matrix corresponding to the unsymmetric frictional stiffness matrixNo need to use congruent meshes between pads & disks

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Formulation (in case of 2D)Noncongruent meshes using 3DBODY Contact approachBased on the same existing assumptions + an uniform pressure distribution assumptionGenerate internally the dummy grids and springs to measure normal forces automaticallyEquivalent to the combination of using MPC and springs

a1- a

u5,u6,f5,f6

Y

u3,u4,f3,f4

u1,u2,f1,f2

k

X

( )( )

( )( )

( ) ( ) ( )( ) ( ) ( )

)(5

ffffff

uuuuuu

ka10ka1a0ka10ka10ka1a0ka10ka1a0ka0ak0

ka1a0ka0ak0ka10ak0k0

ka10ak0k0

6

5

4

3

2

1

6

5

4

3

2

1

2

2

2

2

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−−−

−−−−−−−

−−−

μμμ

μμμ

μμμ

pad grid

rotor grids

dummy grid and spring

Present: Enhanced Linear Approach

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1st Run:Use noncongruent meshesUse BODY CONTACT approach for contact definition between pads and rotors No need to define the normal springs between pads and rotors No need to define the frictional stiffness via DMIG entriesGenerate data for both 2nd and 3rd runs

2nd RunGenerate the spring forces and output them in DMIG format Generate the unsymmetric frictional stiffness matrix and output them in DMIG formatGenerate MPC entries associated with “GLUED” parts

3nd Run:Generate a complex eigenvalute job data and include the DMIG and MPC entries generated in 2nd runPerform complex eigenvalue analysis with data that includes the DMIG and MPC entries generated in 2nd run

Present: Enhanced Linear Approach

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Present: A Simple Demo ExampleA Simple Procedure Demo Model :

Noncongruent meshes for disk, pads, piston, etc.Glued contact

Pistons are glued to pads

Pads are glued to pistons but are in contact with disk

Disk is in contact with pads

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C O M P L E X E I G E N V A L U E S U M M A R YROOT EXTRACTION EIGENVALUE FREQUENCY DAMPINGNO. ORDER (REAL) (IMAG) (CYCLES) COEFFICIENT1 1 0.0 0.0 0.0 0.02 2 0.0 5.273472E+01 8.392991E+00 0.03 3 0.0 5.768647E+01 9.181087E+00 0.04 4 0.0 8.836539E+01 1.406379E+01 0.05 5 0.0 1.051991E+02 1.674296E+01 0.06 6 0.0 1.072297E+02 1.706614E+01 0.07 7 1.897113E+00 1.954785E+02 3.111137E+01 -1.940994E-028 8 -1.897113E+00 1.954785E+02 3.111137E+01 1.940994E-029 10 -1.978217E+00 3.174125E+02 5.051777E+01 1.246464E-02

10 9 1.978217E+00 3.174125E+02 5.051777E+01 -1.246464E-0211 11 0.0 3.935520E+02 6.263575E+01 0.012 12 0.0 4.004129E+02 6.372769E+01 0.013 13 0.0 4.080416E+02 6.494183E+01 0.014 15 -2.100486E+00 4.685824E+02 7.457720E+01 8.965280E-0315 14 2.100486E+00 4.685824E+02 7.457720E+01 -8.965280E-0316 16 0.0 5.598912E+02 8.910944E+01 0.017 17 0.0 6.120831E+02 9.741605E+01 0.018 18 0.0 6.156371E+02 9.798169E+01 0.019 19 0.0 6.248976E+02 9.945554E+01 0.020 20 0.0 6.439939E+02 1.024948E+02 0.0

Present: A Simple Demo Example (Con’t)

List of Complex Eigenvalues

Unstable Modes

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Present: Validation of A Real Model

A real brake model was used to validate the enhanced approach against traditional approach

Case 1Congruent meshesConstant spring coefficients at all contact grids

Case 2Congruent meshesVariable spring coefficients corresponding to the contact area of each grid

Case 3Noncongruent meshesVariable spring coefficients corresponding to the contact area of each grid

Image model .,)1(,)1(,,,, 22 etckaakaakakaakak μμμ −−

.,)1(,)1(,,,, 22 etckaakaakakaakak μμμ −−

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Case 1Exact the same results as that from traditional approach

Case 2Maximum relative error of frequencies is 0.024%

Case 3

0.002%09,003009,0030105

0.033%-0.0069 8,931192.3 -0.0065 8,934181.8 104

0.033%0.0069 8,931-192.3 0.0065 8,934-181.8 103

・・・

0.068%-0.0294 6,440593.9 -0.0293 6,445593.3 70

0.068%0.0294 6,440-593.9 0.0293 6,445-593.3 69

0.038%06,512006,515068

・・・

0.438%0419.12 00417.30 04

0.068%0264.62 00264.44 03

0.010%037.42 0037.42 02

00.0028 000.0004 01

(%)DampingFreq.RealDampingFreq.Real

Relative ErrorCase 3Results by Traditional ApproachMode No.

Present: Validation of A Real Model

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MD R3: Linear/Nonlinear Approach

Goal:Make full use of 3D body contact approach like current enhanced linear approach for easy modeling

Also take various nonlinear effects into accountNonlinear approach

Preserve current enhanced linear approach Linear approach

Improve performance through tightly chaining nonlinear analysis and complex eigenvalue analysis in a single solution in advanced integrated nonlinear solution…Thermal-structural coupling analysis (R3+)Etc.

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MD R3: Nonlinear Approach

Steps of nonlinear approachPerform nonlinear analysis that takes various nonlinear effects such as contact, differential stiffness, etc. into accountCalculate the complex eigenvalue analysis based on the updated matrices of a nonlinearly deformed structure configuration

Linear perturbation analysis by incorporating modal as well as direct complex eigenvalue analysis techniques It is as part of Analysis Chaining Solution Diagram

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Analysis Chaining Diagram

Nonlinear + Linear Analyses Chaining

SUBCASE 1STEP 1

ANALYSIS=NLSTATICS…

STEP 2ANALYSIS=NLSTATICS

…STEP 3

ANALYSIS=MODESNLIC STEP 1, LOADFAC, 0.2……

Nonlinear Analyses ChainingSUBCASE 10

STEP 1ANALYSIS = NLSTATICSLOAD = 1

…STEP 2

ANALYSIS=NLTRANDLOAD= 3…

SUBCASE 20STEP 1

ANALYSIS=NLSTATICS…

STEP 2ANALYSIS=NLSTACTICS…

STEP 3NLIC 1ANALYSIS=NLTRAN

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Nonlinear Analyses ChainingA Simplified Door Opening Model The first step of nonlinear static analysis is followed by a second step of nonlinear transient analysis

SOL 400CENDSUBCASE 1

BCONTACT = 888STEP 100ANALYSIS = NLSTATNLPARM = 1

STEP 200ANALYSIS = NLTRANTSTEPNL= 2

BEGIN BULKPARAM LGDISP 1NLPARM 1 200 FNT 25 YESTSTEPNL 2 1000 0.005 10BCTABLE 888 1

SLAVE 5MASTER 4

BCBODY 4 3D DEFORM 4 0BSURF 4 31 32 33 34 35 36 37. . . . .BCBODY 5 3D DEFORM 5 0BSURF 5 1 2 3 4 5 6 7. . . . .

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Nonlinear + Linear Analyses Chaining

Nonlinear Static Deformation

First Mode

Second Mode Third Mode

Rotating Fan Blade Model: normal modes under Pressure Load + Rotational Force

22

Simple Brake Squeal Test ModelThe same simple model was used to validate the new procedures of both linear approach and nonlinear approach

Pistons are glued to pads

Pads are glued to piston but are in contact with disk

Disk is in contact with pads

23

Preliminary Results from Linear ApproachC O M P L E X E I G E N V A L U E S U M M A R Y

ROOT EXTRACTION EIGENVALUE FREQUENCY DAMPINGNO. ORDER (REAL) (IMAG) (CYCLES) COEFFICIENT1 1 0.0 0.0 0.0 0.02 2 0.0 5.273496E+01 8.393030E+00 0.03 3 0.0 5.768645E+01 9.181084E+00 0.04 4 0.0 8.836546E+01 1.406380E+01 0.05 5 0.0 1.051992E+02 1.674297E+01 0.06 6 0.0 1.072297E+02 1.706613E+01 0.07 8 -1.897122E+00 1.954785E+02 3.111138E+01 1.941003E-028 7 1.897122E+00 1.954785E+02 3.111138E+01 -1.941003E-029 10 -1.978225E+00 3.174125E+02 5.051777E+01 1.246469E-02

10 9 1.978225E+00 3.174125E+02 5.051777E+01 -1.246469E-0211 11 0.0 3.935520E+02 6.263575E+01 0.012 12 0.0 4.004129E+02 6.372769E+01 0.013 13 0.0 4.080415E+02 6.494183E+01 0.014 15 -2.100551E+00 4.685824E+02 7.457720E+01 8.965555E-0315 14 2.100551E+00 4.685824E+02 7.457720E+01 -8.965555E-0316 16 0.0 5.598912E+02 8.910944E+01 0.017 17 0.0 6.120831E+02 9.741605E+01 0.018 18 0.0 6.156371E+02 9.798169E+01 0.019 19 0.0 6.248975E+02 9.945553E+01 0.020 20 0.0 6.439938E+02 1.024948E+02 0.0

The results of new linear approach are very close to present enhanced linear approachThe implementations of linear approach were confirmed

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Preliminary Results from Linear Approach

0.0Hz 31.11Hz

74.58Hz50.52Hz

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Preliminary Results from Nonlinear Approach

C O M P L E X E I G E N V A L U E S U M M A R YROOT EXTRACTION EIGENVALUE FREQUENCY DAMPINGNO. ORDER (REAL) (IMAG) (CYCLES) COEFFICIENT

1 1 0.0 1.283076E+00 2.042079E-01 0.02 2 0.0 4.377497E+01 6.967003E+00 0.03 3 0.0 5.511817E+01 8.772329E+00 0.04 4 0.0 8.595654E+01 1.368041E+01 0.05 6 -1.369358E+00 1.029338E+02 1.638243E+01 2.660657E-026 5 1.369358E+00 1.029338E+02 1.638243E+01 -2.660657E-027 8 -3.490013E+00 1.905830E+02 3.033222E+01 3.662460E-028 7 3.490013E+00 1.905830E+02 3.033222E+01 -3.662460E-029 9 0.0 2.964131E+02 4.717562E+01 0.0

10 10 0.0 3.098099E+02 4.930778E+01 0.011 11 0.0 3.180490E+02 5.061906E+01 0.012 12 0.0 3.586938E+02 5.708788E+01 0.013 13 0.0 3.928318E+02 6.252112E+01 0.014 14 0.0 3.999804E+02 6.365886E+01 0.015 16 -3.269191E+00 4.614810E+02 7.344699E+01 1.416826E-0216 15 3.269191E+00 4.614810E+02 7.344699E+01 -1.416826E-0217 17 0.0 5.396171E+02 8.588273E+01 0.018 18 0.0 5.927913E+02 9.434566E+01 0.019 19 0.0 6.113345E+02 9.729690E+01 0.020 20 0.0 6.161508E+02 9.806345E+01 0.0

The implementations of nonlinear approach were also confirmed

26

Preliminary Results from Nonlinear Approach

0.2Hz 16.38Hz

73.45Hz30.33Hz

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Concluding RemarksBrake Squeal Analysis by MD Nastran

Crawl-walk-run approachProven linear approach: simple but quick and effective alternative studiesNonlinear approach: detailed study to investigate the effects ofvarious nonlinearities

MD Analyses of braking system by MD SolutionsMD Adams:

Motion: Operational Effect, Brake Torque Variation, etc. Coupled motion-structural: obtain and export loads for FEM analyses like brake squeal analysis, etc.

MD Nastran:Linear-nonlinear FEA: import loads from MD Adams for Deformation, Vibration, Brake Squeal, Optimization, Thermal-Structural Coupling, Etc.

MD Nastran

Modes

• Linearized model• Modal coords.

MD Adams

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Anti-Lock Braking, Vibration, Brake Torque Variation, and FEA

Loads Transferring

MD Simulations for Braking Analysis

Deformation, Vibration,

Brake Squeal, Optimization,

Thermal-Structural Coupling

Thank You !