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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.)
7
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
10
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
11
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
12
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
13
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
14
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
15
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 −−
16
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
17
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.
18
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
19
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
20
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. . . . .
21
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
24
Preliminary Results from Linear Approach
0.0Hz 31.11Hz
74.58Hz50.52Hz
25
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
27
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
28
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 !