STOCHASTIC SIMULATION APPROACH FOR … Lifetime Determination of commercial vehicle braking systems 9. 2015 ... Weibull Probability function of lifetime Weibull Full data –ML-Estimation

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Institut für Maschinenelemente, Universität Stuttgart

Technologie Transfer Initiative, Universität Stuttgart

M. Sc. Martin Dazer

Dipl.-Ing. Stefan Kemmler

Prof. Dr.-Ing. Bernd Bertsche

Institute of Machine Components

Reliability Engineering

Technology Transfer Initiative

Dr.-Ing. Tobias Leopold

Dipl.-Ing Jens Fricke

Knorr-Bremse Group

Systeme für Nutzfahrzeuge GmbH

Rob

ust

Se

nsitiv

e

Virtual

Lifetime

Determination

Durability (t-t0) 106 LC

Fa

ilu

re P

rob

ab

ilit

y F

(t)

Sh

ap

e p

ara

me

ter

b

TechnologieTransferInitiative

STOCHASTIC SIMULATION APPROACH FOR

REALISTIC LIFETIME FORECAST AND ASSURANCE

OF COMMERCIAL VEHICLE BRAKING SYSTEMS

12. Weimar Optimization and Stochastic Days 2015

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Research Cooperation

2

Rob

ust

Se

nsitiv

e

Virtual

Lifetime

Determination

Durability (t-t0) 106 LC

Fa

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(t)

Sh

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TechnologieTransferInitiative

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Agenda

1. Motivation

2. Application example: brake caliper

3. Simulation process

4. Design of Experiments

5. Result evaluation

6. Summary and Outlook

3

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1. MOTIVATION

Virtual Lifetime Determination of

commercial vehicle braking systems

4

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Identification of specific product

properties:

Failure mechanism

Lowest and longest cycle times

Failure distribution

Control of correct target-

engineering by requirements

1. Motivation

5

Real durability tests

Systematic and

unsystematic

scattering of

lifetime

tmin

tmax

Scattering produces a

characteristic lifetime distribution

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1. Motivation

6

Load cycles N (log)Str

ess A

mp

litu

de

(lo

g)

Occurring nominal load

amplitude on component

50%Endurable nominal load

amplitude in operation

Nominal load cycles Nnom (log)

cumulativedensity

Str

ess A

mp

litu

de

(lo

g)

10%

90%

10%

50%

90%

Scatter band of the load

amplitudes occur on the

component

Scatter band of endurable load

amplitudes in operation

Motivation:

Realistic lifetime prediction through

stochastic simulation of stress and

strength

Load cycles N (log)

F(t

) Scatter band of load cycles N

Stochastic

Simulation

Source: Bertsche 2004

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1. Motivation

7

Input

Variability

Output

VariabilitySimulation

System

Parameter variations of the product lead to variations in product properties as a

result of internal correlations.

This can cause functional or structural failure and the deterioration of product

quality.

Aim: Realistic forecast of time to failure

Lifetime

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1. Motivation

8

Parametric CAD-Model

Mapping the caliper geometry

Changes in the geometry

FEA-Simulation model

Mapping of damage-related effects

Stress amplitudes

Strain amplitudes

Parameter study

Automatic simulation of

the established DOE

Statistical evaluation

Source: Dynardo 2015

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2. APPLICATION EXAMPLE

BRAKE CALIPER



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10

Brake caliper

Application example:

Brake caliper

Pressure

cylinder dummy

Brake discBrake pads

2. Application example Brake caliper

Piston

Piston

Have to withstand high loads in

case of overload

Therefore tested with high loads

Failures in Low-Cycle-Fatigue

range (LCF)

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,150,

10,

09

99

90

8070605040

30

20

10

5

3

2

1

Standardized lifetime

Pe

rce

nt

Shape 2,89116

Scale 0,465464

A v erage 0,414998

Stdv . 0,155908

Median 0,410044

Statistics

Probability plot for lifetime

Full data - ML-Estimation

Weibull

Probability function of lifetime

Weibull

Full data – ML-Estimation

Standardized lifetime

Pe

rce

nt

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3. SIMULATION PROCESS



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Loadcase simulation

Load

definition

Lifetime distribution

Carrying

Capacity

Material

properties

Werkstoff-

charakterisierung

Bauteil-

geometrie

Beanspruchungs-

Zeitfunktion

Zyklische Spannungs

Dehnungskurve

Dehnungswöhlerlinie

FEM-Analyse Schädigungsparameter

Schädigungsparameter-

wöhlerlinie

SchädigungsrechnungKlassierung des

Lastkollekivs

Bauteillebensdauer

N

εA

elastisch

plastisch

Gesamt

PSWTPB

σA, σm,

ε A

F N

N

εA

120000116000112000108000

12

10

8

6

4

2

0

Belastung

Hä

ufi

gke

it

Belastungsverteilung

La

sta

mp

litu

de

n1 n2 n3N

n1 n2 n3

La

sta

mp

litu

de

Stress / Strain

hysteresis

Definition of claim and the associated

amplitudes Characterization of the relevant

material properties

Deterioration

Stochastic Simulation for variance determination

σa

εa

P =

10 %

50 %

90 %

εa

N

Pü =

10 %50 %

90 %

10 %

50 %

90 %

Pü =


The level of

deterioration related

stress and strain

amplitudes

Stress / Strain Wöhler

curve

Identification and

consideration of non-

homogeneous loading

states

Strength

Tension Pressure Shear Bending Torsion

Characterization of the

strength model

Source: maschinenbau-wissen 2015Source: Haibach 2005

Source: SOFEA 2015

Source: Haibach 2005

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3. Simulation process – load definition

Real force flow in the

brake caliper

Simulated force flow in

the brake caliper

Bolt pretension for lower

FE-calculation time

Contact surface

Contact surface

Clamping

force

system boundary

Clamping force causes

high FE-calculation time

system boundary

Same damage

state

System

simulation to

determine

deterioration

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3. Simulation process – load definition

14

There are variations in the clamping force occuring through

hysteresis effects

Load spectrum is determined from test reports

Best-Fit: Lognormal distribution

1,151,11,0510,95

99

95

80

50

20

5

1

Standardized clamping force

Pe

rce

nt

Test for goodness of fit

Lognormal

A D = 0,242

p-v alue = 0,759

Probability function for clamping force

Lognormal - 95%-KI

1,111,081,051,020,990,96

10

8

6

4

2

0


De

nsit

iy

Shape 0,03424

Scale 0,03219

N 50

Lognormal

Densitiy function of clamping force


Distribution of clamping force

LognormalProbability function of clamping force

Lognormal – 95 % KI


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Loadcase simulation

Load

definition


Carrying

capacity

Material

properties

Werkstoff-

charakterisierung

Bauteil-

geometrie

Beanspruchungs-

Zeitfunktion

Zyklische Spannungs

Dehnungskurve




wöhlerlinie


Lastkollekivs

Bauteillebensdauer

N

εA

elastisch

plastisch

Gesamt

PSWTPB

σA, σm,

ε A

F N

N

εA

120000116000112000108000

12

10

8

6

4

2

0

Belastung

Hä

ufi

gke

it


La

sta

mp

litu

de

n1 n2 n3N

n1 n2 n3

La

sta

mp

litu

de

Stress / Strain

hysteresis



material properties

Deterioration


σa

εa

P =

10 %

50 %

90 %

εa

N

Pü =

10 %50 %

90 %

10 %

50 %

90 %

Pü =



curve

Identification and


homogeneous loading

states

Strength



strength model

The level of


stress and strain

amplitudes



Source: SOFEA 2015

Source: maschinenbau-wissen 2015

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0

100

200

300

400

500

600

700

800

900

1000

0 1 2 3 4 5 6 7 8 9 10

Str

ess σ

[Mpa

]

Strain ε [%]

Material behavior GJS-600-6

Cyclic First load First discharge

3. Simulation process – material properties

16

Re = 370 MPa Significantly higher cyclic

yield strength Re‘

Mathematical modeling using

Ramberg-Osgood-equation:

𝜀𝑎,𝑡 =𝜎𝑎𝐸+

𝜎𝑎𝐾

1𝑛

R‘e = 500 MPa

Material shows clear

solidifying behaviorRe = first load yield strength

Re‘ = cyclic yield strength

General Ramberg-Osgood-

equation:


𝜎𝑎𝐾

1𝑛

Modelling the fist load curve:

𝐾 = 1160𝑛 = 0,19

Modelling the first discharge

curve:

𝐾′′ = 1300𝑛′′ = 0,17

Modelling the cyclic curve:

𝐾′ = 924,9𝑛′ = 0,1

Nominal material behavior

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Re = 370 MPa

Deutlich höhere zyklische

Streckgrenze Re‘

Mathematische Modellierung

mit Ramberg-Osgood-

Gleichung

R‘e = 500 MPa

Werkstoff zeigt deutliches

verfestigendes VerhaltenRe = zügige Streckgrenze

Re‘ = zyklische Streckgrenze

0

100

200

300

400

500

600

700

800

900

1000

0 1 2 3 4 5 6 7 8 9 10

Str

ess σ

[Mpa

]

Strain ε [%]

Material behavior GJS-600-6

First load

How can the scattering

behavior mapped

mathematically?

General Ramberg-Osgood-

equation:


𝜎𝑎𝐾

1𝑛

Modelling the fist load curve:

𝐾 = 1160𝑛 = 0,19

Modelling the first discharge

curve:

𝐾′′ = 1300𝑛′′ = 0,17

Modelling the cyclic curve:

𝐾′ = 924,9𝑛′ = 0,1

Nominal material behavior

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σa

εaεg,min εg,max

Rm

Rm,min

Rm,max

ε0,2

Rp0,2

Rp0,2;min

Rp0,2;max

ε0,2,min ε0,2,max


Only data from the static tensile test available

Mathematical derivation of the scattering of n and k

18

Dis

trib

utio

n o

fsm

alle

ste

xtr

em

e

va

lue

s

Distribution of smallest extreme

values

Determination of the

limits and the

distribution of n, and k:

Using two interpolation

points solving

Ramberg Osgood

iteratively for n and k

𝜀0,2𝑖 =𝑅𝑝0,2𝑖𝐸

+𝑅𝑝0,2𝑖𝐾

1𝑛

𝜀𝑔𝑗 =𝑅𝑚𝑗

𝐸+

𝑅𝑚𝑗

𝐾

1𝑛

σa

εaεgεg,min εg,max

Rm

Rm,min

Rm,max

ε0,2

Rp0,2

Rp0,2;min

Rp0,2;max

ε0,2,min ε0,2,max

σa

εaεgεg,min εg,max

Rm

Rm,min

Rm,max

ε0,2

Rp0,2

Rp0,2;min

Rp0,2;max

ε0,2,min ε0,2,max

Generation of

point clouds using

monte carlo

simulation with m

samplings

1

2

Loop to j, i = m

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Determination of the

variance of K and n using

1000 Monte Carlo

Samplings

Best fit for initial loading K

and n with Weibull

distribution

0,19650,19500,19350,19200,19050,18900,1875

180

160

140

120

100

80

60

40

20

0

n

De

nsit

y

Location 164,4

Scale 0,1952

N 1000

Weibull

Density function n

0,10400,10240,10080,09920,09760,0960

140

120

100

80

60

40

20

0

n'

De

nsit

y

Location 83,37

Scale 0,1026

N 1000

Weibull

Density function n'

0,17700,17550,17400,17250,17100,16950,16800,1665

180

160

140

120

100

80

60

40

20

0

n''

De

nsit

y

Location 143,5

Scale 0,1746

N 1000

Weibull

Density function n''

132813241320131613121308

120

100

80

60

40

20

0

K''

De

nsit

y

Location 435,0

Scale 1326

N 1000

Weibull

Density function K''

1180117611721168116411601156

140

120

100

80

60

40

20

0

K

De

nsit

y

Location 389,9

Scale 1174

N 1000

Weibull

Density function K

940936932928924920916

140

120

100

80

60

40

20

0

K'

De

nsit

y

Location 310,5

Scale 936,2

N 1000

Weibull

Density function K'

Proportionate transfer

of variance at K‘, K‘‘,

n‘ and n‘‘

Density function of n

Density function of n‘

Density function of n‘‘ Density function of K‘‘

Density function of K‘

Density function of K

K‘‘n‘‘

n‘

n K

K‘

Den

sity

De

nsity

Den

sity

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Loadcase simulation

Load

definition


Carrying

Capacity

Material

properties

Werkstoff-

charakterisierung

Bauteil-

geometrie

Beanspruchungs-

Zeitfunktion

Zyklische Spannungs

Dehnungskurve




wöhlerlinie


Lastkollekivs

Bauteillebensdauer

N

εA

elastisch

plastisch

Gesamt

PSWTPB

σA, σm,

ε A

F N

N

εA

120000116000112000108000

12

10

8

6

4

2

0

Belastung

Hä

ufi

gke

it


La

sta

mp

litu

de

n1 n2 n3N

n1 n2 n3

La

sta

mp

litu

de

Stress / Strain

hysteresis



material properties

Deterioration


σa

εa

P =

10 %

50 %

90 %

εa

N

Pü =

10 %50 %

90 %

10 %

50 %

90 %

Pü =



curve

Identification and


homogeneous loading

states

Strength



strength model

The level of


stress and strain

amplitudes



Source: SOFEA 2015

Source: maschinenbau-wissen 2015

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3. Simulation process – deterioration

21

Ört

lich

e K

erb

sp

an

nu

ng

σ [M

Pa]

Örtliche Kerbdehnung ε [%]

First load simulation:


𝜎𝑎𝐾

1𝑛

Correct mapping of the location

of the hysteresis

Mean stress influence

First discharge load simulation -

Masing behavior:


𝜎𝑎𝐾′′

1𝑛′′

𝜎𝑚

Cyclic operation load

simulation:


𝜎𝑎𝐾′

1𝑛′

Abstract representation

Formation of residual

compressive stress

Localnotc

hstr

ess σ

[MPa]

Local notch strain ε [%]

Accelerated Representative Simulation Process (ARSP)

for deterioration calculation at constant load

Nearly elastic

operating load

hysteresis

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Loadcase simulation

Load

definition


Carrying

Capacity

Material

properties

Werkstoff-

charakterisierung

Bauteil-

geometrie

Beanspruchungs-

Zeitfunktion

Zyklische Spannungs

Dehnungskurve




wöhlerlinie


Lastkollekivs

Bauteillebensdauer

N

εA

elastisch

plastisch

Gesamt

PSWTPB

σA, σm,

ε A

F N

N

εA

120000116000112000108000

12

10

8

6

4

2

0

Belastung

Hä

ufi

gke

it


La

sta

mp

litu

de

n1 n2 n3N

n1 n2 n3

La

sta

mp

litu

de

Stress / Strain

hysteresis



material properties

Deterioration


σa

εa

P =

10 %

50 %

90 %

εa

N

Pü =

10 %50 %

90 %

10 %

50 %

90 %

Pü =



curve

Identification and


homogeneous loading

states

Strength



strength model

The level of


stress and strain

amplitudes


Source: SOFEA 2015

Source: maschinenbau-wissen 2015Source: Haibach 2005

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σ a

N

PFailure

90 %50 %10 %

Stress

Level 1

Stress

Level 2

Endurance

Level

k

3. Simulation process – strength

23

εa

N

b2

ε'f,1ε'f,2

σ'f,2 /E

σ'f,1 /Eb1

c2c1

According to Haibach:

modeling of stress levels with

normal distribution

using Database of FKM

Determination of scattering of k:

Monte Carlo Simulation using 3

interpolation points

Determination of scattering of σ‘f , ε‘f ,

b and c: Using uniform distribution

between σ‘f1 and ε‘f1

Modeled with:

𝑁 = 𝑁𝐷 ∙𝜎

𝜎𝐷

−𝑘

Only 2 nominal literature

sources. No large database

Uniform Distribution

Scattering of

endurance strength

ND

Modeled with:

𝜀𝑎 =𝜎𝑓′

𝐸2𝑁 𝑏 + 𝜀𝑓

′ 2𝑁 𝑐

Stress Wöhler curve Strain Wöhler curve

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3. Simulation process – strength

24

-4,5-4,8-5,1-5,4-5,7-6,0-6,3

120

100

80

60

40

20

0

k

De

nsit

y

Average -5,456

Stdv 0,3371

N 1000

Normal

Density function of k

18000001600000140000012000001000000800000

100

80

60

40

20

0

ND

De

nsit

y

Average 1281073

Stdv 193278

N 1000

Normal

Density function of ND

Result for variance of k and

ND of Stress Wöhler curve

Result for variance of σ‘f , ε‘f , b and c of Strain Wöhler curve

812811810809808807806

40

30

20

10

0

Sig(f)

De

nsit

y

Density function of sigf

-0,0654-0,0660-0,0666-0,0672-0,0678-0,0684-0,0690

50

40

30

20

10

0

b

De

nsit

y

Density function of b

Compatibility condition

according to Haibach:

𝑛′ =𝑏

𝑐; 𝑘′ =

𝜎′𝑓

(𝜀′𝑓)𝑛′

Using uniform

distribution

Already

determined

Completely

determined

Density function of k

Normal

Density function of ND

Normal

ND

k

Den

sity

Den

sity

Density function of σf

Uniform

Density function of b

Uniform

b

σf

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4. DESIGN OF EXPERIMENTS



25

X3

X2

X1

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Spacefilling

latinhypercube sampling

Optimal cover of the

parameter space

Design Parameter

Using 3σ Normal Distribution

Material Model

Using Distribution of smallest

extreme values

Strength Model

Stress: Using Normal Distribution

Strain: Using Uniform Distribution

Load Spectra

Using Lognormal Distribution and

ARSP

Source: Siebertz 2010

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aze

r


27

Spacefilling

latinhypercube sampling

Optimal cover of the

parameter space

Load Spectra

Using Lognormal Distribution and ARSP

1,111,081,051,020,990,96

10

8

6

4

2

0


De

nsit

iy

Location 0,03424

Scale 0,03219

N 50

Lognormal

Densitiy function of clamping force

Ört

lich

e K

erb

sp

an

nu

ng

σ [M

Pa]


Localnotc

hstr

ess σ[M

Pa]

Local notchstrain ε [%]

Ört

lich

e K

erb

sp

an

nu

ng

σ [M

Pa]


Local notchstrain ε [%]

Integration of the load

spectrum through

parametric study

Higher

compressive

residual stress

Localnotc

hstr

ess σ[M

Pa]

Source: Siebertz 2010


Distribution of clamping force

Lognormal

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Pro

f. D

r.-I

ng

. B

. B

ert

sch

e; M

. S

c. M

art

in D

aze

r

Sending deterioration

outputs from FE-

Simulation


28

Sampling of

Design Parameter

Material model

Sampling of

Strain strength model

Calculation of lifetime

Sampling of

Stress strength model

Calculation of lifetime

Sending deterioration outputs

from FE-Simulation

Workflow

Step 1

Workflow Step 2

Wo

rkflo

w S

tep

2Workflow

Step 3

Workflow

Step 3

Sending back

results to

Robustness Analysis for

Postprocessing

Workflow

Step 4

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5. RESULT EVALUATION



29

06

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Pro

f. D

r.-I

ng

. B

. B

ert

sch

e; M

. S

c. M

art

in D

aze

r

5. Result Evaluation

30

10,10,01

99,999

90807060504030

20

10

5

32

1

0,1

Standardized time to failure

Pe

rce

nt

FKM approx.

PSWT approx.

Real data

Variable

Probability plot of FKM; PSWT; Real data


Weibull

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Pro

f. D

r.-I

ng

. B

. B

ert

sch

e; M

. S

c. M

art

in D

aze

r

10,10,01

99,999

90807060504030

20

10

5

32

1

0,1

Standardized time to failure

Pe

rce

nt

FKM approx.

PSWT approx.

Real data

Variable

Probability plot of FKM; PSWT; Real data


Weibull

5. Result Evaluation

31

Very good approximation

using stress based FKM

algorithm

Nearley same Weibull shape

parameter b

Small deviation in

characteristic life time T

Larger deviation on upper levels

using strain based PSWT

Pronounced Differences to conservative side

for PSWT Additionally insufficient fit

FKM approximation slightly overestimates the

real life time

Reason for this behavior:

Increasing plastic component of

strain amplitude

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6. SUMMARY AND OUTLOOK



32

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Pro

f. D

r.-I

ng

. B

. B

ert

sch

e; M

. S

c. M

art

in D

aze

r

6. Summary and Outlook

Mapping of systematic uncertainties of

Design Parameter

Material model

Strength model

Load

Development of a Accelerated Representative Simulation Process

Succesfull development of a virtual lifetime distribution

Check of reproducibility

Future investigations to the influence of plastic compontents in

strain amplitudes for deterioration calculation

Investigations to the strain based strength model

33

Increase of maturity level in early design

stages!

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15



THANK YOU FOR YOUR

ATTENTION!

[email protected]

[email protected]

34

Documents

STOCHASTIC SIMULATION APPROACH FOR … Lifetime Determination of commercial vehicle braking systems 9. 2015 ... Weibull Probability function of lifetime Weibull Full data –ML-Estimation