Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

1

Mer om Rainflowcykler

Kurs i Lastanalys för Utmattning3-4 Oktober 2011

SP Bygg och Mekanik

Pär Johannesson

Par.Johannesson@sp.se

Nivå-korsningar

Last-spektrum

Rainflow-matris

Rainflow Cycle Counting: Hysteresis and rate independence

Rainflow counting reflects

– Masing rule and

– Material memory rules

and counts load events leading to local hysteresis cycles.

strain

stress

hanging

standing

Hysteresis model(cyclic stress-strain curve, Masing and Memory rules)

PJ/2011-09-29 2

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

2

From Outer Load to Local Load

• Rainflow cycle counting is motivated by considering local stresses and strains (hysteresis models), but often applied to outer loads.

• When and why do the local arguments apply to outer loads?

• If σ(t) = ϕ(L(t))(*), then load cycles and local stress-strain cycles open and

close at the same time (e.g. (*) holds for forces acting on a stiff component

and stresses calculated from linear FEA and Neuber’s rule)

σ

ε

L

σ, ε

L

ε

∆L

Rainflow counting of external loads is well justified in such cases!

PJ/2011-09-29 3

Definition av rainflowcykler – Rychlik

• För varje lokalt maximum ska man försöka nå upp till samma nivå, baklänges eller framlänges, genom att tappa så lite höjd som möjligt.

• Den k:te rainflowcykeln definieras alltså som (mkrfc,Mk), där mk

rfc=max(mk+,mk

-).

• Definitionen av rainflowcykler av Rychlik (1987):

• Denna definition är ekvivalent med andra definitioner: Endo’s, ASTM, 4-point, ... (även Range-Pair)

• Räknar hysteres-cykler i lasten.

PJ/2011-09-29 4

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

3

last

tid

1. Vrid diagrammet så tiden går nedåt

2. Börja från toppen och låt en droppe per maximum (eller min) rinna neråt

3. Stanna om något av följande gäller

a) Passerar större max (mindre min) än startpunktens

b) Korsar tidigare droppes väg

4. Identifiera slutna loopar

Definition av rainflowcykler – Endo’s Original (i)

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last

tid

Definition av rainflowcykler – Endo’s Original (ii)

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Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

4

Rainflow Cycle Counting: Algorithmic description

Application of the 4-point rule to the discretized turning point signal x

1. Initialize an empty N by N matrix RFM and an empty residual vector RES (r=0).

2. Initialize the 4 point stack (s1, s2, s3, s4) = (x1, x2, x3, x4), and set k = 5 (next point).

3. Apply the counting rule:

if min(s1, s4) ≤ s2, s3 ≤ max(s1, s4),

then store the cycle (s2, s3), RFM(s2, s3) = RFM(s2, s3) +1, delete (s2, s3) from the

stack and refill it:

a) if r = 0: (s1, s2, s3, s4) = (s1, s4, xk, xk+1), k = k+2

b) if r = 1: (s1, s2, s3, s4) = (RESr, s1, s4, xk), k = k+1, r = r - 1

c) if r > 1: (s1, s2, s3, s4) = (RESr-1, RESr, s1,s4), r = r – 2

else, go to the next point: r = r + 1, RESr = s1 , (s1, s2, s3, s4) = (s2, s3, s4, xk), k = k + 1

4. Repeat step 3 until the signal is exhausted.

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� Sequence of 14 turning points with 8 levels.

� Demonstrate the different counting methods.

Simple Example

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Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

5

Rainflow Cycle Counting: Simple example

x = (2 , 7 , 4 , 8 , 2 , 5 , 4 , 6 , 1 , 7 , 4 , 5 , 2 , 5)

0 5 10 15

1

2

3

4

5

6

7

8

Time

Load

Cycles:

(7 , 4) (5 , 4) (2 , 6) (4 , 5)

RES = (2 , 8 , 1 , 7 , 2 , 5)

PJ/2011-09-29 9

Övning: Räkna rainflowcykler

• Räkna rainflowcyklerna i signalen

x = (1, 4, 2, 3, 2, 5, 3, 4, 3, 4)

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Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

6

An example for first and repeated runsStress signal: (0, 360, -200, 400, 240, 440, 0)

– 4-point counting gives

RFM = (400,240),

RES = (0, 360, -200, 440, 0)

– Closed cycles after first run:

(400,240) and (360,-200)

– Closed cycles in second run:

(0, 360), (440, -200) and

(400,240)

Counting results

Rainflow Cycle Counting: The residual

first run second run

0 5 10 15 20-500

0

500

str

ess [

MP

a]

sample

M 1

M 1

M 2

M 2

M 1

0 1 2 3 4 5 6

x 10-3

-200

-100

0

100

200

300

400

500

M 1

M 1

M 2

M 2

M 1

strain

str

ess [

MP

a]

M 1

M 1

PJ/2011-09-29 11

0 2 4 6

x 10-3

-200

0

200

400M 1

M 1

M 2

M 2

M 1

strain

str

ess [

MP

a]

M 1

M 1

An example for first and repeated runs

3

4

51

2

Type of cycles Cycles Algorithm Damage

First run as well

as second run

2, 4

(identical)

4-point count d0

First run only 1 Extra rule on

the residual

d1

Second run only 3 and 5 Extra rule on

the residual

d2

=

= ⋅ + + − ⋅

+⇒ =

− +

0 1 2

0 2

1 2

1/

1 ( 1)

1

d N

N d d N d

d dd

d d

� For short signals: d1 , d2 can’t be neglected

since they may contain large cycles.

� For long signals: d0 >> d1 , d2 (typically)

� d ≈ d0 (4-point-count)

� For HCF, N >> 1 : d ≈ d0+d2

= RFM + 4-point-count(RES,RES)

Rainflow Cycle Counting: The residual (ctd.)

Total damage:

PJ/2011-09-29 12

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

7

Definition of the Markov matrix MM(i,j) = Number of transitions from bin i to bin j

Markov Counting

PJ/2011-09-29 13

Markov Counting (ctd.)Ex 1: Vertical wheel force (country road)

• The Markov matrix contains

the number of transitions in

the discretized turning point

signal from one level (row) to

the next level (column)

0 500 1000 1500 20000

10

20

30

0 500 1000 1500 20000

10

20

30

5 10 15 20 25

5

10

15

20

25

To

From

5 10 15 20 25

5

10

15

20

25

To

From

0 0.5 10

0.2

0.4

0.6

0.8

1

to

fro

m

(b) Markov matrix

50

100

150

200

250

Ex 2: Ramp + noise and sinusoidal + noise

• Both signals have similar Markov matrices but different Rainflow matrices.

• Damage(Markov) << Damage(Rainflow).

• Differences become small for narrow band loads.

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Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

8

Markov Load – Model for Turning Points

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Assumptions:

Markov Model:

Markov Property:

Load Measurement Turning Points Markov Matrix

TP-filter Model

Frequencies of transition

Extract peaks & valleys

� Frequency content not important.

� Stationarity

� Markov Chain of Turning Points.

� Frequency of transitions given by Markov matrix.

� Next value only depends on the current value,

not on complete history of values.

Example: Markov load

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Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

9

Example: Five Simulated Markov loads

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� All 5 simulations are different.

Damage

Exponent = 5

Example: Five Simulated Markov loads

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Level crossings Load spectrum

Blue: five simulated Markov loads

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

10

Limiting rainflow matrix

PJ/2011-09-29 19

� What is the typical shape of the rainflow matrix for a random load?

� Limiting shape of rainflow matrix

� Definition: The shape of the rainflow matrix for a very long observation.

n = 100 n = 1 000 n = ∞n = 10 000

Example: Markov load – Limiting rainflow matrix

PJ/2011-09-29 20

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

11

Example: Five Simulated Markov loads

PJ/2011-09-29 21

Level crossings Load spectrum

Blue: five simulated Markov loads

Red: Obtained from theoretically computed limiting rainflow martix

Rainflow damage: upper & lower bounds

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Upper Bound

Markov matrix

Markov Load Model

Markov count

Level crossings

True value---

Limiting Rainflow matrix

Lower Bound

Example: Previous Markov model

Upper Bound: 0.313

Markov model: 0.306

Lower bound: 0.165

InputExpected

rainflow damage

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

12

1D methods

2D methods Rainflow Markov

Level-crossing

Rangecount

Range-pair count

Time signals

Cycle Counting – Overview of Methods

DamageUpper bound

Lower bound

Rainflowdamage

PJ/2011-09-29 23

Rainflowcykler och multi-input-laster

Kurs i Lastanalys för Utmattning3-4 Oktober 2011

SP Bygg och Mekanik

Pär Johannesson

Par.Johannesson@sp.se

Nivå-korsningar

Last-spektrum

Rainflow-matris

PJ/2011-09-29

24

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

13

Realistic Example – Measured Service Loads

PJ/2011-09-29 25

� Vertical wheel force measured on the front left wheel of a truck.

� Three road types: City, Highway and Country.

Definition av rainflowcykler – Rychlik

• För varje lokalt maximum ska man försöka nå upp till samma nivå, baklänges eller framlänges, genom att tappa så lite höjd som möjligt.

• Den k:te rainflowcykeln definieras alltså som (mkrfc,Mk), där mk

rfc=max(mk+,mk

-).

• Definitionen av rainflowcykler av Rychlik (1987):

• Denna definition är ekvivalent med andra definitioner: Endo’s, ASTM, 4-point, ... (även Range-Pair)

• Räknar hysteres-cykler i lasten.

PJ/2011-09-29 26

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

14

Service load example – Rainflow counting

PJ/2011-09-29 27

� Demonstrate counting methods using realistic service loads.

� Different ways of plotting and presenting the result.

� Discussion and interpretation of results.

Service load example – Level crossing & Range-pair

100

101

102

103

104

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

count

wh

ee

l fo

rce

z f

ron

t le

ft [

]

level crossing

city

highway

country road

100

105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

count

wh

ee

l fo

rce

z f

ron

t le

ft [

]

range pair

city

highway

country road

Range pair & level crossing can be used as display options for rainflowmatrices

– Comparison of different signals by overlaid plotting

– RP and LC hold somewhat complementary information

PJ/2011-09-29 28

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

15

29

Multidimensionella laster – Vändpunkter & Accelerering Multidimensionella laster eller multi-input laster:

– Lasten har flera införingspunkter, eller

– lasten påförs i flera riktningar.

– Hur reducera lasten?

– Hur definiera vänd-punkter för multi-input lasten?

– Hur accelerera lasten?

PJ/2011-09-29

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2D-last – Tidssignal & Vändpunkter

Vändpunkter för 2D-last:

Behåll värden vid de tidpunkter då antingen X1 eller X2

har en vändpunkt.

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

16

PJ/2011-09-29 31

2D-last –Vändpunkter & Rainflowfilter

Vändpunkter för 2D-last:

Vändpunkterna är värdena då antingen X1 eller X2

har en vändpunkt.

PJ/2011-09-29 32

2D-last – Vändpunkter i 4 riktningar

Vändpunkter i 4 riktningar (X1, X2 , X1+X2 och X1-X2) för 2D-last :

För att bättre bevara ”fasen” mellan signalerna studeras linjär-kombinationer.

Behåll värdena då någon av signal-erna X1, X2, X1+X2

eller X1-X2 har en vändpunkt.

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

17

33

2D-last – Fasplan & rainflowfilter

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1(a) TP, 2 riktningar

X1, kraft vänster / kN

X2,

kra

ft h

ög

er

/ kN

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1(b) TP, 4 riktningar

X1, kraft vänster / kN

X2,

kra

ft h

ög

er

/ kN

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1(c) Rainflow−filter, 2 riktningar

X1, kraft vänster / kN

X2,

kra

ft h

ög

er

/ kN

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1(d) Rainflow−filter, 4 riktningar

X1, kraft vänster / kN

X2,

kra

ft h

ög

er

/ kN

PJ/2011-09-29

Multi-input Loads: From Outer Load to Local Load• Rainflow cycle counting is motivated by considering local stresses and

strains (hysteresis models), but often applied to outer loads.

• When and why do the local arguments apply to outer loads?

• For one input: If σ(t) = ϕ(L(t))(*), then load cycles and local stress-strain

cycles open and close at the same time (e.g. (*) holds for forces acting on a

stiff component and stresses calculated from linear FEA and Neuber’s rule)

σ

εL1

σ, ε

L2

L

ε

∆L

Rainflow counting of linear combinations of external loads is well justified in such cases!

PJ/2011-09-29 34

Superposition principle: σ=c1L1+c2L2

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

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35

Rainflow Projection (RP) Method

inputRainflow matrices

c2,1 L1 + c2,2 L 2 + c2,3 L3

c1,1 L1 + c1,2 L 2 + c1,3 L3

projections

projection Rainflow - counting

PJ/2011-09-29

36

RP- visualisation - load-influence-sphere

L1

y

z x

L2

-L1

(- L1- L2 + L3)/√√√√(3)

L3• projektion• Rainflow -

counting• damage-

accumulation

damage - potential

Rainflow Projection (RP) Method

PJ/2011-09-29

Kurs i Lastanalys för Utmattning

3-4 Oktober 2011

Rainflowcykler

19

37

RP- visualisation - histogram

• projektion• Rainflow -

counting• damage-

accumulation

Rainflow Projection (RP) Method

PJ/2011-09-29