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Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Mer om Rainflowcykler
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
SP Bygg och Mekanik
Nivåkorsningar
Lastspektrum
Pär Johannesson
[email protected]
Rainflowmatris
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.
stress
standing
Hysteresis model
(cyclic stress-strain curve,
Masing and Memory rules)
hanging
strain
PJ/2011-09-29
2
1
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
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)
L
L
σ
σ, ε
∆L
ε
ε
Rainflow counting of external loads is well justified in such cases!
PJ/2011-09-29
3
Definition av rainflowcykler – Rychlik
•
Definitionen av rainflowcykler av Rychlik (1987):
• 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 mkrfc=max(mk+,mk-).
•
•
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
2
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Definition av rainflowcykler – Endo’s Original (i)
last
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
tid
PJ/2011-09-29
5
Definition av rainflowcykler – Endo’s Original (ii)
last
tid
PJ/2011-09-29
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3
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
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.
PJ/2011-09-29
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PJ/2011-09-29
8
Simple Example
Sequence of 14 turning points with 8 levels.
Demonstrate the different counting methods.
4
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Rainflow Cycle Counting: Simple example
x = (2 , 7 , 4 , 8 , 2 , 5 , 4 , 6 , 1 , 7 , 4 , 5 , 2 , 5)
8
7
Load
6
5
4
3
2
1
0
5
10
15
Time
Cycles:
(7 , 4)
(5 , 4)
(2 , 6)
(4 , 5)
RES = (2 , 8 , 1 , 7 , 2 , 5)
PJ/2011-09-29
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PJ/2011-09-29
10
Övning: Räkna rainflowcykler
• Räkna rainflowcyklerna i signalen
x = (1, 4, 2, 3, 2, 5, 3, 4, 3, 4)
5
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Rainflow Cycle Counting: The residual
– 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)
stress [MPa]
first run
500
second run
M
M
M
1
M
-500
1
2
1
0
M
0
5
2
10
sample
15
20
500
M
400
M
M
stress [MPa]
An example for first
and repeated runs
Stress signal:
(0, 360, -200, 400, 240, 440, 0)
Counting results
1
M
1
2
1
300
200
100
0
M
2
-100
-200
0
1
2
3
strain
4
5
6
-3
x 10
PJ/2011-09-29
11
Rainflow Cycle Counting: The residual (ctd.)
stress [MPa]
An example for first
and repeated runs
M 1
400
M
M
1
1
2
M
2
4
200
3
1
0
5
M
2
4
strain
Total damage:
d = 1/ N
1 = N ⋅ d0 + d1 + (N − 1) ⋅ d2
⇒d =
d0 + d2
1− d1 + d2
Cycles
Algorithm
Damage
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
2
-200
0
Type of cycles
First run as well
as second run
6
-3
x 10
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)
PJ/2011-09-29
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6
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Markov Counting
Definition of the Markov matrix M
M(i,j) = Number of transitions from bin i to bin j
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)
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.
30
5
(b) Markov matrix
From
20
1
250
200
0.6
150
20
0
0.4
100
30
0.2
50
20
25
0
500
1000
1500
2000
5
10 15 20 25
To
5
10 15 20 25
To
5
From
from
15
10
0.8
0
10
0
0.5
to
1
10
15
10
20
0
25
0
500
1000
1500
2000
PJ/2011-09-29
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7
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Markov Load – Model for Turning Points
Load Measurement
Turning Points
TP-filter
Extract
peaks & valleys
Markov Matrix
Model
Frequencies
of transition
Assumptions:
Frequency content not important.
Stationarity
Markov Model:
Markov Chain of Turning Points.
Frequency of transitions given by Markov matrix.
Markov Property:
Next value only depends on the current value,
not on complete history of values.
PJ/2011-09-29
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PJ/2011-09-29
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Example: Markov load
8
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Example: Five Simulated Markov loads
All 5 simulations are different.
Damage
Exponent = 5
PJ/2011-09-29
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Example: Five Simulated Markov loads
Level crossings
Load spectrum
Blue: five simulated Markov loads
PJ/2011-09-29
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9
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Limiting rainflow matrix
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 = 10 000
n=∞
PJ/2011-09-29
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Example: Markov load – Limiting rainflow matrix
PJ/2011-09-29
20
10
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Example: Five Simulated Markov loads
Level crossings
Load spectrum
Blue: five simulated Markov loads
Red: Obtained from theoretically computed limiting rainflow martix
PJ/2011-09-29
21
Rainflow damage: upper & lower bounds
Input
Level
crossings
Expected
rainflow damage
Example:
Previous Markov model
Upper Bound
Upper Bound:
Markov Load Model
Markov
matrix
True value
--Limiting
Rainflow matrix
Markov model: 0.306
Lower bound:
Markov count
0.313
0.165
Lower Bound
PJ/2011-09-29
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11
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Cycle Counting – Overview of Methods
Time signals
2D methods
1D methods
Damage
Markov
Rainflow
Range-pair
count
Levelcrossing
Range
count
Rainflow
damage
Upper
bound
Lower
bound
PJ/2011-09-29
23
Rainflowcykler och multi-input-laster
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
SP Bygg och Mekanik
Nivåkorsningar
Lastspektrum
Pär Johannesson
[email protected]
Rainflowmatris
PJ/201109-29
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12
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Realistic Example – Measured Service Loads
Vertical wheel force measured on the front left wheel of a truck.
Three road types: City, Highway and Country.
PJ/2011-09-29
25
Definition av rainflowcykler – Rychlik
•
Definitionen av rainflowcykler av Rychlik (1987):
• 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 mkrfc=max(mk+,mk-).
•
•
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
13
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Service load example – Rainflow counting
Demonstrate counting methods using realistic service loads.
Different ways of plotting and presenting the result.
Discussion and interpretation of results.
PJ/2011-09-29
27
Service load example
– Level crossing & Range-pair
Range pair & level crossing can be used as display options for rainflow
matrices
– Comparison of different signals by overlaid plotting
– RP and LC hold somewhat complementary information
level crossing
range pair
1
0.35
city
highway
country road
0.8
0.7
0.6
0.5
0.4
0.25
0.2
0.15
0.1
0.05
0.3
0.2
0
10
city
highway
country road
0.3
wheel force z front left []
wheel force z front left []
0.9
1
10
2
10
count
3
10
4
10
0
0
10
5
10
count
PJ/2011-09-29
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14
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
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 vändpunkter för multi-input
lasten?
– Hur accelerera lasten?
PJ/2011-09-29
29
2D-last – Tidssignal & Vändpunkter
Vändpunkter för 2D-last:
Behåll värden vid
de tidpunkter då
antingen X1 eller X2
har en vändpunkt.
PJ/2011-09-29
30
15
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
2D-last –Vändpunkter & Rainflowfilter
Vändpunkter för 2D-last:
Vändpunkterna är
värdena då
antingen X1 eller X2
har en vändpunkt.
PJ/2011-09-29
31
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 linjärkombinationer.
Behåll värdena då
någon av signalerna X1, X2, X1+X2
eller X1-X2 har en
vändpunkt.
PJ/2011-09-29
32
16
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
2D-last – Fasplan & rainflowfilter
(a) TP, 2 riktningar
(b) TP, 4 riktningar
1
X2, kraft höger / kN
2
X , kraft höger / kN
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
X , kraft vänster / kN
1
0.5
0
−0.5
−1
−1
1
−0.5
0
0.5
X , kraft vänster / kN
1
(c) Rainflow−filter, 2 riktningar
(d) Rainflow−filter, 4 riktningar
1
X2, kraft höger / kN
2
X , kraft höger / kN
1
0.5
0
−0.5
−1
−1
1
−0.5
0
0.5
X , kraft vänster / kN
1
0.5
0
−0.5
−1
−1
1
−0.5
0
0.5
X , kraft vänster / kN
1
1
PJ/2011-09-29
33
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)
L
σ
σ, ε
∆L
ε
L2
L1
ε
Superposition principle: σ=c1L1+c2L2
Rainflow counting of linear combinations of external loads is well
justified in such cases!
PJ/2011-09-29
34
17
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Rainflow Projection (RP) Method
input
projections
c1,1 L1 + c1,2 L 2 + c1,3 L3
projection
Rainflow matrices
Rainflow - counting
c2,1 L1 + c2,2 L 2 + c2,3 L3
PJ/2011-09-29
35
Rainflow Projection (RP) Method
RP- visualisation - load-influence-sphere
y
• projektion
• Rainflow counting
• damageaccumulation
L2
-L1
L1
L3
(- L1- L2 + L3)/√
√(3)
z
x
damage - potential
PJ/2011-09-29
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18
Kurs i Lastanalys för Utmattning
3-4 Oktober 2011
Rainflowcykler
Rainflow Projection (RP) Method
RP- visualisation - histogram
• projektion
• Rainflow counting
• damageaccumulation
PJ/2011-09-29
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