Mer om Rainflowcykler

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1 Mer om Kurs i Lastanalys för Utmattning SP Bygg och Mekanik Pär Johannesson Par.Johannesson@sp.se Nivåkorsningar Lastspektrum 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/

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) L L σ σ, ε L ε ε Rainflow counting of external loads is well justified in such cases! PJ/ 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 (m k rfc,m k ), där m k rfc =max(m k+,m k- ). Denna definition är ekvivalent med andra definitioner: Endo s, ASTM, 4-point,... (även Range-Pair) Räknar hysteres-cykler i lasten. PJ/

3 Definition av rainflowcykler Endo s Original (i) 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 last tid PJ/ Definition av rainflowcykler Endo s Original (ii) last tid PJ/

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=). 2. Initialize the 4 point stack (s 1, s 2, s 3, s 4 ) = (x 1, x 2, x 3, x 4 ), and set k = 5 (next point). 3. Apply the counting rule: if min(s 1, s 4 ) s 2, s 3 max(s 1, s 4 ), then store the cycle (s 2, s 3 ), RFM(s 2, s 3 ) = RFM(s 2, s 3 ) +1, delete (s 2, s 3 ) from the stack and refill it: a) if r = : (s 1, s 2, s 3, s 4 ) = (s 1, s 4, x k, x k+1 ), k = k+2 b) if r = 1: (s 1, s 2, s 3, s 4 ) = (RES r, s 1, s 4, x k ), k = k+1, r = r - 1 c) if r > 1: (s 1, s 2, s 3, s 4 ) = (RES r-1, RES r, s 1,s 4 ), r = r 2 else, go to the next point: r = r + 1, RES r = s 1, (s 1, s 2, s 3, s 4 ) = (s 2, s 3, s 4, x k ), k = k Repeat step 3 until the signal is exhausted. PJ/ Simple Example Sequence of 14 turning points with 8 levels. Demonstrate the different counting methods. PJ/

5 Rainflow Cycle Counting: Simple example x = (2, 7, 4, 8, 2, 5, 4, 6, 1, 7, 4, 5, 2, 5) Load Time Cycles: (7, 4) (5, 4) (2, 6) (4, 5) RES = (2, 8, 1, 7, 2, 5) PJ/ Övning: Räkna rainflowcykler Räkna rainflowcyklerna i signalen x = (1, 4, 2, 3, 2, 5, 3, 4, 3, 4) PJ/

6 M 1 M 2 M 1 M 2 M 1 M 1 M 1 M 1 M 2 M 2 M 1 M 2 M 1 M 2 M 1 Kurs i Lastanalys för Utmattning Rainflow Cycle Counting: The residual An example for first and repeated runs Stress signal: (, 36, -2, 4, 24, 44, ) Counting results 4-point counting gives RFM = (4,24), RES = (, 36, -2, 44, ) Closed cycles after first run: (4,24) and (36,-2) Closed cycles in second run: (, 36), (44, -2) and (4,24) stress [MPa] stress [MPa] 5 first run second run sample strain x 1-3 PJ/ Rainflow Cycle Counting: The residual (ctd.) An example for first and repeated runs stress [MPa] 4 M strain x 1-3 Total damage: d = 1/ N 1 = N d + d + ( N 1) d d + d2 d = 1 d + d Type of cycles Cycles Algorithm Damage First run as well as second run 2, 4 (identical) 4-point count d First run only 1 Extra rule on the residual Second run only 3 and 5 Extra rule on the residual For short signals: d 1, d 2 can t be neglected since they may contain large cycles. For long signals: d >> d 1, d 2 (typically) d d (4-point-count) d 1 d 2 For HCF, N >> 1 : d d +d 2 = RFM + 4-point-count(RES,RES) PJ/

7 Markov Counting Definition of the Markov matrix M M(i,j) = Number of transitions from bin i to bin j PJ/ 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 (b) Markov matrix From from To to From To PJ/

8 Markov Load Model for Turning Points Load Measurement Turning Points Markov Matrix TP-filter Model Extract peaks & valleys Frequencies of transition Assumptions: Markov Model: Markov Property: 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. PJ/ Example: Markov load PJ/

9 Example: Five Simulated Markov loads All 5 simulations are different. Damage Exponent = 5 PJ/ Example: Five Simulated Markov loads Level crossings Load spectrum Blue: five simulated Markov loads PJ/

10 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 = 1 n = 1 n = 1 n = PJ/ Example: Markov load Limiting rainflow matrix PJ/

11 Example: Five Simulated Markov loads Level crossings Load spectrum Blue: five simulated Markov loads Red: Obtained from theoretically computed limiting rainflow martix PJ/ Rainflow damage: upper & lower bounds Input Expected rainflow damage Example: Previous Markov model Level crossings Upper Bound Markov Load Model Markov matrix True value --- Limiting Rainflow matrix Upper Bound:.313 Markov model:.36 Lower bound:.165 Markov count Lower Bound PJ/

12 Cycle Counting Overview of Methods Time signals 2D methods Rainflow Markov 1D methods Range-pair count Levelcrossing Range count Damage Rainflow damage Upper bound Lower bound PJ/ och multi-input-laster Kurs i Lastanalys för Utmattning SP Bygg och Mekanik Pär Johannesson Par.Johannesson@sp.se Nivåkorsningar Lastspektrum Rainflowmatris PJ/

13 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/ 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 (m k rfc,m k ), där m k rfc =max(m k+,m k- ). Denna definition är ekvivalent med andra definitioner: Endo s, ASTM, 4-point,... (även Range-Pair) Räknar hysteres-cykler i lasten. PJ/

14 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/ 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 wheel force z front left [] level crossing city highway country road wheel force z front left [] range pair city highway country road count count PJ/

15 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/ D-last Tidssignal & Vändpunkter Vändpunkter för 2D-last: Behåll värden vid de tidpunkter då antingen X 1 eller X 2 har en vändpunkt. PJ/

16 2D-last Vändpunkter & Rainflowfilter Vändpunkter för 2D-last: Vändpunkterna är värdena då antingen X 1 eller X 2 har en vändpunkt. PJ/ D-last Vändpunkter i 4 riktningar Vändpunkter i 4 riktningar (X 1, X 2, X 1 +X 2 och X 1 -X 2 ) för 2D-last : För att bättre bevara fasen mellan signalerna studeras linjärkombinationer. Behåll värdena då någon av signalerna X 1, X 2, X 1 +X 2 eller X 1 -X 2 har en vändpunkt. PJ/

17 2D-last Fasplan & rainflowfilter 1 (a) TP, 2 riktningar 1 (b) TP, 4 riktningar X 2, kraft höger / kn.5.5 X 2, kraft höger / kn X 1, kraft vänster / kn X 1, kraft vänster / kn 1 (c) Rainflow filter, 2 riktningar 1 (d) Rainflow filter, 4 riktningar X 2, kraft höger / kn.5.5 X 2, kraft höger / kn X 1, kraft vänster / kn X 1, kraft vänster / kn PJ/ 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, ε L L σ L 2 ε ε L 1 Superposition principle: σ=c 1 L 1+ c 2 L 2 Rainflow counting of linear combinations of external loads is well justified in such cases! PJ/

18 Rainflow Projection (RP) Method input projections c 1,1 L 1 + c 1,2 L 2 + c 1,3 L 3 Rainflow matrices projection Rainflow - counting c 2,1 L 1 + c 2,2 L 2 + c 2,3 L 3 PJ/ Rainflow Projection (RP) Method RP- visualisation - load-influence-sphere y L 2 projektion Rainflow - counting damageaccumulation z -L 1 (- L 1 - L 2 + L 3 )/ (3) L 3 damage - potential L 1 x PJ/

19 Rainflow Projection (RP) Method RP- visualisation - histogram projektion Rainflow - counting damageaccumulation PJ/