σ - FATIGUE 2014

Fatigue2014
11th International Fatigue Congress
2-7 March 2014
Accurate prediction of fatigue life
under random loading
Norio Takeda and Tomohiro Naruse
Hitachi research laboratory, Hitachi Ltd.
Background
①
■The accuracy of fatigue life prediction in the frequency domain is
investigated, and methods of improving the accuracy are proposed.
Random vibration
Excitation force
by engine
Design object
Being mounted on engine
ECU※
※1:
Power Spectral Density
※2: Engine Control Unit
Cycle counting method
in the frequency domain
400
mm
Frequency
Design Standard
（ISO 16750-3）
Stress
amplitude
Acceleration
PSD※
Stress PSD
・Dirlik’s method
・Level crossing method
Frequency
Random response
analysis
Fatigue
life
Count
・S-N curve
・Linear
Cycle counting damage rule
Cycle counting method in the frequency domain
②
■The probability density of stress amplitude is defined as the combination
of statistical distributions.
・Dirlik’s method
Z
Z2
Z2 

−
−
−
DK
(s ) = 1  D1 e Q + D22Z e 2 R 2 + D3 Ze 2 
p RFC
σX  Q

R


Exponential
distribution
Rayleigh
distribution
s: stress amplitude, Z = s/ σX
σX, D1, D2, D3, Q and R are evaluated given a stress PSD.
・Level crossing method in the frequency domain
Stress amplitude histogram
(=probability density of stress amplitude
×total number of counts)
Rainflow method
Stress amplitude
Probability density of stress amplitude
⇒Discrete
distribution
Frequency domain
method
⇒Continuous
distribution
Probability density of stress amplitude
(
) (
p LCC (s ) = α2 s / σ X2 exp − s 2 / 2σ X2
Rayleigh distribution
α2 is evaluated given a stress PSD.
)
Counts
Generation of random stress
③
■Random stress having two peaks in the frequency domain is generated
with an artificial structure composed of cantilevers.
・Young’s modulus: 210 GPa
・Poisson’s ratio: 0.3
・Mode damping ratio: 4 or 8%
A
100
RMS = √50E m/s2
900
E
φd1
5 types of
random stress
100
（No. 1～5）
φ20
0
0
50
Frequency (Hz)
φd2
Stress at A, No. 3, d1 = 10 mm，d2 = 60 mm，E = 1200 (m/s2)2/Hz
Stress
PSD
応力 PSD
（MPa2/Hz×105）
400
応力
（MPa）
Stress
(MPa)
Acceleration PSD
((m/s2)2/Hz)
Random acceleration
200
0
-200
5.0
4.0
8.7
3.0
2.0
42.6
1.0
0.0
-400
0.0
0.5
Time (sec.)
1.0
0 10 20 30 40 50 60
Frequency (Hz)
Comparison of probability density and damage
④
■ When using the frequency domain methods,
fatigue damage caused by large stress amplitude is clearly observed.
Damage distribution
Probability density of stress amplitude
(= each count / all counts)
Stress amplitude (MPa)
Stress amplitude (MPa)
Stress at A, No. 3, d1 = 10 mm，d2 = 60 mm，E = 1200 (m/s2)2/Hz
Probability density
Damage by large amplitude
Damage di
The maximum value of stress amplitude is about 400 MPa in the time domain,
and 500 MPa in the frequency domain.
The frequency domain approaches are too conservative
to predict fatigue life.
Estimation of probability density function
⑤
■The upper limit of stress amplitude is estimated from PSD, and
the probability density is adjusted using the limit.
Stress amplitude (MPa)
We can cut the doubtful large amplitude larger than the estimated upper limit.
Limit （ σ max ）
Damage di
Equation for estimating the upper limit σ
max
of stress amplitude from PSD

ε

= σ RMS  2 ln(rN p ) +

2 ln(rN p )





σ RMS ：Root mean squared value, r ：Irregularity factor, Np ：The number of observed peaks
are estimated from stress PSD without stress time history.
Adjustment of probability density
⑥
■Probability density should be zero at the estimated upper limit.
Squared error adjustment
Constant adjustment
p(σ)
σmax
p(σmax)
Original
distribution
Stress amplitude
Stress amplitude
p(σ)
σmax
p(σ)×(σ / σmax)2
Original
distribution
0
0
Probability density
1
 {p (σ ) − p(σ max )}
p adj1 (σ ) =  A

0

(0 ≤ σ ≤ σ max )
(σ max < σ )
A：Constant for normalization
Probability density
 

 1  p(σ ) −  σ
σ
padj 2 (σ ) =  A 
 max



0
2


 p (σ )



(0 ≤ σ ≤ σ max )
(σ max < σ )
Effect of probability density adjustment
■The damage estimated in the frequency domain nears that estimated
in the time domain especially by using squared error adjustment.
Level crossing method
Stress amplitude (MPa)
Stress amplitude (MPa)
Dirlik’s method
Damage di
Damage di
・The stress amplitude above the upper limit is appropriately cut
by using constant adjustment.
・The damage distribtion is well modified by using squared error adjustment.
⑦
Summary of adjustment effect
⑧
■The accuracy in fatigue life prediction is improved for all types of random
stress to be investigated by using the proposed adjustment.
Normalized fatigue life ※
Predicted fatigue life in the time domain = Predicted fatigue life in the frequency domain
0.98
0.77
Avg.
0.57
Dirlik
0.60
Dirlik
＋
adj.1
Dirlik
＋
adj.2
0.45
0.48
Level
crossing
Level
crossing
＋
adj.1
Level
crossing
＋
adj.2
5 types of
random stress
are used for
this investigation.
adj. 1: Constant
adjustment
adj. 2: Squared error
adjustment
(※Fatigue life predicted in the frequency domain ÷ Fatigue life predicted in the time domain)
Summary
⑨
The accuracy of fatigue life prediction in the frequency domain
was investigated. Then, two methods of adjusting the probability density
estimated in the frequency domain were proposed to improve the accuracy.
(1) The maximum value of stress amplitude estimated in the frequency domain
is larger than that estimated in the time domain. As a result, the frequency
domain approach is too conservative to predict fatigue life.
(2) Although the fatigue lives predicted using Dirlik’s method without adjustment
ranged from 53 to 63% of the fatigue lives predicted in the time domain,
the fatigue lives predicted using Dirlik’s method with squared error adjustment
neared those in the time domain and ranged from 90 to 111%.
END
Flow of predicting fatigue life
②
■The probability density and damage distribution of stress amplitude are
compared between time domain approach and frequency domain approach.
Fatigue life evaluation in the time domain
Time
Miner rule
σ1
σ1
n1
σi
ni
nM
σM
Counts
Stress range
Time
Stress range
Rainflow
count
Stress
Acceleration
Numerical
analysis
σi
④
σM
Fatigue life evaluation in the frequency domain
Dirlik’s or Level crossing
method
method
Frequency
Stress range
Frequency
Stress PSD
Acceleration
PSD
Numerical
analysis
σ1
σi
σM
n1
N1
Ni
NM
Cycles to fatigue failure
ni
＜１
i Ni
Damage ratio
M
ni
nM
Counts
∑
Fatigue
life
平均確率密度，損傷度（時間領域）
④
■50個の応力波形を頻度解析し，平均の損傷度を計算
50個の
平均確率密度※
応力
時間
応力振幅
頻度
Rainflow法
50個の
平均損傷度
応力振幅
時間
計20,000
ｶｳﾝﾄ
応力振幅
応力
Rainflow法
応力振幅
時間領域
確率密度
損傷度
・・・
・・・
頻度
波形50個
頻度分布50個
周波数領域で計算した応力振幅の確率密度，
損傷度と比較
※頻度÷全頻度（20,000）＝確率密度
平均確率密度，損傷度（周波数領域）
⑤
応力
■50個の平均PSDを求め，それを頻度解析して損傷度を計算
周波数領域
時間
周波数
Dirlik’s法
or
Level
crossing法
50個の
平均確率密度※
50個の
平均損傷度
応力振幅
応力 PSD
周波数
応力 PSD
50個の
平均PSD
応力振幅
応力 PSD
50個の波形に
FFT適用
確率密度
損傷度
・・・
周波数
PSD 50個
時間領域で計算した応力振幅の確率密度，損傷度と比較
Stress amplitude σa（MPa）
疲労寿命曲線
1000
500
200
R = -1
100
10,000
104
100,000
105
1,000,000
106
10,000,000
107
Number of cycles to failure Nf (cycles)
SS400
材質：
引張強さ： 454 MPa
２乗誤差補正の根拠
Dirlikは応力PSDから確率密度を計算する式を
次式が最小になるように決定
2乗誤差補正
[ p(z ) − p (z )]
=
p (z )
2
2
i
e
p(σ)
i
i
p(z i ) ：Rainflow法で求められる
応力振幅ziでの確率密度
p(σ)×(σ / σmax)2
応力振幅
σmax
i
p (z i ) ：PSDから求められる
応力振幅ziでの確率密度
元の分布
0
確率密度
 

 1  p(σ ) −  σ
σ
padj 2 (σ ) =  A 
 max



0
2


 p (σ )



誤差は確率密度の大きさに比例して含まれ，
２乗誤差が最小となるように係数が定められた
(0 ≤ σ ≤ σ max )
(σ max < σ )
p(σ)×(σ / σmax)2
確率密度の
大きさに比例
２乗誤差が
最小
Dirlikが考慮したPSD
■70種類のPSDを用意し，Rainflow法による確率密度と一致するように
PSDから確率密度を計算する式を決定
56種類
14種類

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