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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