Mar 23, 2015 - Equivalent Stress. Approach. Multiaxial. Evaluation Criteria. Equivalent Stress. Approach. Non Linear Model. (Material). F re qu e n c y. D oma.
3° International conference on material and component performance under Variable amplitude loading – VAL 2015 March 23 – 26, 2015 Prague, Czech Republic
Minisymposium on:
“Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing”
Organizers: Denis Benasciutti, Adam Niesłony, Filippo Cianetti Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
March 23, 2015
Minisymposium on: “Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing”
Topic:
“Dynamic simulation vs. spectral methods”
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Scenario (Dynamic simulation vs. spectral methods) Input
Time Domain Input Array
System Model
Non Linear Model
Stress Recovery
(Material)
Fatigue Evaluation
Stress Tensor
Time Domain
Multiaxial Evaluation Criteria
Inputj
Inputk
Equivalent Stress Approach
Inputi
Linear/Non Linear Behaviour
Frequency Domain
Frequency Domain Input Array
Linearised Non Linear Model
PSD Stress Matrix
Gaussianity Indeces
Multiaxial Evaluation Criteria
Inputj Inputk Inputi
Equivalent Stress Approach
Linear Behaviour (State Space Representation)
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Requests (Dynamic simulation vs. spectral methods)
How can I improve my performances ? Can I anticipate a first evaluation of fatigue ?
Dynamic Simulation (DS)
Approximated evaluation
Full Stress Recovery
Fatigue Evaluation Partial data
Is it possible to manage less data from DS ? What are my essential needs to perform evaluation ?
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Simulation (Dynamic simulation vs. spectral methods) “Stress recovery”
Stress Recovery Idea
CAD Model
System/ Component
Dynamic Simulation
FE Model
Component
FE Modal Model (modal analysis, CMS)
System
MBS Model
FE Dynamic Analysis (time, frequency)
Component
SS Model
Dynamic Simulation
C++ Dynamic Analysis (time, frequency)
Dynamic Simulation
MBS Dynamic Analysis (time, frequency)
Result
Lagrangian coordinates
Result
Lagrangian coordinates
Result
Lagrangian coordinates
Stress Recovery
FE Modal Combination
Stress Recovery
C++ Modal Combination
Stress Recovery
MBS Modal Combination
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Simulation (Dynamic simulation vs. spectral methods) “Stress recovery”
Idea
CAD Model
System/ Component
FE Model
Component
FE Modal Model (modal analysis, CMS)
Dynamic Simulation
Dynamic Analysis (time, frequency)
Result
Lagrangian coordinates
Stress Recovery
Modal Combination
System
MBS Model
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Simulation (Dynamic simulation vs. spectral methods)
Forces Input (MIMO)
Time domain stress recovery
𝝈 𝑡 = 𝚽σ 𝒒(𝑡) Frequency domain stress recovery
𝑮𝜎 = 𝚽σ ∙ 𝑮𝑞 ∙ (𝚽σ )T
Motions Input (MIMO)
Time domain stress recovery
𝝈 𝑡 = 𝚽𝐶𝜎 𝜹𝐵 + 𝚽σ 𝒒(𝑡) Frequency domain stress recovery
𝑮𝜎 = 𝚽σ ∙ 𝑮𝑞 ∙ (𝚽σ )T + 𝚽Cσ ∙
𝑮𝑥 ∙ 𝚽Cσ 4 𝜔
T
− 𝚽 σ ∙ 𝑯𝑞 ∙
𝑮𝑥 ∙ 𝚽Cσ 2 𝜔
T
− 𝚽Cσ ∙
𝑮𝑥 ∙ 𝑯𝑞 𝜔2
T
∙ (𝚽σ )T
(FEA motion input)
(FEA forces input, MBS)
“Stress recovery”
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation”
Dynamic Simulation
Simulation code
Modal Approach
Stress Recovery
Fatigue code
Fatigue Evaluation
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation”
Dynamic Simulation
Simulation code
Modal Approach
Stress Recovery
Fatigue code
Fatigue Evaluation
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation”
Simulation code
Fatigue code
Requested results from Dynamic Simulation
Dynamic Simulation
Modal Approach
Stress Recovery
Fatigue Evaluation
Requested results from Dynamic Simulation
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation”
Simulation code
Fatigue code
Dynamic Simulation
Modal Approach
Stress Recovery
Fatigue Evaluation
Approximated evaluation
Multiaxiality - Preumont Spectral method - Dirlik
Fatigue Strength - Wohler Damage Cumulation - Miner
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation improvement”
Simulation code
Fatigue code
Reference Procedure 1.0
𝑮𝑥
𝑯𝑞 𝜱𝜎𝑗 𝜱𝜎𝐶 𝑗
𝑮𝜎 𝑗
𝑯𝜎 𝑗
𝐺𝐸𝑄𝑉𝑀 𝑗
𝐷𝑗
𝑚𝑛 𝑗
Dynamic analysis
I/O relation
Stress recovery
Multiaxial criteria
Statistical analysis
Damage
Frequency Analysis
Stress Frequency Response Function
Stress PSD evaluation
Uniaxial Synthesis (i.e. Preumont)
Spectral Moments evaluation
Damage evaluation (i.e. Dirlik)
𝑗 th element Procedure 2.0
𝑮𝜎 𝑗
𝑮𝑞
𝑮𝑥
𝐺𝐸𝑄𝑉𝑀 𝑗
𝐷𝑗
𝑚𝑛 𝑗
Dynamic analysis
Matrices Combination Stress recovery
Multiaxial criteria
Statistical analysis
Damage
Frequency Analysis
Stress PSD evaluation
Uniaxial Synthesis (i.e. Preumont)
Spectral Moments evaluation
Damage evaluation (i.e. Dirlik)
𝑗 th element
𝜱𝜎𝑗 𝜱𝜎𝐶 𝑗
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation improvement”
Simulation code
Fatigue code
Procedure 2.0
𝑮𝜎 𝑗
𝑮𝑞
𝑮𝑥
𝐺𝐸𝑄𝑉𝑀 𝑗
𝐷𝑗
𝑚𝑛 𝑗
Dynamic analysis
Matrices Combination Stress recovery
Multiaxial criteria
Statistical analysis
Damage
Frequency Analysis
Stress PSD evaluation
Uniaxial Synthesis (i.e. Preumont)
Spectral Moments evaluation
Damage evaluation (i.e. Dirlik)
𝑗 th element
𝜱𝜎𝑗 𝜱𝜎𝐶 𝑗
Procedure 3.0
𝑯𝑞
𝑮𝑥
𝐷𝑗
𝑚𝑛 𝑗
𝚯𝑛 𝚲𝑛 𝚿𝑛 𝚪𝑛
Dynamic analysis
I/O relation
Statistical analysis
Matrices Combination + Multiaxial Criteria
Damage
Frequency Analysis
Q Frequency Response Function
Spectral Matrices evaluation
Spectral Moments Uniaxial Synthesis evaluation (i.e. Preumont)
Damage evaluation (i.e. Dirlik)
𝜱𝜎𝑗 𝜱𝜎𝐶 𝑗
𝑗 th element
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
And now ? (Dynamic simulation vs. spectral methods)
Dynamic Simulation codes
Fatigue Evaluation codes
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
APPENDIX
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Components Durability – Frequency domain Component
Displacement function
Shape function
Generalised coordinates
(6 x 1)
Fatigue evaluation
Stress recovery
Dynamic Analysis
Hp
Introduction of component with elastic properties (modal approach)
State variables
Linearised Non Linear System
Inputs
(m x m)
(n x n)
(m x n) (n x n) (n x m)
PSD matrix of system’s generalised coordinates
Inputi Inputk
Outputs
Inputi
Stress state
(Multiaxial Fatigue)
PSD function of the “Equivalent von Mises stress” (Preumont et al.)
Probability density function (pdf) of the cycles stress range by Dirlik formula
Wöhler curve
Miner rule
Per time unit Fatigue damage
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation improvement”
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Fatigue (Dynamic simulation vs. spectral methods) “Fatigue evaluation improvement”
𝑯𝑞
𝑮𝑥
𝐷𝑗
𝑚𝑛 𝑗
𝚯𝑛 𝚲𝑛 𝚿𝑛 𝚪𝑛
Dynamic analysis
I/O relation
Statistical analysis
Matrices Combination + Multiaxial Criteria
Damage
Frequency Analysis
Q Frequency Response Function
Spectral Matrices evaluation
Spectral Moments Uniaxial Synthesis evaluation (i.e. Preumont)
Damage evaluation (i.e. Dirlik)
𝜱𝜎𝑗 𝜱𝜎𝐶 𝑗
𝑗 th element
Spectral moments for MIMO condition
𝑚𝑛 = 𝜱𝜎 ∙ 𝚯𝑛 ∙ (𝜱𝜎 )𝑇 + 𝜱𝜎𝐶 ∙ 𝚲𝑛 ∙ 𝜱𝜎𝐶
(Forces Input)
𝑚𝑛 = 𝜱𝜎 ∙ 𝚯𝑛 ∙ (𝜱𝜎 )𝑇
Spectral moments definition
∞
𝑚𝑛 =
0
𝑓 𝑛 𝐺𝜎 𝑓 𝑑𝑓
𝑇
− 𝜱𝜎 ∙ 𝚿𝑛 ∙ 𝜱𝜎𝐶
𝑇
− 𝜱𝜎𝐶 ∙ 𝚪𝑛 ∙ (𝜱𝜎 )𝑇 ∞
Modal coefficients
(Motion Input)
𝚯𝑛 =
0
𝑅𝑒(𝑮𝑞 )𝑓 𝑛 𝑑𝑓
𝑅𝑒 𝑮𝑥 𝑛 𝑓 𝑑𝑓 𝜔4 𝑮𝑥 𝚿𝑛 = 𝑅𝑒(𝑯𝑞 ∙ 2 )𝑓 𝑛 𝑑𝑓 𝜔 𝑮𝑥 T 𝚪𝑛 = 𝑅𝑒 2 ∙ 𝑯𝑞 𝑓 𝑛 𝑑𝑓 𝜔 𝚲𝑛 =
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Non-linear systems Time domain input Reconstruction of a set of time input processes that significantly represent the load conditions hypothesised
Transient analysis and ABCD sampling Analysis of the nonlinear behaviour of the system
Frequency analysis
Frequency analysis
Stress recovery
Generation of the sample composed of the Lagrangian coordinates PSD matrices G q
Evaluation of the Lagrangian coordinates PSD matrix G q which best represents the response of the system
Evaluation of the PSD matrix S of the stress tensor and/or of the PSD function G of the equivalent stress state
Non-linear model
Inputj Inputk Inputi
Sampling of the state-space matrices through step by step system linearization
Evaluation of the Lagrangian coordinates PSD matrix for each element of the sample considered
Evaluation of the Lagrangian coordinates PSD matrix, average of the sampled system states
State Space Model
Frequency domain Input
rs element of the matrix
G Gqq
Definition of the PSD matrix of inputs
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
Non-linear vs. Non-Gaussian Input
Time Domain
Time Domain Array Input
System Model
Non Linear Model
Stress Recovery
(Material)
Fatigue Evaluation
Stress Tensor
Stress state Gaussianity Indeces
Inputj
Inputk Inputi
Frequency Domain
Non Linear Behaviour
Frequency Domain Array Input
States sample
PSD Stress Matrix
Fatigue life assessment with random loadings: spectral methods, dynamic simulations, testing
