S22-1 PAT318, Section 22, March 2005 SECTION 22 VIBRATION FATIGUE ANALYSIS

S22-2 PAT318, Section 22, March 2005

S22-3 PAT318, Section 22, March 2005 Fatigue for Dynamic Loading n All Fatigue analysis requires cycles of stress and/or strain to be identified. n Fortunately, this does not always require a transient dynamic analysis. n Options: u Static (or Pseudo-Static) l fatigue analysis scales static stress or strain tensor by time histories u Transient (direct or modal) l fatigue analysis uses stress or strain data from FEA job results file u Random Vibration (frequency domain PSD of stress) l fatigue analysis converts PSD to expected cycles of stress

S22-4 PAT318, Section 22, March 2005 Time-domain: Static method (with or without inertia relief) Transient method (Direct or Modal) Frequency-domain: Frequency Response Analysis (Transfer Function Method) Random Vibration Analysis Choice of Analysis Domain

S22-5 PAT318, Section 22, March 2005 IS Dynamic Response calculation necessary? frequency Transfer function F n 1 st natural frequency F L Highest loading frequency F L < 1/3 F N Yes, if the highest possible frequency of loading is Yes, if the highest possible frequency of loading is greater than one third of the 1 st natural frequency. greater than one third of the 1 st natural frequency.

S22-6 PAT318, Section 22, March 2005 Identify set of static FE loadcases and constraints to simulate service environment Measure or predict loading histories Pk( t ) Elastic stress histories calculated from linear superposition: where k = loadcase i.d. QUASI-STATIC ANALYSIS

S22-7 PAT318, Section 22, March 2005 Quasi-static method (linear superposition) - repeat for each node/element 1A * L 1 (t) + 2A * L 2 (t) +... = A (t) Stress for Unit Load Cases Local Stress Histories Load Time Histories L2L2 A L1L1 L 1 =1 L 2 =1

S22-8 PAT318, Section 22, March 2005 Static Analysis n Advantages: u Computationally cheap for FE analysis. u Minimizes disk space requirements. u Enables same stress data to be used for different loading events for fatigue analysis. (ie multiple events) u Auto-elimination can be used to select entities prior to fatigue analysis to speed up analysis. n Disadvantages: u Static FE analysis requires some kind of constraint which can be unrealistic. u Insufficient accuracy where the natural frequencies of the system are close to the frequency content of the loading.

S22-9 PAT318, Section 22, March 2005 Transient Analysis (Time Domain) Stress for combined loads calculated by FE point by point. Local Stress Histories Load Time Histories L2L2 A L1L1 L1L1 L2L2 Fatigue Analysis For long time histories, issues with solution time and disk space requirements

S22-10 PAT318, Section 22, March 2005 Types of Transient Analysis n Direct (Integration) Transient Methods u Equations of motion of the complete system must be integrated through each time step u Expensive (in both CPU Time, and Disk Space) u Enable non-linear dynamic problems to be solved n Modal (Superposition) Transient Methods u Dynamics and degrees of freedom of system are reduced to a set of modes and therefore much quicker to solve than direct method. u Requires an appropriate set of modes to be selected. u Restricted to linear problems (and commonly used)

S22-11 PAT318, Section 22, March 2005 Mode 1 × 1.5 Mode 2 × Mode 3 × Response Mode Shapes Modal Coordinates Mode 1 Mode 2 Mode 3 The concept of Modal Superpositioning

S22-12 PAT318, Section 22, March 2005 Transient using Modal Superposition - repeat for each node/element Stress for Mode Shapes Local Stress Histories Modal Responses A r1r1 Mode 2 1A * r 1 (t) + 2A * r 2 (t) +... = A (t) r2r2 Mode 1

S22-13 PAT318, Section 22, March 2005 Modal Superposition Calculate Modal Stresses (using NASTRAN Sol 103) Calculate modal responses using Modal Transient Analysis (NASTRAN Sol 112, SDISP(punch) = ALL)

S22-14 PAT318, Section 22, March 2005 Transient Analysis n Advantages: u Includes dynamic effects where the natural frequencies of the system are close to the frequency content of the loading. u Systems can be analysed dynamically without any artificial constraints. u Modal transient is less computationally intensive than direct transient. n Disadvantages: u Transient is more computationally intensive than static analysis. u Disk space requirements very large to store stress state at each time step. u Each loading event is a separate analysis. u Cannot easily locate, critical elements, before fatigue analysis.

S22-15 PAT318, Section 22, March 2005 Transient using Modal Superposition n The practical method of performing a modal transient analysis for fatigue is to combine modal stress and modal responses within MSC-Fatigue. n This is computationally identical to the procedure performed within NASTRAN and has several advantages including reduced disk space.

S22-16 PAT318, Section 22, March 2005 Enables the dynamic response of the structure to be simulated without the disadvantage of storing the transient response for each node / element of the model. Resonant Effects Accounted Method Is Analogous to the Quasi-Static Approach- Modal Participation Factors Associated With Modal Stresses Used in combination with Multi Body Dynamics solution (MDB) allows effective transient analysis of full assemblies Advantages of Modal Superposition Transient

S22-17 PAT318, Section 22, March 2005 Requires judicious choice of number of modes in the Normal Modes analysis. Use of Residual Vector option in MSC.Nastran is recommended Requires appropriate translation of modal responses into time history format. Currently supported for MSC.Nastran using SDISP from punch file Disadvantages of Modal Superposition Transient

S22-18 PAT318, Section 22, March 2005 FATIGUE IN THE FREQUENCY DOMAIN

S22-19 PAT318, Section 22, March 2005 Frequency Domain WHY USE FREQUENCY DOMAIN? PSD frequency PSD Stress frequency Transfer Function time Wind speed Time Domain OutputInput time Hub Stress

S22-20 PAT318, Section 22, March 2005 BENEFITS OF FATIGUE CALCULATION IN FREQUENCY DOMAIN n Analyse structures with dynamic responses to random loading without requiring full transient analysis n Fatigue analysis is relatively rapid n Analysis can be included much earlier in the design cycle n Ability to analyse what if scenarios interactively

S22-21 PAT318, Section 22, March 2005 Time Domain Frequency Domain Fast Fourier Transform (FFT) (throw away phases) Inverse Fourier Transform (IFT) (create random phases) Time in seconds Response variation TIME TRANSIENT or FREQUENCY DOMAIN? Frequency domain analysis can account for dynamic (resonant) effects

S22-22 PAT318, Section 22, March 2005 Random Vibration (PSD Stress) Stress PSD response for combined PSD loads calculated by FEA for each frequency of interest. Local Stress PSD Load PSD Inputs ^2/Hz L1L1 L2L2 Fatigue Analysis g ^2/Hz Probability Density (Dirlik or Narrow Band)

S22-23 PAT318, Section 22, March 2005 – Resolution of stresses onto Principal planes – Multi input loads – Correlation effects using Cross PSDs – Stress tensor stationarity checks – Calculate fatigue life from PSDs – Uses 7 solution methods including; Dirlik, Steinberg and Narrow Band solutions Vibration Fatigue Fatigue analysis in the frequency domain Vibration Fatigue Fatigue analysis in the frequency domain

S22-24 PAT318, Section 22, March 2005 Random Vibration (PSD Stress) n Advantages u Suitable for loading that can be described by PSD (random, stationary and gaussian loading) i.e. wind gusts and sea states. Also simulating PSD shaker tests u Includes dynamic and resonant effects u More efficient for these types of problems than simulating a very long time history with a transient analysis n Disadvantages u Underlining hypothesis (Gaussian, Stochastic, Ergodic )

S22-25 PAT318, Section 22, March 2005 Summary Method Disk Space CPU Time Hot spot detection before analysis Difficulties? Static SmallQuickOK Realistic constraints, no dynamic effects Transient (Modal)Large Medium- Slow DifficultMode selection MBD Modal Superposition Medium OKModel Reduction Vibration (PSD)Medium Difficult Assumptions of stationary, random loading

S22-26 PAT318, Section 22, March 2005 HOW DO WE CALCULATE DAMAGE? Material (S-N analysis) Geometry (S-N Analysis) Fatigue Analysis (Vibration Fatigue) Post Processing Optimization & Testing Loading (PSD)

S22-27 PAT318, Section 22, March 2005 HOW DO WE CALCULATE DAMAGE? FATIGUE MODELLER BLACK BOX M0M0 M1M1 M2M2 M4M4 Transfer Function PSD Transient Analysis RAINFLOW COUNT TIME HISTORY TIME DOMAIN Steady state or FATIGUELIFE STRESS RANGE HISTOGRAM FATIGUELIFE PDF FATIGUE LIFE STRESS RANGE HISTOGRAM FREQUENCY DOMAIN

S22-28 PAT318, Section 22, March 2005 frequency Area of spike = amplitude of sin wave | FFT | Magnitude of FFT Argument of FFT time A Time history Single sinusoidal eddy of frequency, amplitude A and initial phase angle The argument of the FFT gives the phase angle of the sinusoidal wave FFT WHAT DOES AN FFT TELL US?

S22-29 PAT318, Section 22, March 2005 WHAT DOES AN FFT TELL US? n The FFT is a complex number given with respect to frequency. A sine wave of frequency, amplitude A and initial phase angle is represented in the frequency domain by a spike occurring at along the frequency axis. If the magnitude of the complex FFT is plotted, then the area under the spike is found to be the amplitude A of the sine wave. When the argument of the complex FFT is plotted then the area is found to be initial phase angle of the sine wave.

S22-30 PAT318, Section 22, March 2005 frequency PSD In a PSD we are only interested in the amplitude of each sine wave and are not concerned with the phase relationships between the waves. The area under each spike represents the Mean Square of the sine wave at that frequency We cannot determine what the phase relationships between the waves are any more Definition PSD = def | FFT | 2 1 2T WHAT IS A PSD?

S22-31 PAT318, Section 22, March 2005 (Stress) 2 Hz Frequency, Hz Gk(f)Gk(f) fkfk In practice, m 0, m 1, m 2 and m 4 are sufficient to compute all of the information required for the subsequent fatigue analysis MOMENTS FROM A PSD

S22-32 PAT318, Section 22, March 2005 (Stress) 2 Hz Frequency, Hz Gk(f)Gk(f) fkfk E m m E P m m E E P m mm These statistical parameters are needed for subsequent fatigue analyses. EXPECTED ZEROS, PEAKS AND IRREGULARITY FACTOR FROM A PSD

S22-33 PAT318, Section 22, March 2005 = upward zero crossing = peak time Stress (MPa) 1 second Number of upward zero crossings, E[0] = 3 Number of peaks, E[P] = 6 Irregularity factor, = E[0] E[P] = 3 6 Time History x x x x x x x EXPECTED ZEROS, PEAKS AND IRREGULARITY FACTOR FROM THE TIME SIGNAL

S22-34 PAT318, Section 22, March 2005 The probability of the stress range occurring between To get pdf from rainflow histogram divide each bin height by p(S) P(S i ) Stress Range (S) dS S 2 and S dS 2 ii PSds i (). S S t d S dS = bin width t total number of cycles PROBABILITY DENSITY FUNCTIONS (PDFS)

S22-35 PAT318, Section 22, March 2005 where; A widely applicable solution developed after extensive Monte Carlo simulation of a wide range of likely stress response conditions DIRLIK SOLUTION

S22-36 PAT318, Section 22, March 2005 Dirlik Narrow Band Tunna HancockWirsching Chaudhury & Dover Steinberg The best method in all cases Developed for offshore use Railway engineering (UK) Electronic components (USA) } The original solution OTHER SOLUTION METHODS }

S22-37 PAT318, Section 22, March 2005 SUMMARY OF FEATURES n Calculate fatigue life from PSDs n Uses 7 solution methods including; Dirlik, Steinberg and Narrow Band solutions n Ability to handle multiple, partial and fully correlated loads n Mean Stress Correction n Palmgren-Miner Linear Damage n Material and Component S-N n Model Surface Conditions n Factor of Safety Analysis n Biaxiality Indicators nCode nSoft 8E RMS Power (Volts^2. Hz^) Frequency (Hz.) DISPLAY OF NOISE.PSD

S22-38 PAT318, Section 22, March 2005 PROCESS ALTERNATIVES n Use NASTRAN to calculate PSDs of Stress directly and use directly in MSC.Fatigue u Disadvantage : Only basic stress components available as output (no principals etc.) n Use NASTRAN to calculate complex transfer function between inputs and stress results. MSC.Fatigue combines transfer function with input PSDs and Cross spectra to calculate principal stresses vs freq. u Disadvantage: More data for MSC.Fatigue to process.

S22-39 PAT318, Section 22, March 2005 EXAMPLE PROBLEM: VIBRATION FATIGUE Example of vibration fatigue analysis of a bracket. Three load inputs. Critical area: around circular hole. Using transfer function method of vibration fatigue

S22-40 PAT318, Section 22, March 2005 SINGLE LOAD Time-domain Analysis Frequency-domain Analysis (static FE result) (At Frequency = 0 Hz) Frequency-domain Analysis (one of several frequencies)

S22-41 PAT318, Section 22, March 2005 TIME-DOMAIN LOADING INFO SETUP

S22-42 PAT318, Section 22, March 2005 FREQUENCY-DOMAIN LOADING INFO SETUP

S22-43 PAT318, Section 22, March 2005 FREQUENCY-DOMAIN LOADING INFO - MULTIPLE PSDS -

S22-44 PAT318, Section 22, March 2005 Static case: Combined loads Vibration: correlated loads Vibration: uncorrelated loads RESULTS:

S22-45 PAT318, Section 22, March 2005 EXERCISE n Perform Quick Start Guide Chapter 14 Exercise, Vibration Fatigue n Perform Quick Start Guide Chapter 18, Section 9, Exercise, Modal Superposition n Be sure to ask for help if theres anything you dont understand

S22-46 PAT318, Section 22, March 2005