Random Vibration Fatigue Analysis of a Notched Aluminum Beam

Transcription

1 Int. J. Mech. Eng. Autom. Volume, Number 10, 015, pp Received: August 6, 015; Published: October 5, 015 International Journal of Mechanical Engineering and Automation Random Vibration Fatigue Analysis of a Notched Aluminum Beam Giovanni de Morais Teixeira Research and Development, Dassault Systemes Simulia, Sheffield S10 PQ, UK Corresponding author: Giovanni de Morais Teixeira (giovanni.demorais@3ds.com) Abstract: The purpose of this paper is to present a case study where the fe-safe random vibration fatigue approach has been successfully employed. It describes the FEA (finite element analysis) preparation (an aluminum beam) and the necessary steps in fe-safe to perform a fatigue analysis entirely in frequency domain. The method behind fe-safe combines generalized displacements obtained from SSD (steady state dynamic) finite element simulations to modal stresses to get FRF (frequency response functions) at a nodal level, where stress PSDs are evaluated in order to get spectral moments, which are the building blocks of the PDF (probability density function) used to count cycles and evaluate damage. The loading PSDs are then converted into acceleration time histories that allow fatigue to be evaluated in the time domain likewise. Results show a very good agreement between time and frequency domain approaches. Keywords: Fatigue, random vibration fatigue, high cycle fatigue, multiaxial fatigue, power spectral density, frequency domain fatigue. Nomenclature A Von Mises quadratic operator b Fatigue curve exponent D Fatigue damage E[P] Expected number of peaks (peaks per second) f Frequency (Hz) F Force (N) G Gravity of Earth (m s - ), approximately 9.81 m s - g eqv PSD von Mises equivalent stress (MPa Hz -1 ) g ij Components of the input PSD matrix (G Hz -1 ) G Input PSD matrix (G Hz -1 ) h Stress vector (MPa G -1 ) k Fatigue curve coefficient M n n-th spectral moment (Hz n MPa Hz -1 ) N f Number of cycles p, PDF Probability density function PSD Power spectral density (MPa Hz -1 ) 0 Standard deviation (MPa 1/ ) S Stress component (MPa) S a S R Stress amplitude (MPa) Stress range (MPa) ds R Stress range step (MPa) T Time (s) Z Normalized stress range X m Mean frequency 1. Introduction The random vibration fatigue or frequency domain fatigue is a new approach in fe-safe. It is based on the vibration theory for linear systems subjected to random Gaussian stationary ergodic loadings [1]. When a structure responds dynamically to an input excitations there are two possibilities in terms of FEA (finite element analysis): transient and SSD (steady state dynamic) analysis []. Both can take advantage of the MSUP (modal superposition) technique provided the system is linear or any present non-linearity does not affect the regions of interest. The SSD analysis is much faster than the Transient Analysis and it is one of the building blocks of the random vibration fatigue analysis in fe-safe, shortly called PSD analysis. PSD stands for power spectrum

2 46 Random Vibration Fatigue Analysis of a Notched Aluminum Beam density. Fig. 1 shows the PSD Analysis flowchart that describes the analysis procedure in fe-safe. Finite element modal analysis and SSD analysis are combined to get the FRF (frequency response functions) in terms of stresses for every node in the component or structure. These FRFs are scaled by the input PSDs to get either PSD projections on critical planes or von Mises equivalent PSDs. Whatever the choice, these obtained PSDs are used to evaluate the first four spectral moments to compose the Dirlik s PDF (probability density function) that is integrated to get damage. This paper is organized as follows: Section describes the computer model (discretization in terms of finite element mesh), the loading and boundary conditions; Section 3 shows the modal and steady state dynamic analyses used to obtain the modal stresses and generalized displacements, also known as modal participation factors; Section 4 give the finite element dynamic results which are combined to the loading PSDs to evaluate fatigue damage; in Section 5, we use the modal superposition technique and acceleration time signals equivalent to the given PSDs to perform a transient analysis equivalent to the SSD analysis in Section 3; in Section 6, we apply the scale and combine technique in fe-safe to match modal participation factors and modal results and get stress tensors to evaluate fatigue using a standard time domain algorithm; Section 7 gives conclusions. are inspired on actual experiments [3] for the notched beam sketched in Fig.. In the experiments the region outlined as restrained nodes in Fig. is attached to a vertical rod (Z direction) which is the source of the vibration. The vibrational experiment in the present paper is performed in time and frequency domain so that a fair comparison can be established. It is important to keep the FEM (finite element model) small because the correspondent time domain transient analysis is computationally very expensive. In this study, the mesh contains 1793 second order hexahedral elements and nodes. Fig. 3 shows the von Mises stresses for the beam under 1G of vertical loading. The maximum von Mises stress is 8 MPa, on the edge of notch 1. Static structural analysis is not a requirement for the random vibration fatigue approach. However, they provide useful information about the expected level of stresses as the loading frequency tends to 0 Hz, an information that can be used to calibrate the SSD analysis, also known as harmonic analysis. There are several ways of performing a harmonic analysis. Common types of harmonic loads include forces, moments, pressures, velocities and accelerations. A typical situation in a dynamic analysis is when accelerations are prescribed at the supports of a structure or component. Some finite element packages. Finite Element Modelling The performed simulations and fatigue analysis here Fig. Finite element model used in the studies. Fig. 1 Frequency domain fatigue analysis flowchart. Fig. 3 Static structural analysis 1G of vertical loading.

3 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 47 offer the possibility of defining local acceleration, but usually acceleration is the kind of loading defined globally in a finite element model, i.e., specified at all nodes. Then, to keep the generality, the LMM (large mass method) is employed here. The idea is to attach a very large concentrated mass (the order of 1e7 to 1e10 times the mass of the whole structure) to the supports where the accelerations are supposed to be applied in the model. Examples of lumped masses in finite element packages are Mass1 (ANSYS), *MASS (ABAQUS) and CONM (NASTRAN). Fig. 4 shows the large lumped mass linked to the region of interest using RBEs (rigid body elements). According to LMM principle [4] forces can be used rather than accelerations, with the same effect on the component. The magnitude of the force must be equal to the product of the large mass and the desired acceleration (Fig. 4). In ANSYS Workbench, the user can define a remote point, set its behavior (rigid or deformable) and create a point mass attached to it. Remote Forces and Remote Displacements can be defined at remote points. 3. Frequency Domain FE Analysis The first step in the random vibration fatigue approach is the modal analysis. It is fundamentally important to have the most accurate modal analysis as possible. In this study, an artificial large mass is employed; therefore it is necessary to limit the frequency search range in order to avoid rigid body modes. Finite element packages usually offer the option of defining the number of modes to find and frequency search range. In this simulation, 10 modes were requested and the frequency range was set to 0.1-1e8 Hz. The node associated with the large mass must have all its degrees of freedom removed, except UZ (displacement at Z vertical direction). All the other displacements and rotations are set to 0 (UX = UY = ROTX = ROTY = ROTZ = 0). The reason for not constraining the displacement at Z direction is that this is the loading direction, i.e., in the harmonic analysis the beam will be excited by a harmonic acceleration at Z direction. Stresses are requested as output and no damping is required at this point. Table 1 and Fig. 5 show the results of the modal analysis. The lowest frequency found is Hz. The highest frequency in the searched interval is Hz. The stress results in the modal analysis do not mean anything until the harmonic analysis is performed. There is no special requirement for the number of modes that needs to be evaluated in the modal analysis. They vary from case to case, depending on the loading and boundary conditions. Usually, the first 3 or 4 modes are enough to well represent a dynamic response. Fig. 6 shows the influence or participation of modes 1, and 4 on the response of the notched beam subjected to a vertical acceleration. The 4th mode is orders of magnitude lower than the 1st mode. The nd mode is more than 1 order of magnitude lower than the 1st mode. The 3rd mode can be neglected in this case. The second step in the random vibration fatigue Table 1 Modal analysis results. Fig. 4 analysis. Large mass approach: preparing the modal Fig. 5 Mode shapes from the modal analysis.

4 48 Random Vibration Fatigue Analysis of a Notched Aluminum Beam Fig. 7 Modal participation factor file: file_1.mcf. Fig. 6 Modal participation factors magnitudes. approach is the harmonic analysis. There are essentially two ways of performing a harmonic analysis: (1) through MSUP (modal superposition) analysis and () through full harmonic analysis. The modal superposition harmonic analysis is the chosen approach in fe-safe for the following reasons: (1) MSUP harmonic analysis is faster than full harmonic analysis. () It provides MPFs (modal participation factors), which can be scaled and combined to the modal results to get the steady state response. The MPF weight the contribution of each mode shape included in the analysis. (3) The results files are much smaller and easier to manipulate than the ones generated by full harmonic analysis. It is not necessary to prescribe boundary conditions for the harmonic analysis. The frequency range is set to Hz. A constant damping ratio is assumed to be 1.8e-. Clustering the results around the resonant frequencies is requested and the cluster number is set to 0. Stresses are requested at all frequencies as output. A harmonic force is defined as 9800e10 N (Z direction), which produces the same effect as an acceleration of magnitude 1G (9800 mm s - ). The command HROPT, MSUP, nmodes, 1, YES tells ANSYS to output the modal coordinates to a text file named jobname.mcf, which is an ASCII file as Fig. 7 shows. For every frequency, there is a complex number (rectangular format) representing the contribution of each mode. It is recommended to rename these files to match input channels numbers when the analysis involves multiple channels. For instance, file_1.mcf corresponds to channel 1; file_.mcf corresponds to channel ; and so on. This example is a single channel analysis; acceleration at Z direction on the remote point shown in Fig. 4. When the harmonic analysis is finished FRF (frequency response functions) as the one in Fig. 8 can be evaluated for every node and every stress component in the FEM (finite element model). At this point, the FRFs can be combined to the loading PSD matrix (input) to get stress PSDs in order to evaluate the spectral moments at the nodal level. Premount [5] describes in greater detail how to evaluate von Mises PSDs out of frequency response functions and his method is briefly presented in the next section. Fig. 8 shows the expected magnitudes for the stress component S x at every frequency in the range Hz. The peaks correspond to the resonant frequencies. The response at the frequency 11 Hz is the highest (S x = 71 MPa), confirming the dominance of the first mode in this situation. The magnitude of the responses also depends on the assumed (or measured) damping. The response at the frequency 54.6 Hz is the second highest (S x = 66.9 Hz). Some modes may not be excited in a Harmonic Analysis (as the 3rd mode in this example) or its contribution is so small (compared to the other modes) that it is not perceived in the dynamic response. Obviously, the FRFs change from node to node within the finite element model. 4. Frequency Domain Fatigue Analysis The third step in the random vibration fatigue approach is to define the input PSDs file, named psd_file.psd in this example, Fig. 9. This file must follow the convention described in the fe-safe manual. Number of channels is set to 1 and there are no cross PSDs. The only PSD defined is the auto PSD, characterized only by its magnitudes (9 points in the psd_file.psd file, Hz).

5 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 49 Fig. 8 Dynamic response, stress component Sx at the critical node. generated by the harmonic analysis. Click on Files that provide Power Spectral Density (PSD) data to select the PSD input file described in Fig. 9. The steps 3 to 6 in Fig. 14 load the necessary files in fe-safe. After hitting the OK button a dialog window pops up to ask about pre-scanning. Select YES to get to the window shown in Fig. 15. Be sure only Stresses are selected and click on Apply to Dataset List. In this window, 10 increments are shown, corresponding to the results for the 10 modes defined in the modal analysis. After clicking Okay (step 8) the Units Window appears, Fig. 16. Select MPa as the working unit. The fatigue curve for the Aluminum 6061 T6 is defined by the two points in Fig. 17 (N f = 1e4, S a = 07 MPa; N f = 1e6, S a = 11 MPa). Fig. 9 Input PSD file: psd_file.psd. Fig. 10 shows the histogram of the PSD magnitudes in the PSD file. Notice that this input PSD scales mode more than mode 1, meaning that the output stress PSD must show the highest peak around the frequency 54.6 Hz. Fig. 11 corresponds to the SN fatigue curve for the Aluminum 6061 T6, the material used in the simulations. Having all the input files (FE results and input PSD) prepared, fe-safe can be launched and the project folder created, Fig. 1. In this folder, we have the following files: modal.rst (containing the FE modal results), file_1.mcf (containing the modal participation factors for the unit load harmonic analysis) and psd_file.psd (containing the input PSD for the single channel). In fe-safe interface, go to File > Open Finite Element Model for PSD analysis to get the dialog window shown in Fig. 13. Click on Source FE model to select the file that contains the modal results. Click on Files that provide Modal Participation Factor (MPF) data to select the file(s) Fig. 10 Input PSD representing the acceleration loading. Fig. 11 Fatigue curve for aluminum 6061 T6. Fig. 1 Fe-safe project directory dialog box.

6 430 Random Vibration Fatigue Analysis of a Notched Aluminum Beam Fig. 13 Open finite element model for PSD analysis. Fig. 14 Selecting the files for the PSD analysis. Fig. 15 Selecting datasets in the pre-scan operation.

7 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 431 Fig. 16 Selecting the appropriate units in fe-safe. Fig. 17 Definition of the fatigue SN curve. Select the Aluminum 6061-T6 (that has just been created) and double click Material in the Analysis Settings (step 1 in Fig. 18) to assign the material property to all groups. Go to Loading Settings (Fig. 19) and define the exposure time, setting 60 to Length per repeat in seconds. Then the PSD block is a 60 s loading block, meaning that fe-safe lives represent minutes in the Results File. Go to exports, Fig. 0, and at the Tab List of Items type the element numbers that needs to be further investigated. The critical node belongs to element 1069 shown in Fig. 0. Check Export PSD Items* at Tab Log for Items, as in Fig. 1, step 16. fe-safe is requested to output the spectral moments (m 0, m 1, m, m 4 ) and stress PSDs in the log file during the fatigue analysis for the defined elements and nodes. Obviously, the critical nodes and elements are not known at the time of the fatigue analysis setup, unless Fig. 18 Assigning materials to element groups.

8 43 Random Vibration Fatigue Analysis of a Notched Aluminum Beam real tests are performed prior to the simulations. Eq. (1) evaluates the fatigue damage in Dirlik s method [6, 7]. The summation represents the integration of the PDF, Eq. (), over the range of stress ranges, S R. EPT b D Dirlik = SR psr dsr k 0 (1) Fig. 19 Defining the loading block. 1 D ps R = e + e +DZe M Q R 0 -Z -Z -Z 1 Q DZ R 3 () Fig. 0 Creating a list of items to be analyzed. Fig. 1 Z= Requesting PSD items to be exported. M, x m -γ 1 S R 1-γ -D D=, 1+D1 D= 0 1+γ 1-R M M E P =, 1 M x m = M M 0 M 4 γ -xm -D 1.5 γ -D 1 3-DR Q= (3) D D= D-D, R= 1-γ -D +D, The diagram in Fig. represents the PDF described in Eq. (). Ideally the integration of the PDF (equivalent to the area A) should result 1, meaning that 100% of the possibilities in the process were accounted for. But this would imply ad infinitum summation of Eq. (1). In practice, a good number for the summation upper limit is a number between 10

9 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 433 and 0, that provides a result (PDF integration) close to or higher. This upper limit in fe-safe is defined by the RMS stress cut-off multiple in Fig. 3 (step 0). Go to FEA Fatigue > Analysis Options and choose the PSD Tab. Make sure the PSD Response is von Mises (step 19 in Fig. 3) and the RMS stress cut-off multiple is 10 for the present experiment. The integration domain is the integration upper limit minus integration lower limit, Fig.. Then the field Number of stress range intervals (which defaults to 1000) controls the integration steps (ds R in Eq. (1)) by dividing the integration domain in even segments. These two numbers (upper limit and number of intervals) have an impact on accuracy and computation speed. The bigger they are the slower the calculation and the more accurate the fatigue results. It is recommended to start with fe-safe defaults (Fig. 3) and gradually change these values when needed. The PSD Response in this investigation is von Mises, evaluated according to Eq. (4). The symbol * stands for the complex conjugation. A is the quadratic von Mises operator. h i is the frequency response function for channel i. g ij are terms of the input PSD matrix G. N is the number of channels. g eqv is a scalar representing the von Mises equivalent stress. N N j* i g eqv = h Ah gij i=1 j=1 (4) i T h x y z xy yz xz where, A, g11 L g1n G M O M. g 1 L g N NN NN Next click on Analyze (step in Fig. 4) and continue (step 3). When the Analysis is finished, click on open results folder (Fig. 5, step 4) to get the results file. The worst life-repeats shown in Fig. 5 correspond to 01.7 s (3.363 min). Fig. 6 shows the life contour plot for the notched beam. Node 7 (that belongs to element 1069) is the critical, where life is the lowest. In the output location there is a file named modalresults.log, where detailed Fig. Dirlik s probability density function. Fig. 3 Fe-safe PSD analysis options.

10 434 Random Vibration Fatigue Analysis of a Notched Aluminum Beam Fig. 4 Running the fatigue analysis. Fig. 5 Analysis completed dialog. Fig. 6 Fatigue life contour plot. information about element 1069 can be found. Spectral moments and equivalent stress PSD are the essential additional information related to node element 1069 in the Log File. As the exports (Fig. 19) do not specify the node, all the nodes attached to element 1069 are exposed in the diagnostics. The worst node is shown in Fig. 6. The 0th spectral moment corresponds to the variance of the stress PSD at node 7. The RMS (root mean square) of the variance is the standard deviation represented by 0. In a normal distribution the probability of finding a stress amplitude within 3 times the standard deviation (in this case 3 0 = 94 MPa) is 99.73%. SQRT (M /M 4 ) corresponds to the expected number of peaks per second and SQRT (M /M 0 ) corresponds to the upward mean crossing per second. The equivalent stress PSD for node 7 is plotted in Fig. 7. It is worth mentioning that the frequency range in the diagram is the intersection of the ranges in the following files: file_1.mcf and psd_file.psd. An interesting aspect of this particular Stress PSD is that its highest peak occurs at the second resonant frequency, despite the fact the first mode is dominant

11 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 435 (Fig. 8). The four spectral moments are evaluated from this PSD curve, according to Eq. (5): n N n k Δ (5) M = f PSD k f k =1 It is important to emphasize that the information provided in the log file (written in \jobs\job_01\fe-results\jobname.log) is enough to build Dirlik s PDF. Spectral moments can be extracted from the PSD in Fig. 8. The PDF, Eq. (), can be evaluated from the spectral moments and from Dirlik s derived constants Eq. (3). In his Ph.D. thesis, Benasciutti [8] discusses in great detail the available frequency domain approaches and proposes a new method which is based on a combination of level crossing and range count PDFs, balanced by a factor that weights the narrow band and broad band contribution to the fatigue damage. His work opened the door to a more comprehensive approach were mean and residual stresses could then be incorporated by using a multi-variate distribution concept. Fig. 7 Von Mises PSD for Item e Fig. 8 Fatigue life results. 5. Time Domain FE Analysis In order to check the results obtained by the random vibration fatigue approach, the notched beam is also analyzed in the time domain. The challenge is to guarantee the time domain approach is equivalent to the frequency domain, otherwise the comparison is useless. The first step in this direction is to get an acceleration history that is compatible with the prescribed PSD (Figs. 9 and 10). The problem can be stated as the generation of random time series with prescribed power spectra and there are several ways of solving it [9]. In general lines, the procedure can be summarized as follows: (1) Choose the frequencies f i in the PSD periodogram (Fig. 10); () Choose random phase angles i to match those frequencies; (3) Evaluate the amplitudes from the given PSD A = G Δf, i i i where G i represents the PSD amplitudes and f i is the frequency bandwidth (constant); (4) Sum the individual spectral components for every time t. The sampling rate should be at least ten times the highest spectral frequency. In the equation below Y is the resultant time vector. If the PSD units are (G Hz -1 ), for example, the time history units are G (multiples of the standard gravity acceleration). n sinπ Y t = A ft+φ i=1 i i i (5) Assess the quality of the statistical distribution of the obtained acceleration history. Check its Gaussianity by evaluating skewness, kurtosis, standard deviation, etc. Compare the variance of both PSD and time series and check if the number of peaks and zero crossings are coherent with the spectral moments. Fig. 9 shows the first 3 seconds of the synthetized acceleration history that corresponds to the PSD in Fig. 10. The length of this signal is 10 s. The analysis in the time domain needs to be based on the MSUP technique and LMM approach. The finite

12 436 Random Vibration Fatigue Analysis of a Notched Aluminum Beam Fig. 9 Synthetized acceleration time history. element model is the same (in terms of mesh definition) and the forces exciting the transient analysis have the magnitudes of (ACC x 9800e10 mm s - ). ACC are the acceleration magnitudes in Fig. 9. The acceleration file contains 3767 acceleration records. This is the number of transient simulations that need to be performed. The result of the MSUP transient analyses is the file msuptrans.mcf. It contains scale factors to be multiplied by the modal stresses in order to evaluate the stress history for every node in the model. Fig. 30 shows the components of the stress history for node 7 at element This node is referred in fe-safe as item This example is practically a uniaxial fatigue problem since the component S x (stress in the X direction) is much larger than all the other stress components. S x magnitudes are in the range -300 to 300 (Fig. 31). If the loadings are narrow band there is a good chance to get sensible results using Bendat s approach [10], which tends to be conservative. Dirlik s solution can be used for narrow and broad band processes, therefore chosen to be the approach used in this study. Lalanne has also developed an arbitrary bandwidth approach [11] that has served as the foundation to the latest TB method (Tovo & Benasciutti method). Both TB and Lalanne Methods are as robust as Dirlik, with the advantage of being less empirical. 6. Time Domain Fatigue Analysis The time domain analysis starts with the creation of a project direction, Fig. 3, where the following files need to be copied to: modal_factors_for_msup_analysis.txt (containing the modal factors for the transient analysis) and msuptrans.rst (containing the FE modal results). Fig. 30 Large mass approach for MSUP transient analysis. Fig. 31 Stress components history at element Fig. 3 Fe-safe project directory dialog box. In fe-safe interface, right click on Current FE Models (Fig. 33, step 1) and choose Open Finite Element Model. Select the msuptrans.rst file and click on YES when asked about pre-scanning. Make sure only Stresses are selected and check whether 10 increments are found in the file, Fig. 34. They correspond to the 10 modes requested in the modal analysis. Select MPa as the units for the stresses, Fig. 35, and keep the default for the other units. Right click on Loaded Data Files and select Open Data Files (Fig. 36) and choose the file modal_factors_for_msup_analysis.txt. This file contains rows and 10 columns. Each column scales a particular modal result. Column 1 scales modal

13 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 437 Fig. 33 Opening finite element model for transient analysis. Fig. 34 Pre-scanning the finite element model. Fig. 35 Defining units in fe-safe. Fig. 36 Loading the modal participation factors.

14 438 Random Vibration Fatigue Analysis of a Notched Aluminum Beam stresses in dataset 1, column scales modal stresses in dataset, and so on. Click on the first item under the modal factors for MSUP analysis in the Loaded Data Files and on the fe-safe plot shown in Fig. 37 (steps 7 and 8) to see the diagram for the scale factors in column 1. The material properties must be defined next. It is the same Aluminum 6061-T6 shown in Figs. 11 and 17. Choose von Mises algorithm (no mean stress correction) by following the steps 9 to 13 in Fig. 38. This study is using von Mises as the fatigue method for both frequency and time domain analyses. Right click on Loading Settings panel and clear all loadings according to Fig. 39. Click on Dataset 1 and on load file 1 (steps 15 and 16 in Fig. 40). In loading settings click on add (step 17) and load history (step 18). Follow these steps for Datasets 1 to 10 and load files #1 to #10 to create the block displayed in Fig. 41. This procedure corresponds to the scale and combine technique in the time domain. Click on Analyze and continue (Fig. 4) after checking the fatigue setup displayed. In Fig. 43, the worst element and node is being reported as 4.69, which is equivalent to 46.1 s. Click on Open results folder to see the life contour plot on the notched beam. Fig. 44 shows the life contour plot for the notched beam. Node 7 (that belongs to element 1069) shows a fatigue life of 53 s, since the loading block is equivalent to 10 s. Compare the contour plots in Figs. 6 and 44 (Time and Frequency Domain) and check Fig. 37 Plotting participation factors for mode 1. Fig. 38 Choosing the fatigue algorithm.

15 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 439 Fig. 39 Clearing the loading definitions. Fig. 40 Creating a loading block. Fig. 41 Defining the loading block. Fig. 4 Running the fatigue analysis.

16 440 Random Vibration Fatigue Analysis of a Notched Aluminum Beam Fig. 43 Worst life-repeats result. Fig. 44 Fatigue life contour plot. how close the results are. For node 7 the difference in the reported lives (53 s and 0 s) is 0.%. 7. Conclusions This paper has shown how to perform a Fatigue Analysis in the frequency domain using the software fe-safe. It also presented a counter example in the time domain for comparison. The predicted life at the failure location differs by 0.%. Considering the differences in the FE modelling and in the Fatigue Methodologies a bigger difference (in terms of fatigue lives) can be expected between time and frequency domain approaches. The task of synthetizing time signals compatible with a given PSD can introduce noise and undermine the equivalence of the modal superposition transient analysis. A probability density function in the frequency domain replaces the traditional rainflow counter in the time domain. Therefore there is no such thing as definite number of cycles in the frequency domain, but the probability of finding cycles of given amplitude. Moreover, the random vibration fatigue approach is based on linear vibration theory and statistical assumptions that may not be present in some circumstances. It is relevant to mention that mean and residual stresses have not been discussed in this study, despite its importance. Plasticity correction is another subject that deserves more attention and needs to be addressed separately. Accuracy is the central theme of this paper, but actually speed is what makes frequency domain so attractive. A frequency domain implementation that solves a problem 1000 times faster than an equivalent time domain implementation brings the opportunity to solve much larger problems. Fe-safe can handle multiple channels and therefore addresses multiaxial fatigue problems. If non-proportionality is expected is recommended to switch from von Mises to Critical Plane approach. In summary, the fe-safe random vibration fatigue is a powerful and fast approach that can provide accurate results when compared to equivalent approaches in the time domain. It can also be used to design accelerated tests that may be of high economic importance or used to perform a quick scan on very large problems that would take weeks to be solved in the time domain. The tool allows such problems to be solved faster and allows important adjusts to be made before either a more detailed time domain investigation takes place or prototypes are manufactured. References [1] A. Nieslony, E. Macha, Spectral Method in Multiaxial Random Fatigue, Lecture Notes in Applied and Computational Mechanics, Vol. 33, Springer Berlin Heidelberg, 007. [] G.M. Teixeira, R. Hazime, J. Draper, D. Jones, Random vibration fatigue: Frequency domain critical plane approaches, in: ASME International Mechanical Engineering Congress and Exposition, San Diego, California, Nov. 15-1, 013. [3] V.K. Nagulapalli, A. Gupta, S. Fan, Estimation of fatigue life of aluminium beams subjected to random vibration, in: 007 IMAC-XXV: Conference & Exposition on Structural Dynamics, Orlando, Florida, Feb. 19-, 007. [4] Y.W. Kim, M.J. Jhung, Mathematical analysis using two modelling techniques for dynamic responses of a structure subjected to a ground acceleration time history, in: ASME 010 Pressure Vessels and Piping Division/K-PVP Conference, Washington, USA, Jul. 18-, 010. [5] A. Preumont, Random Vibration and Spectral Analysis, Kluwer Academic Publishers, 009.

17 Random Vibration Fatigue Analysis of a Notched Aluminum Beam 441 [6] G.M. Teixeira, Random vibration fatigue A study comparing time domain and frequency domain approaches for automotive applications, SAE Technical Paper , Detroit, April 014. [7] T. Dirlik, Application of computers in fatigue analysis, Ph.D. Thesis, University of Warwick, [8] D. Benasciutti, Fatigue analysis of random loadings, Ph.D. Thesis, University of Ferrara, Italy, 004. [9] M. Giuclea, A.M. Mitu, O. Solomon, Generation of stationary Gaussian time series compatible with given power spectral density, in: Proceedings of The Romanian Academy, Series A, Vol. 15, 014, pp [10] J.S. Bendat, A.G. Piersol, Measurement and Analysis of Random Data, Wiley, New York, [11] C. Lalanne, Mechanical Vibration and Shock, Volume V, Hermes Penton Ltd, London, 00.