Random Vibration Fatigue Analysis of a Notched · PDF filerandom vibration fatigue analysis in...

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Int. J. Mech. Eng. Autom. Volume 2, Number 10, 2015, pp. 425-441 Received: August 6, 2015; Published: October 25, 2015 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 2PQ, UK Corresponding author: Giovanni de Morais Teixeira ([email protected]) 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 -2 ), approximately 9.81 m s -2 g eqv PSD von Mises equivalent stress (MPa 2 Hz -1 ) g ij Components of the input PSD matrix (G 2 Hz -1 ) G Input PSD matrix (G 2 Hz -1 ) h Stress vector (MPa G -1 ) k Fatigue curve coefficient M n n-th spectral moment (Hz n MPa 2 Hz -1 ) N f Number of cycles p, PDF Probability density function PSD Power spectral density (MPa 2 Hz -1 ) 0 Standard deviation (MPa 1/2 ) S Stress component (MPa) S a Stress amplitude (MPa) S R 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 [2]. 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
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  • Int. J. Mech. Eng. Autom. Volume 2, Number 10, 2015, pp. 425-441 Received: August 6, 2015; Published: October 25, 2015

    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 2PQ, UK

    Corresponding author: Giovanni de Morais Teixeira ([email protected])

    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-2), approximately 9.81 m s-2 geqv PSD von Mises equivalent stress (MPa2 Hz-1) gij Components of the input PSD matrix (G2 Hz-1) G Input PSD matrix (G2 Hz-1) h Stress vector (MPa G-1) k Fatigue curve coefficient Mn n-th spectral moment (Hzn MPa2 Hz-1) Nf Number of cycles p, PDF Probability density function PSD Power spectral density (MPa2 Hz-1)

    0 Standard deviation (MPa1/2)

    S Stress component (MPa) Sa Stress amplitude (MPa) SR Stress range (MPa)

    dSR Stress range step (MPa) T Time (s) Z Normalized stress range Xm 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 [2]. 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

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    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 Dirliks PDF (probability density function) that is integrated to get damage.

    This paper is organized as follows: Section 2 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.

    2. Finite Element Modelling

    The performed simulations and fatigue analysis here

    Fig. 1 Frequency domain fatigue analysis flowchart.

    are inspired on actual experiments [3] for the notched beam sketched in Fig. 2. In the experiments the region outlined as restrained nodes in Fig. 2 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 10036 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

    Fig. 2 Finite element model used in the studies.

    Fig. 3 Static structural analysis1G of vertical loading.

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    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 Mass21 (ANSYS), *MASS (ABAQUS) and CONM2 (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

    Fig. 4 Large mass approach: preparing the modal analysis.

    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 10.95 Hz. The highest frequency in the searched interval is 510.6 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, 2 and 4 on the response of the notched beam subjected to a vertical acceleration. The 4th mode is 2 orders of magnitude lower than the 1st mode. The 2nd 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. 5 Mode shapes from the modal analysis.

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    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 (2)