Linköpings universitet SE–581 83 Linköping
Linköping University | Department of Management and Engineering
Master thesis, 30 ECTS | Mechanical Engineering
Spring 2018 | LIU-IEI-TEK-A--18/03040--SE
Evaluation of Finite Element
simulation methods for High
Cycle Fatigue on engine
com-ponents
Utvärdering av simuleringsmetoder för analys av
högcykelut-mattning på motorkomponenter
Óscar Pacheco Román
Supervisor : Stefan Lindström Examiner : Robert Eriksson
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Abstract
This document reflects the results of evaluating three computational methods to anal-yse the fatigue life of components mounted on the cylinder block; two currently in use at Scania and one that has been further developed from its previous state.
Due to the cost of testing and the exponential increase in computational power through-out the years, the cheaper computational analyses have gained in popularity. When a com-ponent is mounted in a fairly complex assembly such as an engine, simplifications need to be made in order to make the analysis as less expensive as possible while keeping a high degree of accuracy.
The methods of Virtual Vibrations, VROM and VFEM have been evaluated and com-pared in terms of accuracy, computational cost, user friendliness and general capacities. Additionally, the method VFEM has been further developed and improved from its previ-ous state.
A in-depth investigation regarding the differences of the methods has been conducted and improvements to make them more efficient are suggested herein. The reader can also find a decision matrix and recommendations regarding which method to use depending on the general characteristics of the component of interest and other factors.
Two components, which differ in complexity and mounting nature, have been used to do the research.
Keywords: High Cycle Fatigue, HCF, Virtual Vibrations, VROM, VFEM, Virtual Chan-nels, Random Vibrations, Engine Vibrations, FEM, Abaqus.
Acknowledgments
I want to thank Adam Eriksson and Fredrik Birgersson for their continuous support and guidance during the development of this project. It has been a real pleasure to work with such brilliant minds. Although I have learnt a lot from them, it is my wish and hope to, in some years, have the same knowledge of FEM as they do. Special thanks to Adam for his corrections and feedback of the report and for making my time during the thesis enjoyable.
I would also like to thank Fredrik Reuterswärd and Jonas Lenander for their participation in the steering committee of the project as well as to Jerk Svedman for letting me use his numerical model for the Flywheel housing analysis. Also to the rest of the NMBS group, who made my time in Scania memorable. Thanks also to the NMBT department for their assistance to the project.
Express my gratitude also to Scania in general and Fredrik Reuterswärd in particular for giving me the possibility to develop my Master Thesis at the NMBS department. I believe that, especially in engineering, theory is nothing without practice, and the commitment of companies like Scania to offer thesis projects is essential in the development of future engi-neers.
Thanks to Stefan Lindström for his academic support and detailed feedback, which has greatly contribute to give style and a professional appearance to this report.
Me gustaría dedicar este proyecto en particular y mis logros como estudiante en general a Pyly y Victor. Soy consciente de que todo lo que soy es en gran parte gracias a vosotros. Mis éxitos son el reflejo de vuestros éxitos como padres y nunca podré estar lo suficien-temente agradecido por todo lo que habéis hecho por mi. Gracias por no rendiros en los momentos en los que no quería aprender inglés, ni nadar, ni hacer muchas cosas que, desde mi inmadurez en aquellos tiempos, no consideraba importantes. Estoy muy satisfecho de vosotros.
Abbreviations
– NMBS: Base Engine Strength Analysis group at Scania.
– NMDD: Base Engine Dynamics, Acoustics & Tribology group at Scania. – NMBT: Base Engine Strength Testing group at Scania.
– FEA: Finite Element Analysis. – FEM: Finite Element Method. – CAD: Computer Aided Design. – CAE: Computer Aided Engineering.
– PSD: Power Spectral Density. The PSD of a time signal describes the distribution of power in frequency components for that signal.
– FRF: Frequency Response Function. An FRF expresses the relationship in the frequency domain between a linear, time-invariant input signal and the output it causes.
– SN curve: Stress-cycles curve. It is used to plot the fatigue life of a material against the applied stress amplitude.
– RMS: Root Mean Square.
– ROM: Resonance Order Method.
– FROM: Force based Resonance Order Method. – VROM: Virtual Resonance Order Method.
– INP-file: Input Data file. File containing the numeric model set-up for an Abaqus com-putation.
– ODB-file: Output Data Base file. File containing results of an Abaqus computation. – CSYS: Coordinate System.
– RBM: Rigid Body Motion. – RPM: Revolutions Per Minute.
Contents
Abstract iii
Acknowledgments iv
Abbreviations v
Contents vi
List of Figures viii
List of Tables x
1 Introduction 1
1.1 Motivation . . . 1
1.2 Aim and Research questions . . . 2
1.3 Delimitations . . . 2
1.4 Software . . . 2
1.5 Conventions . . . 3
2 Theory 4 2.1 Nature of Engine Vibrations . . . 4
2.2 High Cycle Fatigue (HCF) . . . 5
2.3 Virtual Channels . . . 7
2.4 Virtual Vibrations with DesignLife . . . 10
2.5 VROM . . . 14
2.6 VFEM . . . 16
3 Method 18 3.1 Components . . . 18
3.2 Virtual Vibrations with DesignLife . . . 20
3.3 VROM method . . . 23
3.4 VFEM method . . . 26
4 Results 28 4.1 Moonlander results . . . 28
4.2 Flywheel housing results . . . 33
5 Discussion 36 5.1 Results . . . 36
5.2 Evaluation and comparison between methods . . . 42
5.3 Improvements on the methods . . . 47
5.4 Decision matrix . . . 52
6 Conclusion 56
List of Figures
1.1 Global CSYS used. . . 3
2.1 Example of a waterfall colour map. . . 5
2.2 Typical SN curve. . . 6
2.3 An example of a measuring point (c1). These three accelerometers deliver data to three physical channels. . . 7
2.4 Illustration of the six DOFs of a rigid body. . . 7
2.5 Illustration of the torsion or "twist" DOF. . . 8
2.6 Body with angular velocity ω and a body fixed reference system originating in point B. . . 10
2.7 Rainflow method example. . . 17
3.1 Moonlander assembled in a 5 cylinder engine. . . 18
3.2 Different configurations used for the Moonlander analyses. . . 19
3.3 Flywheel housing (in pink) and the components attached to it. . . 20
3.4 Simulation of a screw and loads applied for a pre-tension step. . . 21
3.5 Component mounted to a simplified cylinder block. . . 23
3.6 FE model of a simplified cylinder block. . . 24
3.8 Exporting ASCII files from DLS. . . 25
3.7 Scheme of primary and secondary bases. . . 25
3.9 VROM Plug-in interface. . . 26
4.1 Eigenfrequencies convergence analysis for the Moonlander. . . 29
4.2 Largest stress cycle amplitude [Pa] for excitation in Y-direction. . . 30
4.3 Largest stress cycle amplitude [Pa] for excitation in Z-direction. . . 31
4.4 Safety factor field obtained by the VFEM method. . . 32
4.5 Largest stress cycle amplitude [Pa] for excitation in Z-direction. . . 33
4.6 Von Mises stress field [Pa] for the most critical region of the flywheel housing. . . . 34
4.7 Safety factor field for the most critical region of the flywheel housing. . . 35
5.1 Accelerometer position. . . 36
5.2 PSDs at the CG of the Moonlander and eigenfrequencies for the first configuration. 37 5.3 Comparison of PSD accelerations measured in this project, in [2] and in testing [7]. 37 5.4 Contact region between Moonlander and Cylinder block (in transparent yellow). . 38
5.5 PSDs at the CG of the Moonlander and eigenfrequencies for the second configura-tion. . . 39
5.6 Accelerations measured in Moonlander. . . 39
5.7 PSDs at the CG of the PTO and eigenfrequencies for the flywheel housing assembly. 40 5.8 Comparison of velocities measured in testing and in the simulation for a node of the PTO. . . 41
5.9 Comparison of accelerations measured in testing and in the simulation for a node of the PTO. . . 41
5.10 Influence of damping in difference between VROM and VFEM responses. . . 43 5.11 Evolution of von Mises stresses with time at a node of the Moonlander when this
is excited with a constant signal of 167 Hz (blue line) and with a sweep from 160 to 180 Hz (red line). . . 44 5.12 Example of time invested for each fe in VFEM for an engine speed range
[900-2,200] RPM with a spacing of∆RPM=100 RPM. . . 44
5.13 Comparison of percentage fatigue life difference, computational time and memory requirements for a study of a component with 0.6% damping ratio. . . 47 5.14 Comparison interpolation methods. . . 48 5.15 Comparison of acceleration amplitude obtained by different interpolation methods. 48 5.16 Comparison of von Mises stress values obtained for the critical engine speeds
when using data with different∆RPMvalues. . . 49
5.17 Displacements, velocities and accelerations for two example nodes using VFEM with 6DOF and with 7DOF with a zero twist amplitude. . . 50 5.18 Von Mises Stresses at the same node using VFEM with 6DOF and with 7DOF with
List of Tables
2.1 Example of a matrix for a DOF p where data is supplied for n engine orders in an
engine speed range [900-2,200] RPMs with a∆RPMof 100 RPMs. . . 14
3.1 Components that are mounted onto the flywheel housing and their weight. . . 20
3.2 Constants to calculate the support factor. . . 22
4.1 Results of the mesh convergence analysis for the Moonlander. . . 28
4.2 Eigenfrequencies of the Moonlander calculated with the Virtual Vibrations method for the first configuration. . . 29
4.3 Fatigue life calculated by DesignLife for the first configuration of the Moonlander. 29 4.4 Computational cost Virtual Vibration method for the first configuration of the Moonlander. . . 30
4.5 Eigenfrequencies of the Moonlander calculated with the Virtual Vibrations method for the second configuration. . . 30
4.6 Fatigue life calculated by DesignLife for the second configuration of the Moonlander. 31 4.7 Computational cost Virtual Vibration method for the second configuration of the Moonlander. . . 31
4.8 Eigenfrequencies of the Moonlander calculated by the VROM method. . . 31
4.9 Computational cost VROM method for the Moonlander. . . 32
4.10 Computational cost VFEM method for the Moonlander. . . 32
4.11 Eigenfrequencies of the Flywheel housing calculated by the Virtual Vibrations method. . . 33
4.12 Fatigue life calculated by DesignLife for the flywheel housing. . . 33
4.13 Computational cost Virtual Vibration method for the Flywheel housing. . . 34
4.14 Eigenfrequencies of the Flywheel housing calculated by the VROM/VFEM method. 34 4.15 Computational cost VROM method for the Flywheel housing. . . 35
4.16 Computational cost VFEM method for the flywheel housing. . . 35
5.1 User time consumption for data obtention and post-processing set-up. . . 45
5.2 Computational time (in minutes) needed for the analyses performed in this project. 46 5.3 Weight assignment . . . 52
5.4 Weighted decision matrix. . . 54
6.1 Average mechanical properties of Aluminium AN-EC 46,000 at 20˝C . . . . 60
6.2 Average mechanical properties of Steel at 20˝C . . . . 60
1
Introduction
1.1
Motivation
At this moment, there are six different methods used by the Base Engine Strength Analy-sis (NMBS) group at Scania to evaluate the fatigue life of engine components with respect to engine vibrations. These methods may be divided into two groups: methods that model component testing at a shaker table and methods that model the full-working engine condi-tions.
There are two methods that mimic the shaker table testing conducted on engine compo-nents at Scania. These methods are the Virtual Vibration method with Root Mean Square (RMS) stress evaluation [11] and the Virtual Vibration method with fatigue life evaluation in nCode DesignLife [10]. In these two methods a very similar Finite Element Analysis (FEA) is conducted, but the complexity of the post FEA fatigue calculations differ. In the first method the fatigue life is evaluated by comparing the calculated RMS stress to the fatigue limit of the material. A more complex fatigue strength evaluation including mean stress influence, the definition of a full SN (stress-cycles) curve and the influence of factors such as surface roughness, surface finish etc. is available using DesignLife in the second method.
The methods that mimic full engine conditions are largely based on the Resonance Order Method (ROM) [12] and its subsequently branched methods Force-based Resonance Order Method (FROM) [5] and Virtual Resonance Order Method (VROM) [13]. Another method named Virtual FEM (VFEM) [2] that was not fully developed at the time of writing this report is under investigation and used sometimes. In these methods, dynamic simulations are used to compute the nodal excitations of a component at specific engine speeds within the engine performance spectrum. The excitations are later Fourier transformed into the frequency do-main (except for the VFEM method which is performed directly in the time dodo-main), allowing the identification of frequency components at, what are called, the engine orders (described in chapter Nature of Engine Vibrations). The set of critical engine speeds are found by com-puting the eigenfrequencies of the component to be analysed and matching them with the linear relations represented by the engine orders.
Having so many different methods leads to unalignment in the way of working of the NMBS team. Not all of the team members know how to use all the methods and the team lacks a clear criterion that establishes which method is the most advantageous to use in each case. Although there are some basic guidelines, currently each team member uses the method
1.2. Aim and Research questions
she/he is more familiar with, sometimes causing loss of time and, in the worst scenario, inaccurate results.
1.2
Aim and Research questions
The purpose of this project is to evaluate the methods of Virtual Vibrations with Design-Life, the VROM and the VFEM methods currently in use at Scania and benchmark them to find which one is more suitable for which kind of component. The project tries to give an-swer to the following questions for the evaluation of HCF of components subjected to engine vibration-induced loads:
1. Taking into account the characteristics of a certain component (geometry, interactions, boundary conditions etc.), which method is more suitable to study its fatigue life? 2. What are the reasons behind selecting one or the other method?
In order to answer the research questions a semi-quantitative estimation of the perfor-mance of the different methods for evaluating fatigue has been done.
Additionally, improvement on the VFEM method has been done as an extra part during its evaluation.
1.3
Delimitations
The number of engine components (two) as well as the number of evaluation methods (three) are the main delimitations of this thesis. These delimitations arise from the time constraints of the project. If there were more time available, either more methods currently in use at Scania would have been evaluated and included in the bench-marking, or more components would have been analysed, resulting in a more complete output.
1.4
Software
The software used and referred in this report is the following: – Abaqus Implicit (v2017). The main FEA solver used at NMBS. – HyperMesh (v2017). A software for pre-processing FEA files. – Simlab (v2017). A software for pre-processing FEA files. – HyperView (v2017). A software for post-processing FEA files.
– nCode DesignLife (v6.13). A software for performing fatigue analysis based on FEA results.
– FEMFAT (v5.3). A software for performing fatigue analysis based on FEA results. – BaseEngine, EasyVibrations and DriveLineSim (DLS): Scania’s in-house developed
soft-ware to obtain the vibration signals in different parts of the engine in different formats by accessing databases of previously measured data.
1.5. Conventions
1.5
Conventions
Coordinate System
In this report, the Coordinate System (CSYS) used has the following properties: its directions are parallel to the traditional engine CSYS, whose origin is at the centre of the flywheel with the X-direction pointing towards the gearbox and the Z-direction parallel to the cylinders with its positive direction pointing upwards. The Y-direction is supplementary to the X and Z directions defined to form an orthogonal system (figure 1.1).
Figure 1.1: Global CSYS used.
Notation
Here under the nomenclature of the most relevant variables used in this report is stated with its corresponding units. Even if in the text sometimes the variables are referred to in another units, all the formulas in this report should be used with SI-units.
f [Hz]: Excitation frequency. fn [Hz]: Natural frequency.
fe [Hz]: Engine speed. Engine frequency of rotation.
o [-]: Engine order.
fx(o) =o ¨ fe [Hz]: Engine order excitation frequency.
w [rad/s]: Angular velocity of a rigid body. ˙
w [rad/s2]: Angular acceleration of a rigid body. p [-]: Degree Of Freedom.
σ [Pa]: Stress. Sf [-]: Safety factor.
2
Theory
2.1
Nature of Engine Vibrations
Numerous components are mounted onto the cylinder block of an engine. Due to their own inertia and attachment properties, the components do not move in phase with the engine. The movement induced to a component mounted in the cylinder block by the vibrations resulting from the combustion cycle can lead to high stress amplitudes. If the excitation frequency, f is the same as the natural frequency (i.e. eigenfrequency), fnof the component, it is said that the
component is in resonance, resulting in the highest stress amplitudes and leading to fatigue mechanical failures in many cases.
Scania manufacture four-stroke engines. This means that a firing occurs for each cylinder every other crankshaft revolution. An engine order, o, indicates how many times an event oc-curs with respect to a reference event. When working with four-stroke engines, a crankshaft revolution is usually considered the "reference event", defining the first engine order (o1acts
once every 360˝). The lowest engine order considered receives the name of "half order" as it
occurs once every two crankshaft turns (o0.5 acts once every 720˝) and in the same fashion
for higher engine orders. Experience has shown that the most critical movements occur at the half and full order multipliers of the engine speed fe. At the same time, the most critical
engine orders are related to the number of cylinders of the engine being the third and sixth the most critical ones for a six cylinder engine and the fourth and eighth for an eight cylinder engine.
If an engine is running at a certain speed, the excitation frequency at which a specific engine order is acting, fx(o)can be easily calculated:
fx(o) =o fe (2.1)
As the reader may have noticed, there is a direct relation between engine speed, order excitation frequency and component responses, which can be accelerations, displacements, stresses etc. The amplitudes of the responses can be plotted with respect to both engine speed and excitation frequency. 3D plots are not very useful in these cases as the visualisation is poor when data density is high. Waterfall colour maps are usually used instead. In these plots, fe and f are represented in the vertical and horizontal axes. The magnitude of the
response is represented by a colour contour. The diagonal lines of high intensity emanating from the origin of the map are called order lines. The natural frequencies can be identified
2.2. High Cycle Fatigue (HCF)
by looking the straight bright colour lines extending perpendicular to the frequency axis (see figure 2.1 [9]). A colour map with response amplitudes plotted in a dB format allows the users to easily distinguish the resonances as vertical lines and order lines as slanted lines owing to the difference in brightness of the colours.
Figure 2.1: Example of a waterfall colour map.
Design engineers should work hand in hand with stress engineers in order to design a component in such a way that its eigenfrequencies do not coincide with the first multipliers of the engine orders, i.e avoiding that order lines and eigenfrequencies lines cross each others, by this increasing its operational life.
2.2
High Cycle Fatigue (HCF)
Fatigue is a phenomenon that can cause a component that is subjected to repeated loading to fail even if the resultant stresses are much lower than the Ultimate Tensile Strength (UTS) of the material. Fatigue is an expensive phenomenon for the society as it is estimated that around 60 to 90 percent of all mechanical failures are caused by it [3].
The fatigue life of a component can be short (i.e the integrity of the component is main-tained during a couple of thousands cycles), or long (life of millions or even infinite number of cycles if the stresses do not reach a threshold value known as fatigue limit). The first fatigue life type is commonly denoted as Low Cycle Fatigue (LCF) whereas the second type is named High Cycle Fatigue (HCF).
High Cycle Fatigue, also known as Stress-based fatigue, is the most common fatigue be-haviour of components in the automotive industry. Fatigue life can be evaluated using a Wöhler or SN curve. In these curves, the stress amplitudes causing failure (on the ordinate axis) with respect to the number of cycles (on the abscissa axis) are plotted for a given ma-terial. The data needed to graph the SN curves are obtained experimentally and typically indicate that 50% of the specimens would fail at a shorter life than at the one traced by the
2.2. High Cycle Fatigue (HCF)
curve. Given a stress amplitude, one can determine the fatigue life of a component, i.e. how many cycles the component can be subjected to at that stress amplitude before failure. The same way, one can know the maximum stress amplitude a component can be subjected to if a desired fatigue life is sought.
Figure 2.2: Typical SN curve.
The parameters that define a SN curve (figure 2.2) are the following: σUTS is the Ultimate Tensile Strength of the material.
SRI1 stands for Stress Range Intercept. This value is taken at 1 cycle. In case of lacking test data, it can be approximated by SRI1=1.65 σUTS.
σ1000 is the stress amplitude for a fatigue life of 1,000 cycles, which is considered the transition
point between LCF and HCF. If testing data is not available, this value can be approxi-mated as σ1000=0.9 σUTS.
σu is the fatigue limit or endurance limit (break point for a zero mean stress alternating load).
For steel usually takes a value of around 0.357 σUTSand for aluminium 0.258 σUTS.
Nc1 is the maximum service life or fatigue transition point (typically 106´108cycles). N f c is the Numerical fatigue cut-off life. Beyond this life, damage is assumed to be zero.
k1 is the first slope constant (above the fatigue limit). The slope takes a value of b1=´1/k1.
k2 is the second slope constant (below fatigue limit). k2 = 2k1´2 for cast components. The
slope takes a value of b2=´1/k2.
b0 can be approximated as b0=´0.085.
The fatigue life of a component is influenced by the mean stress that the component is subjected to. This way, a component subjected to a tensile mean stress would move the curve of the diagram to the left, resulting in a shorter life for a given stress amplitude. Alterna-tively, if the component has a compressive mean stress, the curve would move to the right increasing the life of the component. Also, other characteristics such as surface finish, stress
2.3. Virtual Channels
concentrations or notches, raw material volume etc. can affect the fatigue life of a component. A Haigh diagram is the tool used to find the revised values of the parameters defining the corrected SN curve accounting for mean stress and the rest of the factors affecting life.
2.3
Virtual Channels
Figure 2.3: An example of a measuring point (c1). These three accelerometers de-liver data to three physical channels. The so called Virtual Channels are calculated
from the real channels used to measure acceler-ations, the physical channels. A physical channel is made up of data measured using accelerome-ters, see figure 2.3, and consists of a large amount of acceleration values in the time domain.
The theory of Virtual Channels [1], [4] as-sumes that:
1. The dynamics of a cylinder block can be modelled as a Rigid Body Motion (RBM) with an additional twisting DOF around the X-axis.
2. By measuring the engine block vibrations with at least seven accelerometers, the vi-brations at any position on the cylinder block can be calculated.
Regarding the first point, it is a common practice [8] to simulate the Internal Combustion Engines (ICE) as a RBM-model to find the engine response forces. This has been shown to be a good approximation for frequencies below 200 Hz. The movement of a rigid body can be described by combining six different independent movements; translation and rotations in three coordinate directions, see figure 2.4.
Figure 2.4: Illustration of the six DOFs of a rigid body.
However, it has been found that the engine block also oscillates such that bending and twisting modes are produced [4]. It was found that the addition of only a twisting DOF is enough to simulate the cylinder block motions up to 300 Hz with sufficient accuracy [2]. The major contribution to engine roll and twisting torque comes from the forces coming from the piston in the combustion phase and its corresponding reaction forces at the crankshaft bearings.
2.3. Virtual Channels
Figure 2.5: Illustration of the torsion or "twist" DOF.
Taking the previous into account, a minimum of seven accelerometers are needed in order to compute the motion of the seven DOFs of the engine. The locations of these accelerometers have to be established wisely as it is important that they are linearly independent of each other, i.e. that none of the seven acceleration signals can be calculated with the help of a linear combination of the others. Usually the accelerations are measured on the engine block (without gearbox) in a so-called test cell. Sometimes the measurements are also done in a vehicle, which can result in lower accelerations than those measured in the test cell, but these measurements are not usually used to study HCF for auxiliary components like the ones studied in this thesis. Measurements are conducted at full load (requesting the maximum output torque the engine can offer at each RPM) under an engine speed sweep of 600-2,400 RPMs for approximately three to five minutes.
Once measurement data is available for one particular engine, the Virtual Channels can be used to calculate the accelerations at any point of the engine block. The generalised accelerations aAat a point A of a rigid body with respect to a reference coordinate system O
(see figure 2.6) can be calculated by:
aA =aB+ω r˙ BA+ω(ω rBA), (2.2)
where:
aA is the unknown acceleration vector at a point A. aA=aAx aAy aAzT
aB is the known acceleration vector of point B of the body with respect to the reference CSYS
O.
rBA is the vector from B to A. rBA =rA´rB
ω represents the angular velocity of the rigid body. ˙
ω is the angular acceleration of the rigid body.
However, due to the periodic nature of the motion of an ICE, the magnitude of the term ω (ωrBA)is significantly smaller than the other terms and can be neglected, resulting in the
reduced expression:
aA«aB+ω r˙ BA. (2.3)
Adding the angular accelerations to the previous equation one could define the com-ponents of the acceleration in the six DOFs of the rigid body. However, as said before, in this method the rigid body formulation is extended by adding a seventh DOF, the twist of
2.3. Virtual Channels
the engine block around the X-axis which, besides being time-dependant, is also position-dependant. The angular acceleration of the twist, ˙ωT, is computed as:
˙
ωT(t, x) =γ(t)x, (2.4) where γ is the angular acceleration twist factor. The vector of angular accelerations is then augmented to: ˙ ω= ˙ ωx(t) +ω˙T(t, x) ˙ ωy(t) ˙ ωz(t) . (2.5)
The acceleration vector ¨q containing the accelerations of the DOFs of the engine at the origin is:
¨q=a0x a0y a0z ω˙x ω˙y ω˙z γT, (2.6) which now allows one to rewrite the acceleration aAof point A at position rA = [XA, YA, ZA]
as aA(t) = [XA] ¨q(t), (2.7) where, [XA] = 1 0 0 0 ZA ´YA 0 0 1 0 ´ZA 0 XA ´ZAXA 0 0 1 YA ´XA 0 YAXA . (2.8)
The reader may notice that the vector ¨q has seven unknowns, indicating, as explained before, that a minimum of seven measurement channels are needed to solve the system of equations at each sampling time. Using additional channels and using the least-square method for every sampling time would give more accurate results [4]. Measurement channels used for adapting the model are here denoted as base channels. To describe the nth accelerometer value anlocated at the position rnby use of equation 2.7, the acceleration vector anhas to be
projected onto the direction vector of the accelerometer un.
un =ux uy uzT; |un|=1 (2.9) a˚
n(t) =unT an(t) =unT [Xn] ¨q(t). (2.10)
All base channel values are compiled in the vector abaseas:
abase(t) = a˚ 1(t) a˚ 2(t) .. . a˚ n(t) = uT1 X1 uT2 X2 .. . unTXn ¨q(t) = [L] ¨q(t). (2.11)
The left-hand side of the equation 2.11 is obtained by measurements while the generalised accelerations ˆ¨q are computed using the Moore-Penrose pseudo-inverse as:
ˆ¨q(t) = ([L]T[L])´1[L]T a
base(t). (2.12)
Now it is possible to compute a vibration time signal at any location point A of the engine block by
2.4. Virtual Vibrations with DesignLife
Figure 2.6: Body with angular veloc-ity ω and a body fixed reference sys-tem originating in point B.
Once the generalised accelerations have been found for a point on the engine, the complete six DOFs corresponding to the RBM of the engine block are known. The twist of this block around the X-axis is also known. This motion, derived from measured accelerometer signals, is used in the simulations to prescribe the engine block motion as a function of time. The motion can also be Fourier transformed to the frequency domain if needed.
2.4
Virtual Vibrations with DesignLife
The method of Virtual Vibrations with DesignLife [10] models the physical testing that can be performed on a component with a shaker table. Due to the numberof mounted components in the cylinder block, testing all of them using a shaker table is non-viable, mostly due to economical and time consumption reasons. There is thus a need to use Finite Element Analysis (FEA) to evaluate whether the component can withstand the vibrations it will be subjected to during its working life.
In this method, an FE model is subjected to the vibrations that would be input in a real shaker table testing. The calculation of fatigue life is carried out in the nCode DesignLife software.
This method is an evolution of another method used by Scania [11]. In the previous ver-sion of this method, instead of using DesignLife, the fatigue life was calculated manually using the RMS values of stress output from the FE model and comparing them to the fatigue limit of the material. However, this is not sufficient to accurately calculate the fatigue life of a component if the stress varies with frequency. Another evolution from the previous ver-sion is the possibility to automatically apply correction factors such as those due to surface treatments, temperature gradients etc. There is also the possibility of accounting for mean stresses. Although none of the methods can take into account the stress gradient factor that considers the reduction of stress with depth below the surface, in the new method it is pos-sible to compensate for this by correcting the fatigue data as will be explained later. Even if at Scania, the shaker table tests are performed in the laboratory in three separate directions, X, Y and Z, one at each time, the new simulation method has the potential of evaluating the life when the component is subjected to multi-axial load. As a conclusion, the Virtual Vibra-tions with DesignLife method is a more accurate evolution from the previous method, mainly by virtue of the possibility to use complete SN curves, and not only the fatigue limit of the material, to calculate the fatigue life.
Using Virtual Channels, the accelerations at any point of the engine can be calculated from testing data. The testing data is in the time domain and, for a shaker table test, has to be Fourier transformed in the frequency domain (equation 2.14). Power Spectral Densities (PSD) are used in this method and in real testing as input of accelerations. A signal in the frequency domain can be expressed as a PSD using equation 2.15. PSDs have the units of(m/s2)2/Hz.
ˆaA(f) = 1 2T żT ´T aA(t)e´i2π f tdt. (2.14) PSDaA(f) = |ˆa(f)|2 fs , (2.15)
2.4. Virtual Vibrations with DesignLife
In a shaker table test, accelerations from the Centre of Gravity (CG) of the component to be studied ˆaCG(f)are used. Alternatively, if one wants to study the behaviour of a component
due to the vibrations coming from another component attached to it, the accelerations from the CG of the attached component are to be used. The PSDs of the three components of the extrapolated acceleration for the CG (PSDaCGx(f), PSDaCGy(f) and PSDaCGz(f)) are
scaled by a factor Sv, both in the real testing and in the Virtual vibrations method. In Scania,
components are designed to have an infinite life, which is considered to be above 2 million cycles (achieved after approximately 10,000 hours). The scaling is used to reduce the testing time in the shaker table to β number of hours in each direction which would be equivalent to 10,000 working-condition-hours (depending on the component and material). This is done due to the infeasibility of running a test for 10,000 hours in each direction. Thus, this test aims to verify that the infinite life is achieved for the components. The scaling value, Sv, has been
experienced to recreate fairly good what happens in reality and is calculated as:
Sv=1.5 10, 000
β
0.25
, (2.16)
where:
10,000 is the number of hours for which the fatigue cut-off of the material is reached.
β is the number of hours of the test.
1.5 is a safety factor to consider damping, probability of survival, number of components tested etc.
0.25 is an average value of the first slope of an SN curve, b1.
How does it work
After pre-loading the component in Abaqus and computing its natural frequencies (the pro-cess is explained more in detail in the Method chapter) it is subjected to a sinusoidal constant amplitude vibration of 1 m/s2 in its mounting points. The purpose of this is to obtain the Frequency Response Function (FRF) of the component, i.e. how the response varies for a unit load of excitation along the frequency range. The equation of an FRF is:
ˆ H(f) = Y(ˆ f) ˆ X(f), (2.17) where, ˆ
H(f) is the Frequency Response Function. ˆ
Y(f) is the response of the system in the frequency domain. ˆ
X(f) is the excitation of the system in the frequency domain.
The stress frequency response σ(f), output of the Abaqus job, is combined with the ac-celeration spectrum measured in the engine for the CG of the component, PSDaCG(t), inside
DesignLife in order to obtain the PSD of the resulting stresses, PSDσ(f), in accordance with: PSDσ(f) =σ(f)2PSDaCG(f). (2.18)
In order to conduct a realistic simulation, it is crucial to accurately predict the damping (C) that the component has. The endurance life is heavily affected by the damping ratio (ζ). For example, for a 1D spring, the stresses are calculated as:
σ(f) = k
(2π fn)2
a
(1 ´(f / fn)2)2+ (2ζ f / fn)2
2.4. Virtual Vibrations with DesignLife
where:
fn is the natural, undamped frequency:
fn= 1
2π c
k
m (2.20)
ζ is the ratio of critical damping:
ζ= C Ccrit
(2.21) Ccrit is the critical damping calculated as:
Ccrit=2
?
m k (2.22)
fd is the damped frequency:
fd= fn
b
1 ´ ζ2 (2.23)
m is the mass and k is the stiffness constant.
Once the PSD of the response, PSDσ(f), has been calculated, the parameters needed to evaluate the damage first and the fatigue life later, can be calculated.
The spectral moments are used to predict the potential fatigue damage from the response PSD functions; in this case PSDσ(f). The spectral moments (n = [0 ´ 4]) are computed in accordance with [6] as:
mn= fmax ż fmin PSDσ(f)f nd f . (2.24)
The Root Mean Square (RMS) stress (σRMS) of a node is then obtained as:
σRMS=
?
m0, (2.25)
being m0the zeroth moment of area of the PSD or, in other words, the area beneath the PSD
curve. The expected number of peaks (change of trend in the signal) in the PSD is provided by:
E[P] =c m4
m2. (2.26)
The irregularity factor (the ratio between the number of up-zero crossings with positive derivative and the number of positive peaks per second) is:
γ= ?m2 mom4
. (2.27)
Conversion to a Rainflow frequency range can be done using two different methods, Lalanne and Dirlik. Lalanne is used as default method in DesignLife; it is less empirical than Dirlik’s method and offers similar results. In Lalanne’s theory, the expected number of cycles of stress range∆σ, occurring in t seconds is given as equation 2.28:
N(∆σ) =E[P] t PDFLalanne(∆σ) (2.28)
In which PDFLalanne(∆σ) is the Probability Distribution Function according to Lalanne
2.4. Virtual Vibrations with DesignLife PDFLalanne(∆σ) = 1 2σRMS #c 1 ´ γ2 2π exp ´∆σ2 8(σRMS)2(1 ´ γ2) + ∆σγ 4σRMSexp ´∆σ2 8(σRMS)2 " 1+erf ∆σγ 2σRMS a 2(1 ´ γ2) !#+ , (2.29)
where the error function erf(x) of the stress signal value in the time interval [0,x] is provided by: erf(x) = ?2 π x ż 0 e´t2dt. (2.30)
This function, PDFLalanne(∆σ), together with the SN curve of the material allows one to
calculate the service life, SL, as follows [10]:
SL= D 2σUTS ż 0 PDFLalanne(∆σ) Nc1 ∆σ 2σu ϑ d(∆σ) bm 4 m2 , (2.31) where:
D is the cumulative damage value for failure, usually 0.2-1.0.
ϑ #
´k2 if∆σ ă 2σu
´k1 otherwise
The rest of parameters are the ones describing a typical SN curve defined in the section High Cycle Fatigue (HCF).
As indicated in the theory section of High Cycle Fatigue (HCF), the SN curves are ob-tained experimentally from tests and indicate the maximum number of cycles a component would withstand without breaking with a 50% probability. If the calculation of the fatigue life should be carried out using a different failure probability, one can re-calculate the value for the maximum service life, Nc1:
Nc1new=Nc110Φ
´1(P)S
logN, (2.32)
where:
Φ´1 is the probit function or, in other words, the inverse of the cumulative distribution
func-tion of the standard normal distribufunc-tionΦ.
P is the new probability of survival.
SlogN is the deviation of the logarithmic standard life spread (usually between 0.15 and 0.25)
When this test is performed at a shaker table it is usually rated as pass/fail depending if the component stands for β hours in each direction or not. As the accelerations are scaled, these are not present in real life and the calculation of a safety factor would be unreasonable. If a component does not withstand the test it is usually sent back for redesign.
2.5. VROM
2.5
VROM
The VROM (Virtual Resonance Order Method) aims to calculate the fatigue life of a compo-nent by mimicking its behaviour when mounted directly on the engine, c.f. Virtual Vibra-tions method. The VROM consists on generating Frequency Response FuncVibra-tions (FRF) for the seven DOFs of the cylinder block and then compute the operative stresses combining those FRFs with the vibrations coming from the engine block. For every DOF p and mode r, the FRF Hrp(f)can be built up from the modal generalised displacements GUrp and their
phase angle GUPrp, both outputs of an Abaqus steady state dynamic analysis subjected to a
constant unit acceleration as described in the Method chapter. The FRFs are built as:
Hrp(f) =m(cosθ+i ¨ sinθ), (2.33)
being m=GUrp(f)and θ=GPUrp(f)[rad].
Thus, the system will be formed by a number of FRFs being the product of the number of eigenmodes and the number of DOFs. The idea behind generating the FRFs is to later combine them with the accelerations coming from the engine block ap(f) to obtain the
operational modal displacements qr(f) i.e. the total (taken into account all DOFs) modal
displacements of the component as:
qr(f) = 7
ÿ
p=1
Hrp(f)ap(f). (2.34)
In this method, stresses are only computed for the potentially most harmful engine speeds. Any engine speed fe within the range of interest which any order (lower than the
maximum order to consider) causes the excitation frequency o ¨ fecoincide with any
eigenfre-quency of the component is tagged as critical. To exemplify this lets us assume that one wants to study the fatigue life of a component in a range of engine speeds 900 to 2,200 RPM and that the component has an eigenfrequency of 228 Hz. Twenty engine orders are to be considered (o=0.5, 1, . . . , 10). Using equation 2.35 the engine speeds which excite the eigenfrequencies for every engine order can be calculated. From the calculated feicrit, only the ones that lay in the range [900 - 2,200] are considered for the stress study.
feicrit= fn oi
. (2.35)
In the example, for fn = 228 Hz, the first order (o1) would give a critical engine speed
fe1crit=13, 685 RPM, which is out of the range of interest. However, for the 6.5 order (o6.5) the
critical engine speed would take a value of fcrit
e6.5= 1, 955 RPM and thus would be taken into
account in the analysis.
Seven matrices (one for each DOF) gather the data from the accelerations for a range of engine speeds with a spacing between engine speeds∆RPM. Each matrix line contains the
engine speed and the complex value of acceleration for the orders of interest. The matrices have the size nRPMx nobeing nRPMthe number of engine speeds in the frequency range with
a separation of∆RPMand nothe number of engine orders to consider (as in table 2.1).
Table 2.1: Example of a matrix for a DOF p where data is supplied for n engine orders in an engine speed range [900-2,200] RPMs with a∆RPMof 100 RPMs.
fe[RPM] o0.5 o1 . . . on 900 1+j 2+4j . . . -9-6j 1,000 2-j 0-2.3j . . . -3+j .. . . . . 2,200 5+6j -1-j . . . 2+2j
2.5. VROM
The number of engine speeds and orders has to be the same in every DOF. The accel-erations for the previously identified critical speeds are linearly interpolated using the two nearest provided fe records. In the example matrix of table 2.1, for the 0.5 engine order at
fe = 900 RPM the acceleration has an amplitude of 1+j and an amplitude 2-j at fe = 1, 000
RPM. If fe=950 RPM is one of the critical engine speeds, the acceleration at fecrit=950 RPM
for the 0.5 order would take a value of 1.5+0j and so forth.
Once the acceleration values have been interpolated for each critical engine speed, they can be Fourier transformed to the frequency domain by:
afecrit p (f) = k ÿ o=0.5,1,1.5,... Ao, fecrit p F cos2π ¨ fecrito ¨ t , (2.36)
whereF is the Fourier transform symbol and Ao, fecrit
p is the acceleration, expressed as a
com-plex number, interpolated at the critical engine speed fecrit and engine order o for the DOF p.
Combining equations 2.34 and 2.36, the operational modal displacements in the frequency domain at the critical speeds considering the seven DOFs, qfecrit
r (f)can be computed: qfecrit r (f) = 7 ÿ p=1 Hrp(f)af crit e p (f) = k ÿ o=0.5,1,1.5,... Fcos2π ¨ fecrito ¨ t ÿ7 p=1 Ao, fecrit p Hrp(f) = k ÿ o=0.5,1,1.5,... Fcos2π ¨ fecrito ¨ t r qo, fecrit r , (2.37) whereqr o, fecrit
r is a complex number that magnifies the total modal displacement of mode r at
the engine speed fcrit e .
As the Fourier transform and its inverse are linear operators, calculating the modal dis-placements in the time domain is trivial,
qfecrit r (t) = F´1(qf crit e r (f)) = k ÿ o=0.5,1,1.5,... F´1 Fcos2π ¨ fecrito ¨ t r qo, fecrit r = k ÿ o=0.5,1,1.5,... ˇ ˇ ˇ ˇrq o, fcrit e r ˇ ˇ ˇ ˇcos 2π ¨ fecrito ¨ t+α , (2.38)
where α is the phase angle:
α=atan im(rqo, fecrit r ) re(rqo, fecrit r ) . (2.39)
Once back in the time domain, 50 time discretization points, ti, evenly distributed
be-tween 0 and 2/ fecritare used to discretize two revolutions at any steady state fecritso that the
operative stresses are calculated for those points.
As the structure is linear, the peak amplitude of stresses can be calculated with the modal amplitudes as σf crit e (t i) = N ÿ r=1 qfecrit r (ti)ψr. (2.40)
2.6. VFEM
The modal stress fields ψr are obtained from Abaqus and stresses are computed for the
50 time points that belong to a critical engine speed. Then, the the maximum stress values for each mesh element considered along the 50 time points used to discretize two engine revolutions at the critical speeds are found.
σf crit e max =max(σf crit e (t i)). (2.41)
The highest stress value for the whole engine speed range is found out by comparing the maximum stresses at the critical speeds.
σmax =max(σf
crit e
max). (2.42)
After manually applying all the correction factors to the obtained σmax, the safety factor with
respect to infinite life can now be calculated using the fatigue limit of the material σuas:
Sf =
σu
σmax. (2.43)
2.6
VFEM
The VFEM (Virtual FEM as was named in [2]) method is very similar to the VROM method. The major difference is that, while the VROM method is performed in the frequency domain, and results are given only for the critical engine speeds, VFEM is directly performed in the time domain. For this method, the cylinder block is excited with accelerations values mea-sured in testing in the seven DOFs mentioned. Abaqus then calculates the stresses resulting in the component.
As the loading is not composed by constant amplitude cycles, but by a series of cycles with different stress levels σai, the Rainflow counting method is used to find out the number
of cycles Ni(σai)the component is subjected to for each stress level. Manually, the Rainflow
method would be applied as follows:
1. Depict the loading history as a function of time, rearranging the sequence so that it starts with the smaller minimum or the largest maximum.
2. Rotate clockwise 90˝ the loading history (i.e. with the time axis pointing downwards
like in figure 2.7).
3. Imagine that the start of the loading history is the top of a Japanese pagoda building and that each maximum and minimum define a roof level. Rain drops drip from the top of each roof level. When the drop reaches the end of the roof it has two options: fall to the next roof or stop if any of the following conditions are given:
• Starts at a minimum and passes an equal or smaller minimum (like drop A starting at 0; the amplitude at 2 is smaller than at 0).
• Starts at a maximum and passes an equal or bigger maximum (like drop B starting at 1; the amplitude at 3 is bigger than at 1).
• Merges with the path defined by an earlier drop (like drop F which started at 5 and comes across drop D that started at 3).
4. Pair up half-cycles of same amplitude and opposite sense and count the number of complete cycles.
2.6. VFEM
Figure 2.7: Rainflow method example.
FEMFAT, the post-processing software used in this method, has an algorithm that performs the previous operations math-ematically.
Once the number of cycles at each stress level present in the loading history is known, the Palmgren-Miner rule is used to estimate the life of the component. The fraction of total life consumed LCσaiby a stress level σaiis:
LCσai=
Ni(σai)
Nf(σai)
. (2.44)
At the next stress level σai+1, another fraction of the life
would be consumed. Fatigue failure is expected to occur when the summation of all the life fractions, D, becomes a certain value (usually 1). k ÿ i=1 Ni(σai) Nf(σai) =D, (2.45) where:
Ni(σai) is the number of cycles with stress amplitude σai.
Nf(σai) is the fatigue life at stress amplitude σai.
k is the number of different stress levels present in the spectrum.
D represents the life consumed by the exposures to the cycles at the different stress levels. Normally failure occurs when D=1.
Usually, a factor Wiwhich quantifies the damage caused by each stress level, is defined as
Wi=Ni(σai)σai, i=1 . . . k (2.46)
If the critical damage is assumed to be the same for all stress levels (this is usually the case in engine vibrations considering constant mean stresses caused by components pre-tension and thermal stresses):
WFailure= N1σa1=N2σa2=¨ ¨ ¨=Nkσak, (2.47)
then, equation 2.45 can be rewritten as: řk
i=1Ni(σai)σai
WFailure
=D˚, (2.48)
where in this case D˚represents the proportion of damage with respect to the critical value.
Failure normally occurs when D =1. The safety factor of the component can then be calcu-lated as:
Sf =
1
Nc1 ¨ D˚, (2.49)
3
Method
3.1
Components
Moonlander
The Moonlander (Figure 3.1a) is not a functional part of the engine and it is not mounted on the engines that are installed in the trucks. Instead, it is installed in another part of the truck and it is sometimes used for measuring its different responses for a number of engine excitations and use this data to study the accuracy of different methods of evaluating fatigue life. The Moonlander is a component manufactured in Aluminium AN-EC 46,000, whose mechanical properties are compiled in table 6.1 in Appendix A. It is only attached to the cold side of the radiator cap of the engine with one Steel M8 bolt (Figure 3.1b).
(a) Moonlander (b) Location on the engine Figure 3.1: Moonlander assembled in a 5 cylinder engine.
Moonlander FE model
A mesh convergence analysis was performed in order to select a mesh that offered acceptable results (estimated error under 5%) and was not very expensive regarding computational time.
3.1. Components
(a) First configuration. Moonlander screwed to the shaker table
(b) Second configuration. Moonlander screwed to a part of the cylinder block and this one to the shaker table or to the whole cylinder block.
Figure 3.2: Different configurations used for the Moonlander analyses.
To do so, the component was fixed in X, Y and Z directions in the screw hole and a load of 100 N in the -Y direction was applied in the bottom face. Eigenfrequencies and maximum stresses were analysed to perform the convergence study. Results of the mesh convergence study can be consulted in the Results chapter. The final mesh used for the Moonlander was formed by 78,700 Tetrahedral elements (C3D10I) and had a higher mesh density in the region under the attachment area, where the maximum stresses are expected to occur.
For the method of Virtual Vibrations two mounting configurations were studied. In the first configuration the Moonlander was simulated to be directly attached to the shaker table. In the second configuration, a small part of the cylinder block on which the Moonlander is mounted in reality was included in the FE model (and used to attach the Moonlander to it). The back side of the part of the cylinder block was attached to the shaker table. The reason for doing this was to study the influence that different BCs and damping can have when doing a shaker table test. The second configuration FE model was also used for the VROM and VFEM methods, as for these methods the components need to be attached to the cylinder block by definition.
Flywheel housing
The flywheel housing for the CBE1 F1 generation engine was the second component chosen to be analysed. The flywheel housing is a component made out of Aluminium AN-EC 46,000 with a total weight of approximately 15.64 kg. The function of this component is to serve as cover for the flywheel and as connector case between cylinder block and gearbox. Due to its big dimensions and the free space around it (as consequence of size difference between cylinder block and gearbox) it is used to accommodate several components, some of them considerably bulky and heavy such as the start engine, compressor and the power takeoff unit (PTO) (refer to table 3.1). Fatigue life in the flywheel housing, based on the loads from the PTO, has been studied. The flywheel housing is screwed to the gearbox at one of its sides and to the transmission housing at the other side. The transmission housing is screwed
3.2. Virtual Vibrations with DesignLife
directly to the cylinder block. In figure 3.3 the flywheel housing, coloured in pink, and the components attached to it can be observed.
Flywheel housing FE model
As the FE model was inherited from a previous study, no mesh convergence analysis was performed on this model. The whole model is formed by 3,260,000 elements, which 1,079,000 belong to the flywheel housing. The model is constrained in all DOFs at the contact surface between transmission housing and cylinder block and also at the brackets holding the start engine and fuel pump as they are connected to the cylinder block, which is considered to be very stiff.
(a) Front view (b) Rear view
Figure 3.3: Flywheel housing (in pink) and the components attached to it.
Table 3.1: Components that are mounted onto the flywheel housing and their weight.
Colour Component Mass [kg]
Flywheel housing 15.64 Start engine 12.85 Compressor 29.20 Servo 4.85 Fuel pump 9.90 PTO 40.30 Transmission housing 8.93 Part of gearbox –
3.2
Virtual Vibrations with DesignLife
When using this method, a small change in the mesh has to be done as DesignLife can only handle 2D stresses and strain. Thus, a very thin (1µm in order not to add ex-tra stiffness to the component) layer formed by Membrane elements (M3D6) needs to be generated covering the solid elements of the component or of the region to analyse. As
3.2. Virtual Vibrations with DesignLife
only this set of elements is used by DesignLife to calculate the fatigue life, the FEA re-sults are only requested for the elements contained in this set. Components mounted on the engine block are usually screwed to it. Hence, as well as it is done in a real shaker table, the first step is to simulate the stress contribution of the screws to the system.
Figure 3.4: Simulation of a screw and loads applied for a pre-tension step. This stress field could not be obviated as it has great
influence in the rest of the procedure introducing a mean stress to the component that affects its fatigue life as was explained in the Theory chapter. To do this, FE models of the screws have to be included in the simulation. The mesh of the screws can be coarse as no results are requested for them; their only purpose is to transfer the pre-tension forces to the component. Displacements in X, Y and Z directions are fixed in the end face of the screws and on the face of the com-ponent that is in contact with the table. A pre-tension force is applied to the screws and a Tie contact inter-action is used to connect the bottom face of the head of the screw and the component face. Figure 3.4 dis-plays an example of how to simulate the influence of a screw.
After the stresses caused by the pre-tension forces
are computed, an eigenfrequency step is carried out. The goal of this step is to calculate the natural frequencies of the component. The FEA solver needs this data to conduct the modal analysis. Once the natural frequencies of the model are calculated by the solver, the model is subjected to a sinusoidal unit load via a steady state dynamics step. The reason why this is a steady state dynamics step and not a random response step with a unit PSD as input is simply because the current version of DesignLife (v13.1) cannot handle the results coming from random response analyses. As the PSDs of the accelerations are only measured up to 1,500 Hz, the modal analysis is only performed until that frequency. In this step a damping ratio has to be introduced in the model. If there is no testing information about the damping of the component, a damping ratio of 2% is normally used. A stress field with respect to excitation frequency for the unit load, i.e. an FRF of stresses, is the required input to DesignLife in order to calculate the fatigue life of the component.
The PSDs of accelerations are obtained from experimental data using the in-house developed programs BaseEngine and EasyVibrations:
1. In BaseEngine, the user selects the engine where her/his component is to be mounted from a database named BEVARA. In this database, the accelerations measured experi-mentally in the time domain are stored for a large number of Scania engines and con-figurations.
2. The user then selects the coordinates to where these accelerations should be interpo-lated using the Theory of Virtual Channels. In this method, the accelerations have to be calculated for the coordinates of the Center of Gravity of the component (CG) when this is mounted in the engine block following the CSYS described in section Coordinate System.
3. These accelerations are then exported from BaseEngine and imported into EasyVibra-tions where the acceleraEasyVibra-tions are Fourier transformed to the frequency domain and of-fered in PSD format to the user.
Once the response of the model to a unit constant sinusoidal load is obtained from the Abaqus model, the FRF is known. When the PSDs of the accelerations are available, they are imported
3.2. Virtual Vibrations with DesignLife
into DesignLife and combined together with the Abaqus .odb file in order to calculate the fatigue life of the component. This procedure is described in [10] and has to be performed thrice; once for each direction. DesignLife offers the option of calculating the fatigue life if the accelerations in the three directions occur at the same time. However, as mentioned previously, at NMBT the tests with the shaker table are performed in the three directions separately, and therefore should be performed in DesignLife in order to mimic the real test in the best possible way.
DesignLife has a vast library of materials that can be assigned to the model. If the material is not available in the library, new materials can be created. The most important mechani-cal properties to input in this case are the ones that define the shape of the SN curve. The used version of DesignLife cannot take into account the stress concentration regions such as notches. Hence, for this step it is of great importance to use the recommended equations and parameters values indicated in [10], as its author have corrected the values of the parame-ters in order to take into account stress concentration regions. Correction factors for the SN curve (e.g. surface finish, raw material volume, surface treatments etc.) can be introduced in DesignLife. However, DesignLife does not offer the possibility of applying stress gradi-ent correction during vibration analysis. Hence, a support factor S that takes into account the fatigue life improvement due to the material under the membrane elements has to be introduced. This support factor can be calculated using the following equations [10]:
S $ ’ & ’ % 1+χ ¨10´(aG´0.5+Rm/bG) 0 ă χ ă 0.1 1+?χ ¨10´(aG+Rm/bG) 0.1 ă χ ă 1 1+?4χ ¨10´(aG+Rm/bG) 1 ă χ ă 100 (3.1) where:
χ is the stress gradient slope constant. χ « 2.33 ¨ 10´3[m]/r for radius and χ « 2 ¨ 10´3[m]/t for thickness.
r is the radius in m. t is the thickness in m.
Rm is the UTS of the material in Pa.
aGand bG are two constants that depend on the material class (table 3.2).
Table 3.2: Constants to calculate the support factor.
Material Stainless Steel Other Steel GS GGG GT GG Wrought AL Alloy Cast Al Alloy nCode mat type no. 23-25 13,14,16-22,26,-99 9-12,15 5-8 2-4 1 100-105 106 aG[-] .4 .5 .25 .05 -0.05 -0.05 0.05 -0.05 bG[MPa] 2400 2700 2000 3200 3200 3200 850 3200
One can notice that a value of a radius (r) or a thickness (t) has to be used to calculate the stress gradient factor S. However, for complex geometries, one cannot always know the region where the fatigue life is going to be lower, and thus where this factor has to be applied. A way of dealing with this is to divide the membrane layer into several element sets when setting up the model. This way, it is possible to apply different material properties to different groups of elements. One can create element sets around the radii where high stresses are likely to occur and assign each of them different materials based on the same base material but with different support factors. Another option is to first run an analysis without any
3.3. VROM method
correction factors, see where the fatigue life is lower, calculate the correction factor taking into account the dimensions of that region and run the analysis again, with the corrected SN curve parameters. When the support factors are calculated, the SN curve has to be corrected replacing the previous value of SRI1 with SRI1corrected:
SRI1corrected=SRI1 ¨ S (3.2)
Once the analysis has been run, DesignLife can plot fields of minimum number of cy-cles, equivalent life in any unit of time, safety factor, maximum stress amplitude etc. in an .odb file.
3.3
VROM method
As this method analyses the behaviour of the component when it is directly mounted on the engine block, the component needs to be assembled with the engine block in the FE model. As the block is to be considered almost a Rigid Body, it can be modelled as a very basic rectan-gular prism and with an extremely coarse mesh. However, a small part of the actual cylinder block geometry needs to be included in the model in order to imitate as good as possible the contact and interaction between component and cylinder block, see figure 3.5. It is clear that even if the cylinder block can be considered a Rigid Body, the material this is formed by and the geometry close to where the component is attached influences the behaviour of the com-ponent. Adding this small part allows for considering those phenomena while maintaining a reasonable computational time.
Figure 3.5: Component mounted to a simplified cylinder block.
The way of modelling the cylinder block has some peculiarities considered worthy of a detailed explanation (see figure 3.6). Abaqus has a keyword, *Rigid Body, to assign a component the characteristics of a rigid body, i.e. zero relative displacements between nodes. However, this keyword cannot be used since the cylinder block is to be subjected to a twisting moment, and thus have seven DOFs instead of six. Instead, the cylinder block is assigned a material with a extremely high Young’s modulus. By this, the cylinder block essentially behaves as a rigid body, but still permits the enforcement of a twist motion to it.
3.3. VROM method
Three RBE3 elements are also needed to make the model work. One of them connects all the nodes of the REAR face with the node situated at the origo, named REAR, as the reference node. This set of nodes acts as the primary base where all accelerations except for twist are to be applied. The same thing is done in the FRONT face with the node named FRONT as reference node. This other set of nodes acts as the secondary base where twist is to be applied. The third one links the nodes of the second closest parallel face from the ORIGO with a reference node named REAR_TW IST situated very close to the REAR node in a virtual axis joining REAR and FRONT nodes (see figure 3.6).
A connector element of type CONN3D2 with a "revolute" behaviour connecting the FRONT and REAR_TW IST nodes needs to be created. The purpose of this element is to act as a hinge between those nodes in order to apply a realistic twisting motion to the whole cylinder block. The component together with the part of the real cylinder block is attached by kinematic couplings to the closest nodes of the simplified cylinder block.
Figure 3.6: FE model of a simplified cylinder block.
The inclusion of the cylinder block in the FE model is only needed for components that are not attached in the REAR face, for example the Moonlander. A much simpler FE model without the cylinder block can be used for components attached only to the REAR face where the ORIGO is, like the flywheel housing, as twist is zero in that plane. If a component is attached both at the REAR face and at another point of the cylinder block, the cylinder block model has also to be included.
1. The first step, is to pre-load the component with the forces resulting from the screws used to attach the component to the engine block or by any thermal loads present in the component.
2. The second step is to obtain the eigenfrequencies and eigenvalues of the system. The motions of the vibrations are to be applied to the so called "bases" (figure 3.7). Those "bases" have to be defined in the eigenfrequency extraction step as Boundary Condi-tions (BCs). Requesting the stresses as element output is mandatory in this step in or-der to let the VROM script build the operative stresses. For complex assemblies formed by multiple components it is advised to only request the output for the element set defining the component of interest, saving time and memory.
3. Once the eigenfrequencies have been extracted, the next thing to perform is a steady state dynamic step with a constant unit load in each of the seven DOFs. Since the goal
