Virtual full vehicle durability testing of a coach

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Virtual full vehicle durability testing of a coach

KIM BLADH

Master of Science Thesis Stockholm, Sweden 2012

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Virtual full vehicle durability testing of a coach

Kim Bladh

Master of Science Thesis MMK 2012:17 MKN 055 KTH Industrial Engineering and Management

Machine Design SE-100 44 STOCKHOLM

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Examensarbete MMK 2012:17 MKN 055

Virtuell hållfasthetsprovning av en turistbuss

Kim Bladh

Godkänt

2012-06-05 Examinator

Ulf Sellgren Handledare

Ulf Sellgren

Uppdragsgivare

Scania CV AB

Kontaktperson

Peter Eriksson

Sammanfattning

Konkurrensen inom fordonsindustrin har under lång tid föranlett förbättringar i produktutvecklingen hos tillverkare. Effektivisering av produktutveckling inbegriper ofta åtgärder för att öka produktkvalitet och samtidigt minska kostnader och time-to-market. I takt med att kapaciteten och möjligheterna med Computer-Aided Engineering har ökat har så även konceptet med simuleringsdriven produktutveckling fått allt större genomslag. Virtuell helfordonsprovning för utmattningsutvärdering är en av många utmaningar som tillverkarna står inför när datorbaserade simuleringar ges en alltmer framträdande roll i fordonsutvecklingen.

Det här examensarbetet har genomförts som ett första steg mot att implementera dynamisk virtuell provning i utvecklingen av bussar på Scania. Målet med arbetet har varit att utvärdera hur väl en helfordonsmodell av en buss kan representera verkligheten och till vilken noggrannhet påkänningar i bussens struktur kan predikteras. Tidigare utförd provning av en Scania Touring turistbuss har varit utgångspunkt för modelleringen och de simuleringar som genomförts i detta arbetet.

En virtuell modell har skapats i multi-body systems (MBS)-verktyget MSC.Adams. Chassiram och karosstruktur har implementerats som en flexibel kropp i modellen för att kunna återskapa strukturens dynamik och resulterande påkänningar. Approximationer av strukturdämpningen har tagit fram med hjälp av inversmodellering. Modellen analyserades utifrån två olika helfordonssimuleringar som benämns Virtuell Skakrigg respektive Virtuell Väg. I den förstnämnda simuleringen har provriggsmjukvaran RPC Pro använts i kombination med Adams/Car för att generera en drivsignal till en skakrigg som modellen kopplats till.

Drivsignalen itereras fram utifrån uppmätta lastsignaler från den fysiska provningen. Den sistnämnda simuleringen innebär istället att modellen körs över en virtuell version av vägprofilen från provbanan.

Modellen har utvärderats utifrån dess korrelation med mätdata från den fysiska provningen.

Resultaten från simulering på virtuell väg uppvisade bra överrensstämmelse för vertikala navkrafter men sämre för laterala och longitudinella. Accelerationsresponser i strukturen var påtagligt beroende av strukturdämpningen som förväntat. Erhållna töjningsresponser var icke- konservativa för samtliga framtagna strukturdämpningar. I den virtuella skakriggen visade sig alla accelerationsresponser i strukturen vara möjliga att reproducera med hög noggrannhet. Två av det fyra utvärderade töjningsresponserna visade god korrelation till den fysiska mätningen.

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Master of Science Thesis MMK 2012:17 MKN 055

Virtual full vehicle durability testing of a coach

Kim Bladh

Approved

2012-06-05 ^ExaminerUlf Sellgren Supervisor

Ulf Sellgren

Commissioner

Scania CV AB

Contact person

Peter Eriksson

Abstract

The competitive nature of the automotive industry has always implied a necessity to improve product development concerning time-to-market, cost and product quality. As capacity of computer-aided engineering tools has evolved, so has the strive for simulation-driven design.

Virtual durability testing using full vehicle models is one of many challenges posed in front of vehicle manufacturers when computer simulations are given a key role in product development.

This thesis has been initiated as a preliminary step towards implementing dynamic virtual durability testing in the development of buses and coaches at Scania. The objective has been to assess the predictability of a full vehicle coach model, and specifically to what level of precision structural loads can be predicted. Previously performed proving ground testing of a Scania Touring coach has been the basis for the modelling and simulations carried out in this thesis.

A virtual model of the Scania Touring coach has been created in multi-body simulation software package MSC.Adams. The chassis frame and body structure of the coach has been incorporated as a flexible body to depict its dynamic properties and structural loads. Approximations of the coach’s structural damping were derived by means of reverse engineering via design of experiment. The model was analysed using two different types of full vehicle simulations, in this paper referred to as Virtual Test Rig and Virtual Proving Ground. In the first mentioned simulation procedure, test rig software RPC Pro has been used in conjunction with Adams/Car to generate displacement inputs at the wheel spindles. These displacements are back-calculated from response signals measured during the physical test on the proving ground. In the latter simulation, the unconstrained model was instead driven over a digitized version of a proving ground road profile.

The model performance has been evaluated against the measured data from the physical test.

Results from virtual proving ground simulations show good correlation of vertical spindle loads but not as well for spindle loads in lateral and longitudinal directions. Acceleration responses in the coach structure demonstrated evident damping dependency as expected. The evaluated strain responses were non-conservative for all derived structural damping approximations. Simulations in the virtual test rig has shown that accelerations in the coach body structure are possible to replicate with high accuracy. The results from the virtual test rig demonstrated well-correlated strain responses for two of the four evaluated locations.

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ACKNOWLEDGEMENT

The author would like to express his sincerest gratitude to the people at the dynamics and strength analysis group, RBRA, at Scania bus chassis development and also all others involved from Scania for their support throughout this thesis work.

A special thanks to Peter Eriksson at Scania bus chassis development for his valuable guidance and advice during this thesis. Thanks are also directed to Robin Wagman for his work on the FE- model of the coach.

The author also thanks Igor Maletin for his help in providing test data and answering questions regarding the performed physical test.

Anders Ahlström and Anders Anbo are gratefully acknowledged for their help regarding software and simulation procedures. Furthermore, the author wishes to thank Niklas Karlsson and Martin Linderoth for their feedback and advice concerning test rigs and RPC Pro.

The author also wishes to express his appreciation to Ulf Sellgren at KTH for his valued feedback and suggestions during the thesis work.

Kim Bladh Stockholm, June 2012

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NOMENCLATURE

This chapter presents the nomenclature used in this thesis paper. Additional definitions and symbols with dissimilar meanings in different contexts are further explained in the text.

Mathematical style Notation Description

w scalar variable, italic

{𝐰} vector, bold, enclosed in curly brackets [𝐰] matrix, bold, enclosed in square brackets {−}^𝑇, [−]^𝑇 vector transpose, matrix transpose

[−]⁻¹ matrix inverse

[−]^𝐻 matrix complex conjugate transpose

𝑤̇ time derivative, dot

𝑤̈ second time derivative, double dot

Symbols

Symbol Description

{𝐮} Deformation vector of a structure [𝐌] Mass matrix of a structure

[𝐊] Stiffness matrix of a structure [𝐂] Damping matrix of a structure {𝐅} Force vector of a structure

[𝐑] Rigid body transformation matrix

{𝐮} Eigenvector

{𝛗} Normalized eigenvector

𝜔𝑖 Eigenfrequency to the 𝑖th mode shape of a structure

[𝚽] Modal matrix

{𝐙} Modal coordinate vector α, β Rayleigh damping constants

ξ Damping ratio

[𝐖] Reduction basis matrix {𝐪} Generalized coordinates

{𝐚} Modal coordinate vector, Craig-Bampton fixed-interface normal modes

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[𝚽n] Craig-Bampton fixed-interface normal mode matrix [𝚿c] Craig-Bampton constraint mode matrix

[𝐍] Orthonormalization transformation matrix {𝐔} Frequency domain drive signal vector {𝐘} Frequency domain response signal vector [𝐇] Frequency response function

[𝐒_𝑎𝑎] Auto power spectral density matrix of signal vector {𝐀}

[𝐒_𝑎𝑏] Cross power spectral density matrix of signal vectors {𝐀} and {𝐁}

[𝐉] Pseudo inverse frequency response function 𝐺(𝑗) Time domain discrepancy index for signal 𝑗 𝑇𝐷𝐷𝐼𝑚 Mean time domain discrepancy index 𝑎𝑖,𝑗 𝑖:th sample in signal 𝑗 of measurement 𝑎 𝑎�_𝑗 Mean value of signal 𝑗 of measurement 𝑎

𝑞 Number of compared signals

𝑠_𝑖 Load amplitude level of the 𝑖th block 𝑁_𝑓,𝑖 Number of cycles to failure under load 𝑠_𝑖 𝐶, 𝛽 Basquin’s law material parameters

𝑛𝑖 Number of applied cycles in the 𝑖:th load block 𝑘 Total number of load blocks

𝐷 Damage (Pseudo-damage)

𝐷𝑟𝑒𝑙 Relative pseudo-damage

Abbreviations

CAE Computer-Aided Engineering CCF Central Composite Face-Centered

CMS Component Mode Synthesis

CoG Centre of Gravity

CRG Curved Regular Road

DFT Discrete Fourier Transformation

DOE Design of Experiment

d.o.f. Degree(s) of Freedom

DP Development Process

FEM Finite Element Method

ILC Iterative Learning Control

MAM Mode Acceleration Method

MBS Multi-body Simulation (Multi-body System)

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MDM Mode Displacement Method

NVH Noise, Vibration and Harshness PSB Persistent Slip Bands

PSD Power Spectral Density RPC Remote Parameter Control

SOD Start of Development

SOP Start of Production

TDDI Time Domain Discrepancy Index VPG Virtual Proving Ground

WFT Wheel Force Transducer

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1 INTRODUCTION

This chapter describes the background and purpose of the thesis. The delimitations in which the work has been confined and the methods employed when performing the work are also presented.

1.1 Background

Road induced fatigue is one of the most common failure modes in the automotive industry. At heavy vehicle manufacturer Scania CV AB, henceforth mentioned as Scania, numerous methods have been adopted to test the durability of their product. Such methods include physical vehicle and component testing on public roads, proving grounds and in various types of test rigs. In addition to physical testing, computer simulations using finite element method (FEM) are employed throughout the design process. With the advent of simulation-driven design in the automotive industry, virtual testing is being given a far more influential role in product development. Replacing physical durability testing with virtual testing procedures necessitates analyses able to consider full vehicle surroundings. As a consequence, the dynamics and strength analysis group at Scania bus chassis development has initiated this thesis as a preliminary step towards implementing virtual full vehicle durability testing.

1.2 Aim

This thesis aims at evaluating the possibilities of virtual full vehicle simulations for durability assessments on the basis of how satisfactory a full vehicle coach model can be made to replicate reality. The thesis work has been intended as a first advance towards establishing more advanced virtual testing procedures in the development of buses and coaches at Scania. The purpose has therefore been to investigate how well the structural loads in a coach, experienced during physical testing, can be predicted by full vehicle testing of a virtual model.

1.3 Delimitations

The thesis work was limited to include only two types of virtual full vehicle analyses. Where the focus has been on the execution and assessment of these two analyses and how the virtual model performs compared to physical test data. No concluding fatigue life analysis of the coach body structure was decided to be performed, only comparative studies of structural responses. The virtual full vehicle model was of a Scania Touring 4x2 coach as measured data from previously performed tests of this particular coach were available. To carry out the analyses, available software at Scania has been used. These include multi-body simulation (MBS) software MSC.Adams (MSC Software Corporation, 2012) and FEM software Altair Hyperworks (Altair Engineering, Inc., 2012) and Nastran (MSC Software Corporation, 2012). Furthermore, test rig software MTS RPC Pro (MTS Systems Corporation, 2012) has also been utilized.

1.4 Method

The beginning of the thesis work involved a pre-study of vehicle testing, both physical and virtual. This information retrieval phase also concerned attaining necessary understanding of available software and what simulation procedures these software would accommodate.

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A virtual full vehicle model was created based on the specifications of the tested Scania Touring coach. The model was created in Adams/Car, consisting of subsystems such as suspensions and driveline from the current established Scania vehicle dynamics simulation library. A flexible body was generated from a detailed FE-model of the chassis and body structure to complete the vehicle assembly. Creating this FE-model has not been a part of the thesis due to the time consuming work such modelling entails. Instead a model which was started on before the commencing this thesis work has been used and modified to necessary extents. The complete full vehicle model was then verified to match the specifications of the real coach regarding weight, centre of gravity position, damper and air spring properties, etc.

One of the analyses that have been performed is called Virtual Test Rig and is carried out using Adams/Car together with the test rig software RPC Pro. Similar to physical test rigs, the model is spindle-coupled with a virtual test rig and measured responses from the physical testing are reproduced in the model by an iterative deconvolution technique. The second analysis performed is called Virtual Proving Ground, solely carried out in Adams/Car. It involves driving the model over a virtual 3D-road corresponding to the road profile of the test track segment at which the physical testing was performed.

The model performance was evaluated with respect to the physical test data. Different measurements of the models capability to imitate the physical vehicle responses were used in assessing the models performance.

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2 FRAME OF REFERENCE

This chapter presents the theoretical reference frame in which essential knowledge and previous research are introduced to lay basis for the remainder of the report.

2.1 Fundamentals of metal fatigue

A common failure mode of metal structures, vehicles in particular, is material fatigue. This failure phenomenon occurs by cumulative material damage brought on by repetitive loading over a certain period of time. For vehicles, this could for instance be the variable amplitude loading induced from a road or vibrations from a driveline. Fatigue is realized by the forming and growth of microscopic cracks in the material and is traditionally summarized in three succeeding phases:

• Crack nucleation. Persistent slip bands (PSB) are formed in the material grains from dislocation pile-ups. The dislocation accumulation occurs at regions of high surface stress concentrations when the structural component is subjected to several loading cycles. The PSBs can rise up and extrude or intrude the component surface and thereby forming microscopic notches and imperfections in the material surface which in turn allows for cracks initiation as explained by Frost, et al. (1974). Internal defects or surface imperfections from manufacturing can also become initiation points for cracks.

• Crack propagation. The propagation rate of a crack is often divided into three different regions I, II and III. These regions are dependent on the change of the stress-intensity factor at the crack tip (Pokluda & Šandera, 2010). Intrinsic microstructural damage mechanisms ahead of the crack tip acts to promote the propagation while extrinsic mechanisms at the crack wake impedes its growth. These mechanisms are more thoroughly discussed in (Ritchie, 1999).

• Final failure. When the crack reaches critical size the material’s capacity to sustain the applied load is compromised and ultimately fracture occurs.

As mentioned, in the crack nucleation phase dislocation pile-ups traditionally take place at regions of high stress. This explains why fatigue cracks typically emanate from geometrical and material discontinuities in a structure; for instance, holes, notches, fillets, welds and other structural joints which are all known to give rise to stress concentrations.

2.2 Durability testing in vehicle design

Several methods exist for testing of a new vehicle design. The objective is often to analyse the vehicle in terms of either noise, vibration and harshness (NVH), handling or durability. For the case of vehicle durability, the test methods are usually performed by means of:

• Testing on public roads and proving grounds

• Laboratory test rigs

• Computer simulations (FEM, MBS)

With the advent of virtual prototyping, early design stages tend to utilize more advanced computer simulations to provide load predictions and indications of design flaws concerning strength and fatigue. Performance analysis of virtual prototypes is what is commonly referred to as Virtual testing. As mentioned in Ferry, et al. (2002), conducting early design iterations by virtual testing without the need of a physical prototype is regarded as highly advantageous due to

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the decreased time to market and saved physical prototyping costs. The reduced necessity for expensive prototype manufacturing and time consuming tests has brought an incentive to further advance the concept of virtual testing.

Physical durability tests are ideally carried out at later stages of the development process (DP) for validation purposes. In other words, to confirm that the physical component meet the expected fatigue design life suggested by the initial numerical computer simulations. The above described development process was put in contrast to the more traditional development practice for the application of engine design by Rainer (2003). The two development processes were illustrated as in Figure 1.

a) b)

Figure 1.a) Traditional DP, b) Virtual testing DP. Adopted from (Rainer, 2003)

The traditional product development process in Figure 1.a has been known to put large emphasis on physical testing in the main design steps. Simple simulations have been used to support specifications in early concept stages while more complex model simulations have been used in parallel with real testing during the development of prototypes. The process in Figure 1.b integrates virtual testing as the key component in the design stages. Complex simulations with consideration to complete system environments can be employed at early design stages to support design decisions. The role reversal is evident, where in the traditional development process physical testing was backed up by computer simulations, the more modern development process suggests design based on virtual testing with backup from few physical prototype tests (Zwaanenburg, 2002).

The phasing out of physical testing and introduction of pure simulation-based design is however usually met with some reluctance. The reliance of virtual testing results is often questioned. Until full confidence can be given to virtual testing, physical testing will still be an important part of the product design stage.

Since physical testing has been adopted for several decades in the vehicle industry, the methods used today have been refined progressively over the years. As mentioned previously, physical tests tend to be time consuming. This is a direct consequence of fatigue damage accumulation being dependent of the number of loads cycles, and thus by extension, becoming highly time dependant. Reducing the time it takes to carry out physical durability testing has therefore been deemed vital and over the years different accelerated testing procedures have been developed.

System Assemblies

Components

Strategic

decision Concept Prototype

development Series Increasing complexity of simulation models

Decreasing of influencing parameters

Development time

SOD SOP

System Assemblies

Components

Strategic

decision Concept Prototype

development Series Increasing complexity of simulation models

Decreasing of influencing parameters

Development time

SOD SOP

Specification

Virtual testing Virtual testing Virtual testing

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All durability test inputs are in some sense derived from a Mission Profile. The Mission Profile consists of a number of loading events assumed representative of customer service loads. Power Spectral Density (PSD) spectra or signal time histories have in most cases been measured for each such event together with an approximation of how often or how long the vehicle is expected to experience each event during its service life (Halfpenny, 2006).

Accelerating the test procedure is performed in the Test Synthesis phase. During this stage the test is shortened without reducing the fatigue damage content from that of the Mission Profile.

This stage is applicable for all types of accelerated test methods, for instance when generating drive signals to test rigs, load input to FE-analyses or even when designing proving ground obstacles courses. Accelerated durability testing follows three different principal test synthesis methods as suggested by Halfpenny (2006).

• Compressed time. Excluding of non-damaging load sequences from the Mission Profile leaving only the most predominant damaging content left.

• Load amplification. Utilize the exponential relationship between stress amplitude and fatigue damage. By amplifying the load and thus overstressing the component yields in decreased time to achieve same fatigue damage accumulation as the Mission Profile.

• Combined. A common approach is to combine the above test acceleration methods.

2.3 Physical durability testing

Leaving the aforementioned computer simulations to be treated in the next chapter, the different physical durability test procedures are discussed more closely below.

2.3.1 Long-term testing

Long-term testing is often performed on either public roads or test tracks. The purpose of the test is to achieve mile coverage in a full vehicle with similar loading to what is experienced during customer usage. It is therefore not to be considered as an accelerated test. Long-term tests will ensure accurate replication of the loads present during customer usage, but with the downside of being extremely time consuming.

2.3.2 Proving ground testing

Proving ground testing also classifies as a full vehicle test although with the difference of being an accelerated test method. A vehicle proving ground often includes several durability tracks, obstacle courses, corresponding to various real-life rough road loading situations, see Figure 2.

In many cases, these obstacles are exaggerated regarding their dimensions with the intent of accelerating the test according to the load amplification approach. The fact that the testing is limited to repetitive runs over a concatenated series of these durability tracks also imply that only the severely damaging content which is captured by these obstacles is being considered (Halfpenny & Pompetzki, 2011). It can therefore be classified as an accelerated test method by combination of the load amplification and compressed time approach.

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Figure 2. Scania city bus on proving ground track for NVH testing 2.3.3 Road simulation test rigs

Further acceleration of durability tests has meant that the proving grounds would have to be moved in to a laboratory setting. Dodds and Plummer (2001) mention that road simulation test rigs were introduced in the 1950’s and that it has over the years developed into a highly sophisticated durability testing method. The reason why this type of testing has seen such wide acknowledgement in the vehicle industry is the advantages of a controlled environment, and consequently superior repeatability.

Road simulation test rigs are today predominantly based on a mathematical principal called frequency domain iterative learning control (ILC). Several software programs became available after the introduction of this load reproduction technique as discussed by Yudong, et al. (2012).

For example, Remote Parameter Control (RPC) by MTS System Corporation, Iterative Transfer Function Compensation (ITFC) proposed by Shenk Coporation, Multi-Input Multi-Output Iterative Control (MIMIC) by Tiab Corporation and also Time Waveform Replication (TWR) created by Instron Corporation and LMS Corporation. The iterative control technique is used to reproduce operational loads with high precision. The operational loads, or target responses, are often measured signals from proving ground testing. A more detailed description of the ILC algorithm will be presented in a succeeding chapter.

The test rig approach enables continuous accelerated durability testing with minimal human involvement and thereby further reducing testing times compared to that of proving ground testing. This is because the drive signal of the test rig can easily be combined to simulate several proving ground durability tracks sequentially without any significant delay time. The drive signal can then be repeated continuously until fatigue damage is realized by evident crack forming.

Road simulation test rigs with this kind of load replication technique are most commonly used for full vehicle testing, but subsystems of all sizes are possible to test, as shown in Figure 3.

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a) b)

Figure 3.a) Full vehicle test rig (Halfpenny & Pompetzki, 2011), b) Bus rear section in test rig 2.3.4 Components and subsystem testing

Components or subsystems with very high endurance limits are sometimes not feasible to test by means of full vehicles testing. The above mentioned load replication technique is occasionally used with smaller test rigs to perform tests on subsystems like suspensions or individual chassis components. More common however, is to use constant amplitude test rigs or vibration test rigs for component or subsystem testing, see Figure 4. The test specimen is constrained in a test rig and loaded repeatedly by a predetermined displacement, or frequency spectrum in the case of vibration test rigs, until failure occurs. By principle, several specimens are tested to map the variation of durability. This in turn can be used as a base for a reliability prognosis of the component or subsystem being tested.

Figure 4. Chassis frame in constant amplitude test rig

2.4 Virtual durability testing

The concept of virtual prototyping and the advances in Computer-Aided Engineering (CAE) over the years have incited the development of a large quantity of different analysis techniques for durability testing of vehicles in a virtual environment. Virtual durability testing is generally characterized in three steps.

• Load history prediction

• Stress/strain analysis

• Fatigue life assessments

Performing fatigue life assessments, whether it being in fatigue post-processing software or calculated analytically, necessitate that the experienced stress or strain distributions from loading are known. A preceding stress/strain analysis is therefore always required, traditionally involving

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FEM. Furthermore, a prerequisite for achieving realistic stress and strain results and succeeding fatigue predictions is that loads acting on the structure are known or can be estimated with sufficient precision. Virtual durability testing can therefore, if necessary, incorporate stress/strain analysis with preceding or combined MBS analysis for load predictions.

Extensive literature is available on the subject of fatigue life estimation, both regarding modelling and assessment, and will for that reason be left to references (Bishop & Sherratt, 2012) (Niemi, 1995). Attention will instead be on the aspects of load prediction and stress/strain analysis, the first two listed steps in virtual durability testing. The following chapters present the most commonly suggested stress/strain analyses for virtual durability testing of vehicles.

Stress/strain analyses using FEM are described as well as methods with combined MBS load predictions.

2.5 Static stress/strain analysis by FEM

As mentioned, virtual vehicle durability testing aspires at achieving fatigue assessments of vehicle subsystems or components. The most simplistic fatigue predictions are based on static analysis of such subsystems or components. Equivalent static load cases, derived from different dynamic load histories, are separately applied to a FE-model. The structural responses can then be superimposed in order to estimate the fatigue strength (Lee, et al., 2011).

Static analysis can be employed using two different techniques. The boundary conditions of the structure being analysed often determine which technique is best suited. The conventional static analysis necessitates a constrained structure whereas the second inertia relief technique is used for unconstrained structures. The inertia relief technique is described further below.

The drawback with the static analysis is that the translation of real dynamic loads to equivalent static load cases tends to lack accuracy as consideration to local vibrations and dynamic effects is non-existent as stated by Kuo and Kelkar (1995) and Haiba, et al. (2002).

2.5.1 Static analysis with inertia relief

Common in both the automotive and aerospace industry are analyses of unconstrained structures subjected to constant or quasi-static external loads. Lee, et al. (2011) give the example of a vehicle driven on a road or an airplane in flight. Unconstrained, in this sense, implies a system on to which no motion constraints have been enforced.

The traditional finite element static analysis cannot be employed in the case of unconstrained structures due to the singularities in the stiffness matrix introduced from rigid body motion.

However, using static analysis with the inertia relief technique overcomes this predicament. The technique considers the applied external loads and calculates the resulting rigid body accelerations with respect to a reference point. These accelerations together with the mass matrix give in turn the inertial forces at every nodal degree of freedom. Combining these inertial forces with the external loads, balanced at the mentioned reference point, yields a static equilibrated formulation of the problem (Lee, et al., 2011).

The basic equations in the inertia relief method are derived from the dynamics theory. Explained in this section is the inertia relief theory described by Lee, et al. (2011). The total deformation {𝐮𝐭} experienced by the unconstrained structure includes a term describing the rigid body motion {𝐮𝐫} and also a contribution from the flexible deformation {𝐮}.

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{𝐮_t} = {𝐮_r} + {𝐮} (1)

The static formulation through inertia relief of an unconstrained structure with rigid body modes with respect to its centre of gravity is an approximation given by the dynamic equilibrium equation below:

[𝐌]{𝐮̈r} + [𝐌]{𝐮̈} + [𝐊]{𝐮} = {𝐅} (2) Where the mass and stiffness matrix have been denoted [𝐌] and [𝐊] respectively and the external force vector is represented as {𝐅}. The damping term has been neglected due to the assumption of steady-state structural response. Furthermore, the inertial forces arising from flexible deformations are considered small in comparison to those from the rigid body motions.

These are hence neglected, yielding in the following equation:

[𝐌]{𝐮̈_r} + [𝐊]{𝐮} = {𝐅} (3)

Eq. (3) expresses the basic static analysis with inertia relief for an unconstrained structure with its centre of gravity as the centre of its rigid body motions. The rigid body modes of the unconstrained structure can be expressed with respect to an arbitrary reference point for the finite element application. A rigid body transformation matrix [𝐑] correlating the rigid body motions of the reference point �𝐮̈r,0� to motions at the structural nodes can be expressed as:

{𝐮̈_r} = [𝐑]�𝐮̈_r,0� (4)

The rigid body transformation matrix also describes the relation between the applied load vector {𝐅} and the resultant force vector at the reference point {𝐅0}. This equation is given as:

[𝐑]^T{𝐅} = {𝐅0} (5)

In the same way, the nodal inertial forces from the rigid body motion [𝐌]{𝐮̈_r} can be expressed as inertial forces with respect to the reference point according to:

[𝐑]^T[𝐌]{𝐮̈r} = [𝐑]^T[𝐌][𝐑]�𝐮̈r,0� (6) The equilibrium formulation where the resultant external loads are balanced by the inertial forces at the reference point is then given as:

[𝐑]^T[𝐌][𝐑]�𝐮̈r,0� = [𝐑]^T{𝐅} (7) Solving for the rigid body acceleration from the above equation yields:

�𝐮̈_r,0� = ([𝐑]^T[𝐌][𝐑])⁻¹[𝐑]^T{𝐅} (8) This balancing acceleration field is applied back to the structure. The nodal displacements {𝐮}

relative to the reference point, in other words the flexible deformation of the structure, can then be calculated by the following equation, which is given by substituting Eq. (4) in Eq. (3).

[𝐌][𝐑]�𝐮̈r,0� + [𝐊]{𝐮} = {𝐅} (9)

Due to the stiffness matrix being singular as a consequence of the unconstrained structure, additional efforts have to be made in order to actually solve for the relative nodal displacement.

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Although several methods exist for solving for the displacements, the technique described here takes advantage of the nodal displacement vector being orthogonal to the rigid body eigenvectors in [𝚽r]. Because the rigid body transformation [𝐑] can be expressed as a linear combination of these rigid body mode shapes, {𝐮} can be uncoupled with the rigid body motion according to:

[𝚽r]^T[𝐌]{𝐮} = 0 → [𝐑]^T[𝐌]{𝐮} = 0 (10) Combining Eqs. (6), (8) and (9) with Eq. (10) yields the following equation from which the flexible nodal displacements can be solved:

� {𝐮}

�𝐮̈_r,0�� = � [𝐊] [𝐌][𝐑]

[𝐑]^T[𝐌] [𝐑]^T[𝐌][𝐑]�

−1� [𝐅]

[𝐑]^T[𝐅]� (11)

The application of the inertia relief technique in the case of analysing the structural stresses and strains in a coach is somewhat limited. The method is only valid when considering a structure in steady-state or when the frequency content of the applied excitations are well below the natural frequencies of the structure as mentioned by Kuo and Kelkar (1995), that is to say, it can be considered as quasi-static.

Eriksson (2002) describes how typical bus and coach structures exhibits the majority of response energy in two distinct frequency bands when looking at the vertical vibration response. The first frequency band at 1-2 Hz corresponds to rigid body modes such as pitch and bounce. The enhanced vibrations in the latter frequency band around 8-12 Hz emanates from a number of factors. One being the dynamics of a typical bus rear suspension, which when considered as a single-d.o.f. spring-mass system displays an eigenfrequency within the 8-12 Hz frequency region. Another factor is the engine and gearbox assembly which has distinct eigenfrequencies in this region when on its isolation mounts. Furthermore is the fact that a coach body structure often also exhibits a number of free vibration frequencies at this mentioned frequency band.

With this in mind, using inertia relief when analysing the influence of road induced loads on a coach structure can be unsuitable since the quasi-static approximation will neglect the significant dynamic effects mentioned above.

2.6 Dynamic stress/strain analysis by FEM

Dynamic analyses are commonly used to calculate the transient response of a structure when subjected to time-dependant loading. The governing equation of a dynamic finite element problem consists of a system of second order linear differential equations given as:

[𝐌]{𝐮̈} + [𝐂]{𝐮̇} + [𝐊]{𝐮} = {𝐅} (12)

This complete dynamic equilibrium equation includes the damping matrix [𝐂], which was neglected in the previously covered static inertia relief technique. Today several structural dynamics calculation methods have been implemented in the finite element analysis application together with reduction methods to decrease the number of system d.o.f. for beneficial gains in computational efficiency. These calculation methods aid in answering different questions about the structural dynamics of the problem at hand. The most common dynamic analyses available in FE software, as mentioned by Cook, et al. (2002), are:

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• Modal analysis. Determines the natural frequencies and mode shapes of a structure by solving the structural eigenvalue problem.

• Harmonic response analysis. Also known as frequency response or forced vibration analysis. Calculates the sustained cyclic response (subsequent to initial transients) of a structure from a sinusoidal loading with known amplitude and frequency.

• Direct transient response analysis / Modal transient response analysis. Determines the structural transient response, often called response history, to any general time- dependant loading. This is calculated by time integration of the differential equation of motion. Model order reduction techniques are commonly used with this type of analysis.

• Response spectrum analysis. An inexpensive approach to approximate the maximum structural response to a given excitation spectrum assuming linear system response, without regard to when it appears during the response history. Normally, the peak responses of a number of the lowest structural modes are combined to estimate the peak linear response of the structure.

• Random vibration analysis. Similar to the response spectrum analysis but with the difference of being probabilistic as stochastic excitation described by statistical properties is used as input. Both the aforementioned response spectrum analysis and the random vibration analysis are often used for loading conditions such as seismic loads (earthquakes), wind loads, rocket motor vibrations and so on.

Narrowing this assortment of analyses down to those appropriate for determining the fatigue life of vehicle structures is necessary. If the purpose is to analyse the amplitude varying road loads with interest of the full response time history of a vehicle structure, the transient response analyses is considered best suited.

2.6.1 Direct transient response analysis

The ordinary transient analysis is performed by direct integration. This refers to calculating the responses by incremental time integration of Eq. (12) without changing its form. The responses in the structure are calculated at different time steps, separated by time increments ∆𝑡𝑖. The practice of direct integration uses the equation of motion, a time integration method and known conditions at one or several previous time steps as mentioned by Cook, et al. (2002). The integration algorithms used in FE software are either explicit or implicit depending on how this integration method is formulated. An explicit algorithm uses a difference expression with information only from preceding time steps according to:

{𝐮}n+1 = 𝑓({𝐮}n, {𝐮̇}n, {𝐮̈}n, {𝐮}n−1, … ) (13) Whereas the implicit method uses a difference expression which is combined with the equation of motion at time step 𝑛 + 1:

{𝐮}n+1= 𝑓({𝐮̇}n+1, {𝐮̈}n+1, {𝐮}n, {𝐮̇}n, {𝐮̈}n, … ) (14) The methods are often also classified according to how far the information dates back in the difference expression. Single-step indicates that only information from step 𝑛 is included whereas if information dates back to step 𝑛 − 1 the method is classified as a two-step method (Cook, et al., 2002). Two common integration methods are the Central difference method (two- step explicit) and Newmark method (single-step implicit).

In practical use, without going into specific details, it is recommended that the explicit method is used for wave propagation problems where high-frequency modes have to be regarded. These types of analyses often span over a small period of time. This is a consequence of the method being conditionally stable, meaning that there is a quite small critical time step ∆𝑡_𝑐𝑟 which

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should not be overstepped, otherwise the calculations become unstable (Cook, et al., 2002). In the vehicle industry, the explicit integration is for the most part used for crash simulations and roll-over tests. The implicit method is better suited for structural dynamics problem where loads are varying more slowly and only the lower modes are dominant in the structural response, as in the case of road induced loads acting on a vehicle. The method is unconditionally stable and ∆𝑡 is therefore not limited by stability but by consideration to accuracy. In contrast to the explicit method, it is however more computationally expensive each time step (Cook, et al., 2002). The computational cost becomes apparent when large d.o.f. systems like full vehicles finite element models are analysed. Adopting modal methods is sometimes considered more feasible to allow for model order reduction, as suggested by Huang, et al. (1998).

2.6.2 Modal transient response analysis

Transient response analysis by modal approach is based on mode superposition, a procedure where normal mode shapes are superimposed to characterize the dynamic response of a linear structure. Model order reduction is generally performed with this type of analysis to reduce computational cost, the reduction is realized by modal truncation (Lee, et al., 2011). Figure 5 illustrates how an original system model can be decoupled and reduced using modal superposition and truncation.

Figure 5. Schematic of modal analysis and system reduction (Qu, 2004)

The mode superposition method is performed by initially solving the system eigenfrequencies and corresponding mode shapes. The solution to the undamped free vibration structure in Eq.

(12) with a total of 𝑛 d.o.f. is:

{𝐮(t)} = {𝐮} sin 𝜔𝑡 (15)

where

{𝐮} - The eigenvector, natural mode shape of the structure 𝜔 – The eigenvalue, natural frequency of the structure

Substituting Eq. (15) in Eq. (12) yields the 𝑛:th order eigenproblem as:

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([𝐊] − 𝜔²[𝐌]){𝐮} = [𝟎] (16)

Solving the eigenvalues ω_n² and corresponding eigenvectors from the above equation and normalizing the eigenvectors with respect to the mass matrix yields the orthonormal mode shapes which are further on noted {𝛗}_n. The modal matrix is then defined with each eigenvector in its columns:

[𝚽] = [{𝛗}1, {𝛗}₂, … , {𝛗}_n] (17) The mode superposition refers to the introduction of a coordinate transformation to generalized d.o.f. 𝑍𝑖, often called modal coordinates, to help in decoupling the system of equations.

{𝐮(t)} = [𝚽]{𝐙} = �{𝛗}iZi(t)

n i=1

(18)

The equation explains how the displacement in every d.o.f. is a linear summation of the system modal shapes where every mode contribution is defined by the modal coordinate. Substituting Eq. (18) and its derivatives into Eq. (12) yields:

[𝐌][𝚽]�𝐙̈� + [𝐂][𝚽]�𝐙̇� + [𝐊][𝚽]{𝐙} = {𝐅} (19) To decouple the equations, the orthogonal properties of the modal matrix can be used by premultiplying by its transpose [𝚽]^T.

[𝚽]^T[𝐌][𝚽]�𝐙̈� + [𝚽]^T[𝐂][𝚽]�𝐙̇� + [𝚽]^T[𝐊][𝚽]{𝐙} = [𝚽]^T{𝐅} (20) The orthogonal properties will diagonalise the stiffness and mass matrices and due to the normalization of the eigenvectors, the mass matrix becomes the identity matrix [𝐈]. The orthogonal properties of the eigenvectors are described by:

�{𝛗}ⁿ^T[𝐌]{𝛗}m= 0 m ≠ n

{𝛗}_n^T[𝐌]{𝛗}n = 1 m = n �{𝛗}ⁿ^T[𝐊]{𝛗}m= 0 m ≠ n

{𝛗}_n^T[𝐊]{𝛗}n= ω_n² m = n (21) To decouple the system it is also a prerequisite that the damping matrix can be diagonalised. The eigenvectors, although orthogonal to the mass and stiffness matrices, are generally not orthogonal to the damping matrix. This implies that the structural damping has to be expressed in a certain way to allow for it to be diagonalised. The necessary condition for the diagonalisation of the damping matrix is that the matrix [𝐌]⁻¹[𝐂] commutes with [𝐌]⁻¹[𝐊], that is:

[𝐂][𝐌]⁻¹[𝐊] = [𝐊][𝐌]⁻¹[𝐂] (22)

One common case which satisfies this condition is the proportional damping (Rayleigh, 1877), also referred to as Rayleigh damping. This viscous damping representation is expressed as proportional to the mass and stiffness matrices according to:

[𝐂] = α[𝐌] + β[𝐊] (23)

Besides proportional damping, FE software often offers direct modal damping as an alternative.

The diagonal modal damping matrix is simply defined by describing the n-th diagonal coefficient as:

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[𝐂] = diag(2ξ1ω1, … , 2ξnωn) (24) Where 𝜉_𝑛 is the damping ratio for mode 𝑛 prescribed by the user. By this approach, it becomes possible to set individual damping ratios for each mode, either estimated or experimentally determined. By using either of these viscous damping models, the system can be reduced to 𝑛 decoupled motion equations according to:

M_φ,nZ̈_n+ C_φ,nŻ_n+ K_φ,nZ = F_φ,n(t) (25) Where the coefficients in Eq. (25), the diagonal elements in the modal coefficient matrices, are:

M_φ,n= {𝛗}_n^T[𝐌]{𝛗}_n= 1 K_φ,n = {𝛗}_n^T[𝐊]{𝛗}n = ω_n²

C_φ,n = {𝛗}_n^T[𝐂]{𝛗}n = 2ξ_nω_n or C_φ,n = {𝛗}_n^T[𝐂]{𝛗}n = α + βω_n² F_φ,n(t) = {𝛗}nT{𝐅(t)}

2.6.2.1 Modal truncation

Mode truncation assumes that a sufficiently accurate solution is obtainable using only a subset of modes. By only retaining this reduced set of modes the number of decoupled system equations is reduced, which in turn provides gains in computational efficiency (Qu, 2004). In most cases, not all computed modes are noticeably excited and can thereby be excluded without any significant loss in accuracy. Although, it should be kept in mind that truncating modes in a particular frequency range may consequently truncate response behavior in the same frequency range.

Cook, et al. (2002) mention that it is not only important to consider the frequency content of the transient loading, but also its spatial distribution when choosing at what cut-off frequency modes should be truncated. When performing mode truncation by only retaining 𝑚 modes, Eq. (18) becomes:

{𝐮(t)} = �{𝛗}_iZ_i(t)

m i=1

𝑚 ≪ 𝑛 (26)

The mode superposition method described above is known as the mode displacement method (MDM). It is evident that the contributions from the omitted modes are ignored with this method.

Additional correction methods have therefore been suggested to increase the accuracy for a given number of modes, or equivalently, to obtain similar accuracy by fewer retained modes. Two of these correction methods are the mode acceleration method (MAM) and modal truncation augmentation, both based on the concept of static correction. The general idea of MAM is that loading represented by the non-retained modes will only contribute to a quasi-static response but not to any acceleration or velocity responses (Cornwell, et al., 1983). The response may be expressed as before but with an additional correction term according to:

{𝐮(t)} = �{𝛗}iZ_i(t)

m i=1

+ {𝐪_𝑐𝑜𝑟} (27)

The correction term is obtained by rewriting Eq. (12) in to the following form:

{𝐮(t)} = [𝐊]⁻¹𝐅(t) − [𝐊]⁻¹[𝐂]{𝐮̇} − [𝐊]⁻¹[𝐌]{𝐮̈} (28)

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Substituting the derivatives of Eq. (26) into the above equation presents the mode acceleration approximation of the physical displacements:

{𝐮(t)} = [𝐊]⁻¹𝐅(t) − [𝐊]⁻¹[𝐂] �{𝛗}iŻi m

i=1

− [𝐊]⁻¹[𝐌] �{𝛗}iZ̈i m

i=1

(29)

Only the velocity and acceleration terms have been transformed by modal transformation while the first term contribution corresponds to the pseudo-static displacement due to 𝐅(t). The name mode acceleration method has originated from the fact that the method involves superposition of the modal accelerations rather than displacements. The modal truncation augmentation method is more or less an extension of the modal acceleration method and is left to reference (Dickens, et al., 1997).

2.6.3 Component mode synthesis

When a complex structural system is to be analysed with regards to its dynamic response, it is common to adopt substructuring methods such as component mode synthesis (CMS).

Substructuring refers to the subdivision of a complete structure into substructures, or superelements, whose boundaries may be arbitrarily specified. The order reduced synthesized structure, composed of its assembled substructures, can then be analysed using any structural dynamics analysis. Employing this type of method often yields in large managerial and computationally economical advantages as mentioned by Cook, et al. (2002). CMS also plays a key role in flexible MBS analyses. Its application becomes apparent if one would consider the bodies of a MBS model as individual substructures. Using CMS on a FE-model of such a body allows for it to be formulated and imported as a flexible body in a MBS model. This in turn gives the possibility to analyse the dynamic system considering component flexibility, but also to analyse the specific component when in its system environment, coupled to various MBS elements with linear and nonlinear properties.

Two general types of CMS methods exist, the fixed-interface and free-interface approaches.

Only the fixed-interface Craig-Bampton method (Craig & Bampton, 1968) will be presented here as it is a widely used method and the single most common condensation technique adopted for flexible MBS. Just as with any reduction techniques, the general idea is to find a low-dimension subspace [𝐖] which sufficiently estimates the displacement vector {𝐮} in Eq. (12). Such an approximation can be written as:

{𝐮} ≈ [𝐖]{𝐪} (30)

Where the physical coordinates {𝐮} are expressed in terms of the component generalized coordinates {𝐪} and the reduction basis matrix [𝐖]. In the aforementioned modal superposition approach, the reduction basis vectors were the free vibration eigenvectors of a reduced set of modes, and the generalized coordinates were the so called modal coordinates.

For the Craig-Bampton method, a partitioned form of Eq. (12) is used. In its partitioned form, Eq. (31), the system has been divided into a set of boundary d.o.f. {𝐮b} and a set of interior d.o.f.

{𝐮_i}. Subscript 𝑏 denotes the interface boundary set and 𝑖 represents the interior set. The boundary set contains the d.o.f. that later on might be constrained or coupled to another substructure.

�𝐌^bb 𝐌bi

𝐌_ib 𝐌_ii� �𝐮̈𝐮̈^b_i� + �𝐂^bb 𝐂bi

𝐂_ib 𝐂_ii� �𝐮̇𝐮̇^b_i� + �𝐊^bb 𝐊bi

𝐊_ib 𝐊_ii� �𝐮_b

𝐮_i� = �𝐅𝐅^b_i� (31)

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The Craig-Bampton method defines its reduction base by a truncated subset of fixed-interface normal modes and a set of constraint modes determined by the number of defined boundary points. The fixed-interface normal modes are the vibration modes of the structure with all boundary points fixed. The constraint modes are the resulting static displacement shapes when unit displacements are individually applied to each single boundary d.o.f. while the remaining ones are kept fixed. As such, the finite element structure is transformed from its original physical coordinates {𝐮} to a hybrid set of physical coordinates at the boundary {𝐮b} and modal coordinates {𝐚} of the interior set. The relation between these hybrid coordinates and the physical coordinates are:

{𝐮} = �𝐮_b

𝐮_i� = [𝐖] �𝐮𝐚 � = [𝚿^b ^c 𝚽_n] �𝐮𝐚 � ^b (32) In the above equation the constraint mode matrix [𝚿_c] and the fixed-interface normal mode matrix [𝚽n] are to be determined. The fixed-interface normal modes are computed from the eigenvalue problem of the interior set. As these are calculated with fixed boundaries {𝐮b} = {𝟎}, the equation becomes:

([𝐊_ii] − 𝜔²[𝐌_ii]){𝐮_i} = [𝟎] (33) After solving Eq. (33) and retaining only the first 𝑚 < 𝑛 modes, the eigenvectors can be collected as:

[𝚽im] = [{𝛗}1, {𝛗}2, … , {𝛗}m] (34) With accordance to the partitioned formulation in Eq. (31), the fixed-interface normal mode matrix with respect to all coordinate in the substructure becomes:

[𝚽_n] = � 𝟎𝚽im� (35)

The constraint modes are the static displacement patterns obtained when imposing a unit displacement at an interface d.o.f. in the set 𝑏 while others are kept fixed. The static form of Eq.

(31) with both sets included and assuming zero inertia effects {𝐅i} = {𝟎} becomes:

�𝐊^bb 𝐊_bi 𝐊_ib 𝐊_ii� �𝐮_b

𝐮_i� = �𝐑𝟎� (36)

Obtained from the lower partition of Eq. (36) is:

{𝐮_i} = −[𝐊_ii]⁻¹[𝐊_ib]{𝐮_b} (37) The unit matrix [𝐈] describes the unit displacement of each boundary d.o.f. in turn when the constraint modes are to be obtained, i.e. {𝐮_b} = [𝐈]. It is therefore realized that the constraint modes contribution on the interior set is given by:

{𝚿} = −[𝐊ii]⁻¹[𝐊ib][𝐈] = −[𝐊ii]⁻¹[𝐊ib] (38) The constraint mode matrix with respect to the entire substructure can then be written as:

[𝚿_c] = � 𝐈𝚿� = � 𝐈

−[𝐊ii]⁻¹[𝐊ib]� (39)

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The complete Craig-Bampton reduction base, or transformation, is given by assembling Eq. (35) and Eq. (39).

{𝐮} = [𝐖] �𝐮𝐚 � = [𝚿^b ^c 𝚽_n] �𝐮𝐚 � = �^b 𝐈 𝟎

−[𝐊ii]⁻¹[𝐊ib] 𝚽im� �𝐮𝐚 � ^b (40) The Craig-Bampton reduced mass, damping and stiffness matrices are obtained after substituting Eq. (40) into Eq. (31) and then projecting Eq. (31) onto the subspace [𝐖]:

�𝐌� � = [𝐖]^𝑇[𝐌][𝐖] = � 𝐈 𝟎

𝚿 𝚽im�^T�𝐌^bb 𝐌_bi

𝐌ib 𝐌ii� � 𝐈 𝟎

𝚿 𝚽im� = �𝐌�_CC 𝐌�_CN

𝐌�_NC 𝐈 � (41)

�𝐂�� = [𝐖]^𝑇[𝐂][𝐖] = � 𝐈 𝟎

𝚿 𝚽_im�^T�𝐂^bb 𝐂bi

𝐂_ib 𝐂_ii� � 𝐈 𝟎

𝚿 𝚽_im� = �𝐂�_CC 𝟎

𝟎 2𝛏𝛚� (42)

�𝐊�� = [𝐖]^𝑇[𝐊][𝐖] = � 𝐈 𝟎

𝚿 𝚽im�^T�𝐊^bb 𝐊_bi

𝐊ib 𝐊ii� � 𝐈 𝟎

𝚿 𝚽im� = �𝐊�_CC 𝟎

𝟎 𝛚^𝟐� (43)

Subscripts 𝐶 and 𝑁 denotes constraint modes and normal modes respectively. The Craig- Bampton dynamic equation of motion can be written as follows, under the assumption that forces are only applied at boundary nodes, {𝐅i} = {𝟎}:

�𝐌�_CC 𝐌�_NC

𝐌�_CN 𝐈 � �𝐮̈𝐚̈ � + �^b 𝟎 𝟎 𝟎 2𝛏𝛚� �𝐮̇_b

𝐚̇ � + �𝐊�CC 𝟎

𝟎 𝛚^𝟐� �𝐮𝐚 � = �^b 𝐅_b

𝟎 � (44)

Similar considerations as explained in the modal transient response analysis should be taken into account when truncating fixed-interface normal modes. A general rule of thumb suggested by Young (2000) is that mode shapes with frequencies at least 1,5 times higher than the frequency content of the structural loads should be retained.

2.7 Load prediction & Stress/strain analysis by MBS

The previously discussed finite element analyses are based on one crucial prerequisite, that is, the input loads at the interfaces of the structure are known. This however is not always the case.

An early vehicle design may imply that no test data are available or only exist as spindle loads or possibly structure acceleration responses. To overcome this predicament, multi-body dynamics has been established in the vehicle industry to perform load predictions in virtual durability testing.

2.7.1 Rigid multi-body simulation

Rigid multi-body simulation is characterized by the simulation of systems constituted by rigid bodies interconnected via motion constraint elements such as joints and force elements like springs, dampers and bushings. Early use of multi-body dynamics in vehicle durability simulations allowed the user to obtain and extract loads acting at the component interfaces within a vehicle subsystem. For example, Conle and Mousseau (1991) derived loads acting on a suspension control arm using MBS, which were then used as input loads to a separate FE-model of that same component. Stresses, strains and ultimately a fatigue prediction could then be computed.

Extracted loads from a MBS simulation can be used as input to any of the aforementioned static or dynamic analysis techniques. It should however be noted that since the rigid MBS approach

Virtual full vehicle durability testing of a coach

Virtual full vehicle durability testing of a coach

KIM BLADH

Virtual full vehicle durability testing of a coach

Kim Bladh

Sammanfattning

Abstract

ACKNOWLEDGEMENT

NOMENCLATURE

Mathematical style Notation Description

Symbols

Symbol Description

Abbreviations

TABLE OF CONTENTS

1 INTRODUCTION

1.1 Background

1.2 Aim

1.3 Delimitations

1.4 Method

2 FRAME OF REFERENCE

2.1 Fundamentals of metal fatigue

2.2 Durability testing in vehicle design

2.3 Physical durability testing

2.4 Virtual durability testing

2.5 Static stress/strain analysis by FEM

2.6 Dynamic stress/strain analysis by FEM

2.7 Load prediction & Stress/strain analysis by MBS