Virtual full vehicle durability testing of a coach

KIM BLADH

Master of Science Thesis Stockholm, Sweden 2012

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

I

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.

II

III

Master of Science Thesis MMK 2012:17 MKN 055

Virtual full vehicle durability testing of a coach

Kim Bladh

Approved

2012-06-05 Examiner

Ulf 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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V

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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VII

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

�� time derivative, dot

�� 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

VIII

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

𝑠𝑖 Load amplitude level of the 𝑖th block

𝑁𝑓,𝑖 Number of cycles to failure under load 𝑠𝑖

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)

IX

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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XI

TABLE OF CONTENTS

1 INTRODUCTION 1

1.1 BACKGROUND 1 1.2 AIM 1 1.3 DELIMITATIONS 1 1.4 METHOD 1

2 FRAME OF REFERENCE 3

2.1 FUNDAMENTALS OF METAL FATIGUE 3 2.2 DURABILITY TESTING IN VEHICLE DESIGN 3 2.2.1 ACCELERATION OF PHYSICAL DURABILITY TESTS 5 2.3 PHYSICAL DURABILITY TESTING 5 2.3.1 LONG-TERM TESTING 5 2.3.2 PROVING GROUND TESTING 5 2.3.3 ROAD SIMULATION TEST RIGS 6 2.3.4 COMPONENTS AND SUBSYSTEM TESTING 7 2.4 VIRTUAL DURABILITY TESTING 7 2.5 STATIC STRESS/STRAIN ANALYSIS BY FEM 8 2.5.1 STATIC ANALYSIS WITH INERTIA RELIEF 8 2.6 DYNAMIC STRESS/STRAIN ANALYSIS BY FEM 10 2.6.1 DIRECT TRANSIENT RESPONSE ANALYSIS 11 2.6.2 MODAL TRANSIENT RESPONSE ANALYSIS 12 2.6.3 COMPONENT MODE SYNTHESIS 15 2.7 LOAD PREDICTION & STRESS/STRAIN ANALYSIS BY MBS 17 2.7.1 RIGID MULTI-BODY SIMULATION 17 2.7.2 FLEXIBLE MULTI-BODY SIMULATION 18 2.8 VIRTUAL PROVING GROUND 20 2.9 VIRTUAL TEST RIG 21 2.9.1 ITERATION ALGORITHM 22 2.10 MODEL CORRELATION MEASURES 24 2.10.1 POWER SPECTRAL DENSITY AND TIME HISTORY 24 2.10.2 TIME DOMAIN DISCREPANCY INDEX 24 2.10.3 RELATIVE PSEUDO-DAMAGE 25 2.11 TEST PROCEDURE 27 2.11.1 VEHICLE PROPERTIES 27 2.11.2 SENSORS 27

3 MODELLING AND SIMULATION PROCEDURES 31

3.1 VIRTUAL COACH MODEL 31 3.1.1 TIRE MODEL 33 3.2 VIRTUAL TEST RIG MODEL 33 3.3 MODE TRUNCATION 33 3.4 DAMPING 36 3.5 SIMULATION PROCEDURES 39 3.5.1 VIRTUAL PROVING GROUND 39 3.5.2 VIRTUAL TEST RIG 40

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4 RESULTS 41

4.1 VIRTUAL PROVING GROUND 41 4.2 VIRTUAL TEST RIG 52

5 CONCLUSIONS AND DISCUSSION 59

5.1 CONCLUSIONS 59 5.1.1 VIRTUAL PROVING GROUND 59 5.1.2 VIRTUAL TEST RIG 59 5.2 DISCUSSION 60

6 FUTURE WORK 63

7 REFERENCES 65

APPENDIX A: SENSOR AND SIGNAL DESIGNATION 67

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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

Spec

ifica

tion

Virtual testing

Virtual testing

Virtual testing

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2.2.1 Acceleration of physical durability tests 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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{𝐮t} = {𝐮r} + {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]{��r} + [𝐌]{��} + [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]{��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 ���r,0� to motions at the structural nodes can be expressed as:

{��r} = [𝐑]���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]{��r} can be expressed as inertial forces with respect to the reference point according to:

[𝐑]T[𝐌]{��r} = [𝐑]T[𝐌][𝐑]���r,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]���r,0� = [𝐑]T{𝐅} (7)

Solving for the rigid body acceleration from the above equation yields:

���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]���r,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}���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]{��} + [𝐂]{��} + [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, {��}n, {��}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({��}n+1, {��}n+1, {𝐮}n, {��}n, {��}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 ∆𝑡𝑐𝑟 which

12

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} sin𝜔𝑡 (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 ωn2 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]���� + [𝐂][𝚽]���� + [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]���� + [𝚽]T[𝐂][𝚽]���� + [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}nT[𝐌]{𝛗}m = 0 m ≠ n{𝛗}nT[𝐌]{𝛗}n = 1 m = n

�{𝛗}nT[𝐊]{𝛗}m = 0 m ≠ n{𝛗}nT[𝐊]{𝛗}n = ωn

2 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 𝜉𝑛 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:

Mφ,nZn + Cφ,nZn + Kφ,nZ = Fφ,n(t) (25)

Where the coefficients in Eq. (25), the diagonal elements in the modal coefficient matrices, are: Mφ,n = {𝛗}nT[𝐌]{𝛗}n = 1 Kφ,n = {𝛗}nT[𝐊]{𝛗}n = ωn

2 Cφ,n = {𝛗}nT[𝐂]{𝛗}n = 2ξnωn or Cφ,n = {𝛗}nT[𝐂]{𝛗}n = α + βωn

2 Fφ,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}iZi(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}iZi(t)m

i=1

+ {𝐪𝑐𝑜𝑟} (27)

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

{𝐮(t)} = [𝐊]−1𝐅(t)− [𝐊]−1[𝐂]{��} − [𝐊]−1[𝐌]{��} (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}iZi

m

i=1

− [𝐊]−1[𝐌]�{𝛗}iZi

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. {𝐮i}. 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.

�𝐌bb 𝐌bi𝐌ib 𝐌ii

� ���b��i� + �𝐂bb 𝐂bi

𝐂ib 𝐂ii� ���b��i

� + �𝐊bb 𝐊bi𝐊ib 𝐊ii

� �𝐮b𝐮i � = �𝐅b𝐅i

� (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} = �𝐮b𝐮i � = [𝐖] �𝐮b𝐚 � = [𝚿c 𝚽n] �𝐮b𝐚 � (32)

In the above equation the constraint mode matrix [𝚿c] 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:

([𝐊ii]− 𝜔2[𝐌ii]){𝐮i} = [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:

[𝚽n] = � 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:

�𝐊bb 𝐊bi𝐊ib 𝐊ii

� �𝐮b𝐮i � = �𝐑𝟎� (36)

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

{𝐮i} = −[𝐊ii]−1[𝐊ib]{𝐮b} (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. {𝐮b} = [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:

[𝚿c] = � 𝐈𝚿� = � 𝐈−[𝐊ii]−1[𝐊ib]� (39)

17

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

{𝐮} = [𝐖] �𝐮b𝐚 � = [𝚿c 𝚽n] �𝐮b𝐚 � = � 𝐈 𝟎−[𝐊ii]−1[𝐊ib] 𝚽im

� �𝐮b𝐚 � (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 𝟎𝚿 𝚽im

�T�𝐌bb 𝐌bi𝐌ib 𝐌ii

� � 𝐈 𝟎𝚿 𝚽im

� = �𝐌� CC 𝐌�CN

𝐌�NC 𝐈� (41)

�𝐂�� = [𝐖]𝑇[𝐂][𝐖] = � 𝐈 𝟎𝚿 𝚽im

�T�𝐂bb 𝐂bi𝐂ib 𝐂ii

� � 𝐈 𝟎𝚿 𝚽im

� = �𝐂�CC 𝟎𝟎 2𝛏𝛚�

(42)

�𝐊�� = [𝐖]𝑇[𝐊][𝐖] = � 𝐈 𝟎𝚿 𝚽im

�T�𝐊bb 𝐊bi𝐊ib 𝐊ii

� � 𝐈 𝟎𝚿 𝚽im

� = �𝐊�CC 𝟎𝟎 𝛚𝟐� (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� ���b��� + �𝟎 𝟎

𝟎 2𝛏𝛚� ���b��� + �𝐊

�CC 𝟎𝟎 𝛚𝟐� �

𝐮b𝐚 � = �𝐅b𝟎 � (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

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does not consider the flexibility of components, the retrieved loads can, to certain extent, be inaccurate.

2.7.2 Flexible multi-body simulation With the computational capacity evolving, the complexity of the analyses has followed. The rigid body assumption has lost much of its necessity as condensed finite element models, flex bodies, are now frequently introduced into the MBS environment to replace rigid bodies. The term flexible multi-body simulation is frequently used for such analyses. Flexible multi-body simulation is not only adopted to achieve accurate load predictions at component interfaces, it also facilitates stress/strain analysis of the flex body.

2.7.2.1 Flex body theory in MBS Flex bodies in MBS software Adams are based on the previously described Craig-Bampton substructuring technique. However, as described by Óttarsson (2000), minor modifications have been necessary to the flex body formulation in order to increase their suitability for dynamic multi-body simulations. The modification is performed to eliminate the following issues:

• 6 rigid body modes are retained in the Craig-Bampton constraint modes. These have to be disabled because Adams applies its own rigid body d.o.f.

• The constraint modes in the original Craig-Bampton technique are derived from a static condensation. As a result, these modes do not express any contributing frequency content to the flexible body.

• Constraint modes in the original Craig-Bampton modal basis cannot be disabled, as this would imply an added constraint to the structure.

The listed problems above associated with the original Craig-Bampton modal basis are solved by mode shape orthonormalization. The eigenvalue problem of the Craig-Bampton representation in Eq. (44) is initially solved:

�𝐊��{𝐪} = 𝜔𝐶𝐵 2 �𝐌� �{𝐪} where {𝐪} = �𝐮b𝐚 � (45)

A transformation matrix [𝐍] is constructed from the acquired eigenvectors. This matrix allows for transformation of the Craig-Bampton reduction basis to an orthogonal basis with corresponding modal coordinate vector {𝐪∗} according to:

{𝐪} = [𝐍]{𝐪∗} (46)

The coordinate transformation can hence be written as:

{𝐮} = [𝐖]{𝐪} = [𝐖][𝐍]{𝐪∗} = [𝐖∗]{𝐪∗} (47)

Where [𝐖∗] denotes the orthonormalized Craig-Bampton mode shapes. The interpretation of the physical meaning of these modes is in some cases difficult. The fixed interface normal modes are transformed to an approximation of the normal modes for the unconstrained structure whereas the constraint modes are transformed to boundary eigenvectors, more thoroughly explained in (Óttarsson, 2000). Moreover, some modes will lack any kind of physical interpretation. Calculation of stresses and strains in flexible bodies is performed by modal stress and strain recovery. Computing the orthonormalized modal stress and strain matrices during the generation of the flex body gives the possibility to retrieve the stresses and strains when combined with modal coordinates from a MBS simulation. Additional theory of flexible bodies in MBS simulations is left to references (Géradin & Cardona, 2001) and (Óttarsson, 2000).

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The classification of different MBS analyses for vehicle durability testing is based on how the structural loads are derived. A categorization made in Tebbe, et al. (2006) divides analyses in semi-analytical and fully analytical approaches. The expression analytical is however unfortunate as people often have different interpretations of what is to be considered as analytical. For the sake of clarification, the approaches are here instead classified as semi-virtual and fully virtual.

2.7.2.2 Semi-virtual analysis The semi-virtual approach combines physical measurements with a virtual vehicle model, hence the name semi-virtual. Measurements from a proving ground test, such as spindle loads, suspension displacements or structural accelerations have been used in different types of semi-virtual analyses. These analysis techniques, mentioned in Tebbe, et al. (2006) and Liu, et al. (2007), are for instance the approach of applying measured displacements directly to an unconstrained vehicle model. This assures the control of the spindle positions, although only limited control of the sprung mass and its relative position to the suspension. The more acute issue is however the difficulties in obtaining measurements of spindle displacements. Often, they are derived from spindle accelerometers, which in many cases can introduce large errors and imply other difficulties. A technique where spindle loads from wheel force transducers (WFT) are applied directly to an unconstrained model was furthermore mentioned in Tebbe, et al. (2006). This method is said to often cause unstable simulations manifested in phenomenon such as drift and rollover in space by the unconstrained model. These occur as consequences of dissimilarities between the virtual MBS model and the physical vehicle. Errors and noise in measurement data and missing initial states also influence these erroneous vehicle motions. Another suggested technique is to apply similar loads as described above but to a constrained vehicle model in order to remove any unwanted spatial motions of the model. This however eliminates the inertia effects of the sprung mass and consequently often mean that the quality of the loadings becomes untrue. The basic principles of these methods are illustrated in Figure 6.

a)

b)

Figure 6.a) Free-Free, b): Body-locked

The slightly more intricate semi-virtual technique of back-calculating displacements to obtain the same measured responses from proving ground testing in the virtual model is considered superior. The analysis is sometimes referred to as Virtual Test Rig and uses the same concept as a physical road simulation test rig. A separate chapter below has been devoted for the virtual test rig analysis, see section 2.9.

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2.7.2.3 Fully virtual analysis The fully virtual analysis is independent of physical vehicle specific measurements. This poses a large advantage compared to the semi-virtual analysis since lack of physical measurement data in early design stages does not matter. A fully virtual analysis typically involves a MBS vehicle model with a tire model and a digitized road profile. The unconstrained vehicle model is driven over the virtual road and by doing so, the structural loads can be assessed. This type of analysis is commonly referred to as Virtual Proving Ground (VPG) or Virtual Road and is discussed further in the following chapter.

2.8 Virtual proving ground An early virtual proving ground (VPG) technique was suggested by Zhang and Tang (1996). Their concept of virtual proving ground incorporated creating finite element vehicle and tire models, and combining these with a finite element road model to perform nonlinear dynamic structural analyses using LS-DYNA finite element solver. Zhang, et al. (1999) further discuss the applications of the VPG techniques, mentioning possibilities in not only durability analysis, but also NVH, handling, crashworthiness and safety analysis. With the focus on vehicle durability analysis, the VPG approach entails driving the unconstrained vehicle model over a digitized proving ground, allowing for subsequent fatigue assessments, see Figure 7.

Figure 7. Virtual Proving Ground (eta, Engineering Technology Associates, Inc., 2012)

Today the term virtual proving ground is frequently used to denote a more general analysis technique. The concept is still the same, however, it is not strictly confined to finite element modeling of tires and road profiles. Virtual proving ground analyses have today been implemented in the majority of MBS software with vehicle application. Because this analysis uses a road profile as input, which is invariant to the vehicle driving upon it, the test can be performed without any need of vehicle specific physical measurement. However, shifting the load input from the wheel spindles, as in most semi-virtual analysis, to road level, introduces more complexity and uncertainties to the model. A driver module and an advanced tire-model are both necessities for this type of analysis. One of the most crucial components in a VPG analysis is the tire model. Achieving realistic results is highly dependent on how well the tire model can represents the actual tire characteristics (Daeoh, et al., 2009). Several tire models have been developed over the years with various degrees of complexity. Theory behind such tire models are not discussed in this paper.

21

2.9 Virtual test rig The virtual test rig analysis is based on the same procedure as its physical counterpart. The principal idea is to simulate road testing, hence the name road simulator test rigs. Apart from the name Virtual Test Rig, the analysis is also sometimes referred to as Virtual Test Laboratory or Virtual Iteration due to its iterative approach for generating drive signals. It should of course be noted that test rig analysis, physical or virtual, is not only viable for full vehicle models. Analyses on subsystem levels have also been conducted, for example Dannbauer, et al. (2006) and You, et al. (2003) used virtual MBS test rigs to analyse a suspension subsystem and a truck trailer refrigerator unit. The method involves an actuator driven test rig to which a vehicle or subsystem is coupled. Displacement signals for each actuator are back-calculated in an iterative process to match similar responses to those measured from proving ground testing. In other words, measured proving ground signals are replicated in the test rig specimen by deriving displacement drive signals from the same proving ground signals. The term drive signals signify the collection of signals used to drive all the test rig actuators. The measured proving ground responses which are attempted to be replicated in the test rig vehicle can for instance be wheel forces and moments measured by wheel force transducers (WFT), accelerations in the vehicle structure or strain gauge measurements. These proving ground signals are often mentioned as target response or desired responses. The vehicle model can be connected to a test rig in several different ways, most common is by spindle-connection as performed by You and Joo (2006), see Figure 8. The complex test rig in the figure is able to simulate loads in all 6 d.o.f at each wheel spindle. The actuators in the test rig can be driven by either displacement of force. This gives the possibility of simulating using a number of different control modes, full displacement control, full force control or combined force-displacement control.

Figure 8. Virtual test rig, 6 d.o.f. spindle coupling (You & Joo, 2006)

Another automotive industry example is from DaimlerChrysler, presented by Dressler, et al. (2009), where a tire-coupled 12-channel test rig was used, shown in Figure 9. A customized RMOD-K 30 tire model was used in the MBS vehicle model. Since the contact between the tire-bowl walls and tire carcass could not be described by the original tire model, a finite element model was used to estimate the lateral tire stiffness. A spring-damper model was then implemented in the MBS model to simulate the lateral behaviour of the tire when in contact with the tire-bowl sidewalls.

22

Figure 9. Tire-coupled 12-axis test rig (Dressler, et al., 2009) In comparison to the typical tire-coupled test rig, the four-post test rig, which only applies vertical excitations, this 12-axis test rig allows for replication of lateral, longitudinal forces and spindle moment. The advantage of this type of tire-coupled test rig is that the load input is applied at road level, making the input loading invariant of the vehicle being tested. The implication of this is that a variety of vehicle configurations could possibly be tested with the same drive signals. The obvious downside is however the need for a complex tire model.

2.9.1 Iteration algorithm The replication of proving ground loads in a laboratory test rig is typically accomplish by the aforementioned ILC control method. In this chapter the RPC technique developed by MTS Systems will be described. The method is used in both physical and virtual test rigs and involves a iterative deconvolution technique to reproduce measured responses. The basics of this iteration technique, as presented by Dodds and Plummer (2001), are described in this chapter. The iterative control algorithm adjusts the drive signal from one iteration-cycle to the next with the aim to achieve a response target profile. The intention is to find a drive signal which reproduces the field-measured responses with acceptable accuracy. With physical testing, when a satisfactory drive signal has been achieved it is replayed numerous times to perform the repetitive durability test. In the virtual test rig, the final drive signal only has to be simulated once to extract data for subsequent fatigue analysis. The basic theory behind the iterative drive signal generation is discussed below. The traditional iterative control techniques in road simulation test rigs use a frequency domain model of the system. The system in this case refers to the mechanical system represented by the vehicle. The initial stage involves system identification through excitation of the system by a drive signal vector {𝐮si(𝑡)}m×1 consisting of uncorrelated white-pink noise defined by the user in a frequency domain vector {𝐔si(ωk)}m×1. Subscript 𝑚 denotes the number of actuators (drive signals) in the test rig. After playing out this drive signal, measured responses in the vehicle are arranged in a vector {𝐲si(𝑡)}n×1. A existing prerequisite is that the number of responses 𝑛 is either equal or larger than the number of drive signals, that is to say, 𝑚 ≤ 𝑛. If this condition is not satisfied, the system would be undetermined. The frequency domain {𝐘si(ωk)}n×1 is then calculated by Discrete Fourier Transformation (DFT) of the obtained response time history signal vector. Where 𝜔𝑘 corresponds to a distinct frequency range determined by the user.

23

The goal of the system identification stage is to estimate the system frequency response function (FRF) [𝐇(𝜔𝑘)]. The FRF is a linear approximation of the vehicle in the frequency domain and relates the signal input to the system response as:

{𝐘(𝜔𝑘)} = [𝐇(𝜔𝑘)]{𝐔(𝜔𝑘)} (48)

The technique used to estimate [𝐇(𝜔𝑘)] can vary, one of the most common techniques is the 𝐻1 estimator, which calculates the FRF according to:

[𝐇(𝜔𝑘)] = �𝐒yu(𝜔𝑘)�[𝐒uu(𝜔𝑘)]−1 (49)

Where [𝐒uu(𝜔𝑘)] denotes the input auto power spectral density matrix and �𝐒yu(𝜔𝑘)� the cross power spectral density matrix. These spectral density estimates are calculated by Eq. (50), which estimates the spectral density between two general DFT vectors {𝐀(ωk)} and {𝐁(ωk)}:

[𝐒ab(𝜔𝑘)] =𝑇

2𝜋{𝐀(𝜔𝑘)}{𝐁(𝜔𝑘)}𝐻 (50)

In the above equation, 𝑇 is the signal time length. With vectors {𝐔si(ωk)} and {𝐘si(ωk)} from the noise excitation, the FRF is thus calculated substituting Eq. (50) in Eq. (49):

[𝐇(𝜔𝑘)] = {𝐘si(ωk)}{𝐔si(ωk)}𝐻[{𝐔si(ωk)}{𝐔si(ωk)}𝐻]−1 (51)

With the FRF estimated, the initial drive signal can be calculated by rearranging Eq. (48). The desired responses {𝐖(𝜔𝑘)} from the proving ground testing are used to obtain an initial drive signal estimate. The following equation applies when the number of target responses is equal to the number of input signals.

{𝐔(𝜔𝑘)} = [𝐇(𝜔𝑘)]−1{𝐖(𝜔𝑘)} (52)

When target signals exceed the number of input signals a pseudo inverse calculation achieves the estimation using least square fit:

{𝐔(𝜔𝑘)} = [𝐉(𝜔𝑘)]{𝐖(𝜔𝑘)} (53)

Where the inverse pseudo model is given by:

[𝐉(𝜔𝑘)] = �[𝐇(𝜔𝑘)]𝑇[𝐇(𝜔𝑘)]�−1[𝐇(𝜔𝑘)]𝑇 (54)

The subsequent iteration procedure is shown in Figure 10. The error, i.e. the difference between the desired and achieved response, is back-calculated as an input signal correction which is added to the previous drive signal. Gain values between 0 and 1 are manually defined for each input signal and added in [𝛂𝑖] = 𝑑𝑖𝑎𝑔(𝛼𝑖,1, … ,𝛼𝑖,𝑚). Gain values can also be specified for each response signal, in that case, the weighing is added before the error back-calculation. The gain values enable the user to prioritize certain signals by weighing the contributions to the subsequent drive signal.

24

Figure 10. Iteration scheme

In addition to the above iteration scheme, different test rig software allows the system model [𝐇(𝜔𝑘)] to be updated throughout the iterative process. Updating the system model entails adjusting the inverse frequency response function at each iteration step. This is deemed desirable in order to localize the linearisation of the often non-linear vehicle behaviour and achieve better convergence against the target response (Plummer, 2007).

2.10 Model correlation measures 2.10.1 Power spectral density and time history Signal time history and power spectral density (PSD) are two basic means to evaluate responses in vehicle testing. The PSD describes how the average power of a response signal is distributed over the frequency domain. It can therefore be used to get an idea of in which frequency bands the response exhibits most energy. The time history in conjunction with the PSD spectrum give a good indication of how well-correlated a model is when compared to physical measurements as differences in amplitudes, frequency content and phase shifts become evident.

2.10.2 Time domain discrepancy index Time domain discrepancy index (TDDI) was suggested by Forsén (1999) according to the following formulation when comparing a simulated signal 𝑎𝑗, against an experimental one 𝑓𝑗:

⇒ [𝐇(𝜔𝑘)]

System identification {𝐔si(ωk)} & {𝐘si(ωk)}

{𝐔i(𝜔𝑘)} = [𝐉(𝜔𝑘)]{𝐖(𝜔𝑘)} {𝐔i(𝜔𝑘)} → {𝐮i(𝑡)}

Calculate initial drive, 𝑖 = 0

⇒ {𝐲i(𝑡)}

Drive test rig measure responses

{𝐞i(𝑡)} = {𝐰(𝑡)} − {𝐲i(𝑡)} {𝐞i(𝑡)} → {𝐄i(𝜔𝑘)}

Calculate error

Error acceptable?

∆{𝐔i(𝜔𝑘)} = [𝐉(𝜔𝑘)]{𝐄i(𝜔𝑘)} ∆{𝐔i(𝜔𝑘)} → ∆{𝐮i(𝑡)}

{𝐮i+1(𝑡)} = {𝐮i(𝑡)} + [𝛂𝑖]∆{𝐮i(𝑡)}

Update drive signal

Final drive signal yes

no

𝑖 = 𝑖 + 1

25

𝐺(𝑗) =∑ �𝑎𝑖,𝑗 − 𝑓𝑖,𝑗�

2𝑖

∑ �𝑎𝑖,𝑗 − 𝑎�𝑗�2

𝑖

(55)

where

𝑎�𝑗 =1𝑁�𝑎𝑖,𝑗𝑖

(56)

𝐺(𝑗) is the discrepancy index for the signal denoted by index 𝑗 and subscript 𝑖 represents the sample number within the signal. The simulated and measured signals are assumed to have the same sampling frequency and length 𝑁. Averaging the discrepancy indices from each signal comparison yields the mean TDDI value, quantifying the overall correlation between the physical and simulated tests. The total number of compared signal channels is represented by 𝑞. The equation describing the overall average TDDI thus becomes:

𝑇𝐷𝐷𝐼𝑚 =1𝑞�𝐺(𝑗)𝑗

(57)

The discrepancy index becomes zero when two identical signals are compared, whereas two sine signals 180 degrees out of phase will yield a TDDI value of 2. The TDDI measure is foremost meant as a measure considering phase alignment of two compared signals, however, amplitude differences are naturally also considered. A quantified target TDDI value of 0,3 or less has been suggested (Forsén, 1999). This value indicates the accuracy that is possible to achieve with repeated physical test using the same test vehicle. The TDDI measurement is generally unsuitable for lengthier virtual proving ground simulations. This is a consequence of the phase shifts that occur between measured and simulated response signals as slight deviations in vehicle velocity are often present. The measurement is however very much applicable on the virtual test rig simulations.

2.10.3 Relative pseudo-damage Fatigue and durability performance is heavily dependent on load amplitudes and number of loading cycles. A measure of the potential fatigue damage, pseudo-damage, represented in each response signal from the model is compared to the pseudo-damage of the physical measurements. Calculation of pseudo damage is based on a synthetic Wöhler curve, correlating the load magnitude to the number of cycles until fatigue occurs. Basquin’s law describes this relationship as a straight line in a log-log diagram (Basquin, 1910):

𝑁𝑓,𝑖(𝑠𝑖) = 𝐶𝑠𝑖−𝛽 (58)

Where 𝑁𝑓,𝑖(𝑠𝑖) denotes the number of cycles with amplitude 𝑠 to failure. Parameters 𝐶 and 𝛽 are material constants, 𝛽 describes the slope of the Wöhler curve. Traditional Wöhler curves, S-N curves, possess a knee-point signifying either a fatigue limit or a change in the relationship between load magnitudes and loading cycles, see Figure 11.

26

Figure 11. General Wöhler curve

In this thesis, calculation of pseudo-damage is done without consideration to any knee-point and with a slope factor of 6. In order to calculate the cumulative damage, cycle-counting has to be performed on the signal to be analysed. This is commonly achieved by a rainflow counting algorithm. This allows a complex variable amplitude load signal to be condensed into blocks of constant amplitude load cycles. The damage can then be estimated using the Palmgren-Miner linear damage rule (Miner, 1945), seen in Eq. (59) below.

𝐷 = �𝑛𝑖𝑁𝑓,𝑖

𝑘

𝑖=1

(59)

The number of cycles 𝑛𝑖 at a certain load amplitude block in the examined load signal is divided by 𝑁𝑓,𝑖, the suggested fatigue-life at the same amplitude level. Summating each ratio of damaging contribution at every load amplitude level block 𝑖 in the signal gives the total accumulated damage. Combining Eq. (58) and Eq. (59) yields:

𝐷 = 𝐶−1�𝑛𝑖

𝑘

𝑖=1

𝑠𝑖𝛽 (60)

The relative pseudo-damage is hence given by:

𝐷𝑟𝑒𝑙 =𝐷𝑠𝐷𝑚

=∑ 𝑛𝑠,𝑖𝑘𝑠𝑖=1 𝑠𝑠,𝑖

𝛽

∑ 𝑛𝑚,𝑗𝑘𝑚𝑗=1 𝑠𝑚,𝑗

𝛽 (61)

Subscripts 𝑚 and 𝑠 stands for measured signal and simulated signal respectively. Note that the material constant 𝐶 is eliminated in the equation above. Target values for the relative-pseudo damage are quantified by considering an acceptable scatter for the magnitude of an equivalent constant amplitude load. A ± 10% load amplitude scatter yields threshold values of approximately 0,53 to 1,8, as a consequence of the power-law dependency in Eq. (58).

log(N)

Haibach slope factor 2β-1

Basquin slope factor β

log(s)

Fatigue limit

27

2.11 Test Procedure Performed tests of test vehicle Scania Touring K400EB4x2NI, see Figure 12, has been the basis for the modelling and virtual test simulations performed in this thesis work.

Figure 12. Test vehicle Scania Touring K400EB4x2NI

2.11.1 Vehicle properties The test vehicle was a two axle touring coach equipped with an individual front suspension. The centre of gravity (CoG) position in longitudinal and lateral direction is identified from measured axle loads and wheel force transducer data from testing. The vertical CoG position of this particular coach has also previously been measured at three different occasions.

2.11.2 Sensors Accelerations in the body structure have been measured in three directions at 12 different positions throughout the coach. Four strain gauges were fitted to rectangular profiles in the body structure. Damper strokes at both the front and rear suspensions have also been measured. Wheel force/torque transducers measuring force and torque loads between rim and hub at each wheel were mounted during testing. An illustration depicting the arrangement and naming conventions of the sensors of interest for this thesis work is shown in Figure 13. All accelerations have been expressed with positive directions coinciding with the vehicle-design coordinate system as seen in the figure. Force and torque measurements at the wheels are expressed in wheel-fixed coordinates.

28

Figure 13. Sensor arrangement, right hand side

For illustrative purposes, Figure 13 only shows the locations of the accelerometers and wheel transducers mounted on the right-hand side of the coach. A close to identical sensor arrangement was used on the left side. The naming convention refers to Swedish descriptions of the sensor properties and locations. The last letter describes the measured direction and the second to last letter describes whether the sensor is mounted on the right or left side of the coach. Additional letters explain the type of measurement and position in the x-z-plane. AC identifies acceleration, F force, M moment, D damper stroke and T strain. See Appendix A for a complete description of all sensors. Two of the four strain gauge measurements fitted to the body frame were mounted on the right side, one measuring the strain at the rear overhang and one at a sidewall frame member between the front and rear wheels. On the left side, both the remaining two strain gauges were fitted to the opposite sidewall member between the front and rear wheels. Positions for the four strain gauges are illustrated more specific in Figure 14. It should be noted that sensor signals ACBNHZ, ACBOHX, ACBOHY and ACBOHZ were not recorded during the particular test that has been analysed in this thesis. Therefore, these four signals will not be included in the results evaluation presented later on.

29

Figure 14. Detailed strain gauge locations

30

31

3 MODELLING AND SIMULATION PROCEDURES

This chapter describes the modelling of the coach and how the simulations were carried out. The subsystems included in the vehicle model are presented shortly together with a mode truncation investigation and the deriving of structural damping approximations.

3.1 Virtual coach model The coach model was created in the multi-body simulation software MSC.Adams using subsystem models established in the Scania vehicle dynamics library. These subsystem models have in some cases been modified and adapted with data representative to the specific vehicle studied. The model is comprised of subsystems detailing the front and rear suspensions, seen in Figure 15. The triangle and vertical links in the front suspension system are modelled as flexible parts whereas the rear suspension consists purely of rigid parts. Connections such as different idealized joints and bushings with stiffnesses in all 6 d.o.f. are used to couple parts within each subsystem as well as connecting subsystems to one another. Air spring, damper and bushing characteristics are based on physical measurements or data provided by manufacturers.

a)

b)

Figure 15.a) Individual front suspension (including anti-roll bar and steering subsystem), b) Rigid rear axle suspension (including anti-roll bar subsystem)

A system detailing the powertrain was also included in the full vehicle model, see Figure 16 below. Modelling the suspension and inertial properties of the powertrain is necessary due to its large influence on the structural dynamics of the coach. The parts in the powertrain have been modelled as rigid bodies with mass and inertia properties correlating to the coach being studied. Engine and gearbox mounts are modelled by bushings with measured stiffnesses.

Figure 16. Powertrain subsystem interfaced to rear suspension

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The remainder of the vehicle was implemented as a flexible body which included the chassis frame, body structure, panels, windows, auxiliary systems and various interior components. The finite element model, see Figure 17, has been modelled with relatively high detail in order to achieve close representation of the actual stiffness and mass distribution of the studied coach. The majority of the model consists of first order element. A element size of around 5-10 mm have been used for the chassis frame and body structure, while body panels, windows and floors have been meshed with element sizes ranging from 40 mm up to 140 mm. Performing durability assessments by full vehicle simulations will require detailed modelling with regions of small element sizes to allow for accurate fatigue life estimations. Even though it has not been within the scope of this work to perform such fatigue life predictions, it has been of interest to investigate the possibilities and limitations when using this kind of large and comprehensive FE-model.

a)

b)

Figure 17. Finite element model, a) with body and window panels, b) chassis and body structure The vehicle mass and CoG position were obtained from previously mentioned measurements. To achieve the same mass and CoG location in the virtual model, the mass distribution of the finite element model was tuned. The model, before substructuring, had approximately 13,5 million d.o.f. and was condensed using the aforementioned Craig-Bampton method to create the flexible body. The final coach model assembled from these subsystems is shown in Figure 18, here fitted with a tire model. The flex body was defined with a total of 38 interface nodes at which the subsystems were connected.

a)

b)

Figure 18. Flexible MBS coach model, a) Flex body rendered, b) Flex body transparent

33

3.1.1 Tire model In the case of virtual proving ground simulations, a flexible structure tire model (FTire) has been used. No FTire models of the specific tires used during the proving ground testing were available during the thesis work, instead a FTire model of a 315/80R22.5 truck tire has been used. This is a larger tire dimension than the 295/80R22.5 tires actually mounted on the real coach during testing. An operating inflation pressure of 8 bar has been assumed.

3.2 Virtual test rig model The test rigs which has been used in the virtual test rig simulations is the 10 channel test rig utilized by Kjellsdotter (2011). This test rig, seen in Figure 19, is designed with vertical and longitudinal translational actuation on all four spindles, while only using a single lateral actuator at the right-hand spindles on the front and rear suspensions. This is because a kinematics error can otherwise occur if two opposing actuators were to impose out-of-phase motion at each spindle on a rigid axle beam. Since the studied coach uses an individual front suspension, this kinematic error cannot occur as the suspension has some compliance. A 11 channel test rig using lateral input at both front spindles was therefore created, however no significant difference was observed in the accuracy of the drive signal iterations. In the succeeding chapters, simulation results will only be presented for simulation performed with the 10 channel test rig.

Figure 19. The 10 channel virtual test rig

The simple nature of the test rig means that no torque loads at the spindles can be replicated. Moreover, replication of lateral and longitudinal spindle loads has been found difficult in this test rig, likely because all the actuators are being run individually and by pure displacement control. Plummer (2007) explains how motions of a test specimen in a structural test rig can cause large changes in actuator loads. When actuators are driven individually, it cannot be guaranteed that the motion of the specimen is accurately considered by the drive signals on each iteration, which can yield in large reaction forces when the actuators impose displacements on the moving test specimen. Most physical multi-axial test rigs therefore use two control loops for opposing actuators, where one displacement-controlled actuator determines the specimen motion and the other actuator is used to control the compression/tension forces on the specimen (Plummer, 2007). This type of control strategy is traditionally employed for opposing lateral and longitudinal actuator pairs. Implementing such control techniques in this test rig has not been included in this thesis work.

3.3 Mode truncation The effect on the dynamics of the coach body structure when truncating different amount of fixed-interface normal modes in the flexible body has been necessary to evaluate as close representation of the actual structure dynamics is vital. The size and detail of the finite element model has implied limitations in the number of modes possible to retain. Many local modes at

34

body panels and windows exist within the frequency span of interest. This means that a lot of modes are retained when stating that fixed normal modes up to 40-50 Hz should be included in the flex body formulation. Moreover, with strain modes included for element sets at strain gauge locations, limitations of scratch disc space have been found to be easily exceeded. A coloured noise drive signal with frequency content from 0 to 50 Hz was played out on all test rig actuators with the bus model attached. The default damping in Adams was used for the flex body. Three simulations were performed with 100, 170 and 290 fixed-interface normal modes retained in the flex body. The upper frequency limits of the fixed-interface normal modes were 23 Hz, 31 Hz and 42 Hz respectively. Below are the PSD spectrums of vertical acceleration responses at accelerometer positions ACFNHZ and ACDNHZ, see Figure 20 and Figure 21. These are the locations at which the responses displayed largest differences.

Figure 20. PSD acceleration response ACFNHZ from noise excitation

Figure 21. PSD acceleration response ACDNHZ from noise excitation

The frequency response from the noise signal matches closely below 20 Hz, with some differences for the 100 mode structure. Between 20-30 Hz the flex bodies with 170 and 290 modes still match with only minor differences. The 100 mode flex body is however showing a very different frequency response. The reason is likely because its retained fixed normal modes only spans to 23 Hz and the dynamics of the flex body above this frequency is left to be described by a few fixed constraint modes. Interesting in this case is the fact that greater frequency response is experienced by the 100 mode flex body compared to the others at frequencies above its 23 Hz. Similar behaviour occurs for the 170 mode flex body, its response starts to depart significantly from the 290 mode above its 31 Hz limit, some differences are noted at lower frequencies as well. Similar behaviour was found in the upper part of the body structure in longitudinal direction. Acceleration responses ACFOHX and ACDOHX are shown in Figure 22 and Figure 23.

0 10 20 30 40 50 60Frequency [Hz]

PS

D v

ertic

al a

ccel

erat

ion

[g2 /

Hz] PSD spectrum from noise signal - ACFNHZ

100 modes (23 Hz)170 modes (31 Hz)290 modes (42 Hz)

0 10 20 30 40 50 60Frequency [Hz]

PS

D v

ertic

al a

ccel

erat

ion

[g2 /

Hz] PSD spectrum from noise signal - ACDNHZ

100 modes (23 Hz)170 modes (31 Hz)290 modes (42 Hz)

35

Figure 22. PSD acceleration response ACFOHX from noise excitation

Figure 23. PSD acceleration response ACDOHX from noise excitation

The maximum amount of fixed-interface normal modes possible to retain when strain modes have been included has been 170. This means that the models ability to represent the structure dynamics above approximately 30 Hz may be inaccurate. However, it should be noted that the dominating frequency content of the road induced loads from the physical measurements are all below 30 Hz, see Figure 24, therefore 170 modes has been considered adequate. Hence, all subsequent simulations have been performed with 170 fixed-interface normal modes retained, counting the constraint modes at the 38 interface points, the total number of modes in the flex body formulation then amounts to 398.

a)

b)

Figure 24. Measured spindle loads, a) FFAVX, b) FFAVZ

0 10 20 30 40 50 60Frequency [Hz]

PS

D lo

ngitu

dina

l acc

eler

atio

n [g

2 /H

z] PSD spectrum from noise signal - ACFOHX

100 modes (23 Hz)170 modes (31 Hz)290 modes (42 Hz)

0 10 20 30 40 50 60Frequency [Hz]

PS

D lo

ngitu

dina

l acc

eler

atio

n [g

2 /H

z] PSD spectrum from noise signal - ACFOHX

100 modes (23 Hz)170 modes (31 Hz)290 modes (42 Hz)

0 10 20 30 40 50

PSD - FFAVX

Frequency [Hz]

PSD

long

itudi

nal s

pind

le fo

rce

[N2 /Hz]

Measured

0 10 20 30 40 50Frequency [Hz]

PSD

ver

tical

spi

ndle

forc

e [N2 /H

z]

PSD - FFAVZ

Measured

36

3.4 Damping Another important aspect that had to be considered when modelling the coach was the damping of the body structure. Energy dissipation in the material and structural joints has large influence on the dynamic behaviour and is therefore necessary to consider. The structural damping was described by modal damping ratios in Adams. The damping ratios were derived by means of reverse engineering, which imply model parameter estimation based on physical test measurements. The approach was to generate drive signals so that responses not affected by structural damping were replicated with good match to the measured responses. Drive signals were generated in the virtual test rig which replicated physical test responses close to the structures excitation points. These response signals were vertical spindle forces, damper strokes and also lateral and longitudinal accelerations at positions close to the front and rear suspensions. Figure 25 illustrates these responses in green, it should be noted that all four vertical spindle loads were iterated towards, not only the two seen in the figure. The vertical spindle forces and damper stroke responses are more or less independent of the structural damping, meaning that they do not change noticeably when the structural damping is modified. Replicating these responses in the virtual test rig ensures that the load inputs in vertical direction in to the structure are similar to that of the physical test.

Figure 25. Sensors used during drive signal generation (FFAVZ and FDAVZ not shown)

Ideally, invariant loading in lateral and longitudinal directions should also be replicated. For instance, lateral and longitudinal spindle loads or accelerations on the suspensions. Spindle loads in these directions are, however, not possible to replicate in the current test rig and no suspension accelerations were available from the physical test. To describe the longitudinal and lateral loads sufficiently, the accelerometers in the structure closest to the front and rear suspensions had to be

37

used instead. This is of course not ideal as they are influenced by the structural damping to a certain extent. A design of experiment (DOE) approach was adopted for deriving an approximation of the structural damping. Adams allows the damping to be expressed as either a single scalar damping ratio applied uniformly for all modes or as a function expression. Using function expression variables like FXFREQ or FXMODE enables the possibility to apply different levels of damping for specific modes or mode frequencies. FXFREQ denotes the mode frequency whereas FXMODE signify the mode number. The default damping ratio function for flex bodies in Adams is: CRATIO = IF(FXFREQ-100:0.01,0.1,IF(FXFREQ-1000:0.1,1.0,1.0)) This function describe a 1% damping ratio for modes below 100 Hz, 10% for modes between 100-1000 Hz and 100% for modes larger than 1000 Hz. Two modified damping functions for the flex body were defined according to: CRATIO = STEP(FXFREQ, 0, DAMP_RATIO, DAMP_RANGE, 1) CRATIO = IF(FXFREQ-10.0:DAMP_RATIO_1, DAMP_RATIO_1, IF(FXFREQ-20.0:DAMP_RATIO_2, DAMP_RATIO_2, IF(FXFREQ-40.0:DAMP_RATIO_3, DAMP_RATIO_3, IF(FXFREQ-200.0:DAMP_RATIO_4, DAMP_RATIO_4, 1.0)))) The first damping ratio is constituted by a step function ranging from a parameter DAMP_RATIO to 100% over the mode frequency span 0 Hz to DAMP_RANGE. The parameters DAMP_RATIO and DAMP_RANGE are the two design factors included in the first DOE study. The second function uses nested IF functions with four different damping ratios DAMP_RATIO_1, DAMP_RATIO_2, DAMP_RATIO_3 and DAMP_RATIO_4 as design factors, each describing the damping within fixed mode frequency spans, 0-10 Hz, 10-20 Hz, 20-40 Hz and lastly 40 - 200 Hz. Damping for modes above 200 Hz are given 100 % damping. These damping expressions are illustrated in Figure 26.

a)

b)

Figure 26. a) Damping expression #1, b) Damping expression #2

A two factor and a four factor central composite face-centred (CCF) experiment design were defined using Adams/Insight, a DOE and optimization software interfaced with Adams/Car and other products in the MSC.Adams suite of software. A general two factor CCF design, as used in the DOE study for the first damping expression, is illustrated in Figure 27, it is comprised of a factorial design augmented with a set of star point and centre point experiment runs. The design factor levels for both damping expressions were defined as presented in Table 1 and Table 2.

0 50 100 150 200 2500

20

40

60

80

100

Damping expression #1 - STEP function

Mode frequency [Hz]

Dam

ping

ratio

[%]

0 50 100 150 200 2500

20

40

60

80

100

Damping expression #2 - Nested IF functions

Mode frequency [Hz]

Dam

ping

ratio

[%]

38

Figure 27. Two factor CCF experiment design

Table 1. Design factor levels in first damping expression

- 0 + DAMP_RATIO 1% 8% 15% DAMP_RANGE 100 Hz 200 Hz 300 Hz

Table 2. Design factor levels in second damping expression - 0 +

DAMP_RATIO_1 1% 6% 11% DAMP_RATIO_2 4% 11% 18% DAMP_RATIO_3 9% 18% 27% DAMP_RATIO_4 12% 24% 36%

The procedure to derive the design variables in each damping function is illustrated in Figure 28. For each experimental run, the acceleration responses which had not been iterated towards when generating the drive signal were extracted and compared to the physical test measurements. The TDDI value for each response signal and the mean TDDI value of all signals were calculated and used as response variables in the experiment. Quadratic response surfaces were then fitted to each of these response variables. The response surfaces are approximation functions of the actual vehicle model response and provide an inexpensive representation of the model behaviour for changes in damping. An optimization is performed based on these response surfaces with the objective function to minimize each response variable, in other words to achieve as low TDDI values as possible on the studied responses in the structure. Each response variable was weighed equally.

Figure 28. Schematic of the approach for obtaining damping variables

From the DOE and response surface optimization analysis the following damping design variables were obtained, see Table 3 and Table 4. The suggested optimized value of design factor DAMP_RATIO_3 is its upper boundary value defined in the DOE test. This indicates that an optimum for this design factor is probably located beyond the limit value, however, no efforts were made to further search for this optimum. Another interesting outcome is that DAMP_RATIO_4 is suggested to be lower than DAMP_RATIO_3, meaning that modes between 40-200 Hz are given less damping than modes within the 20-40 Hz frequency span.

39

Table 3. Optimized damping design variables in first damping function

Optimized value DAMP_RATIO 11,6 % DAMP_RANGE 128 Hz

Table 4. Optimized damping design variables in second damping function

Optimized value DAMP_RATIO_1 3,4% DAMP_RATIO_2 9,2% DAMP_RATIO_3 27,0% DAMP_RATIO_4 21,1%

The obtained damping expressions together with the default damping are shown in Figure 29. There are apparent differences between the three damping expressions. Expression two has, when compared to the first expression, significantly lower damping in the regions at 0-20 Hz and also above 40 Hz. However, the default damping describes by far the lowest energy dissipation.

Figure 29. Optimized and default damping expressions

3.5 Simulation procedures 3.5.1 Virtual proving ground A 3D virtual road of the studied proving ground obstacle track has been used when performing the VPG simulations. The digitized road stretch was a curved regular grid (CRG) road (OpenCRG, 2008) containing high resolution 3D road data generated from laser scanning. The coach model on the virtual road is shown in Figure 30. Both optimized damping expressions and the default damping setting in Adams were studied. Appropriate settings for the steering PID controller and trajectory preview distance have been derived to keep the coach in a straight path during simulations and to eliminate heavy steering corrections.

0 50 100 150 200 2500

20

40

60

80

100

Derived damping expressions

Mode frequency [Hz]

Dam

ping

ratio

[%]

Damping expression #1Damping expression #2Default damping

40

Figure 30. Coach model on the virtual road

3.5.2 Virtual test rig The vehicle model was connected to the 10 channel test rig and simulations were performed with the objective to reproduce all 32 acceleration responses in the structure measured during the physical test. The coach model mounted by its spindles is shown in Figure 31. The target signals are from a single run over a part of the test track. Only the structural damping defined by the second damping expression was used. The simulations were performed within the frequency band of 0-45 Hz, meaning that a cut-off frequency of 45 Hz was used when filtering and evaluating signals.

Figure 31. Coach model spindle-coupled to the 10 channel test rig

41

4 RESULTS

In this chapter the results from the performed simulations are presented and evaluated. Signal time histories and PSD spectra are only presented for a selection of responses.

4.1 Virtual proving ground Virtual proving ground simulations were carried out for both derived damping expressions and the default damping in Adams. Only relative pseudo-damage is presented as TDDI values are highly sensitive to small deviations in cruise control speed and therefore of minor interest to evaluate when comparing lengthier physical tests with simulations. Seen in Figure 32 and Figure 33 are the achieved relative pseudo-damages for the damper strokes, spindle loads and acceleration responses for the different structural damping settings. All signals have been filtered and edited identically before calculating the relative pseudo-damage.

Figure 32. Relative pseudo-damage of damper strokes and spindle loads with different damping

Figure 33. Relative pseudo-damage of acceleration responses with different damping

There are several important things that can be noted from the figures above. First of all, responses in longitudinal direction at spindles and in the lower part of the structure all exhibit small values. Below in Figure 34 are the PSD spectra of longitudinal spindle loads FFAHX and

00,5

11,5

22,5

DDAH

BZ

DDAV

BZ

DDAH

FZ

DDAV

FZ

DFAH

Z

DFAV

Z

FFAV

X

FFAV

Y

FFAV

Z

FDAV

X

FDAV

Y

FDAV

Z

FFAH

X

FFAH

Y

FFAH

Z

FDAH

X

FDAH

Y

FDAH

Z

Damper strokes and spindle loads - Relative pseudo-damage

Damping function #1 (Step) Damping function #2 (Nested IF) Default damping

6 10 6 10 8 9 6 4 3 5 4 10 30 8 10 52 5

0

0,5

1

1,5

2

2,5

3

Mea

nAC

BNHX

ACBN

HYAC

BNVX

ACBN

VYAC

BNVZ

ACBO

VXAC

BOVY

ACBO

VZAC

DNHX

ACDN

HYAC

DNHZ

ACDN

VXAC

DNVY

ACDN

VZAC

DOHX

ACDO

HYAC

DOHZ

ACDO

VXAC

DOVY

ACDO

VZAC

FNHX

ACFN

HYAC

FNHZ

ACFN

VXAC

FNVY

ACFN

VZAC

FOHX

ACFO

HYAC

FOHZ

ACFO

VXAC

FOVY

ACFO

VZAccelerations - Relative pseudo-damage

Damping function #1 (Step) Damping function #2 (Nested IF) Default damping

42

FDAHX at the front and rear right side wheels, the y-axis scale in Figure 34.b is 10 times the magnitude of the axis in Figure 34.a. Only the signal from one simulation is shown since there is no significant difference between spindle forces from the simulations with the various damping functions, as seen in Figure 32.

a)

b)

Figure 34. PSD spectra of longitudinal spindle forces, a) FFAHX, b) FDAHX Frequency energy content between 0-25 Hz seems poorly represented by the simulated model. It is likely that the virtual tire model has been unable to reproduce the same longitudinal loads that have been measured. Short segments from the time histories of force signals FFAHX and FDAHX are shown in Figure 35.

a)

b)

Figure 35. Time histories of longitudinal spindle loads a) FFAHX, b) FDAHX

The inability to represent longitudinal loads reflects on the longitudinal accelerations in the lower parts of the structure, which consequentially also gets a lower relative pseudo-damage. The PSD spectra of the longitudinal accelerations ACFNHX and ACDNHX in the lower part of the structure are shown in Figure 36. It is noticeable that the frequency content between 10-25

0 10 20 30 40

PSD - FFAHX

Frequency [Hz]

PSD

long

itudi

nal s

pind

le fo

rce

[N2 /Hz]

MeasuredSim. Damp. #1

0 10 20 30 40

PSD - FDAHX

Frequency [Hz]P

SD lo

ngitu

dina

l spi

ndle

forc

e [N2 /H

z]

MeasuredSim. Damp. #1

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Time history - FFAHX

Time [s]

Long

itudi

nal

spin

dle

forc

e [N

]

MeasuredSim. Damp. #1

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Time history - FDAHX

Time [s]

Long

itudi

nal

spin

dle

forc

e [N

]

MeasuredSim. Damp. #1

43

Hz which is missing in the longitudinal spindle forces is also unrepresented in these acceleration responses.

a)

b)

Figure 36. PSD spectra of longitudinal accelerations in the lower part of the body structure

a) ACFNHX, b) ACDNHX Figure 33 suggests that lateral acceleration responses in the upper part of the body show more significant dependency on the damping prescribed in the flex body. Even though longitudinal input loads to the structure are lower than in reality, the simulated upper acceleration responses are correlating good for the second damping function and quite good for the first, whereas the standard damping function demonstrates a lot higher pseudo-damage. PSD spectra for the longitudinal accelerations ACFOHX and ACDOHX in the upper part of the structure, are shown in Figure 37. The y-axis scales in these plots are 5 times larger than in Figure 36, as indicated by the bold markers on the y-axes.

a)

b)

Figure 37. PSD spectra of longitudinal accelerations in the upper part of the body structure

a) ACFOHX, b) ACDOHX A rigid body pitch mode at approximately 1 Hz becomes more evident in the upper acceleration responses as the locations where these responses are measured are positioned further away from the pitch centre. The structure of the virtual model seems to be prone to excitation at 10 Hz and between 20-30 Hz, something that is not seen in the measured structure responses. The derived

0 10 20 30 40

PSD - ACFNHX

Frequency [Hz]

PSD

long

itudi

nal a

ccel

erat

ion

[g2 /Hz]

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40

PSD - ACDNHX

Frequency [Hz]

PSD

long

itudi

nal a

ccel

erat

ion

[g2 /Hz]

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40

PSD - ACFOHX

Frequency [Hz]

PSD

long

itudi

nal a

ccel

erat

ion

[g2 /Hz]

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40

PSD - ACDOHX

Frequency [Hz]

PSD

long

itudi

nal a

ccel

erat

ion

[g2 /Hz]

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

44

damping functions manage to damp out most of these excitations, while the default damping display considerably higher peaks. Turning the attention to the lateral spindle loads, Figure 32 imply that all the simulated lateral spindle loads contain lower pseudo-damage than the measured signals. The front lateral spindle loads are well below the desired range and the rear spindle loads differs only slightly to the desired 0,53 threshold value. The PSD of the front and rear lateral spindle forces on the right-hand side wheels are illustrated in Figure 38, the y-axis scale is 2,5 times higher in Figure 38.b as in Figure 38.a. Spindle force FDAHY show similar characteristics to the measured signal, with some peak value difference.

a)

b)

Figure 38. PSD spectra of lateral spindle forces, a) FFAHY, b) FDAHY

Segments from the time histories of force signals FFAHY and FDAHY are shown in Figure 39. The simulated rear lateral spindle force FDAHY demonstrates quite good correlation to the measured signal as suggested by its PSD spectrum in Figure 38.b. The lateral force FFAHY at the right front wheel does not match nearly as good, this is a consequence of the driver control module enforcing steering corrections to keep the virtual vehicle on a straight path. Steering PID controller values and trajectory settings have been tuned to the best of the author’s ability to get as good a correlation as possible between simulated and measured signals. The steering corrections made by driver module explain to some extent the quite poor value of relative pseudo-damage calculated for FFAHY and FFAVY in Figure 32.

a)

0 10 20 30 40Frequency [Hz]

PSD

late

ral s

pind

le fo

rce

[N2 /Hz]

PSD - FFAHY

MeasuredSim. Damp. #1

0 10 20 30 40Frequency [Hz]

PSD

late

ral s

pind

le fo

rce

[N2 /Hz]

PSD - FDAHY

MeasuredSim. Damp. #1

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7Time [s]

Late

ral

spin

dle

forc

e [N

]

Time history - FFAHY

MeasuredSim. Damp. #1

45

b)

Figure 39. Time histories of lateral spindle loads a) FFAHY, b) FDAHY

Lateral acceleration responses in the lower parts of the structure are shown in Figure 40, acceleration in the rear overhang section has also been included to get an understanding of how the damping affects the lateral motions of the rear overhang. The same y-axis scale is used in all three plots.

a)

b)

c)

Figure 40. PSD spectra of lateral accelerations in the lower part of the body structure

a) ACFNHY, b) ACDNHY, c) ACBNHY The virtual model matches the rear overhang lateral acceleration in Figure 40.c quite well for all damping expressions. The model displays worse correlation for responses ACDNHY and ACFNHY. Where the two derived damping expressions both result in too much energy

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Time history - FDAHY

Time [s]

Late

ral

spin

dle

forc

e [N

]

MeasuredSim. Damp. #1

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACFNHY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACDNHY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACBNHY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

46

dissipation. PSD spectra for the lateral accelerations ACFOHY, ACDOHY and ACBOVY, located in the upper part of the structure, are shown in Figure 41. Y-axis scales are 6 times larger compared to Figure 40, as indicated by the bold axis tick markers.

a)

b)

c)

Figure 41. PSD spectra of lateral accelerations in the upper part of the body structure

a) ACFOHY, b) ACDOHY, c) ACBOVY As indicated by the relative pseudo-damage in Figure 33, the models lateral acceleration responses in the upper parts of the structure correlates quite well for the second damping expression. From the spectra in Figure 41 it is evident that the rigid body motion, the first peaks around 1 Hz, matches good with what has been measured. The obtained flexible acceleration responses in the region of 9 Hz correlates quite well for the second damping expression, although, there are some noticeable differences in peak values and in some cases slightly shifted peaks when compared to the measured responses. The vertical spindle load responses FFAHZ and FDAHZ are illustrated in Figure 42 and Figure 43. The values from the calculated relative pseudo-damage suggest that these loads were reproduces with very good accuracy, which can also be confirmed by the following figures.

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACFOHY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACDOHY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

late

ral a

ccel

erat

ion

[g2 /Hz]

PSD - ACBOVY

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

47

a)

b)

Figure 42. PSD spectra of vertical spindle forces, a) FFAHZ, b) FDAHZ

a)

b)

Figure 43. Time histories of lateral spindle loads a) FFAHZ, b) FDAHZ

The excellent replication of vertical spindle loads also means that damper strokes are expected to match with decent accuracy, time histories of damper strokes DFAHZ, DDAHFZ and DDAHBZ are presented in Figure 44. As seen in the figure, simulated damper strokes displays matching behaviour with what has been measured at both the front and rear suspensions.

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

spi

ndle

forc

e [N2 /H

z]PSD - FFAHZ

MeasuredSim. Damp. #1

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

spi

ndle

forc

e [N2 /H

z]

PSD - FDAHZ

MeasuredSim. Damp. #1

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7Time [s]

Ver

tical

spin

dle

forc

e [N

]

Time History - FFAHZ

MeasuredSim. Damp. #1

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7Time [s]

Ver

tical

spin

dle

forc

e [N

]

Time History - FDAHZ

MeasuredSim. Damp. #1

48

a)

b)

c)

Figure 44. Time histories of damper strokes a) DFAHZ, b) DDAHFZ, c) DDAHBZ

The relative pseudo-damage in Figure 33 suggests that simulated vertical acceleration responses show reasonably similar damaging content to what has been measured, especially for the second damping expression. The main reason for this is of course because vertical spindle forces and damper strokes are correlating well with measured signals. The PSD spectra of the vertical acceleration responses in the lower parts of the structure are shown in Figure 45. With the default damping, energy content in the acceleration responses around 8-12 Hz is overestimated. The derived damping functions provide better estimation of the energy dissipation, with slight advantage to the second damping expression.

2 3 4 5 6 7 8 9 10Time [s]

Dam

per s

troke

[mm

]

Time history - DFAHZ

MeasuredSim. Damp. #1

2 3 4 5 6 7 8 9 10Time [s]

Dam

per

stro

ke [m

m]

Time history - DDAHFZ

MeasuredSim. Damp. #1

2 3 4 5 6 7 8 9 10

Time history - DDAHBZ

Time [s]

Dam

per s

troke

[mm

]

MeasuredSim. Damp. #1

49

a)

b)

c)

Figure 45. PSD spectra of vertical acceleration in the lower part of the body structure

a) ACFNHZ, b) ACDNHZ, c) ACBNVZ Similarly, PSD of vertical accelerations in the upper parts of the structure are presented in Figure 46. The y-axes use the same scale as in the preceding figure of the lower situated vertical acceleration responses. The energy content in the responses is very similar when compared to the responses in Figure 45. It should be noted that the peak in the 1-2 Hz frequency band matches well in the front of the coach while larger differences in peak values are apparent above the rear axle and most noticeably at the rear overhang. This indicates that the rigid body modes influencing vertical motion, pitch and bounce, are not completely replicated in the simulations.

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACFNHZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACDNHZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACBNVZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

50

a)

b)

c)

Figure 46. PSD spectra of vertical acceleration in the upper part of the body structure

a) ACFOHZ, b) ACDOHZ, c) ACBOVZ With the knowledge of how the simulated input loads from the tire model have been reproduced and how the resulting acceleration responses in the structure matches measured responses, it is of interest to evaluate how the strain measurements also correlates. The relative pseudo-damage of the strain responses in the model are shown in Figure 47. The default damping achieves the best relative pseudo-damage. As seen in most of the presented PSD spectra however, the default damping demonstrates excessive response energy in the frequency band at 8-12 Hz. It is likely that the lower energy dissipation of the default damping compensates for the road induced loads in lateral and longitudinal directions being slightly smaller than measured, and thus yielding in strain pseudo-damage more close to the proving ground measurements.

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACFOHZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACDOHZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACBOVZ

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

51

Figure 47. Relative pseudo-damage of strain responses

From Figure 47 it can be concluded that the strain predictions from the simulations are non-conservative. In other words, the potential fatigue damage is less than what the measured strain gauge signals suggests. The reason for this is in some extent due to the tire models inability to replicate some of the load inputs to the structure with sufficient accuracy. In Figure 48 to Figure 51 are the PSD and time history signals for all four strain responses.

Figure 48. PSD and time history of strain response TCX03_HB

Figure 49. PSD and time history of strain response TCX04_VB

0

0,1

0,2

0,3

0,4

0,5

0,6

Mea

n

TCX0

3 _H

B

TCX0

4_VB

TCX0

4_VF

TCX0

9_BA

K

Strains - Relative pseudo-damage

Damping function #1 (Step) Damping function #2 (Nested IF) Default damping

0 5 10 15 20 25

PSD - TCX03 HB

Frequency [Hz]

PS

D s

train

[ µs2 /

Hz]

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

2 2.5 3 3.5 4 4.5 5

Time history - TCX03 HB

Time [s]

Stra

in [ µ

s]

MeasuredSim. Damp. #2

0 5 10 15 20 25Frequency [Hz]

PS

D s

train

[ µs2 /

Hz]

PSD - TCX04 VB

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

2 2.5 3 3.5 4 4.5 5

Time history - TCX04 VB

Time [s]

Stra

in [ µ

s]

MeasuredSim. Damp. #2

52

Figure 50. PSD and time history of strain response TCX04_VF

Figure 51. PSD and time history of strain response TCX09_BAK Just as all the acceleration responses, the strain responses show distinct contributions from the two characteristic frequency bands. Strain gauge response TCX03_HB differs remarkably from the measured signal and achieves a very low relative pseudo-damage as a result, whereas simulated strain response TCX04_VF displays the best correlation with the measured strain gauge signal.

4.2 Virtual test rig Simulations in the virtual test rig were carried out with the goal of reproducing all the accelerations measured on the proving ground with satisfactory accuracy. Therefore, only the 32 available acceleration responses have been considered during iteration. The results presented below are from simulations using the 10-channel test rig and with the second derived damping function. The final drive signals that achieve the results presented below were accomplished after 36 drive signal iteration loops in the virtual test rig. TDDI values of the acceleration responses in the model using the virtual test rig are presented in Figure 52. All 32 accelerations responses in the body frame structure of the model have been possible to reproduce with TDDI values below the limit of 0,3. A mean TDDI value of 0,101 was attained.

0 5 10 15 20 25Frequency [Hz]

PS

D s

train

[ µs2 /

Hz]

PSD - TCX04 VF

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

2 2.5 3 3.5 4 4.5 5

Time history - TCX04 VF

Time [s]

Stra

in [ µ

s]

MeasuredSim. Damp. #2

0 5 10 15 20 25Frequency [Hz]

PS

D s

train

[ µs2 /

Hz]

PSD - TCX09 BAK

MeasuredSim. Damp. #1Sim. Damp. #2Sim. Damp. default

2 2.5 3 3.5 4 4.5 5

Time history - TCX09 BAK

Time [s]

Stra

in [ µ

s]

MeasuredSim. Damp. #2

53

Figure 52. TDDI values of acceleration responses

The lateral accelerations in the lower parts of the rear overhang, accelerations ACBNHY and ACBNVY, were found most difficult to converge. Consequently, the TDDI values for these accelerations responses are the highest. Many acceleration responses in lateral direction are showing slightly higher TDDI values compared to other directions. This is mainly because lateral accelerations started diverging somewhat after iteration loop 31, when vertical acceleration responses were given more priority. Relative pseudo-damage for the same acceleration responses are shown in Figure 53. Most responses are within the specified target range, only a few response channels show slight deviation from the desired range. The mean value pseudo-response is about 0,94, which can be considered as good, seeing that a value of 1 indicates identical damaging content in the model responses compared to the proving ground measurements.

Figure 53. Relative pseudo-damage of acceleration responses

The PSD and time history signals for two of the responses in the figure above are shown in Figure 54 and Figure 55 to give an idea of the accuracy at which the acceleration responses have been reproduced.

0

0,05

0,1

0,15

0,2

0,25

0,3

Mea

nAC

BNHX

ACBN

HYAC

BNVX

ACBN

VYAC

BNVZ

ACBO

VXAC

BOVY

ACBO

VZAC

DNHX

ACDN

HYAC

DNHZ

ACDN

VXAC

DNVY

ACDN

VZAC

DOHX

ACDO

HYAC

DOHZ

ACDO

VXAC

DOVY

ACDO

VZAC

FNHX

ACFN

HYAC

FNHZ

ACFN

VXAC

FNVY

ACFN

VZAC

FOHX

ACFO

HYAC

FOHZ

ACFO

VXAC

FOVY

ACFO

VZ

Accelerations - TDDI

0

0,5

1

1,5

2

2,5

Mea

nAC

BNHX

ACBN

HYAC

BNVX

ACBN

VYAC

BNVZ

ACBO

VXAC

BOVY

ACBO

VZAC

DNHX

ACDN

HYAC

DNHZ

ACDN

VXAC

DNVY

ACDN

VZAC

DOHX

ACDO

HYAC

DOHZ

ACDO

VXAC

DOVY

ACDO

VZAC

FNHX

ACFN

HYAC

FNHZ

ACFN

VXAC

FNVY

ACFN

VZAC

FOHX

ACFO

HYAC

FOHZ

ACFO

VXAC

FOVY

ACFO

VZ

Accelerations - Relative pseudo-damage

54

Figure 54. PSD and time history of response ACFNHX

Figure 55. PSD and time history of response ACBOVZ TDDI values for responses that were disregarded during the iteration are shown in Figure 56. Damper stroke responses demonstrate quite low TDDI values, between 0,43 and 0,68, even though not having been considered when generating the drive signal. The aforementioned problem of not being able to reproduce spindle loads in lateral and longitudinal direction is also manifested here. The large TDDI values for these responses are a consequence of actuators being driven individually and by pure displacement control, yielding in large reaction forces as actuator displacements are not able to follow the motion of the vehicle properly.

Figure 56. TDDI values of damper stroke and spindle load responses (non-prioritized)

Since these responses have been neglected during iteration, they are not expected to correlate well with measured signals. This is because exact load introduction into the structure cannot be

0 10 20 30 40

PSD - ACFNHX

Frequency [Hz]

PSD

long

itudi

nal a

ccel

erat

ion

[g2 /Hz]

MeasuredSimulated

2 2.5 3 3.5 4 4.5 5

Time history - ACFNHX

Time [s]

Long

itudi

nal a

ccel

erat

ion

[g]

MeasuredSimulated

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

acc

eler

atio

n [g2 /H

z]

PSD - ACBOVZ

MeasuredSimulated

2 2.5 3 3.5 4 4.5 5

Time history - ACBOVZ

Time [s]

Ver

tical

acc

eler

atio

n [g

]

MeasuredSimulated

186 6 31 261 46 49 128

00,5

11,5

22,5

33,5

4

DDAH

BZ

DDAV

BZ

DDAH

FZ

DDAV

FZ

DFAH

Z

DFAV

Z

FFAV

X

FFAV

Y

FFAV

Z

FDAV

X

FDAV

Y

FDAV

Z

FFAH

X

FFAH

Y

FFAH

Z

FDAH

X

FDAH

Y

FDAH

Z

Damper strokes and spindle loads - TDDI

55

guaranteed when iterating towards accelerations in the structure. This becomes apparent when looking at a vertical wheel force signal, see Figure 57. Here the low frequency peak indicates that the rigid body motions are reproduced quite well, however, much high frequency content are introduced via the vertical actuator in order to get the accelerations to match well with the measurements. Further presentation of the pseudo-damage for these signals are for this reason not of interest since the conclusion is that rigid body modes are represented well but high frequency content is also introduced to a higher extent in the model.

Figure 57. PSD spectrum and time history signal of vertical spindle force FFAHZ Since structural acclerations have been possible to reproduce in the model with good accuracy, it is of interest to assess how well the strains in the model are predicted. The TDDI values and relative pseudo-damage when comparing the strains in the virtual model to the measured strain gauge signals are presented in Figure 58 and Figure 59.

Figure 58. TDDI values of strain responses

Figure 59. Relative pseudo-damage of

strain responses TDDI values for the strain responses in the model all exceeds the limit of 0,3. Best correlation is found for the strain measurement at the rear overhang, TCX09_BAK and also side wall member strain TCX03_HB. Relative pseudo-damage above 1 indicates that the model is conservative in its prediction, in other words, it shows a larger fatigue damage than in reality, which is the case for three of the four strain responses. The PSD spectra and time history signals of the simulated strain responses and the physical strain gauge measurements are presented below in Figure 60 to Figure 63.

0 10 20 30 40Frequency [Hz]

PSD

ver

tical

spi

ndle

forc

e [N2 /H

z]

PSD - FFAHZ

MeasuredSimulated

2 2.5 3 3.5 4 4.5 5Time [s]

Ver

tical

spin

dle

forc

e [N

]

Time history - FFAHZ

MeasuredSimulated

1,73

2,49

1,89 1,87

0,66

0

0,5

1

1,5

2

2,5

3

Mea

n

TCX0

3 _H

B

TCX0

4_VB

TCX0

4_VF

TCX0

9_BA

K

Strains - TDDI

5,16

2,22

6,61 11,40

0,42

012345678

Mea

n

TCX0

3 _H

B

TCX0

4_VB

TCX0

4_VF

TCX0

9_BA

K

Strains - Relative pseudo-damage

56

Figure 60. PSD spectrum and time history signal of strain response TCX03_HB

Figure 61. PSD spectrum and time history signal of strain response TCX04_VB

Figure 62. PSD spectrum and time history signal of strain response TCX04_VF

0 5 10 15 20 25

PSD - TCX03 HB

Frequency [Hz]

PSD

str

ain

[ µs2 /H

z]

MeasuredSimulated

10 10.5 11 11.5 12 12.5 13

Time history - TCX03 HB

Time [s]

Stra

in [ µ

s]

MeasuredSimulated

0 5 10 15 20 25Frequency [Hz]

PSD

str

ain

[ µs2 /H

z]

PSD - TCX04 VB

MeasuredSimulated

10 10.5 11 11.5 12 12.5 13

Time history - TCX04 VB

Time [s]

Stra

in [ µ

s]

MeasuredSimulated

0 5 10 15 20 25Frequency [Hz]

PSD

str

ain

[ µs2 /H

z]

PSD - TCX04 VF

MeasuredSimulated

10 10.5 11 11.5 12 12.5 13

Time history - TCX04 VF

Time [s]

Stra

in [ µ

s]

MeasuredSimulated

57

Figure 63. PSD spectrum and time history signal of strain response TCX09_BAK The 1-2 Hz response peaks, induced from rigid body modes, are very prominent in the first three simulated strain measurements. The rigid body modes produced in the test rig clearly have larger influence on the strain responses in the model than what the rigid body modes active during the proving ground test had on the real vehicle. All four strains from the simulation show higher magnitudes for the second peak at around 9 Hz when compared to the measurements. Comparing strain TCX03_HB and TCX04_VB, both measuring the strain at identical places on the two opposite side wall members, it is evident that TCX03_HB, which is positioned on the right side of the coach, have higher contribution from the 9 Hz mode. This is seen in both the simulated and measured responses, and is most likely related to the differences in stiffness and load paths around the doorways on the right side of the coach. The simulations using the virtual test rig approach have shown that replicating structural acceleration responses with high accuracy does not necessarily assures that strain responses are in the same vicinity of accuracy. The “transformation” from acceleration responses to strain loads within the structure is seemingly poorly represented by the model.

0 5 10 15 20 25Frequency [Hz]

PSD

str

ain

[ µs2 /H

z]

PSD - TCX09 BAK

MeasuredSimulated

10 10.5 11 11.5 12 12.5 13

Time history -TCX09 BAK

Time [s]

Stra

in [ µ

s]

MeasuredSimulated

58

59

5 CONCLUSIONS AND DISCUSSION

In this chapter a discussion of the employed methods and obtained results is presented along with the conclusions that have been made from the thesis work.

5.1 Conclusions The predictability of a full vehicle coach model has been evaluated in this work using two different full vehicle durability analyses. A MBS model of a Scania Touring 4x2 coach has been modelled in Adams/Car, taking into consideration the flexibility of the coach chassis frame and body structure. The model has been simulated by means of a virtual proving ground and a virtual test rig. Measurements from a proving ground test have been used as desired target responses when evaluating the model performance. Below are the conclusions made from the performed studies in this thesis work.

5.1.1 Virtual proving ground • Spindle loads displayed lower magnitude in longitudinal direction than what had been

measured, and to some extent so did lateral spindle loads. This is likely a consequence of the tire model being unable to represent correct loading. At the same time, vertical forces and damper strokes were reproduced well.

• Front lateral forces are heavily dependent on the steering corrections made by the driver module. Achieving simulated lateral spindle forces that matches measurements closely is therefore difficult. Poor PID controller settings have been found to yield significant differences in accelerations close to the front suspension as heavy steering corrections are imposed.

• Responses in the structure show great dependence on the prescribed damping of the flex body. The second derived damping expression with nested IF functions achieved the best relative pseudo-damage for the structural acceleration responses. The default damping in Adams describes to low energy dissipation in the structure and consequently displays very large relative pseudo-damage for most responses, whereas the first derived damping expression dissipates too much energy.

• Strain responses suggest that VPG simulations for all three damping settings are non-conservative, most likely a consequence of the lower input loads to the structure. The standard damping setting achieves strain responses with pseudo-damage closest to the measured strain gauge signals. This is, however, probably due to the lower energy dissipation of the damping compensating the fact that some input loads to the structure are of smaller magnitudes than in reality.

5.1.2 Virtual test rig • Acceleration responses in the coach body structure have been possible to reproduce with

high accuracy. A TDDI mean value of 0,101 and a mean relative pseudo-damage of 0,94 were achieved.

• It could be concluded that the derived drive signals contained slightly more high frequency content than what the actual measured road induced loads contained. This in turn might indicate that the second damping expression, which was used in the test rig simulations, dissipates slightly more energy than in reality. It could also indicate that the dynamics of the wheel suspensions were not correctly modelled for frequencies above a

60

certain point. The implication of this is that simulation in the virtual test rig is less dependent on the structural damping or other model deviations, as it will in any case introduce as much energy, frequency content, as necessary to replicate the target responses.

• Simulated strain responses in the model matched well for only two of the four evaluated strain gauge locations, even though acceleration responses in the structure had been replicated with high accuracy. The other two strain response achieved remarkably high relative pseudo-damage values.

• Three of the four strain responses had a relative pseudo-damage above 1. A final conclusion of simulation responses being conservative is therefore not completely valid since one strain measurement demonstrated a value of 0,42.

• Comparing the strain responses from the simulation in the virtual test rig with the responses from VPG simulations show large differences in strain response energy in both the 1-2 Hz frequency band related to rigid body modes and in the 8-12 Hz frequency band.

5.2 Discussion There are naturally several possible reasons to why simulated and measured responses exhibit differences. Before discussing such reasons related to the virtual model it should be pointed out that physical measurements does not necessarily always depict the exact truth since equipment inaccuracies and human mistakes can influence. In this study, the measured responses have been considered definite target responses, one should therefore bear in mind that discrepancies may not only be introduced from the virtual model, but also from minor errors associated with physical measurements. FE-models of this kind of large structure as used in this study can easily incorporate modelling errors with varying impact on the simulated model behaviour. The size and complexity of the model makes such modelling errors hard to notice. The author has to the best of his ability tried to verify that the model was free from any severe modelling errors. It is also necessary to point out that the FE-model is naturally a simplification of the actual coach. There are several components that have been either simplified geometrically, represented by approximated point mass elements or even left out during the modelling. The effect of these simplifications on the final results are of course unclear, but should nonetheless be kept in mind. Another uncertainty which is likely to have influence on the results is the positions of the markers used for measurement requests in the virtual model. Attaining correct measurements during simulations necessitates that specific sensor location and orientation coordinates used during the physical test have been well documented. In this study, some of the specified sensor coordinates were found to deviate from structural members in the coach body structure of the model. Sensor positions had to be estimated in these cases. Generalizing the damping of the structure by the derived damping expressions is quite a rudimentary method for approximating the actual energy dissipation in the structure. The second damping expression could be further improved by adding more nested IF functions and dividing the mode frequency ranges in even smaller segments. There is also a degree of incertitude involved in the generation of the drive signals that were used in the DOE experiments since consideration had to be made to lateral and longitudinal structural responses in the lower parts of the structure, which possess some damping dependence. A more accurate approach would be to generate individual drive signals for each DOE experiment run, however, this was considered far

61

too time-consuming to carry out within the time frame of this thesis. The best method would of course be to have accelerometer measurements on the suspension assembly, close to the interface to the chassis frame, or to improve the test rig to enable the possibility of reproducing lateral and longitudinal spindle loads as well. Longitudinal spindle loads in the virtual proving ground approach were significantly lower when compared to the physical test measurements. Lateral spindle loads were also found somewhat lower than measured. The precision of the tire model can rightfully be questioned as tire models in general are known to occasionally lack accuracy in load predictability. However, there are other factors which may also influence the accuracy at which spindle loads are reproduced. Before concluding that the lower spindle loads are solely a consequence of an inapt tire model, studies should be performed to investigate the influence of the compliance in the suspension and steering systems. The sensitivity to different tire operating pressures and how the tire model performs on a different virtual road are also of interest. To make any final conclusions of the predictability of the model regarding durability, physical tests with more strain gauges fitted throughout the coach structure are necessary. It would also be of interest to evaluate strains in the model using the virtual test rig when several acceleration responses in very close proximity to the strain gauge positions have been measured. This would allow for local acceleration responses to be replicated and possibly increase the accuracy of the predicted strain responses in the model.

62

63

6 FUTURE WORK

In this chapter suggestions and recommendations for future work are presented. The virtual testing performed in this thesis has shown that additional studies are necessary to gain further knowledge about the limitations and possibilities of the two full vehicle durability analyses. In the case of virtual proving ground testing, it is recommended that sensitivity analyses are carried out for compliance parameters in the suspension subsystems in order to investigate how spindle loads are affected. Other parameter studies are also suggested to determine whether the low spindle load are a direct consequence of the tire model being unable to represent these loads. To derive a more accurate approximation of the structural damping, it is advised, as mention previously, that the virtual test rig is updated to include more sophisticated control methods for pairwise actuators. It is otherwise recommended that measurements which include longitudinal and lateral accelerations on the suspensions are used for deriving damping expressions. For the virtual test rig analysis, it is recommended that a sensitivity study is carried out to see how much results differ when, in the FE-model, extracting strain data from different nodes in proximity of a suggested strain gauge position. Another recommendation is to perform a more local evaluation using the virtual test rig. The approach would be to perform a physical test with several accelerometers fitted in very close proximity to strain gauge positions. With this approach, local acceleration responses could be replicated and possibly increase the accuracy of the predicted strain responses in the model.

64

65

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APPENDIX A: SENSOR AND SIGNAL DESIGNATION Ch. Label Unit Positive dir. Signal

1 ACBNHX g rearwards Longitudinal acc. at back section, lower, right side 2 ACBNHY g rightwards Lateral acc. at back section, lower, right side 3 ACBNVX g rearwards Longitudinal acc. at back section, lower, left side 4 ACBNVY g rightwards Lateral acc. at back section, lower, left side 5 ACBNVZ g upwards Vertical acc. at back section, lower, left side 6 ACBOVX g rearwards Longitudinal acc. at back section, upper, left side 7 ACBOVY g rightwards Lateral acc. at back section, upper, left side 8 ACBOVZ g upwards Vertical acc. at back section, upper, left side 9 ACDNHX g rearwards Longitudinal acc. at drive axle, lower, right side 10 ACDNHY g rightwards Lateral acc. at drive axle, lower, right side 11 ACDNHZ g upwards Vertical acc. at drive axle, lower, right side 12 ACDNVX g rearwards Longitudinal acc. at drive axle, lower, left side 13 ACDNVY g rightwards Lateral acc. at drive axle, lower, left side 14 ACDNVZ g upwards Vertical acc. at drive axle, lower, left side 15 ACDOHX g rearwards Longitudinal acc. at drive axle, upper, right side 16 ACDOHY g rightwards Lateral acc. at drive axle, upper, right side 17 ACDOHZ g upwards Vertical acc. at drive axle, upper, right side 18 ACDOVX g rearwards Longitudinal acc. at drive axle, upper, left side 19 ACDOVY g rightwards Lateral acc. at drive axle, upper, left side 20 ACDOVZ g upwards Vertical acc. at drive axle, upper, left side 21 ACFNHX g rearwards Longitudinal acc. at front axle, lower, right side 22 ACFNHY g rightwards Lateral acc. at front axle, lower, right side 23 ACFNHZ g upwards Vertical acc. at front axle, lower, right side 24 ACFNVX g rearwards Longitudinal acc. at front axle, lower, left side 25 ACFNVY g rightwards Lateral acc. at front axle, lower, left side 26 ACFNVZ g upwards Vertical acc. at front axle, lower, left side 27 ACFOHX g rearwards Longitudinal acc. at front axle, upper, right side 28 ACFOHY g rightwards Lateral acc. at front axle, upper, right side 29 ACFOHZ g upwards Vertical acc. at front axle, upper, right side 30 ACFOVX g rearwards Longitudinal acc. at front axle, upper, left side 31 ACFOVY g rightwards Lateral acc. at front axle, upper, left side 32 ACFOVZ g upwards Vertical acc. at front axle, upper, left side 33 DDAHBZ mm rebound stroke Damper stroke, drive axle, rear, right side 34 DDAVBZ mm rebound stroke Damper stroke, drive axle, rear, left side 35 DDAHFZ mm rebound stroke Damper stroke, drive axle, front, right side 36 DDAVFZ mm rebound stroke Damper stroke, drive axle, front, left side 37 DFAHZ mm rebound stroke Damper stroke, front axle, right side 38 DFAVZ mm rebound stroke Damper stroke, front axle, left side 39 FFAVX N w-fix rearwards Longitudinal force, front axle, left wheel 40 FFAVY N w-fix rightwards Lateral force, front axle, left wheel 41 FFAVZ N w-fix upwards Vertical force, front axle, left wheel 42 FDAVX N w-fix rearwards Longitudinal force, drive axle, left wheel 43 FDAVY N w-fix rightwards Lateral force, drive axle, left wheel 44 FDAVZ N w-fix upwards Vertical force, drive axle, left wheel 45 FFAHX N w-fix rearwards Longitudinal force, front axle, right wheel

68

46 FFAHY N w-fix rightwards Lateral force, front axle, right wheel 47 FFAHZ N w-fix upwards Vertical force, front axle, right wheel 48 FDAHX N w-fix rearwards Longitudinal force, drive axle, right wheel 49 FDAHY N w-fix rightwards Lateral force, drive axle, right wheel 50 FDAHZ N w-fix upwards Vertical force, drive axle, right wheel