Fatigue analysis of engine brackets subjected to road induced loads

IN THE FIELD OF TECHNOLOGY DEGREE PROJECT

VEHICLE ENGINEERING

AND THE MAIN FIELD OF STUDY MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2016 ,

Fatigue analysis of engine brackets subjected to road induced loads

THERESE EK

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

www.kth.se

Abstract

In this master thesis, methods for fatigue analysis of front engine brackets subjected to road induced gravity loads (g-loads) are studied. The objective of the thesis is to investigate the possibility to improve simulation and test analysis for the components. The powertrain is modeled with varying degrees of complexity and the different models are compared to each other and to Scania’s models for analysis of the engine suspension. The analysis begins with g-loads and proceeds with time-dependent loads. It is investigated how simulated strains in the cylinder block correspond to measured strains from the test track at Scania. Finally, it is investigated how component tests corresponds to actual loads by comparing the results.

The results from the first part of the thesis indicate that worst load case is loading in the negative

z -direction and the model of the powertrain with isolators modelled as spring elements is the best for

g-loads lower than -3g and the model is sufficient for loads lower than -8g. The results from the second

part of the thesis indicate that the simulated strains generally correspond to the measured strains, but

with a slight difference in strain amplitude.

Sammanfattning

I detta examensarbete studeras ber¨ akningsmetoder f¨ or utmattning av motorf¨ asten som uts¨ atts f¨ or v¨ agin- ducerade gravitationslaster (g-laster). Syftet med examensarbetet ¨ ar att unders¨ oka f¨ orb¨ attringsm¨ oj- ligheter f¨ or komponenternas ber¨ aknings- och provmetoder. Drivlinan modelleras med olika grader av kom- plexitet och de olika modellerna j¨ amf¨ ors med varandra, samt med Scanias modell av motorupph¨ angningen.

Ber¨ akningsanalysen b¨ orjar med g-laster och fortskrider med tidsberoende laster. Unders¨ okning av hur v¨ al simulerade t¨ ojningar i cylinderblocket motsvarar uppm¨ atta t¨ ojningar fr˚ an Scanias provbana genomf¨ ors.

Slutligen unders¨ oks hur komponentprovning motsvarar verkliga laster genom att j¨ amf¨ ora resultat.

Resultaten fr˚ an den f¨ orsta delen av examensarbetet indikerar att det v¨ arsta lastfallet ¨ ar f¨ or last i den negativa z -riktningen och att modellen av drivlinan d¨ ar isolatorerna har modellerats med fj¨ aderelement

¨ ar den b¨ asta f¨ or laster l¨ ager ¨ an -3g och den ¨ ar tillr¨ acklig f¨ or laster l¨ agre ¨ an -8g. Resultaten fr˚ an den andra

delen av examensarbetet indikerar att de simulerade t¨ ojningarna st¨ ammer v¨ al ¨ overrens med de uppm¨ atta

t¨ ojningarna, dock med en liten skillnad i t¨ ojningsamplitud.

Acknowledgements

I would like to express my sincere gratitude to my supervisor Jonas Lenander for giving me the opportunity to write this thesis, for the interesting discussions, for his enthusiasm, encouragement and expertise. I could not have asked for a better supervisor at Scania. I thank everybody at NMBS for welcoming me and giving me insight about how great it is to work at Scania, and alongside such an amazing group.

Besides my supervisor at Scania, I would like to thank my supervisor at KTH, Prof. S¨ oren ¨ Ostlund for his interest in my progress, meticulousness, immense knowledge for being a truly inspiring role model in the field of solid mechanics.

Last but not least, I give all my love to my wonderful family, for always believing in me, for your incredible patience, honesty, wisdom and unconditional love.

Stockholm, August 2016

Therese Ek

Contents

1 Introduction 5

1.1 Problem description . . . . 6

1.2 Software . . . . 6

1.2.1 Catia V5 . . . . 6

1.2.2 FE-Software . . . . 6

1.2.3 MATLAB 2014b . . . . 7

1.3 Outline of thesis . . . . 7

2 Background 8 2.1 Theoretical background . . . . 8

2.1.1 Fatigue . . . . 8

2.1.2 Equivalent loads . . . . 9

2.1.3 Gravity loads . . . . 10

2.2 Historical background . . . . 10

2.2.1 Test methods . . . . 11

2.2.2 Statistical methods . . . . 11

2.2.3 Acceptance criteria . . . . 12

2.2.4 Models . . . . 12

3 Models 14 3.1 Geometry . . . . 14

3.2 Materials . . . . 17

3.3 Linear model . . . . 18

3.4 Nonlinear model No.1 . . . . 20

3.5 Nonlinear model No.2 . . . . 20

3.6 Time-dependent model . . . . 20

4 Method 23 4.1 Linear model . . . . 23

4.2 Nonlinear model No.1 . . . . 24

4.3 Nonlinear model No.2 . . . . 24

4.4 Time-dependent model . . . . 25

5 Results and analysis 26 5.1 Model analysis . . . . 26

5.2 Time-dependent analysis . . . . 33

6 Discussion 38

7 Summary 40

Nomenclature

α Material constant β Material constant

 Strain

ν Poisson’s ratio

Φ Normal distribution function

ρ Density

Θ Central safety factor

C

Mooney-Rivlin material constant C

Mooney-Rivlin material constant

D Damage

d Pseudo damage

D

Mooney-Rivlin material constant D

Design life

E Young’s modulus e

Absolute error f

Sample frequency

L Load

M The total mass for each component m The mass for each component N One cycle

P

Reliability R Resistance S

Amplitude stress S

Endurance limit S

Maximum stress S

Mean stress S

Minimum stress t Time increment t

Safety index

V

Variance coefficient for the resistance R

V

Variance coefficient for the load L

CONTENTS CONTENTS

N

Fatigue life

M Moment vector

R Distance vector

r Distance vector

g-loads Gravity loads

p Percent of survival

DL6 6 - cylinder engine

DOF Degrees of freedom

FE Finite element

NCG New Cab Generation

ODB Output database

SPC Single point constraint

SPCLB Left back node set

SPCLF Left front node set

SPCRB Right back node set

SPCRF Right front node set

STR Scania Technical Report

TKX71 Name of strain gauge

TKX73 Name of strain gauge

TKX74 Name of strain gauge

TKX75 Name of strain gauge

Chapter 1

Introduction

The powertrain is the most vital system of a vehicle. The related components in the system are designed to withstand influencing factors such as combustion pressure, high temperatures, noise and vibrations.

Some components are designed for road induced loads due to uneven roads which may cause fatigue damages. Among the components that are exposed to such loads are the engine brackets, the cylinder block and the flywheel housing. This study focus on the front engine suspension of a truck and mainly on the engine brackets.

In year 2010, at a R&D Technology meeting in association with the New Cab Generation (NCG) project, recommendations for new isolators were introduced. The primary function of the isolators is to connect the powertrain to the chassis. The secondary function is to reduce transmitted vibrations from, for example, the engine, unbalanced shafts, gear contact and ignition of fuel. The isolators also reduce vibrations due to road induced loads. Since the Technology meeting in 2010, the isolators developed by Scania have changed significantly. They are made of a different rubber material and the geometry has changed. Furthermore, the test track has been rebuilt and, thus, the loads that the truck is subjected to have changed. This report investigates how the new isolator configuration affect the loads that the engine brackets and the cylinder block are subjected to.

Previous analysis of the engine suspension at Scania has been evaluated as well as test methods. The

powertrain was modeled with varying degrees of complexity and the different models were compared to

each other. The analysis began with g-loads and proceeded with time-dependent loads. Finally, it was

investigated how test results corresponded to actual loads from the test track at Scania by comparison

of strains and load levels.

1.1. PROBLEM DESCRIPTION CHAPTER 1. INTRODUCTION

1.1 Problem description

Due to the new isolator configuration there is need for evaluation of load changes on the engine suspension, particularly the engine brackets and their connection with the isolators and the cylinder block. The change of loads might change the fatigue life and in order to evaluate this, models of the powertrain are needed.

A finite element model (FE-model) of the powertrain was created and modified with varying degrees of complexity. The modifications were made in order to create a model that possibly better corresponds to the reality.

The objectives of the thesis are subdived into two analysis parts. In the first part, different models of the powertrain are analysed, using g-loads that are applied in different directions. In the second part, time-dependent loads that are applied on the isolators are analysed. The objectives of the first part were to:

• determine how the constraint point configuration in a linear model influences the results.

• determine if the FE-model needs to be nonlinear or if a linear model is sufficient.

• compare fatigue life using the chosen model to fatigue life from Scania’s previous analysis model.

The objectives of the second part were to:

• evaluate if it is possible to perform analysis of quasi-static time-dependent loads in a time effective manner.

• compare simulated and measured strain results, from the test track, in the cylinder block in order to validate results.

1.2 Software

1.2.1 Catia V5

Component geometries created in Catia V5 R24 [1] were used in the development of the FE-model.

1.2.2 FE-Software

In this section, the software packages used in order to construct models, perform simulations and evaluate results are presented.

Pre-processors

Two pre-processors were used when constructing geometries, discretizing the model and managing contact areas. The pre-processors both belong to Altair [2]. SimLab 14.0 [3] was used for discretization of each component in the powertrain and Hypermesh [4] was used for assembling the components into one model.

Hypermesh was also used to create contact areas. The software generates geometry data, which are then called from an input data file, created in EditPad Pro 7 [5]. In the input data file, material parameters, contacts, loads and load steps were defined.

Solver

Abaqus 6.14-2 [6], was the software that carried out the FE-analysis of the model defined in the input file.

Reaction forces, displacements, stresses and strains are all computed and output is saved to an output

database (ODB)-file which can be further processed and analysed by a post-processor.

CHAPTER 1. INTRODUCTION 1.3. OUTLINE OF THESIS

Post-processors

In order to perform a fatigue analysis the software FEMFAT 5.1.1 [7] was used to analyse the results from the FE-analysis. FEMFAT estimates the safety factor and stress amplitude of the analysed model by applying conventional fatigue theory such as the stress-life approach further described in subsection 2.1.1.

Abaqus was also used to visualize the results from the FE-analysis and the fatigue analysis.

1.2.3 MATLAB 2014b

In this thesis, Matlab 2014b [8] is used for general numerical calculations, but it is a software that also has capabilities of for instance, data analysis and visualization, programming and algorithm development.

1.3 Outline of thesis

This thesis starts with a background, which contains information about how loads and fatigue can be defined and analysed together with the terminology that will be used throughout the report. The chap- ter proceeds with a historical background on how analysis and testing of fatigue previously have been performed and evaluated at Scania.

Chapter 3 introduces geometry and materials of the components in the powertrain. Furthermore, the FEM-model and its linear and nonlinear variants are presented. The models described in this chapter are then used in Chapter 4. In order to meet the objectives of the thesis, Chapter 4 describes the methods used for development of a final model and how to compare the results from this model with test results.

Results and analysis are presented in Chapter 5. In this chapter the methods described in Chapter 4

are used in order to determine the final model. Chapter 6 discusses the evaluation of the models, the

simulation analysis and results. Chapter 7 summarizes results and conclusions. This chapter also contains

a recommendation for further development of acceptance criteria, analysis and test methods.

Chapter 2

Background

This chapter introduces theory in the area of solid mechanics that are relevant for the thesis and a historical background of analysis and test methodology at Scania.

2.1 Theoretical background

2.1.1 Fatigue

In the automotive industry, it is of importance to consider fatigue. Fatigue failure occurs at a lower load than the yield limit of the material due to the cyclic nature of the loading. In ASTM E1823 [9] the definition of fatigue is:

“The process of progressive localized permanent structural change occurring in a material sub- jected to conditions that produce fluctuating stresses and strains at some point or points and that may culminate in cracks or complete fracture after a sufficient number of fluctuations.”

This process of structural change is a result of cyclic loading. A cycle, shown in Figure 2.1, is one complete sequence of values of force that is repeated under constant amplitude loading.

Time

Stress

0

Samp

N

Smax

Smin

Figure 2.1: Constant amplitude loading for two cycles with the amplitude S

. N = 1 is marked with a horizontal arrow.

The mean stress (S

) is derived according to Eq. (2.1),

CHAPTER 2. BACKGROUND 2.1. THEORETICAL BACKGROUND

S

= S

+ S

2 (2.1)

where the maximum and minimum stresses are shown in Figure 2.1.

Fatigue life, denoted N

, is the number of cycles that the specimen can sustain at a constant amplitude loading before failure occurs. In the stress life approach, specimens are tested at different load levels until failure. The graphical representation of the results is a logarithmic diagram where the stress amplitude (S ) versus number of cycles until failure (N

) are presented. The construction of this S-N diagram , also called W¨ ohler diagram [10], is a method developed in 1870 to present fatigue data. One test of a specimen loaded at constant amplitude until failure gives one failure point in the S-N diagram. The S-N curve is the average curve drawn through these failure points, see Figure 2.2 [10].

Figure 2.2: S-N diagram, each test gives one failure point and an average curve is drawn through them. The endurance limit (S

), also called fatigue limit, is the stress magnitude for an infinite number of cycles, theoretically speaking.

If nothing else is stated, the S-N curve refers to the fatigue life that 50 % of the specimens can sustain [11]. The specimens are the population in the quote from ASTM E1823:

“Fatigue life for p percent survival - an estimate of the fatigue life that p percent of the population would attain or exceed under a given loading. The observed value of the median fatigue life estimates the fatigue life for 50 percent survival. Fatigue life for p percent survival values, where p is any number, such as, 95, 90, and so forth, also may be estimated from the individual fatigue life values.”

N

can be expressed according to Basquins equation, where the constants α and β are material parameters [12].

N

= αS

(2.2)

2.1.2 Equivalent loads

The fluctuations mentioned in the first ASTM E1823 quote, caused by different kinds of loading are not constant. For example, pot holes of varying sizes, curbstones and speed bumps all cause different kinds of loads that a truck can be subjected to. By deriving a constant amplitude that would give the same fatigue damage per kilometer as the measured damage, the load amplitude can be simplified to be constant. This constant load amplitude is the equivalent load (S

) [13].

The rainflow method developed by professor Tatsuo Endo in the 1960s transforms a complicated load sequence to damage equivalent cycles (N

) for constant load amplitudes (S

) [14]. The damage according to Palmgren-Miner’s damage rule [15] for the measured load over time is expressed as,

D = α

X

k

S

k = 1, 2, 3... (2.3)

2.2. HISTORICAL BACKGROUND CHAPTER 2. BACKGROUND

The damage can also be expressed as in Eq. (2.4) where the parameter (N

) comes from Eq. (2.2).

D = X

k

1

N

(2.4)

Failure occurs if D is greater than or equal to 1. The pseudo damage (d) in Eq. (2.5) is the damage without the constant α.

d = X

k

S

(2.5)

The design life (D

) is the damage that the component is designed for. The constant K in Eq. (2.6) is the number of repetitions that gives the design life.

D

= K · D (2.6)

When the equivalent damage in Eq. (2.7) is damage equivalent to D

, the S

equals to Eq. (2.8).

D

= N

· α

· S

(2.7)

S

=  K · d N



(2.8) The advantage of using S

is that it can be used for comparison of measured loads on different kinds of roads or for different vehicles [11].

2.1.3 Gravity loads

There are different kinds of loads that can be applied when analysing a structure. Gravity load is a dead load, which is defined according to the quote:

“Dead loads: refer to loads that typically don’t change over time, such as the weights of materials and components of the structure itself...” [16]

The definition of acceleration of gravity in the Science Dictionary, quoted:

“The acceleration of a body falling freely under the influence of the Earth’s gravitational pull at sea level. It is approximately equal to 9.806 m (32.16 ft) per second per second, though its measured value varies slightly with latitude and longitude. Also called acceleration of free fall.” [17]

Thus, gravity (g) is a constant acceleration that is included in the term weight of a structure. The mass of a structure is always subjected to a gravity load (g-load) of g. In static finite element analysis, the magnitude of the g-load that the structure is subjected to has to be defined. It can be defined to be smaller or greater than g and the direction of the acceleration has to be defined as well as the densities of the components of the structure. It is an advantage to define the load that the structure is subjected to as a g-load since all elements in the model is subjected to the g-load. This load definition reduce high stress gradients at points, lines or surfaces where a load with a corresponding amplitude could have been defined since the g-load act on the whole structure.

It was described in Section 2.1.1 that in fatigue analysis, the equivalent load is used as a simplification.

Gravity loads can also be used as equivalent loads and the fatigue analysis is performed in the same manner, with the difference that the load amplitude is defined as a g-load.

2.2 Historical background

In this section, the information is mainly from the report Operativa m˚ al avseende utmattning. L˚ angtid-

sprov och skakprov [18].

CHAPTER 2. BACKGROUND 2.2. HISTORICAL BACKGROUND

2.2.1 Test methods

Scania performs different kinds of test, two examples are full vehicle tests and rig tests for a system of components or single components. Full vehicle tests are performed at the Scania test track. The test track has passages, corresponding to different road conditions. The engine suspension has historically been tested in a verifying shake test where the load sequence is simulated and more recently the front engine suspension is tested in a W¨ ohler test [19]. The W¨ ohler test performed at Scania is described in Design guidelines [20]. The purpose of performing tests is to ensure the quality of the products and to develop component designs, targets, requirements as well as acceptance criteria and reliability.

The two test methods complement each other in a sense how time consuming they are and how well they corresponds to reality. Components tested in a full vehicle test have accurate installation, bound- ary conditions and environment, such as temperature and weather conditions. Therefore, this method correspond well to reality. The full vehicle test is very time consuming and expensive, the fatigue tests performed at Scania are in general destructive. Usually only one specimen of each component in the vehicle is tested at the same time, therefore, it is impossible to estimate the statistical spread for the components from a single test.

A rig test performed on a big system of components, a partly assembled vehicle, is less time consuming than a full vehicle test, but it is more complicated to recreate the real boundary conditions and envi- ronment. Interaction between between air forces, road conditions, corrosion and rust are examples of parameters that are not reproduced in rig tests. Rig tests performed on single components are even less time consuming and the statistical spread and component characteristics can effectively be analyzed.

2.2.2 Statistical methods

The fatigue life of a component until failure generally has a log-normal distribution. This assumption of distribution is consistent with general fatigue analysis for high cycle fatigue found in the literature and with experiences at Scania. High cycle fatigue is when N

is greater than 10

cycles [21]. The meaning of a log-normal distribution is that the logarithm of N

has a normal distribution.

The reliability (P

) is a parameter that describes how strong the component is for a specified load and resistance [11]. The parameter p in Eq. (2.9) is the percent age of survival and the reliability is defined as,

P

= 1 − p

P

= Φ(t

)

(2.9)

t

= Θ − 1 q

V

+ (Θ · V

)

(2.10)

where Φ is the normal distribution function, Θ is the central safety factor and t

is the safety index. In Eq. (2.10) V

and V

are variance coefficients defined as standard deviation divided by expectancy for the load (index L) and resistance (index R). Solving Eq. (2.10) for Θ with p ≤ 50%, the safety factor can be expressed as,

Θ = 1 +

r 1 − 

1 − (t

· V

)



· 

1 − (t

· V

)



1 − (t

· V

)

(2.11)

thus, the safety factor can be estimated when V

and V

are determined from the performed rig tests.

2.2. HISTORICAL BACKGROUND CHAPTER 2. BACKGROUND

2.2.3 Acceptance criteria

When designing components, every component have to meet an acceptance criterion. The acceptance criterion can be targets or requirements. In revision 2 of STR6000001 [18], all acceptance criterion are denoted as targets, but with different levels of fulfillment of the acceptance criterion. The engine bracket has an old acceptance criteria of a safety factor > 1 with p = 50 % for 10

cycles [22]. The engine bracket is a component with the reliability ”higher”, which means that when failure occurs it is very costly and may lead to personal injury.

With the assumption that equivalent loads corresponds to real loads and that the fatigue loads have a log-normal distribution, the loads can be described as the product of road conditions, speed, spring stiffness among other factors. If the acceptance criterion is not met, the component design need to be revalued. When there is a change of the component design or a change of design in adjacent components, the acceptance criteria may need to be redefined as well. The isolator design has changed and therefore, the acceptance criterion for the front engine bracket is under development.

2.2.4 Models

According to Design guidelines, the W¨ ohler test performed at Scania is simulated. An example of a model that has been used in a W¨ ohler test simulation is shown in Figure 2.3 [22] and it is not only the engine brackets that are of interest when performing the W¨ ohler test simulation. Other components of interest are for instance, the cylinder block, the main bearing caps and the flywheel housing.

Figure 2.3: Model used in the simulated W¨ ohler test performed at Scania. The front cover is illustrated in blue, the left engine bracket in grey, the ladder frame in purple and the cylinder block is illustrated in beige.

In order to develop acceptance criteria and to determine the strength of a new component design of the

powertrain, in 2015 different departments at Scania initiated a common analysis model of the powertrain

[23]. The model of the powertrain, shown in Figure 2.4 is an example of a reference model used when

developing a model of the powertrain for fatigue analysis of the flywheel housing.

CHAPTER 2. BACKGROUND 2.2. HISTORICAL BACKGROUND

Figure 2.4: Reference model of the powertrain for development of a powertrain model for fatigue analysis of the flywheel housing.

Note that the V-engine cylinder block in Figure 2.4, and other components can be substituted by other

designs, therefore, the powertrain has various model variants. Another example of a model used in

fatigue analysis is where only a part of the cylinder block, a part of the front cover and the engine

bracket is modeled [24]. The boundary conditions and load cases depends on the analysis. W¨ ohler test

simulation, general fatigue analysis with a load sequence from measurement data and fatigue analysis

with equivalent loads all have different analysis methods. The constraint point configuration has not

always been consistent at Scania, although the analysis has been the same or similar and this is one of

the reasons to why the analysis in the first part of this thesis has been performed.

Chapter 3

Models

This chapter contains information about the different model geometries, defined contact surfaces and the material properties of the components. There are four types of models: one linear model, two nonlinear models and one time-dependent model. Linear model has three different configurations for boundary conditions that will be described. Nonlinear model No.1 contains nonlinear springs and Nonlinear model No.2, in addition, contains front isolators meshed with solid elements.

Time-dependent model has four configurations, two of the configurations have different strain gauges of different lengths, with the same contact surfaces as in all previous models. One configuration of the model has a decreased contact surface between the engine brackets and the cylinder block and the last configuration has an increased contact surface. This is described further in Section 3.6.

3.1 Geometry

To create a more realistic interface for the engine brackets the chosen components are the outer adjacent components of the powertrain. The components used in the geometry and the global coordinate system can be seen in Figure 3.1 and are listed below:

1. front cover 2. engine brackets 3. ladder frame

4. cylinder block 5. transmission plate 6. flywheel housing

7. front gearbox housing 8. back gearbox housing 9. suspension beam

Figure 3.1: Exploded view of the components used in the model of the powertrain.

CHAPTER 3. MODELS 3.1. GEOMETRY

Screw joints are not included in any model since sliding between contact surfaces tend to be small enough that it can be neglected for the 6 - cylinder (DL6) engine [25]. All components were descretized in SimLab separately and were then imported to Hypermesh, where all contact surfaces were defined. All components, aside from the cylinder block, were discretized with second order elements. The cylinder block is discretized with first order elements in order to save computation time. When the powertrain is mounted to chassis, there is a 5

angle between the powertrain’s local x -axis and the ground, therefore, the model was rotated 5

around the y-axis according to the arrow θy in Figure 3.2.

Figure 3.2: There are six degrees of freedom (DOF). The first DOF is movement along the x -axis. The second DOF is movement along the y-axis and the third DOF is movement along the z -axis. The fourth, fifth and sixth DOF is the rotational movement around the x, y and z -axes, respectively. The arrows illustrate the positive direction for each degree of freedom.

The contacts surfaces between the different components in the powertrain model are shown in Figure 3.3.

Figure 3.3: Defined contact surfaces are illustrated in pink.

The geometry in Figure 3.3 is used in Linear model and Nonlinear model No.1. In Nonlinear model No.2

the only difference is that the geometry of the isolators are added to the model geometry. A principal

sketch of the left isolator is shown in Figure 3.4.

3.1. GEOMETRY CHAPTER 3. MODELS

Figure 3.4: The light grey illustrates the rubber of the isolator and the dark grey

illustrates the top and bottom metal parts of the isolator. The rubber part of

the isolator consist of three different rubber components, a compression rubber, a

rebound rubber and a main rubber.

CHAPTER 3. MODELS 3.2. MATERIALS

3.2 Materials

Material properties for each component except the isolators are given in Table 3.1.

Table 3.1: Material properties for components used in the model.

Component Material E [GP a] ρ [kg/m

] ν [-]

Front cover Aluminum 46000 75 2750 0.33

Engine brackets Nodular cast iron 167.5 7200 0.26

Ladder frame Aluminum 46000 75 2750 0.33

Cylinder block Grey cast iron 105 7200 0.26

Transmission plate Steel 208 7800 0.3

Flywheel housing Aluminum 46000 75 2750 0.33

Front gearbox housing Aluminum 46000 75 2750 0.33

Back gearbox housing Grey cast iron 110 7100 0.23

Suspension beam Nodular cast iron 167.5 7200 0.26

The rubber parts of the isolator follow a Mooney - Rivlin material model which is an incompressible hyper elastic model. The contact surfaces between aluminum and rubber is assumed to have a kinematic friction coefficient of 0.5 [26]. In FEMFAT, fatigue analysis is performed with the stress life approach described in subsection 2.1.1. The material properties of the engine brackets are defined in FEMFAT according to Table 3.2 and the parameters are the same for all models.

Table 3.2: Parameters used in fatigue analysis of the engine brackets.

Parameters Value Unit

Endurance limit 170 MPa

Ultimate strength 500 MPa

Yield strength 320 MPa

Surface roughness 200 µm

Cycles 1 · 10

-

Survival probability 50 %

All components of the powertrain are not included in the geometry, and thus the model’s center of gravity

and total mass need to be modified in order to correspond to the engines real center of gravity. This

was done by modifying the densities of all components except for the engine brackets’ since they are the

components of interest in the analysis. First, all component mass centers and volumes were extracted

from the geometry in Hypermesh and with known material properties the model’s total mass could be

determined. The model’s total mass and center of gravity in global coordinates were determined by use of

the Center of Mass for Particles formula [27]. One center of mass coordinate for the system is the sum of

the components individual mass m

multiplied with their individual center of gravity in that coordinate

and then divided by the total mass (M ) of the system. Eq. (3.1) show how the center of gravity in the

global coordinates x, y and z were determined.

3.3. LINEAR MODEL CHAPTER 3. MODELS

x = P

i=1

m

x

M

y = P

i=1

m

y

M

z = P

i=1

m

z

M

(3.1)

The model’s center of gravity and total mass were then modified to correspond to the powertrain’s center of gravity and total mass by changing the component densities. Due to lack of components that could affect the center of gravity in some coordinates it was assumed that the most important center of gravity coordinates to get right were x and z.

An object in space rotates around the center of gravity if the extended load vector does not cross the center of gravity. It is therefore of importance to modify the model’s center of gravity to make the model’s movement correspond to the real system’s movement. Newton’s first law of motion, also called the law of inertia, relates to the importance of the system’s total mass. Inertia is the resistance of an object to change in its state of motion and the inertia increases with the object’s total mass. Therefore, the total mass of the model also need to correspond to the real system’s total mass.

3.3 Linear model

The geometry used in Linear model can be seen in Figure 3.1 and the materials used in the model are found in Table 3.1. The isolators are not included in the geometry. The nodes around the engine brackets’

screw holes on the bottom surface are connected to a node on each side by a single point constraint (SPC), see Figure 3.5.

Figure 3.5: Nodes around the left engine bracket’s screwholes are connected to a SPC node, the white node in the middle.

The white SPC node in Figure 3.5 is located on the left engine bracket, but the nodes around the screw holes on the right engine bracket and the suspension beam were connected to a node in the same manner.

These four SPC nodes are where the boundary conditions were applied. The engine brackets’ SPC nodes

are the front SPC nodes and the suspension beam’s SPC nodes are the back SPC nodes and the three

model configurations of Linear model are shown in Table 3.3.

CHAPTER 3. MODELS 3.3. LINEAR MODEL

Table 3.3: Constraint point configurations, the constraint points for the engine brackets (the front SPC nodes) and the suspension beam (the back SPC nodes) are either moved or unmoved.

SPC configurations Front Back

Configuration 1 Unmoved Unmoved

Configuration 2 Moved Unmoved

Configuration 2 Moved Moved

In order to evaluate how much the constraint points influence the results when using this model, these three configurations of the model were compared to each other. The unmoved position is when the nodes were close to the screw holes according to Figure 3.5 and the moved position is when the nodes were moved a distance from the screw holes. The moved positions corresponds to a calculated load position in the isolators [23]. The unmoved position for the SPC nodes can be seen in Figure 3.6(a) and Fig. 3.7(a).

The moved SPC nodes are shown in Figure 3.6(b) and Figure 3.7(b).

(a) Unmoved position (b) Moved position

Figure 3.6: Unmoved and moved front SPC nodes.

(a) Unmoved position (b) Moved position

Figure 3.7: Unmoved and moved back SPC nodes.

The moved SPC node in Figure 3.6(a) and Fig. 3.7(a) were positioned at the point where the grey lines

meet.

3.4. NONLINEAR MODEL NO.1 CHAPTER 3. MODELS

3.4 Nonlinear model No.1

The same geometry and material properties were used in this model as in Linear model, but instead of having the nodes around the screw holes connected to one node, that node was connected to three nonlinear spring elements for each of the x, y and z -directions. The spring elements for the left engine bracket are illustrated in Figure 3.8.

Figure 3.8: Spring connection, the blue point connecting the springs illustrate the moved position described in section 3.3 and the red points are the constrained nodes.

The node connecting the springs in Figure 3.8 is only a connection and the other red nodes in Figure 3.8 are the nodes that were constrained. There are three springs for every SPC node, one spring with stiffness in the x -direction, one spring with stiffness in the y-direction and one spring in the z -direction. The spring stiffness for each spring comes from measurements. The two x -springs have the same stiffness, the two y-springs have the same stiffness and the two z -springs have the same stiffness. The spring stiffness for the back springs were defined in the same manner. However, the springs constraining the back SPC nodes have a lower stiffness compared to the springs constraining the front SPC nodes.

3.5 Nonlinear model No.2

In this model the front spring elements in Nonlinear model No.1 and Figure 3.8 were replaced with the front isolators modeled with solid elements. The nodes at the isolator’s bottom surface were connected to a new SPC node by a kinematic coupling. The same springs that were used to connect the suspension beam in Nonlinear model No.1 were used in this model.

3.6 Time-dependent model

The time-dependent model is model Configuration 3 of Linear model with four strain gauges added to

the geometry. The strain gauges were modeled with T3D2 elements, which is a three dimensional 2-node

truss element. In January 2016, load measurements were performed on the powertrain and the strain

gauges in the model were positioned according to Appendix C in the associated load measurement report

[28]. The strain gauges were positioned on the left side of the cylinder block, close to the left engine

bracket. The strain gauges are listed and schematically illustrated in Figure 3.9

CHAPTER 3. MODELS 3.6. TIME-DEPENDENT MODEL

1. TKX71 2. TKX73 3. TKX74 4. TKX75

Figure 3.9: The strain gauges, positioned on the left side of the cylinder block are illustrated in yellow.

Instead of two node sets for the front and back SPC nodes, the time-dependent model had four node sets, one node set for each constraint point. The node sets are itemized and described below:

1. SPCLF is the Left Front constraint point 2. SPCLB is the Left Back constraint point 3. SPCRF is the Right Front constraint point 4. SPCRB is the Right Back constraint point

Each constraint point was connected to the nodes around the screw holes on the bottom surface on the engine brackets and the suspension beam. Two configurations of the strain gauges in the model were used according to Table 3.4.

Table 3.4: Length of strain gauges.

Gauge configuration TKX71 TKX73 TKX74 TKX75 Unit

Configuration 1 4.33 2.47 2.74 5.03 mm

Configuration 2 2.5 2.5 2.5 2.5 mm

Three configurations of the contact surfaces between the engine brackets and the cylinder block were used, while the length of the strain gauges was kept constant. The contact surfaces used for Linear model and Nonlinear model No.1 were common for all models. The contact surfaces between the left and right engine bracket and the cylinder block were altered by first decreasing the contact surface and then increasing the surface. Therefore, the total number of configurations is four. The three contact surface configurations for the left engine bracket are shown in Figure 3.10.

(a) Original contact surface (b) Decreased contact surface (c) Increased contact surface

Figure 3.10: The different contact surface configurations used in the time-dependent model.

3.6. TIME-DEPENDENT MODEL CHAPTER 3. MODELS

All configurations are summarized in Table 3.5.

Table 3.5: Gauge and contact surface configurations.

Configurations Description

Configuration 1 Model where the length of the strain gauges were in an interval of 2.47 mm to 5.03 mm with original contact surfaces

Configuration 2 Model where the length of the strain gauges were 2.5 mm

Configuration 3 Model with decreased contact surfaces for the contact between the engine brackets and the cylinder block, the length of the strain gauges were kept constant

Configuration 4 Model with increased contact surface for the contact between the engine

brackets and the cylinder block, the length of the strain gauges were kept

constant

Chapter 4

Method

In this chapter, the methods used to attain the results for each model are described as well as the methods used to compare the results. Stresses, strains, reaction forces and safety factors are the results evaluated when applying loads, constraints, and performing fatigue analysis. Quasi-static analysis was performed on all models.

4.1 Linear model

As described in Section 3.3 the linear model has three configurations for the position of the front and back SPC nodes where the boundary conditions were applied according to Table 3.3. In order to evaluate how the position of the constraints influences the results, the same load was applied for each one of the three configurations in the x, y and z -directions, respectively. The applied load was a g-load, described in Section 2.1.3. The model was subjected to a g-load of +10g in the x and y-direction and -10g in the z -direction. Boundary conditions with constrained degrees of freedom (DOF) for the different load cases are presented in Table 4.1 [23].

Table 4.1: Constrained DOFs for each load case.

Load case Front DOF Back DOF

X 2, 3 1, 2, 3

Y 2, 3 1, 2

Z 2, 3 1, 2, 3

Two node sets were created for each engine bracket. One node set contain 7 nodes in a point close to the fillets where the stress concentration was the highest. There was one node set for the engine bracket’s upper surface and one node set for the corresponding point on the lower surface. The arithmetic mean was derived for the results in each node set according to Eq. (4.1) [29], where x

represents the result in each node.

¯ x = 1

n

n

X

i=1

x

(4.1)

The mean 1st Principal stress for the two node sets in the three different configurations were compared by deriving the absolute error (e

) [30] according to,

e

= x

− x (4.2)

4.2. NONLINEAR MODEL NO.1 CHAPTER 4. METHOD

The parameter x

in Eq. (4.2) is the mean 1st Principal stress in Configuration 2 and x is the mean 1st Principal stress in Configuration 1. This absolute error is the first item in the list below, the derived differences between the three configurations:

1. the difference between Configuration 1 and Configuration 2.

2. the difference between Configuration 1 and Configuration 3.

3. the difference between Configuration 2 and Configuration 3.

Fatigue analysis was performed in FEMFAT for Configuration 3 with the parameters defined in Table 3.2 and the extracted fatigue results were:

• the amplitude stress

• the safety factor

In the fatigue analysis, S

was -g for loading in z and 0g when loading in y. This is discussed further in Chapter 6. The absolute mean for the fatigue results were also derived according to Eq. (4.1). The results were then compared in order to meet the objectives of the thesis, to determine how the constraint point configurations influence the results and how to determine which model is to be preferred.

4.2 Nonlinear model No.1

In Section 3.4, Figure 3.8 show the SPC nodes and the node connecting the springs to the nodes around the isolator’s screw holes. The SPC nodes were constrained in all DOFs for all load cases. The applied loads are both negative and positive g-loads. There were four load cases, the first load case is zero gravity to -10g in y for one load step and an increment size of 0.1. All load cases had the same step and increment size. The second load case is zero gravity to +10g in y. The third load case was zero gravity to -10g in z and the fourth load case was zero gravity to +8g in z. The load cases and boundary conditions are given in Table 4.2.

Table 4.2: Load cases and constrained DOFs for each SPC node.

Load case Negative load Positive load DOF

Y 0g to -10g 0g to 10g 1, 2, 3

Z 0g to -10g 0g to 8g 1, 2, 3

The same node sets as in Linear model was used in the analysis and the fatigue analysis was performed in the same manner, including calculation of the mean value of the node sets. The extracted results were:

• the 1st Principal stress

• the 3rd Principal stress

• the amplitude stress

• the safety factor

4.3 Nonlinear model No.2

The SPC nodes and loads were applied according to Table 4.2 and the fatigue analysis was performed in

the same manner as in the previous section with the same extracted results.

CHAPTER 4. METHOD 4.4. TIME-DEPENDENT MODEL

4.4 Time-dependent model

The two gauge configurations and the three contact surface configurations were used in the quasi-static analysis, previously described in Section 3.6. The first strain gauge configuration shared the same model configuration as one of the tie configurations, therefore there are in total four configurations. Each derived load component in the x, y and z -direction in the time domain from the load measurements [28] were applied to each one of the four node sets: SPCLF, SPCLB, SPCRF and SPCRB. The chosen load sequence come from measurements on the truck over the toughest passage at the test track. It was assumed that the powertrain was subjected to loads that would cause the most damage at this passage.

The greatest load amplitude in the load sequence was chosen to be in the middle of a 4 seconds long signal. The sequence start before the greatest load amplitude since the load is zero at the beginning of the load sequence.

If the loads are applied to the nodes in the node sets, the constraints can not be defined at the same nodes. Thus, in the static stress analysis performed in Abaqus, inertia relief is used [31]:

“Inertia relief: involves balancing externally applied forces on a free or partially constrained body with loads derived from constant rigid body accelerations”

Density or mass needs to be specified for computing inertia relief loads and the altered component densities were used in the analysis. The step was static and having a static step means that the inertia relief loading varies with the applied external loading, the model is in equilibrium in every step. The measurement sample rate, f

[Hz], was used to derive the time increment (t) according to Eq. (4.3), which was used in the analysis.

t = 1 f

(4.3)

The strain results in the four strain gauges were extracted with a code written in Matlab by Jonas

Lenander [32]. The strain results from the measurements and the analysis were compared.

Chapter 5

Results and analysis

This chapter contains the results for the first and second part of the thesis, described in Section 1.1.

The first part is a comparison of models and the second part include the time-dependent results and analysis.

5.1 Model analysis

Comparison between the constraint point configurations

The greatest stress in the points on the upper and lower surface on the engine bracket seen in Figure 5.1 occurred in the load case where the load was applied in the negative z -direction with a mean acceleration load of -g.

(a) (b)

Figure 5.1: The critical stress points on the left engine bracket. (a) is the stress point located on the upper surface and (b) is the stress point located on the lower surface.

This load case resulted in the highest stress concentration in the point shown in Figure 5.1(b). Therefore,

only the 1st Principal stress is presented for this point on the left engine bracket and the corresponding

point on the right engine bracket. The results for the three configurations in Table 3.3 are described in

Section 3.3 and the results are presented in Figure 5.2 with the same axis limits and scaling.

CHAPTER 5. RESULTS AND ANALYSIS 5.1. MODEL ANALYSIS

0 2 4 6 8 10

Gravity load

1stPrincipalstress

Left engine bracket loaded in negative Z Configuration 1

Configuration 2 Configuration 3

(a)

0 2 4 6 8 10

Gravity load

1stPrincipalstress

Right engine bracket loaded in negative Z Configuration 1

Configuration 2 Configuration 3

(b)

Figure 5.2: Trends in 1st Principal stress for different g-loads with applied load in the negative z -direction. (a) is the results for the left engine bracket. (b) is the results for the right enginge bracket.

When comparing Figure 5.2(a) and Figure 5.2(b) it can be seen that the stress in the left engine bracket was greater than the stress in the right engine bracket for all configurations. In Figure 5.2 there is no difference between the results in Configuration 2 and Configuration 3 for each engine bracket. However, Configuration 1 results in a greater stress. It is therefore concluded that moving the back SPC nodes will not influence the result in the points where the stress concentration is the highest in the left and right engine bracket.

In order to get a better understanding of why the stress was higher in Configuration 1, the moment (M) around the stress point in Figure 5.3 was derived for Configuration 1 and Configuration 3. Note that Configuration 3 has the same front SPC node configuration as Configuration 2.

Figure 5.3: R and r are the distance in three dimensions from the stress point A illustrated in red and the SPC nodes for the left engine bracket. B represent the SPC node in Configuration 1 and C represent the SPC node in Configuration 3.

The moment is derived by taking the cross product of the direction vector (r or R) and the reaction force

(F) in point B or C. In Eq. (5.1) the moment is derived to show the how the moment varies with the

direction vector. The positive moment directions is illustrated by the rotation arrows in Figure 3.2.

5.1. MODEL ANALYSIS CHAPTER 5. RESULTS AND ANALYSIS

R = [R

, R

, R

]

F = [F

, F

, F

]

M = R × F

(5.1)

The expression in Eq. (5.1) in matrix form is,

M =











(R

· F

) − (R

· F

) (R

· F

) − (R

· F

) (R

· F

) − (R

· F

)











(5.2)

According to Table 4.1 the SPC nodes were not constrained in the x -direction and thus, there were no reaction forces in the x -direction and Eq. (5.2) becomes;

M =











(R

· F

) − (R

· F

)

−(R

· F

) (R

· F

)











(5.3)

The components r

and R

were very small and they were almost the same in Configuration 1 and Configuration 3. Therefore, changing the configuration from r

to R

, will not influence the results. The reaction force component F

was almost the same in Configuration 1 and Configuration 3, however, the magnitude of the component F

increased in Configuration 3. The resultant of the reaction forces on the left and right engine bracket are schematically illustrated in Figure 5.4.

Figure 5.4: The cylinder block, seen from the front of the truck. The left engine bracket is illustrated in green and the right engine bracket in blue. The reaction forces starting at point B are the results from Configuration 1 and the reaction forces starting at point C are the results from Configuration 3.

The greatest moment component in point A, shown in Figure 5.3, was M

for both Configuration 1

(M

) and Configuration 3 (M

). The difference between M

for the left and the right engine

bracket, and the difference between the configurations are presented in Table 5.1.

CHAPTER 5. RESULTS AND ANALYSIS 5.1. MODEL ANALYSIS

Table 5.1: Difference between the absolute value of M

in point A for the left and right engine bracket, and between Configuration 1 and Configuration 3.

Description Left [%] Right [%]

Difference between M

and M

10.3 16

M

[%] M

[%]

Difference between Left and Right

4.37 10.4

5.1. MODEL ANALYSIS CHAPTER 5. RESULTS AND ANALYSIS

Comparison between the models

The greatest stress in the points on the upper and lower surface on the engine bracket seen in Figure 5.1 occurered in the load case where the load was applied in the negative z -direction. The presented results are the 1st Principal stress and the safety factor for different g-loads in the negative z -direction. The results for the right and left engine bracket are presented in the same figure where the results for the right engine bracket is the dashed line. The 1st Principal stress is shown in Figure 5.5.

0 2 4 6 8 10

Gravity load

1s t P ri n ci p al st re ss

Loaded in negative Z Right solid

Left solid Right spring Left spring Right linear Left linear

Figure 5.5: Trends in 1st Principal stress for the left and the right engine bracket. Nonlinear model No. 1 is illustrated in blue and Nonlinear model No. 2 is illustrated in red and Linear model with Configuration 3 in black.

Linear model with Configuration 3 was used for comparison since the SPC nodes had the same placement as Nonlinear model No. 1. This SPC node configuration corresponds to when the reaction forces acts in the isolator in Nonlinear model No. 2. The results for Nonlinear model No. 1 and Nonlinear model No.

2 were almost the same until -3g, but then the results were diverging. The stress in Linear model is just above 30 % of the stress in Nonlinear model No. 1 and 50 % of the stress in Nonlinear model No. 2.

The reaction forces in point C, seen in Figure 5.4, were compared for Linear model with Configuration 3 and Nonlinear model No. 1. The reaction forces were linear for both models. According to Table 4.2, Nonlinear model No. 1 was constrained in the x -direction. In comparison to Linear model, F

is not equal to zero in Nonlinear model No. 1, but F

was small in comparison to F

and F

. M

in the engine bracket is dependent on F

and F

according to Eq. (5.3). F

was the same for both models, but F

was considerably higher in Linear model. F

reduces the F

contribution to M

. Therefore, Linear model has a lower moment in point A (the stress point) than Nonlinear model No. 1, thus, the difference in stress. The reason to why F

was lower in Nonlinear model No. 1 may be due to the definition of the spring stiffness in the y-direction. The model is allowed to move in the y-direction, but in Linear model, the constraints in the y-direction may cause Linear model to be too rigid. Thus, creating a greater reaction force in the y-direction.

The stress results were higher for the left engine bracket for all models. The center of gravity is 4 mm closer to the left engine bracket (4 mm in the negative y-direction) in the model compared to reality and this may influence the results. The difference between the left engine bracket and right engine bracket is around 14 % for Linear model, 10 % for Nonlinear model No. 2 and 5 % for Nonlinear model No.

1. The difference between the center of gravity in the model compared to reality is very small to draw

the conclusion that this is the only factor that influence the stress differences between the left and the

right engine bracket. Design difference between the left and right engine brackets may also influence the

results.

CHAPTER 5. RESULTS AND ANALYSIS 5.1. MODEL ANALYSIS

When the isolator is compressed at -3g, the compression rubber came in contact with the top aluminum part of the isolator. This means that the reaction force point changes from being in the middle of the main rubber to the compression rubber. As described in Section 5.1, this results in a change of the reaction force components and thus, the bending moment in the stress point, shown in Figure 5.3, decreases. The stress decreases with a decreased moment.

The safety factors are shown in Figure 5.6 and it is a very big difference between Linear model and the

other two models. The safety factors for Nonlinear model No. 1 and Nonlinear model No. 2 are similar,

but in Linear model the safety factor is very high for lower loads.

5.1. MODEL ANALYSIS CHAPTER 5. RESULTS AND ANALYSIS

5 6 7 8 9 10

0 2 4 6 8

Gravity load

S af et y fac tor

Loaded in negative Z

Right solid Left solid Right spring Left spring Right linear Left linear

Figure 5.6: Safety factors for the left and the right engine bracket. Nonlinear model No. 1 is illustrated in blue and Nonlinear model No. 2 is illustrated in red and Linear model with Configuration 3 in black.

The absolute difference in safety factor at -5g is 0.56 between Nonlinear model No. 1 and Nonlinear model No. 2 which is small compared to difference in safety factor for Linear model at the same load.

According to acceptance criteria, the safety factor should be greater than 1 [22] and this is fullfilled for a load lower than -8g for all models. However, Nonlinear model No. 1 is very close to 1 at -8g since the stress is higher for this model compared to Nonlinear model No. 2 for loads greater than -3g.

It is assumed that Nonlinear model No. 2 is closer to reality due to the changes of the reaction force points. This model captures the real movements of the powertrain and the isolator when the isolator is compressed when loaded in the negative z -direction and extended when loaded in the positive z -direction.

This model is used as a reference for the comparisons. However, Nonlinear model No. 1 should correspond well to reality too since the nonlinear spring deformation comes from measurements, but the springs do not capture the moment in the isolator when loaded. Nonlinear model No. 1 is however recommended to use for loads lower than -3g in the negative z -direction, which is the most critical load case, since the stress results were the same for Nonlinear model No. 2. Nonlinear model No. 1 can be used for loads lower than -8g in the negative z -direction, but the stress is possible to be higher in the model than in reality which should be taken into account.

The reason to why Nonlinear model No. 1 is preferred to Nonlinear model No. 2 is due to the reduction of computation time when using this model. The isolator in Nonlinear model No. 2 is complicated compared to the springs in Nonlinear model No. 1. The isolator in Nonlinear model No. 2 has a rubber material which is nonlinear, contact surfaces, friction and this may result in convergence problems since the model would be more complicated for the software to solve.

If the Linear model is used, Configuration 1 would be the configuration that best correlate with the

results for Nonlinear model No. 1 and Nonlinear model No. 2. This configuration resulted in the highest

stresses and would reduce the difference in results more than using Configuration 3 that was used in the

model comparison.

CHAPTER 5. RESULTS AND ANALYSIS 5.2. TIME-DEPENDENT ANALYSIS

5.2 Time-dependent analysis

The results are presented for each strain gauge in Figure 3.9 and the model configurations are described in Table 3.5. An interval around one of the highest load amplitudes with the corresponding strain results in the measured time signal was used in the analysis. The simulated strain results () in time (t) were compared to measured strain results and the results for each strain gauge are presented with the same axis limits and scaling, note that the measured result is the same for each strain gauge, for all its configurations. Configuration 1 can be seen as the original model result for each strain gauge which the other configurations can be compared to. The strain results for the strain gauge TKX71 are presented in Figure 5.7.

t

0

TKX71

Simulation

(a)

t

0

TKX71

Simulation

(b)

t

0

TKX71

Simulation

(c)

t

0

TKX71

Simulation

(d)

Figure 5.7: Results for TKX71, (a) is the configuration where the length of the strain gauges was different. (b) is the configuration where the length of the strain gauges was the same.

(c) is the configuration with a decreased contact surface and (d) is the configuration with an increased contact surface.

The whole signal is not shown in Figure 5.7(a) in any of the figures containing Configuration 1 because the simulation was not completed. There was no apparent reason to why the simulation did not go through since the solution was converging and this is discussed further in Chapter 6. However, the part of the solution that was completed is still included in order to compare the different configurations.

The simulated results were very similar to the measured results, aside from magnitudes at the greatest

peaks which is smaller for all configurations. When comparing Configuration 1 and Configuration 2 in

Figure 5.7(b) it can be seen that the there is no difference at all between the results when changing

lengths of the strain gauges. Therefore, Configuration 2 which contains the full solution was preferred

when comparing with the remaining configurations. There is barely any difference between Configuration

5.2. TIME-DEPENDENT ANALYSIS CHAPTER 5. RESULTS AND ANALYSIS

2 and Configuration 3 in Figure 5.7(c), but there is a slight decrease in magnitude between Configuration 4 in Figure 5.7(d) and Configuration 2.

The strain results for the strain gauge TKX73 are presented in Figure 5.8.

t

0

TKX73

Simulation

(a)

t

0

TKX73

Simulation

(b)

t

0

TKX73

Simulation

(c)

t

0

TKX73

Simulation

(d)

Figure 5.8: Results for TKX73, (a) is the configuration where the length of the strain gauges was different. (b) is the configuration where the length of the strain gauges was the same.

(c) is the configuration with a decreased contact surface and (d) is the configuration with an increased contact surface.

When comparing the results for strain gauge TKX73, there is no difference between Configuration 1 in

Figure 5.8(a) and Configuration 2 in Figure 5.8(b) when changing lengths of the strain gauges. However,

there is a difference between Configuration 2 and Configuration 3 in Figure 5.8(c), the magnitude in

Configuration 3 has increased. There is also a difference in magnitude in Configuration 4 in Figure 5.8(d),

but in this case the magnitude decreased. Furthermore, the simulated strain results were very similar to

the measured strain results.

CHAPTER 5. RESULTS AND ANALYSIS 5.2. TIME-DEPENDENT ANALYSIS

The strain results for the strain gauge TKX74 are presented in Figure 5.9.

t

0

TKX74

Simulation

(a)

t

0

TKX74

Simulation

(b)

t

0

TKX74

Simulation

(c)

t

0

TKX74

Simulation

(d)

Figure 5.9: Results for TKX74, (a) is the configuration where the length of the strain gauges was different. (b) is the configuration where the length of the strain gauges was the same.

(c) is the configuration with a decreased contact surface and (d) is the configuration with an increased contact surface.

For strain gauge TKX74, there is no difference between Configuration 1 in Figure 5.9(a) and Configuration

2 in Figure 5.9(b). There is a slight increase in magnitude in Configuration 3 in Figure 5.9(c) and there

is a decrease in Configuration 4 in Figure 5.9(d). Furthermore, the magnitude for all configurations have

a magnitude lower than the measured strain amplitudes.