,
STOCKHOLM SWEDEN 2020
Modelling of Automotive
Suspension Damper
VENKATA DINESH RAJU JONNALAGADDA
SAURABH VYAS
Modelling of Automotive
Suspension Damper
Venkata Dinesh Raju Jonnalagadda
Saurabh Vyas
Master of Science in Engineering
Master programme in Vehicle Engineering KTH Royal Institute of Technology
Supervisors at Volvo Cars: Mohit Asher, Johan Hultqvist Supervisor at KTH: Lars Drugge
Examiner at KTH: Lars Drugge
Date of presentation: 23rd October 2020 TRITA-SCI-GRU 2020:350
We would like to thank our thesis advisors Mohit Asher and Johan Hultqvist at Volvo Cars and Assoc. Prof. Lars Drugge from Engineering Mechanics department at Royal Institute of Technology. They were always available to guide us during the thesis work. Despite their huge contribution, they allowed this to be our independent work and supported us with the required resources. We would also like to extend our gratitude to the following experts who were involved in helping us during the modelling, testing, implementation and validation process for this thesis:
• Carl Sandberg, Manager Vehicle Dynamics CAE, Volvo Cars • Ajay Daniel, Vehicle Dynamics Test Engineer, Volvo Cars
Without their contribution and input, the testing at Volvo Hällered Proving Ground could not have been successfully conducted.
Along with this, we are also grateful to VI-Grade GmbH, for providing us with the required software and Volvo Cars Group for presenting us this great opportunity.
A
BSTRACTA hydraulic damper plays an important role in tuning the handling and comfort characteristics of a vehicle. Tuning and selecting a damper based on subjective evaluation, by considering the opinions of various users, would be an inefficient method since the comfort requirements of users vary a lot. Instead, mathematical models of damper and simulation of these models in various operating conditions are preferred to standardize the tuning procedure, quantify the comfort levels and reduce cost of testing. This would require a model, which is good enough to capture the behaviour of damper in various operating and extreme conditions.
The Force-Velocity (FV) curve is one of the most widely used model of a damper. This curve is implemented either as an equation or as a look-up table. It is a plot between the maximum force at each peak velocity point. There are certain dynamic phenomena like hysteresis and dependency on the displacement of damper, which cannot be captured with a FV curve model, but are required for better understanding of the vehicle behaviour.
This thesis was conducted in cooperation with Volvo Cars with an aim to improve the existing damper model which is a Force-Velocity curve. This work focuses on developing a damper model, which is complex enough to capture the phenomena discussed above and simple enough to be implemented in real time simulations. Also, the thesis aims to establish a standard method to parameterise the damper model and generate the Force-Velocity curve from the tests performed on the damper test rig. A test matrix which includes the standard tests for parameterising and the extreme test cases for the validation of the developed model will be developed. The final focus is to implement the damper model in a multi body simulation (MBS) software.
The master thesis starts with an introduction, where the background for the project is de-scribed and then the thesis goals are set. It is followed by a literature review in which few advanced damper models are discussed in brief. Then, a step-by-step process of developing the damper model is discussed along with few more possible options. Later, the construction of a test matrix is discussed in detail followed by the parameter identification process. Next, the validation of the developed damper model is discussed using the test data from Volvo Hällered Proving Ground (HPG). After validation, implementation of the model in VI CarRealTime and Adams Car along with the results are presented. Finally the thesis is concluded and the recommendations for future work are made on further improving the model.
En hydraulisk stötdämpare spelar en viktig roll för att fordonets hantering och komfort. Att justera och välja en stötdämpare baserat på subjektiv utvärdering, genom att beakta olika an-vändares åsikter, skulle vara en ineffektiv metod eftersom användarnas komfortkrav varierar mycket. Istället föredras matematiska modeller av stötdämpare och simulering av dessa modeller under olika driftsförhållanden för att standardisera inställningsförfarandet, kvantifiera komfort-nivåerna och minska testkostnaden. Detta skulle kräva en modell som är tillräckligt bra för att fånga upp stötdämparens beteende under olika drifts- och extrema förhållanden.
Force-Velocity (FV) -kurvan är en av de mest använda stötdämparmodellerna. Denna kurva implementeras antingen som en ekvation eller som en uppslagstabell. Det är ett diagram som redovisar den maximala kraften vid varje maxhastighetspunkt. Det finns vissa dynamiska fenomen som hysteres och beroende av stötdämparens förskjutning, som inte kan fångas med en FV-kurvmodell, men som krävs för att bättre förstå fordonets beteende.
Denna avhandling genomfördes i samarbete med Volvo Cars i syfte att förbättra den befintliga stötdämparmodellen som är en Force-Velocity-kurva. Detta arbete fokuserar på att utveckla en stötdämparmodell, som är tillräckligt komplex för att fånga upp de fenomen som diskuterats ovan och tillräckligt enkel för att implementeras i realtidssimuleringar. Avhandlingen syftar också till att upprätta en standardmetod för att parametrisera dämparmodellen och generera Force-Velocity-kurvan från de test som utförts på stötdämpartestriggen. En testmatris som innehåller standardtest för parametrisering och extrema testfall för validering av den utvecklade modellen kommer att utvecklas. Det sista fokuset är att implementera stötdämparmodellen i en multi-body simulation (MBS) programvara.
Examensarbetet inleds med en introduktion, där bakgrunden för projektet beskrivs och därefter definieras målen med arbetet. Det följs av en litteraturöversikt där några avancerade stötdämparmodeller diskuteras i korthet. Därefter diskuteras en steg-för-steg-process för att utveckla stötdämparmodeller tillsammans med några fler möjliga alternativ. Senare diskuteras konstruktionen av en testmatris i detalj följt av parameteridentifieringsprocessen. Därefter diskuteras valideringen av den utvecklade stötdämparmodellen med hjälp av testdata från Volvo Hällered Proving Ground (HPG). Efter validering presenteras implementeringen av modellen i VI CarRealTime och Adams Car tillsammans med resultaten. Slutligen avslutas rapporten med slutsatser från arbetet och rekommendationer för framtida arbete görs för att ytterligare förbättra modellen.
N
OMENCLATUREm : Moving mass in the damper (kg)
Kf riction: Friction stiffness (N/m)
Fviscous: Viscous friction coefficient (N/m2/sec2)
Fstatic: Static coulomb friction coefficient (N)
Ff riction: Frictional force in the damper (N)
vel : Damper input velocity (m/sec)
Kparallel: Parallel spring stiffness (N/m)
Kseries: Series spring stiffness (N/m)
Cparallel : Parallel damper coefficient (N/m/sec)
Cseries: Series damper coefficient (N/m/sec)
Vpeak: Peak velocity (m/sec)
A : Amplitude of the input excitation (m) Cr1: Damping coefficient for rebound section 1
Cr2: Damping coefficient for rebound section 2
Cr3: Damping coefficient for rebound section 3
Cc1: Damping coefficient for compression
section 1
Cc2: Damping coefficient for compression
section 2
Page
List of Tables xi
List of Figures xii
1 Thesis Introduction 1
1.1 Background . . . 1
1.2 Scope and objectives . . . 2
1.3 Project timeline . . . 3
2 Literature Review 4 2.1 Physical damper models . . . 4
2.2 Empirical damper models . . . 5
3 Model Development 9 3.1 Damper Model . . . 9
3.2 Friction Models . . . 10
3.2.1 Coulomb Friction Model . . . 11
3.2.2 Stribeck Model . . . 11
3.2.3 Simplified Stribeck Model . . . 13
3.2.4 Friction Model Comparison . . . 13
3.3 Hysteresis Study and Analysis . . . 14
3.3.1 Hysteresis plots . . . 14
3.4 Spring Models . . . 17
3.4.1 Series spring . . . 17
3.4.2 Parallel Spring . . . 17
3.5 Model initial conditions and setup . . . 22
3.5.1 Integration over Differentiation . . . 22
3.5.2 Initial Position . . . 24
4 Testing 27 4.1 Damper test rig . . . 27
4.2 Test Matrix . . . 28
4.3 Parameter Identification . . . 28
4.3.1 Standard Test Data . . . 28
TABLE OF CONTENTS
4.3.3 Generating FV curves . . . 31
4.3.4 Data fitting . . . 32
4.3.5 Effect of lateral/side forces on damper . . . 33
4.3.6 Interface for manual parameter identification . . . 33
5 Validation 35 5.1 Validation of the model . . . 35
5.1.1 Validation using test data of non-Volvo car dampers . . . 35
5.1.2 Validation at varying frequency input . . . 37
5.1.3 Validation of Volvo car damper . . . 40
5.2 Accuracy of the model . . . 41
5.2.1 Error Matrix . . . 41
5.2.2 Error plots . . . 43
6 VI-Car RealTime implementation 48 6.1 Implementation . . . 48
6.2 Simulation runs . . . 51
6.2.1 Bump test . . . 51
6.2.2 Step steer test . . . 52
6.2.3 Sine steer test . . . 54
6.2.4 Track test . . . 54
6.3 Conclusion from VI-Car RealTime . . . 56
7 Adams Car Implementation 57 7.1 Model development . . . 57
7.1.1 Series spring and damper . . . 58
7.1.2 Mass . . . 58
7.1.3 Parallel spring and primary damper . . . 59
7.1.4 Friction Model . . . 60
7.1.5 Bushings . . . 60
7.2 Damper test rig . . . 61
8 Conclusions 64 9 Future Work 66 9.1 Including additional parameters . . . 66
9.1.1 Effects of heating on damping coefficient . . . 66
9.1.2 Effect of volumetric change of the damper cylinder . . . 66
9.1.3 Effect of the side forces . . . 66
9.1.4 Effect of bushings . . . 66
9.2 Frequency selective damper (FSD) . . . 67
9.3 Advance tests in Adams Car . . . 67
9.4 Correlation of the damper parameters and the physical properties . . . 67
Bibliography 68
A Appendix 69
A.1 Validation and Error plots: Damper - 1 . . . 69
A.2 Validation plots . . . 77
L
IST OFT
ABLESTABLE Page
FIGURE Page
1.1 Force Velocity characteristic curve of a passive damper . . . 2
1.2 Jounce and Rebound Strokes . . . 2
1.3 Force-velocity lookup table . . . 2
1.4 Project timeline . . . 3
2.1 Detailed Dymola model of a damper model from AutomotiveDemos library . . . 5
2.2 Hybrid model which combines a piece-wise linear function and a hysteresis model . . 6
2.3 Polynomial and first order lag model . . . 6
2.4 Fluid stiffness as a function of velocity . . . 7
2.5 Fluid stiffness as a function of velocity . . . 7
2.6 Parametric model schematic . . . 8
3.1 Damper model layout . . . 10
3.2 Model implementation in Simulink . . . 10
3.3 Coulomb friction . . . 11
3.4 Coulomb friction with viscous effect . . . 11
3.5 Stribeck friction . . . 12
3.6 Coulomb friction with viscous effect . . . 12
3.7 Friction parameters . . . 12
3.8 Velocity and Displacement signals . . . 13
3.9 Coulomb friction . . . 14
3.10 Coulomb and stribeck friction . . . 14
3.11 Ideal FV Curve. . . 15
3.12 Ideal FV curve with measured data . . . 15
3.13 Hysteresis measurement. . . 15
3.14 Hysteresis Plot. . . 16
3.15 Hysteresis shape variation. . . 16
3.16 Shape low velocity (2.58 mmps). . . 16
3.17 Series spring implementation in Simulink model. . . 17
3.18 Linear parallel spring implementation . . . 17
3.19 Output with a linear parallel spring. . . 18
3.20 Non-linear parallel spring implementation. . . 18
3.21 Non-linear parallel spring output. . . 19
LIST OF FIGURES
3.23 Linear vs Non-linear parallel spring output. . . 20
3.24 Spring stiffness profile . . . 21
3.25 Linear spring . . . 21
3.26 Non-linear lookup spring . . . 21
3.27 Force-velocity output . . . 22
3.28 Force-displacement output . . . 22
3.29 Series spring and damper block . . . 23
3.30 Derivative of displacement signal . . . 23
3.31 Integration of velocity signal . . . 23
3.32 Parallel spring-damper block . . . 24
3.33 Parallel spring force . . . 24
3.34 Force-displacement output . . . 25
3.35 Force-velocity output . . . 25
3.36 Correct integration condition . . . 26
3.37 Corrected parallel spring force . . . 26
3.38 Corrected force-displacement result . . . 26
3.39 Corrected force-velocity result . . . 26
4.1 Damper Test Rig. . . 27
4.2 Force Velocity Displacement cross plot . . . 30
4.3 Steps involved in parameter identification . . . 30
4.4 Stitched data used to parameterize the model . . . 31
4.5 Effect of adding friction data in FV . . . 32
4.6 Comparison of FV curves . . . 32
4.7 Friction force vs Side force . . . 33
4.8 Side force on a damper . . . 34
4.9 Implementation of side force with the damper model . . . 34
4.10 GUI for manual parameter identification . . . 34
5.1 Force-Velocity plot: Damper 2 . . . 35
5.2 Force-Displacement plot: Damper 2 . . . 35
5.3 3D Force-Velocity-Displacement plot: Damper 2 . . . 36
5.4 3D Force-Velocity-Displacement plot for low frequency data . . . 36
5.5 Constant amplitude displacement . . . 37
5.6 Varying amplitude velocity signal . . . 37
5.7 FVD cross-plot after fitting . . . 38
5.8 Varying amplitude displacement . . . 38
5.9 Constant amplitude velocity signal . . . 38
5.10 FVD cross-plot for new input . . . 39
5.11 Force-Velocity comparison . . . 39
5.12 Force-Displacement comparison . . . 39
5.13 FV plot for Volvo damper . . . 40
5.15 FVD Cross plot of a Volvo damper model for a standard test . . . 40
5.16 FV plot - friction test . . . 41
5.17 FD plot - friction test . . . 41
5.18 FV plot Test 4 . . . 42
5.19 FD plot Test 4 . . . 42
5.20 FV plot for Test 8 . . . 42
5.21 FD plot for Test 8 . . . 42
5.22 FVD cross-plot - Standard test . . . 43
5.23 Error cross-plot - Standard test . . . 43
5.24 Error force-displacement plot . . . 44
5.25 Error force-velocity plot . . . 44
5.26 FVD cross-plot . . . 45 5.27 Error cross-plot . . . 45 5.28 Force-displacement plot . . . 45 5.29 Error-displacement plot . . . 45 5.30 Force-velocity plot . . . 46 5.31 Error-velocity plot . . . 46 5.32 FVD cross-plot . . . 46 5.33 Error cross-plot . . . 46
6.1 VI-Car RealTime implementation . . . 49
6.2 VI-Car RealTime implementation . . . 49
6.3 Results with different time steps . . . 50
6.4 Results from initial test . . . 51
6.5 Front left wheel - FV plot . . . 51
6.6 Front left wheel - FV plot . . . 51
6.7 Chassis displacement from bump test . . . 52
6.8 Front right wheel - FV plot . . . 53
6.9 Rear right wheel - FV plot . . . 53
6.10 Chassis displacement - step steer . . . 53
6.11 Chassis acceleration - step steer . . . 53
6.12 FV crossplot for sine steer . . . 54
6.13 FV crossplot for sine steer . . . 54
6.14 Front left crossplot - track test . . . 55
6.15 Rear left crossplot - track test . . . 55
6.16 Chassis displacement from track test . . . 55
6.17 Operation region . . . 56
7.1 Damper Model Layout . . . 57
7.2 Damper Model in Adams Car . . . 57
7.3 Series Spring . . . 58
7.4 Series Damper . . . 58
LIST OF FIGURES
7.6 Spline lookup for non-linear parallel spring . . . 59
7.7 Parallel Spring . . . 59
7.8 Primary Damper . . . 59
7.9 Friction Model . . . 60
7.10 Spline lookup table for friction model . . . 60
7.11 Top and bottom mount bushings . . . 61
7.12 Damper model implementation with the damper test rig model in Adams Car . . . 62
7.13 Test setup with the damper test rig model in Adams Car . . . 62
7.14 Comparison of Adams Car and Simulink models with Force Velocity curve . . . 63
8.1 Improvement in the damper model compared to a FV curve . . . 64
A.1 Force-Velocity: Standard test . . . 69
A.2 Force-Displacement: Standard test . . . 69
A.3 Error-Velocity: Standard test . . . 69
A.4 Error-Displacement: Standard test . . . 69
A.5 Force-Velocity: 40mm test . . . 70
A.6 Force-Displacement: 40mm test . . . 70
A.7 Error-Velocity: 40mm test . . . 70
A.8 Error-Displacement: 40mm test . . . 70
A.9 Force-Velocity: 30mm test . . . 71
A.10 Force-Displacement: 30mm test . . . 71
A.11 Error-Velocity: 30mm test . . . 71
A.12 Error-Displacement: 30mm test . . . 71
A.13 Force-Velocity: 25mm test . . . 72
A.14 Force-Displacement: 25mm test . . . 72
A.15 Error-Velocity: 25mm test . . . 72
A.16 Error-Displacement: 25mm test . . . 72
A.17 Force-Velocity: 15mm test . . . 73
A.18 Force-Displacement: 15mm test . . . 73
A.19 Error-Velocity: 15mm test . . . 73
A.20 Error-Displacement: 15mm test . . . 73
A.21 Force-Velocity: 5mm test . . . 74
A.22 Force-Displacement: 5mm test . . . 74
A.23 Error-Velocity: 5mm test . . . 74
A.24 Error-Displacement: 5mm test . . . 74
A.25 Force-Velocity: 2mm test . . . 75
A.26 Force-Displacement: 2mm test . . . 75
A.27 Error-Velocity: 2mm test . . . 75
A.28 Error-Displacement: 2mm test . . . 75
A.29 Force-Velocity: 1mm test . . . 76
A.30 Force-Displacement: 1mm test . . . 76
A.32 Error-Displacement: 1mm test . . . 76
A.33 Force-Velocity plot: Damper 3 . . . 77
A.34 Force-Displacement plot: Damper 3 . . . 77
A.35 Force-Velocity plot: Damper 4 . . . 77
A.36 Force-Displacement plot: Damper 4 . . . 77
A.37 Force-Velocity plot: Damper 5 . . . 78
A.38 Force-Displacement plot: Damper 5 . . . 78
A.39 Force-Velocity plot: Damper 6 . . . 78
A.40 Force-Displacement plot: Damper 6 . . . 78
A.41 Force-Velocity plot: Damper 7 . . . 79
A.42 Force-Displacement plot: Damper 7 . . . 79
A.43 Force-Velocity plot: Damper 8 . . . 79
A.44 Force-Displacement plot: Damper 8 . . . 79
A.45 Force-Velocity plot: Damper 9 . . . 80
A.46 Force-Displacement plot: Damper 9 . . . 80
A.47 Force-Velocity plot: Damper 10 . . . 80
A.48 Force-Displacement plot: Damper 10 . . . 80
A.49 Force-Velocity plot: Damper 11 . . . 81
C
H A P T E R1
T
HESISI
NTRODUCTION1.1 Background
Vehicle manufacturers such as Volvo have a great reputation for manufacturing comfortable vehicles with predictable characteristics. A large role in achieving these standards is played by vehicle simulations. These vehicle simulations allow engineers to design and optimize vehicle performance faster. With improved computational capabilities, today these simulations can provide more information and can be used for various applications.
Simulations play an important role in vehicle design. So, accurate models which have a good correlation with the real-world results are desired. To achieve this, the vehicle components are divided into systems and sub-systems, which can be modelled individually. This is done to simplify the problem. This allows the developed model to be isolated from the effects from other vehicle system interactions and understand their limitations. However, on the other hand one limitation of this approach is the loss of installation effects of the component in the sub-system.
A shock-absorber, dampens oscillations produced by the spring or any other excitation from the vehicle. The damping force can be related to the relative velocity between the upper and lower mounts of a damper or sprung and unsprung mass between which the damper is mounted (1). So, a Force-Velocity plot describes the behaviour of the damper as shown in the Figure 1.1. A damper has two strokes: Compression stroke or Jounce and Expansion stroke or Rebound. ’Jounce’ occurs when the top and bottom mounts of the damper move towards each other and ’Rebound’ occurs when the top and bottom mounts move away from each other. In other words, jounce occurs when the compression of damper takes place and the rebound occurs when the damper expands as shown in Figure 1.2.
Figure 1.1: Force Velocity characteristic curve of
a passive damper Figure 1.2: Jounce and Rebound Strokes
regions.
Figure 1.3 shows the commonly used damper model, this works as a lookup table with velocity as the input producing the corresponding force. This model is generally created using maximum velocity - force data points from damper measurement data. Each circle in the plot represents the force at maximum velocity of a test which are then interpolated to get a curve.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Velocity (m/sec) -2000 -1000 0 1000 2000 3000 4000 Damper force (N)
Generic force-velocity curve
Figure 1.3: Force-velocity lookup table
This model is not sufficient to include the characteristics of a damper completely. Being generated using a small dataset this lookup method does not hold well in other damper operation conditions. This requires a more advanced model which can make use of the information from the lookup table and add certain dynamic characteristics which make the model valid for a larger operation region.
1.2 Scope and objectives
CHAPTER 1. THESIS INTRODUCTION
• Development of a damper model which exhibits force-velocity and force-displacement char-acteristics similar to real hydraulic shock absorbers.
• Collect measurement data for various dampers and develop a method to identify model parameters.
• Development of a test matrix, including the measurement data required from manufacturers to parametrize the model.
• Implementation of the model in ADAMS/Car and VI Grade Car Realtime for CAE simula-tions.
These objectives define the goals for the project. However, each target is accompanied with constraints. The model should be real-time capable making it essential to balance complexity with solve time. Similarly, the test matrix used for parameter identification should be limited and practical, since this will be used by manufacturers to provide the measurements.
1.3 Project timeline
Before beginning the project, it is essential to create a project timeline with smaller milestones and targets. This was done using the project management software - Projectplace.
Figure 1.4 shows the timeline created for the project. This includes the main tasks with sub-tasks and important dates such as mid-term presentations and final presentation. Management plays a very important role in projects and need constant supervision and updates in the time-plan when required. To ensure that the time-frame for each task was sufficient a weekly review with the thesis supervisors was conducted to update and discuss the project progress.
2
L
ITERATURER
EVIEWDamper models can be categorised into two major categories: Physical models and Empirical models. To get a better idea on existing models and understand the improvements required, a literature review is carried out in the initial weeks of the project. In this chapter, different damper modelling techniques and their limitations will be discussed and a suitable model type for this thesis work will be selected.
2.1 Physical damper models
Physical damper modelling involves representing each component and phenomenon, that effects the damper behaviour, individually in the model. A damper behaviour is ideally influenced by the physical elements and processes in it (2) such as:
• Valve flow characteristics • Oil characteristics • Seal friction • Piston • Gas force
• Expansion of the damper cylinder
• Influence of thermodynamic properties such as density with change in heat
Figure 2.1 shows a damper model in Dymola, developed using libraries provided by Modelon (2). The problem with these kind of models is complexity. Parameterisation of the damper would be a tedious process. It would be difficult to accurately formulate the thermodynamic influences on the cylinder, piston material density, oil density and the remaining components of a damper. Also, physical models would consume a lot of computational power in real-time simulations, because of complexity of the model.
CHAPTER 2. LITERATURE REVIEW
Figure 2.1: Detailed Dymola model of a damper model from AutomotiveDemos library
2.2 Empirical damper models
Empirical modelling in contrast to physical modelling does not need all the information about the system. This is generally represented with equations which produce the output for a certain input similar to the real-world but does not necessarily explain much about the system by the way it is modelled. A good example of an empirical model is Pacejka’s Magic Formula for tyre data. This includes a set of equations which represent the tyre data correctly and in certain simulations this is enough. However, this method requires a lot of input measurement data and is not reliable outside the range of the measurement data. Some empirical methods are studied and discussed in this section.
An ideal FV curve model, as shown in Figure 1.3, is a non-linear curve. In this curve, the positive force represents jounce and the negative force represents rebound stroke (2). An actual damper deviates from the FV curve and produces hysteresis as shown in Figure 1.1. Hysteresis is the vertical or the horizontal gap between the jounce and rebound strokes. So, a realistic damper model should have the non-linearity of a FV curve, along with the hysteresis. To capture the non-linearity, Barethiye V et al. (3) has proposed a method, which uses a piece-wise linear function model for capturing the non linearity and a neural network model to capture the hysteresis. The Equations 2.1 and 2.2, represents the rebound and the compression strokes respectively of a piece-wise linear function.
A hybrid model which combines the piece-wise linear function and the hysteresis model is developed as shown in the Figure 2.2 (3)
Figure 2.2: Hybrid model which combines a piece-wise linear function and a hysteresis model This model was accurate enough to capture most of the damper properties. The problem with this model is, there is no correlation between the obtained coefficients and the actual damper. The hysteresis model, which is a neural network model, can produce the hysteresis accurately in most of the cases. However if some design study has to be undertaken, there would be no parameter to tune, so that the result of tuning and the effect of parameters can be understood. This hybrid model is less intuitive compared to a physical model.
Some empirical models introduce methods to add hysteresis using a 1st order lag term to induce delay which effectively in a cross-plot results in hysteresis. One such method is discussed in (4), where a polynomial function is first used to define the base force-velocity curve and a first order filter is applied later to add hysteresis, as shown in Figure 2.3.
Figure 2.3: Polynomial and first order lag model
Here the fluid stiffness if first viewed as a function of velocity, higher lag at low velocity and no lag at high velocity shows that the fluid can be considered stiffer at higher velocity and less stiffer at lower velocities (4). This can be seen in Figure 2.4.
From Figure 2.4 it can be seen that the time constant changes suddenly as a step function, this can cause unstable results. To account for this a ramp function can be used as shown in Figure 2.5.
This concept of representing the oil and gas compressibility is further used in the spring modelling of the thesis where one spring is modelled as a nonlinear spring with varying stiffness. The spring in the model will act as a delay as shown in Figure 2.3.
CHAPTER 2. LITERATURE REVIEW
Figure 2.4: Fluid stiffness as a function of velocity
Figure 2.5: Fluid stiffness as a function of velocity
C
H A P T E R3
M
ODELD
EVELOPMENT3.1 Damper Model
The model developed in this thesis has physical and empirical aspects in it. The shock-absorber is represented using spring and damper elements, which represents air and oil compressibility. From the literature review, it was observed that using a stiffness induces hysteresis in the FV curve by producing a delay in the force output with respect to the velocity input. This forms the basic idea for the model development. To differentiate between the hysteresis produced by oil and air compressions, two different springs are introduced, which are separated by a mass. Each of the spring and damper elements, gives the model certain properties such as hysteresis and slope. Figure 3.1 shows the damper layout with series spring, series damper, primary spring, parallel damper, mass and friction model. All the components are named according to their connection with the primary damper as shown in Figure 3.2
• Series Spring: Represents the compressibility of the gas in the damper. It is called series
spring since it is in series with respect to the primary damper.
• Series Damper: Added to dampens oscillations produced by series spring. This damper
component is in series with the primary damper.
• Mass: Represents the moving parts inside the damper, such as, mass of piston rod, oil and
air
• Parallel Spring: Represents the oil compressibility. This component is in parallel
connec-tion with the primary damper.
• Primary damper: It has Force-Velocity characteristic curve
• Friction Model: The entire damper friction is modelled in the friction model.
Figure 3.1: Damper model layout Figure 3.2: Model implementation in Simulink
3.2 Friction Models
Friction in a complete suspension system comes due to various physical reasons. Since the objective for this thesis is to develop a damper model and carryout component level validation the friction phenomenon included is parameterised to account for the friction in the damper alone. The friction in a damper arises from the contact between piston and cylinder walls, seals and viscous force due to the damping fluid and valve design.
It is essential to select the right friction model to include every possible dynamic characteristics and at the same time be real-time capable. When used in simulations, the friction force abruptly changes sign when crossing zero velocity. This can cause the simulation to become unstable and have incorrect results. To ensure the model’s real-time capability without becoming unstable, a low time step for the simulation is selected along with a simplified friction model. Three models were picked based on the literature review and implemented to study their pros and cons, namely:
• Coulomb friction • Stribeck model
• Simplified stribeck model
CHAPTER 3. MODEL DEVELOPMENT 3.2.1 Coulomb Friction Model
This is the most simple and widely used friction model. This type of friction is generally used to model dry surface contacts. The underlying equation for this model is as below (6)
Ff riction= Fstatic.tanh(Kf riction.vel) (3.1)
Equation 3.1 shows the basic equation used for modelling friction in Simulink. This is fairly simple to parameterize using friction tests done on a damper but do not completely represent the damper characteristics in mind and high velocity regions since it does not include the viscous effect exhibited by the fluid. Equation 3.1 is extended to include the viscous friction effect as a function of the damper velocity squared.
Ff riction= Fstatic.tanh(Kf riction.vel) + sign(v).Fviscous.vel2 (3.2)
The results produced by Equation 3.1 and 3.2 are shown in Figure 3.3 and 3.4,
-1.5 -1 -0.5 0 0.5 1 1.5 Velocity (m/s) -100 -80 -60 -40 -20 0 20 40 60 80 100 Friction force (N) Coulomb Friction Static Friction Slope - Friction Stiffness
Figure 3.3: Coulomb friction
-1.5 -1 -0.5 0 0.5 1 1.5 Velocity (m/s) -200 -150 -100 -50 0 50 100 150 200 Friction force (N)
Coulomb friction with viscous effects
Figure 3.4: Coulomb friction with viscous effect The static friction values can be used to define the max friction values (without viscous part). The slope can be defined using the friction stiffness. The viscous coefficient defines the curve at high velocities.
3.2.2 Stribeck Model
Also known as stick-slip friction, this friction model represents friction in a damper better than coulomb friction due to it’s ability to capture higher order dynamics at very low velocity. This increases the complexity of the model and to have a stable solution the time-step has to be small. The equation defining a stribeck model are derived from (7),
Ff riction= Fcoulomb− Fstatic− Ft.(1 −|vel/velf s|c)d+ Fviscous.vel2.sign(vel) (3.3)
Ff riction= Fcoulomb− Fstatic− Fcoulomb.(1 −|vel/vels|c)d+ Fviscous.vel2.sign(vel) (3.4)
velocity. These equations are used to run the Simulink model. From Figure 3.5 it can be observed that the low velocity characteristics include a characteristic where the force spike occurs at zero-velocity crossing, this is not seen in the Coulomb friction model.
-1.5 -1 -0.5 0 0.5 1 1.5 Velocity (m/s) -80 -60 -40 -20 0 20 40 60 80 Friction force (N) Stribeck friction Fs Ft a, c, d
Figure 3.5: Stribeck friction
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Velocity (m/s) -80 -60 -40 -20 0 20 40 60 80 Friction force (N) Stribeck friction Vfs
Figure 3.6: Coulomb friction with viscous effect The magnitude of this spike can be controlled using the friction transition coefficient (Ft) and the static friction (Fs) value defines the value to which the force drops to. The a, c and d coefficients define the shape of the curve in the region where the force has an overshoot. Figure 3.6 shows the break-away velocity (Vfs) which defines the velocity range within which the friction force must reach the defined transition value and return to the static friction. Similarly, Vs in Eq.(3.4) is defined to control the shape of the curve after the break-away velocity (Vfs). In order to understand the effect of each parameter better a simple GUI with sliders for each parameter is made as shown in Figure 3.7. These GUIs are also used to find a reasonable initial guess to begin the parameter identification.
CHAPTER 3. MODEL DEVELOPMENT 3.2.3 Simplified Stribeck Model
From the equations and the model in the previous section, it can be seen that this stick-slip model although being more flexible increases the complexity by increasing the number of unknown parameters. In the coulomb model three parameters are to be identified, whereas in the stribeck model nine parameters need to identified which increases the complexity. To find a good balance between accuracy and complexity a third model is implemented, this is a simplified version of the stribeck model. The underlying equation can be found from (6),
Ff riction=�2.e .(Fbrk− Fc).exp(−(vel/velst)2).vel/velst+ Fc.tanh(vel/velcoul) + Fv.vel2 (3.5)
vst= vbrk.�2 (3.6)
vcoul= vbrk/10 (3.7)
These equations produce results similar to the full stribeck model but since Equation 3.5 define constants, the number of variables to be identified reduces to four parameters which is just one more than Coulomb friction and also includes the dynamic characteristics at low velocities.
Similar to the stribeck model this simplified version has a break-away velocity which defines the velocity at which the transition friction (Fbrk) is reached and returned to static friction (Fc). The fine tuning coefficients which define the shape of the overshoot are not available in this model making it more rigid but the result comparison in the following sub-section shows that this is better than the Coulomb model without that added complexity.
3.2.4 Friction Model Comparison
To make some initial observations a data-set for friction test conducted on the rig was used. Friction tests for dampers are conducted at low frequencies at very low velocities since friction is more dominant in this region. Figure 3.8 shows the velocity and displacement measurement signals from the test rig.
0 1 2 3 4 5 6 7 Time (m/s) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Velocity (m/sec) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Displacemnt (m)
Low Velocity Test
Figure 3.8: Velocity and Displacement signals
profile from friction test is run through the model with friction and a simple spring for hysteresis, which is elaborated upon in the next section on springs. Force is plotted against velocity in a cross-plot as shown in Figure 3.9 and 3.10.
Figure 3.9: Coulomb friction Figure 3.10: Coulomb and stribeck friction
From the results it can be seen that the real measurement data features characteristics that the coulomb model can not account for due to the model limitation, however, with the same base parameters the simplified-stribeck model shows the effects required at low speeds.
A similar trend is observed in other test data, since the simplified stribeck model also produces a result with better low-speed characteristics and given that it requires only four parameters this model is considered in the following text and models.
3.3 Hysteresis Study and Analysis
As discussed in section 2.2, the actual behaviour of a damper deviate from an ideal FV characteristics and produce hysteresis, which is the vertical or the horizontal gap between the jounce and rebound strokes. Figure 3.11 shows the desired FV characteristic and in Figure 3.12 the blue curve shows the measured data from a damper test rig.
The factors which contribute to hysteresis are air compressibility, oil compressibility and volumetric expansion of the damper cylinder. The volumetric expansion of the damper cylinder is ignored as its effect on damper behaviour is negligible.
3.3.1 Hysteresis plots
In the model, since the springs account for the hysteresis, it is important to study the hysteresis characteristics to decide on the type of springs. To understand how the hysteresis varies during jounce and rebound strokes, a hysteresis plot is made by considering the vertical difference between the force at each velocity point as marked in Figure 3.13.
CHAPTER 3. MODEL DEVELOPMENT
Figure 3.11: Ideal FV Curve. Figure 3.12: Ideal FV curve with measured data
Figure 3.13: Hysteresis measurement.
velocities and it gradually reduces as the velocity increases. Also, it can be observed that the increase and decrease in the hysteresis value is not linear. The hysteresis value reduces with reduce in peak velocities at particular amplitudes and is not symmetric about the y-axis. So it is difficult to say which non-linear curve can fit this kind of data.
Further, to clearly understand the shape of hysteresis variations, the area under each of jounce and rebound parts of FV curve are plotted. The area under the FV curve is an accurate measure of hysteresis. This area is obtained by integrating the force with respect to velocity. Figure 3.15 shows how the hysteresis value changes with velocity. In Figure 3.16, it can observed the hysteresis variations at very low velocity of 2.58 mm/s. These plots form basis for the non-linear springs, which will be further discussed in the sub-section 3.4.2.2.
Figure 3.14: Hysteresis Plot.
CHAPTER 3. MODEL DEVELOPMENT
3.4 Spring Models
3.4.1 Series spring
As mentioned in section 3.1, a series spring in the damper model represents the air compress-ibility. Since the impact of air compression in a damper is almost negligible, a very high spring stiffness value is introduced for the series spring. This makes the series spring stiff and contribute less for the deviation from the ideal FV. The high stiffness series spring is implemented in the model as shown in Figure 3.17. The velocity input is given to the model which is integrated to get the displacement and then multiplied with series spring stiffness (p.kir) to get the force output of the series spring.
Figure 3.17: Series spring implementation in Simulink model.
3.4.2 Parallel Spring
The parallel spring is used to represent the oil compressibility, since the effect of oil compress-ibility is very high compared to air in a hydraulic damper. So this spring should account for most of the hysteresis. The parallel spring has been experimented with different springs such as, linear spring, non-linear spring of various types, and look-up tables which are discussed in this section. 3.4.2.1 Linear Parallel Spring
To start with, a linear spring is used as a stiffness which adds hysteresis to the model. The spring is implemented in the model as shown in Figure 3.18. The velocity output from the mass is integrated and multiplied with parallel spring stiffness (p.kpr) to get the force output from the parallel spring.
Figure 3.18: Linear parallel spring implementation
The model produces the output as shown in Figure 3.19 with a linear spring. The blue curve represents the measurement data and the model output is shown with the orange curve.
3.4.2.2 Non-Linear Parallel Spring
Figure 3.19: Output with a linear parallel spring.
CHAPTER 3. MODEL DEVELOPMENT
In this case, the input is same as in Figure 3.18. The velocity is integrated and then squared to get the displacement squared. It is then multiplied with a ’sign’ function in-order to keep the positive and negative signs of displacement. This whole term is then multiplied with the non-linear parallel spring stiffness ’K’ to get the force output of the non-linear parallel spring. The result produced using the non-linear parallel spring is shown in Figure 3.21
Figure 3.21: Non-linear parallel spring output.
3.4.2.3 Linear Vs Non Linear Spring
Further, the output from linear and non-linear springs are compared to check if the non-linear springs improves the model. The output of both the models is compared with measurement data in Figure 3.22.
From Figure 3.22 it can be seen that both linear and non-linear springs capture the damper properties well in a Force-Velocity-Displacement cross plot. But, on closer observation, in Figure 3.23, it can be observed that non-linear spring gives a better output compared to linear spring because of it’s ability to produce varying offset from a FV curve. Changing the non-linear spring from K ∗ x2to K ∗ x3or K ∗ x4did not improve the output. So, K ∗ x2has been used for the parallel
spring in the final model.
3.4.2.4 Non-Linear Parallel Spring - Look-up tables
Figure 3.22: Linear vs Non-linear parallel spring
output. Figure 3.23: Linear vs Non-linear parallel springoutput.
equations as mentioned in the previous section. This lookup table method provides more flexibility and is unique for every damper since it is made using damper data.
The hysteresis is measured as the difference in the force value between compression and rebound for the same velocity points as described in sub-section 3.3.1. The results for this can be seen in Figure 3.14 where the hysteresis is higher at low velocities and decreases as the velocity increases. It is also essential to note that the magnitude of the hysteresis curve over the velocity range changes based on the frequency of the test. However, the shape of the curve is very similar for all frequencies, this means for every frequency the hysteresis is maximum at zero velocity and reduces with increasing velocity in a non-linear fashion. This variation is not well defined since it changes from damper to damper. Since the parallel spring is a direct contributor to hysteresis the shape of the hysteresis plot can be used to define the spring’s stiffness profile.
CHAPTER 3. MODEL DEVELOPMENT
Figure 3.24: Spring stiffness profile
Figure 3.24 shows the hysteresis as Y-offset in the loading and unloading cycle. The advantage of using this method over higher power springs is the unsymmetrical spring behaviour which is derived from a damper test. This makes the spring look-up more accurate since it is made from measured data for every damper. The Simulink implementation is shown in Figure 3.25 and 3.26,
Figure 3.25: Linear spring Figure 3.26: Non-linear lookup spring
The Simulink models show the implementation with a linear spring gain and the lookup table, with the changes marked in circles.
Figure 3.27: Force-velocity output Figure 3.28: Force-displacement output against measurement data from the test rig. The linear spring has a constant spring stiffness which does not allow the hysteresis to reduce in the non-linear fashion, the output from the non-linear spring shows the effect of varying stiffness since the hysteresis changes with velocity. Figure 3.28 shows a zoomed view of Figure 3.27 to show the effect of the two outputs better against the measured data.
Although the model has more flexibility in terms of non-symmetric spring derived from the damper test data, the k-x squared formulation is used since it is easier to run and identify the value for. Being non-generic it is also difficult to assume this for dampers that are still being designed. The look-up table generation requires more test data and processing for which a more efficient method can be developed in future work.
3.5 Model initial conditions and setup
This section explains certain decisions taken while developing the model and how to set it up correctly with initial conditions to get correct results. Initial conditions are essential in order to get useful results which will be discussed in the following subsections.
3.5.1 Integration over Differentiation
CHAPTER 3. MODEL DEVELOPMENT
Figure 3.29 shows the block architecture where the input velocity is used and integrated to get the input displacement. The difference between the two methods is that differentiating a signal is similar to predicting the future state whereas integration uses the current state to get the other state. In this case the velocity determines the displacement for the current step, while differentiating the displacement we use the rate of change of displacement to get a velocity which can cause spikes and unrealistic results at the beginning of the simulations and at zero values of displacement.
Figure 3.29: Series spring and damper block
Figure 3.30 shows the results comparing the measured velocity signal from the test rig and the derivative of the displacement signal from measurement.
0 0.05 0.1 0.15 0.2 0.25 Time (sec) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Velocity (m/sec)
Derivative of displacement vs measured velocity
Measured velocity Derivative of displacemnt
Figure 3.30: Derivative of displacement signal
0 0.05 0.1 0.15 0.2 0.25 Time (sec) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Displacement (meter)
Integration of velocity vs measured displacement
Measured displacement Integration of velocity
It can be seen from the plot that the signal produced after differentiating produces unwanted noise in the input signal and can cause the simulation to become unstable in certain conditions. The velocity curve from the measurement is a smooth curve shown in blue in Figure 3.30 and the red signal denotes the derivative output. Similarly, Figure 3.31 shows the displacement signals where the blue signal represent the measurement signal and the red dashed line denotes the output after integrating the measured velocity. Here it can be seen that the result from integration is better and perfectly overlays on the measurement signal without any oscillations. This shows that the integration block is better in these models instead of differentiating signals. Therefore, integration of velocity is chosen over differentiating the displacement to calculate the velocity since integration is more stable and robust in simulations and real-time application.
3.5.2 Initial Position
From the previous subsection it can be concluded that the integration block is a better option so the measured velocity is used and integrated to get the displacement to use in this model. However, it is important to note that integration blocks need the correct initial conditions set in order to produce correct results. The results in Figure 3.31 were produced by setting the initial condition in the integration block to 50mm (0.05m) since that is the initial position of the damper. If the initial condition is set to zero, the displacement plot would be shifted down by 0.05 m starting from zero. The simulation would still run but the results obtained would not be correct. This subsection discusses the effects of having incorrect initial conditions and how it should be set to avoid incorrect results.
Figure 3.32 shows the parallel block with the spring and damper part as a gain and a force-velocity lookup table. The force-velocity of the connecting mass is integrated here to find the force output from the parallel spring. For simplicity the linear spring is used here to demonstrate the effect of incorrect initial integration conditions. Figure 3.33 shows the force produced by the parallel spring, when the initial condition is set to zero the displacement is negative which produces an entirely negative spring force.
CHAPTER 3. MODEL DEVELOPMENT
This spring force contributes to the hysteresis produced in the force-velocity crossplot output from the simulations. If the parallel spring force is always negative it will only add hysteresis below the standard FV curve instead of distributing the hysteresis equally on either sides.
Figure 3.34 and 3.35 show the force-displacement (FD) and force-velocity (FV) results for increasing value of parallel spring stiffness. The spring stiffness are in ascending order denoted by blue, red and yellow lines respectively. Due to the addition of only negative spring force the force displacement has an x-axis offset and with the variation of the stiffness only the left-hand side of the plot is affected with numerically reduced force values.
Figure 3.34: Force-displacement output Figure 3.35: Force-velocity output Similarly, the hysteresis variation shown in Figure 3.35 can be directly linked to the result from Figure 3.33 since the x-axis is velocity for both plots. As the stiffness is increased the negative spring force increases and adds the hysteresis only under the reference F-V plot. The arrows in the figure denote the direction in which the hysteresis increases with spring stiffness. This is not correct since the force-velocity results of a damper can not be matched if the hysteresis is added only on one side of the reference force-velocity line.
In order to resolve this issue, the initial condition in the integration block must be set to the initial position of the damper piston. In the damper rig this can be directly seen from the displacement signal, when implementing in an actual vehicle this has to be calculated by finding out the position of the damper piston when in static-loaded condition. Since the damper data from a test rig is being used, the initial displacement value from the signal is used as the initial condition for the integration block.
Figure 3.36 and 3.37 show the results with the correct initial conditions and how the hysteresis is now evenly distributed on either sides of the reference F-V line.
When the conditions are set correctly the parallel spring force is centered on the y-axis at zero, this shows that the spring force will be added correctly to the final results. This can be seen in Figure 3.37.
Figure 3.36: Correct integration condition Figure 3.37: Corrected parallel spring force
Figure 3.38: Corrected force-displacement result Figure 3.39: Corrected force-velocity result by adding on either sides.
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ESTING4.1 Damper test rig
The developed model is to be parameterised and validated with the data from real-time tests. The damper test rig is most widely used to perform different tests on an actual damper. A damper test rig (8), as shown in Figure 4.1, typically consists of a mechanical structure, lower and upper grippers which hold the damper between them. Bushings may or may not be included during testing. Bushings, if included with the damper during the test, produces additional damping effect which must be considered in the damper model. Since the thesis focused on modelling the damper behaviour alone, bushings were not included in the tests. Either of the grippers may be used to give input to damper. Various kinds of inputs such as sine, step, triangular, or even the actual road profile can be given in the damper test rig.
Figure 4.1: Damper Test Rig.
damper test rig at Volvo Hällered Proving Ground (HPG) has been used.
4.2 Test Matrix
The parameterisation and validation of the damper model becomes more accurate as the number of tests are increased. At the same time, it will increase the complexity at both the testing phase as well as the parameterization phase. While testing in the rig, at high frequencies, the damper gets heated because of the friction inside the damper. This would make the damper fluid less denser thereby changing the damping properties. If different dampers are used for testing, they may behave in a different way since it is not guaranteed that all the manufactured dampers have same tolerances and same amount of fluid in them. So, to prevent this situation, it is desired to have least number of test done without compromising on the accuracy. To estimate the optimal number of tests required and the frequency at which the test are to be performed, a test matrix is developed which covers most of the operating frequency range starting from 0.014 Hz to 25.03 Hz. The maximum and minimum frequencies for the tests are set, keeping the rig limitations in mind. The amplitude range lies between 50 mm and 1 mm. The peak velocity range is 52 mm/second and 0.4 mm/second. The frequency (f ) of particular test can be calculated using Equation 4.1
f =2 ∗Vpeak
π∗ A (4.1)
The test matrix with various combinations are shown in Table 4.1. Tests beyond frequency 25.03 Hz are out of scope for the test rig. These are marked ’-’ in the matrix. It can be observed that in each test, amplitude is kept constant and the peak velocity is changed to produce different frequencies. From ’Test 1’ to ’Test 8’, the peak velocity variations are kept constant and the amplitudes are changed from 50 mm to 1 mm. Test 9 is has low frequency cycles between 0.041 Hz and 0.006369 Hz.
Test 1 is considered to be the standard test, since, the data from this test is used to param-eterise the damper model. At high frequencies, the damper elements’ influence is much higher compared to the damper friction. Test 9 is used to parameterise friction model, since friction is predominant at low frequencies, as discussed in the above chapters. All the other tests are used for validation of the parameterized model in different regions. The force, velocity and displacement data from the standard test or the test 1 is plotted in 3D as shown in Figure 5.7. Each curve of one particular color represents one particular frequency.
4.3 Parameter Identification
To get the properties of a particular physical damper in the developed model, the model needs to be parameterised using the damper test data. The steps followed for parameter identification are shown in Figure 4.3.
4.3.1 Standard Test Data
CHAPTER 4. TESTING
Table 4.1: Test Matrix.
Test 1 Amplitude [mm] 50 50 50 50 50 50 50
(Standard Test) Frequency [Hz] 0.17 0.42 0.83 1.25 1.67 3.33 5.00
-Figure 4.2: Force Velocity Displacement cross plot
Figure 4.3: Steps involved in parameter identification
CHAPTER 4. TESTING 4.3.2 Stitching the raw data
The damper model is parameterized using the test data from the damper test rig. If all the required parameters are estimated using one particular frequency data, the model behaves well in the frequencies around which parameterisation is done. This would deteriorate the damper model behaviour at frequencies which are very different from the parameterised frequency. To prevent this particular problem, all the measurement data from one particular test set is stitched to get an input to the damper model which covers all frequencies as shown in Figure 4.4
Figure 4.4: Stitched data used to parameterize the model
4.3.3 Generating FV curves
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Velocity(m/s) -1500 -1000 -500 0 500 1000 1500 2000 2500 3000 3500 Force (N)
Friction data included Friction data excluded 2.58mm/second data
Figure 4.5: Effect of adding friction data in FV
-8 -6 -4 -2 0 2 4 6 Velocity(m/s) 10-3 -140 -120 -100 -80 -60 -40 -20 0 20 Force (N)
Friction data included Friction data excluded 2.58mm/second data
Figure 4.6: Comparison of FV curves
4.3.4 Data fitting
Once the standard test data, friction test data and FV curves are ready, the next step is to fit the model to the measurement data. For fitting the model to the data, least squares fitting is used. Least squares fitting is a mathematical procedure to find an optimal curve which minimises the residuals, which are the squares of the deviation or offset of the points from the curve. This method is used instead of just using the absolute value of the offsets, because the residuals can be treated as a quantity which is continuous differentiable (9).
The FV curve generated will be used in the ’Primary Damper’ element of the model. Using the friction test data, the four friction parameters: ’Coulomb friction’, ’Peak friction’, ’Peak friction velocity’ and ’Gas force’ are estimated. The Viscous friction, alone is estimated using the data from standard test. This is because, the viscous friction coefficient is multiplied with the velocity square as shown in the Equation: 3.5. If estimated at low values, the effect of viscous friction on the force would be very low. So, viscous friction coefficient is estimated using high frequency data, where the velocities are relatively much higher at a fixed amplitude.
CHAPTER 4. TESTING 4.3.5 Effect of lateral/side forces on damper
Along with the longitudinal forces acting along the axis, lateral/side forces as shown in Figure 4.8 also affect the behaviour of a damper. To understand the effect of side forces, data from a previously tested damper with side forces (Fside) was studied. The effect of side force is implemented as
increase in the friction force. 30 friction tests were performed on this damper, with side forces ranging from 0 N to 300 N. The result from each test and the line connecting the average of friction force at each side force is plotted as shown in Figure 4.7
Figure 4.7: Friction force vs Side force
The slope of line connecting the average of 30 tests, gives the proportionality constant kavg. It
is multiplied to the side force, Fside, and added to side force load measured at ’0’ N,(F0), to obtain
the overall friction force (Ff), as shown in Equation 4.2.
Ff = F0+ kavg∗ Fside (4.2)
The side force has been implemented in the developed damper model as shown in Figure 4.9
4.3.6 Interface for manual parameter identification
To manually fit the data to the model and find parameters, an interface in MATLAB has been developed. The linear and non-linear parallel spring, Coulomb friction model, Stribeck friction model and simplified Stribeck model and the remaining damper parameters are included in this interface as shown in Figure 4.10.
Figure 4.8: Side force on a damper
Figure 4.9: Implementation of side force with the damper model
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ALIDATION5.1 Validation of the model
5.1.1 Validation using test data of non-Volvo car dampers
The developed damper model is validated using the test data from Volvo Hällered Proving Ground, after parameterising the dampers according to the steps explained in section 4.3. This validation step would help in understanding if the current model is sufficient to capture most of the passive damper characteristics before ordering the test for Vovlo dampers. The validation plots for high frequency data, which has velocities ranging from 52 mm/second to 1571 mm/second at constant amplitude of 50 mm of a damper (Damper 2) are shown in Figures 5.1, 5.2 and 5.3. In the plots, the measurement data is represented using the blue curve and the simulation output is shown as the red curve. From Figure 5.3,it can be observed that the model captures the damper property well in most of the regions.
Figure 5.3: 3D Force-Velocity-Displacement plot: Damper 2
Next, a low frequency data (2.58 mm/second at 50 mm amplitude), is used to validate the friction model as shown in Figure 5.4. The spike in the graph is produced at the starting condition when running the simulation.
Figure 5.4: 3D Force-Velocity-Displacement plot for low frequency data
CHAPTER 5. VALIDATION 5.1.2 Validation at varying frequency input
Table 4.1 shows the test matrix which contains the standard test points conducted on the damper test rig. The matrix is read and the test are conducted according to the rows. This means the frequency is varied by changing the maximum velocity while keeping the displacement amplitude constant for a single test. However, it is essential to verify if the results would be the same if the tests were performed using test points according to columns instead. When we read the matrix column wise the maximum velocity remains fixed and the frequency is now varied by changing the maximum amplitude. Since hysteresis and the force plots heavily depend on frequency it would be interesting to investigate if it matters how the frequency is controlled. This section involves discussion including the two methods to vary frequency and comparing results for the two to find if there is any correlation.
A simulation test was run with constant maximum amplitude of 30 mm, at which the model parameters are identified. The frequency varies as the maximum velocity changes. This test can also be found in Table 4.1 named as Test 3. To understand the test better the input velocity and displacement are plotted first. Figure 5.5 and 5.6 show the input signals to the damper from Test 3 in the test matrix.
Figure 5.7: FVD cross-plot after fitting
Once verified that simulation output and the measured force signals match in the cross-plot it is verified if the same set of parameters identified from this test with constant amplitude and varying velocity would produce the same result when the velocity is made constant and the displacement is varied. A signal is generated to have such characteristics of constant maximum velocity and varying maximum displacement. The newly generated input signals are as shown Figure 5.8 and 5.9,
Figure 5.8: Varying amplitude displacement Figure 5.9: Constant amplitude velocity signal Now the model is run with these input signals with the same model parameters as found earlier from the Standard test. The FVD cross-plot is shown in Figure 5.10
CHAPTER 5. VALIDATION
Figure 5.10: FVD cross-plot for new input Figure 5.11: Force-Velocity comparison in 3D plots. To make it easier to view the 2D plot for force-velocity is shown in Figure 5.11. Here we can see that the characteristic force-velocity trend for the two cases matches very well. From Figure 5.12 we can see that because the displacement was not the same for the two simulation output this plot looks different but the force velocity plot in Figure 5.11 have a good correlation since the frequency is the same.
Figure 5.12: Force-Displacement comparison
5.1.3 Validation of Volvo car damper
Once the damper model is validated as discussed in the sections 5.1.1 and 5.1.2, test data is ordered from Volvo Hällered Proving Ground, for a Volvo car damper (’Damper 1’). All the tests as discussed in Table 4.1 are performed. As concluded from section 4.3, the damper model is parameterised with the standard test data and friction model from Test 9. After obtaining the damper and friction parameters, simulations are performed in Simulink with the similar input as given in the test rig. Figures 5.13 to 5.15 show the validation with the standard test data. The blue curve represents the test data, the red curve represents the fitted data and the FV curve is shown using the green curve. In this case the damper model captures most of the hysteresis properties, except in few velocity regions.
Figure 5.13: FV plot for Volvo damper Figure 5.14: FD plot for Volvo damper
CHAPTER 5. VALIDATION
The friction model, with the estimated parameters, is validated against the friction test data. From the results shown in Figures 5.16 and 5.17, it can be concluded that even the friction model is good enough to capture the Stribeck or stick-slip effect.
Figure 5.16: FV plot - friction test Figure 5.17: FD plot - friction test In Figures 5.13 to 5.15, the model is parameterised using the standard test data and validated against the same data. So, the result is expected to be good. The goodness of the results will be discussed in detail in section 5.2. Further, to check if the model produces good results at tests which are at lower amplitudes compared to the standard tests, the model is tested with data from all the remaining tests of the test matrix and is included in Appendix A, along with the validation plots for the test data of non-Volvo dampers. The results from Test 4 and Test 8 are shown in Figures 5.18, 5.19, 5.20 and 5.21. It can be observed that, at Test 4, which is the 25 mm amplitude test, the model performs well. At very low amplitude such as 1 mm, in Figure 5.20 and 5.21, the model doesn’t capture the hysteresis accurately, but compared to a FV curve, the model is much better. Also at such low amplitudes, the results from the damper test rig are not completely accurate and reliable.
5.2 Accuracy of the model
5.2.1 Error Matrix
To quantify how good the model is, the error is calculated. The mean error, Errormean, which is the
average of difference between the measured data (Fmeasured) and simulation output (Fsimulation)
at each velocity point of a Force-Velocity curve, is calculated and presented in Table 5.1. Error percentage is calculated with respect to the maximum force in each test (Fmax). This is because,
Figure 5.18: FV plot Test 4 Figure 5.19: FD plot Test 4
Figure 5.20: FV plot for Test 8 Figure 5.21: FD plot for Test 8
Errormean=Σ(FTotalnumberof pointssimulation− Fmeasured) (5.1)
Error%=Σ(Fmax− Errormean)
Fmax (5.2)
CHAPTER 5. VALIDATION
Table 5.1: Error Matrix. Mean Error (N) Error % w.r.t Maximum Force Error in Region 1 (N) Error in Region 2 Test 1 Standard Test 148.1041 4.2726 141.668 6.4362 Test 2 149.7678 4.2178 143.2456 6.5222 Test 3 142.2287 4.0449 135.7386 6.49 Test 4 145.4056 4.1071 138.9616 6.444 Test 5 141.5526 4.1212 135.4052 6.1474 Test 6 139.7643 8.802 138.3788 1.3855 Test 7 149.4565 11.2116 149.4565 0 Test 8 156.7276 28.4648 156.7276 0 Test 9 131.8962 64.3886 131.8962 0 0.006369 Hz Test 134.1533 76.7313 134.1533 0 5.2.2 Error plots
After developing the model and setting up a parameter identification method it is essential to study if this new method is better than the generally used FV curve. One way of doing this is mentioned in section 4.3 where the error matrix is created and the error is found as a single number expressed as a percentage value for various test points. This section uses a similar approach where instead of using the average value of error percentage, it will be estimated for the entire operation range of the damper. To generate these results, Equation 5.2 is used for all the velocity-displacement points in a test section. To begin the analysis, the Standard test from the matrix is chosen. One reason to choose this test first is because it is used to generate the FV curve as well as parametrise the new model. This is a good starting point since both the models are developed using this dataset and this provides a good way to compare the results from the two models. Figure 5.22 shows the FVD crossplot from the new model and the test data. Figure 5.23 shows the error as a percentage for the new model and the FV curve model.
