Eindhoven University of Technology
MASTER
Quick and accurate modeling of shock absorbers using a hydraulic test-rig
de Bakker, S.
Award date:
2020
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Quick and accurate modeling of shock absorbers using a hydraulic test-rig
MSc. thesis S. de Bakker
0841765
Eindhoven University of Technology
Department of Mechanical Engineering
Dynamics & Control
Supervisors:
dr.ir. T.P.J. van der Sande prof.dr. H. Nijmeijer
DC 2020.058
Eindhoven, June 2020
Abstract
The department of Mechanical Engineering of the Eindhoven University of Technology (TU/e) does a lot of research in the field of vehicle dynamics, part of which is done on the behaviour of (active) shock absorbers. Therefore, a hydraulic-rig is converted into a shock absorber test-rig.
This test-rig can be used to perform dynamic tests on shock absorbers. The aim of this thesis is to optimize and streamline the measurement and modeling protocol of the TU/e shock absorber test-rig. It should then be possible to use the test-rig to quickly measure and model the dynamic force behaviour of different shock absorbers.
To reach this goal, different improvements are made to the test-rig itself. First, an improved way of estimating the shock absorber velocity is implemented. This is done by using an additional acceleration sensor and a Kalman filter type of observer that estimates the shock absorber velocity based on the measured position and acceleration. This new velocity estimator results in a velocity signal with much less noise than before. Next to the velocity estimation, also the test-rig control is updated and slightly improved and some other measures are taken to further reduce measurement noise.
To be able to use the test-rig to quickly model the dynamic behaviour of shock absorbers, different non-parametric shock absorber models are developed that can be fitted to measurement data from the test-rig without knowledge of the internals of the shock absorber. In this way, it is not needed to take the shock absorber apart to study its internals. These different shock absorber models range from the simplest linear damping model to more complex models like an hyperbolic tangent and a 4th order polynomial model. These different shock absorber models are then fitted to a data set that is a combination of measurements using sines with different frequencies as reference trajectory. The models are fitted to the data set using an optimization algorithm with a weighted error function. The different shock absorber models are then compared to measurements to see how accurate they are. The hyperbolic tangent models is able to estimate the shock absorber force with an accuracy of up to 3.53 % at a 6 Hz frequency. The extended polynomial has the lowest average error for all studied frequencies and the linear model has the highest average error for the studied frequencies. It can be concluded that the more complex the shock absorber model gets, the more accurate it is able to estimate the shock absorber forces.
Nomenclature
E Estimation error N
F Shock absorber force N
N Number of samples −
a Shock absorber acceleration m
s2
c Shock absorber damping coefficient N s
m
kv Velocity observer gain −
kz Position observer gain −
v Shock absorber velocity m
s
xi Shock absorber model fitting coefficients −
z Shock absorber position m
Table of Contents
Abstract i
1 Introduction 1
1.1 Aim and objectives . . . 2
1.2 Report outline . . . 2
2 Literature review 3 2.1 Shock absorber test-rigs . . . 3
2.1.1 Electromechanical test-rigs . . . 3
2.1.2 Hydraulic testers . . . 4
2.1.3 Shock absorber test cycles . . . 4
2.2 Shock absorber designs. . . 6
2.2.1 Mono-tube shock absorbers . . . 6
2.2.2 Dual-tube shock absorbers. . . 6
2.3 Shock absorber models . . . 7
2.3.1 Physical models. . . 7
2.3.2 Empirical models (black-box models). . . 8
2.4 Summary and conclusions . . . 8
3 TU/e shock absorber test-rig 9 3.1 Test-rig hardware. . . 10
3.2 Velocity measurement . . . 12
3.2.1 Velocity estimation - Zhu Observer . . . 13
3.2.2 Velocity estimation - Kalman filter . . . 16
3.3 Test-rig control . . . 20
3.3.1 Plant identification. . . 20
3.3.2 Feedback controllers . . . 20
3.3.3 Feed-forward control . . . 21
3.4 Summary and conclusions . . . 25
4 Shock absorber models and model fitting 27 4.1 Shock absorber characteristics. . . 27
4.2 Shock absorber models . . . 28
4.2.1 Linear model . . . 28
4.2.2 Positive/negative velocity linear model. . . 28
4.2.3 Extended polynomial model. . . 29
4.2.4 Hyperbolic tangent model . . . 30
4.3 Measurements and model fitting . . . 32
4.3.1 Measurement procedure . . . 33
4.3.2 Model fitting procedure . . . 37
4.3.3 Fitting of the linear shock absorber model. . . 40
4.3.4 Fitting of the positive/negative velocity linear model . . . 41
4.3.5 Fitting of the positive/negative velocity linear model with hysteresis . . . . 43
4.3.6 Fitting of the extended polynomial shock absorber model . . . 45
4.3.7 Fitting of the hyperbolic tangent model with hysteresis . . . 48
4.4 Comparing the different shock absorber models . . . 52
4.5 Summary and conclusions . . . 53
5 Conclusions and recommendations 55 5.1 Conclusions . . . 55
5.2 Recommendations . . . 56
Bibliography 57 A Acceleration sensor data sheet 59 B Acceleration sensor calibration 77 C Hardware improvements 79 C.1 Position signal . . . 79
C.2 Force signal . . . 80
C.3 Temperature signal . . . 84
D TU/e Code of Scientific Conduct 85
1. Introduction
Already since the 1900’s, passive hydraulic shock absorbers have been used in automobiles, [1], their main use is to decouple the vehicle body from road inputs in order to improve comfort (pleas- antness) and road holding (safety), [2]. Automotive shock absorbers are part of the suspension system. The suspension system is needed to guarantee handling and comfort. For a good hand- ling and braking, the tire-road contact forces need to be as stable as possible. Each wheel should remain in contact with the ground at all times. Comfort means that vibrations, induced by road profiles during driving, are of a minimal nuisance to the passengers, [3]. Since shock absorbers have such big influence on both the safety and the pleasantness off the ride as experienced by the driver and the passengers, shock absorbers are the subject of constant research and development.
The department of Mechanical Engineering of the Eindhoven University of Technology (TU/e) does a lot of research in the field of vehicle dynamics, part of this research is done on the behaviour of (active) shock absorbers. Because of this research, a hydraulic-rig formerly used to perform tensile tests on materials is converted into a shock absorber test-rig. This shock absorber test-rig can be used to perform dynamic tests on (semi-active) suspension components. A shock absorber can be vertically clamped in the test-rig. The bottom of the shock absorber is attached to a movable hydraulic cylinder, which exerts a force on the shock absorber, causing the shock absorber to compress or extend. The shock absorber movement can be prescribed and the corresponding shock absorber force can be measured, [4].
When tuning the suspension components of a vehicle, generally a trade-off has to be made between ride (e.g. comfort on uneven roads) and handling (e.g. response on steering inputs). A smoother ride typically asks for softer shock absorbers, while better handling typically asks for stiffer shock absorbers, [2]. The dynamic properties of suspension components such as shock absorbers and springs have an influence on both the ride and handling of a vehicle. The tuning of the properties of these suspension components is often done by test drivers, [5] [6]. These test engineers rate the performance of the vehicle on a test track subjectively. Since this rating is done subjectively, it can change over time and it can vary from engineer to engineer. Nowadays, computers are widely used for virtual prototyping and as a result, computer simulations are becoming an integral part of the design process. Virtual prototyping is far less subjective, also less hardware prototypes are required, therefore the design process of new vehicles is accelerated and can be done in a cheaper way. However, virtual prototyping requires accurate modeling of the vehicle’s suspension components. These component models can not be generated and verified without doing hardware tests, therefore hardware testing of these suspension components will always be required. This is one of the reasons why research is being done at the Eindhoven University of Technology on the determination of dynamic properties of springs and shock absorbers as used in vehicles and the shock absorber test-rig has been built, [7].
The suspension component that is the subject of this report is the shock absorber. The main purpose of a shock absorber is the dissipation of energy, energy that is stored in the spring when this spring is compressed. The shock absorber is placed in between the sprung and unsprung masses to suppress oscillation and control the motion of the sprung mass due to longitudinal and lateral acceleration, [1]. It does this by developing a force that works in the opposite direction of the movement of the shock absorber. This shock absorber force is developed by the movement of a piston through a cylinder filled with oil. When relative motion occurs between the sprung and unsprung masses, the piston is forced through the oil in the cylinder and, by directing the fluid through different valves and orifices, the kinetic energy that is stored in the spring can be damped before it is transmitted to the sprung mass, [8]. Because the oil has to move from one side of the piston to the other, the shock absorber force is mainly velocity dependent. The larger the absolute velocity with which the piston moves, the larger the resisting force will be. The workings of shock absorbers will be further explained in Section2.2.
1.1 Aim and objectives
The aim of this master thesis, as the title might suggest, to be able to quickly and accurately model shock absorbers using the TU/e shock absorber test-rig.
The aim is to be able to use the test-rig to quickly measure and model the dynamic force behaviour as function of known inputs of different shock absorbers. The result of the measurement and modeling protocol of the test-rig should then be a model that describes the force behaviour of such shock absorber based on the inputs like shock absorber position, velocity and acceleration.
Improvements should be made to the test-rig that lead to a shock absorber test-rig that is suitable for a ”production”-like testing and modeling procedure of these shock absorbers. In order to get to this ”production”-like testing and modeling procedure, the test-rig should be made more reliable and easy to use than it was at the start of this project. Next to that, shock absorber models should be developed that can be fitted to the measurement data in a quick way while still being accurate. Since dismantling of a shock absorber to examine and measure its internals is not always possible or preferable and probably takes quite some time, the shock absorber models should not need details of the internals of the shock absorbers in order to fit the models to the measurement data.
This leads to the following research question: What changes to the shock absorber test-rig and which shock absorber models are needed for fast and reproducible shock absorber testing and modeling and how can these be implemented?
1.2 Report outline
This report consists of several chapters, Chapter 2 is the literature review which gives a brief explanation about the TU/e shock absorber test-rig, shock absorbers in general, different shock absorber test-rigs, shock absorber designs and shock absorber models. Then, Chapter3 explains the TU/e shock absorber test-rig in more detail and explains the improvements that have been made to come to a more reliable test-rig. After that, Chapter4gives the different non-parametric shock absorber models that have been developed and describes the measurements that are used to fit the shock absorber models to, after that, it describes the fitting of these different models to the measurement data. And finally, Chapter5gives some conclusions and recommendations that can be drawn and made based on this report.
2. Literature review
The shock absorber is one of the most complex parts of the suspension system to model. In general, the shock absorber behaves in a nonlinear and time-variant way. Dampers are typically characterized by the force-velocity diagram, also referred to as the characteristic diagram. Some information can also be extracted by plotting force as a function of displacement. This results in a work diagram, a resistance curve or a control diagram, [3].
In order to model the dynamic behaviour of a shock absorber, hardware tests are needed to measure the forces a shock absorber develops under certain inputs. Shock absorber test-rigs are used to perform these hardware tests, different types of shock absorber test-rigs and different test cycles that are widely used in industry will be explained in Section2.1.
Shock absorbers can be designed in different ways. Section2.2discusses two of the most common types of shock absorbers, namely the mono-tube style shock absorbers and the dual-tube style shock absorbers. It discusses the internal working of these two shock absorber types and explains how they work.
To describe the behaviour of a shock absorber based on a certain position/velocity input, multiple different shock absorber models have been developed over the years. A brief discussion of these can be found in Section2.3.
Finally, Section2.4gives a short summary and draws some conclusions of this chapter.
2.1 Shock absorber test-rigs
Two types of shock absorber testers can be distinguished in literature: electromechanical and hydraulic testers. Both will be discussed in the following paragraphs. After that, the shock absorber test-rig of the Eindhoven University of Technology will be discussed.
2.1.1 Electromechanical test-rigs
The first electromechanical testers used a slider-crank mechanism with a connecting rod to re- ciprocate the shock absorber in a roughly sinusoidal manner. This reciprocating movement is delivered by an electric motor with a flywheel mounted on the drive axle, [9]. A schematic repres- entation of a reciprocating shock absorber tester using a slider-crank mechanism can be seen in Figure2.1. Later, a new type of electromechanical testers was developed
The inclination of the connecting rod in the slider-crank mechanism introduces a substantial harmonic into the damper motion, which is non-sinusoidal. This harmonic can be eliminated by using a Scotch Yoke mechanism, which gives a true sinusoid. A schematic representation of a Scotch Yoke drive shock absorber tester can be seen in Figure2.2. The American company MTS produces such Scotch Yoke type shock absorber test-rigs, called the SYD (Scotch Yoke Dyno) system, [10].
The frequency of these electromechanical testers that use slider-crank or Scotch Yoke mechanisms can be adjusted either by use of a variable speed DC motor or by a variable ratio gearbox. Variation of stroke may be possible by disassembly of the apparatus, so the stroke is set to give the desired maximum speed, within the limits of the damper and test apparatus. Because of the limitations described above, these types electromechanical testers are usually limited to small low-powered units. These are suitable for limited testing and low-speed comparative work, including matching at low speeds. For larger testers it is usually preferred to use hydraulic drive, [9].
Another type of electromechanical testers is the so called electromagnetic actuated test-rig. Boggs [11] uses a Roehrig Electromagnetic Actuator (EMA) shock dynamometer. These test-rigs use an EMA to actuate the shock absorber and no reciprocal actuator like the slider-crank and the Scotch Yoke, this allows a wide variety of inputs, including sine waves, triangle waves, random
inputs, track data, or any other user-defined input, [11]. The American company MTS produces these electromechanical testers as well, [10].
Figure 2.1: Reciprocating tester with a slider-crank mechanism, [9].
Figure 2.2: Scotch Yoke drive tester, [9].
2.1.2 Hydraulic testers
The first (electro)hydraulic test-rigs were introduced in the 1960’s, [12]. Hydraulically driven shock absorber test-rigs are capable of delivering high power inputs and are flexible controllable. The hydraulic ram in these hydraulic testers is usually double acting, typically acting at a pressure of around 1 MPa with a force capability of 10 kN. Very-high-quality valves are required to control the oil supply accurately, with a sophisticated control system and a large pump, making the system expensive, [9].
Cafferty [13], uses a Zonic 1107-4-T 8900 N hydraulic actuator with dual-loop master controller as actuator for a hydraulic shock absorber test-rig. The American company MTS produces several different types of shock absorber test-rigs, including hydraulic test-rigs. These hydraulic test-rigs are generally used for durability tests, [10]. The position of the hydraulic ram is controlled by a voltage input in these test-rigs. This voltage input is used to create different reference signals such as sinusoidal waves, triangular waves, square waves, random motions or an external input, [9].
2.1.3 Shock absorber test cycles
Over the years, different test cycles have been developed to test shock absorbers and to measure their characteristics. A well known test cycle that is used in the automotive industry is stand- ardized by the German Association of the Automotive Industry (VDA). The reference trajectory consists of 7 merged sines with an amplitude of 50 mm. The individual sines are excited for 4
periods and between them a delay of 1 s is added in the bottom dead center. The period of the 7 different sines are decreasing in time, which results in an increase in maximum velocity of the trajectory, [2]. The VDA cycle is plotted in Figure2.3.
0 10 20 30 40 50
Time [s]
-50 -40 -30 -20 -10 0 10 20 30 40 50
Position [mm]
Figure 2.3: Position reference of the VDA cycle.
Another type of test cycle that is known in literature is the triangular test, also called a waveform, this uses a triangular displacement waveform instead of a sinusoidal one. A symmetric triangular displacement waveform gives nominally constant velocity over the whole stroke. This is used in [14] and is called isokinetic excitation which gives a uniform distribution of experimental data throughout the velocity-displacement domain, which is not the case with harmonic excitation, [14].
A way of testing shock absorbers that is used in [13] is by using random excitation as a reference signal. In here, Cafferty states that random testing of a shock absorber offers a useful alternative to the more commonly used repetitive harmonic testing. Advantages are, amongst others, a much better coverage of the position/velocity plane is achieved compared to standard repetitive harmonic tests, characterization of the absorber is much faster than in repetitive harmonic testing and no control of the input or output is needed beyond setting the overall level of excitation and upper and lower frequency limits, [13].
In [15], different test cycles are discussed like sine-on-sine excitation, multiple frequency input excitation and road excitation. The sine-on-sine excitation uses a carrier wave which is a sine with a high amplitude but low frequency and adds a sine with a low amplitude but a high frequency as a rider wave to it. Multiple frequency input excitation is comparable to the random excitation as used in [13]. Road excitation can be seen as real life use of a shock absorber, the reference signal is obtained from a test vehicle driving on different types of road surfaces. This results in a test cycle that is as close to the real use of a shock absorber as possible, [15].
2.2 Shock absorber designs
The main task of a shock absorber in a vehicle suspension system is absorbing energy. The actual shock in a suspension system that is caused by road irregularities for instance is absorbed by the spring. The shock absorber then damps the spring motion, it does this by converting the kinetic energy that is put into the system by the road irregularities into heat.
This report only focuses on telescopic (linear) shock absorbers. Telescopic shock absorbers come in many different variants, but two main categories can clearly be distinguished: mono-tube and dual-tube shock absorbers. These two types of shock absorbers are very clearly described by Dixon [9]. A short and brief explanation of the two types of shock absorbers will be given in next subsections.
2.2.1 Mono-tube shock absorbers
A mono-tube shock absorber, as shown in Figure2.4, consists of two chambers filled with oil which are separated by a piston, connected to this piston on one side is the piston rod. When the shock absorber is compressed, the piston rod pushes the piston in and oil flows from the compression chamber to the rebound chamber through a set of orifices and valves in the piston, hereby creating a damping force. By the movement of the piston the volume of one of the two oil-filled chambers is reduced, in compression this chamber is called the compression chamber. The other oil-filled chamber is called the rebound chamber and will grow in volume. Extensions of the shock absorber will yield in the opposite of that previously described. The piston rod will move outwards, by which it decreases the rebound chamber’s volume and increasing that of the compression chamber. Oil will flow from the rebound side of the piston to the compression side and again, create a damping force. The third chamber, called the gas chamber, compensates for the volume changes that are caused by the piston rod moving into the shock absorber during compression. This makes use of the fact that oil is nearly incompressible but gas is not. When the piston rod enters the rebound chamber, the gas in the gas-chamber is compressed to compensate for the change in volume. Some mono-tube shock absorbers have a free floating gas piston that prevents the gas in the gas-chamber to mix with the oil in the compression chamber, other mono-tube shock absorbers don’t have such free floating gas piston, these are also known as emulsion dampers, [9]. Another function of the gas-chamber is to compensate for the change of volume of the oil when it warms up, [16].
Figure 2.4: Schematic representation of a mono-tube style shock absorber, [17].
2.2.2 Dual-tube shock absorbers
Dual-tube shock absorbers, as shown in Figure 2.5, exist in two different types. The first one consists of two tubes that are placed concentrically to each other, as the one shown in Figure2.5.
The second type has a secondary, separate tube that is placed outside the shock-absorber itself, this secondary tube is connected to the shock absorber by a hose through which the oil can flow
between the primary and secondary tube, this type of dual-tube shock absorber is also known as a piggyback style shock absorber, [17].
Dual-tube shock absorbers have, just as the mono-tube shock absorbers, two oil-filled chambers that are separated by a piston in the primary tube. A set of valves and orifices in the piston allow oil to flow from one side of the piston to the other upon compression or extension of the shock absorber. Under compression, the shock absorber shortens, the piston moves and oil is forced to flow from the compression chamber through the piston orifices and valves into the rebound chamber. The volume corresponding to the immersed piston rod volume is thereby pushed into the secondary tube (reserve chamber in Figure 2.5), [16]. The opposite holds for a rebound motion (extension) of the shock absorber, oil is forced to flow from the rebound chamber into the compression chamber, as well as from the secondary tube (reserve chamber) via the base valve assembly into the compression chamber. Part of the secondary tube is filled with gas, the gas- chamber, to account for changes in volume caused by moving in and out the piston rod, just as for the mono-tube style shock absorber. Because of the base valve assembly, the manufacturer or user of the shock absorber has more possibilities to change the shock absorber characteristics, which is a major advantage of the dual-tube over mono-tube shock absorbers, [9].
Figure 2.5: Schematic representation of a dual-tube style shock absorber, [17].
2.3 Shock absorber models
Over the years, multiple different shock absorber models have been developed. Two different types can be distinguished, physical and empirical models, these will be discussed shortly in the following paragraphs. A broader study of different shock absorber models can be found in [3].
The main goal of all these different shock absorber models is to model the dynamic behavior of shock absorbers, thus the shock absorber force, as accurately as possible. This means that an ideal shock absorber models describes the behavior for both low as well as high velocities and both low as well as high frequencies.
2.3.1 Physical models
Physical shock absorber models are models that contain a physical model of all the shock absorber components. The physical model describes the dynamic behaviour of all these components as well as the hydraulic fluid flow. These models generally use measurements to estimate parameters of the internal shock absorber architecture and valve assemblies, [3].
One of the earliest physical models is that of Lang [18], [19]. Lang’s model was developed in 1977 and was the first physical model that actually aims to describe the behaviour of the shock absorber
in a broad range of operating conditions. The mechanistic model calculates the shock absorber force from a system of differential equations, [3]. Back in the days, Lang’s model could not be readily identified from force data only. The internal pressures had to be measured or predicted numerically and this involved time consuming procedures, to give an idea, the simulation of one period took about 7 hours, [3].
An attempt to present a readily identifiable model was the development of an explicit physical model by Reybrouck, [20]. Reybrouck’s model gives an explicit expression for the shock absorber force as a function of displacement, velocity and acceleration.
2.3.2 Empirical models (black-box models)
Empirical models (black-box or non-parametric models) are not based on physical modeling. These models consist of algebraic equations without any physical meaning to describe the dynamic beha- viour of the shock absorber. These models fully rely on measurements to tune the mathematical formulations and parameters. These parameters do not necessarily have a definite meaning but they are strongly correlated with the measurements, [3].
An example of an empirical shock absorber model is presented by Barethiye [21], which uses a combination of piecewise linear fitting to describe the nonlinear behaviour of the shock absorber and a neural network model for the hysteresis characteristics. Another example is the hyperbolic tangent model as described by Cafferty [13]. This model uses a hyperbolic tangent to model the damping characteristics of a shock absorber. Higher-order polynomials can also be used to model shock absorber forces, Boggs [11] models shock absorber forces using such higher-order polynomial. A so called grey-box model is introduced by Beghi in [22]. It is a combination of a non-linear parametric model and a black-box neural-network-based model. The parametric part is a combination of three contributions, namely the friction, gas and hydraulic forces. The black- box, neural-network-based part of the system is introduced to describe the highly non-linear high frequency behaviors that are not captured by the parametric model, [22].
An advantage of these empirical models compared to the physical models is that no knowledge of the internals of a shock absorber is necessary. These models can be fitted to the measurement data without the need of taking apart the shock absorber itself or knowing the dimensions of the valves and orifices inside, which can make the whole fitting process much easier and quicker.
2.4 Summary and conclusions
In this chapter, existing literature on the subject of shock absorbers and shock absorber modeling and testing is studied. Section 2.1 studies the different ways of testing shock absorbers. First, the different types of shock absorber test-rigs are studied, namely electromechanical test-rigs and hydraulic test-rigs. The pros and cons of the different types of test-rigs are discussed as well. After that, different shock absorber test cycles that are used in literature and their pros and cons are studied. This section also gives some ideas for what test-cycles to use to test a shock absorber.
Section2.2studies two main shock absorber designs, namely mono- and dual-tube shock absorbers.
It describes their working and the differences between the two designs. Lastly, Section2.2studies different types of shock absorber models that can be found in literature. It studies their workings and what is needed to fit the models. From this, it can be concluded that a type of black-box (or grey-box) model is most suited for the purpose of this project, since it’s not needed to take apart the shock absorber itself or to know the dimensions of the valves and orifices inside to fit this model to measurement data.
3. TU/e shock absorber test-rig
A Zwick-Rel 1852 material tester, formerly used by the Polymer Technology section of the Mech- anical Engineering Department of Eindhoven University of Technology to perform tensile tests, was converted into a hydraulic shock absorber test-rig by Nick Feijen, [7]. He also developed the first controller and performed the first successful tests on a shock absorber. An overview picture and a more detailed picture of the TU/e shock absorber test-rig can be found in Figure3.1.
(a) Overview (b) Detailed
Figure 3.1: The TU/e shock absorber test-rig, [17]. With (1): height adjustment block, (2) force sensor, (3) upper shock absorber clamp, (4) bottom shock absorber clamp, (5) height adjustment controls, (6) rubber mounts and (7) shock absorber.
After the work done by Nick Feijen, Jamie de Blok further improved both the usability as well as the user safety of the shock absorber test-rig by adding a Graphical User Interface (GUI), [17].
He also improved the control protocol of the test-rig. In 2018, Rob Goris further improved the GUI and the control protocol of the test-rig, he updated the feedback controller and added a feed-forward controller which is used to counteract known disturbances, e.g. inertial, gravitational and friction forces, [2]. Goris also programmed a gas and friction test to measure the gas and friction forces of the shock absorber.
In this chapter, different improvements that have been added to the TU/e shock absorber test-rig will be discussed. First, in Section 3.1, the improvements that have been made to the hardware of the test-rig will be explained. Then, in Section 3.2, an improved way to measure the shock absorber velocity is will be discussed. After that, in Section 3.3the test-rig’s control protocol is discussed. Lastly, Section 3.4 gives a short summary and draws some conclusion based on this chapter.
3.1 Test-rig hardware
In order to improve the velocity measurement, as will further be explained in Section 3.2, an acceleration sensor is added to the shock absorber test-rig. Such accelerometer translates the rectilinear acceleration of the hydraulic cylinder into an electrical signal. The accelerometer used in the shock absorber test-rig is the ADXL326 [23], of which the data sheet can be found in AppendixA.
An accelerometer consists of a structure that is micro-machined and built on top of a silicon wafer stage. Polysilicon springs suspend the structure over the surface of the wafer and provide a resistance against acceleration forces. Deflection of the structure is measured using a differential capacitor. Acceleration then deflects the moving mass and unbalances the differential capacitor resulting in a sensor output whose amplitude is proportional to the acceleration, [23].
The sensor that is used came of the suspension of the 2001 BMW 318i as used by Tom van der Sande in his PhD thesis, [24]. The sensor that is used is a 3-axis, ±16 g accelerometer. Since the sensor was used in the BMW, it is converted to run on a 12V DC input, therefore an external power supply and measurement amplifier are needed, these are shown in Figure 3.2. The sensor itself is placed on the lower shock absorber clamp. Since the bottom of the tested shock absorber is directly mounted to the lower shock absorber clamp, it has the same acceleration as the shock absorber itself.
Figure 3.2: Measurement amplifier (top) and external power supply (bottom) for the added accelerometer.
Since the acceleration sensor has voltage as output, it is necessary to translate this output voltage into a physical acceleration. In order to get to this physical acceleration, the sensor needs to be properly calibrated. The two parameters that have to be calibrated are the zero g bias and the 1 g tuning parameter. The values that have been found for this sensor are: 1.566869 V for the zero g bias and 0.0615 V/g for the 1 g tuning parameter. The explanation of the acceleration sensor calibration can be found in AppendixB.
Figure 3.3: Acceleration sensor placed on the bottom shock absorber clamp.
Next to the acceleration sensor, some additional hardware is added to the shock absorber test-rig in order to reduce the amount of measurement noise present in the measurement signals coming from the test-rig. This hardware mainly consists of shielding the measurement hardware and ferrite cores to filter the measurement signals. The hardware measures to reduce this measurement noise in the signals are further explained in AppendixC.
Figure3.4 gives a schematic representation of the TU/e shock absorber test-rig. As can be seen, the test-rig consists of a hydraulic cylinder on which a shock absorber clamp is placed, this clamp is called the lower shock absorber clamp. Above that, the upper shock absorber clamp is connected to the height adjustment block, which can be used to raise or lower the upper shock absorber clamp to compensate for the difference in length of different shock absorbers. The shock absorber can be placed between these two shock absorber clamps and can then be excited by the hydraulic cylinder. The LVDT (Linear Variable Differential Transformer) that measures the position of the hydraulic cylinder, and therefore the shock absorber position, is built into the hydraulic cylinder.
The acceleration sensor is mounted on the lower shock absorber clamp and shown in green in Figure3.4. The force sensor is placed between the height adjustment block and the upper shock absorber clamp and shown in red in Figure3.4.
Force sensor (red) Height adjustment block
Hydraulic cylinder with built-in LVDT Acceleration sensor (green)
Upper shock absorber clamp
Lower shock absorber clamp Shock absorber position (z)
Rubber mounts (black)
Figure 3.4: Schematic representation of the TU/e shock absorber test-rig with in red the force sensor, in green the acceleration sensor and in black the rubber mounts on which the test-rig hangs.
3.2 Velocity measurement
A problem of the shock absorber test-rig at the start of this project was that the velocity meas- urement was very noisy, which can be seen in Figure3.6. The reason for this is that at the start of this project, the shock absorber velocity was determined by taking the numerical derivative of the position signal provided by the LVDT in the test-rig. Since this position signal already is a noisy measurement, its numerical derivative becomes even more noisy, therefore an alternative way to determine the velocity is required. A schematic representation of the control and measurement of the shock absorber test-rig excluding the force and temperature sensors is shown in Figure3.5. In here, also the feedback and feed-forward controllers can be seen, these will be further explained in Section3.3.
In this section, two observers that can be used to determine the shock absorber velocity will be described and compared. First, in Section3.2.1, an observer that is based on the one introduced by Wen-Hong Zhu in [25] will be discussed. Then, in Section3.2.2, a Kalman filter based observer that estimates the velocity will be given and the results of both observers will be compared. A low-pass filter could also be used to filter the velocity signal, however this will come with a phase delay in the filtered signal [26], which is not desirable.
Shock absorber acceleration
Test-rig Shock absorber position Position
reference
Shock absorber velocity
Feedback controller
Velocity estimator Feed-forward
controller
Measurement output +
-
++
Figure 3.5: Schematic representation of the control and measurement loop of the shock absorber test-rig excluding the force and temperature sensors.
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-1.5 -1 -0.5 0 0.5 1 1.5
Velocity [m/s]
Figure 3.6: Example of the noisy velocity signal as acquired by taking the numerical derivative of the position signal. A sine with an amplitude of 10 mm and a frequency of 10 Hz is used as reference.
3.2.1 Velocity estimation - Zhu Observer
The first velocity observer that is created is based on the observer as developed by Wen-Hong Zhu, [25] [27]. The observer as designed by Wen-Hong Zhu in [25] is based on basic kinematics.
It uses a state-space system where the state vector x consists of the position and velocity,
x =z v
. (3.1)
Then, the measured acceleration (a∗) is used as input for the velocity state. The error between the measured position (z) and estimated position (ˆz) is then used as a feedback term, multiplied by an observer gain vector. This results in the following observer, given as:
˙ˆz(t)
˙ˆv(t)
=0 1 0 0
ˆz(t) ˆ v(t)
+
0 a∗(t)
+kz
kv
(z(t) − ˆz(t)) (3.2)
in which a∗ is the acceleration as measured by the accelerometer. ˆz and ˆv are the estimated position and velocity respectively. And the observer gains are given by kz> 0 and kv> 0 and can be combined into one observer gain vector K:
K =kz kv
. (3.3)
Equation (3.2) can then be rewritten into the following:
˙ˆz(t)
˙ˆv(t)
=−kz 1
−kv 0
| {z }
A
ˆz(t) ˆ v(t)
+kz 0 kv 1
| {z }
B
z(t) a∗(t)
(3.4)
with matrices A, the state matrix and B the input matrix. This continuous time state space system is then converted into a discrete time state space system with sampling time ∆t using Euler’s method, [28]:
Ad= I + A · ∆t (3.5)
Bd= B · ∆t (3.6)
This results in the following discrete time state space system:
ˆzk+1
ˆ vk+1
=1 − kz· ∆t ∆t
−kv· ∆t 1
| {z }
Ad
ˆzk
ˆ vk
+kz· ∆t 0 kv· ∆t ∆t
| {z }
Bd
zk
a∗k
(3.7)
The working of the velocity observer is first tested in simulations and after that tested on the test-rig. For this, a sine with a frequency of 10 Hz and an amplitude of 10 mm is used as a position reference, sampling time ∆t is set to 1/1000 s. Figure3.7 shows a comparison between the measured position and the position as estimated by the velocity observer with three different sets of observer gains. These different sets of observer gains are tuned by hand, such that the estimated position and velocity are as close to the measurement as possible, but with less noise.
Figure3.8shows a comparison between the measured velocity (numerical derivative of the position) and the velocity as estimated by the velocity observer with the same three sets of observer gains.
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-15 -10 -5 0 5 10 15
Position [m/s]
Measurement kz = 5, k
v = 700 kz = 50, k
v = 400 kz = 5, k
v = 400
Figure 3.7: Comparison between the measured position and the position as estimated by the velocity observer with different observer gains.
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-1.5 -1 -0.5 0 0.5 1 1.5
Velocity [m/s]
Measurement kz = 5, k
v = 700 kz = 50, k
v = 400 kz = 5, k
v = 400
Figure 3.8: Comparison between the measured velocity and the velocity as estimated by the velocity observer with different observer gains.
Looking at the position estimates in Figure 3.7, one can see that some of the gains show some phase lag compared to the measurement. Also, the amplitude of the observer estimates doesn’t show the same values as the measurement. Looking at the velocity estimates in Figure 3.8, not much difference can be seen between the three sets of observer gains. The normalized position and velocity RMS errors of the observer with the different observer gain sets are given in Table 3.1. The normalized RMS error is calculated as:
EN RM S= ERM S xmax− xmin
(3.8)
with xmax− xmin the range of the measured data and ERM S the RMS error defined as:
ERM S = v u u t 1 n
n
X
i=1
(xi− ˆxi)2 (3.9)
with xi the measured value (position or velocity) at time instance i, ˆxi the position or velocity as estimated by the observer at the same i-th time instance and n the number of data-points in the measurement. The RMS errors are normalized so that the RMS position errors and RMS velocity errors can be compared.
Table 3.1: Normalized RMS errors of the estimated position and velocity by the observer for different observer gains.
Observer gains NRMS position error [-] NRMS velocity error [-]
kz= 5, kv= 700 0.1034 0.0691
kz= 50, kv= 400 0.0453 0.0687
kz= 5, kv= 400 0.1198 0.0690
As can be seen in Table3.1, from the three sets of observer gains plotted in Figure 3.7, kz = 50 and kv= 400 gives the lowest NRMS position error compared to the measurement. The difference in NRMS velocity error between the three different sets of observer gains is minimal, but also here, the observer with gains kz= 50 and kv = 400 gives the lowest gives the lowest error of the different sets of observer gains. Therefore, kz= 50 and kv= 400 are chosen as the observer gains for the velocity observer. Although it looks like the observer with said gains is able to estimate the shock absorber velocity, it cannot be said that the observer gains that are chosen are optimal
since the observer gains are tuned by hand. Because of that, a type of observer that automatically finds these observer gains is looked for, this type of observer will be explained in next section.
3.2.2 Velocity estimation - Kalman filter
Another type of observer is a Kalman filter. This type of observer has the advantage that it automatically tunes its gain matrix, compared to the observer of Section3.2.1this is advantageous because the observer gains for the previous described observer are tuned by hand.
Consider the Kalman filter that is given by the following discrete linear time-invariant system
ˆ
xk+1= Aˆxk+ Buk+ L(yk− Ckxˆk) (3.10) for which the following state vector xk is chosen:
xk =zk
vk
, (3.11)
where z is the shock absorber position and v is the shock absorber velocity. ukis the known input vector, the acceleration measured by the accelerometer in this case, yk is the measurement vector, the position measured by the LVDT. In (3.10) A is the system matrix, B is the input vector, L the gain matrix and C the output matrix.
The test-rig is then modeled in discrete time as xk+1=1 ∆t
0 1
| {z }
A
zk vk
| {z }
xk
+0 1
|{z}
B
0 a∗k
| {z }
uk
·∆t +σz σa
| {z }
wk
(3.12)
yk=1 0
| {z }
C
zk vk
| {z }
xk
+ σz
|{z}
vk
(3.13)
and input u is the acceleration measurement and feedback y is the position measurement. The noise characteristics are given by
Q = wwT =
σ2z σzσa
σaσz σ2a
, R = vvT = σ2z, (3.14)
in here σa and σz are the standard deviation of a measurement at stand still of the acceleration and position signal respectively. Because this measurement is done at stand still, so the test-rig had zero input, everything that is measured is considered to be noise. On top of that, it is also assumed that this noise can be considered as white measurement noise with mean ¯wk = ¯vk = 0.
The noise levels that are measured are: σz = 0.0104 mm and σa = 0.0401 mm/s2, which results in the following noise matrices
Q =
0.01042 0.0104 · 0.0401 0.0401 · 0.0104 0.04012
, R = 0.01042. (3.15)
Using these noise matrices Q and R, the optimal gain L can then be found by solving the Ricatti equation.
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-10 -5 0 5 10
Position [mm]
Measurement Observer Kalman filter
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-1000 0 1000 2000
Velocity [mm/s]
Figure 3.9: Comparison between the measured velocity, the velocity estimated by the observer and the velocity estimated by the Kalman filter.
Figure3.9compares the result of both the Kalman filter with gains L as well as the observer from previous section with gains K as given in (3.16) to the measured position and velocity.
L =6.4249 6.4830
K = 50 400
(3.16)
As can be seen, both the velocity result of the observer and that of the Kalman filter show similar behaviour as the velocity measurement, however with much less noise. When looking at the position signals however, a bigger difference is visible. As can be seen, the Kalman filter position estimate is much closer to the measurement than the observer position estimate. This can also be seen in the position error plot shown in Figure3.10. Table3.2compares the normalized RMS position and velocity errors for both the Kalman filter and the observer, these are calculated by (3.8). As can be seen, the normalized RMS position error is much lower for the Kalman filter then for the observer, the normalized RMS velocity error is almost the same. Because the Kalman filter automatically tunes its gain matrix where the gain matrix of the observer of Section3.2.1is hand tuned, the Kalman filter velocity estimate will be used as the velocity signal.
Table 3.2: Normalized RMS errors of the estimated position and velocity by the observer and the Kalman filter.
NRMS position error [-] NRMS velocity error [-]
Kalman filter 0.0062 0.0689
Observer 0.0453 0.0687
1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 Time [s]
-1.5 -1 -0.5 0 0.5 1 1.5 2
Error [mm]
Observer Kalman filter
Figure 3.10: Comparison between the position error of both the observer as well as the Kalman filter.
Figure 3.11 shows the Bode diagrams of both the observer as well as the Kalman filter for the two different inputs and two different outputs. The shock absorber position is both measured as well as estimated by the observer and Kalman filter. A magnitude of 0 dB for the measured position to the estimated position means good position tracking. Looking at the magnitude plot of the measured position to estimated position Bode diagram, one can see that the magnitude of the observer is equal to 0 dB up to around 5 Hz, the magnitude of the Kalman filter is equal to 0 dB up to round 100 Hz. This shows that the Kalman filter has good position tracking up to a much higher frequency than the observer. Looking at the measured position to estimated velocity magnitude plot, one can see that both the observer as well as the Kalman filter show a derivative action for low frequencies up to round 1 Hz. This can be seen by the +1 slope of the magnitude plot. Looking at the measured acceleration to estimated velocity magnitude plot, one can see that both the observer as well as the Kalman filter show an integrating action for high frequencies from around 1 Hz, which can be seen by the -1 slope of the magnitude plot. From this, it can be concluded that the observer and the Kalman filter both estimate the velocity based on the position measurement for low frequencies and based on the acceleration measurement for high frequencies, with a mixed area from around 0.5 Hz to 1 Hz.
-150 -100 -50 0
To estimated position z
From measured position z
-180 -90 0 90 180
To estimated position z
-50 0 50
To estimated velocity v
10-2 100 102
-180 -90 0 90
To estimated velocity v
From measured acceleration a
100 102
Observer Kalman
Bode Diagram
Frequency (Hz)
Magnitude (dB) ; Phase (deg)
Figure 3.11: Bode diagrams of both the observer as well as the Kalman filter for the two different inputs and two different outputs.
3.3 Test-rig control
The test-rig is controlled using both feedback control and feed-forward control. Figure3.5 gives a schematic representation of the control and measurement loop of the shock absorber test-rig.
The control loop consists of a feedback controller and a feed-forward controller. The feedback controller is developed by Rob Goris [2], this controller is checked and if possible improved during this project in Section 3.3.2. To do this, a plant identification has been performed, the result of which can be found in Section 3.3.1. The feed-forward controller, also developed by Rob Goris is checked as well in Section3.3.3. A more detailed schematic representation of the feed-forward controller will also be given in that section.
3.3.1 Plant identification
An indirect closed loop measurement using the three-point method as explained by Gert Witvoet in [29] is used to identify the plant dynamics of the hydraulic test-rig. This is done by applying a known disturbance d to the test-rig, the signal used for this disturbance is band limited white-noise with a noise power of 0.001. Then, since disturbance d is known and both the plant input u and position error e can be measured, the plant dynamics of plant H can be calculated by
H = P S(f )
S(f ) , (3.17)
with P S(f ) the process sensitivity which is the transfer from d to e and is calculated by e
d = −P S(f ). (3.18)
And with S(f ) the sensitivity which is the trasnfer from d to u and is calculated by u
d = S(f ). (3.19)
The results of this are shown in the Bode plot in Figure3.13. The plant is a first order system, but shows a peak in magnitude around 160 Hz.
Figure 3.12: Schematic representation of the closed loop used for the plant identification, [29].
3.3.2 Feedback controllers
Using the frequency response of the uncontrolled test-rig as shown in previous section, the feedback controller as described in [2] and given by (3.20) can be studied. Based on this, it became clear that with a higher P-action, a higher bandwidth could be achieved while still having a modulus
100 101 102 -40
-20 0 20 40
Magnitude [dB]
100 101 102
Frequency [Hz]
-180 -90 0 90 180
Phase [°]
Figure 3.13: Plant dynamics of the hydraulic test-rig.
margin that is low enough. This resulted in an improved feedback controller based on the old one and it is given by (3.21).
The two open loop Bode plots of the plant with the two feedback controllers previously mentioned are shown in Figure 3.14. Comparing the two feedback controllers, one can find that the old feedback controller has a bandwidth of 18.61 Hz with a modulus margin of 4.9 dB. The new feedback controller has a bandwidth of 22.28 Hz and a modulus margin of 6.0 dB. So with this new feedback controller a higher bandwidth is achieved while still having a modulus margin of 6.0 dB.
Cold= 0.12
|{z}
P-action
+0.12 · 1 · 2π s
| {z }
I-action
+
1 2π4s + 1
1 2π36s + 1
| {z }
lead-filter
+
1 2π100s + 1
1 2π20s + 1
| {z }
lag-filter
(3.20)
Cnew = 0.142
| {z }
P-action
+0.12 · 1 · 2π s
| {z }
I-action
+
1 2π4s + 1
1 2π36s + 1
| {z }
lead-filter
+
1 2π100s + 1
1 2π20s + 1
| {z }
lag-filter
(3.21)
3.3.3 Feed-forward control
The feed-forward controller used in the test-rig was already designed by Nick Feijen [7]. A schem- atic representation of this feed-forward controller is given in Figure 3.15. In order to make sure this was done correctly and to try to possibly improve this feed-forward controller, additional feed- forward parameter tuning has been performed. For this, the high bandwidth feedback controller is changed for one with a much lower bandwidth because otherwise, no real position error would occur. The position reference signal that is used to tune the feed-forward parameters is shown in Figure 3.16. This feed-forward tuning is done without a damper placed in the test-rig, so no
100 101 102 -60
-40 -20 0 20
Magnitude [dB]
Old controller New controller
100 101 102
Frequency [Hz]
-180 -90 0 90 180
Phase [°]
Figure 3.14: Open loop frequency responses of the test rig with the old feedback controller as designed by Goris [2], and the newly designed feedback controller.
external forces other than gravity are working on the hydraulic cylinder. Figure3.16also shows the position tracking without any feed-forward control, here, relatively large errors in position tracking can be seen. In order to further reduce these errors, the feed-forward parameters are tuned again.
The four feed-forward parameters that are tuned are: Kf st(static feed-forward), Kf a(acceleration feed-forward), Kf v(velocity feed-forward) and Kf c(Coulomb friction feed-forward). By following the procedure described by Gert Witvoet [29], these parameters are tuned and are changed slightly compared to the values Nick Feijen found.
The final values that are used as the feed-forward parameters are shown and compared to the old values in Table3.3. The units of the four feed-forward parameters are given in this table as well. In here, Kv is a gain that converts the feed-forward control inputs into a voltage output.
However, this gain Kvis unknown, it should be possible to determine it by measuring the relation between the voltage output of the control system and the force developed by the hydraulic ram.
Unfortunately, due to the closing of the university, this has net been done during this project.
The resulting position tracking with fully tuned feed-forward control can be seen in Figure 3.17, as can be seen, the large errors that were present in Figure3.16without any feed-forward control are gone. Figure3.18shows a comparison between the error without feed-forward control and the error with full feed-forward control. Again, this figure shows a considerable decrease in position tracking error for the full feed-forward control compared to no feed-forward control.
Table 3.3: New and old feed-forward parameters.
Feed-forward parameter Old New Kf st [kg · m/s2· Kv] 0.05 0.05 Kf a [kg · Kv] 0.00002 0.000026 Kf v [kg/s · Kv] 0.0034 0.00324 Kf c [kg · m/s2· Kv] 0.0 0.007
K
f stK
f cK
f aK
f v+ +
+ +
+ + sign(v)
a
v
Feed-forward output
Figure 3.15: Schematic representation of the feed-forward controller with the feed-forward para- meters Kf st, Kf c, Kf a and Kf v and the shock absorber velocity and acceleration as v and a respectively.
0 2 4 6 8 10 12 14
Time [s]
-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
Position [m]
Actual position Reference
Figure 3.16: Position tracking without any feed-forward control.
0 2 4 6 8 10 12 14 Time [s]
-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Position [m]
Actual position Reference
Figure 3.17: Position tracking with full feed-forward control.
0 2 4 6 8 10 12 14
Time [s]
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025
Position error [m]
Full feed-forward No feed-forward
Figure 3.18: Comparison of the position tracking errors with and without feed-forward control.
3.4 Summary and conclusions
In this chapter two ways to better estimate the shock absorber velocity have been created. First, an observer is created that estimates the shock absorber velocity based on hand tuned observer gains.
After that, a Kalman filter is used to estimate the shock absorber velocity with automatically tuned gains. The results of both the observer as well as the Kalman filter are compared and it is decided to use the Kalman filter for the shock absorber velocity estimation. Also, the acceleration sensor that is added to the test-rig as well as the extra hardware needed for this sensor are discussed.
Finally, the test-rig controllers are updated and compared to the previous controllers.
4. Shock absorber models and model fit- ting
In previous projects on the TU/e shock absorber test-rig the main aim was to develop physical models of different shock absorbers, [2] [17]. As discussed in Section2.3.1, these physical models generally use measurements to estimate parameters of the internal shock absorber architecture and valve assemblies. Because it is not always possible to get or measure these parameters of the internal shock absorber architecture and valve assemblies for every shock absorber, empirical (or non-parametric) models as explained in Section2.3.2can be helpful. These empirical models are not based on physics and therefore don’t need any measurements of the shock absorber’s dimensions and its internals.
First, this chapter describes the basic characteristics of a shock absorber in Section 4.1. After that it describes different non-parametric shock absorber models in Section 4.2. These models range from the simplest estimation of a linear damping coefficient to more complex models like a higher-order polynomial that is able to describe the hysteresis behaviour of a shock absorber and a hyperbolic tangent including hysteresis. Then, in Section4.3, the measurements and the fitting of the different models to the measurement data is explained and Section4.4compares the different shock absorber models to each other. Finally, Section4.5 gives a brief summary and draws some conclusions based on this chapter.
4.1 Shock absorber characteristics
A shock absorber’s characteristics consist of two main parts, firstly the general dimension data of a shock absorber (e.g. stroke, minimum and maximum length, diameters and the way it’s mounted to the vehicle). Secondly, the force characteristics of the shock absorber, which is the aim of this project to model. The force characteristics of a shock absorber are determined by its internals and the oil used in it. The main damping force of a shock absorber is caused by the oil flowing through the orifices and valves in the piston. This force can be measured by exciting the shock absorber with different velocities.
Next to that, another part of the shock absorber internals that causes a form of damping is mechanical friction between the moving parts. This sliding friction arises between different parts of the shock absorber, think of friction between the rod and the rod oil seal and friction between the piston oil seal and the shock absorber wall, [9]. These friction forces are difficult to measure because in order to minimize the velocity dependent oil flows in the shock absorber, slow movements of the shock absorber are required and these slow movements are hard to control accurately by the shock absorber test-rig, [2].
Next to these velocity dependent forces, static characteristics should also be considered. These static characteristics are, amongst others, a gas-force caused by the insertion of the rod into the shock absorber tube. Because this rod takes up volume in the shock absorber tube, the total volume of that tube decreases and that is compensated for by compressing the gas present in the gas chamber. This gas-force acts like a spring with stiffness. Another static characteristic is the Coulomb friction arising from rod and piston friction, [9]. All of these static characteristics can also only be determined by very low velocity measurements, [2].
Another shock absorber characteristic is hysteresis, this hysteresis is the results of the compress- ibility of the oil inside the shock absorber. The effects of this compressibility will only be seen in transient operation, so when acceleration of the shock absorber piston is large. For sinusoidal excitation, the highest accelerations occur at the end of the strokes, so when the position is the highest and the velocity is zero. The effect of the compressibility of then is to introduce hyster- esis into the sinusoidal force/velocity curve with the greatest spread at zero velocity. Because of this, generally speaking, the higher the frequency of the sine excitation, the higher the maximum acceleration, the bigger the amount of hysteresis, [9].
4.2 Shock absorber models
In this section, multiple different shock absorber models will be described that are developed during this project. First, a simple linear damping shock absorber model is given. Second, a linear model that uses two different damping coefficients for both positive and negative velocities is explained. After that, an extended polynomial models that uses a 4th order polynomial to describe the shock absorber force is introduced. And lastly, a hyperbolic tangent shock absorber model is discussed.
4.2.1 Linear model
The simplest way to describe the damping force of a shock absorber is by using a constant damping coefficient. Then, the damping force is described as a regular linear equation with the damper velocity times this damping coefficient plus a force offset, as in (4.1). Here, F is the shock absorber force, cd is the estimated damping coefficient, v is the shock absorber velocity and Foffset is the force offset.
F = cd· v + Foffset (4.1)
An example of a shock absorber force measurement performed on the TU/e shock absorber test-rig with this model fitted to it is shown in Figure4.1.
0 Velocity [m/s]
0
Force [N]
Estimated Measured
Figure 4.1: Example of a shock absorber force measurement performed on the TU/e shock absorber test-rig and the linear shock absorber model fitted to it.
4.2.2 Positive/negative velocity linear model
Most shock absorbers have different damping values for positive and negative velocities, [9]. This difference in damping for positive and negative velocities can be seen in Figure4.2as well. There- fore, this model uses linear damping coefficients just as the one described in Section 4.2.1, but
