Modeling of a World Rally Championship Car Damper and Experimental Testing of Its Components

Master’s Thesis

Modeling of a World Rally Championship Car Damper and Experimental Testing of Its Components

Author:

Richard Chahine rchahine@kth.se

Examiner: Lars Drugge, KTH Vehicle Dynamics Mentor: Erik Lönnqvist, Öhlins Racing AB Project Commissioner: Magnus Danek, Öhlins Racing AB

Executed at: Öhlins Racing AB

University: The Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering, Vehicle Dynamics Division

Version 6

Stockholm, June 2011

KTH Department of

Vehicle Dynamics

ii

Please note that most figures in this report are not drawn to scale or with high precision. They are

only used as an aid to make the text easier to understand.

iii

Abstract

Rally cars are driven on many different types of surfaces. Each type of surface demands a special type of damper setup. In order to achieve optimum performance on the snow covered and icy Swedish roads, the gravel of the Spanish rallies and the smooth tarmac of the German rallies, a large flexibility in the possible damper settings is required. Prodrive, a British motorsport group, has been racing two Mini Countryman as factory team cars for BMW Mini as of Rally D’Italia in Sardinia in May 2011 and has requested that Öhlins Racing AB equips these cars with dampers. Öhlins Racing AB has been developing a damper for rally applications called the TPX. This damper is equipped with an Active Rebound Control system (ARC). The ARC allows for high levels of grip to be achieved together with good chassis control.

The TPX damper with its ARC system is quite complex in structure. As there are many parts in the damper which can be altered, optimizing the damper would require a very large number of tests. A physical model of the TPX damper with its ARC system would reduce the time spent in the lab and help speed up the development of the damper. Prodrive would also like to a have a model of the damper that they can use in their model of the entire car which they use to setup the cars for races and to develop the car.

The goal of this Master’s Thesis was therefore to create a model based in MatLab Simulink that qualitatively but not necessarily quantitatively replicated the dampers behavioral trends.

Components which are very difficult to model, such as shim stacks, needed not be modeled. Their characteristics could be measured in the lab.

During this Master’s Thesis project a model for the TPX damper was created using Simulink to model

most of the physical parts of the damper. The rest of the model including its inputs and control were

taken care of by a GUI. The model functions so simulations can be performed. Plotting the results of

the simulations together with data from experimental tests was also made easy by the GUI. The

results from comparisons between the simulated damper and the real object indicate that

refinements need to be made to the model before it can be put to use as a tool for helping in

optimizing the TPX damper’s construction. Hysteresis in the form of friction as well as damper

flexibility does not seem to be negligible. The variation of the oil’s compressibility and the dynamic

behavior of the check valves also need to be looked into. The graphs from the simulations seem to

replicate the real dampers performance trends as intended. The numerical magnitudes of the data

produced by the simulation are however not accurate. Overall the model produced during this

master’s thesis seems to be a good step forward on the path to producing a useful model. Some

suggestions for the next steps in improving the model are provided.

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v

Table of Contents

1 Introduction ... 1

2 Problem Formulation: Project definition and boundaries ... 3

3 TPX Construction Functioning Principles ... 5

3.1 Rebound Stroke ... 6

3.2 Compression Stroke ... 7

3.3 PDS ... 8

3.4 ARC ... 8

3.5 Oil Reservoir ... 9

3.6 Gas Reservoir ... 10

4 Model Principles ... 11

4.1 Flow Restrictors ... 11

4.2 Shim Stacks ... 11

4.3 Chamber Characteristics ... 14

4.4 Gas Reservoir ... 15

4.5 Interaction between parts ... 16

5 Model Construction ... 17

6 GUI Control ... 29

6.1 Reservoir Needle Control ... 29

6.2 Low Speed Rebound Damping Control... 30

6.3 GUI as a Control Panel ... 32

6.3.1 Controls ... 32

6.3.2 Inputs ... 32

6.3.3 Shim stacks ... 33

6.3.4 Results display ... 33

6.3.5 Output plots ... 33

6.3.6 Error codes ... 34

6.3.7 Starting a simulation... 34

6.3.8 Save function ... 34

6.3.9 Load function ... 34

7 Shim Stacks Modeled vs. Shim Stacks Measured ... 35

7.1 The Experimental Setup ... 36

7.2 List of Performed Tests ... 36

7.3 Validation and Analysis... 37

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8 ARC Implementation ... 45

8.1 Test 1: ΔP vs. Q for the shim stack as a function of ARC chamber pressure ... 45

8.2 Test 2: ARC chamber stiffness: flow into the chamber versus change in ARC pressure. ... 46

8.3 Model Implementation ... 47

9 Conclusions ... 49

10 Suggestions for Future Work ... 51

11 Acknowledgements ... 52

12 References ... 53

1

1 Introduction

Rally cars are driven on many different types of surfaces. Each type of surface demands a special type of damper setup. In order to achieve optimum performance on the snow covered and icy Swedish roads, the gravel of the Spanish rallies and the smooth tarmac of the German rallies, a large flexibility in the possible damper settings is required. This results in the need for dampers with many variable parameters. In order to know how the damper behaves when each parameter is varied a large number of tests is required.

Öhlins Racing AB has been developing a damper for rally applications called the TPX. Prodrive, a British motorsport group, has been racing two Mini Countryman as factory team cars for BMW Mini as of Rally D’Italia in Sardinia in May 2011. Öhlins has received the order to equip these cars with dampers. The TPX dampers have been chosen for this application. The front dampers on these cars are equipped with Active Rebound Control (ARC); a system which causes the damping force on the rebound stroke to be increased when the rebound chamber pressure is high for a long period of time. This allows for high levels of grip to be achieved by having low rebound damping at frequencies around 16 Hz. It is the motion of wheel in the 16 Hz region that has the largest effect on the cars grip.

At the same time larger rebound damping forces can be used for motional frequencies of approximately 1.5 Hz. The cars motion in the 1.5 Hz region has the greatest affect on how the driver perceives the car. Low rebound damping in this region would result in a feeling of the car wobbling about. A high damping in this region would make the car feel more stable, responsive and predictable for the driver thereby improving its chassis control. In this way the ARC system allows for both high grip and high chassis control to be achieved.

The dampers have been tested together with the team drivers on several occasions and the results were good. The drivers liked the new Active Rebound Control system. As there are many parameters in the Active Rebound Control unit and the damper that can be altered, optimizing a damper with the ARC system would require a very large number of tests. A model of the damper with the ARC would reduce the amount of tests needed by making it possible to judge what effects certain changes to the damper settings have. A model could therefore save considerable time in the optimization process.

This Master’s Thesis was therefore commissioned by Öhlins Racing to create such a model that could be used to study the damper’s performance trends when it settings are altered. When a good setting is found experimental tests can be performed with similar settings to find the best one. This way the number of tests can be reduced.

Erik Lönnqvist, who is the mentor for this Master’s Thesis, has worked on a Simulink model for a TT44 damper as his Master’s Thesis dated year 2000 [1]. This report was used together with the Öhlins’

manual for the TT44 damper [2] to understand how the model was constructed. Although the TT44

and the TPX dampers are very different both in their construction and their working principles the

basic approach to modeling the damper could be used as a start off point for building a model for the

TPX.

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3

2 Problem Formulation: Project definition and boundaries

The main purpose of this Master’s Thesis is to create a model that can be used to predict an Öhlins TPX damper’s behavior when its components’ parameters are altered. The boundaries set for this Master’s Thesis are as stated:

 The model shall be built it MatLab Simulink.

 The model shall be a qualitative and does not necessarily have to be quantitatively correct. In other words, one should be able to read from the simulations the damper’s behavioral trends but not necessarily know how much damping one gets at a certain damper velocity.

 Components which are very difficult to model, such as shim stacks, need not be modeled.

Their characteristics will be measured in the lab.

 The Positional Damping System (PDS) is disregarded in this model.

 The patent pending ARC valve mechanism’s characteristics will be partially experimentally attained and partially theoretically modeled.

 Hysteresis (flexibility) in the damper construction will be experimentally measured not modeled.

 Hysteresis (compressibility) in the damper oil due to its constituents and air mixture will also be taken forth experimentally.

 Friction is very difficult to model in this case and will be initially disregarded. Compensation factors may be considered further on in the project.

The secondary goal of this Master’s Thesis is to provide a model that Prodrive, a British rally team,

can use in their model of the entire rally car.

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5

3 TPX Construction Functioning Principles

The new TPX damper system makes use of two pistons, see Figure 1. Each piston has holes in it.

Some of these holes are blocked by a shim stack and others by a check valve. The shim stack and the check valve are located on opposite sides of the piston. The rebound shim stack is located on the rebound piston and the compression shim stack on the compression piston. The rebound shim stack opens during the rebound stroke and the compression shim stack during the compression stroke. The check valve located on the rebound piston is referred to as the rebound check valve. However this rebound check valve opens during the compression stroke. The compression check valve which is located on the compression piston opens during the rebound stroke.

The two piston construction divides the main structure of the damper into three parts: a rebound chamber, a compression chamber and a middle chamber. The middle piping and the middle chamber are connected by holes in the piston shaft. These holes are large enough for the middle chamber and middle piping to be considered as one entity for the modeling purposes.

Figure 1: Overall dissected view of a TPX damper [3].

Compression chamber

Rebound chamber Middle chamber

ARC Rebound shim stack

Compression check valve located here

Compression shim stack located here

Rebound check valve located here

Compression piston

Rebound piston PDS

Middle piping

(inside rod)

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3.1 Rebound Stroke

Low speed

During a low speed rebound stroke, the rebound chamber pressure is too low to force the rebound shim stack to open. The oil from the rebound chamber can then go through a hole in the piston shaft to the middle chamber and compression chambers, as shown in Figure 2. The holes between the rebound chamber and middle piping and the compression chamber and middle piping act as restrictions to the flow in between these chambers. This occurs at low damper speeds.

Figure 2: Oil flow during a Low speed Rebound stroke [3].

High Speed

If high pressures are reached in the rebound chamber during a rebound stroke, the rebound shim

stack will open allowing the damper oil to move to the middle chamber. The oil flow in this scenario

is shown in Figure 3. If the middle chamber pressure rises, the compression check valve will open

sending oil into the compression chamber. This occurs at higher damper speeds in parallel to the low

speed flow which continues to take place at high speeds.

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Figure 3: Oil flow during a High speed Rebound stroke [3].

3.2 Compression Stroke

Low speed

In a similar manner to the events during a rebound low speed stroke during a low speed compression stroke, the compression chamber pressure is too low to force the compression shim stack to open.

The oil from the rebound chamber can then go through a hole at the top of the piston shaft to the middle and rebound chambers as shown in Figure 4. The holes between the rebound chamber and middle piping and the compression chamber and middle piping act as restrictions to the flow in between these chambers. This occurs at lower damper speeds.

Figure 4: Oil flow during a Low speed Compression stroke [3].

8 High Speed

If high pressures are reached in the compression chamber during a compression stroke, the compression shim stack will open allowing the damper oil to move to the middle chamber. If the middle chamber pressure then rises, the rebound check valve will open sending oil into the rebound chamber. This occurs at higher damper speeds in parallel to the low speed flow which continues to take place at high speeds. The direction of the oil flow in such a case is shown in Figure 5.

Figure 5: Oil flow during a High speed Compression stroke [3].

3.3 PDS

The Positional Damping System (PDS) provides extra damping when the damper is nearing its maximum compressed position. As this system does not have an impact on how the damper functions under ‘normal’ conditions it can be disregarded in this damper model.

3.4 ARC

The ARC system, shown in Figure 6, provides high damping forces at low motional frequencies

together with low damping forces at higher frequencies of motion. The ARC chamber is a separate

chamber located within the middle chamber. A tight canal is present between the ARC chamber and

the rebound chamber. As rebound pressure increases during a rebound stroke the pressure

difference between the ARC chamber and the rebound chamber causes damper oil to flow from the

rebound chamber to the ARC chamber. A shim with a hole in it, located between the canal and the

ARC chamber, serves as the main flow restrictor. The ARC chamber has flexible walls made of shims

and o-rings. As the pressure in the chamber builds up these flex outwards. One of the shims is

connected to a disk which in turn rests on the rebound shim stack. So as the pressure in the rebound

chamber builds up, the pressure in the ARC chamber increases causing its walls to flex. One of the

ARC chamber’s walls in turn applies a force on to the rebound shim stack thereby increasing the

pressure difference between the middle chamber and the rebound chamber required to open the

rebound shim stack.

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Figure 6: The ARC chamber [3].

3.5 Oil Reservoir

The middle chamber is connected to the oil reservoir via a pipe located within the piston rod.

Between the middle chamber and the oil reservoir there is a low speed needle that limits the flow in- between the two chambers. The needle position can be controlled by an external adjuster. If the pressure in the middle chamber is high enough a shim stack between the middle chamber and the oil reservoir opens and allows oil to flow into the reservoir. The preload, which controls the threshold for when the shim stack first opens, can be adjusted via an external adjuster. If the pressure in the middle chamber drops below a certain level a check valve also located between the oil reservoir and the middle chamber opens allowing oil to flow from the oil reservoir to the middle chamber. A picture of the oil and gas reservoirs is shown in Figure 7.

Figure 7: Oil and gas reservoirs [3].

Gas Reservoir

Oil Reservoir

Separator Piston

ARC chamber

Rebound shim stack Figure removed at the

request of Öhlins Racing AB

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3.6 Gas Reservoir

The gas reservoir applies a pressure onto the oil reservoir and thereby sets the base pressure for the

entire system. As mentioned earlier the oil reservoir is connected to the middle chamber which in

turn is connected to the rebound and compression chambers. This allows the gas reservoir pressure

to be applied to the entire damper. This type of construction prevents cavitations in the damper

which is a great asset. The presence of a gas reservoir allows for the compensation of the piston shaft

volume which enters the damper during a compression stroke and exits the damper during a

rebound stroke. The damper oil in itself is not compressible enough to compensate for the piston

shaft volume entering and exiting the damper. The oil displaced by the piston shaft upon it entering

the damper is indirectly sent to the oil reservoir where it causes the separator piston to move

compressing the gas in the gas reservoir. This will cause the system pressure to be slightly higher

when the piston is further inside the damper compared to when the damper is at its fully extended

position.

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4 Model Principles

Mathematical models are created for the flow restrictors, shim stacks, chamber characteristics and the gas reservoir. Each of these mathematical models is used in several places within the Simulink model. The data specific to each damper part is used together with a combination of these four basic mathematical models to create the blocks that model the part.

4.1 Flow Restrictors

Flow restrictors such as the low speed holes between the rebound chamber and middle chamber and the compression chamber and the middle chamber as well as the oil reservoir needle have their opening areas calculated from damper drawings found in the Öhlins database [3]. These opening areas are then used in the Bernoulli equation. The loss term in the generalized Bernoulli equation corresponding to an area decrease followed by an area increase is represented by equation 1 according to a fluid mechanics handbook [4]. This equation is used to calculate the flow past a restriction for the instantaneous difference in pressure between the two chambers that the flow restrictor links. Turbulent flow is assumed at all times.

(1)

where:

is the flow past the restriction orifice

is the discharge coefficient, here considered at a 90° edge

is the pressure drop over the restriction orifice, in other words, the difference in pressure between the two chambers the restriction orifice links

is the damper oil’s density

In the Master’s Thesis report by Erik Lönnqvist [1], Erik mentions that H. Lang, in his PhD dissertation, A Study of the Characteristics of Automotive Hydraulic Dampers at High Stroking Frequencies, at the University of Michigan in 1977, developed analytical and empirical expressions for discharge coefficients. The conclusion was that a model is too complex and a constant value of 0.7 resulted in good correlation with experimental results.

4.2 Shim Stacks

The shim stacks and the check valves function in similar ways as the check valves are actually simple

shim stacks in the case of the TPX damper. The shim stack characteristics are too complex to allow

for theoretical modeling. To get the shim stacks’ characteristics into the model they were placed in a

flow bench and the flow past the shims was measured as a function of the pressure drop over the

shim stack. The flow was first increased and then decreased. The results from the Flow bench test

are summarized in Figure 8.

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Figure 8: The flow past the shim stacks as a function of the pressure drop over the shim stacks for 5 of the shim stacks present in the damper.

As one can see in Figure 9, the same pressure drop over the shim stack results in two different flows past the shim stack depending on whether the flow is being increased or decreased.

Figure 9: The flow past the rebound shim stack as a function of the pressure drop over it.

The data from the flow bench was therefore divided into two portions, one for when the flow is increasing and one for when the flow is decreasing. The data was filtered and the red and the green lines attained. This is shown in Figures 10 and 11, where Figure 11 is an enlarged picture of the central portion of Figure 10. In a case in which the flow increases for a short time and then drops again the two curves might be closer to one another. This is however not taken into consideration in the model.

0 10 20 30 40 50 60 70 80 90

0 0.5 1 1.5 2 2.5

ALL 5

Flow [l/min]

Delta Pressure [MPa]

Comp shims Reb Shims ARC Comp CV Reb CV Reservoir Shims

0 10 20 30 40 50 60 70 80 90

0 0.5 1 1.5 2 2.5

Rebound Shimstack ARC

Flow [l/min]

Delta Pressure [MPa]

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Figure 10: The flow past the rebound shim stack as a function of the pressure drop over it with the filtered data and the linear model added. The filtered data is too close to the original data to be seen

in this figure. Therefore, an enlargement is provided in Figure 11 below.

Figure 11: An enlargement of a portion of Figure 10, showing the green, the blue, and the red lines.

A linear model, adapted to the data by the author, is represented by the blue line positioned in between the red and the green ones as shown in Figures 10 and 11. When the model operator selects, “shim stacks measured” in the GUI control panel either the red or the green lines are used to decide the flow past a shim stack for a certain pressure drop over the shim stack depending on whether the flow is increasing or decreasing. If “shim stacks modeled” is selected the blue line is used to decide the flow past a shim stack for a certain pressure drop.

0 0.5 1 1.5

x 10^-3 0

0.5 1 1.5 2

2.5x 10⁶ Rebound Shimstack ARC

Flow [m³/s]

Delta Pressure [Pa]

7 7.1 7.2 7.3 7.4 7.5

x 10^-4 1.25

1.3 1.35 1.4 1.45 1.5 1.55 1.6

x 10⁶ Rebound Shimstack ARC

Flow [m³/s]

Delta Pressure [Pa]

Measured data Measured

data Filtered data:

Green line

Blue line:

Linear model Filtered data:

Red line Blue line:

Linear model

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This is done for all the shim stacks and check valves in the model. In the case of the check valves which are also shim stacks only the linear model is used; this being the red line shown in Figure 12.

Both the rebound and compression check valves, not only the rebound check valve shown in Figure 12, displayed what seems to be a very dynamic behavior. This was suspected by Öhlins’ engineers.

The exact data from the flow bench measurement for the check valves was therefore not used but the linear model shown in Figure 12 was used instead. The reservoir check valve could not be placed in the flow bench without the construction of a special adapter. The reservoir check valve data was set to be the slope of the line for the rebound check valve together with half the preload of the compression check valve. This should be an acceptable temporary solution.

Figure 12: The flow past the rebound check valve as a function of the pressure drop over the check valve.

4.3 Chamber Characteristics

The rebound, compression and middle chambers along with the oil reservoir are all modeled as fixed volumes with flows entering and exiting them. Using the expression for fluid compressibility [5], equation 2, the pressure in a chamber is calculated using the flows entering and exiting the chamber together with the oil’s compressibility.

(2)

where:

is the change in chamber pressure

0 0.5 1 1.5

x 10^-3 0

0.5 1 1.5 2 2.5 3 3.5

4x 10⁵ Rebound check valve (opens during compression)

Flow [m³/s]

Delta Pressure [Pa]

Linear model:

Red line

Measured

data

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is the change in chamber’s volume

is the damper oil’s compressibility

is the chamber’s volume

The change in the chamber’s volume can be calculated using equation 3.

(3) where:

t is the time

is the flow out of the chamber

is the flow into the chamber

By inserting equation 3 into equation 2 and then integrating, with the initial gas pressure set to

, the instantaneous pressure in a chamber can be calculated as shown by equation 4.

(4)

where:

is the gas pressure in the damper at the start of simulation

No tests have been performed to study the oils compressibility. The value for the oils compressibility was taken from Erik Lönnqvist’s Master’s Thesis report [1] and should, according to the engineers at Öhlins Racing, not vary that much from the compressibility of the oil in the TPX damper. There are, however, suspicions that the oils compressibility varies with temperature and pressure. The value used for the oil’s compressibility

is

[1/Pa] and for the oil’s density

is [kg/m

].

4.4 Gas Reservoir

Nitrogen gas is used to pressurize the oil in the damper. The change in gas pressure when the damper is in motion can be modeled under dynamic conditions, according to [1], by a reversible, almost adiabatic, polytropic process described by equation 5.

(5)

where:

is the initial gas pressure is the initial gas volume

is the final or current gas pressure is the final or current gas volume

is the polytropic exponent of nitrogen = 1.3

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The initial gas pressure can be used together with the oil reservoir pressure to calculate the force on the piston which separates the gas and the oil according to equation 6.

(6) where:

is the force on the separator piston

is the oil reservoir pressure

is the pressure in the gas chamber

is the area of the separator piston

The separator piston’s velocity and position can then be calculated by integrating its acceleration which can be calculated from the force on the separator piston according to equation 7. The initial conditions for both the separator piston’s velocity and the separator piston’s position are considered to be their values at the beginning of the simulation when the damper is standing still and can therefore be set to zero.

(7) where:

is the separator piston’s mass

is the separator piston’s acceleration

The separator piston’s velocity can then be used in equation 8 to calculate the rate of change of oil reservoir volume. This rate of change of volume is the equivalent of a flow into or out of the oil reservoir.

(8) where:

is the oil flow in or out of the oil reservoir

is the separator piston’s velocity.

The separator piston’s position can then be used to calculate the change in gas reservoir volume which in turn can be used with equation 5 to calculate the new instantaneous gas pressure in the gas reservoir. This process is then continuously looped to calculate the gas pressure at each point in the simulation timeline.

4.5 Interaction between parts

Signal lines join the different parts of the model. Plenty of loops are present as pressures in different

chambers affect the flow in between them and the flow in between two chambers affects the

pressure in both chambers.

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5 Model Construction

The model of the TPX damper is built in Simulink. The different levels in the model are described below. Figure 13 shows a tree diagram containing only the major subsystems of the model. The input to the damper model is the damper velocity curve that the model is to follow. The main output from the model is the damping force that the damper gives. The highest level in the simulink model is shown in Figure 14. Figures 15 to 34 show the contents of the subsystem that is discussed in the corresponding section.

Figure 13: A tree diagram showing the major subsystems of the model.

Figure 14: Overall model view, the highest level of the model.

Model

GUI input

control TPX with ARC

Volume

Calculations Compontents

ARC Chamber Compression

Chamber Compression

check valve Compression

low speed Gas and oil

reservoirs

Gas Reservoir

Gas Reservoir Separator

Piston Oil Resrvoir

Reservoir Shim Stacks and Reservoir

Pipes Main Compression

Shim Stacks

Main Rebound Shim

Stacks Middle Piping Rebound

Chamber Rebound

check valve Rebound low

speed Pressure sum

on main piston =>

Damping force

damping_force

To Workspace2

position

To Workspace1

Damper_velocity

To Workspace Damper velocity

Damping force

Piston position

TPX with ARC Results

Damper velocity curve GUI input control

18 A GUI input control

This subsystem contains the blocks necessary to make it possible for the GUI to control the damper velocity input signal to the Simulink model.

Figure 15: The contents of the GUI input control subsystem

B TPX damper

This subsystem contains the blocks that make up the physical model of the damper. It contains three of its own subsystems: Volume Calculations, Components subsystem, and Force on Main Pistons.

Figure 16: The contents of the TPX Damper subsystem

B.A Volume calculations

Here the damper velocity is used to calculate the main piston’s position and the rebound and compression chamber volumes as well as the rate of change of their volumes.

1 Damper velocity curve Multiport

Switch GUI_input_signal

-ve step -ve sinus

+ve step +ve sinus

2 Piston position

1 Damping force Damper velocity= piston velocity

Cormpression volume Compression volume dot Rebound volume Rebound volume dot Piston position [m]

Volume calculations Simulation time

Compression chamber oil pressure

Rebound chamber oil pressure

Force

Presssure sum on main piston

=> Damping force Cormpression volume

Compression volume dot Rebound volume Rebound volume dot

Compression chamber oil pressure

Rebound chamber oil pressure Components

Clock

1 Damper velocity

19

Figure 17: The contents of the Volume Calculations subsystem.

B.B Components

This is the main subsystem at this level. Its contents represent the dampers actual components and include: rebound check valve, compression check valve, rebound low speed, compression low speed, main shim stacks, compression chamber, rebound chamber, middle piping, gas and oil reservoirs, as well as the ARC chamber.

Piston Position

Piston Velocity

5 Piston position [m]

4 Rebound volume dot

3 Rebound volume

2 Compression volume dot

1 Cormpression volume -1

-1 -K-

Piston_area_rebound1 -K-

Piston_area_rebound -K-

Piston_area_compression1 -K- Piston_area_compression 1/s

Integrator Limited

-C- Initial Rebound chamber volume

-C- Initial Compression

chamber volume

Add1 Add 1

Damper velocity=

piston velocity

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Figure 18: The components subsystem.

Compression check valve opens during rebound and rebound check valve open during compression! 2 Rebound ch oil pressu1 Compression oil pressu Middle_chamber_pressure To Workspace2Rebound_chamber_pressure To Workspace1

Compression_chamber_pressure To Workspace Middle piping pressure Rebound Chamber pressureFlow to middle piping from rebound chamber Rebound low speed Middle piping pressure Rebound Chamber pressureFlow to rebound chamber from middle chamber via check valve Rebound check valve

flow past rebound shim stack flow to ARC Rebound volume Rebound volume dot Flow to middle piping from rebound chamber Flow to rebound chamber from middle chamber via check valve Rebound chamber oil pressure Rebound Chamber

Flow out to oil reservoir Flow past compression check valve Flow to middle piping from compression chamber Flow past compression shim stacks to middle chamber Flow past rebound shim stacks to middle chamber Flow to middle piping from rebound chamber Flow past rebound check valve

Pressure in middle piping Middle Piping Middle chamber pressure ARC pressure Rebound chamber oil pressureFlow past rebound shim stacks Main Rebound Shim Stacks Compression chamber oil pressure Middle chamber oil pressureFlow past compression shim stacks Main Compression Shim Stacks

Pressure in middle pipingOil flow into reservoir Gas and oil reservoir Compression Chamber pressure Middle piping pressureFlow to middle piping from compression chamber Compression low speed

Compression Chamber pressure Middle piping pressureFlow to compression chamber from middle chamber via check valve Compression check valve Flow to compression chamber from middle chamber via check valve Flow to middle piping from compression chamber Compression volume Compression volume dot flow past compression shim stack

Compression chamber oil pressure Compression chamber Rebound Chamber pressureARC Pressure Flow to ARC ARC Chamber 4 Rebound volume dot

3 Rebound volume 2 Compression volume dot

1 Cormpression volume

21 B.B.A Rebound check valve

This box contains the implementation of the linear shim stack model, discussed in section 4.2 above, for the rebound check valve.

Figure 19: The contents of the Rebound check valve subsystem.

B.B.B Compression check valve

This box is similar in structure to the one for the rebound check valve but here the data for the compression check valve is implemented.

Figure 20: The contents of the Compression Check valve subsystem.

B.B.C Rebound low speed

This subsystem contains the flow restrictor equation discussed in section 4.1. The opening area of the restriction and other data corresponding to this specific restriction are used. The opening area of the restriction can be varied by changing the number of clicks in the GUI control panel.

1

Flow to rebound chamber from middle chamber

via check valve -C-

rebound cv preload

Flow is only allowed if the preload pressure

is overcome 0

-K-

1/k compression cv 2

Rebound Chamber pressure

1 Middle piping pressure

1

Flow to compression chamber from middle chamber

via check valve -C-

compression cv preload

Flow is only allowed if the preload pressure

is overcome 0

-K-

1/k compression cv 2

Middle piping pressure 1

Compression Chamber pressure

22

Figure 21: The contents of the Rebound low speed subsystem.

B.B.D Compression low speed

This box is similar to the one for the rebound low speed but contains the data corresponding to the compression low speed. The flow restrictor equation is also used here.

Figure 22: The contents of the Compression low speed subsystem.

B.B.E Rebound shim stacks

This box contains the implementation of two look up tables with the shim stack characteristics discussed in section 4.2 as well as the linear approximation of the shim stack characteristics also discussed in section 4.2 above. These are implemented here for the rebound shim stacks.

Representation of Q=Cd*A*sqrt(2*delta_p/rho_oil)

1 Flow to middle piping from rebound chamber rho_oil

rho_oil

u Sqrt

-1 -K-

Gain Divide

-K- Cd_rebound

Allows the flow to be -ve (ie reverse direction) if delta_p is negative even though sqrt and abs are used when calculating the flow Add

|u|

Abs -K-

A_LS_rebound 2

Rebound Chamber pressure

1 Middle piping pressure

Representation of Q=Cd*A*sqrt(2*delta_p/rho_oil)

1 Flow to middle piping from compression chamber rho_oil

rho_oil

u Sqrt

-1 2

Gain Divide

-K- Cd_compression

Allows the flow to be -ve (ie reverse direction) if delta_p is negative even though sqrt and abs are used when calculating the flow Add

|u|

Abs -K-

A_LS_compression 2

Middle piping pressure 1 Compression Chamber pressure

23

Figure 23: The contents of the Rebound shim stacks subsystem.

B.B.F Compression shim stacks

This box is similar to the one for the rebound shim stacks in structure but in this case it contains the data for the compression shim stacks.

Figure 24: The contents of the Compression shim stacks subsystem.

B.B.G Compression Chamber

This subsystem contains the chamber characteristics described in section 4.3 above. As discussed earlier these are used to calculate the pressure in the chamber. The data in this block is specific to the compression chamber.

1 Flow past rebound shim stacks -C-

rebound preload

Switch3

Switch1 -C-

Slope control rebound side 0 Scope2

Multiport Switch2

Multiport Switch1

Lookup Table3 - Lookup Table2 +

Lookup Table (2-D)

ARC_on_off

Divide1 du/dt

Derivative Shimstack_measured_modeled

Add2 Add3 3

Rebound chamber oil pressure

2 ARC pressure 1

Middle chamber pressure

1

Flow past compression shim stacks -C-

compression preload

Switch2 Switch 0 -C-

Slope control compression side

Multiport Switch

Lookup Table1 - Lookup Table +

Divide

du/dt Derivative1

Shimstack_measured_modeled Add2

Add 2

Middle chamber oil pressure 1

Compression chamber oil pressure

24

Figure 25: The contents of the Compression chamber subsystem.

B.B.H Rebound chamber

This subsystem contains the chamber characteristics described in section 4.3 above. As discussed earlier these are used to calculate the pressure in the chamber. The data in this block is specific to the rebound chamber.

Figure 26: The contents of the Rebound chamber subsystem.

delta_pc/dt

1

Compression chamber oil pressure

p_gas_initial p gas initial beta_oil

beta_oil

xo1/s Integrator Divide

Add 5

flow past compression shim stack 4

Compression volume dot

3 Compression volume

2

Flow to middle piping from compression chamber

1

Flow to compression chamber from middle chamber via check valve

delta_p/dt

1 Rebound chamber

oil pressure p_gas_initial

p gas initial

beta_oil beta_oil

1 xos Integrator Divide

Add 6

Flow to rebound chamber from middle chamber via check valve 5

Flow to middle piping from rebound chamber 4

Rebound volume dot

3 Rebound volume

2 flow to ARC

1

flow past rebound shim stack

25 B.B.I Middle piping

This subsystem contains the chamber characteristics described in section 4.3 above. As discussed earlier these are used to calculate the pressure in the chamber. The data in this block is specific to the middle chamber.

Figure 27: The contents of the Middle piping subsystem.

B.B.J Gas and oil reservoirs

This subsystem contains the gas reservoir and oil reservoir subsystems.

Figure 28: The contents of the Gas and Oil Reservoirs subsystem.

B.B.J.A Gas reservoir and separator piston

The gas reservoir subsystem makes use of the equations presented in section 4.4 to calculate the change in gas pressure and the rate of change of the oil reservoir volume.

delta_p_m/dt

1 Pressure in middle piping p_gas_initial

p_gas_initial

beta_oil beta_oil middle_piping_volume

Middle piping volume

1 xos Integrator Divide

7

Flow past rebound check valve 6

Flow to middle piping from rebound chamber

5

Flow past rebound shim stacks to middle chamber 4

Flow past compression shim stacks to middle chamber 3

Flow to middle piping from compression chamber

2

Flow past compression check valve 1

Flow out to oil reservoir

1 Oil flow into reservoir Oil_reservoir_pressure

To Workspace Oil reservoir pressure

Pressure in middle piping Oil flow into reservoir Reservoir shim stacks

& reservoir pipes Oil reservoir volume

Oil reservoir volume dot

Oil flow into reservoir

Oil Reservoir Pressure

Oil Reservoir Oil reservoir pressure

Gas Reservoir volume

Oil reservoir volume dot Gas Reservoir & separator piston

1 Pressure in middle piping

26

Figure 29: The contents of the Gas Reservoir and Separator Piston subsystem.

B.B.J.A.A Gas reservoir

This subsystem contains equation 5 which depicts the change in gas pressure as a function of the displacement of the separator piston.

Figure 30: The contents of the Gas Reservoir subsystem.

B.B.J.A.B Separator piston

This subsystem contains the equations describing the motion of the separator piston as discussed in section 4.4.

Figure 31: The contents of the Separator Piston subsystem.

Separator piston velocity

Separator piston position net force on separator piston

gas pressure

Gas reservoir volume 2

Oil reservoir volume dot

1 Gas Reservoir volume -K-

separator piston area1 -K-

separator piston area

-K- separator piston

area2 -1

Gas_reservoir_pressure To Workspace

Net force on separator pistonSeparator piston velocity Separator piston position Separator piston

-C- Initial oil volume -C-

Initial gas volume

Gas reservoir volume Initial gas volumeGas presssure

Gas reservoir Add2

Add1 Add

1 Oil reservoir pressure

1 Gas presssure -C-

polytropic exponent for Nitrogen

Product uv

Math Function

-C- Initial gas pressure Divide

2 Initial gas volume

1 Gas reservoir volume

separator piston acceleration

2 Separator piston

position 1 Separator piston

velocity -C- 0

Separator piston mass

1 xo s Integrator1 1/s

Integrator Divide

1

Net force on separator piston

27 B.B.J.B Oil reservoir

The oil reservoir subsystem houses the chamber characteristics described in section 4.3. As discussed earlier these are used to calculate the pressure in the reservoir.

Figure 32: The contents of the Oil Reservoir subsystem

B.B.J.C Reservoir shim stacks and reservoir pipes

The subsystem named Reservoir shim stacks and reservoir pipes contains a model of the reservoir shim stacks. These shim stacks are modeled in the same way as the compression and rebound shim stacks mentioned earlier. The reservoir low speed flow was calculated using the restriction equation in a similar way to the compression low speed flow and the rebound low speed flow.

Figure 33: The contents of the Reservoir Shim stacks and Reservoir pipes subsystem.

delta_p_res/dt

1 Oil Reservoir

Pressure p_gas_initial

p_gas_initial beta_oil

beta_oil

1 xos Integrator Divide

Add 3

Oil flow into reservoir 2

Oil reservoir volume dot

1 Oil reservoir volume

delta_p

Shimstacks

Bypass

Oilflow into reservoir past shim stacks

Oilflow into reservoir through low speed oil reservoir piping

Check valve

Oilflow into reservoir via check valve (this value is either negative or zero as the check valve only opens to allow flow out of the reservoir)

Reservoir cv preload

both r plus since it works in the opposite direction

1 Oil flow into reservoir -K-

slope control

rho_oil rho_oil -C- reservoir preload

Switch3

Switch1 Switch

u Sqrt

Scope1

Multiport Switch

Lookup Table1 - Lookup Table +

-1

-1 -K-

Gain Divide

du/dt Derivative1

Shimstack_measured_modeled

-C-

0 Constant1

0 Constant

-K-

Cd_reservoir

Allows the flow to be -ve (ie reverse direction) if delta_p is negative even though sqrt and abs are used when calculating the flow

|u|

Abs -K-

A_LS_reservoir

-K-

1/k_reservoir_cv 2

Pressure in middle piping 1 Oil reservoir pressure

28 B.B.K ARC chamber

This block contains the information relevant to the ARC system. This will be discussed in detail later in section 8.3.

B.C Force on main piston

The force on the main piston is calculated from the pressure executed by the compression chamber, middle chamber and rebound chamber on the main piston. This is equal to the damping force provided by the damper.

Figure 34: The contents of the Force on Main Piston subsystem.

1 Force

-K- Shaft Area

-K-

Piston_area_rebound -K-

Piston_area_compression

-C- Atmospheric Pressure

Add 2

Rebound chamber oil pressure

1 Compression chamber

oil pressure