Conceptualisation and Analysis of an

Automotive Shock Absorber

with Integrated Hydraulic Mount

by

Christopher Roman Urbaniak

A thesis

presented to the University of Waterloo

in fulfillment of the

thesis requirement for the degree of

Master of Applied Science

in

Mechanical Engineering

Waterloo, Ontario, Canada, 2006

© Christopher Roman Urbaniak 2006

ii

I hereby declare that I am the sole author of this thesis.

I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the

purpose of scholarly research.

Signature

I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other

means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly

research.

Signature

iii

Abstract

In the development of an automotive suspension system, ride comfort and handling often present

conflicting dynamic stiffness and dynamic damping requirements. It is not enough to simply increase

system dynamic stiffness or damping to deal with body and wheel mode resonance problems because

high frequency noise and vibration will be accentuated. To address the inherent difficulties in

meeting these needs, a passive shock absorber with integrated hydraulic mount components is

presented. A hydraulic mount has the benefit of producing a tuned mass absorber effect, which can

be tuned to increase dynamic stiffness at a particular frequency without adverse effects at other

frequencies.

Two classes of conceptual designs are studied: the first combines a shock absorber with a hydraulic

mount attached externally, and the second integrates a hydraulic mount decoupler device with the

internal workings of the shock absorber. Several physical embodiments are presented, with a detailed

analysis performed on three designs in total.

When compared to the road handling properties of standard shock absorbers, the two internal

integration designs presented show more than double the improvement of the external design, with no

negative effect on ride comfort. The two favourable designs exhibit similar quantitative

improvements, but different qualitative behaviour. The first, Model A, has a narrow frequency band

of increased dynamic stiffness, suitable for targeting particular behaviour without affecting other

frequency ranges. The second, Model B, has a much wider frequency band of increased dynamic

stiffness in the 1-30 Hz range, but also has a large decrease in dynamic stiffness at frequencies greater

than 30 Hz. This high frequency effect is very beneficial to ride comfort.

Finally, future considerations presented include nonlinear damper orifice modelling, physically

increasing allowable decoupler travel, and creating a semi-active version of the modified shock

absorber. It is recommended that further study be performed on Models A and B in an effort to

commercialise the designs.

iv

Acknowledgements

So after two years’ worth of “blood, sweat and tears”, and a little Frisbee thrown in for some fun, my

thesis is complete; it’s been a good ride. However, I could never have done it without the guidance,

assistance, and support of so many. At the risk of leaving out a name or two, I would like to mention

those who have been especially inspirational.

First and foremost, a big Thank You goes out to my beautiful wife, Andrea, for being steadfast

beside me through my schooling. You’ve always been there for me through the fun times and the

tough times. I love you so much.

Thank you Mom and Dad for constantly encouraging me, asking the “right questions”, and

generally keeping me on the straight and narrow. You’re probably saying I make you so proud, but I

think the truth is actually the other way around. Thank you, as well, to my mom and dad “in law” for

your home away from home.

I have been very fortunate to be the recipient of excellent professional and academic advice. Prof.

Amir Khajepour, my supervisor, has always been very approachable, willing and able to guide me

through the technical issues inherent in researching and writing a thesis. My colleague Orang Vahid

has been tremendous, mentoring me right from the get-go. Prof. John McPhee, whose selfless act two

and a half years ago was critical to my starting grad school, has been backing me for more than half a

decade now. Thank you Prof. McPhee and Prof. Fathy Ismail for reading my thesis in a prompt

manner.

Thank you to Joe Mihalic, Joseph Liu, and the rest of the guys at Cooper Standard in Mitchell for

helping me learn about four-posters and vibration testing; to Phong Vo and his team at General

Dynamics Land Systems – Canada for encouraging me and illustrating that a Master’s is within reach;

to Brett McAllister for that important help in fall ’04, and your continued friendship; and to Joe,

Sami, Saleh, and the rest of the office for making me laugh and keeping me sane.

Be sure to keep in touch, everyone!

v

Table of Contents

Chapter 1

Introduction.........................................................................................................1

1.1 Literature Review ......................................................................................................................... 3

1.2 Research Goals ............................................................................................................................. 8

1.3 Thesis Outline............................................................................................................................... 9

Chapter 2

Background ...................................................................................................... 11

2.1 Shock Absorbers......................................................................................................................... 12

2.1.1 Basic Structure..................................................................................................................... 12

2.1.2 Shock Absorbers vs. Struts .................................................................................................. 13

2.1.3 Road Inputs.......................................................................................................................... 15

2.1.4 Modelling and Dynamics..................................................................................................... 16

2.1.5 Testing ................................................................................................................................. 19

2.2 Tuned Mass Absorbers ............................................................................................................... 21

2.3 Hydraulic Mounts....................................................................................................................... 25

2.4 Parameter Optimisation .............................................................................................................. 28

2.5 Sensitivity Analysis .................................................................................................................... 29

Chapter 3

Shock Absorber with Hydraulic Mount ........................................................ 31

3.1 Concept....................................................................................................................................... 31

3.2 Equations of Motion ................................................................................................................... 33

3.3 Parameter Optimisation .............................................................................................................. 36

3.4 Simulation Results...................................................................................................................... 37

vi

3.5 Summary..................................................................................................................................... 42

Chapter 4

Shock Absorber with Internal Decoupler ..................................................... 43

4.1 Concept....................................................................................................................................... 43

4.2 Embodiments.............................................................................................................................. 45

4.3 Development Process Overview................................................................................................. 47

4.4 Model A: Radial Decoupler........................................................................................................ 47

4.4.1 Model Description ............................................................................................................... 47

4.4.2 Lumped Parameter Model ................................................................................................... 48

4.4.3 Equations of Motion ............................................................................................................ 50

4.4.4 Parameter Optimisation ....................................................................................................... 51

4.4.5 Shock Absorber Dynamic Characteristics ........................................................................... 53

4.4.6 Simulation Results............................................................................................................... 58

4.4.7 Relaxed Constraints............................................................................................................. 61

4.4.8 Physical Construction .......................................................................................................... 69

4.5 Model B: Vertical Decoupler ..................................................................................................... 70

4.5.1 Model Description ............................................................................................................... 70

4.5.2 Lumped Parameter Model ................................................................................................... 71

4.5.3 Equations of Motion ............................................................................................................ 71

4.5.4 Parameter Optimisation ....................................................................................................... 72

4.5.5 Shock Absorber Dynamic Characteristics ........................................................................... 73

4.5.6 Simulation Results............................................................................................................... 76

4.5.7 Relaxed Constraints............................................................................................................. 79

4.5.8 Physical Construction .......................................................................................................... 83

4.6 Other promising models ............................................................................................................. 84

4.7 Summary..................................................................................................................................... 87

Chapter 5

Future Considerations..................................................................................... 89

5.1 Damper Orifice Modelling ......................................................................................................... 89

vii

5.2 Increased Decoupler Travel........................................................................................................ 91

5.3 Semi-Active Shock Absorber ..................................................................................................... 92

5.3.1 Concept................................................................................................................................ 92

5.3.2 Sensitivity Analysis ............................................................................................................. 93

5.3.3 Variable Orifice ................................................................................................................... 95

5.4 Summary..................................................................................................................................... 97

Chapter 6

Conclusions and Recommendations............................................................... 98

viii

List of Figures

Figure 1-1: Effect of increasing damping of 1-DOF system by 10%..................................................... 2

Figure 1-2: Effect of integrated hydraulic mount and shock absorber ................................................... 3

Figure 1-3: Thesis roadmap.................................................................................................................. 10

Figure 2-1: Component concepts.......................................................................................................... 11

Figure 2-2: Two popular shock absorber configurations [35] .............................................................. 13

Figure 2-3: Typical automotive strut [37] ............................................................................................ 14

Figure 2-4: Typical shock absorber mounted on a pickup truck rear axle [38].................................... 14

Figure 2-5: Composite sinusoidal road input [33]................................................................................ 15

Figure 2-6: Frequency content and respective amplitude of sinusoidal road input [33] ...................... 16

Figure 2-7: Four distinct zones of a shock absorber F-v curve [41]..................................................... 17

Figure 2-8: Comparison of actual, piecewise linear, and linear shock absorber F-v curves [41] ........ 18

Figure 2-9: Suspension linkages........................................................................................................... 18

Figure 2-10: F-p curve for an actual shock absorber [41] .................................................................... 19

Figure 2-11: Comparison of actual and piecewise linear shock absorber F-p curves [41] .................. 20

Figure 2-12: Shock absorber F-v curve, all values used....................................................................... 20

Figure 2-13: Different mass positions with the same instantaneous velocity ...................................... 21

Figure 2-14: 1-DOF system [44] .......................................................................................................... 23

Figure 2-15: Mass-spring-damper system response with and without TMA ....................................... 23

Figure 2-16: TMA on quarter-car model.............................................................................................. 24

Figure 2-17: Sprung mass acceleration of TMA-tuned quarter-car model........................................... 24

Figure 2-18: Unsprung mass acceleration of TMA-tuned quarter-car model ...................................... 25

Figure 2-19: A typical hydraulic engine mount, or hydromount [46] .................................................. 27

Figure 2-20: A typical hydromount (photograph) [47] ........................................................................ 27

Figure 2-21: Lumped parameter hydromount model [46].................................................................... 28

Figure 3-1: External hydraulic mount integration ................................................................................ 31

Figure 3-2: Different configurations for external hydromount suspension system.............................. 32

Figure 3-3: Engine and suspension mount orientation comparison...................................................... 33

ix

Figure 3-4: Mechanical system model [48].......................................................................................... 34

Figure 3-5: Wheel-ground relative position, external design ............................................................... 38

Figure 3-6: Body acceleration, external design .................................................................................... 38

Figure 3-7: Decoupler motion, external design .................................................................................... 39

Figure 3-8: Wheel-ground relative position (more decoupler travel), external design ........................ 40

Figure 3-9: Body acceleration (more decoupler travel), external design ............................................. 40

Figure 3-10: Decoupler motion (more decoupler travel), external design ........................................... 41

Figure 4-1: Internal hydraulic mount integration ................................................................................. 43

Figure 4-2: Simple sketches of internal integration design, Model A.................................................. 45

Figure 4-3: Summary of model embodiments...................................................................................... 46

Figure 4-4: Lumped parameter model, Model A.................................................................................. 48

Figure 4-5: Close-up of decoupler lumped parameter model............................................................... 50

Figure 4-6: Mechanical equivalent of decoupler model....................................................................... 50

Figure 4-7: Dynamic stiffness, Model A.............................................................................................. 54

Figure 4-8: Dynamic damping, Model A ............................................................................................. 55

Figure 4-9: Decoupler movement, Model A ........................................................................................ 55

Figure 4-10: Shock absorber force vs. velocity, amplitude = 0.005 m, Model A ................................ 57

Figure 4-11: Shock absorber force vs. velocity, amplitude = 0.01 m, Model A .................................. 57

Figure 4-12: Quarter-car system model................................................................................................ 59

Figure 4-13: Wheel absolute position, Model A .................................................................................. 59

Figure 4-14: Wheel absolute acceleration, Model A............................................................................ 60

Figure 4-15: Wheel-ground relative position, Model A....................................................................... 60

Figure 4-16: Dynamic stiffness, mL increased, Model A...................................................................... 62

Figure 4-17: Dynamic damping, mL increased, Model A..................................................................... 62

Figure 4-18: Wheel absolute position, mL increased, Model A............................................................ 63

Figure 4-19: Wheel absolute acceleration, mL increased, Model A...................................................... 63

Figure 4-20: Wheel-ground relative position, mL increased, Model A................................................. 64

Figure 4-21: Decoupler movement, decoupler travel limit increased, Model A.................................. 65

Figure 4-22: Dynamic stiffness, decoupler travel limit increased, Model A........................................ 66

Figure 4-23: Dynamic damping, decoupler travel limit increased, Model A....................................... 66

Figure 4-24: Wheel absolute position, decoupler travel limit increased, Model A.............................. 67

Figure 4-25: Wheel absolute acceleration, decoupler travel limit increased, Model A........................ 67

x

Figure 4-26: Wheel-ground relative position, decoupler travel limit increased, Model A................... 68

Figure 4-27: Comparison of simulation results, Model A.................................................................... 69

Figure 4-28: External view of possible physical construction of Model A.......................................... 69

Figure 4-29: Lumped parameter model, Model B................................................................................ 71

Figure 4-30: Dynamic stiffness, Model B ............................................................................................ 74

Figure 4-31: Dynamic damping, Model B............................................................................................ 75

Figure 4-32: Decoupler movement, Model B....................................................................................... 75

Figure 4-33: Shock absorber force vs. velocity, Model B.................................................................... 76

Figure 4-34: Absolute position, Model B............................................................................................. 78

Figure 4-35: Absolute acceleration, Model B ...................................................................................... 78

Figure 4-36: Wheel-ground relative position, Model B ....................................................................... 79

Figure 4-37: Dynamic stiffness, mL increased, Model B...................................................................... 80

Figure 4-38: Dynamic damping, mL increased, Model B ..................................................................... 80

Figure 4-39: Wheel absolute position, mL increased, Model B ............................................................ 81

Figure 4-40: Wheel absolute acceleration, mL increased, Model B...................................................... 81

Figure 4-41: Wheel-ground relative position, mL increased, Model B................................................. 82

Figure 4-42: Comparison of simulation results, Model B .................................................................... 83

Figure 4-43: Typical dynamic stiffness for Model E ........................................................................... 85

Figure 4-44: Typical dynamic stiffness for Model F............................................................................ 85

Figure 4-45: Typical dynamic stiffness for Model H........................................................................... 86

Figure 4-46: Typical dynamic stiffness for Model I............................................................................. 86

Figure 4-47: Improvements of the three designs .................................................................................. 88

Figure 5-1: Nonlinear extension........................................................................................................... 89

Figure 5-2: Shock absorber force vs. velocity, piecewise linear damping, Model A........................... 90

Figure 5-3: Shock absorber force vs. velocity close up, piecewise linear damping, Model A............. 91

Figure 5-4: Modified Model A schematic ............................................................................................ 92

Figure 5-5: Sensitivity summary for dynamic stiffness, Model A ....................................................... 94

Figure 5-6: Sensitivity summary for dynamic damping, Model A....................................................... 94

Figure 5-7: Sensitivity summary for decoupler movement, Model A.................................................. 95

Figure 5-8: Shock absorber force vs. velocity, semi active valve control, Model A............................ 96

Figure 5-9: Shock absorber force vs. velocity close up, semi active valve control, Model A ............. 96

Figure 5-10: Orifice control signal (larger value corresponds to larger opening) ................................ 97

xi

List of Tables

Table 1-1: Metrics used for suspension evaluation ................................................................................ 8

Table 1-2: Road inputs used for suspension evaluation ......................................................................... 8

Table 3-1: State variable definitions..................................................................................................... 35

Table 3-2: Parameter values for non-mount components..................................................................... 36

Table 3-3: Hydromount parameter values ............................................................................................ 36

Table 3-4: Tuneable parameter information......................................................................................... 37

Table 3-5: Improvements to system behaviour, external design .......................................................... 39

Table 3-6: Improvements to system behaviour (more decoupler travel), external design ................... 41

Table 3-7: Tuneable parameter information (more decoupler travel) .................................................. 41

Table 3-8: Improvements to system behaviour (chassis grounded), external design ........................... 42

Table 4-1: Tuneable parameter information, Model A......................................................................... 53

Table 4-2: Improvements to system behaviour, Model A .................................................................... 61

Table 4-3: Improvements to system behaviour, mL increased, Model A.............................................. 64

Table 4-4: Improvements to system behaviour, decoupler travel limit increased, Model A................ 65

Table 4-5: Improvements to system behaviour, mL and decoupler travel limit both increased, Model A

...................................................................................................................................................... 68

Table 4-6: Tuneable parameter information, Model B......................................................................... 73

Table 4-7: Improvements to system behaviour, Model B .................................................................... 77

Table 4-8: Improvements to system behaviour, mL increased, Model B.............................................. 82

xii

Nomenclature

AL Effective cross-sectional area of lower decoupler [m2]

Ad Effective cross-sectional area of hydraulic mount decoupler [m2]

Ap Cross-sectional area of main shock absorber piston, or effective pumping area of hydraulic

mount (context-specific) [m2]

Ar Cross-sectional area of shock absorber piston rod [m2]

b12 Shock absorber damping (unmodified) between sprung and unsprung masses [N-s/m]

bd Mechanical equivalent of Rd [N-s/m]

bL Mechanical equivalent of RL [N-s/m]

br Damping of hydraulic mount rubber [N-m]

C1 Compliance of hydraulic mount upper chamber [m5/N]

C2 Compliance of hydraulic mount lower chamber [m5/N]

CL Compliance of lower decoupler chamber [m5/N]

f1 Net force exerted by shock absorber onto sprung mass [N]

f2 Net force exerted by shock absorber onto unsprung mass [N]

f12 Net force exerted by shock absorber onto sprung and unsprung mass (if f1 = f2) [N]

IL Effective inertia of fluid in lower decoupler [kg/m4]

k12 Spring stiffness between sprung and unsprung masses [N-m]

k2r Spring stiffness between unsprung mass and road (i.e., tire stiffness) [N-m]

kB Effective stiffness of gas below bottom dividing piston [N-m]

k1 Mechanical equivalent of 1

1

C [N/m]

k2 Mechanical equivalent of 2

1

C [N/m]

kL Mechanical equivalent of L

C

1 [N/m]

kr Stiffness of hydraulic mount rubber [N-m]

kT Effective stiffness of gas above top dividing piston [N-m]

xiii

m1 Mass of sprung mass (vehicle chassis) [kg]

m2 Mass of unsprung mass (wheel and tire) [kg]

mb “Mass” of base of hydraulic mount (mb = 0) [kg]

mL Effective total mass of lower decoupler [kg]

md Effective total mass hydraulic mount decoupler [kg]

mLf Effective total mass of fluid in lower decoupler [kg]

mLs Effective total mass of steel embedded in lower decoupler [kg]

PB Pressure in bottom shock absorber chamber [Pa]

PL Pressure in lower decoupler chamber [Pa]

PT Pressure in top shock absorber chamber [Pa]

Qh Net flow of fluid from bottom shock absorber chamber to top shock absorber chamber

through orifices in main piston [m3/s]

QL Flow from shock absorber through lower decoupler into lower decoupler chamber [m3/s]

Rd Effective resistance of hydraulic mount decoupler [N-s/m5]

Rh Effective resistance to flow of fluid from bottom shock absorber chamber to top shock

absorber chamber through orifices in main piston [N-s/m5]

RL Effective resistance of lower decoupler [N-s/m5]

x1 Vertical displacement (positive up) of top portion of shock absorber, and

vertical displacement (positive up) of sprung mass [m]

x2 Vertical displacement (positive up) of bottom portion of shock absorber, and

vertical displacement (positive up) of unsprung mass [m]

xB Vertical displacement (positive up) of bottom dividing piston [m]

xb Vertical displacement (positive up) of base of hydraulic mount [m]

xd Displacement (positive up) of hydraulic mount decoupler [kg]

xL Displacement (positive out if radial, positive up if vertical) of lower decoupler [m]

xr Vertical displacement (positive up) of road surface [m]

VT Volume of lower shock absorber chamber [m3]

VB Volume of upper shock absorber chamber [m3]

ρ Density of fluid in shock absorber [kg/m3]

Capital letters represent the Laplace transform of time-dependent variables unless otherwise noted.

1

Chapter 1

Introduction

There are many different suspension systems in use on today’s automobiles. These systems can be

classified as passive, adaptive or active, with basic passive systems commonly consisting of a parallel

spring and damper [1, 2]. Two primary purposes of a suspension system are to maintain good ride

comfort and road holding [1, 2, 3]. Ride comfort can be quantitatively described as the absolute

acceleration of the vehicle chassis, with a lower acceleration being preferred [3, 4, 5, 6, 7]. Road

holding can be quantitatively described as the relative position of the tire and the road [3, 5, 6, 7]. A

constant relative position indicates a constant normal force between the tire and the road, which is

desirable for vehicle control purposes.

As is often the case with engineering problems, there is a trade off between ride comfort and road

holding. Several researchers [1, 3, 8, 9, 10] indicate that minimizing displacements or acceleration

due to lower frequency inputs requires higher dynamic stiffness, whereas minimizing displacements

or acceleration due to higher frequency inputs requires lower dynamic stiffness. Specifically, system

excitation at the body and wheel resonant frequencies can be particularly troublesome. This

resonance directly increases chassis acceleration and changes the relative tire/road displacement, thus

deteriorating both road holding and ride comfort.

In an effort to combat this resonant response, the system damping may be increased. However,

with conventional passive automotive shock absorbers, this has the effect of increasing the dynamic

stiffness at all frequencies, to the detriment of ride comfort. As seen in the mass-spring-damper

example in Figure 1-1, if the damping is increased by 10%, the resonant excitation decreases at the

frequency of interest; however, the dynamic stiffness increases at all frequencies, with a greater

increase at higher frequencies. This is especially undesirable at much high frequencies (50-150 Hz)

because the stiffer system will transmit more noise, vibration, and harshness. This concept is clearly

supported in [9] and [11].

2

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

Mass p

ositio

n /

Base e

xcitation)

Baseline

Modified

0 5 10 15 20 25 300

2

4

6

8

10

12x 10

4

Frequency (Hz)

Magnitude (

Dam

per

dynam

ic s

tiff

ness /

Base e

xcitation)

Baseline

Modified

a) Sprung mass position b) Damper dynamic stiffness

Figure 1-1: Effect of increasing damping of 1-DOF system by 10%

It is apparent that basic passive systems cannot effectively deal with these conflicting dynamic

stiffness vs. frequency requirements [1, 12]. To circumvent this issue, some designs have

implemented active or semi-active shock absorbers. Fully active applications often utilize an actuator

applied directly to the chassis and/or wheel hub. While very effective, these actuators often require a

great deal of energy input and are very expensive, often making them infeasible in real world

applications [13, 14]. Semi-active applications, on the other hand, use much less energy; they only

change the characteristics of the system. For example, the damping coefficient may be altered by

adjusting a valve in the shock absorber, which in turn changes the effective damping as the control

system computer deems necessary. However, semi-active systems can still be quite costly, and are

currently available primarily on high-end luxury automobiles [13]. Several approaches are discussed

in Section 1.1.

The principal designs proposed and investigated herein are neither active nor semi-active, but

passive, with a possible extension to semi-active. However, unlike basic conventional passive

systems, these designs are frequency dependent, and are capable of meeting the conflicting

requirements discussed above.

All of the designs integrate a hydraulic mount, or components thereof, with a shock absorber. A

hydraulic mount by design has excellent high-frequency isolation properties. It also behaves as a

tuned mass absorber (TMA) because of the fluid movement within the mount. By taking advantage

of the inherent isolation and tuned mass absorber properties, it is possible to increase the damping at a

3

desired frequency without increasing the dynamic stiffness over too large of a range of frequencies.

Figure 1-2 shows this concept, which will be discussed more in later chapters. Notice how the change

in position of the sprung mass is almost identical to that of Figure 1-1; however, the increase in

dynamic stiffness of the integrated design is localized to the desired frequency, with slight decreases

at higher frequencies. Furthermore, it is even possible to increase low frequency damping and

decrease high frequency dynamic stiffness, both significantly, as will be discussed in Section 4.5.

By deciding on the correct embodiment and tuning the parameters accordingly, it is possible to

create a passive shock absorber capable of increasing or decreasing its damping effect at

predetermined frequencies. This in turn can have a positive effect on both ride comfort and road

holding.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified

0 5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

10x 10

4

Frequency (Hz)

Magnitude (

F12 /

Xr)

Baseline

Modified

a) Sprung mass position b) Damper dynamic stiffness

Figure 1-2: Effect of integrated hydraulic mount and shock absorber

1.1 Literature Review

There are several methods of dealing with the inherent trade-offs in automotive suspension design.

Most solutions are either active or semi-active in nature, using many different actuation and control

techniques; a selection are discussed below. Also discussed are some examples that incorporate a

hydraulic strut mount into the suspension design, as well as a summary of metrics used in suspension

evaluation. Section 1.2 will outline how the current work differs from and builds upon these

examples.

4

Tan and Bradshaw [1] developed a high-fidelity quarter-car model of an active suspension system

using a hydraulic actuator. Their primary goal was to isolate and identify the important design

parameters, such as hydraulic, valve, friction, tire, and bushing parameters. Because Tan and

Bradshaw focused on frequencies greater than or equal to the wheel hop frequency (approximately

12-15 Hz), their model should be used with caution for low frequency or body mode events. Kim and

Won [3] used a similar quarter-car model of an active hydraulic suspension system. Their purpose,

however, was to treat the hydraulic actuator as asymmetric, or single-rod. They show that their

equations can be simplified for the symmetric case by setting the piston areas of both sides equal.

Ando and Suzuki [2] also used a quarter-car model with hydraulic actuator. Recognizing the

conflicting high and low frequency requirements, they decomposed the system into fast and slow

subsystems, relating to ride quality and road holding respectively. Although their model is nonlinear,

they linearized it for each of the two subsystems using the singular perturbation method. The final

controller is a composite of the controllers for the two subsystems. Anakwa et al [15], on the other

hand, document the design, construction, and testing of a pneumatic active suspension. Their primary

method for selecting a pneumatic actuator or a hydraulic actuator was to avoid possible oil spills;

otherwise, their system is comparable to a hydraulic system. They designed their controller using the

pole placement method. No comparisons to a passive system, simulated or otherwise, are presented.

Campos et al [12] aim to control the heave and pitch motions of a vehicle using a half-car model,

as well as isolate road input vibrations. Their primary method is to “roll off” the damping coefficient

as the frequency exceeds the wheel hop frequency. Their closed-loop controller is set up with an

inner loop to reject road disturbances at the front and rear wheels and an outer loop to control heave

and pitch. To control these four variables with only two inputs (the front and rear actuators), they

decouple the inputs via a so-called Butterfly input-decoupler transformation. Ikenaga et al, in an

apparently collaborative effort with Campos et al, perform similar experiments on both quarter-car

[16] and full-car [17] models. Their full-car models adds vehicle roll to the list of metrics.

Several researchers [14, 18, 19, 20] have proposed fuzzy-logic, neural networks, and genetic

algorithms for active and semi-active suspension systems. For example, Yoshimura et al [14] aim to

optimize passenger comfort with hydraulic actuators and a half-car model. By augmenting a linear

control system with fuzzy-logic, they are able to more accurately capture the nonlinear dynamics

associated with the vehicle. They define their fuzzy-logic rules in a linguistic manner, and then use

the product-sum-gravity method to determine control parameters. Their method requires several

iterations to determine the most suitable parameter values for the scaling and weighting functions.

5

Yoshimura et al [5] also studied the control methods of a pneumatic actuator and a quarter-car

model. Instead of applying fuzzy logic or neural networks to the control system as per [14], this time

they implemented a sliding mode controller. They propose that their sliding mode controller is

simpler than a fuzzy logic or neural network controller, yet more accurate than a linear quadratic (LQ)

controller. Although they minimize a weighted performance index of tire deflection, suspension

deflection, wheel velocity, and body velocity, they also examine the root mean square values of body

acceleration, velocity, and displacement. When they compare the results of their sliding mode control

system to an LQ control system and a passive system, all metrics improve from passive to LQ to

sliding mode. The one exception is that the actuator effort is greater for the sliding mode controller

than for the LQ controller.

Lu and DePoyster [6] focus not on active or semi-active applications, but rather the control of both.

They use a full-car model and generalize the (semi-)active force as either active suspension force or

semi-active damper force. They study both the time and frequency domains. Their twist is the use of

a so-called mixed H2/H∞ controller, which simultaneously minimizes peak frequency responses (H∞

norm) and variances with respect to white noise (H2 norm). Naem et al [7] employ a more traditional

optimisation method to an electro-rheological / magneto-rheological (ER/MR) damper. They discuss

the construction of and governing principles behind this semi-active device. They optimise the

control of their quarter-car model by minimizing a weighted objective function of body acceleration

and tire force.

A popular [21] class of semi-active damper involves a continuously variable valve controlling fluid

flow between internal chambers. Park et al [21] and Kim et al [22] collaborated and presented papers

on their work. Kim et al discuss the entire development of a continuously variable shock absorber for

Mando Corporation. They focus on the control system and vehicle integration, ensuring that comfort

is optimized and that the system properly interacts with existing systems such as antilock brakes and

electronic stability programs. Park et al, on the other hand, discuss the operation and analytical

model of the shock absorber. By clearly illustrating and modelling the internal workings, including

the solenoid, they are able to reduce the shock absorber response time. Rather than study a

continuously variable damper, Mo and Sunwoo [13] examine the design of a simpler two-state, or

bistate, semi-active hydraulic damper. Their damper is similar in nature to that of [21] and [22].

However, they adjust the valve only between two settings. Although experimental results on a

quarter-car model are favourable, concerns are raised about guaranteed stability of the system.

6

Thompson et al [23] illustrate the combination of a hydroelectric actuator and a tuned mass

absorber to a quarter-car suspension model. They simultaneously determine the controller feedback

gains as well as the optimum TMA spring and damper rates. They determine that the TMA will have

the greatest effect if the mass ratio (absorber mass to unsprung mass) is in the range of 0.2 to 0.5.

Perhaps their most important conclusion is that the addition of the tuned mass absorber has little

effect on vehicle behaviour, with one exception, when compared to the active system without a TMA.

As the absorber mass is increased, the energy expended by the actuator is decreased. In other words,

the tuned mass absorber is taking over for the actuator.

Going back almost 40 years, Shotwell [24] is perhaps one of the first publications dealing with the

application of tuned mass absorbers to reduce body bounce; his list of referenced works is sparse.

Shotwell studied the effect of adding a TMA to control the body acceleration of a heavy construction

equipment. Because of the unique nature of the suspension setup and dynamics, the vehicle’s radiator

package was used as the absorber mass. Real-world time-domain tests indicate noticeable

improvement with a mass ratio in the range of 0.2 to 0.3.

Other valid approaches to improving suspension quality involve the use of hydraulic suspension

mounts, similar in nature to hydraulic engine mounts. Nakajima et al [10] apply a first-principles

approach to the development of a hydraulic suspension mount. The problem with elastomeric mounts

is one of increased dynamic stiffness with increased frequency, which is contrary to the mount

requirements. They theoretically and experimentally examine several different configurations of

hydraulic chambers, passages, partitions, and effective fluid inertia packets before deciding on the

best overall configuration. When installed on a real vehicle, a road noise reduction of 1 dB is realized

on the frequency range of 100-300 Hz.

Shaw and Darling [9] also studied the development of a hydraulic suspension mount in lieu of

conventional elastomeric bushings. They highlight how stiffer conventional mounts will decrease

body accelerations at the body bounce and wheel hop frequencies, but will increase overall vibration

and harshness at frequencies above the wheel hop resonance. They emphasize that “the ideal

suspension would be relatively soft around the resonant frequencies only, thus providing control of

body and wheel modes, but isolating higher frequencies” [9]. Increasing general suspension damping

to control wheel-hop motion will have a definitively negative effect of vibration isolation. One major

benefit of the hydraulic mount is a confined region of high dynamic stiffness, focused near the wheel-

hop resonance, similar to the effect illustrated in Figure 1-2. In their quarter-car model, Shaw and

Darling were able to show a reduction of 25% in wheel hop vibration transmitted to the vehicle body.

7

On four-post rig and road tests, the improvements at wheel-hop resonance were significant near the

strut tower, but became less noticeable as the measurement location approached the seat track. Also,

there was some increased vibration response at the body bounce mode. Subjective real-world tests on

ride quality indicate improvements on the range of 5-10%, which they deem significant.

As with [9] and [10], Tsujiuchi et al [25] target suspension system rubber mounts and bushings in

an effort to minimize the trade-off between ride, control, and noise. However, Tsujiuchi et al do not

consider hydraulic mounts, but illustrate a method for optimizing the dynamic stiffness of the many

rubber bushings used through the suspension system. Their primary goal is to reduce noise

transmission to the passenger compartment around 160 Hz without adverse affect on ride quality,

around 22 Hz. Notably, they investigate the mounts on the front and rear of the lower control arm of

a front suspension, not the strut tower mount. A coherence study indicates that they can reduce the

noise by reducing the lateral movements of the front suspension cross member, which in turn

correspond to the bending mode of the shock absorber. A sensitivity analysis indicates which

particular mount locations affect the 160 Hz mode and which affect the 22 Hz mode, allowing

Tsujiuchi et al to adjust the dynamic stiffness on the appropriate mounts.

In addition to research publications, there also exist several patents on the topic of hydraulic

suspension mounts. Smith et al [26], Jung [27], Hodumi [28], and Graeve [29] have all patented

variations on the theme discussed by Nakajima et al [10] and Shaw and Darling [9]. That is, they

have all found slightly different embodiments of directly replacing an elastomeric strut mount with a

hydraulic strut mount. Most of the differences relate to the physical layout of the mount and its

internal fluid passages. Dreff [11], on the other hand, applies the hydraulic mount principles to the

internal design of a shock absorber, similar to the work presented herein. Discussed are two main

embodiments: parallel tube and concentric tube shock absorbers. The second tube in each case is

designed to take advantage of the fluid resonance properties. The effect on high frequency vibration

isolation with an increase in overall system damping is discussed, along with how the frequency-

dependent damping of the patented design mitigates the trade-off.

While there are many different inputs and metrics used to evaluate suspension design, several are

common across many sources. Their mention has hitherto been largely omitted from this section.

Instead, general classes of metrics and inputs, along with their respective sources, are listed in Table

1-1 and Table 1-2 respectively. Because of their popularity and simplicity, body acceleration and tire

deflection, or wheel-ground relative displacement, were selected as the primary metrics for use in the

current work. Sinusoidal inputs were also selected, and are described in more detail in Section 2.1.3.

8

Table 1-1: Metrics used for suspension evaluation

Metric Source(s)

Body movement (usually acceleration) [2, 3, 4, 5, 6, 7, 9, 12, 14, 16,

17, 18, 19, 20, 23, 24, 30]

Tire force or tire vertical deflection

(relative wheel-ground displacement)

[2, 3, 5, 7, 13, 14, 19, 23]

Suspension deflection (relative wheel-body

displacement)

[5, 18, 19]

Sound pressure / noise [10, 25]

Absorbed power [6, 30]

Table 1-2: Road inputs used for suspension evaluation

Road Input Source(s)

Sinusoidal [3, 5, 8, 9, 15, 16, 17, 19, 24, 31, 32, 33]

Random or white noise [2, 5, 6, 9, 13, 14, 18, 32, 34]

Road swell or bump [3, 13, 32]

Step [7, 23]

Impulse [12]

1.2 Research Goals

As discussed in Section 1.1, there exists a wide array of approaches to balancing the trade-off

between ride comfort and control, many of which focus on active or semi-active solutions. Although

various control strategies are employed, the active and semi-active systems predominantly use a

hydraulic or pneumatic actuator, or a variable valve or magnetorheological fluid, respectively, to

initiate real-time changes in the system. Most passive systems swap the elastomeric strut mount with

a hydraulic mount.

The primary goal of the current research is to investigate and determine the potential of combining

some of the aforementioned approaches, namely, integrating a hydraulic mount into the design of a

passive shock absorber in order to achieve results similar to those of the semi-active systems. By

integrating the mount directly with the damper, there is the potential to offer a cost-effective, off-the-

shelf, commercially-viable solution to automotive manufacturers. There will be no need for the

manufacturers to tune separate dampers and mounts, but one unit as a whole. There will also be no

need to install any electronic control system, or integrate the control with existing vehicle electronic

systems. Overall, the proposed system reduces complexity while improving both ride and handling.

9

At first glance there may appear to be a patent conflict with Dreff [11]. However, closer inspection

reveals definitively different approaches to the same problem. Whereas Dreff employs two separate

concentric or parallel fluid tubes, the focus here is on incorporating the hydraulic mount, and

specifically the decoupler, into the primary tube design.

1.3 Thesis Outline

This thesis is laid out in six chapters, with the middle four represented by the shaded areas in Figure

1-3. Chapter 1 introduces the work and direction, including a literature review. Chapter 2 provides

the background information necessary, including shock absorbers, tuned mass absorbers, hydraulic

mounts, parameter optimisation and sensitivity analysis. Chapter 2 is represented by the leftmost

shaded area of Figure 1-3.

The next three chapters, represented by the horizontal shaded areas Figure 1-3, from the top down,

detail the integration of the various components. In Chapter 3, a hydraulic mount is added to the

shock absorber in an external location, connecting the shock absorber to the chassis or wheel hub.

The shock absorber and mount have separate fluid systems, linked only by mechanical components.

While several possibilities are represented, one embodiment is presented in detail. Equations of

motion are presented, as are optimisation and simulation results.

Chapter 4 incorporates the hydraulic mount decoupler into the internal design of a shock absorber.

As such, both the “mount”, which is no longer a mount per se, and the shock absorber share one fluid

system. Several embodiments of this concept are presented, with the most effective embodiments

presented in detail.

Future extensions are considered in Chapter 5. Based on one of the most promising embodiments

from Chapter 4, discussions are presented on how to improve the modelling accuracy, including

nonlinear internal fluid resistance and increasing the allowable decoupler travel. A semi-active

application is also offered.

Finally, Chapter 6 outlines the overall conclusions and recommendations.

10

Shock

Absorbers

Tuned Mass

Absorbers

Hydraulic

Mounts

Parameter

Optimisation

Sensitivity

Analysis

Future

Considerations

Internal

ConfigurationsAnalysis

External

ConfigurationsAnalysis

Chap 2 Chap 3 Chap 4 Chap 5

Legend:

Figure 1-3: Thesis roadmap

11

Chapter 2

Background

The purpose of Chapter 2 is to provide the reader with the background information necessary to better

understand the work presented in subsequent chapters. Introductory material is presented on shock

absorbers, tuned mass absorbers, hydraulic mounts, parameter optimisation, and sensitivity analysis

as indicated in Figure 2-1.

Figure 2-1: Component concepts

12

2.1 Shock Absorbers

2.1.1 Basic Structure

A shock absorber, or automotive damper, can be loosely described as a fluid-filled cylindrical

chamber separated by a piston, effectively creating two chambers. The outer casing of the cylinder is

connected to either the body or chassis of the vehicle, and the piston is connected via a piston rod to

the other of the chassis or body. Valves in the piston head allow the fluid to flow between the

chambers. The resistance force created by the fluid flowing through the valves acts to dampen the

relative motion between the body and chassis.

An incompressible fluid is used within the shock absorber; therefore, to avoid cavitation, an

allowance must be made for the volume of the piston rod as the piston moves up and down. Two

popular classes of shock absorber, twin tube and monotube, shown in Figure 2-2, have expandable

internal volume [35, 36].

The monotube shock absorber has a second piston called the dividing piston. The chamber below

the dividing piston is filled with compressible gas, allowing the dividing piston to move vertically

when the main piston also moves vertically. The twin tube shock absorber has a pair of concentric

tubes. The inner tube is similar in nature to the monotube shock absorber; however, instead of

employing the use of a dividing piston, the fluid moves through a set of valves from the inner to the

outer tube. The top portion of the outer tube has compressible gas, allowing the incompressible fluid

to flow into and out of the outer tube as necessary. The gas used in each configuration yields a light

spring effect; as such, shock absorbers are not pure dampers, but can often be modelled as such. The

dynamics of both shock absorbers are similar. The monotube design is selected for use herein

because of its simpler design.

13

Rod Seal

Gas

Oil

Rod

Piston and Valves

Dividing Piston

Valves

Gas

Twin Tube Monotube

Figure 2-2: Two popular shock absorber configurations [35]

2.1.2 Shock Absorbers vs. Struts

A shock absorber is not the same as a strut. A strut, as shown in Figure 2-3, is a coil spring

wrapped around a shock absorber [37]. The spring and damper of the strut are in parallel. A strut is

often used on the front wheels of front wheel drive automobiles, due to space envelope considerations

involving the suspension and driveline components. Notice, however, the large amount of space

surrounding the shock absorber in the wheel well; this space is available for the design modifications

described in Chapter 4.

A shock absorber is also often mounted in parallel with a spring. However, it need not be a

concentric, or even a coil spring. One common use of shock absorbers is on the rear axle of pickup

trucks, mounted in parallel with a leaf spring, as shown in Figure 2-4 [38].

14

Because of the similar parallel spring/damper characteristics of a strut and shock absorber/spring

combination, the design and analysis presented herein can easily be considered valid for struts as well

as shock absorbers. However, the focus is on shock absorbers.

Figure 2-3: Typical automotive strut [37]

Figure 2-4: Typical shock absorber mounted on a pickup truck rear axle [38]

15

2.1.3 Road Inputs

To properly analyse the designs, it is important to have a good understanding of the road inputs that

excite the system. Although several different input sources are used (see Table 1-2), sinusoidal road

inputs, described in detail in [33], are selected here. The final road input discussed is a composite of

several specific sinusoidal inputs summed together to create a seemingly random input, as shown in

Figure 2-5 [33]. The amplitude and frequency of the six individual sinusoidal curves are shown in

Figure 2-6 [33]. It is clear that all inputs of frequency greater than or equal to 1 Hz have an amplitude

of at most 5 mm; therefore, 5 mm is used as the expected maximum input displacement for the work

herein.

It is also worth noting that the random road profiles generated in [34] are consistent with Figure

2-5. For example, in [34], three different road profiles have maximum displacements of 10, 20, and

30 mm respectively.

0 10 20 30 40 50 60 70 80 90 100-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Dis

pla

cem

ent

(m)

Time (s)

Figure 2-5: Composite sinusoidal road input [33]

16

0 5 10 152

3

4

5

6

7

8

9

10

Frequency (Hz)

Dis

pla

cem

ent

(mm

)

Figure 2-6: Frequency content and respective amplitude of sinusoidal road input [33]

2.1.4 Modelling and Dynamics

As discussed in Section 2.1.1, there is a damping force created by the movement of fluid through

valves in the piston. In the simplest of terms, this resistance is a function of flow, which is a function

of the relative velocity between the piston (and rod) and the outer casing. As such, the force equation

for a shock absorber is often written as:

bvF = (2.1)

where

b = damper coefficient

v = relative velocity of damper ends

However, a real damping curve usually has four identifiable sections: low- and high-speed, each

for compression and extension. The slope of the F-v (force-velocity) curve is steeper at low speeds

than at high speeds. The rates for the extension or rebound direction are approximately double or

triple the respective rates for the compression or jounce direction [35, 39, 40, 41]. This feature is

illustrated in Figure 2-7 [41].

17

0 0.05 0.1 0.15

-1500

-1000

-500

0

500

Velocity (m/s)

Fout

(N)

Low Velocity,

Compression

Low Velocity,

Extension

High Velocity,

Compression

High Velocity,

Extension

Actual Curve

Figure 2-7: Four distinct zones of a shock absorber F-v curve [41]

When modelling the damping effect of a shock absorber, it may be possible to use four piecewise

linear curves. In other words, the force may be defined as:

0,

0,

0,

0,

,2

,2

,

,

2

1

2

1

<′>

<′<

>′>

>′<

=

vvv

vvv

vvv

vvv

b

b

b

b

b (2.2)

where

changeover velocity High/Low

n)compressio indicates (positiveelocity absorber vshock Relative

rate dampingity High veloc

rate damping velocity Low

2

1

=′

=

=

=

v

v

b

b

The results of such a calculation can be seen in Figure 2-8 [41]. To further simplify matters, the

composite b in Equation (2.2) can be approximated by a single, linear value as first seen in Equation

(2.1), also illustrated in Figure 2-8 [42].

18

It is also important to note that the actual damping force generated by a shock absorber will be

different than the force exerted by a modelled damper on the wheel because of the linkages L in the

suspension system, as seen in Figure 2-9. Because the linkages form a lever, actualmodelled

bb < . This

indicates that the damper value used for modelling purposes should be less than the damper value

seen in actual applications.

0 0.05 0.1 0.15

-1500

-1000

-500

0

500

Velocity (m/s)

Fout

(N)

Actual

Piecewise linear

Linear

Figure 2-8: Comparison of actual, piecewise linear, and linear shock absorber F-v curves [41]

vdamper

Facutal

bactual

vwheel

Fdesired

L1 L2

vwheel

Fmodelled bmodelled

a) Actual setup b) Modelled setup

Figure 2-9: Suspension linkages

19

2.1.5 Testing

To determine the actual F-v curve of a shock absorber, the shock absorber is tested on a

dynamometer. This rig cycles the shock absorber in a sinusoidal wave with constant amplitude and

increasing frequency. For each frequency, this generates a curve of force vs. relative displacement or

position (F-p), as shown in Figure 2-10. The force at maximum velocity is recorded from each curve

and then pieced together to generate the F-v curves described previously. The maximum velocity is

recorded for both directions, compression and extension, which occurs at zero relative position,

indicated in Figure 2-10. This procedure and results are in line with those discussed in [36, 43].

Using the piecewise linear damper rate structure from Equation (2.2), the F-p curve can be reasonably

reproduced, as seen in Figure 2-11.

The potential risk to this approach lies in the behaviour of the damper in regions of non-maximum

velocity. If all of the points from the F-p curve are collected and plotted, the F-v curve becomes

somewhat of an oval, as seen in Figure 2-12. Although this oval-like shape appears to represent

hysteresis, this is not possible because the system is linear.

-0.0152 -0.0102 -0.0051 0 0.0051 0.0102 0.0152

-1500

-1000

-500

0

500

Position (m)

Fout

(N)

Figure 2-10: F-p curve for an actual shock absorber [41]

20

-0.0152 -0.0102 -0.0051 0 0.0051 0.0102 0.0152

-1500

-1000

-500

0

500

Position (m)

Fout

(N)

Actual

Piecewise linear

Figure 2-11: Comparison of actual and piecewise linear shock absorber F-p curves [41]

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-300

-200

-100

0

100

200

300

Velocity (m/s)

Forc

e (

N)

Figure 2-12: Shock absorber F-v curve, all values used

21

The gas present in a shock absorber can be treated as a spring parallel to the primary damping

mechanism. As such, the behaviour observed in Figure 2-12 can be explained using a mass-spring-

damper example, shown in Figure 2-13. Two positions are shown, both with positive (vertical)

instantaneous velocities. In Figure 2-13 a), the mass is above the equilibrium point; therefore, both

the spring and damper forces are in the downward direction. The total force acting on the mass is the

sum of the two sub-forces. In Figure 2-13 b), the mass is below the equilibrium point. In this

situation, the spring force is in the upward direction, but the damper force is still in the downward

direction. As such, the total force acting on the mass is the difference between the two sub-forces. It

is obvious that the magnitude of the force in a) is greater than that in b). The force difference at these

two positions is represented by the upper and lower portions of the curve in Figure 2-12. Since only

the damper is being modelled, the spring effects are removed by plotting the forces only at the

maximum absolute velocities, not at all points.

a) x > 0 b) x < 0

Figure 2-13: Different mass positions with the same instantaneous velocity

2.2 Tuned Mass Absorbers

The purpose of a tuned mass absorber (TMA), or tuned vibration absorber, is to dampen the system

response at the resonant frequency without adversely affecting other frequencies. A TMA is an

additional mass suspended (with or without damping) from the main system mass. Two systems are

shown in Figure 2-14, one without and one with a TMA [44]. Of course, the 1-DOF system becomes

a 2-DOF system with the addition of the TMA.

22

By tuning the resonant frequency of the TMA subsystem with sinusoidal base excitation xin to

correspond with the resonant frequency of the original system, the maximum absolute value of the

main system response x1 can be significantly decreased, as seen in Figure 2-15. Notice how the one

large peak is replaced by a pair of lower peaks. The precise shape of these new peaks can be

controlled by tuning the spring and damper characteristics of the TMA; the lower the damping, the

narrower but deeper the “valley”. The two peaks are produced because the system is now in fact 2-

DOF. While detailed mathematical theory on the tuning of a TMA is presented in [44], all tuning

herein was performed using the optimisation routine described in Section 2.4.

A key characteristic of a TMA is what is known as the mass ratio. The mass ratio is the ratio of

TMA mass to primary system mass. Generally, the greater this ratio, the greater the system

improvement. By increasing the mass of the TMA, the throw, or travel, of the TMA is also reduced.

However, it is often unrealistic to expect a high mass ratio due to space or size constraints. In an

automobile, for example, it is obviously beneficial to decrease the overall vehicle mass. A suitable

mass ratio is often approximately 0.2 to 0.3 [23, 24, 45]. If the primary system mass is considered to

be the vehicle unsprung mass, this implies that a suitable traditional TMA mass would be

approximately 5-10 kg.

To further understand the effects of a tuned mass absorber in an automotive application, a TMA

was attached to the unsprung mass of a quarter-car model as shown in Figure 2-16 a). The resulting

mass ratio is greater by attaching the TMA to the unsprung mass than it would be if it were attached

to the sprung mass, as in Figure 2-16 b). Preliminary results obtained by tuning the TMA parameters

in Figure 2-16 a) to minimize the vertical acceleration of the sprung mass within the optimisation

range of 12-15 Hz are very promising, as indicated in Figure 2-17. Figure 2-18 shows that the

behaviour of the unsprung mass is also improved. A sinusoidal road input excitation xr was used.

This process and its application to more detailed models are presented in Chapter 3 and Chapter 4.

23

m1

m2

a) Without TMA b) With TMA

Figure 2-14: 1-DOF system [44]

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (rad/s)

Magnitude (

X1 /

Xin

)

Without TMA

With TMA

Figure 2-15: Mass-spring-damper system response with and without TMA

24

m1

m2

TMA

m1

TMA

m2

a) TMA attached to unsprung mass b) TMA attached to sprung mass

Figure 2-16: TMA on quarter-car model

0 5 10 15 20 25 30 350

100

200

300

400

500

600

700

800

900

Frequency (Hz)

Magnitude (

X1 /

Xr)

Without TMA

With TMA

Opt Range

Figure 2-17: Sprung mass acceleration of TMA-tuned quarter-car model

25

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Without TMA

With TMA

Opt Range

Figure 2-18: Unsprung mass acceleration of TMA-tuned quarter-car model

2.3 Hydraulic Mounts

The desired frequency vs. dynamic stiffness characteristics for an engine mount are similar in nature

to the desired characteristics of a suspension system [8, 9]. A hydraulic mount, or hydromount, is a

suitable device for attaining these characteristics when mounting an engine. A typical hydraulic

engine mount is shown in Figure 2-19 [46], with a corresponding photograph in Figure 2-20 [47].

The hydraulic suspension mounts discussed in Section 1.1 have identical working characteristics to

engine mounts, but are merely tuned to different frequencies. The operation of a hydromount is

briefly described below.

As mentioned previously, high dynamic stiffness is desired at low frequency, and low dynamic

stiffness is desired at high frequency. However, traditional rubber or elastomeric mounts increase in

dynamic stiffness as the frequency of vibration increases. Because this is contrary to the desired

behaviour, hydraulic mounts were introduced [31].

A hydraulic mount “consists of two fluid-filled chambers connected through a decoupler and

inertia track” [8], with fluid flowing between the chambers as the mount vibrates. The decoupler and

26

inertia track are two passages designed to interact in such a manner as to decrease the dynamic

stiffness at higher frequencies and increase the dynamic stiffness at lower frequencies.

The inertia track is a relatively long, narrow passage. As the excitation frequency increases and the

amplitude decreases, the fluid in the inertia track becomes essentially stationary, forcing all fluid

through the decoupler. The decoupler, a plate floating in a cage, allows the fluid to flow through with

minimal resistance. However, as the frequency drops and the amplitude increases, the decoupler

begins to “bottom out” in its cage, blocking fluid, and forcing the fluid to travel through inertia track

with its relatively higher resistance [8, 9].

Figure 2-21 shows the lumped parameter model of the hydromount [46]. Included in the model are

the stiffness and damping of the outer rubber (kr and br respectively); the compliance of the two

chambers (C1 and C2 respectively); and the flow, inertia, and resistance of the inertia track and

decoupler (Qi(t), Ii, Ri, Qd(t), Id, and Rd respectively). When installed in its intended manner to mount

an engine, the hydromount communicates with the chassis through XT(t) and with the engine through

X(t). Ap represents the effective pumping area of the upper chamber; P1(t) and P2(t) represent the

internal pressure of the upper and lower chambers respectively. Often, the hydromount is modelled

as two separate linear systems: a low frequency model for the operation of the inertia track

(disregarding the decoupler), and a high frequency model for the operation of the decoupler

(disregarding the inertia track). Both models have the same equations of motion, and differ only in

the parameter values selected [48].

The fluid moving back and forth through the decoupler has an effective mass that is analogous to

the TMA mass described in Section 2.2. Because of this, the hydromount can be treated as a tuned

mass absorber with an absorber mass of approximately 100-300g. The benefits are twofold: first, the

system resonance response can be decreased, and second, the high frequency dynamic stiffness can be

decreased. It is these two fundamental concepts that drive the designs in the following chapters.

27

Upper

Chamber

Decoupler

Cage

Lower

Chamber

Decoupler

Inertia

Track

Rubber

Engine Side

Chassis Side

Figure 2-19: A typical hydraulic engine mount, or hydromount [46]

Figure 2-20: A typical hydromount (photograph) [47]

28

AP

kr

br C

1

C2

Id

,Rd

P1(t)

P2(t)

Qi(t)

Qd(t)

Ii ,R

i

X(t)

XT (t)

FT(t)

Figure 2-21: Lumped parameter hydromount model [46]

2.4 Parameter Optimisation

Parameter optimisation refers to the process of determining the “best” values for a set of parameters

by minimizing a desired objective function. In the current work, the MATLAB function fmincon is

used for parameter optimisation. By using a sequential quadratic programming (SQP) method with a

quadratic programming (QP) sub-problem solved at each iteration, fmincon minimizes a user-

specified function subject to upper and/or lower bounds, as well as linear and/or nonlinear constraints

[49]. A detailed description of the operational method used by fmincon is available in the

MATLAB documentation [49].

The limitation with the fmincon function is the possibility of getting “stuck” in a local minima.

It is therefore important to run the optimisation from different initial conditions. It is also helpful to

examine how the objective function changes through the process; for example, if the objective

function is still changing at the time that the user-specified maximum number of iterations is met,

then the optimisation should be rerun for more iterations.

Specific objective functions, parameters, bounds and constraints are described with each model

simulated in the subsequent chapters.

29

2.5 Sensitivity Analysis

The purpose of a sensitivity analysis is to determine which parameters have the greatest effect on the

system behaviour. Knowing how the system behaviour reacts to a change in each parameter allows

for the selection of one or two key parameters for use in a semi-active application. Also, it allows for

a better understanding and selection of the manufacturing tolerances for the system components. A

sensitivity analysis is performed in Chapter 5. The method presented in [46] and [50] is summarized

briefly here.

First, a quality index J is defined. Several different indices are used in Chapter 5, and are described

there. The sensitivity of the quality index to a parameter at a particular operating point can be

measured as a percent change in J with respect to changes in that parameter, again at the particular

operating point. Mathematically, this can be represented by:

( )( )

%100×=i

i

J

UJ µ

α

αδ (2.3)

where

( )( )

point operating respect towith variation Parameter

at evaluatedfunction y sensitivitorder First

parameterspoint operating at the evaluatedindex Quality

indexquality in changePercent

=

=

=

=

i

iU

J

J

µ

αα

αα

δ

The first order sensitivity function i

U is the Jacobian of J, or the change in J with respect to a

change in α at some operating point α .

( ) ( )

ααα

αα

=∂

∂=

i

i

JU (2.4)

To better understand the sensitivity function, rewrite Equation (2.3):

( )( )

%100×=α

µαδ

JUJ i

i (2.5)

30

Equation (2.5) is analogous to a simple “rise-over-run” linear equation, where ( )αi

U represents the

slope, ( )α

µ

J

i represents the normalised “run”, and Jδ represents the percentage “rise”. By

calculating the terms independently and then substituting the results back into Equation (2.5), the

sensitivity can be easily determined.

31

Chapter 3

Shock Absorber with Hydraulic Mount

As highlighted in Figure 3-1, Chapter 3 focuses on the use of a hydraulic mount along with the

suspension spring and damper to connect the sprung and unsprung masses. While many locations and

orientations are possible, only one embodiment is analysed here.

Figure 3-1: External hydraulic mount integration

3.1 Concept

There are several different possible configurations for incorporating a hydraulic mount into the

suspension design, some of which are shown in Figure 3-2. The hydraulic “engine” mount is

represented by the semi-circle icon; the curved and flat parts represent the “engine” and chassis sides

respectively, as described in Section 2.3. For all diagrams, m1 and m2 represent the chassis (sprung

32

mass) and wheel (unsprung mass) respectively. xr represents the road input. The model analysed is

that in Figure 3-2 a). This model was selected over b), d), and f), because of the respective mass

ratios, as discussed in Section 2.2. c) was not selected because of the difficulty in implementation

when compared to the chosen model. Finally, a) was selected over e) because of the mount

orientation, as shown in Figure 3-3. The hydraulic engine mount isolates the vehicle chassis from

engine excitation; to isolate the chassis from road vibration in a), the mount is positioned in the same

orientation relative to the excitation input. I.e., the “engine” side is analogous to the “tire” side.

a) b) c)

d) e) f)

Figure 3-2: Different configurations for external hydromount suspension system

33

Figure 3-3: Engine and suspension mount orientation comparison

3.2 Equations of Motion

Using the system diagram in Figure 3-2 a) and the procedure introduced in [48] (including the use of

a mechanical lever to denote a ratio), the system can be modelled using entirely mechanical

components, as seen in Figure 3-4. This mechanical model represents the decoupler model of the

hydromount. The inertia track model can be constructed by simply replacing all decoupler-related

parameters with inertia track-related parameters; the schematic will remain the same.

The equations of motion can be derived from Figure 3-4. The four equations of motion are:

bb

xbxkxbxkxm &&&&121211211211

++−−= (3.1)

( )rrb

p

d

rrd

p

d

rrbr xkxA

Akkkxbx

A

Akxkkkxbxm 21121212222 +

−++−

+−−−=− &&& (3.2)

( ) ( )b

p

d

p

d

ddd

p

d

p

d

bddd xA

A

A

Akkxbx

A

Akkx

A

Akxbxm

−

++−

+−=−

2

21

2

2121&&&& (3.3)

( ) ( )br

p

d

rd

p

d

rbrdd xkkkA

Akxbxbx

A

Akxkkxkxbbxm

−−−+++

−++=++ 1211211212111212

&&&& (3.4)

34

xr

k2r

m1

k12 b12

m2

kr br

x1

xb

x2

xd

bd k2

k1

Ap-Ad Ad

mb = 0

md

Figure 3-4: Mechanical system model [48]

Equations (3.1) through (3.4) can be combined into state space form according to the variable

definitions in Table 3-1. The matrix form of the state space equations is thus

uxx BA +=& (3.5)

where

[ ]T

bddxxxxxxx &&&&&&&&&&&

2121x = (3.6)

[ ]T

bddxxxxxxx &&&

2121x = (3.7)

[ ]r

x=u (3.8)

( )2

1

1AAA

−= (3.9)

( )1

1

1BAB

−= (3.10)

35

and

+

−

−

−=

rd

dd

r

bbm

bm

bm

bmA

12

2

1211

00000

00000

00000

00000

0000100

0000010

0000001

(3.11)

( )

( ) ( )

( )

−−−−+

−+−+−

−+−

−−−

−−

=

12111211212

12

2

212

2

211

11112

121212

2

0

000

000

0000

0100000

0010000

0001000

kkkA

Akbb

A

Akkkk

A

Ak

A

Akkb

A

Akk

A

Ak

A

Akkkb

A

Akkkk

kbk

A

r

p

d

r

p

d

r

p

d

p

d

d

p

d

p

d

p

d

rr

p

d

rr (3.12)

[ ]T

rkB 000000

21= (3.13)

Table 3-1: State variable definitions

State Variable Definition

1x

1x

2x

2x

3x

dx

4x

1x&

5x

2x&

6x

dx&

7x

bx

1u

rx

36

3.3 Parameter Optimisation

The parameters for the non-mount components are listed in Table 3-2. These parameters are

reasonable when compared to the parameters found in [2, 3, 12, 16, 51]. The original parameters for

the mount, based on [48], are shown in Table 3-3. By tuning a selection of these variables using the

MATLAB function fmincon according to the process outlined in Section 2.4, the system behaviour

can be improved. While several optimisation routines were performed, only two are reported herein.

The reported results were obtained by varying only the two parameters listed in Table 3-4. These

parameters and their ranges were selected because a hydraulic mount can be easily manufactured with

the specified allowable changes.

Table 3-2: Parameter values for non-mount components

Parameter Value Units

m1 300 kg

m2 30 kg

k12 35000 N/m

b12 500 N/(m/s)

k2r 190000 N/m

Table 3-3: Hydromount parameter values

Parameter Description Value Units

Rd Resistance of fluid flow through decoupler 1.17e7 kg/(s-m4)

C1 Compliance of mount upper chamber Tuneable m5/N

C2 Compliance of mount lower chamber 2.6e-9 m5/N

kr Stiffness of mount rubber 225000 N/m

br Damping of mount rubber 100000 kg/s

Ap Effective pumping area of mount rubber 0.0025 m2

Ad Decoupler area 0.00066 m2

bd Damper equivalent of Rd Rd*Ad^2 kg/s

k1 Spring equivalent of C1 Ap^2/C1 N/m

k2 Spring equivalent of C2 Ap^2/C2 N/m

The objective function is the maximum absolute value of the frequency response of the wheel-

ground relative position, taken to the fourth power, and calculated on the range of 10-17 Hz

(opt_range). Although the optimisation function deals specifically with road handling near the

wheel hop frequency, ride comfort (as denoted by body acceleration) is also improved as a result.

37

The fmincon function optionally accepts linear and/or nonlinear constraints, as well as upper and

lower parameter bounds. This is important to guarantee reasonable parameter values and behaviour.

Table 3-4 lists the bounds on the parameters, as well as their optimized values.

Table 3-4: Tuneable parameter information

Parameter Description Low High Optimum Units

C1 Compliance of mount upper chamber 3e-11 3e-8 5.0098e-10 m5/N

md Effective mass of decoupler 0.001 0.5 0.1264 kg

3.4 Simulation Results

The results of the model simulation with sinusoidal road excitation xr are shown in Figures 3-5 to 3-7.

While no actual constraint was placed on decoupler movement, repeated experimentation and

investigation indicated that the longer the optimisation routine ran, the better the results were, but the

more the decoupler moved. It was decided to manually halt the optimisation when the decoupler-to-

input-displacement ratio was approximately 10. This value was selected in order to observe some

non-negligible improvement to system behaviour, but also to limit the decoupler movement.

However, this ratio is unrealistic because, for a road amplitude of 5 mm (at the wheel hop frequency),

the decoupler would be required to travel 50 mm. Even so, the system improvements are still only

approximately 4%, as summarized in Table 3-5.

The poor performance of this model can be attributed to the small TMA mass ratio. As mentioned

in Section 2.2, the mass ratio should be approximately 0.2-0.3. In the current example, the ratio

between decoupler mass and unsprung mass is only about 0.004.

38

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified

Opt range

Figure 3-5: Wheel-ground relative position, external design

0 5 10 15 20 25 300

100

200

300

400

500

600

700

800

900

Frequency (Hz)

Magnitude (

X1 /

Xr)

Baseline

Modified

Opt range

Figure 3-6: Body acceleration, external design

39

0 5 10 15 20 25 300

2

4

6

8

10

12

Frequency (Hz)

Magnitude (

Xd /

Xr)

Figure 3-7: Decoupler motion, external design

Table 3-5: Improvements to system behaviour, external design

Measure Improvement

Body Acceleration 4.3%

Wheel Relative Position 3.6%

To further understand the behaviour of the system, results are presented when the decoupler is

permitted even more movement, as shown in Figures 3-8 to 3-10, and summarized in Table 3-6. For

this new case, the tuneable parameter values are as listed in Table 3-7. It is immediately apparent that

by allowing the decoupler more travel, the effects on the system behaviour are significantly

improved. When the decoupler travel increases by a factor of approximately 1.5, the improvement

increases by a factor of 5.

40

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified

Opt range

Figure 3-8: Wheel-ground relative position (more decoupler travel), external design

0 5 10 15 20 25 300

100

200

300

400

500

600

700

800

900

Frequency (Hz)

Magnitude (

X1 /

Xr)

Baseline

Modified

Opt range

..

Figure 3-9: Body acceleration (more decoupler travel), external design

41

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

Frequency (Hz)

Magnitude (

Xd /

Xr)

Figure 3-10: Decoupler motion (more decoupler travel), external design

Table 3-6: Improvements to system behaviour (more decoupler travel), external design

Measure Improvement

Body Acceleration 21.6%

Wheel Relative Position 17.5%

Table 3-7: Tuneable parameter information (more decoupler travel)

Parameter Description Low High Optimum Units

C1 Compliance of upper mount chamber 3e-11 3e-8 2.0509e-

10 m

5/N

md Effective mass of decoupler 0.001 0.5 0.3193 kg

For the purpose of comparing the present results with the results from Chapter 4, the current design

was simulated with the chassis position grounded. The improvement to wheel-ground relative

position is very similar to that illustrated in Figure 3-5 and Table 3-5, and highlights the validity of

grounding the chassis if the unsprung mass is the focal point of study. See Table 3-8.

42

Table 3-8: Improvements to system behaviour (chassis grounded), external design

Measure Improvement

Unsprung Mass Position 3.6%

Unsprung Mass Acceleration 4.6%

Wheel Relative Position 3.8%

3.5 Summary

In this chapter, several configurations were introduced in which a hydraulic mount was connected

externally to a shock absorber. One promising configuration was studied in detail, with equations of

motion and simulation results presented. The improvement in handling and comfort of approximately

4% can be significantly increased if the decoupler is allowed to travel further.

43

Chapter 4

Shock Absorber with Internal Decoupler

This chapter introduces several models of a shock absorber incorporating hydraulic mount

components, as indicated in Figure 4-1. The most effective models are analysed in detail.

Figure 4-1: Internal hydraulic mount integration

4.1 Concept

The internal integration of hydraulic mount components with a shock absorber can be viewed as an

analogy to the mechanical tuned mass absorber systems discussed in Section 2.2. The benefits of the

internal design are numerous:

• A complete “shock absorber package” solution

• Simplicity of design

44

• Simplicity of construction

• Appreciable effect on system performance

• Applicable to several different shock absorber sizes and models

Because the shock absorber fluid is used as the mass for the tuned mass absorber effect, the

direction of mass movement need not be parallel to the length of the shock absorber, as would be the

case with a mechanical TMA.

As discussed in Section 2.1, there are two main types of shock absorbers, monotube and twin tube.

Both have similar concepts and base equations; the monotube has been selected for analysis because

it can be easily combined with the basic concept of the internal integration.

The basic conceptual design is shown in Figure 4-2. Attached to the shock absorber is a thick ring

housing a decoupler device. This device is essentially a rubber balloon, or diaphragm, that fills and

empties with fluid as the shock absorber is compressed and extended. This oscillatory movement

creates an additional degree of freedom in the system. By tuning the resonant mode accordingly, the

system peak resonant response can be reduced without affecting dynamic stiffness at higher

frequencies.

The compliance of the diaphragm and the resistance of several holes admitting the fluid can be

tuned in the same manner as with a hydraulic mount. The size and shape of the diaphragm can be

adjusted, as can the limits of travel. Metal weights may be embedded in the diaphragm to increase

the effective mass if necessary.

The various embodiments are created by moving the diaphragm package or the gas spring to

different locations (such as above or below the piston). These are discussed in the following section.

45

a) Side view cutaway b) Top view

Figure 4-2: Simple sketches of internal integration design, Model A

4.2 Embodiments

Figure 4-3 shows all of the embodiments studied; they are presented here as lumped parameter

models. In general, there are two major components that are rearranged to form the various

embodiments. First, the decoupler “package” can be mounted above and/or below the main shock

absorber piston. In one case, it is even mounted in a vertical direction near the bottom of the shock

absorber. Second, the dividing piston and its corresponding gas spring may be relocated above and/or

below the main piston. Some models incorporate minor variations, as well, including fluid leakage

and piston mass. The models in Figure 4-3 a) and b) are the most promising, and are presented in

detail in Sections 4.4 and 4.5 respectively.

46

a) Model A: Radial decoupler b) Model B: Vertical decoupler c) Model C: No gas spring

d) Model D: Decoupler leakage e) Model E: Dividing piston mass f) Model F: Upper decoupler

g) Model G: Upper gas spring h) Model H: Two decouplers i) Model I: Two gas springs

Figure 4-3: Summary of model embodiments

47

4.3 Development Process Overview

The models in this chapter were all analysed using the development process described below. Due to

space limitations, however, the steps will only be presented for two embodiments; select information

will be discussed for other promising embodiments.

• Model Description: A brief description and background of the model being discussed.

• Lumped Parameter Model: Based on the initial concept, a lumped parameter model is

discussed.

• Equations of Motion: The initial and final equations from the lumped parameter model

are reported.

• Parameter Optimisation: The parameters are optimised using MATLAB. The resulting

values must be within reasonable limits.

• Shock Absorber Dynamic Characteristics: The dynamic stiffness, 12

12

XX

F

−, and

dynamic damping, ( )

12

12

XXs

F

−, of the shock absorber are analysed, as are the force-

velocity properties, ( )1212

vs. xxf && − . The force-velocity simulation analysis is conducted in

the same manner that physical shock absorbers are tested.

• Simulation Results: Various effects are considered: wheel position relative to ground

(related to maintaining constant tire normal force), absolute position, and absolute

acceleration.

• Relaxed Constraints: To better understand the constraints, some key constraints are

relaxed and the optimisation process is repeated.

• Physical Construction: A discussion on how the model can be realized.

4.4 Model A: Radial Decoupler

4.4.1 Model Description

Model A is one of the first models developed, and is originally shown in Figure 4-2. This model uses

a radial decoupler package and a dividing piston, both below the main piston. Model A is one of the

two best performing models.

48

4.4.2 Lumped Parameter Model

The lumped parameter model for Model A is shown in Figure 4-4. The shock absorber piston has an

internal surface area of Ap [m2]. The piston rod has an area of Ar [m

2]. The orifice in the piston that

allows fluid to flow between the upper and lower chambers has effective resistance Rh [N-s/m5], with

flow Qh [m3/s] being positive into the upper chamber. The pressure and volume of the top and bottom

shock absorber chambers is PT [Pa], VT [m3], PB [Pa], and VB [m

3], respectively. The stiffness of the

compressed gas below the lower chamber is modelled as a spring with stiffness kB [N/m]. The

absolute displacements of the piston and cylinder are x1 [m] and x2 [m], respectively (positive is up).

The absolute displacement of the dividing piston is xB [m] (positive is up). The force applied to the

piston due to the shock absorber internal workings is f1 [N] (positive is up). The force applied to the

cylinder due to the shock absorber internal workings is f2 [N] (positive is down). Because f1= f2 for

Model A, the net force is simply referred to as f12 [N] (positive is in the extension direction).

F1

F2

x2

x1

xb

x2

Qh, RhAp

Ar

PT

PB

kB

xL, mL, RL

AL, QL PL

CL

Figure 4-4: Lumped parameter model, Model A

49

Although the decoupler package is added concentrically around the side of the cylinder, it can be

modelled as a single device attached to the side of the shock absorber, as shown in Figure 4-4. The

movement of the rubber diaphragm is modelled as xL [m] (positive away from the cylinder). The

effective cross-sectional area of the diaphragm is AL [m2]. QL [m

3/s] is the flow of fluid into the

decoupler package, with IL [kg/m4] representing the effective fluid inertia and mLf [kg] being the

mechanical system equivalent to IL. CL [m5/N] is the diaphragm compliance. RL [N-s/m

5] represents

the effective resistance of the diaphragm movement and of the fluid entering and leaving the

diaphragm area. Extra steel may be embedded in the diaphragm; this extra mass is represented by mLs

[kg]. The sum of mLf and mLs is mL [kg].

Alternatively, the diaphragm area can be viewed as having an inner pressure PL [Pa] with

compliance CL. A fluid package with effective inertia IL (or effective mass mLf) represents the system

boundary between the lower chamber and the diaphragm chamber. There is no leakage across this

system boundary, only the movement of the boundary as the fluid oscillates in and out of the

diaphragm chamber. The extra diaphragm mass and effective fluid inertia mass can be lumped

together as mL for the purpose of analysis. Figure 4-5 shows a close-up view of the lumped parameter

model for the diaphragm package. Figure 4-6 shows the mechanical equivalent, where 2

dLLARb =

[N-s/m] and L

d

LC

Ak

2

= [N/m].

50

Lower

chamber QL

AL

PL

xL

Diaphragm

chamber

Fluid parcel /

system boundary

Compliance area

of diaphragm

Limit of diaphragm

movement

Figure 4-5: Close-up of decoupler lumped parameter model

Figure 4-6: Mechanical equivalent of decoupler model

4.4.3 Equations of Motion

The equations of motion are derived entirely in the Laplace (s) domain. All Laplace domain variables

are designated by the capital letter of the corresponding time domain variable. The root equations for

Model A, as derived from Figure 4-4, are as follows:

( ) ( ) ( )hhTBrpTBLrpUpL

RQPPAAPXXkFAAPAPF =−−−−=−−= ,,221

(4.1)

51

( )( )sC

QQRsQIPsVQAAsXsXsVFF

L

L

LLLLBThrpT++==−−== ,,,

1221 (4.2)

( ) ( )LLpBrLLLLLL

AXAXXXXAAImsAXQ +−=−==212

2,, (4.3)

Assuming that x1(t) and x2(t) are displacement inputs to the shock absorber, the transfer function

between (X2-X1) and F12 is

( )

+++

++

+−=−

2

22

22

2

12

12

p

BL

L

L

LL

L

L

p

L

L

LL

L

L

Br

rph

A

kA

C

AsARs

A

mA

C

AsARs

A

mkA

sAARXX

F (4.4)

Equation (4.4) represents the dynamic stiffness of the modified shock absorber, Model A. The

dynamic damping, ( )sXX

F

12

12

−, can be obtained by dividing both sides of Equation (4.4) by s. The

transfer function between (X2-X1) and XL, representing the decoupler movement, or decoupler-to-input

ratio, is

+++

=−

2

2212

p

BL

L

L

LL

L

L

p

BrL

A

kA

C

AsARs

A

mA

kA

XX

X (4.5)

4.4.4 Parameter Optimisation

In order to achieve the best improvement from the integrated system, the system parameters must be

appropriately tuned. This is accomplished via the MATLAB function fmincon, along with suitable

parameter bounds and constraints. The final tuned parameters can then be used to derive physical

dimensions, as outlined in Section 4.4.8. The optimisation routine minimizes a desired objective

function. The objective function used herein is the maximum absolute value of the frequency

response of the wheel-ground relative position for quarter-car model (see Section 4.4.6), calculated on

the range of 12-16 Hz.

52

The objective function can also be easily modified to use the maximum absolute value of the

frequency response of the wheel absolute position or absolute acceleration. Instead of using the

maximum absolute value of the frequency response, it is also possible to use the summation of the

absolute value of the frequency response over a desired optimisation range. Additionally, the

objective function may be taken to any power in order to amplify the larger absolute values.

The fmincon function optionally accepts linear and/or nonlinear constraints, as well as upper and

lower parameter bounds. This is important to guarantee reasonable parameter values and behaviour.

Table 4-1 lists the bounds on the parameters, as well as their optimized values. The selected

nomenclature allows for the various embodiments described in Section 4.2. Two additional

constraints are applied as follows:

pr

AA8

5≤ (4.6)

212

≤− XX

XL (4.7)

Equation (4.6) ensures that the cross-sectional area of the piston rod is not too large. Equation

(4.7) ensures that the movement of the decoupler is restricted to twice the movement of the input

(decoupler-to-input ratio is 2:1). This ratio of 2:1 is reasonable given the desire to restrict decouple

movement to approximately 0.01 m, and given that road input displacement in the frequency range of

interest is at most 0.005 m, as discussed in Section 2.1.3 [33].

53

Table 4-1: Tuneable parameter information, Model A

Parameter Description Low High Optimum Units

AL Effective cross-sectional area of lower

decoupler 0.001 0.1 0.0015388 m

2

Ap Cross-sectional area of main shock

absorber piston 7e-4 0.004 0.0023939 m

2

Ar Cross-sectional area of shock absorber

piston rod 2e-5 0.001 0.0010000 m

2

CL Compliance of lower decoupler

chamber 2e-10 0.02 0.0013102 m

5/N

kB

Effective stiffness of gas below bottom

dividing piston 10 10000 9882.7 N/m

mL Effective total mass of lower decoupler 0 0.5 0.5 kg

RL Effective resistance of lower decoupler 150 4e8 6284800 N-s/m5

ρ Density of fluid in shock absorber 1000 1000 1000 kg/m3

4.4.5 Shock Absorber Dynamic Characteristics

To determine the dynamic characteristics of the shock absorber, two approaches were used, all with x1

fixed; i.e., x1(t) = 0 for all t. Equations (4.4) and (4.5) thus become

( )

+++

++

+−=

2

22

22

2

2

12

p

BL

L

L

LL

L

L

p

L

L

LL

L

L

Br

rph

A

kA

C

AsARs

A

mA

C

AsARs

A

mkA

sAARX

F (4.8)

+++

=

2

222

p

BL

L

L

LL

L

L

p

BrL

A

kA

C

AsARs

A

mA

kA

X

X (4.9)

The dynamic stiffness of the modified shock absorber is represented by Equation (4.8), and is

shown in Figure 4-7. Equation (4.9) represents the decoupler movement. The dynamic damping can

be found by dividing Equation (4.8) by s:

54

( )

+++

++

+−=

sA

kA

C

AsARs

A

mA

C

AsARs

A

mkA

AARsX

F

p

BL

L

L

LL

L

L

p

L

L

LL

L

L

Br

rph

2

232

22

2

2

12 (4.10)

The dynamic damping of the modified shock absorber is shown in Figure 4-8. It is obvious that the

dynamic stiffness and damping both increase by approximately 10% near 14 Hz. The peak frequency

can be tuned according to the routine discussed in Section 4.4.4. Also, as the frequency increases

beyond 25 Hz, the dynamic stiffness and damping decrease slightly. This decrease is beneficial

because it will allow the system to better isolate high-frequency noise.

Figure 4-9 shows that the decoupler motion is indeed within the tolerance specified by the

optimisation routine. It is also observable that the decoupler movement increases around 14 Hz and

decreases beyond 25 Hz, corresponding to the dynamic stiffness and damping behaviour.

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

4

Frequency (Hz)

Magnitude (

F1

2 /

Xr)

Baseline

Modified

Figure 4-7: Dynamic stiffness, Model A

55

0 5 10 15 20 25 30490

500

510

520

530

540

550

560

Frequency (Hz)

Magnitude (

F1

2 /

Xr)

Baseline

Modified

Figure 4-8: Dynamic damping, Model A

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

Frequency (Hz)

Magnitude (

XL /

Xr)

Decoupler

Max allowed

Figure 4-9: Decoupler movement, Model A

56

In addition to the dynamic stiffness and damping, the force-velocity curves are also analysed.

Because force-velocity curves are measured and analysed in the time domain, but the work herein

deals with the frequency domain, it is important to perform the force-velocity curve simulations

appropriately. These curves are determined by cycling the compression and extension cycles of the

shock absorber at different frequencies. As discussed in Section 2.1, the maximum speed for each

cycle, occurring at the zero-compression/zero-extension point, is plotted with the corresponding

force. Shock absorbers are by nature velocity dependent and frequency independent; therefore, as

long as the amplitude-frequency product is the same, the maximum velocity and corresponding force

will be the same, because

( )( )frequencyamplitudevelocitymax

= (4.11)

and

( )( )rate dampingvelocityforcemax

= (4.12)

This is not the case with the modified shock absorber discussed herein, as the force is frequency

dependent. Obviously the amplitude and frequency of the tests must be properly coordinated.

Figure 4-10 shows the force-velocity curve of the modified shock absorber, with a test amplitude of

0.005 m and a frequency range of 1-32 Hz. There is an obvious increase in damping rate near the

velocity of +/- 0.4 m/s, or 12.7 Hz. When the amplitude is increased to 0.01 m, the shape of the

force-velocity curve in Figure 4-11 is the same that in Figure 4-10, although the velocities and forces

observed are doubled. The damping rate increase still occurs at 12.7 Hz. This highlights the

frequency-dependent nature of the system.

57

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-600

-400

-200

0

200

400

600

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 4-10: Shock absorber force vs. velocity, amplitude = 0.005 m, Model A

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1000

-500

0

500

1000

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 4-11: Shock absorber force vs. velocity, amplitude = 0.01 m, Model A

58

4.4.6 Simulation Results

To study the effects of the shock absorber on wheel motion, a modified quarter-car model was

examined, as shown in Figure 4-12. By fixing the chassis of the quarter-car model, it is possible to

focus directly on the wheel hop motion. The parameters shown are as indicated in Table 3-2, where

applicable.

The absolute position and absolute acceleration transfer functions for the quarter-car model are,

respectively:

2

12

212

2

2

22

X

Fkksm

k

X

X

r

r

r +++

= (4.13)

2

2

12

212

2

2

2

2

2

sX

Fkksm

k

sX

X

r

r

r

+++

= (4.14)

The relative position transfer function may be calculated as follows:

122 −=−

rr

r

X

X

X

XX (4.15)

11

2

12

212

2

2

22 −

+++

=−

X

Fkksm

k

X

X

r

r

r

(4.16)

The frequency response of Equations (4.13), (4.14), and (4.16) can be computed by first

substituting in Equation (4.8). When simulated over a range of 0.1-30 Hz, the results are as shown in

Figures 4-13 to 4-15. It is obvious that when the dynamic damping increased, as discussed in Section

4.4.5, the absolute and relative wheel position and the absolute wheel acceleration decreased at the

resonant frequency only. Each of the three measures improved by approximately 11%, as stated in

Table 4-2.

59

m2

xr

k2r

k12

F12

x2

Chassis

F12

Modified shock

absorber

Figure 4-12: Quarter-car system model

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified

Opt range

Figure 4-13: Wheel absolute position, Model A

60

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified

Opt range..

Figure 4-14: Wheel absolute acceleration, Model A

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified

Opt range

Figure 4-15: Wheel-ground relative position, Model A

61

Table 4-2: Improvements to system behaviour, Model A

Measure Improvement

Absolute Position 9.4%

Absolute Acceleration 9.5%

Relative Position 9.2%

4.4.7 Relaxed Constraints

Further simulation results are presented in this section to highlight the compound effect of relaxing

certain key parameters. Implementation of the relaxed parameters may or may not be feasible, or

may introduce negative effects such as increased fuel consumption if the total system mass is

increased. This section is intended to give an understanding of the sensitivity to certain parameters.

Full system sensitivity analyses are presented in Section 5.3.

The first constraint to be relaxed was the upper limit on the total mass of the lower decoupler, mL.

When mL was allowed to vary up to 1 kg, the localized damping increases, with a positive effect on

the system dynamics, as shown in Figures 4-16 to 4-20. The overall effect is an improvement of

approximately 14%, as shown in Table 4-3. This is approximately 5 percentage points better than the

original optimised case.

62

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

4

Frequency (Hz)

Magnitude (

F1

2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-16: Dynamic stiffness, mL increased, Model A

0 5 10 15 20 25 30490

500

510

520

530

540

550

560

570

580

590

Frequency (Hz)

Magnitude (

F12 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-17: Dynamic damping, mL increased, Model A

63

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-18: Wheel absolute position, mL increased, Model A

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-19: Wheel absolute acceleration, mL increased, Model A

64

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-20: Wheel-ground relative position, mL increased, Model A

Table 4-3: Improvements to system behaviour, mL increased, Model A

Measure Improvement

Absolute Position 13.9%

Absolute Acceleration 14.6%

Relative Position 14.1%

The next constraint relaxed was the limit on decoupler travel, as expressed in Equation (4.7),

instead of the decoupler mass constraint. This limit was increased from a factor of 2 to a factor of 3:

312

≤− XX

XL (4.17)

The improvements are approximately 15%, as indicated in Table 4-4. Figures 4-21 to 4-26 show

the improvements graphically.

65

Table 4-4: Improvements to system behaviour, decoupler travel limit increased, Model A

Measure Improvement

Absolute Position 14.7%

Absolute Acceleration 15.6%

Relative Position 15.1%

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Frequency (Hz)

Magnitude (

XL /

Xr)

Decoupler (relaxed)

Decoupler (not relaxed)

Max allowed

Figure 4-21: Decoupler movement, decoupler travel limit increased, Model A

66

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10x 10

4

Frequency (Hz)

Magnitude (

F12 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-22: Dynamic stiffness, decoupler travel limit increased, Model A

0 5 10 15 20 25 30480

500

520

540

560

580

600

Frequency (Hz)

Magnitude (

F12 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-23: Dynamic damping, decoupler travel limit increased, Model A

67

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-24: Wheel absolute position, decoupler travel limit increased, Model A

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-25: Wheel absolute acceleration, decoupler travel limit increased, Model A

68

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-26: Wheel-ground relative position, decoupler travel limit increased, Model A

Finally, if both the decoupler travel limit is increased to 3 and mL is allowed to vary up to 1 kg, the

improvements are as noted in Table 4-5. To conserve space, the graphical results are not presented.

For visual comparison purposes, Figure 4-27 illustrates the results reported in Tables 4-2 to 4-5. It is

apparent that increasing the mass or decoupler travel have similar affects on the system, and that

increasing both results in a significant increase in system improvement.

Table 4-5: Improvements to system behaviour, mL and decoupler travel limit both increased, Model A

Measure Improvement

Absolute Position 18.4%

Absolute Acceleration 21.5%

Relative Position 21.8%

69

1 2 30

5

10

15

20

25

30

35

% I

mpro

vem

ent

Com

pare

d t

o B

aselin

e

Absolute Position Absolute Acceleration Relative Position

Normal constraints

Increased mL (1kg)

Increased decoupler travel (factor of 3)

Increased mL and decoupler travel

Figure 4-27: Comparison of simulation results, Model A

4.4.8 Physical Construction

The proposed shock absorber has been specifically designed to allow for ease of construction. A

possible external view of Model A is shown in Figure 4-28. The design is very similar to that of a

standard shock absorber, with the additional of a band around the middle section.

Figure 4-28: External view of possible physical construction of Model A

70

According to the optimum parameter values as reported in Table 4-1, the inner diameter of the

shock absorber is 0.0552 m, and the outer diameter of the piston rod is 0.0357 m. The fluid density is

left unchanged at 1000 kg/m3, such as water or ethylene glycol. The value for CL can be attained by

constructing a simple rubber diaphragm or bellows with the required radial movement. The value for

RL can be attained by constructing a grate restricting the flow of fluid from the shock absorber main

chamber to the decoupler changer. The gas spring effect kB can be altered by adjusting the pressure of

the gas. In extreme cases, the gas may even be replaced by a physical spring. The effective total

mass mL can be adjusted by embedding steel into the diaphragm. Although mL = 0.5 kg, the amount

of steel necessary will be less than 0.5 kg because the fluid itself has inertia.

Finally, the effective cross-sectional area AL of the decoupler can be attained by adjusting the

height and diameter of the radial decoupler band around the shock absorber, as well as the gap

between the inner diameter of the shock absorber and the rest position of the decoupler. The

decoupler will have an approximate circumference and height according to:

( )πgapdiameterinner nceCircumfere += (4.18)

nceCircumfere

Height LA

= (4.19)

For example, if the decoupler “rests” 0.01 m away from the inner diameter of the shock absorber,

the circumference and height will be 0.236 m and 0.00651 m respectively. If the gap is 0.005 m, the

circumference and height will be 0.205 m and 0.00751 m respectively. Obviously, there is some

inherent degree of flexibility in final construction.

4.5 Model B: Vertical Decoupler

4.5.1 Model Description

Model B is a modification of Model A, as shown in Figure 4-3 b). This model uses a vertical

decoupler package below the main piston, in the location normally reserved for the dividing piston.

The dividing piston is relocated above the main piston.

71

4.5.2 Lumped Parameter Model

The labelled lumped parameter model is shown in Figure 4-29. The parameter definitions are the

same as those of Model A (described in Section 4.4.2), with the notable exceptions being the vertical

nature of the decoupler package, and the relocation of the dividing piston to the upper section of the

shock absorber.

x2

x1

xb

x2

F1

F2

Ap

Ar

PT

PB

Qh, Rh

PL, CL

kT

AL, QL, xL, mL, RL

Figure 4-29: Lumped parameter model, Model B

4.5.3 Equations of Motion

As with Model A, the equations of motion for Model B are derived entirely in the Laplace (s) domain.

The root equations for Model B, as derived from Figure 4-29, are as follows:

( )rpTpB

AAPAPFFF −−=≠121

, (4.20)

72

( )( ) ( ) ( )222

xxkxAQRAAPAAPPFTTLLLLpLLpLB

−−+++−−= & (4.21)

( )LLLLLLLLLLLLB

xAQQPCxAQRQIPP &&&& ==++=− ,,2

(4.22)

( ) ( ) ( )( )rpTTLLpBBT

AAxxvAxxAxxvvv −−=−+−==+1221

,,0 &&&&&&&&&& (4.23)

( ) ( )ThhhTBTTrpT

vQQRPPxxkAAP &−==−−=− ,,2

(4.24)

Assuming that x1(t) and x2(t) are displacement inputs to the shock absorber, the transfer functions

between X1 and X2, and XL and F2 are

( )( ) ( )

( )( ) ( )

( ) ( )( ) ( ) ( )( )

( )( ) ( )2

222

2

1

222

32

X

C

AsARsAIAAsAARkA

ksARAAAAsAARkTAA

X

C

AsARsAIAAsAARkA

sAARsAARkAX

L

L

LLLLrprphTL

TLLrprprphLp

L

L

LLLLrprphTL

rphrphTr

L

++−+−−

+−−−−−−+

++−+−−

−+−−−=

(4.25)

and

( )( )

( )( ) ( )( )

( ) ( )( )

( ) L

rphT

L

L

LLLLrpT

L

Lp

LLLLLLp

rphT

LLrpTT

TLLLLLp

rphT

rphT

XsAARk

C

AsARsAIAAk

C

AAsARsRsIAAA

XsAARk

sARAAkkksARARAA

XsAARk

sAARkF

−−

++−

−+++−+

−−

−+−++−+

−−

−=

2

2

22

22

2

2

12

2

2

(4.26)

4.5.4 Parameter Optimisation

The parameter optimisation procedure for Model B was the same as for Model A. Although the same

general objective function was used, there were two small changes. First, the optimisation range was

expanded to 1-29 Hz. Second, F2 was used in place of F12. Because the effective absorber mass

moves in a direction parallel to the input and output forces, the reactionary forces F1 and F2 are no

73

longer necessarily equal. The constraints are the same as with Model A. Final optimised parameter

values for Model B are listed in Table 4-6.

Table 4-6: Tuneable parameter information, Model B

Parameter Description Low High Optimum Units

AL Effective cross-sectional area of lower

decoupler 0.001 0.1 0.0018211 m

2

Ap Cross-sectional area of main shock

absorber piston 7e-4 0.004 0.0025799 m

2

Ar Cross-sectional area of shock absorber

piston rod 2e-5 0.001 0.0000200 m

2

CL Compliance of lower decoupler

chamber 2e-10 0.02 0.0057348 m

5/N

kT

Effective stiffness of gas below bottom

dividing piston 10 10000 5011.6 N/m

mL Effective total mass of lower decoupler 0 0.5 0.5 kg

RL Effective resistance of lower decoupler 150 4e+8 3.9990e+7 N-s/m5

ρ Density of fluid in shock absorber 1000 1000 1000 kg/m3

4.5.5 Shock Absorber Dynamic Characteristics

Assuming that x1(t) = 0 for all t, as with Model A, the decoupler travel, dynamic stiffness, and

dynamic damping for Model B are respectively:

( ) ( )( ) ( ) ( )( )

( )( ) ( )

++−+−−

+−−−−−−=

L

L

LLLLrprphTL

TLLrprprphLpL

C

AsARsAIAAsAARkA

ksARAAAAsAARkTAA

X

X

222

2

2

(4.27)

( )( ) ( )( )

( ) ( )( )

( ) 2

2

2

22

2

2

2

2

2

X

X

sAARk

C

AsARsAIAAk

C

AAsARsRsIAAA

sAARk

sARAAkkksARARAA

X

F

L

rphT

L

L

LLLLrpT

L

Lp

LLLLLLp

rphT

LLrpTT

TLLLLLp

−−

++−

−+++−+

−−

−+−++−=

(4.28)

74

( )( ) ( )( )

( ) ( )( )

( ) 2

2

2

22

2

2

2

2

2

X

X

sAARk

C

AsARsAIAAk

C

AAsARsRsIAAA

sAARk

sARAAkkksARARAA

sX

F

L

rphT

L

L

LLLLrpT

L

Lp

LLLLLLp

rphT

LLrpTT

TLLLLLp

−−

++−

−+++−+

−−

−+−++−=

(4.29)

The dynamic stiffness and dynamic damping of the Model B modified shock absorber are shown in

Figure 4-30 and Figure 4-31 respectively. There is a significant increase in dynamic stiffness

damping across much of the 0-30 Hz range, accompanied by a significant decrease in dynamic

stiffness and damping beyond 30 Hz. This “roll-off” at higher frequencies is also seen in [12].

Figure 4-32 indicates that the decoupler movement does not exceed a ratio of 2:1, as required.

0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

Frequency (Hz)

Magnitude (

F2 /

Xr)

Baseline

Modified

Figure 4-30: Dynamic stiffness, Model B

75

0 20 40 60 80 100 120 140 160150

200

250

300

350

400

450

500

550

600

Frequency (Hz)

Magnitude (

F2 /

Xr)

Baseline

Modified

Figure 4-31: Dynamic damping, Model B

0 5 10 15 20 25 300.8

1

1.2

1.4

1.6

1.8

2

2.2

Frequency (Hz)

Magnitude (

XL /

Xr)

Decoupler

Max allowed

Figure 4-32: Decoupler movement, Model B

76

The force-velocity curves are also analysed for Model B as they were for Model A. The results can

be seen in Figure 4-33, with an excitation amplitude of 0.005 m and a maximum frequency of

approximately 64 Hz. There is an obvious increase in the force for 0-1 m/s in both directions, and a

significant decrease in force at larger velocities in both directions compared to the baseline. Although

the model used is linear, it more closely exhibits actual shock absorber behaviour than the baseline

due to the high-speed effects. The levelling off at higher speeds is achieved with the valves in an

actual shock absorber; it is conceivable that by combining these valves and the Model B decoupler,

the further levelling off can attained.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1000

-800

-600

-400

-200

0

200

400

600

800

1000

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 4-33: Shock absorber force vs. velocity, Model B

4.5.6 Simulation Results

Quarter-car simulations were performed on Model B in the same manner as with Model A (see Figure

4-12), also with the parameters as indicated in Table 3-2, where applicable. The equation for relative

displacement remains unchanged, and is indicated in Equation (4.16). However, Equations (4.13) and

(4.14) were slightly modified to account for the new force calculations:

77

2

2

212

2

2

22

X

Fkksm

k

X

X

r

r

r +++

= (4.30)

2

2

2

212

2

2

2

2

2

sX

Fkksm

k

sX

X

r

r

r

+++

= (4.31)

The results of the quarter-car simulation for Model B are listed in Table 4-7 and shown in Figures

4-34 to 4-36. The improvements are between 9% and 12%, and are the result of an increase in

damping across most of the 0-30 Hz range. However, this increase in also accompanied by the

previously discussed drop in damping at higher frequencies. If higher damping is not required from

0-30 Hz, the entire dynamic damping curve may be shifted down by adjusting the valve

characteristics of the shock absorber.

Table 4-7: Improvements to system behaviour, Model B

Measure Improvement

Absolute Position 12.0%

Absolute Acceleration 9.1%

Relative Position 11.1%

78

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified

Opt range

Figure 4-34: Absolute position, Model B

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified

Opt range

Figure 4-35: Absolute acceleration, Model B

79

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified

Opt range

Figure 4-36: Wheel-ground relative position, Model B

4.5.7 Relaxed Constraints

Following the outline set with Model A, further Model B simulation results are presented in this

section to highlight the effect increasing the effective decoupler mass. When this mass, mL, was

allowed to vary up to 1 kg, the results are dramatic. As shown in Figures 4-37 and 4-38, the dynamic

stiffness and dynamic damping are increased in the low frequency range (0-30 Hz), decreased in the

middle-high frequency range (30-110 Hz), and increased again in the very high frequency range

(greater than 110 Hz). The former two are desirable traits, and the latter is undesirable. However, the

behaviour is the very high frequency range is still a significant improvement over the baseline.

The increase in dynamic stiffness and dynamic damping in the low frequency range is responsible

for the significant improvements of approximately 24% to the unsprung mass behaviour, as seen in

Figures 4-39 to 4-41 and as reported in Table 4-8. Figure 4-42 illustrates the results reported in

Tables 4-7 and 4-8.

80

0 20 40 60 80 100 120 140 1600

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

Frequency (Hz)

Magnitude (

F2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-37: Dynamic stiffness, mL increased, Model B

0 20 40 60 80 100 120 140 160100

200

300

400

500

600

700

800

Frequency (Hz)

Magnitude (

F2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Figure 4-38: Dynamic damping, mL increased, Model B

81

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-39: Wheel absolute position, mL increased, Model B

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5x 10

4

Frequency (Hz)

Magnitude (

X2 /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-40: Wheel absolute acceleration, mL increased, Model B

82

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (Hz)

Magnitude (

(X2-X

r) /

Xr)

Baseline

Modified (relaxed)

Modified (not relaxed)

Opt range

Figure 4-41: Wheel-ground relative position, mL increased, Model B

Table 4-8: Improvements to system behaviour, mL increased, Model B

Measure Improvement

Absolute Position 26.0%

Absolute Acceleration 22.6%

Relative Position 23.7%

83

1 2 30

5

10

15

20

25

30

35

% I

mpro

vem

ent

Com

pare

d t

o B

aselin

e

Absolute Position Absolute Acceleration Relative Position

Normal constraints

Increased mL (1kg)

Figure 4-42: Comparison of simulation results, Model B

4.5.8 Physical Construction

As with Model A, Model B has been specifically designed to allow for ease of construction. Because

the decoupler moves vertically within the shock absorber, there would be no observable difference to

the exterior when compared to a standard monotube shock absorber, except with the possibility of

being slightly longer.

According to the optimum parameter values as reported in Table 4-6, the inner diameter of the

shock absorber is 0.0573 m, and the outer diameter of the piston rod is 0.0050 m. The fluid density is

left unchanged at 1000 kg/m3, such as water or ethylene glycol.

As with Model A, the values for CL, RL, kT, and mL may be created by a rubber diaphragm or

bellows, decoupler cage or grate, gas pressure or physical spring, or embedded steel, respectively.

Because the decoupler movement is vertical, a flat plate of area AL may be used. If circular, this plate

would have a diameter of 0.0482 m, which is naturally smaller than the inner diameter of the shock

absorber. With a density of 7.85 g/cm3, a steel decoupler of mass 0.5 kg would be 0.0350 m thick.

However, the decoupler mass would be less than 0.5 kg because the parameter mL includes the

effective mass of the fluid passing through the decoupler cage. Additionally, the thickness can be

84

reduced by using a heavier material. Finally, the dividing piston accompanying the gas spring must

have a hole to allow the main piston to pass through.

While there is not as much flexibility in constructing Model B as there is with Model A, Model B

does not require any external modifications and as such will more easily fit within the space envelope

currently used by standard monotube shock absorbers.

4.6 Other promising models

While the Models A and B are the most effective, there are four additional models that yield

promising qualitative results. Models E, F, H, and I all exhibit an appreciable decrease in dynamic

stiffness at a particular frequency range. Figures 4-43 to 4-46 show the four models and their

corresponding dynamic stiffness, respectively. Model I maintains a low dynamic stiffness at all

higher frequencies, whereas the other three models eventually return to the baseline dynamic

stiffness.

The important point to note here is that although the drop in dynamic stiffness occurs at too low of

a frequency, it is conceivable that a more detailed study could “push” the change to a higher

frequency. In this case, the models would behave as Model B. The dynamic stiffness graphs are

presented in a logarithmic scale in order to give a clearer indication of the qualitative system

behaviour.

85

10-2

10-1

100

101

102

103

104

101

102

103

104

105

106

107

108

Frequency (Hz)

Magnitude (

F1/X

r)

Baseline

Modified

a) Schematic b) Damper dynamic stiffness

Figure 4-43: Typical dynamic stiffness for Model E

10-2

10-1

100

101

102

103

104

101

102

103

104

105

106

107

108

Frequency (Hz)

Magnitude (

F1/X

r)

Baseline

Modified

a) Schematic b) Damper dynamic stiffness

Figure 4-44: Typical dynamic stiffness for Model F

86

10-2

10-1

100

101

102

103

104

101

102

103

104

105

106

107

108

Frequency (Hz)

Magnitude (

F1/X

r)

Baseline

Modified

a) Schematic b) Damper dynamic stiffness

Figure 4-45: Typical dynamic stiffness for Model H

10-2

10-1

100

101

102

103

104

101

102

103

104

105

106

107

108

Frequency (Hz)

Magnitude (

F1/X

r)

Baseline

Modified

a) Schematic b) Damper dynamic stiffness

Figure 4-46: Typical dynamic stiffness for Model I

87

4.7 Summary

In this chapter, several configurations were introduced in which the decoupler component of a

hydraulic mount was integrated with a monotube shock absorber. Two configurations, Models A and

B, yielded favourable results. Model A decreased tire deflection by approximately 9% at the wheel

hop frequency without appreciable change to behaviour at other frequencies. Model B decreased tire

deflection by approximately 11% by increasing the entire low-frequency dynamic stiffness, but with a

significant decrease in high frequency dynamic stiffness. The improvements to wheel-ground relative

displacement for Model A and Model B are compared alongside the external design from Chapter 3

in Figure 4-47. Models A and B were also examined with relaxed parameter constraints. It was

shown that by allowing more decoupler travel and/or increasing the decoupler mass, the tire

deflection could be reduced by approximately 14-24%.

It is important to note that the behaviour of Model A has a narrower target frequency. Model B,

however, has a much wider increase in low frequency dynamic stiffness, but is accompanied by a

very significant decrease in high frequency dynamic stiffness. Therefore, Model B is better suited for

reducing high frequency noise, vibration and harshness, whereas Model A is better at reducing

vibration at a particular frequency without adversely affecting other system behaviour at other

frequencies.

Four other configurations from Chapter 4, Models E, F, H, and I, exhibited promising qualitative

behaviour in the form of a large decrease in dynamic stiffness over a defined frequency range. This

behaviour could potentially be tapped to reduce high frequency vibration.

Physical construction of both Models A and B was briefly discussed; Model A has more design

flexibility, but Model B has a more traditional external shape. However, both models could be easily

constructed.

88

1 2 30

2

4

6

8

10

12

14

16

% I

mpro

vem

ent

Com

pare

d t

o B

aselin

e

Absolute Position Absolute Acceleration Relative Position

External Design

Model A

Model B

Figure 4-47: Improvements of the three designs

89

Chapter 5

Future Considerations

The intent of this chapter is to highlight some of the possible extensions to the work discussed herein.

In particular, a nonlinear damper orifice, permitting more decoupler movement, and the ability to

create a semi-active shock absorber are considered. Model A is used for the discussions.

Figure 5-1: Nonlinear extension

5.1 Damper Orifice Modelling

One method for more accurately capturing the dynamics of the shock absorber is to use a detailed

fluid flow model as in [36, 43, 52]. However, another method that still captures more behaviour, but

is simpler than the detailed fluid flow method, is to model the damper orifice using four piecewise

90

linear curves, as originally discussed in Section 2.1.4. For example, Figure 5-2 compares two force-

velocity curves: a baseline piecewise linear shock absorber similar to that shown in Figure 2-8, and a

modified curve based on Model A (incorporating the same four damper rates as the baseline). The

damping rate of both Model A and the baseline decrease at higher speeds, more closely resembling an

actual shock absorber. Even so, the peak in the curve for the modified shock absorber is still evident,

and is more clearly shown in Figure 5-3.

-1.5 -1 -0.5 0 0.5 1 1.5-1200

-1000

-800

-600

-400

-200

0

200

400

600

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 5-2: Shock absorber force vs. velocity, piecewise linear damping, Model A

91

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65100

150

200

250

300

350

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 5-3: Shock absorber force vs. velocity close up, piecewise linear damping, Model A

5.2 Increased Decoupler Travel

As illustrated in Section 4.4.7, increasing the allowable decoupler travel for Model A yields

significant improvements in the results. The main limiting factor in the decoupler movement is the

overall diameter of the radial band; this diameter is limited by the space envelope surrounding a

shock absorber. Correspondingly, the extreme outermost position of the decoupler is in turn

restricted.

One possible manner of increasing the allowable decoupler travel is to redirect the fluid to flow

vertically instead of radially. However, this would make the model almost identical to Model B, with

the vertical decoupler travel. To eliminate the effect of the vertical movement, the fluid should be

directed in both up and down directions, as illustrated in Figure 5-4. The result is a twin-tube-like

construction with a pair of decouplers, one each above and below the opening into the outer tube.

The outer tube is concentric with the main shock absorber body, just as with the unmodified Model A.

This modified Model A will allow greater decoupler travel, yet will not significantly expand the outer

dimensions of the shock absorber.

92

Figure 5-4: Modified Model A schematic

5.3 Semi-Active Shock Absorber

5.3.1 Concept

The main premise behind making the modified shock absorber semi-active is to increase the effective

frequency range, or to allow the user to adjust the response on demand. The former uses a computer

and various sensors and actuators to dynamically adjust one or more parameters, such as the orifice

opening or the compliance stiffness. This method also allows the system to adjust as system

parameters change naturally with age, thus limiting any potential increases in vibration level due to a

mistuned vibration absorber [53, 54]. Alternatively, if an input device allows the user to adjust the

parameters, a complex system of sensors is not required because the system is changing on demand,

not in real-time response to road conditions.

For example, according to [10], a larger orifice opening in a hydraulic suspension mount increases

the frequency at which the dynamic stiffness suddenly increases. However, a larger opening also

increases the dynamic stiffness in the region preceding the sudden increase. As such, a variable

orifice may be able to maintain a lower dynamic stiffness for a larger frequency range. As another

example, [55] use a magnetorheological fluid as a variable damper element within a tuned mass

absorber. In [53], the absorber dynamic stiffness is adjusted to maintain the appropriate resonant

frequency.

93

The following two subsections discuss the sensitivity of Model A parameters, as well as a semi-

active application using a variable orifice.

5.3.2 Sensitivity Analysis

A sensitivity analysis was performed on Model A to determine which parameters have the greatest

effect on the system. In particular, the sensitivity of three functions to a 1% change in each tuneable

parameter was determined for a range of frequencies. The functions analysed are F (dynamic

stiffness), F/s (dynamic damping) and XL (decoupler movement). Summaries of the effects on F, F/s

and XL are presented respectively in Figures 5-5 to 5-7. Each indicates the maximum and minimum

value of the seven sensitivity figures associated with each case.

A 1% change in any of the parameters except decoupler resistance and compliance resulted in a

change of between 0.2% and 0.4% in dynamic stiffness and dynamic damping. Without changing the

structure of the shock absorber itself, changing the decoupler mass and decoupler area result in the

largest changes. These same tuneable parameters have the most effect on decoupler movement,

although the resulting changes are between 1% and 2.6%.

In general, if one parameter is changed, then at least one other must also be changed so as to

maintain the correct resonant frequency. This procedure actually occurs during the MATLAB

optimisation routine fmincon. However, there is a potential benefit to the resonant frequency shift

associated with not changing a second parameter. By being able to shift the peak damping effect, the

damper may be able to better react to changing road conditions, or even be adjusted manually by a

mechanic to account for vehicle resonant frequency changes as the vehicle ages.

94

1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

% C

hange in D

ynam

ic S

tiff

ness

Ap A

r k

B m

L R

L A

L C

L

Max sensitivity

Min sensitivity

Figure 5-5: Sensitivity summary for dynamic stiffness, Model A

1 2 3 4 5 6 7-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

% C

hange in D

ynam

ic D

am

pin

g

Ap A

r k

B m

L R

L A

L C

L

Max sensitivity

Min sensitivity

Figure 5-6: Sensitivity summary for dynamic damping, Model A

95

1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

1

2

3

% C

hange in D

ecouple

r M

ovem

ent

Ap A

r k

B m

L R

L A

L C

L

Max sensitivity

Min sensitivity

Figure 5-7: Sensitivity summary for decoupler movement, Model A

5.3.3 Variable Orifice

A simple way to create a semi-active shock absorber is to use a control valve to open or close the

main piston orifice. This action may be binary (open or closed) or continuously variable. In general,

the larger the orifice opening, the lower the resistance [56]. Figure 5-8 shows the force vs. velocity

curve for a semi-active version of Model A, with a close up shown in Figure 5-9. In this example, the

damping rate is piecewise linear in the same manner as discussed in Section 5.1. To make the model

semi-active, a control valve has been added to the orifice to further open or close it. By opening the

valve at high velocities, the force is decreased. Figure 5-10 shows the control signal; a larger value

corresponds to a larger orifice opening. In this setup, the peak from the passive Model A is combined

with a high speed drop similar to Model B to generate an effective overall result.

96

-1.5 -1 -0.5 0 0.5 1 1.5-1200

-1000

-800

-600

-400

-200

0

200

400

600

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 5-8: Shock absorber force vs. velocity, semi active valve control, Model A

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1100

150

200

250

300

350

400

Velocity (m/s)

Forc

e (

N)

Baseline

Modified

Figure 5-9: Shock absorber force vs. velocity close up, semi active valve control, Model A

97

-1.5 -1 -0.5 0 0.5 1 1.50.9

1

1.1

1.2

1.3

1.4

1.5

Velocity (m/s)

Contr

ol S

ignal

Figure 5-10: Orifice control signal (larger value corresponds to larger opening)

5.4 Summary

In this chapter, several concepts for improving the design were introduced. Discussions and

preliminary results were presented on piecewise linear orifice modelling, increasing available

decoupler travel, and a semi-active application. System behaviour is most sensitive to decoupler

mass and decoupler area if overall structural changes are not permitted. A more “traditional” semi-

active approach involving the damper orifice also yields good results.

98

Chapter 6

Conclusions and Recommendations

The main goal of this thesis was to develop and analyse the conceptual design of a cost-effective,

easy-to-manufacture, frequency-dependent shock absorber. By incorporating either an entire

hydraulic mount or components therefrom, it is possible to target specific frequency bands of interest.

This in turn allows for an increase in damping at the wheel hop frequency accompanied by a decrease

in damping at much higher frequencies, even with a passive device. A literature review highlighted

the fact that most efforts in this area have focused on semi-active or active applications. Some

passive applications have applied similar concepts as those discussed herein, but with significantly

different physical embodiments.

Before discussing the designs presented, background information was provided on several relevant

topics. The basic structure, dynamics, and testing of shock absorbers was presented, along with the

difference between shock absorbers and struts. Tuned mass absorbers, hydraulic mounts, parameter

optimisation, and sensitivity calculations were also discussed.

The first major design, examined in Chapter 3, combined an external hydraulic mount with a

standard shock absorber, between the suspension and the wheel hub. An improvement of

approximately 4% was observed with respect to sprung and unsprung mass acceleration, and tire

deflection. Improvements could be increased significantly if allowable decoupler movement is

increased.

The second and third major designs were presented in Chapter 4. Models A and B incorporated a

hydraulic mount decoupler into the internal workings of a shock absorber. The decoupler in Model

A had a radial direction of movement, whereas the decoupler in Model B moved vertically. System

improvements due to Models A and B were approximately 9% and 11% respectively. Model A

exhibited a narrower band of increased dynamic stiffness, without significant change at other

frequencies. This setup is preferred if there is one particular frequency of interest. Model B exhibited

a much wider band of increased dynamic stiffness in the low frequency range (1-30 Hz), and is

therefore not as capable at targeting a specific frequency. However, a very significant decrease in

dynamic stiffness in the high frequency range (30-150 Hz) was observed, indicating that Model B

99

may be better suited for attenuating high frequency noise and vibration. The potential physical

construction of Models A and B was discussed; both designs would be relatively easy and affordable

to manufacture.

All three major designs showed marked improvements as the parameter constraints were relaxed.

By permitting more decoupler mass and/or more decoupler travel, the system improvements could be

as high as 26%. Because not every combination of relaxed constraints was examined, it is

conceivable that further improvements may still be possible.

Chapter 4 also included a brief discussion on other promising models. Models E, F, H, and I

exhibited favourable behaviour of a qualitative nature. The dynamic stiffness associated with these

models all had large decreases spanning approximately three orders of magnitude, such as 1-1000 Hz

in the case of Model E. If the low frequency “drop point” can be increased beyond 30 Hz or 50 Hz,

these models may also be useful for attenuating high frequency noise and vibration.

Finally, Chapter 5 addresses future considerations, and forms the basis for the recommendations

herein. Brief discussions were presented on the effect of fluid resistance in the shock absorber

modelling accuracy, on how to physically allow more decoupler travel, and how a semi-active

application might be approached. A sensitivity analysis indicated that changing the structure of the

shock absorber, or changing the decoupler mass or travel allowance, would have the greatest effect on

system performance.

Because Models A and B perform much better than the external design, it is recommended that the

external design be disregarded in the future. Models A and B should be the primary focal point for

future work. The modelling of these two designs should be revisited and expanded to include more

accurate components. For example, fluid flow within the decoupler should be examined in a

nonlinear manner. Furthermore, modifications to Model A should be considered to allow more

decoupler travel, as shown Figure 5-4. A semi-active version of either Model A or B should also be

examined in more detail. In conclusion, it is recommended that both Models A and B be extensively

considered for commercialisation.

100

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