ap-6-10.dvi


Acta Polytechnica Vol. 50 No. 6/2010

A Model of Active Roll Vehicle Suspension

I. Čech

Abstract

This paperdescribes active suspensionwith active roll for four-wheel vehicle (bus) bymeans of an in-series pumpactuator
with doubled hydropneumatic springs. It also gives full control law with no sky-craping. Lateral stiffness and solid axle
geometry in the mechanical model are not neglected. Responses to lateral input as well as responses to statistical
unevennesses show considerable improvement of passengers comfort and safety when cornering.

Keywords: active suspension, roll-yaw model of a four-wheel vehicle, cross control, solid axle, hydraulic control, pump
actuator.

1 Introduction
A mathematical description of a model of active roll
vehicle suspension ismade in steady state of the har-
monic input in symbolic form. The complex am-
plitudes and effective values are denoted by capi-
tal letters and the instantaneous values, and also
the constant values, are denoted in lower-case let-
ters. Differentiations are substituted by the operator
s = i2πf, second differentiations by s2, and so on.
As our model includes two more stages of differenti-
ations than in usual mechanical models, a complex
symbolic form is necessary to handle the model.
Only linear relations are used.
A complete 4-wheel vehicle is dealtwith here, but

no heave input is presumed.
The parameters relate to a tall bus.
The computedanswers are comparedwithpassive

suspension.
The results are given for both vertical and hori-

zontal input, in spectral form and in impulse form.
Unlike other proposed solutions, e.g. [1, 2, 3] our

project includes the static load control, and should
be low-powered. This is because of the in-series char-
acter of the control, so that static displacement com-
pensation takes no power. In addition, the controller
uses no throttling and works against no static load.
The control scheme is simple, so that there are no
problems with stability.
There is also the possibility of achieving active

suspension using a source of force parallel to air sus-
pension (which includes the static load displacement
control). This source of force can be implemented
by a linear electric motor. However, the power pro-
ducedby the linearmotor is small in comparisonwith
a rotational electric motor with gearing.

2 Mechanical model
Dealing only with linear relations, we first presume
the unevennesses decompositiondiagram inFig. 1. It

showsananalysis of thevertical input (unevennesses)
z1–z4 of the four wheels 1–4 of the model into input
zks1, zks2 of the heave-pitchmodel [4], whichwill not
be dealt with here, and input zkp1, zkp2 of the four-
wheel rollmodel. This input will be marked zk1, zk2.
Indices 1, 2 relate to front and rear axles. The me-
chanical schema of this roll-yaw model is shown in
Fig. 2.

Fig. 1: Analysis of vertical inputs

This figure shows a model of a suspension with
a solid rear axle. We will consider the radii of gyra-
tion of the body rbx, of the seat rsx, and of the axle
rwx. (The mass of the axle is assumed to be concen-
trated in the wheels.) Then there are the heights of
the mass centers of the body ztb, of the seat zts and
of the wheel ztw. The track is denoted by yw, the
lateral distance of the seats by ys, and the distance
of the spring settings by yc. The height of the joint
of the solid axle ismarkedby zq, and the lateral stiff-
ness is marked by ky. The lateral displacements of
thewheels at stiffnesses ky1, ky2 are denoted by Yky1 ,
Yky2 . Vertical antiphased displacements of the body
(above the wheel), seat and wheel are noted by Zb,
Zs, and Zw1, Zw2.

16



Acta Polytechnica Vol. 50 No. 6/2010

Fig. 2: Mechanical scheme

The most important input of this model is the
lateral acceleration Ac.
We will take into account the lateral sliding of

the rolling tyre. Sliding acts similarly to damper. Its
damping is proportional to the weight of the vehicle
and reciprocally proportional to the travelling veloc-
ity and the constant of the tyre tgα. The stiffness
of a damper is its damping multiplied by operator s.
So

kypk = s · ag(mb + ms + mw)/vx/tgα

The lateral stiffness ky consists of this sliding stiff-
ness kypk and the lateral stiffness of the tyre kwy act-
ing in series. So

1/ky =1/kwy +1/kypk,

We will use

kwyre =0.6kw, tgα =0.1.

The lateral input of themodel is due to the lateral
acceleration Ac of the body.

The solid axle rotates around the center of the
anti-phasedunevennesses C, thebody rotates around
joint Q and translates in lateral direction with the
lateral displacement of the joint.
Only small angular displacements are assumed.
The lateral stiffness in front and rear, and also

the wheel masses front and rear correspond to the
position of the mass centre, i.e.

ky1/ky2 = mw1/mw2 = xw2/xw1

The roll model rotates around the on-axis, but its
angle α

α =arctg(zq/xw)

is small and approximately taken as cosα =1. Nev-
ertheless, the efficient value of the axle joint height
under the mass centre of the body is

zqt = xw1/xw · zq

17



Acta Polytechnica Vol. 50 No. 6/2010

3 The equations of the
mechanical model

Two auxiliary constants are used to enable the equa-
tions to be used also for independent suspension,
namely tn =1, nz =0 when with solid axle, tn =0,
nz = 1 when no solid axle is used. (With indepen-
dent suspension the inertia forces of the wheel are
supported by the body.)
If Y z1, Y z2 are the lateral displacements of the

yaw motion, then the lateral force originating from
the body mass mb is due to the acceleration of cor-
nering, the rollmovement of the body, the roll of the
joint, and the lateral shift of the mass center, so

Sqb =4mbAc+

4mbs
2[(ztb − zqt)Zb + zqtZw2] ·2/yw−

4mb · s2 · (xw2Yky1 + xw1Yky2)/xw
(3.1)

and the lateral force of the seat

Sqs =4msAc+

4mss
2[(zts − zqt)Zb + zqtZw2] · 2/yw−

4ms · s2 · (xw2Yky1 + xw1Yky2)/xw
(3.2)

The vertical forces in the suspension between
body and wheel, front and rear

Sbw1 = kb1(Zb − Zw! − Zr1)+ (3.3)
kst1(Zb − Zw!),

Sbw2 = kb2(yc/yw · (Zb − Zw2) − Zr2)+ (3.4)
kst2yc/yw · (Zb − Zw2)

Nowthe equation ofmoments to thebody around
the on-axis:
moments of the weight of the displacedmass cen-

tres

0= −8[(ztb − zqt)mb +(zts − zqt)ms]ag/ywZb+

moment of the inertia force from the roll motion

+8s2mbr
2
bx + msr

2
sx]/yw · Zb

moments of the forces in the suspension

+ywSbw1 + ycSbw2

moment from the suspension of the seat

−2ysks(Zs − ysZb/yw)

moments from the lateral forces

+(ztb − zqt)Sqb +(zts − zqt)Sqs
moments of the roll motion of the wheels with inde-
pendent suspension

+2 · (mw1 + nzmw2)[2s
2(z2tw + r

2
w)/yw · Zb−

2ztwag/yw · Zb]

moments from the lateral motion of the wheels

+2ztw[(mw1 + nzmw2)Ac−
s2(mw1Yky1 + nzmw2Yky2)]

(3.5)

Equation of the forces vertical to the seat

0= s2msZs + ks(Zs − ys/yw · Zb) (3.6)

Equation of vertical forces to the front wheels

0= s2mw1Zw1 + kw1(Zw1 − Zk1) − Sbw1 (3.7)

Equation of moments to the solid rear axle:
From the inertia of the axle

0= 2s2mw2(y
2
w/4+ tn · z

2
tw + tn · r

2
wx)·

2/yw · Zw2

from the unevennesses

+ywkw2(Zw2 − Zk2)

from the suspension

−ycSbw2

from the lateral forces in the joint Sqb, Sqs

+zqxw1/xw · (Sqb + Sqs)

from the lateral motion of the axle

−2tn · mw2(ztw − zq)(Ac − s
2 · Yky2)

from the weight of the displaced mass centre

−2tn · mw2ztw ·2agZw2/yw

fromtheweight of the body and seat to the displaced
axle

+8(mb + ms)agzqZw2/yw · xw1/xw (3.8)

Equation of lateral forces
Lateral forces between the wheels and the road

0= −2ky1Yky1 −2ky2Yky2

inertia forces of the body and seat

+Sqb + Sqs

from solid axle roll

+s2 · mw2 · ztw · 2/yw · (tn · Zw2 + nz · Zb)

18



Acta Polytechnica Vol. 50 No. 6/2010

from the lateral shift of the wheels

+2(mw1 + mw2)Ac−
s2 · (mw1Yky1 + mw2Yky2)

(3.9)

Finally, the equation of moments to the body
around the z-axis passing through the center of grav-
ity of the body: inertia moments of the body and
seat

0=s2(4mbr
2
bz +4msr

2
sz)·

(Yky1 − Yky2 − zq · 2/yw · Zw2)/xw

plus the inertia moments of the wheels

+s2[2mw1(r
2
wz + x

2
w1
+ y2w/4)+

2mw2(r
2
wz + x

2
w2
+ y2w/4)]·

(Yky1 − Yky2 − zq · 2/yw · Zw2)/xw

and themoment of the forces between thewheels and
the road

+2xw1ky1Yky1−2xw2ky2(Yky2+zq·2/yw·Zw2) (3.10)

4 The control
Fig. 3 shows the control scheme of the suspension. In
it, sensors are marked by their sensitivity constants.
We will assume that active suspension will be

used only on the rear axle.

On the left side is the scheme of the front axle,
with its spring kb1 and the damping bb1 and with
static level control consisting of sensor fz, very low
pass VLP, hydraulic valve V , central accumulator
CA and drain DR. The dynamics of this control
will be not dealt with here, i.e. Zr1 =0.
Some explanation at first.
Elementsmarked SP are pressure sensors. A ring

with minus in it produces the difference of two sig-
nals.
A ringwith N in itmeans anegator—anelement

converting the phase by 180 deg.
A ring with P in it means an amplifier, with am-

plification proportional to the control input marked
by an arrow.
Hydraulic linkage is shown by double lines, and

the signal paths are shownby single lines. The direc-
tions of the signals are marked by triangular arrows.
No feed-back through them is assumed. Signals com-
ing to a common point add together. Crossing signal
paths are not connected.
On the left side, the equipment of front wheels

1, 2 is shown. On the right side, the equipment of
rear wheels 3, 4 is shown. For reasons of stability,
active suspension can be used only on one axle— let
it be the rear axle. (Cars with front drive can use
the front axle.) The lateral acceleration meter (with
sensitivity constant tar) ismounted on the body at a
height of zta.

Fig. 3: Control scheme

19



Acta Polytechnica Vol. 50 No. 6/2010

The active suspension on the rear axle (right part
of the figure) consists of the sensors of the displace-
ment wheel-body marked by fz, producing the mean
value of the front and rear displacements, and of the
sensor of the vertical acceleration, marked by 1/fa,
and of the actuator embodied by a pump driven by
an electric motor. The actuator is marked A, and its
control amplification is ζ2, the common amplication
of signals is ζ. There is also a control velocity sen-
sor, marked by its constant ζr, to develop a source
of velocity by means of a negative feedback. Thus,
the force of the actuator (proportional to the mass
mb + ms) is

Sa =ζ2sζr · Zr2 + ζζ2ζd[ζh · s
2/(2πfa) · Zb+

2πfzζb(Zb − Zw1/2− Zw2/2)−
(Ac −s2Yky2 +2s

2((zta − zqt)Zb+
zqtZw2)/yw −2agZb/yw)tar]b0·
(mb + ms)/m0

(4.1)

where b0 is a formal constant valued 1N/ms and m0
is a constant with dimension kg. and where suit-
able damping with damping coefficient βb is intro-
duced by the element, marked by its transmission
ζb =1+s ·2βb/(2πfz)
Two filters are also included: a low pass with

transmission ζd and cut-off frequency fd, which helps
to prevent undamping of the wheel, and a high-pass
with transmission ζh and cut-off frequency fh, which
eliminates the false signal of the acceleration sensor
of the tilted body. Namely

ζd =1/(1+s/(2πfd)), ζh = s/(1+s/(2πfh))

To the negative feedback: This control equa-
tion makes the actuator approximately a source of
hydraulic current, i.e. the control displacement is
not dependent on the force Sbw2. (This can also be
achieved by very high inner stiffness of the actuator,
but with great losses of power.)
The force of the actuator is at the same time

Sr = krZr2 + k0Zr2 − kb2(Zb2 − Zw2 − Zr2) (4.2)

where Zr2 is the control displacement and k0 is the
stiffness of the balancing spring in which the same
static pressure is maintained as that in spring kb
by means of sensors SP , differentiator D, very low
pass V LP and valve V . (In the computed examples
k0 = kbre.) The inner stiffness of the actuator kr
consists of the damping of pump br and the inertia
of the pump and the electric motor, so

kr = sbr +s2mr

Combining equations 4.1 and 4.2, a single control
equation can be written, as follows

0=ζ2sζrZr2 + ζζ2ζd[ζh ·s
2/(2πfa) · Zb+

2πfzζb(Zb − Zw1/2− Zw2/2)−
(Ac −s2Yky +2s2((zta − zqt)Zb+
zqtZw2)/yw −2agZb/yw)tar]−
[(sbr +s

2 · mr)Zr + k0Zr−
kb · (Zb − Zw − Zr)]m0/(mb + ms)/b0

(4.3)

Cross control is provided on the front axle. This
cross control uses the measurements of the pressure
transducers, marked SP (in Fig. 3). The sum of the
loads in the rear (after filtering through a very low
pass V LP to get static values sst1, sst2) controls the
amplification of the proportional amplifier P of the
load difference of the front wheel load. The outputs
of the two proportional amplifiers are compared, and
theirdifference controls thepumpsof the frontwheels
with the appropriate phase.
An equation to fulfil this aim can be written as

follows

0=sZr1 + ζdpvcc[kbre2(Zb+

yw/yc · Zr2 − Zw2)/(xw1/xw)/sst1−
kbre1(Zb + Zr1 − Zw1)/(xw2/xw)/sst2]

(4.4)

where vcc is the control constantandwhere the trans-
mission of the low path

ζdp =2πfdp/(2πfd +s)

with the characteristic frequency fdp delays the an-
swer to antiphased unevennesses. The influence of
the inner stiffness of the actuator is not included here
(a source of displacement is assumed).
Equations 3.1 to 3.10, 4.3, 4.4make a set of equa-

tions for unknownquantities Sbw1, Sbw2, Sqb, Sqs, Zb,
Zs, Zw1, Zw2, Yky1, Yky2, Zr1, Zr2.

5 Input and output

The spectrum of a surface of medium roughness is
used for the statistical formof the roadunevennesses
(ride velocity vx =60 km/h):

Zf k =
√
(vx)/f ·

√
(1/2−

1/2/(1+(2π · f /4.5 · yw/v4x))/
(1+(rpn · f /vx)2) · 0.64 mm.

(5.1)

where the track yw = 1.8 m. A correction to the
tire diameter rpn is added. The effective value of this
spectrum is 7.7 mm.
The impulse input of the shape 1-cos (also “trans-

lated sinusoidal”) was used in its spectral form

τ f =sin(2πf ti)/(2πf)+

ti/2 · [sin(τ1)/τ1 +sin(τ2)/τ2],
(5.2)

20



Acta Polytechnica Vol. 50 No. 6/2010

where

τ1 = π −2πf ti, τ2 = π +2πf ti

and where ti is the duration of the impulse at its
half-height.
The maximum value of the impulse unevenness

zki is assumedtobeproportional to its duration (pro-
gressive impulses).
These inputs are shown in the graphs of Zf bw or

Zbw.
The lateral input (lateral acceleration) is a sort of

trapezoid-shaped impulse, but with rounded edges.
It has been obtained by adding the two above-
mentioned curves 1−cos with ti/2 (double cosinus),
the second delayed by ti from the first.
This input will be attached to the graphs of lat-

eral seat acceleration.
The graphs of the frequency characteristics Hf

have the effective values attached. These effective
values follow the formula√

2
∫

H2f df within limits 0 : fmax

The pulse-effective values, given with time histories
ht, are the effective values of the whole answer di-
vided by the pulse duration, ie.√∫

h2tdt/ti) within limits t =0− inf

The vertical statistical input is due to the central
path between the wheels.
The time interval between the front and the rear

of the vertical input depends on the axle base xw and
riding speed vx

tx = xw/vx.

So the relation of the vertical input front and rear is

Zk2 = Zk1(cos(stx)+ isin(stx)

6 Criteria
Lateral acceleration in the rear seat

Asy2 =Ac −s
2 · Yky2+

s2[(ztb − zqt)Zb + zqZw2] ·2/yw−
2agZb/yw

Vertical acceleration in the seat

As = s
2Zs

Vertical accelerationof the body at radius yw/2 yw/2

Ab = s
2Zb

Body-wheel displacement front and rear

Zbw1 =Zb − Zw1,
Zbw2 =Zb − Zw2.

Load transfer ratio front and rear

Sw1/sst1 = kw1(Zw1 − Zk1)/sst1,
Sw2/sst2 = kw2(Zw2 − Zk2)/sst2.

where the static forces are

sst1 = 2(mb + ms + mw)agxw2/xw,

sst2 = 2(mb + ms + mw)agxw1/xw

Seat-body displacements front and rear

Zsb1 =Zs1 − Zb,
Zsb2 =Zs2 − Zb

Dynamic-to-static ratio of lateral forces between
wheels and road, front and rear

Sy1/sst1 = ky1Yky1/sst1,

Sy2/sst2 = ky2Yky2/sst2.

Spring displacements, front and rear

Zp1 = Zb − Zw1 − Zr1,
Zp2 = (Zb − Zw2)yw/yc − Zr2

The forces in the suspension, front and rear, are

Sbw1 = kb1(Zb1 − Zw1),
Sbw2 = kb2(Zb2 − Zw2 − Zr2)

7 Universal parameters and
used parameters

The universal parameters (natural frequencies, dam-
ping coefficients) defined for the heave-pitch model
are also used in the roll model, but we must also
consider the anti-roll bar.
If the anti-roll bar stiffness-to-spring-stiffness kb

ratio κstab is given, then

kb =kbre +sbb =4π
2f2b (mb + ms)(1+ κstab)+

s · 4πβbfb(mb + ms)

Parameters used with a high bus V variant X − A,
i.e. cross control front and active control rear

21



Acta Polytechnica Vol. 50 No. 6/2010

fb =1 Hz βb =0.4 mb =1000 kg ztb =0.9 m rbx =0.6 m yw =1.8 m
fw =10 Hz βw =0.05 mw =150 kg ztw =0.5 m rwx =0.3 m yc =1.2 m
fs =3 Hz βs =0.28 ms/mb =0.5 zts =2.7 m rsx =0.4 m ys =1.2 m

ζ =3 fz =0.4 Hz βa =1 fa =0.3 Hz fd =1 Hz fh =0.1 Hz tar =0.075 s
ζ2 =100 ζr =0.9br = bb/1000mr = mb/1000 b0 =1 Ns/m m0 =1 kg zta =0.9 m

vcc =1 m/s fcc =3 Hz κstab1 =0 κstab2 =0 zq =0.3 m xw =5.5 m
xw1/xw2 =2 rbz =2 m rsz =2 m rwz =0.3 m

Fig. 4: Time histories due to the impulse of lateral ac-
celeration. A thick line means an active variant (act),
a thin line means a passive variant (pas). The attached
numbers are imp-eff. values. A semicolon is used tomark
an index. In the graph of power, the attached value refers
to energy consumption [Ws/1000 kg]

Stiffnesses kb, kw are assumed to be proportional
to the static load front and rear:

kb1 =kb · 2xw2/xw, kb2 = kb ·2xw1/xw,
kw1 =kw ·2xw2/xw, kw2 = kw · 2xw1/xw

For variant P − P , i.e. for a vehicle with passive
suspension front and rear, the parameters differ as
follows:

κstab1 =1.2 κstab2 =1.4

zq =0.9 m ζ2=100

The parameters of actuators mr and br are only
estimated.

8 Example of modelling
results

In the graphs in Figs. 4, 5, the active variant act
(passive suspension front, active rear) is indicated by
more prominent lines (lines with dots) than those
of the passive variant (pas–pas). In some suitable
graphs, theappropriate input is shownby interrupted
lines.
Effective values (with frequency characteristics)

or impulse-effective values (effective values of the an-
swersdivided by the impulse duration,with timehis-
tories) are attached to the labels.

Fig. 5: Frequency characteristics due to unevennesses. A
thick linemeans an active variant (act), a thin linemeans
a passive variant (pas). The attached numbers are imp-
eff. values. A semicolon is used to mark an index

22



Acta Polytechnica Vol. 50 No. 6/2010

active passive

Steady-state values due to ac =1 m/s
2

Lateral acceleration in the seat asy [mm/s
2] 0.77 1.35

Body-wheel travel zbw2 [mm] 25 25

Vertical dynamic force to static force ratio, front sw1/sst1 12 20

Vertical dynamic force to static force ratio, rear sw2/sst2 14 20

Steady-state values due to in-phase unevenness zk1 = zk2 =0.01 m

Lateral acceleration in the seat Asy [m/s
2] 0.11 0.15

Vertical dynamic force to static force ratio, front Sw1/sst1 0.02 0.022

Vertical dynamic force to static force ratio, rear Sw2/sst2 0.017 0.019

Steady-state values due to cross unevenness zk1 =0.01 m, zk2 = −0.01 m
Lateral acceleration in the seat Asy [m/s

2] 0.028 0.023

Vertical dynamic force to static force ratio, front Sw1/sst1 0.00042 −0.07
Vertical dynamic force to static force ratio, rear Sw2/sst2 0.009 0.03

9 Stability

Stability against self-exciting oscillation is an impor-
tant criterion of the system. It is most important
with the roll model, where there is also the possi-
bility of roll-over. Asymptotic stability is used, i.e.
the real part of the eigenvalues of the equation ma-
trix must be negative. Stability was checked for all
parameters of a simplified model. The parameters
of nominal value hnom were varied between 1/100 to
100 multiples of its given value hjm. When no insta-
bility was found, output 100was given. If instability
was found at parameter value hcrit, the stability rate
hcrit/hnom was put out. A minimal stability rate of
about 2.2 of the passive model increases to 5 with
active suspension.

10 Conclusions based on the
example

With vertical statistical input, the lateral acceler-
ation in the seat is slightly increased and the lat-
eral force is substantially decreased. The lateral load
transfer is slightly deteriorated, but the effective val-
ues incorporating the values from the heave-pitch
model are little influenced. It has been shown that
it is possible by means of cross control to distribute
the load changesdue to corneringproportional to the
static load of the axle. The lateral input is the main
advantage of active suspension, thanks to the active
roll:
From the viewpoint of ride comfort, active roll

substantially suppresses the lateral force during cor-
nering. From the viewpoint of ride safety, active roll
also substantially suppresses the load transfer to the
outside wheels in a curve. Thus the vehicle can take

curves at a higher speed or with more safety.
With active roll suspension there is noneed touse

a solid axle for anti-roll purposes. This means more
comfort for the passenger and there is reduced dam-
age to the road, due to a big reduction in the lateral
forces betweenwheel and road. A solid axlewas used
in our examples, but with reduced height of the roll
axis.
With active suspension there is also no need for

anti-roll bars. Active roll technology also enables the
design of a slender passenger car for two passengers
sitting in tandem (we can call this a tandemo) with,
e.g., an 0.8 m track.

References

[1] Sampson, D. J. M., Cebon, D.: Active Roll
Control of Single Unit Heavy Road Vehicles,
http://www.cvdc.org/recent papers/
SampsonCebon VSD02.pdf

[2] Vacuĺın, O., Valášek, M., Svoboda, J.: Influence
of Vehicle Tilting on Its Performance, Conf. In-
teraction and Feedbacks, Praha, 2005.

[3] Masashi, Yamashita: Application of H-Y Con-
trol to Active Suspension Systems, Automatica,
Vol. 30, 1994.

[4] Čech, I.: A Pitch-Plane Model of a Vehicle with
Controlled Suspension. Vehicle System Dynam-
ics, Vol. 23, 1998, pp. 133–148.

Ing. Ilja Čech
E-mail: cech10600jes11@seznam.cz
Jesenická 11, 106 00 Prague 10, Czech Republic

23



Acta Polytechnica Vol. 50 No. 6/2010

List of symbols

Complex amplitudes, spectral values and instantaneous values
Ac Af c ac lateral (centripetal) acceleration [m/s

2]
As Af s as vertical acceleration of the seat [m/s

2]
Asy Af ty asy lateral acceleration in the seat above the center of gravity [m/s

2]
Ax Af x ax longitudinal acceleration input [m/s

2]
Sa Sf a sa force developed by the actuator [N]
Sbw Sf bw sbw force between body and wheel [N]
Sq Sf q sq lateral inertia force due to yaw motion [N]
Sp Sf p sp force on the spring [N]
Sqb Sf qb sqb lateral force in the joint due to the body [N]
Sqs Sf qs sqs lateral force in the joint due to the seat [N]
Ssb Sf sb ssb vertical force between seat and body [N]
Sw Sf w sw vertical force between wheel and road surface [N]
Sy Sf y sy lateral force between wheel and road surface [N]
Vr Vf r vr velocity of movement of the actuator [m/s]
Yar Yf ar yar lateral displacement of the body at lateral acceleration meter height [m]
Yky Yf ky yky lateral displacement in lateral stiffnesses ky1, ky2
Zb Zf b zb vertical displacement (for the roll model at radius yw/2) [m]
Yz Yf z yz lateral displacement of the body due to yaw motion [m]
Zbw Zf bw zbw body-wheel displacement [m]
Zk Zf k zk vertical input displacement [m]
Zr Zf r zr control displacement [m]
Zs Zf s zs vertical seat displacement [m]
Zw Zf w zw vertical wheel displacement [m]

Quantities without physical dimension

βa relative damping rate of the control
βb relative damping rate of the body
βs relative damping rate of the seat
ζ common amplification of signals
ζ2 amplification of the actuator
ζb corrective damping element
ζd transmission of low pass filter
ζh transmission of high pass filter
ζr sensitivity constant of feedback sensor
κstab ani-roll bar stiffness
κyw lateral stiffness to radial stiffness ratio
nz, tn nz =0, tn =1 for solid axle, nz =1, tn =0 for no solid axle

Other quantities

ag gravity acceleration [m/s
2]

bb damping of passive suspension [Ns/m]
bpk slip damping of the tyre [Ns/m]
b0 formal constant of the actuator b0 =1 [Ns/m]
br damping of the actuator [Ns/m]
bs seat damping [Ns/m]
bw wheel damping [Ns/m]
f frequency [Hz]
fa 1/(2πf a) sensitivity constant of transducer of vertical acceleration [Hz]
fb natural frequency of passive suspension [Hz]
fd, fh characteristic frequency of the filter [Hz]

24



Acta Polytechnica Vol. 50 No. 6/2010

fr sensitivity constant of the control velocity sensor [Hz]
fz 2πfz constant of the displacement transducer [Hz]
fε 1/(2πfε) constant of the roll acceleration transducer [Hz]
hj nominal value of the parameter
hkrit critical value of the parameter
k0 spring rate of the balancing spring [N/m]
kb stiffness of the body-wheel suspension [N/m]
kbre spring rate of the body-wheel spring [N/m]
kr inner stiffness of the actuator [N/m]
ks stiffness of the seat-body spring [N/m]
kst stiffness of the roll-bar [N/m]
kw radial stiffness of the tyre [N/m]
kwyre real lateral stiffness of the tyre [N/m]
ky lateral stiffness [N/m]
kypk sliding stiffness [N/m]
m0 formal constant m0 =1 kg
mb quarter body mass [kg]
ms quarter seat mass [kg]
mc total mass of vehicle mb + ms + mw [kg]
mr actuator mass transferred to the pump radius [kg]
mw wheel mass [kg]
p power [W]
rpn tyre radius [m]
rbx gyration radius of the body to the longitudinal axis [m]
rbz gyration radius of the body to the vertical axis [m]
rsx gyration radius of the seat to the longitudinal axis [m]
rsz gyration radius of the seat to the vertical axis [m]
rwx gyration radius of the wheel to the longitudinal axis [m]
rwz gyration radius of the wheel to the vertical axis [m]
s operator i2πf
t time [s]
tar sensitivity constant of the lateral acceleration sensor [Hz]
tgal constant of inverse proportionality between slip rotation and load
ti duration of input impulse at half-height [s]
vx travelling speed [m/s]
yc spring distance [m]
ys seat distance [m]
yw wheel track [m]
za height of the lateral acceleration meter [m]
zki maximum value of the vertical input impulse [m]
zq height of the joint [m]
zqt height of the axis on under the center of gravity [m]
zta height of the lateral acceleration transducer [m]
xw axle base [m]
xw1 distance between the mass centre of the body and the front axle [m]
xw2 distance between the mass centre of the body and the rear axle [m]
ztb height of the body mass centre [m]
zts height of the seat mass centre [m]
ztw height of the wheel mass centre [m]

Indices

1 front axle
2 rear axle
b body
f spectral values
re real part

s seat
x longitudinal direction, axis
y lateral direction, axis
w wheel

25