Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 3, pp. 715-731, Warsaw 2010

STUDY OPERATION OF THE ACTIVE SUSPENSION

SYSTEM OF A HEAVY MACHINE CAB

Grzegorz Tora

Cracow University of Technology, Institute of Machine Design, Kraków, Poland

e-mail: tora@mech.pk.edu.pl

This study is a part of research on active suspension systems of cabs in
heavymachinesandtrucks,used for suppressing low-frequencyand large-
amplitude vibrations. The suspension system incorporates two platform
mechanisms placed one upon the other. The lowermechanism is respon-
sible for maintaining the cab in the vertical position whilst the upper
mechanism controls the cab movements in the vertical direction. Mo-
tion of the cab is described using versors associatedwith themechanism
links. Relationships are derived that yield the instantaneous velocities of
the drives that lead to reduction of the cab vibrations in selectedDOFs.
The procedure is shown for calculating the loads acting on the drives
of the active suspension during the specified movement of the machine
frame. The mathematical model is further utilised in simulations of the
suspension operation.

Key words: active vibration reduction, cab suspension, platformmecha-
nism

1. Introduction

When heavy machines and tractors move in rough terrain, vibrations of the
operator cab are generated in its certain DOFs. After a while, high-amplitude
vibrations (up to 0.5m in the vertical direction) of low frequency (up to 5Hz)
make the machine operator tired and less efficient whilst the work safety fe-
atures tend to deteriorate. In order to improve the operator’s comfort while
at work, an active suspension of the cab can be incorporated in the machine
structure. Such an active suspension system incorporates an actuator mecha-
nism that handles the drives and the energy sources, the system measuring
the machine vibrations and the control system. The measurement system is
responsible for collecting real-time information about the movements of the



716 G. Tora

machine frame and the drives (Cardou andAngeles, 2008). The computer uses
this information to derive instantaneous velocities of hydraulic drives which
shall induce the movements of the actuator mechanism placed between the
frame and the cab to reduce low-frequency vibrations of the cab in selected
DOFs.

Active vibration reduction systems were first introduced in the suspen-
sions of the operator’s seats, they were then incorporated in the mounts of
the driver’s cabs in trucks, tractors and heavy machines. Typically, effective
vibration control of the driver’s seat can be achieved in the vertical direction
only. Passive, semiactive and active solutions in the suspensions of farming
tractors and trucks normally involve a greater number of DOFs, whilst the
amplitudes of motion are relatively small (Nakano et al., 1999). The active
suspension of a cab with two degrees of mobility is applied in heavy trucks
used in forestry. Cab displacements (especially lateral displacements) in heavy
trucks, where they are positioned relatively high, are considerable while the
machine travels in rough terrain. Therefore, the author proposes a structure
of the actuator mechanism that would fully answer the case (Tora, 2008).

2. Structure of the mechanism

The structure of themechanism and the number of drives are decided arbitra-
rily, depending on the number of DOFs to be handled. While a heavy truck
moves in rough terrain, the largest vibrations of a high-positioned cab are
registered along the lateral axis of the machine yr, along the vertical axis z
and around the axes xr and yr. Vibration reduction around the zr-axis se-
ems unnecessary as they prove negligible during the ride. When veering, the
cab should rotate with themachine frame. For simplicity, vibrations along the
xr-axis are neglected, too (Fig.1).
The active suspension mechanism incorporates two platform mechanisms

stacked serially one upon the other. The lower mechanism (see Fig.2) ought
to, inter alia, maintain the cab in the direction of the gravity force. The lower
mechanism consists of platform 6 suspended on boards 2 and 3 with the use
of spherical pairs D and E. Boards 2 and 3 are connected tomachine frame 1
by revolving pairs. Along the section DE, machine frame 1, boards 2 and 3,
and platform 6 form a planar mechanism working in the plane yrzr and set
in motion by cylinder 4. Cylinder 5, connected to platform 6 and frame 1
by spherical pairs A and G, induces rotating motion of platform 6 around
the direction defined by points D and E. The mechanisms presented above



Study operation of the active suspension... 717

Fig. 1. Coordinate system. DOFs to be handled by the vibration reduction system

Fig. 2. Lower platformmechanismmaintaining the cab in the vertical position, link
versors, systems of driving forces, inertia and gravity forces



718 G. Tora

has 2 degrees of mobility. The dimensions lOD, lDE, lBE, lOB can be chosen
such that one cylinder 4 should suffice to reduce the vibrations of platform 6
in 2 DOFs (rotation around the xr-axis and translation along the axis yr)
(Tora, 2008).
The upper platformmechanism (also referred to as the Sarrusmechanism)

is placed upon platform 6. It consists of 6 boards (four of them are shown in
Fig. 3) connected in twos by revolvingpairswhilst theneighbouringboardsare
positioned transverse one to another. Thismechanism, therefore, has 1 degree
ofmobility and the cabmoves perpendicularly toplatform6. It is set inmotion
by the cylinder hitched in the joints F and H.

Fig. 3. Upper platformmechanism elevating the cab in the vertical direction

3. Coordinate systems

Four systems of coordinates are considered. The first immobile system xgygzg
is associated with the distance travelled by the machine. The zg-axis is pa-
rallel to the vector of acceleration of gravity g. In this system, we define the
function of road profile on which the machine travels. The system xryrzr is
mobile and is associated with themachine frame. The centre point of the fra-
me is at the point O (Fig.1, Fig.2), the versors of axes are: ix1, i

y
1, i
z
1. Another

mobile system xmymzm is in-between the xgygzg and xryrzr. The origin of
the system xmymzm is at the point O and the zm-axis is parallel to the acce-
leration of gravity vector g whilst the axes zm, xm and xr are coplanar. The
last condition satisfies the requirement that the direction of the gravity for-
ce g in the system xryrzr should be taken into consideration. The coordinate



Study operation of the active suspension... 719

system xmymzm should be utilised to show the versor of the gravity force:
mig = [0,0,−1], which shall be expressed in the system xryrzr using the
transition matrix rmR from the coordinate system xmymzm to xryrzr

ig =
r
mR
mig (3.1)

The direction of the gravity force is of primary importance for two reasons:
firstly – the active suspension ought to position the cab along ig, secondly –
the actuator load is imposed by the gravity force acting upon the cab. The
fourth coordinate system xkykzk is associated with the cab, the versors of its
axes are: ix6, i

y
6, i
z
6. The origin of this coordinate system Q is in the centre of

mass of the cab. While modelling the operation of the active suspension, the
rotation matrix rkR has to be defined, from xkykzk (the cab) to xryrzr (the
frame). The elements of the matrix rkR are obtained basing on versors in the
system associated with the cab expressed in that associated with the frame.
Accordingly, we get

r
kR=



ix6i
x
1 0 i

z
6i
x
1

ix6i
y
1 i

y
6i
y
1 i

z
6i
y
1

ix6i
z
1 i

y
6i
z
1 i

z
6i
z
1


 (3.2)

The second element in the first row is equal to zero because i
y
6⊥i
x
1 (Fig.2).

4. Kinematics of the actuating mechanism

Basic quantities used in kinematic equations are versors associated with par-
ticular links of the mechanism. The versors are expressed in the system asso-
ciated with the frame. This study is limited in scope, so the kinematics of link
position for the given mechanism involves only solving the inverse problem
where the lengths of cylinders s4 and s5 have to be found for the predeter-
mined horizontal position of platform 6. The versor of the cab vertical axis
iz6 = [i

z
6x, i

z
6y, i
z
6z]
⊤ should have the opposite direction to that of the gravity

versor

iz6 =−ig (4.1)

The coordinates of versors of the mechanism links: i
y
6 = [0, i

y
6y, i
y
6z]
⊤,

i2 = [0, i2y, i2z]
⊤, i3 = [0, i3y, i3z]

⊤, i4 = [0, i4y, i4z]
⊤, i5 = [i5x, i5y, i5z]

⊤

and instantaneous lengths of the cylinders s4 and s5 are obtained by solving
equations that are derived recalling (Fig.2):



720 G. Tora

—orthogonality of versors of platform 6

i
y
6i
z
6 =0 (4.2)

— triangle COD
lCOi

y
1 + l2i2 = s4i4 (4.3)

— quadrilateral ODEB

l2i2+ l6i
y
6 + l3i3 = l1i

y
1 (4.4)

— polygon KAGFDO

lKAi
y
1 +s5i5+ lGFi

x
6 = lKOi

x
1 + l2i2+

1

2
l6i
y
6 (4.5)

Equations (4.2) and (4.3)-(4.5) projected onto the directions of the coordinate
system associated with the frame axes xryrzr can be easily transformed into
the system of polynomial equations of the second degree. In this case, an
explicit solution is found though an arbitrary decision is still required and
one solution has to be selected which meets the requirements imposed by the
configuration of the mechanism. For the upper mechanism, the equation is
used that defines the position of point O in the coordinate system associated
with the frame xryrzr

lOQ = l2i2+
1

2
l6i
y
6+(s6+ lHQ)i

z
6 (4.6)

The quantity s6 present in (4.6) is obtained using the integral of the rate of
change of the cylinder length (during the vertical motion of the cab) v6. That
is whywe can eliminate cumbersome fieldmeasurements of the frame position
with respect to the immobile system xgygzg.
As platform6 rotating round frame 1mightmove around the axes of joints

definedby theversors ix1 and i
y
6, we can easily predict the formof the equation

of the angular velocity vector of platform 6

ω6 =ω1+ω61pi
x
1 +ω61bi

y
6 (4.7)

where: ω61p, ω61b – velocity component of the platform 6 with respect to the
frame 1 in the direction ix1 and i

y
1, respectively.

The vectors of angular velocity of links 2, 3 and 4 are represented as the
sum of angular velocity vector of frame 1 and the product of the directional
vector ix1 and themodulus of relative velocity ωj1

ωj =ω1+ωj1i
x
1 j=2,3,4 (4.8)



Study operation of the active suspension... 721

Derivatives of equations (4.3)-(4.5) are utilised to find ω61p and ω61b

lCOω1× i
y
1+ l2ω2× i2 = v4i4+s4ω4× i4

l2ω2× i2+ l6ω6× i
y
6+ l3ω3× i3 = l1ω1× i

y
1 (4.9)

lKAω1× i
y
1 +s5ω5× i5+v5i5+ lGFω6× i

x
6 =

= lKOω1× i
x
1 + l2ω2× i2+

1

2
l6ω6× i

y
6

Obtaining the dot product of i4 and Eq. (4.9)1, and substituting (4.8) into
(4.9)1 yields

ω21 =
v4

l2i
x
1 × i2i4

=
v4
r1

(4.10)

Obtaining the dot product of i3 and Eq. (4.9)2, and substituting (4.7), (4.8)
and (4.10) into (4.9)2 yields

ω61p =−
v4

r1l6(i
x
1×i

y
6
i3)

l2i
x
1×i2i3

=
v4
r3

(4.11)

Obtaining the dot product of i5 and Eq. (4.9)3, and substituting (4.7), (4.8),
(4.10), (4.11) into (4.9)3 yields

ω61b =
v4

lGFi
z
6i5

ix1×

[
(lGF i

x
6
−

1
2
l6i
y
6
)

r3
−
l2i2
r1

]
i5

+
v5

lGFi
z
6i5
=
v4
r4
+
v5
r5

(4.12)

Finally, the vector of angular velocity of the cab becomes

ω6 =ω1+v4
(ix1
r3
+
i
y
6

r4

)
+v5
i
y
6

r5
=ω1+v4E4+v5E5 (4.13)

Linear velocity of the point Q derived from Eq. (4.6) is

vQ =vO+ l2ω2× i2+ω6×
(1
2
l6i
y
6 +(s6+ lHQ)i

z
6

)
+v6i

z
6 (4.14)

Substituting Eq (4.7), (4.8), (4.11), (4.12) into Eq (4.14) yields

vQ =vO+ω1× lOQ+v4C4+v5C5+v6C6 (4.15)

where

C4 =
( l2
r1
i
x
1 × i2+E4× lDQ

)
C5 =E5× lDQ C6 = i

z
6



722 G. Tora

Equations of angular velocity of platform 6 and cab (4.13) and velocities of
the point Q are now used to find the angular and linear acceleration

ε6 = ε1+a4E4+a5E5+εΩ
(4.16)

aQ =aO+ε1× lOQ+a4C4+a5C5+a6C6+aΩ

where

εΩ = v4
d

dt
E4+v5

d

dt
E5

aΩ = v4
d

dt
C4+v5

d

dt
C5+v6

d

dt
C6+ω1×

d

dt
lOQ

5. Conditions of motion of the actuating mechanism

The active suspension mechanism of the cab controls its angular velocity in
the directions ix1 and i

y
6, where the following conditions can be imposed upon

angular velocity of the cab

ω6i
x
1 =0 ω6i

y
6 =0 (5.1)

Conditions (5.1) in conjunction with (4.13) yield instantaneous velocities in
the cylinders of the lower mechanism

v4 =−r3ω1i
x
1 v5 =−r5ω1

(
i
y
6− i

x
1

r3
r4

)
(5.2)

When the cylinders move at the speed given by (5.2), the cab shall perform
rotarymotion only in the direction ix1×i

y
6. The cylinder s6 controls the linear

motion of point Q along the direction of the versor iz6. Such velocity v6 can
be induced that the point Q should be immobile in the direction of the zm
axis

vQig =0 (5.3)

Substituting (5.2) into (4.15) yields

v6 =−

(
vO+ω1× lOQ+v4C4+v5C5

)
ig

C6ig
(5.4)

Conditions (5.2) imposed upon the angular velocity of platform 6, when diffe-
rentiated, are transformed into acceleration formulas

ε6i
x
1 +ω6(ω1× i

x
1)= 0 ε6i

y
6 =0 (5.5)



Study operation of the active suspension... 723

Recalling (4.16)1, we get

a4E4i
x
1 +a5E5i

x
1 +(ε1+εΩ)i

x
1 +(v4E4+v5E5)(ω1× i

x
1)= 0

(5.6)
a4E4i

y
6+a5E5i

y
6 +(ε1+εΩ)i

y
6 =0

On solving the linear system of equations we get accelerations of cylinders 4
and 5. Similarly, differentiating (5.3) and recalling dig/dt = −ω1 × ig and
(4.16)2 yields the formula expressing the acceleration of the cylinder respon-
sible for the cab motion in the vertical direction

a6 =
vQ(ω1× ig)− (aO+ε1× lOQ+a4C4+a5C5+aΩ)ig

C6ig
(5.7)

6. Dynamics of the actuating mechanism

Thedynamicmodel assumes that themechanismcontainsholonomic two-sided
constraints. Of major interest are gravity forces, inertia and the driving force
whilst link deformations and drive elasticity are neglected. The framemass is
assumedtobemuch larger than that of the cabwhilst the cabmass isdecidedly
larger than the masses of the actuating mechanism links. The relationship
between themasses reveals that in the dynamicmodel the excitation inducing
motion of the cab can be treated as kinematic and that the forces of gravity
and inertia of themechanism links can beneglected (Frączek et al., 2008). The
inertia force and moment are obtained from Newton’s and Euler’s equations
(Morecki et al., 2002)

Pb =−mkaQ Mb =−(Ikε6+ ω̃6Ikω6) (6.1)

The mass moment of the cab inertia Ik expressed in the coordinate system
associated with the frame is given as

Ik =
r
kR
k
Ik
r
kR
⊤ (6.2)

where kIk is the moment of the cab inertia in the system associated with the
cab.
For the considered mechanism, the equation of instantaneous power ba-

lance can be formulated, where the sum of power expended by the drives and
power associated with the force andmoment of inertia and the gravity of the
cab should be equal to zero

Q⊤drv+mk(g−aQ)
⊤(vQ−vO−ω1×lOQ)+(−Ikε6−ω̃6Ikω6)

⊤(ω6−ω1)= 0
(6.3)



724 G. Tora

where

Qdr = [Qdr4,Qdr5,Qdr6]
⊤

v= [v4,v5,v6]
⊤

Recalling (4.13) and (4.15) with respect to (6.3), we get

Q⊤drv+mk(g−aQ)
⊤Cv+(−Ikε6− ω̃6Ikω6)

⊤Ev=0 (6.4)

where

C= [C4,C5,C6]
⊤

E= [E4,E5,0]
⊤

It follows from (6.4) that

Qdr =mkC
⊤(aQ−g)+E

⊤(Ikε6+ ω̃6Ikω6) (6.5)

7. Simulation of performance of the active suspension system of

the cab

Equations of kinematics and dynamics of the mechanism of the cab active
suspension become the starting point for simulations, assuming the kinematic
model of excitations acting upon the frame as shown in Fig.4. The frame
is represented by a front bridge (PP ,PL), a longitudinal frame (P,T) and
point O atwhich the active suspension is connected. The rear bridge (TL,TP)
is connected to the longitudinal framevia a rotating joint.Thedirection of the
longitudinal axis is represented by the versor mix1 and that of the front bridge
is defined by the versor mi

y
1. It is assumed that velocity components along the

axis xg of extreme points of the front and rear bridge should be constant and
equal to vp whilst velocity components in the direction yg should equal zero.
Velocity components in the direction zg are associated with the machine’s
travel on curvilinear profiles

zTP = zmax sin
(
2π
vPxt

L

)
zPP = zmax sin

(
2π
vPxt+d

∗

L

)

(7.1)

zTL = zmax sin
(
2π
vPxt

L
+ϕ
)

zPP = zmax sin
(
2π
vPxt+d

∗

L
+ϕ
)

where L is thewavelength, zmax – amplitude, w – bridgewidth, d – distance
between the front and rear bridge, ϕ – phase shift angle of the road profile
between the left- and right-hand side, d∗ = dmix1

mix ≈ d.



Study operation of the active suspension... 725

Fig. 4. Model of kinematic excitation inducing motion of the frame

The versor coordinates of the frame expressed in the xmymzm coordinate
system are

m
i
x
1 =



√

1−
(zP −zT
d

)2
,0,
zP −zT
d




(7.2)

mi
y
1 =


0,

√

1−
(zPP −zPL

w

)2
,
zPP −zPL
w


 miz1 =

mix1 ×
mi
y
1

where: zP =(zPP +zPL)/2, zT =(zTP +zTL)/2.
The versor coordinates of the coordinate system associated with the frame

yield amatrix

r
mR=




mix1x 0
mix1z

0 mi
y
1y

mi
y
1z

miz1x
miz1y

miz1z


 (7.3)

The angular velocity of the frame in the coordinate system xmymzm is given
as

m
ω1 = [

mω1x,
mω1y,0] (7.4)

The ngular velocity components are derived from the system of equations

żP − żT = d(
mω1×

mix1)
z żPP − żPL =w(

mω1×
mi
y
1)
z (7.5)



726 G. Tora

Accordingly, the desired angular velocity components of the frame are

mω1x =
żPP − żPL
w mi

y
1y

mω1y =−
żP − żT
dmix1x

(7.6)

Theparameters required for simulations, i.e. angular velocity of the frame and
linear velocity of the point O should be determined in the coordinate system
associated with the frame

ω1 =
r
mR
mω1 vO =

r
mR
mvP +ω1×rPO (7.7)

where: mvP = [żP ,0,vPx], rPO is a known, constant vector defined in the
coordinate system of the frame.

Simulations are performed for various values of the velocity vPx. Itsmaxi-
mum value is associated with the fact that the wheels must remain in contact
with the road. Comparing the acceleration of gravity to the maximal accele-
ration of an arbitrary extreme point on the frame bridges, we get

vPxmax =
L

2π

√
g

zmax
(7.8)

To evaluate the effects of the active suspension on linear acceleration of the
point Q of the cab, we use the acceleration reduction factor λ relating to
effective accelerations in the selected direction (xm,ym)(λz = 0), the active
suspension system being on and off

λ=

√√√√√√√√

tk∫
0
a2
Q
dt

tk∫
0
a2Qout dt

(7.9)

Evaluation of the systemperformance takes into account the energy ”costs” re-
quired to power-supply the active suspension. In the present study, this aspect
of the system operation is addressed by considering the energy expended to
support the work of the cab active suspension related to the energy difference
between the vibrating motion of the cab in the on and off states

∧

N4,N5,N6,Nout,Non>0

τ =

tk∫
0
(N4+N5+N6) dt

tk∫
0
Nout dt−

tk∫
0
Non dt

(7.10)



Study operation of the active suspension... 727

where: Nout =mkvQout(−aQout+g)+ω1(−Ikε1−ω̃1Ikω1) is the power expen-
dedbythedriveduringthecabmotionwith theactive suspension in theoffsta-
te, vQout,aQout –velocity andacceleration of the cab centre of gravitywith the
active suspension in theoff state. Non =mkvQ(−aQ+g)+ω6(−Ikε6−ω̃6Ikω6)
is the power expended by the drive during cabmotion with the active suspen-
sion in the on state, N4 = Qdr4v4, N5 = Qdr5v5, N6 = Qdr6v6 are power
ratings of individual drives while the active suspension system is on.

Simulation data were gathered for the road profile defined by the parame-
ters: L = 2m, zmax = 0.1m, ϕ = 1.57rad. The maximum velocity derived
from (7.8) is equal to: vmax = 3.15m/s. Simulation time is tk = 120s. The
computer model of the cab yields the cab mass (mk = 469kg) and the ma-
trix of inertia moment values of the cab expressed in the coordinate system
associated with the cab

k
Ik =



133.05 −0.28 −21.47
−0.28 197.37 0.29
−21.47 0.29 148.04


 kgm2

The simulation results are shown in the form of graphs and all relevant qu-
antities are expressed in function of the quotient of velocity in the horizontal
motion and its maximum value: kx = vPx/vPxmax.

Fig. 5. Maximal andminimal forces (a) and velocity (b) in the cylinders

Themaximum and minimum values of drive loading and velocity (Fig.5)
do not exceed the admissible levels for typical hydraulic drives. The average
loading of the drive in the vertical motion (Fig.5a) corresponds to the force
of gravity acting on the cab. Larger power consumption by cylinder 4 (Fig.6)



728 G. Tora

Fig. 6. Maximum power developed in the cylinders

Fig. 7. Vibration reduction factor during cabmotion in the direction (x,y). Energy
consumption factor

is attributable to the fact that it has to handle two DOFs. Although the
active suspension mechanism is not intended for vibration reduction in the
direction coinciding with ride of the cab, vibration damping still occurs λx ≈
0.65, see Fig.7, being the side effect of vibration reduction in the rotating
motion of the cab around i

y
6 (5.1)2. Control of velocity of cylinder 4 is the

consequence of reduction of rotation around ix1 (5.1)1, yet the proper size of
links of the suspension mechanisms ensures considerable vibration reduction
in the direction transverse to the truck ride λy ≈ 0.14, Fig.7. The active
suspension system, when in service, reduces acceleration of the cab, which in
turn lowers the loading due to inertia. Thus, energy required by the driving



Study operation of the active suspension... 729

system to move the cab is lower, too, which is corroborated by the energy
consumption pattern τ (Fig.7). For kx > 0.42, the total energy expended
by the driving system and the active suspension to move the cab is less than
energy required to move the cab whilst the active suspension mechanism is
off, τ < 1.

8. Conclusions

The results have to be treated as approximate as in the underlyingmodel the
drives shall instantaneously implement the computed velocity and the mo-
tion of themachine frame is well known beforehand. In practical applications,
however, the system for motion control will be of key importance and the im-
portant measurable excitation comes in the shape of the framemotion. That
would require a control strategy to compensate for the effects of these distur-
bances. In order that the system for measuring the frame motion should be
autonomous, the system of acceleration sensors can be applied. To determi-
ne instantaneous velocities of the drives (5.2) and (5.4), it is required that
linear and angular velocity of the framemotion should be known. That poses
certain problems, however, as constant has to be precisely determined and
the precision of acceleration signal integration has to be reliably established.
Cylinders 4 and 5, maintaining the cab in the vertical position, operate wi-
thout the risk of exceeding the motion range. Cylinder 6, responsible for the
cab motion in the vertical direction, might quickly reach its critical length
while the cab begins its ride upwards or downwards. The method of finding
the length of cylinder 6 should be such that its motion range should not be
exceeded. Condition (5.1)1 relating to the rotary motion of the cab (rotation
round ix1)might be replaced by the condition formulated for the linearmotion
of the cab: vQi

y
1 = 0. It is reasonable to expect that energy consumption by

cylinder 4 should be smaller. The axis of the load band of cylinder 6 is shifted
upwards by the value of the statistical gravity load (Fig.5a). This load, and
hence power demand, might be reduced when the upper mechanism is provi-
ded with an incorporated system of relieving springs, reducing the loading of
drive 6 by the cab weight. The major step in the calculation procedure invo-
lves finding the transition matrix from the xmymzm to xryrzr systems –

r
mR.

In this study, the matrix is derived basing on the model of the road and the
machine suspension. In practical applications, the matrix elements should be
determined frommeasurements of two angles of the frame deflection from the
vertical direction.



730 G. Tora

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Study operation of the active suspension... 731

Analiza pracy aktywnego zawieszenia kabiny maszyny roboczej

Streszczenie

Artykuł stanowi etap prac dotyczących aktywnego zawieszenia kabiny maszyny
roboczej, służącego do redukcji drgań niskoczęstotliwościowych o dużej amplitudzie.
Zawieszenie składa się z dwóch mechanizmów platformowych umieszczonych jeden
na drugim. Dolny mechanizm jest odpowiedzialny za utrzymanie kabiny w pionie.
Górnymechanizm odpowiada za ruch kabiny w kierunku pionowym. Do opisu ruchu
wykorzystano wersory związane z ogniwami mechanizmu. Wyprowadzono zależno-
ści na chwilowe prędkości napędów, powodujących redukcję drgań kabiny maszyny
w wybranych stopniach swobody. Przedstawiono sposób obliczania obciążeń napę-
dów aktywnego zawieszenia kabiny dla znanego ruchu ramy maszyny. Na podstawie
matematycznegomodelu wykonano symulacje pracy zawieszenia.

Manuscript received October 27, 2009; accepted for print February 26, 2010