Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 2, pp. 627-638, Warsaw 2012
50th Anniversary of JTAM

MODELING AND ANALYSIS OF A TWIN-TUBE

HYDRAULIC SHOCK ABSORBER

Urszula Ferdek
Jan Łuczko
Cracow University of Technology, Faculty of Mechanical Engineering, Kraków, Poland

e-mail: uferdek@mech.pk.edu.pl; jluczko@mech.pk.edu.pl

In this paper, a physical and mathematical model was created for a
twin-tube hydraulic shock absorber, using oil as theworkingmedium.To
analyze themodel, methods of numerical integrationwere incorporated.
The effect of the amplitude and frequency of the excitation, as well as
the parameters describing the flow rate of oil through the valves, were
examined. The basic characteristics of the damping force were obtained.

Keywords:hydraulic damper, shockabsorber, damping force,modelling,
vehicle suspension

1. Introduction

Due to irregularities in the surface of the road, amoving car is often subjected
to vertical, longitudinal and transversal displacements, which in turn greatly
reduce the comfort of passengers. Additionally, periodical detachment of the
wheels of the vehicle from the road surface lowers the effectiveness of both
transmissionofpowerandbrakingsystems. Insuchacase, themanoeuvrability
of the car, as well as the safety of the ride, is also worsened. In order to lower
the amplitude of vibration, all sorts of shock absorbers are used.The so-called
“hard” shock absorbers, provide better traction and stability in curves during
turning,while the “soft” ones allow for the comfort of the ride to be improved.
The hydraulic dampers that are used nowadays have characteristics which are
generally unsymmetrical and nonlinear (Alonso and Comas, 2006; Cui et al.,
2010; Dixon, 2001; Dzierżek et al., 2008).

The choice of these characteristics depends generally on the type of the
vehicle, as well as the requirements given by the manufacturers (which differ



628 U. Ferdek, J. Łuczko

for standard cars, SUVs, sports cars, motorcycles etc.) (Audenino and Belin-
gardi, 1995; Dixon, 2001). Optimization of the characteristics is the result of
both experimental (Cui et al., 2010;Dzierżek et al., 2008;Gardulski, 2009) and
theoretical analysis. Theoretical research is typically based on the analysis of
the quarter- or half-carmodel (Lee and Singh, 2008; Prabakar et al., 2009). In
order to evaluate the performance of the shock absorber used in the suspen-
sion system, it is best to consider the parameter that corresponds to vibration
of the upper car body, as well as the wheel-ground traction (e.g. EUSAMA
index). However, the vast majority of study in this field is directed at active
and semi-active dampers (Lee and Singh, 2008; Maciejewski, 2010; Prabakar
et al., 2009; Spencer et al., 1996), forwhich the description of characteristics is
given through the use of equivalent models, such as Spencer’smodel (Spencer
et al., 1996). This is not only due to the fact that in the case of a passive dam-
per, one cannot directly modify its parameters, also because of the relatively
complicated mathematical description of the hydraulic damper.

In this paper, an attemptwasmade to create amodel for a hydraulic dam-
per that would additionally consider both the inertia and compressibility of
the fluid. It was assumed that the control of the oil flow depends nonlinearly
on the pressuredifference.Thebasic characteristics of the shock absorberwere
found based on the simulation data, in accordance with the parameters corre-
sponding to the structural solutions used in the dampers produced byDelphi
company. For shock absorbers of this type, the characteristics obtained from
both indoor and outdoor experiments were presented in a paper by Dzierżek
et al. (2008).

Themodel that has themost similarity to the one shown in this paper,was
presented by Lee andMoon (2006). However, in their work, the control of the
oil flowdepends on the displacement.Ramos et al. (2005) studied the influence
of the thermal effect on characteristics of the model, paying less attention to
the problem of flow control. An interesting model is also given byAlonso and
Comas (2006), in which they analyzed to which degree the fluid cavitation
influenced the characteristics of the system. A simplifiedmodel, that includes
the compressibility of the fluid and its influence, is presented in the study of
Ferreira et al. (2009). There are other works (Cui et al., 2010; Liu and Zhang,
2002) with equivalent models, in which shock absorber characteristics are de-
scribed using nonlinear functions of velocity and displacement. For example,
Liu and Zhang (2002) considered a piecewise linear damping force-velocity
curve of the absorber, while Cui et al. (2010) approximated the strength with
the polynominal functions of speed, differing for the compression and rebound
process. The choice of coefficients of these polynominals is based on experien-



Modeling and analysis of a twin-tube... 629

ce.However, by using the last ones of the presented approaches to the problem
of modeling the behavior of the hydraulic damper, it is not possible for the
influence of structural parameters on the characteristics of the damper to be
studied.

2. Model of the hydraulic damper

In this study, the most basic characteristics of the hydraulic damper using
oil as a working medium, were determined. For this purpose, a model of a
twin-tube shock absorberwas created, as shown in Fig.1. Typically, the upper
mounting of the shock absorber is attached to the sprung part of the vehicle
mass, while the lower one – to the unsprung parts (such as wheels, axles,
bearings, brakes and some of the elements in the drive transmission system).

Fig. 1. Model of the hydraulic damper

The system, consists of two cylinders: external one (compensatory cham-
ber) (1) and the internal (working) one (2). Inside, the working cylinder, a
piston (3) is located, attached to the rod (4). The piston rod is equippedwith
the rod guide (6) that limits its movement in relation to the cylinders, to the
longitudinal direction only. In the lower part of the internal cylinder, a basc
valve (5) is located. Fourmain chambers can be distinguished: chamber above
the piston “Extension Chamber” (A), chamber below the piston “Compres-
sion Chamber” (B) and the compensatory tank “Reservoir Chamber”, which
consists of twoparts: chamber (C) located above the oil level and chamber (D)
located below it. Chamber (D) is filled with nitrogen under low pressure (4-
8bar). The initial pressure of the nitrogen is defined by the parameter p0,
while its volume by the parameter V0.



630 U. Ferdek, J. Łuczko

Theprinciple behind the operation of the hydraulic shock absorber is quite
simple.When the piston rod is pressed (andmoves towards the bottomvalve),
the pressure in chamber B increases and the resistance force arises, which in
turn causes the oil to flow through appropriate canals into the chamberA. The
excess oil, passes through the bottom valve to the reservoir chamber, causing
the pressure in chamber D to increase. In the rebound process, both piston
and the rodmove in the opposite direction, causing an increase of pressure in
chamber A. The oil flows through the system of canals, back from the space
above the piston, to the chamber below it. At the same time, oil from the
compensatory tank flows through the bottom valve, in order to equalize the
pressure in chamberB. The volume of the pumpedfluid depends primarily on
thepistondisplacement x,while the intensity of flow–on thepiston velocity v
in relation to the cylinder. The resistance force of the damper depends on the
resultant force from pressure acting on the piston and from both dry and
viscous friction.While trying to determine the characteristics of the hydraulic
shock absorber, the analysis shall be limited to the variable of the pressure
force Fp, given by the following equation

Fp =(pA−p0)A1− (pB −p0)A2 (2.1)

in which A1 and A2 define the size of the top and bottom area of the piston
valve,while pA and pB are thevalues ofpressure in chamberAandchamberB,
respectively.

In order to determine the equations which could describe variation of oil
pressure in chambersA,B andC, a relation needs to be considered: ρV =m,
which links the density ρ, volume V and mass m for each one of the shock
absorber chambers. After its differentiation and inclusion of the relation

dρ

dp
=
1

K
ρ (2.2)

the equations take the following form

1

K
ṗAVA+ V̇A =

ṁA
ρA

1

K
ṗBVB + V̇B =

ṁB
ρB

1

K
ṗCVC + V̇C =

ṁC
ρC

(2.3)

where VA, VB, VC are the volumes of the chambers, ρA, ρB, ρC are the values
of oil pressure inside them, while K stands for the compressibility modulus.



Modeling and analysis of a twin-tube... 631

The changes in mass ṁA, ṁB, ṁC are described by using of the mass flow
equations

ṁA = ṁBA− ṁAB ṁB = ṁAB − ṁBA+ ṁCB − ṁBC

ṁC = ṁBC − ṁCB
(2.4)

The lower indices of the flow rates are associated with the chamber names
between which the flow of oil occurs. For example, the mass flow rate ṁAB
determines the flow of oil from chamber A to B, has a non-zero value only if
pA >pB and differs from themass flow rate ṁBA, mostly due to the different
cross-sectional area size of the flow canal (ABA >AAB). These different area
sizes are introduced in the construction of the valves in order to provide a
higher resistance force during rebound (for v > 0), in comparison to the force
that emerges during the compression process (for v< 0).
By introducing Heaviside’s function, the discussed flow rates can be sub-

stituted by the following equations

ṁAB = ρAβAABH(pA−pB)

√

2(pA−pB)

ρA

ṁBA = ρBβABAH(pB −pA)

√

2(pB −pA)

ρB

(2.5)

where β is the flow rate coefficient. Themass flow rates between chambersB
andC can be described in a similar manner

ṁBC = ρBβABCH(pB −pC)

√

2(pB −pC)

ρB

ṁCB = ρCβACBH(pC −pB)

√

2(pC −pB)

ρC

(2.6)

The integration of general equation (2.2), allows for the relations between the
density and pressure of oil in each damper chamber, to be written

ρA = ρ0exp
(pA−pa
K

)

ρB = ρ0exp
(pB −pa
K

)

ρC = ρ0exp
(pC −pa
K

)

(2.7)

where ρ0 is the density of oil under the influence of atmospheric pressure pa.
The characteristics of the damper depend to a significant degree on the pa-

rameters ABA, AAB,ABC, ACB, which influence the rate of mass flow (2.5),



632 U. Ferdek, J. Łuczko

(2.6). The construction of valves shows that these parameters are actually
functions, as they can bemodified during the operation time. Since the inlets
of the canals are covered by the set of discs, which are susceptible to deforma-
tion from respective pressure forces, the highest impact is from the difference
of oil pressure between the adjacent chambers. Additionally, some of the in-
lets are uncovered only after the critical value of the pressure force is reached.
In order for the functions ABA, AAB, ABC, ACB, to be properly described,
the exact geometry of the valve, as well as physical relationships between the
force acting on the disc and its deformations, are required (Czop et al., 2009).
The problem is simplified by assuming that the analyzed functions can be
described using one constant and two variable components. The constant va-
lues represent permanently open canals. To describe the variable components,
a new function ϑ(∆p,pk) is introduced, that is defined by the pressure dif-
ference ∆p and constant pressure pk, referred to as “critical pressure”. The
functions ABA,AAB,ABC,ACB canbedescribedby the equations of a similar
structure

AAB =A
const
AB +A

max
AB [δ1ϑ(pA−pB,0)+ δ2ϑ(pA−pB,pk)]

ABA =A
const
BA +A

max
BA [δ1ϑ(pB −pA,0)+ δ2ϑ(pB −pA,pk)]

ABC =A
const
BC +A

max
BC [δ1ϑ(pB −pC,0)+ δ2ϑ(pB −pC,pk)]

ACB =A
const
CB +A

max
CB [δ1ϑ(pC −pB,0)+ δ2ϑ(pC −pB,pk)]

(2.8)

inwhich theparameters labeledby the indices constand maxare respectively
the size of the cross-section area of the permanently open canals (including
leakage) and the maximal size of the cross-section area of the canals with
variable diameter.

The parameters δ1 and δ2 define the percentage of area size of the inlets,
opened in a continuousmanner from ∆p=0and from ∆p= pk (it is assumed
that δ1 + δ2 = 1). The function ϑ(∆p,pk) is approximated using arctan’
function in a following way

ϑ(∆p,pk)=
2

π
arctan[µ(∆p−pk)]H(∆p−pk) (2.9)

The above definition shows that for the pressure difference ∆p exceeding
the critical pressure pk, function (2.9) rises in away similar to the exponential
fromthevalue of zero to one. If theparameter µ, that defines the susceptibility
of a disc covering a particular canal, is high enough, function (2.9) begins to
resemble a unit step function.



Modeling and analysis of a twin-tube... 633

In order to obtain the final form of the differential equations describing
the pressure changes, values in equations (2.3) need to be substituted by the
following relations

VA =A1(l−x) VB =A2(l+x) (2.10)

An assumption is made here that complete length of the internal cylinder
equals Lw = 2l+h, where h is the width of the piston valve, and that the
displacement of the piston is given by the harmonic function x= asinωt. To
determine the volume of oil in the external cylinder of the shock absorber, a
constant volume relation of the reservoir chamber can be utilized VC +VD =
const, fromwhich

V̇C =−V̇D (2.11)

By assuming that the volume of gas VD in the reservoir chamber satisfies
the equation of polytropic processes

pDV
n
D = p0V

n
0 = const (2.12)

where n is the coefficient of the politrope and by stating that the pressure of
both oil and gas in chambers C and D is equal pD = pC, by differentiating
equation (2.12), it is possible to calculate the change in volume VD

V̇D =−
ṗDVD
npD

(2.13)

After the additional use of equations (2.11)-(2.13)

V̇C =
ṗDV0p

1

n

0

np
1+1
n

D

(2.14)

Finally, the equations that describe the change of oil pressure in each chamber,
take the following form

ṗA =
K

VA

(ṁA
ρA
+A1ẋ

)

ṗB =
K

VB

(ṁB
ρB
−A2ẋ

)

ṗC =
ṁC
ρC

Knp
1+1
n

C

VCnp
1+1
n

C +KV0p
1

n

0

(2.15)

Equations (2.15) together with relations (2.4)-(2.10) and expression (2.1) pro-
vide the basis for determining the characteristics of the hydraulic damper
resistance force.



634 U. Ferdek, J. Łuczko

3. Calculation results

The effect of the amplitude and frequency of the excitation, as well as selected
parameters characterizing the flow of oil through valves, on the basic charac-
teristics of the system is presented below. Based on the analysis of the specific
solution for a damper manufactured by Delphi (Dixon, 2001)obtained using
numerical computation, the following values of parameters were used: length
l = 0.195m, area size [cm2]: A1 = 5.03, A2 = 6.16, A

max
BA = 2A

max
AB = 0.204,

AmaxCB =A
max
BC =0.0679, A

const
BA =A

const
AB = δ0A

max
BA ,A

const
BC =A

const
CB = δ0A

max
CB ,

coefficients δ0 =0.1, δ1 =0.4, δ2 =0.6, β=0.8, volume V0 =90cm
3, oil den-

sity ρ0 = 980kg/m
3, nominal pressure p0 = 6bar, compressibility modulus

K = 1.5GPa, coefficient µ= 1bar−1, politrope index n= 1.4. The solution
to equations (2.15) was found by making use of a variable-step Fehlberg in-
tegration algorithm, based on the Runge-Kutta methods of the 4th and 5th
order.
Figure 2 illustrates the relation between the damping force and relative

displacement for different values of amplitude (Fig.2a for f =2Hz) and fre-
quency (Fig.2b for a = 2cm) of the excitation. The extreme values of the
force are achieved for the extreme velocities (close to the initial position of
the piston) and depend nonlinearly on the value ωa. The ratio between the
maximal andminimal force depends primarily on the ratio of the canal areas

A
(k)
BA
/A
(k)
AB
and A

(k)
BC
/A
(k)
CB
(k = 1,2), responsible for the flow of oil during

compression and rebound.Because in these calculations the ratio between the
considered areas equals 2, the maximal forces (in the case of rebound) are
around twice the minimal ones (in the case of compression).

Fig. 2. Force-displacement diagram (pk =2bar): (a) influence of amplitude
(f =2Hz), (b) influence of frequency (a=2cm)

With the increase in either amplitude or frequency, the characteristics of
the force undergo certain changes – the effect of strong increase in oil pressure



Modeling and analysis of a twin-tube... 635

in chamberA, with a suddendrop of the pressure in chamberB (Fig.3) causes
the maximal forces to be increased.

Fig. 3. Time histories of pressures pA, pB and pC (a=5cm, f =2Hz, pk =2bar)

Figure 3 shows time characteristics of the pressure in each shock absorber
chamber, as well as velocities used to separate the process of compression and
rebound. The graph of pressure pC in the reservoir chamber is similar to a
harmonic function, and is subjected to relatively small changes. More intere-
sting are the graphs showing the pressure in chambersA andB. By studying
these graphs, and especially the one showing pressure pA, some characteristic
points can be spotted (marked by arrows), inwhich the difference between the
appropriate pressures equals the same as the critical pressure. These points
correpond to the moments in which the additional holes in valves are opened
and closed. During compression, for v(t) < 0, the canals are opened at first
in the top valve, and then in the bottom one in order to prevent the oil from
becoming too compressed in chamber B. The closing of the canals is perfor-
med in the inverse order. The similar situation takes place during the rebound
process (for v(t)> 0).Theorder of opening and closing the canals is the same,
but in this case the oil flows into chamberB.

Most useful information concerning the dynamical properties of the consi-
dered dampers can be acquired from Fig.4 which shows the relation between
the force and the relative velocity of the piston. A narrow hysteresis loop can
be seen in the graphs, which proves theminor effect of the inertia of oil. These
characteristics are asymmetrical since the damper putsmore resistance during
the rebound, which is actually desirable, especially in the case of the wheel
falling into a deep hole or overriding a high obstacle.

In the case of absence of additional channels, opened onlywhen the pressu-
re difference exceeds the given value pk (Fig.4a for pk =0), the characteristic
of the damper is progressive throughout the velocity range. For the value
pk 6= 0, the characteristics behave differently if the velocity range is high or
low. In the presented graphs, several points of inflection can be seen, whoose
locations play significant role inproper operation of thedamper.However, the-



636 U. Ferdek, J. Łuczko

Fig. 4. Force-velocity diagram (a=5cm, f =2Hz): (a) influence of parameter pk
(δ0 =0.1), (b) influence of parameter δ0 (pk =2bar)

se locations dependmostly on the value of parameter pk (Fig.4a) and design
parameters of the valve (especially the appropriate areas – Fig.4b), and to a
much lesser degree, on the parameters of excitation (amplitude, frequency).

The optimal characteristic of the damper depends also on the parameters
of the suspension system and the inertial parameters which characterize the
vehicle, inwhich thedamper is tobemounted.Therefore, in order todetermine
this characteristic, it is necessary to analyze the appropriate vehicle model.

By analyzing the graph in Fig.4b, an adverse effect of increasing the area
size of the permanently open canals (including leakage) can be observed, mo-
stly due to the decrease in difference between themaximal forces during com-
pression and during rebound, and the fact that the characteristic is similar to
degressive (e.g. for δ0 = 0.3) throughout the whole velocity range. The ad-
verse effect of increased leakage is also confirmed by the experimental results,
discussed in the paper Lee and Singh (2008).

4. Conclusions

The analysis of themodeled hydraulic shock absorber, allows for several conc-
lusions to be drawn:

• The largest impact on the characteristics of the damping force is from
the parameters that depend on the geometrical and physical properties
of the design of top and bottom valves of the shock absorber.

• The most desirable (for the comfort of ride) asymmetry of the force
characteristic can be obtained only through the application of a specific



Modeling and analysis of a twin-tube... 637

design of the valve, which would provide a lower mass flow rate during
rebound than during compression.

• Agood convergence of the results in comparison with the results descri-
bed in literature proves that the assumptionsmade during themodeling
of the system where chosen correctly and appropriately.

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Modelowanie i analiza dwururowego tłumika hydraulicznego

Streszczenie

Wpracywprowadzonomodel fizyczny imatematycznydwururowegoamortyzato-
ra hydraulicznego,w którym czynnikiem roboczym jest olej amortyzatorowy.Do jego
analizy wykorzystanometody numerycznego całkowania. Zbadano wpływ amplitudy
i częstości wymuszenia oraz parametrów opisujących przepływ oleju przez zawory.
Wyznaczono podstawowe charakterystyki siły tłumienia.

Manuscript received September 15, 2010, accepted for print November 17, 2011