Theoretical evaluation of the influence of the thermodynamic processes on the selection of shock absorbers for sports cars.
Nagurnas, S. ; Zuraulis, V. ; Skackauskas, P. 等
1. Introduction
Oscillations of sprung and unsprung masses have negative influence
on the dynamics and the safety of driving [1], therefore shock absorbers
are used in vehicles suspensions of all types. Shock absorbers used in
serial cars have to ensure the safety and the comfort of driving, i.e.,
to not allow the oscillations of the car's body from road roughness
grow and last for a long time, which can cause harmful loads for
passengers and damage fragile shipments. The suspension and its shock
absorbers in a sports car have to ensure optimal car's handling
characteristics and suitable grip of the wheels to the road surface. It
doesn't matter for what type of competition (drift, circuit, drag,
rally, etc.) a car is designed. In all sports cars a shock absorber is
the most important and the most complex part of suspension, which
carries the heaviest load and which plays the decisive role in
maintaining a car's stability on the race track. Though the
operating of the twin-tube and monotube shock absorbers is based on the
hydraulic fluid's (oil) resistance against the existing load, these
shock absorbers, due to different constructions, have different
characteristics which influence the choice what construction of shock
absorbers may be used in cars which are designed for different purposes.
In literature sources [2-4] monotube shock absorbers are
accentuated as the most suitable and most commonly used shock absorbers
in sports cars. In the source of literature [5] opportunities of
optimising the suspension of sports cars using monotube shock absorbers
are examined. In the publication it is indicated that constructions of
monotube shock absorbers' valves and pistons, body, etc. are more
easily modified trying to obtain the needed suspension's
characteristics and they are also more suitable for intensive
exploitation and heavier loads. The manufacturer "Ohlins" of
shock absorbers indicates that more than 80% of "Cart formula"
class cars use the monotube shock absorbers manufactured by this firm
[6], and the manufacturer "Tein" of sports shock absorbers had
published a work where constructional advantages and drawbacks of
monotube and twin-tube shock absorbers are examined and compared as well
as suitability for races and everyday exploitation. The manufacturer
points out that twin-tube shock absorbers are more suitable for everyday
exploitation in a city because of the two cylinders body construction
which allows to gain better protection against external damages, i.e.,
if the outer cylinder of such a shock absorber is damaged, it does not
influence the work of a shock absorber's piston and rod, and the
performance of the shock absorber is not affected. Furthermore, low gas
pressure in such shock absorbers ensures smaller loads on seals, smaller
friction in the shock absorber and more comfortable driving which is why
exploitation expenses of such shock absorbers are lower. The
manufacturer "Tein" emphasises that the main factor, which
makes monotube shock absorbers more suitable for sports cars and
intensive loads, is the emitted temperature [7]. Warner and Takheja also
emphasise the importance of the influence of the temperature's
factor [8]. In this work it is indicated that the worsening
characteristics of hydraulic fluid due to high temperature have
influence on the car's suspension's height too if springs of
improper stiffness are used. That has an especially great significance
in the "Indy Light Formula" class cars. In the experimental
tests it was established that a change of 0.76 mm in the
suspension's height can determine a 0.5 s longer time of finishing
a race track when its length is 1.6 km [8]. During the exploitation,
depending on the loads, shock absorbers can heat up to approximately
120[degrees]C [9]. In the sources of literature [10] and [11] by using
experimental methods it is confirmed that the thermodynamic processes
occurring in shock absorbers have influence on their working efficiency.
In these works during the experiments with shock absorbers in the
laboratory it was established that the too high temperature of a shock
absorber's body can cause certain deformations of the shock
absorbers and their parts, and a shock absorber's construction
influences most the insufficient heat transfer to the atmosphere and the
shock absorber's heating. Taking into account that in all of the
discussed works there are presented only the results, obtained during
the practical experiments, the aim of this work is to analyse and
clarify the presumptions presented in various academic publications that
a shock absorber's construction has great influence on the heat
transfer and distribution in a shock absorber's body, by means of
theoretical modelling of the thermodynamic processes which occur in
shock absorbers.
2. Methods of modelling and research
For the investigation and modelling of the heat transfer to the
atmosphere and temperature distribution in the shock absorbers' of
different constructions, these shock absorbers were chosen:
"Bilstein MDS860" coil-over monotube shock absorber which is
meant for formula class cars, and short stroke sports type twin-tube
"Sachs S27" shock absorber. The monotube "Bilstein
F4" shock absorbers which were used in serial "Mercedes-Benz
C" cars and twin-tube "Monroe" shock absorbers which were
used in "Fiat Panda" cars were chosen for comparison. The
models of the shock absorbers were formed and the investigating was done
using SolidWorks software package. The investigated shock absorbers
bodies main geometrical data, which were determined based on the
literature sources [9, 12], are presented in Table 1.
The modelling of the indicated shock absorbers' temperature
distribution and heat transfer to the atmosphere was done in a four
stages:
1. During the first stage, an additional cooling of the shock
absorber was not modelled, i.e. natural convection was happening and the
conditions of shock absorbers' testing in laboratories were
imitated. In the case of natural convection the heat transfer
coefficient a from the atmosphere to the shock absorber is equal 5-25
W/[m.sup.2]K [13], (accepted value--20 W/[m.sup.2]K).
2. During the second stage, an additional cooling of the shock
absorber was modelled, i.e. forced convection (imitating the air flow
which cools the shock absorber while the car is moving on a track) was
introduced. When the forced convection is created by the airflow over
the shock absorber, then [alpha] changes in a range of 20-350
W/[m.sup.2]K [14], (accepted values--100, 225 and 350 W/[m.sup.2]K).
While imitating the forced and natural convections, these modelling
conditions were accepted: Oil temperature values: 0, 15, 30, 45, 60,
90[degrees]C; Atmosphere temperature--22[degrees]C; Oil heat transfer to
the body coefficient size: 50, 150 and 300 W/[m.sup.2]K. The modelling
of all shock absorbers and processes is done repeatedly with different
oil temperatures, different heat transfer coefficient from oil to body
and heat transfer coefficient from atmosphere to body.
3. During the third stage, the estimation of shock absorber's
body alloy influence on the thermal conductivity through the wall of the
shock absorber's body and its transfer from the wall to the
atmosphere, is made. Shock absorbers cylinders (bodies) are usually made
of steel or aluminium alloys, for example, high quality 7075-0 or 6061
aluminium alloy is used. The thermal conductivity coefficient of a shock
absorber depends on the material of the cylinder. In order to ascertain
the influence of the thermal conductivity coefficient on the thermal
conductivity through the wall of the shock absorber body and its
transfer to the atmosphere, modelling of temperature distribution and
heat transfer was done with these shock absorbers made of different
alloys. Thermal conductivity coefficient of 7075-0 aluminium alloy is
173, steel alloy--50 and oil--0.152 W/mK. The values of thermal
conductivity coefficient are accepted based on the provided accurate
materials characteristics in the SolidWorks software package's
materials' library, choosing the alloys from which the shock
absorbers are made.
Due to the shock absorber body being a cylindrical surface, the
intensity of the heat transfer between the fluids through the wall is
described by linear heat flux density. Heat transfer between fluids
through a wall can be divided into three stages: 1) Heat transfer from a
fluid (oil) to a wall (shock absorber body); 2) Thermal conductivity
through a body wall; 3) Heat transfer from the wall to the fluid
(atmosphere).Taking into account all these stages, the linear heat flux
density for the monotube and twin-tube shock absorbers can be calculated
[15]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)
where [[alpha].sub.i] heat transfer coefficient; [T.sub.fi] is
fluid (oil and atmosphere) temperature; [d.sub.i] is diameters of shock
absorber cylinders; [lambda] is thermal conductivity coefficient.
The amount of heat transferred through the shock absorber body to
the atmosphere during a certain time, is described as the heat flow.
Heat flow is calculated [14, 15]:
Q = l [q.sub.l], (2)
where l is a shock absorber cylinder height.
In a simplified case, the temperature in any intermediate point of
shock absorber cylindrical body can be calculated [14, 15]:
[t.sub.n+1] = [t.sub.n] - [q.sub.l]/2[pi][[lambda].sub.i] ln
[r.sup.n+1]/[r.sub.n], (3)
where [t.sub.n] is shock absorber inner wall temperature;
[r.sub.n+1] is radius to the point, where the temperature is calculated;
[r.sub.n] is inner radius of shock absorber body.
4. Fourth stage: investigation of the influence of the temperature
on the oil viscosity, density and damping force. This investigation is
carried out by mathematically evaluating the processes occurring in the
shock absorber and by deriving the dependencies of fluid's
(oil's) viscosity, density and shock absorber's damping force
on the changing oil temperature during exploitation. Additionally,
experimental tests were done, during which the dependencies of values of
oil viscosity and density on the change of the oil temperature are
determined. Experimental tests are done using "Motorex Fork Oil
W10" oil for shock absorbers, which has nominal kinematic viscosity
of 52.50 [mm.sup.2]/s (43.25 mPa x s) at the temperature of
40[degrees]C, 100[degrees]C--9.90 [mm.sup.2]/s (7.70 mPa x s), and
density at the temperature of 20[degrees]C -840 kg/[m.sup.3] [16]. The
experiments were carried out by heating the oil and using a rotary
viscometer "Brookfield DV-I Prime" to evaluate the oil
viscosity at the temperatures of 20, 30, 40, 50, 60, 70, 80,
90[degrees]C. Density is also determined by heating the oil and
evaluating it using the hydrometer at the temperatures of 20, 30, 40,
50, 60, 70, 80, 90, 100, 110, 120[degrees]C.
3. Modelling results
In Table 2 the results obtained during the modelling are presented,
when the oil's thermal conductivity coefficient is [[alpha].sub.1]
= 300 W/[m.sup.2]K. These results show that in cases of both natural and
forced convections, the smaller temperature of the body settles and the
bigger linear heat flux density is transferred through the monotube
shock absorbers. In Table 2 the tendency of changing values of settling
bodies' temperature and linear heat flux density is presented, when
the shock absorber's cooling conditions are changing. The tendency
of the changes of examined values is analogous to the other cases of
modelling.
[FIGURE 1 OMITTED]
Taking into account the results of modelling, which were obtained
while imitating the conditions of shock absorbers' testing in
laboratories, it is seen that in the case of natural convection, the
accumulated amount of heat in the shock absorber and its body
temperature depends only on the oil heat transfer coefficient, because
when it is getting larger, the value of the linear heat flux density
grows also, and the influence of the atmosphere temperature is minimal
(except the cases when it is very low or very high). During the process
of natural and forced convections, the main (inner) cylinder of the
twin-tube shock absorber is not cooled at all due to the fact that in
the twin-tube shock absorbers the cylinder is surrounded by oil from
both sides, and the linear heat flux density there is minimal or
non-existent.
Contrary to the case of natural convection, the air flow which
cools the shock absorber when the car is moving on a track is imitated,
the forced convection is formed, and the accumulated amount of heat in
the shock absorbers and the settled body temperature depend not only on
the oil heat transfer coefficient but also on the value of atmosphere
heat transfer to the body coefficient, i.e., on the intensity of
cooling. The differences between twin-tube and monotube shock absorbers,
regarding the settled temperature of bodies and the transferred linear
heat flux, when different cooling cases of shock absorbers are being
modelled, are presented in Fig. 1. In Fig. 1 it is seen that when the
oil temperature in the shock absorber working chamber is lower than the
atmosphere temperature, the air flow that circulates round the shock
absorber does not cool the shock absorber body but heats it, depending
on the atmosphere heat transfer coefficient. Because of this reason,
since instead of the heat transfer to the atmosphere there is the
obtainment of heat, the linear heat flux density is negative. In this
case, the lower the heat transfer coefficient and the larger the
atmosphere heat transfer coefficient are, the more the shock absorber
heats up. A modelling example of the temperature distribution in the
shock absorbers' bodies is presented in Fig. 2.
[FIGURE 2 OMITTED]
In Fig. 3, there is a graph which shows the differences between the
settled bodies' temperatures and the transferred linear heat flux
density in all of the modelled shock absorbers when they are made from
different aluminium and steel alloys, the oil temperature ranges from 0
to 90[degrees]C, its heat transfer coefficient being 300 W/[m.sup.2]K
and the atmosphere heat transfer coefficient being 350 W/[m.sup.2]K.
Based on the presented graph and the obtained results, it is seen that
when the shock absorber body is made from an alloy with lower thermal
conductivity coefficient, the linear heat flux density is decreasing,
therefore the shock absorbers body accumulates a larger amount of heat
and the settling temperature in the body increases. Using Eq. (1) it is
determined that in the cases modelled in this work the maximal change of
linear heat flux density is not big--7.5 W/m (the maximal temperature
change between the uniform shock absorbers made of different alloys is
5[degrees]C). If a shock absorber body is made of the 1060-H16 aluminium
alloy, with thermal conductivity coefficient equal to 230 W/mK, under
the conditions that are presented in Fig. 3, the "Bilstein F4"
shock absorber maximal linear heat flux density increases from 1277 to
1279 W/m. If the shock absorber's body is made of the AISI 321
steel alloy with thermal conductivity coefficient equal to 16.1 W/mK,
the linear heat flux density decreases to 1256 W/m. That is why it can
be stated that the influence of shock absorber body material thermal
conductivity coefficient on the linear heat flux density is not large
compared to the influence of the oil and atmosphere heat transfer
coefficient.
[FIGURE 3 OMITTED]
Based on Eq. (2), the overall amount of heat, which the shock
absorber transfers to the atmosphere, depends not only on the linear
heat flux density but also on the geometric shapes of the shock
absorber--its body (cylinder) height. Evaluating the heights of
cylinders of the tested shock absorbers, the biggest amount of heat,
under the conditions presented in Fig. 3, is transferred into the
atmosphere by the "Bilstein F4" shock absorber--428 W
("Bilstein MDS 860"--175 W, "Sachs S27"--40 W,
"Monroe"--55 W). However, this indicator is not suitable to
evaluate shock absorbers because the geometric parameters of the tested
shock absorbers are different.
In this thesis the modelling is done accepting that the conditions
of convection forming and occurring are ideal. During realistic
conditions, a number of external factors have influence on the
thermodynamic processes occurring in shock absorbers: protective
elements of shock absorbers, technical condition of shock absorbers,
wind direction and speed, temperature of vehicle engine. Under realistic
conditions, the tendency of the temperature changes and distribution in
the shock absorbers' body remains the same as the presented one
because all the named factors determine the value of the atmosphere
temperature surrounding a shock absorber. Because of this reason, the
accepted values of the atmosphere heat transfer coefficients, which are
applied while modelling, are affected [15, 17], and, during the
modelling, wide-ranging limits of these coefficients are evaluated,
which encompass the named external factors.
4. Temperature influence on the oil viscosity, density and the
shock absorber damping force
Thermal motion of fluids (oil) is described as vibration of their
molecules about certain temporary equilibrium positions. After a certain
period of time t the molecule equilibrium position gives a jump. These
periods of time t are very different and constantly disorderly change.
Such periods of time are called relaxation time. During low temperatures
the fluids' viscosity is high because the periods of time t are
relatively long, the distances between molecules are small and their
interaction is strong. While the fluid heats, time t decreases and the
fluid molecules become more mobile. Because of this, while the
temperature rises, the fluids viscosity exponentially decreases. Dynamic
fluid viscosity dependence on the oil's temperature can be
evaluated using the Arrhenius-type equation [18]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)
where [[eta].sub.n] is constant oil viscosity during a certain
temperature; [T.sub.n] is nominal oil temperature at which the constant
oil viscosity is determined; e is Euler number; [T.sub.i] is oil
temperature; C is coefficient which is approximately equal to the ratio
between the oil molecules activation energy and the universal gas
constant [18]:
C = E/R, (5)
where E is oil molecules activation energy, i.e. the minimal energy
needed for the fluid molecule to sever the bonds with adjacent molecules
and to jump from one balance position to the other, J/mol (for example,
water activation energy is approximately 15950 J/mol, methanol--10060
J/mol [18]); R--the universal gas constant.
The oil viscosity change dependency on the temperature can be most
easily determined using the Eq. (4). The initial oil temperature
[T.sub.1] is inserted into this formula and it is divided by an
analogical formula (all of the values in the formula do not change).
When oil temperature after change T2 is inserted into Eq. (4), an
expression is obtained which shows how many times the oil viscosity
decreases (equations are made and solved using the Maple package):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
The oil density, depending on the temperature and the oil
characteristic, changes as well. Based on the source literature [19],
the oil density dependency can be described as a decreasing linear
temperature function and calculated:
[rho] = [rho]n/1 + [alpha] ([T.sub.i] - [T.sub.n1]), (7)
where [[rho].sub.1] is constant oil density during a certain
temperature; [alpha] is coefficient of volumetric thermos expansion. For
the oils used in shock absorbers this coefficient value is [alpha]
[approximately equal to] 0.001 [K.sup.-1] [[approximately equal to] 1000
ppm/K [19]; [T.sub.1] is nominal oil temperature at which the constant
oil density is determined.
In order to properly and accurately determine the shock absorber
damping force it is needed to evaluate a number of various shock
absorber construction parameters and indicators: friction between the
moving shock absorber elements, precise pressure in the working and gas
chambers, piston construction, valve construction and the
characteristics, the geometric shapes of valve plates, etc. The
evaluation is very complex and it requires many theoretical and
experimental tests. In the source of literature [20] a relatively
accurate monotube shock absorber mathematical model is being formed and
examined, and it is indicated that the overall shock absorber damping
force can be expressed as a non-linear shock absorber rod shift x(t),
rod velocity [??](t) and rod acceleration [??](t) function. To evaluate
the influence of the temperature on the monotube shock absorber's
damping characteristics, using Eqs. (4)-(7), the presented model of the
shock absorber is remade [20], an expression is obtained:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where [A.sub.g] is average cross-section area of the piston with
regard to the cross-section area of the rod; Z(t) is relative
displacement (velocity, acceleration); [C.sub.d] is dynamic discharge
coefficient; m is mass of the system; [A.sub.n] is oil orifice area;
[L.sub.1] is length of the oil orifice; [F.sub.f] is estimation of the
friction of the system.
In Fig. 4 a graph is presented in which the dependencies of the
values of the "Motorex Fork Oil W10" oil density and viscosity
are compared, when the oil temperature is changing. These dependencies
were measured and theoretically calculated during an experiment. In the
graph it can be seen that the obtained values do not entirely match with
the theoretically calculated ones, using Eqs. (4) and (7). The
mathematically estimated oil dynamic viscosity decrease, when the oil
temperature changes in the indicated interval, was approximately 10%
smaller than the actual viscosity decrease during the experiment. The
measurement errors, which according to the viscometer readings ranged
from 0.1 to 1.5%, had the influence on the forming of the difference.
However, based on the source literature [18], the difference between the
theoretically and practically determined viscosity values is mostly
influenced by the oil molecules activation energy because it is a very
unstable and constantly changing value which depends on the ambient
humidity, pressure, the intensity of processes, and other various
indicators. Using Eq. (4) and the information which is provided by the
analysed oil manufacturer [16], it was determined that this oil
molecules activation energy is around 27900 J/mol. However, after
analysing the obtained practical results, it was determined that during
the experiment the oil molecules activation energy was around 33900
J/mol.
[FIGURE 4 OMITTED]
The obtained graphs of the viscosity and density characteristics
are almost identical to the theoretical ones, so a conclusion can be
made that the method of calculating the influence of the temperature on
the oil viscosity is formed correctly. The estimation of the oil density
and viscosity change using the Eqs. (7) and (4) is expedient and it can
be applied during the theoretical modelling of the shock absorber
characteristics.
5. Conclusions
Based on the results obtained from the investigation of the
influence of the thermodynamic processes occurring in the shock
absorbers of different constructions, these conclusions, which are
important for the selection of the shock absorbers for sports cars, can
be made:
1. In all modelled cases the lower body temperature settles and the
larger value of the linear heat flux density is gained in the monotube
shock absorbers.
2. Damping force and oil viscosity directly depend on the oil
temperature in the working chamber of the shock absorber. When the
temperature is rising, oil viscosity and the shock absorber damping
force decrease. Because of this reason, monotube shock absorbers, due to
their characteristics to transfer heat to the atmosphere more
effectively, ensure a smaller decrease of damping force than twin-tube
shock absorbers, under the same experiment conditions.
3. In the case of natural convection, the accumulated amount of
heat in the shock absorber and its body temperature depend only on the
oil's heat transfer coefficient. In the case of forced convection,
the accumulated amount of heat in the shock absorber and the body
settled temperature depend on the oil heat transfer coefficient and on
the atmosphere heat transfer to the body coefficient.
4. The influence of the thermal conductivity coefficient of the
shock absorber body material on the linear heat flux density is not
large compared to the influence of the oil and the atmosphere heat
transfer coefficients.
5. The calculation method of the influence of the temperature on
the oil viscosity and density is formed properly, because the difference
between the theoretically calculated and during the experiment measured
values of oil density is [less than or equal to]2.5%, and the difference
between viscosity values is [less than or equal to]10%. The formed
method can be applied during the various theoretical modelling of the
shock absorbers.
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S. Nagurnas, Vilnius Gediminas Technical University, Basanaviciaus
str. 28, LT-03224 Vilnius, Lithuania, E-mail: saulius. nagurnas@vgtu.lt
V. Zuraulis, Vilnius Gediminas Technical University, Basanaviciaus
str. 28, LT-03224 Vilnius, Lithuania, E-mail: vidas.zuraulis@vgtu.lt
P. Skackauskas, Vilnius Gediminas Technical University,
Basanaviciaus str. 28, LT-03224 Vilnius, Lithuania, E-mail:
paulius.skackauskas@vgtu.lt
A. Rimkus, Vilnius Gediminas Technical University, Basanaviciaus
str. 28, LT-03224 Vilnius, Lithuania, E-mail: alfredas.rimkus@vgtu.lt
J. Gargasas, Vilnius Gediminas Technical University, Basanaviciaus
str. 28, LT-03224 Vilnius, Lithuania, E-mail: justinas.gargasas@vgtu.lt
Received July 02, 2015
Accepted March 15, 2016
crossref http://dx.doi.org/10.5755/j01.mech.22.2.12529
Table 1
The main geometrical data of shock absorbers bodies
Inner/outer Inner/outer
cylinder cylinder Cylinder wall
Model height, mm diameter, mm thickness, mm
Bilstein 132/- 42/- 3
MDS860
Bilstein F4 335/- 39.40/- 2
Monroe 268.50/299 30/44 2
Sachs S27 144/162 29/38.30 1.5
Table 2
Modelling results
Bilstein MDS860
The heat transfer coefficient, created
by the airflow over the shock absorbers
[[alpha].sub.2], W/[m.sup.2]K
Initial oil 20 100
temperature
[T.sub.0], [T.sub.1], [q.sub.1], [T.sub.1], [q.sub.1],
[degrees]C [degrees]C W/m [degrees]C W/m
0 2.5 -53.8 8.05 -208
15 15.6 -17.1 17.24 -66.3
30 29.3 19.6 27.07 75.8
45 42.9 56.7 36.58 218
60 56.6 92.9 46.08 360.2
90 83.9 166.3 65.1 644.5
[T.sub.0] Monroe
0 1.1 -31 1.8 -53
15 15 -9.9 15.6 -16.9
30 30 11.4 29.9 19.3
45 45 32.7 44.9 55.4
60 60 53.9 59.9 91.6
90 90 96.5 89.9 163.9
Bilstein MDS860
The heat transfer coefficient, created
by the airflow over the shock absorbers
[[alpha].sub.2], W/[m.sup.2]K
Initial oil 350 20
temperature
[T.sub.0], [T.sub.1], [q.sub.1], [T.sub.1], [q.sub.1],
[degrees]C [degrees]C W/m [degrees]C W/m
0 13.9 -428 2.7 -50.7
15 19.4 -136 15.9 -16.1
30 24.9 155.8 29.4 18.4
45 31.4 447.9 43.4 52.9
60 35.9 740.1 57.4 87.5
90 49.8 1324.4 85.3 156.6
[T.sub.0] Monroe Sachs S27
0 1.8 -61 1.4 -34
15 15.6 -19 15.5 -10.7
30 29.9 22.1 29.9 12.3
45 44.9 63.3 44.9 35.3
60 59.9 104.6 59.8 58.3
90 89.9 187.2 89.7 104.3
Bilstein F4
The heat transfer coefficient, created
by the airflow over the shock absorbers
[[alpha].sub.2], W/[m.sup.2]K
Initial oil 100 350
temperature
[T.sub.0], [T.sub.1], [q.sub.1], [T.sub.1], [q.sub.1],
[degrees]C [degrees]C W/m [degrees]C W/m
0 7.25 -198 13.2 -413
15 17.31 -63.1 19.2 -132
30 27.83 72.1 25.5 150.3
45 38.77 207.3 32.1 432.2
60 49.71 342.5 38.6 714.1
90 71.59 612.9 51.7 1277
[T.sub.0] Sachs S27
0 2.1 -67 2.3 -81.4
15 15.6 -21.3 15.7 -25.9
30 29.9 24.4 29.8 29.6
45 44.7 70.1 44.5 85.1
60 59.6 115.8 59.2 140.6
90 89.2 207.3 88.5 251.7
where [T.sub.1] is settling mono tube shock absorber's
body temperature (twin-tube shock absorber inner body
temperature); [q.sub.1] is overall linear heat flux
density transferred through the shock absorbers bodies.
