ICAME-2006

1. International Conference on Advances In Mechanical Engineering
SRM Institute of Science and Technology Deemed University, Chennai, India: 14-16 December 2006
*
Lecturer in Mechanical Engineering Department, AISSMS’S College of Engineering, Kennedy Road, Pune, India – 411001,
Email: csdharankar2@yahoo.co.in , csdharankar2@rediffmail.com
**
Head Training Workshop, Defense Institute of Advanced Technology, Girinagar, Pune, India – 411025,
Email: mkhada@hotmail.com
***
Formerly Professor and Head Department of Mechanical Engineering,
Walchand College of Engineering, Sangli , India – 416415,
Email: shridhar.joshi@rediffmail.com
1
Analysis of Heat Transfer Effects on Spring Characteristic
of Hydro-Pneumatic Struts
*
C. S. Dharankar
Lecturer in Mechanical Engineering Department, AISSMS’S College of Engineering, Pune, India
and
**
M. K. Hada
Head Training Workshop, Defense Institute of Advanced Technology, Pune, India
and
***
S. G. Joshi
Formerly Professor and Head Department of Mechanical Engineering
Walchand College of Engineering, Sangli, India
A thermal time constant model was developed by solving the energy equation of a gas in a closed container
using real gas approach, to analyze the effect of irreversible heat transfer on the dynamic spring force
characteristic of hydro-pneumatic struts used in the vehicle suspension systems.
In the present paper this model is reviewed and the analysis is modified by defining a non-dimensional frequency
parameter. It is shown by the numerical simulation that the amount of energy dissipated by the hydro-
pneumatic spring and its spring characteristic depends only on the magnitude of this parameter and a single
parameter will properly characterizes the heat transfer effects and inherent damping of the hydro-pneumatic
spring. Also the pneumatic spring is shown equivalent to the Zener’s anelastic model and the existence of this
non-dimentional parameter is determined analytically.
Keywords: Anelastic model, Hydro-pneumatic spring, Non-dimensional frequency parameter, Thermal time
constant.
Nomenclature
Ao, a, Bo, b, CO, c = BWR constants
Aw = Area of cylinder wall in contact with gas (m2
)
Cv = Specific heat of gas at constant volume (J/kg-0
K)
C = Damping coefficient of dashpot (N-s/m)
E = Energy dissipated per cycle (J)
F = Spring force (N)
Fmax = Maximum force across the Anelastic model
Fmax = Minimum force across the Anelastic model
f = Frequency of excitation (Hz)
h = Overall heat transfer coefficient of hydro-pneumatic
spring unit (J/kg-0
K)
K = Stiffness of spring in series with dashpot (N/m)
K1 = Stiffness of spring in parallel with dashpot (N/m)
mg = mass of gas (kg)
P = Absolute pressure of gas (Pa)
Po = Initial absolute pressure of gas (Pa)
Q = Heat transfer from surrounding to gas (J)
T = Absolute temperature of gas (0
K)
Tw = Absolute temperature of cylinder wall and
surrounding (0
K)
Tmax = Absolute temperature of gas at the end of rapid
compression (0
K)
Tτ= Absolute temperature of gas at t = τ sec. (0
K)
t = Time (s)
U = Internal energy of gas (J)
Vo = Initial volume of gas at (m3
)
v = Specific volume of gas (m3
)
W = Work done by the gas (J)
X = Amplitude of excitation displacement (m)
x = Displacement of excitation (m)
α , γ = BWR constants
τ = Thermal time constant (s)
ω = Circular frequency of excitation (rad /sec)
ωτ = Non-dimensional frequency parameter
Introduction
Now days, the hydro-pneumatic struts are becoming
more popular in vehicle suspension system for effective
shock and vibration isolation. A hydro-pneumatic strut
consists of a pneumatic spring (known as hydro-
pneumatic spring) and a hydraulic damper as a single
unit in a cylindrical vessel with a floating piston or
diaphragm separating oil from the gas (usually
nitrogen). The gas serves the purpose of spring to store
energy and returning it to the damper oil during
expansion. The damper oil dissipates this energy in the

2. DHARANKAR, HADA AND JOSHI
2
form of heat due to restriction to flow of oil through the
orifice.
It is well known fact that spring characteristic of
hydro-pneumatic strut shows a hysteresis loop of energy
dissipation (inherent damping) that depends on the
excitation frequency and thermal time constant of the
charged gas. Conventionally, the spring force of hydro-
pneumatic spring is modeled using reversible polytropic
gas process assuming ideal gas behavior of the charged
gas. This approach results in large errors because it fails
to account for the irreversible heat transfer between the
gas and it’s surrounding during compression-expansion
cycle and the charged gas nitrogen cannot be treated as
an ideal gas.
Initially the thermal time constant model has been
developed for the gas process in a gas charged hydraulic
accumulators. Els and Grobbelaar [1] have extended the
application of thermal time constant model to determine
time-temperature dependent (dynamic) spring force
characteristic of hydro-pneumatic struts. This model
predicts the dynamic spring characteristic of hydro-
pneumatic strut more accurately along with the
irreversible heat transfer during the compression-
expansion cycle. Therefore a thermal time constant
model along with real gas approach replaces the
conventional approach of reversible polytropic process
of ideal gas (P Vn
= constant).
Otis and Pourmovahed [2] have suggested an algorithm
for computing non-flow gas processes based on
Benedict-Webb-Rubin (BWR) equation of state and
thermal time constant to describe heat transfer process
in the gas springs and hydro-pneumatic accumulators.
Also it was shown that a constant value of thermal time
constant fits the experimental data very well and the
analysis is fairly insensitive to its value.
Pourmovahed and Otis [3] have suggested the
experimental and analytical methods to determine the
thermal time constant for gas charged hydraulic
accumulators. The analytical method uses experimental
correlation for cylindrical accumulators oriented in
vertical and horizontal position. The correlation was
developed by treating the gas as a real gas and by
approximating the heat convection process in a gas to
heat conduction a solid.
Pourmovahed and Otis [ 4] have shown that for small
changes in gas volume (5 to 10 %), the linearized
thermal time constant model of a pneumatic spring is
equivalent to a special case of the Zener’s Anelastic
model which consists of two springs and one dashpot as
shown in Fig. 1
In the present paper the analysis of the spring
characteristic of hydro-pneumatic spring is presented by
defining a non-dimensional frequency parameter (ωτ).
The hydro-pneumatic spring is analyzed for both
sinusoidal and triangular wave excitation and it is
shown by the numerical simulation that a single
parameter ωτ properly characterizes its spring force.
The existence of the parameter (ωτ) is determined
analytically by solving the Zener’s Anelastic model for
sinusoidal excitation.
Modelling of Irreversible Heat Transfer in the
Hydro-Pneumatic Spring
Real Gas Approach
In the present study the Benedict-Webb-Rubin
(BWR) equation of state is used for real gas behavior of
the charged gas (nitrogen) used in the hydro-pneumatic
spring. The Benedict-Webb-Rubin equation of state for
gas is,
( )
2
0
0 0 2 2 3
v
6 2 3 2
b R T - aCR T 1
P B R T - A -
v T v v
a 1
c 1 e (1)
v v v T
−γ
 
= + + 
 
 γα  
+ + +  
   
( )
2
3
o o
2 2
v
v
3 3 2
B R 2 C / TP R b
1
T v v v
2 c
1 e (2)
v T v
γ 
− 
 
+∂   
= + +   
∂   
γ 
− + 
 
The Thermal Time Constant Model
Following assumption are made in deriving the
thermal time constant model for the hydro-pneumatic
spring.
• Homogeneous quasi static gas compression process
• Inertia effects of gas like shock waves are ignored
• Constant gas mass process i.e. closed system is
assumed
• Heat storage in the cylinder wall of the strut is
neglected.
• Effect of compressibility of oil is neglected
A hydro-pneumatic spring can be modeled as a gas in
a closed container with one movable boundary as
piston. The energy equation for a gas in a closed
container can be written as,
Rate of change of internal energy of gas = Rate of heat
transfer to gas – Rate of work done by the gas
dU dQ dW
dt dt dt
= − (3)
Rate of heat transfer to gas from surrounding is,
T)-(TwAh
dt
dQ
w= (4)
Rate of work done by the gas is,
dt
dv
mP
dt
dW
g= (5)
Rate of change in internal energy for real gas is,














−





∂
∂
+=
dt
dv
P
T
P
T
dt
dT
Cm
dt
dU
V
Vg (6)
Substituting equations (4), (5) and (6) in (3), one gets
dt
dv
T
P
C
T)TT(
dt
dT
vv
w






∂
∂
−
τ
−
= (7)
Where, τ =
w
vg
Ah
Cm
(8)
τ is called as thermal time constant. It has unit of time
as seconds.
The differential equation (7) represents the thermal time

3. DHARANKAR, HADA AND JOSHI
3
constant model, which can be used to predict the spring
force characteristic of hydro-pneumatic spring.
The Thermal Time Constant
From equation (8) it can be seen that, the thermal time
constant (τ) is not a constant in real sense but varies as
the wall area (Aw) and heat transfer coefficient (h)
varies due to piston motion during the cycle of
excitation. However, as shown by Otis and
Pourmovahed [2] a constant value of τ fits experimental
data quite well. A constant value of thermal time
constant τ can be determined experimentally or
analytically as suggested by Pourmovahed and Otis [3].
In the present analysis a constant value of τ is calculated
by using analytical method.
Consider that the gas is compressed rapidly from its
initial temperature Tw to Tmax. At the end of compression
the gas volume is held constant where the temperature
of the gas decreases from Tmax.
For constant volume process, 0
dt
dv
= and equation (7)
is reduces to
)TT(
dt
dT w
τ
−
= (9)
This is the differential equation of gas process at
constant volume during which temperature of gas
decreases from temperature Tmax to the final
equalization temperature Tw.
Integrating above equation between limits of
temperature Tmax to T and time 0 to t,
( )wmaxw TTTT −=− e (- t/τ )
(10)
Above equation (10) gives the law of gas cooling at
constant volume, which can be used to find temperature
of gas at any instant of time.
At t = τ , T = Tτ
( )
e
TT
TT wmax
w
−
+=τ
( Tmax – Tτ ) = 0.6321 ( Tmax – Tw ) (11)
Decrease in peak temperature in τ sec. =
(0.6321) × (Difference between peak and final
equilibrium temperature)
Thus, the thermal time constant can be defined as the
time needed for the temperature or pressure to decrease
by 63.21% of the difference between the peak and final
equilibrium values under constant volume processes.
Above definition forms the basis for the experimental
determination of the value of thermal time constant.
The Non-Dimensional Frequency Parameter
The non-dimensional frequency parameter (ωτ) is
defined as the product of the circular frequency of
excitation (ω) and the thermal time constant (τ).
ωτ = ω × τ = 2 π f × τ (12)
From equation (10) it can be seen that as τ → ∞ the
process is adiabatic (no heat transfer) and as τ → 0 the
process is isothermal. Also as ω or f → ∞ the process is
adiabatic and as ω → 0 the process is isothermal.
Therefore it can be concluded that, as ωτ → ∞ the
process is adiabatic and as ωτ → 0 the process is
isothermal.
Following are the advantages of defining the
parameter ωτ -
• A single value of the parameter (ωτ) properly
characterizes the spring force and heat transfer
effects of hydro-pneumatic spring.
• It is shown that the energy dissipated by the hydro-
pneumatic spring is maximum near ωτ = 1, which
enables to identify the range of excitation
frequencies near which the energy dissipation is
predominant. Such identification helps in
developing active or semi-active suspension
systems.
The Anelastic Model
Fig. 1 shows the special case of Zener’s Anelastic
model which can be used to represent the material or
hysteretic damping in metals. This model can also be
used to represent the inherent damping in pneumatic
springs when changes in gas volume are small. The
thermal time constant (τ) can be considered equivalent
to the factor C/K [4].
At very low excitation frequencies the dashpot C will
not transmit any force to the spring K resulting in no
energy dissipation and the entire spring force is due to
the spring K1 and this condition corresponds to the
isothermal compression of gas. At very high excitation
frequencies the dashpot C will be locked and behaves
like a rigid element due to very high damping force and
again there is no energy dissipation and the entire spring
force is due to the parallel combination of spring K1 and
K and this condition corresponds to the adiabatic
compression of gas.
The Response of Anelastic Model to Sinusoidal
Excitation
The force developed across the Anelastic model,
F =K1 x + K y (12)
Considering force balance at point A,
0Ky
dt
dx
dt
dy
C =+





− (13)
For sinusoidal input excitation,
F
F
x (t)
y (t)
K1
K
C
A
Gas
Tw
P, v, T
Fig. 1: Representation of the pneumatic spring
using the Anelastic model.

4. DHARANKAR, HADA AND JOSHI
4
x = X sin (ω t), t)(cosX
dt
dx
ωω= (14)
The equation (13) can be rewritten as,
t)(cosXy
C
K
dt
dy
ωω=





+ (15)
The equation (15) is a first order linear differential
equation, which can be solved using standard methods
for solution of linear differential equation. The steady
state solution of equation (15) can be assumed of the
form,
y = Y sin (ω t + φ) (16)
Substituting equation (16) in (15),
X cos φ = Y, X sin φ = Y K /Cω (17)
The amplitude and phase angle of displacement of point
A are obtained as,
2
)C(K /1
X
Y
ω+
= (18)






ω
=φ
C
K
tan 1-
(19)
The energy dissipated per cycle is,
∫
ωπ
=
/2
0
dxFE (20)
Using equations (12), (14), (16) and substitution ωt = θ,
[ ]
][2sinYXK
2
1
dcosX)(sinYKsinXKE
2
0
1
πφ=
θθφ+θ+θ= ∫
π
Using equations (17) and (18),
ω
+
ω
π
=
C
K
K
C
XK
E
2
(21)
For maxima dE / dω = 0,
Therefore, the energy dissipated per cycle will be
maximum when,
1
K
C
or
C
K
=
ω
=ω (22)
From equation (21) it can be seen that the value of
energy dissipated per cycle is same for two values of
excitation frequencies say ω1 and ω2 such that,
2
21
C
K






=ωω (23)
The Transient Response of Anelastic Model
When the Anelastic model is subjected to the step
displacement of amplitude X, the point A gets displaced
through X. The step input is,
x = 0, t < 0
= X, t > 0
At time t = 0, the force across the Anelastic model is
maximum and is given by,
Fmax = K1X + K X (24)
After time t > 0, the point A starts from y = X and
returning to its equilibrium position y = 0, during this
period the force across the Anelastic model is,
F = K1X + K y (25)
When the point A reaches to its equilibrium position y =
0, the force across the Anelastic model is minimum and
is given by,
Fmin = K1X (26)
From equation (25) and (26)
y = (F - Fmin) / K (27)
After time t > 0, dx/dt = 0, the equation (13) can be
written as,
y
C
K
-
dt
dy






= (28)
Substituting equation (27) in (28),
)K/C(
)FF(
dt
dF min −
= (29)
The equation (29) is the differential equation of force
relaxation in the Anelastic model when it is subjected to
the step input. Comparing the equation (29) with (9), it
can be concluded that the force relaxation in the
Anelastic model is equivalent to the temperature or
pressure relaxation in the hydro-pneumatic spring at
constant volume.
Since,
K
C
,
K
C ω
≡ωτ≡τ
From equation (22), the energy dissipated by the is
maximum when the parameter ωτ = 1 or ω = 1/τ
Computer Simulation
A computer program in MATLAB is developed as per
the guidelines suggested by Otis and Pourmovahed [2]
for numerical solution of the nonlinear differential
equation (7). The solution of this differential equation
gives the gas temperature at different instants, which is
used to determine the gas pressure using BWR equation
of state of gas. Then the gas pressure is converted in to
spring force using the relation, Spring Force = Gas
Pressure × Piston area.
Fig. 2: The sinusoidal and triangular wave excitations
t
x
+X
-X
0
1/4f 1/4f
Sinusoidal
Triangular wave

5. DHARANKAR, HADA AND JOSHI
5
-80 -60 -40 -20 0 20 40 60 80
10
20
30
40
50
60
Displacement (mm)
Force(kN)
f = 0.2 Hz
For Sinusoidal Excitation
τ = 6.9 sec.
f = 0.002 Hz
f = 0.03 Hz
-80 -60 -40 -20 0 20 40 60 80
10
20
30
40
50
60
Displacement (mm)
Force(kN)
τ = 15 sec.
For Sinusoidal Excitation
f = 0.1 Hz
τ = 3.5 sec.
τ = 0.5 sec.
Fig. 3: Effect of excitation frequency on the dynamic spring characteristic of hydro-
pneumatic strut.
Fig. 4: Effect of thermal time constant on the dynamic spring characteristic of hydro-
pneumatic strut.

6. DHARANKAR, HADA AND JOSHI
6
-80 -60 -40 -20 0 20 40 60 80
10
20
30
40
50
60
Displacement (mm)
Force(kN)
For Sinusoidal Excitation ωτ = 6
ωτ = 1
ωτ = 0.08
-80 -60 -40 -20 0 20 40 60 80
10
20
30
40
50
60
Displacement (mm)
Force(kN)
For Triangular Wave Excitation
ωτ = 6
ωτ = 1
ωτ = 0.08
Fig. 6: Effect of parameter (ωτ) on the dynamic spring characteristic of hydro-pneumatic
strut for triangular wave excitation.
Fig. 5: Effect of parameter (ωτ) on the dynamic spring characteristic of hydro-pneumatic
strut for triangular wave excitation.

7. DHARANKAR, HADA AND JOSHI
7
Specificatios for a Spring Unit
A hydro-pneumatic spring with following
specifications is used in the present study.
Gas used: Nitrogen
Initial gas pressure Po = 6 MPa,
Initial gas volume Vo = 0.62 liter,
Temperature of cylinder wall or surrounding Tw =35 o
C,
Diameter of piston D = 70 mm,
Amplitude of excitation displacement X = 70 mm.
The constant values of thermal time constant (τ) for
different values of initial gas pressure (Po) are calculated
analytically using Otis method [3] as given in table-1.
Table- 1: Constant values of the thermal time constant
Po (MPa) 4.5 6 7.5
τ (sec.) 6.4 6.9 7.2
Types of Excitations Used
Two types of excitations sinusoidal and triangular
wave excitations are used for the purpose of analysis as
shown in Fig.2
1. Sinusoidal excitation (For single cycle) :
x = X sin(ω t), 0 ≤ t ≤ 1/f
2. Triangular wave excitation (For single cycle) :
x = (4f X) t, 0 ≤ t ≤ 1/4f
x = -(4f X) t + 2X, 1/4f < t ≤ 3/4f
x = (4f X) t - 4X, 3/4f < t ≤ 1/f
Results
Fig. 3 to Fig. 6 shows the spring force characteristics
of a hydro-pneumatic spring. The hysterisis loop in the
spring force characteristic shows that the hydro-
pneumatic spring posses some amount of inherent
damping.
Fig. 3 shows the effect of excitation frequencies on
the spring force characteristic of the hydro-pneumatic
spring. It can be seen that at lower excitation
frequencies the hysteresis loop is small and the process
is approximated to isothermal because there is sufficient
time for heat transfer between the gas and it’s
surrounding. On the other hand at higher excitation
frequencies the hysteresis loop is also small and the
process is approximated to adiabatic.
Fig. 4 shows the effect of thermal time constant on the
spring characteristic. It can be seen that a longer thermal
time constant has the same effect as that of higher
excitation frequency while a shorter thermal time
constant has the same effect as that of lower excitation
frequency.
Fig. 5 and Fig. 6 show the effect of parameter ωτ on
the spring characteristic for both sinusoidal and
triangular wave excitations. It can be seen that at very
small values of the parameter ωτ (less than 0.1) the
hysteresis loop is small and the process is approximated
to isothermal. On the other hand at higher values of the
parameter ωτ (greater than 15) the hysteresis loop is
also small and the process is approximated to adiabatic.
Fig. 7 shows the variation of percentage of input
energy lost (heat dissipated) with ωτ for both sinusoidal
and triangular wave excitations. It can be seen that the
heat dissipation is maximum near ωτ = 1, but not
exactly at ωτ = 1 due to non-linearity in the gas force.
This is very important prediction as it is possible to
identify the range of excitation frequencies near which
the energy dissipation or inherent damping is
predominant.
Conclusions
From the results of the computer simulation and
analytical discussion of the Anelastic model it can be
concluded that a single parameter ωτ properly
characterizes the heat transfer effects and spring force of
hydro-pneumatic springs.
The inherent damping of the hydro-pneumatic spring
may affect the performance of active or semi-active
hydro-pneumatic suspension systems. The analysis will
help in developing active or semi-active suspension by
realizing that the inherent damping of hydro-pneumatic
spring is a function of the parameter ωτ.
The proposed analysis will simplify the process of
designing of the hydro-pneumatic spring, as it is
possible to compare the performance of different hydro-
pneumatic spring units in the design stage.
References
1. Els P.S., Grobbelaar B., “Investigation of the time-
temperature dependency of hydro-pneumatic
suspension systems”, SAE technical paper series
930265, March 1993, pp.318-327.
2. Otis D. R., Pourmovahed A., “An algorithm for
computing nonflow gas processes in gas springs
and hydro-pneumatic accumulators”, Transactions
of the ASME, Journal of Dynamic systems,
Measurement and Control, Vol. 107, March 1985,
pp. 93-96.
3. Pourmovahed A., Otis D. R., “An experimental
thermal time constant correlation for hydraulic
accumulators”, Transactions of the ASME, Journal
of Dynamic systems, Measurement and Control,
Vol. 112, March 1990, pp. 116-121.
4. Pourmovahed A., Otis D. R., “Effects of Thermal
Damping on The Dynamic Response of a Hydraulic
Motor-Accumulator Systems”, Transactions of the
ASME, Journal of Dynamic Systems, Measurement
and Control, Vol. 106, March 1984, pp. 21-26.
Non-Dimensional Frequency Parameter (ωτ)
%ofInputEnergyLost
Sinusoidal Excitation
Triangular wave Excitation
Fig. 7: Percentage of input energy lost in the form
of heat during a cycle of excitation.