Proceedings of the 10th World Congress on Mechanical, Chemical, and Material Engineering (MCM'24)
Barcelona, Spain –August 22-24, 2024
Paper No. HTFF 135
DOI: 10.11159/htff24.135

Pressure Drop and Flow Loss Approximation by Comparison between
Mathematical Model and Numerical Model of a Simplified Fluid Mount.
Alfiya Ashraf1, Nader Vahdati1, Yap Yit Fatt1

Khalifa University, Mechanical and Nuclear Engineering Department,
SAN Campus, Abu Dhabi, UAE 127788
100062575@ku.ac.ae, nader.vahdati@ku.ac.ae, yap.fatt@ku.ac.ae

1

Abstract – Vibration isolators like hydraulic engine mounts are employed in the automotive industry to mitigate the effect of vibrationand noise produced in the vehicle by the engine. Several analyses on the transient response of the hydraulic engine mounts have been
carried out in literature to improve the performance of the device. In this paper, efforts have been taken to study the oscillatory fluid flow
behaviour of the working fluid inside these mounts. A simplified fluid mount model is considered, and its respective numerical andmathematical models are developed to study the effect of input parameters such as frequency and displacement on pressure loss acrossthe inertia track. Comparison of the inertia track pressure loss obtained from both the models provides validation on the modelling
approach. This approach can be further utilized to develop a correlation between, the input displacement, the inertia track geometry, thefluid density and viscosity and the pressure loss across the inertia track in hydraulic engine mounts and thereby enhancing the damping
efficiency of these mounts.
Keywords: Hydraulic engine mounts, fluid mounts, hydraulic dampers, oscillatory fluid flow, computational fluid dynamics.

1. Introduction

A fully articulated main rotor system of a helicopter has three hinges per blade, pitch hinge, flapping hinge, and a leadlag hinge. The lead-lag hinge allows the blade to move forward and backward in the plane of the rotor disk. Ground resonanceis a mechanical instability affecting helicopters on the ground. It occurs at certain rotor speeds, where the lead–lag motionof the rotor blades couples with the motion of the fuselage, creating a self-excited oscillation, which is very violent andgenerally can result in total destruction of the helicopter, if the pilot is not careful and well trained. To control the groundresonance, passive hydraulic dampers are used in between the consecutive rotor blades of the main and tail rotors, as shownin Figure 1.

Hydraulic Dampers

Fig. 1: NH90 helicopter with hydraulic dampers in between the main rotor blades[1]The internal design, of the hydraulic dampers, is shown in Figure 1 [1]. The hydraulic dampers, used in helicopters,
consist of: (1) two fluid chambers (shown in pink color) connected to one another via a carefully designed orifice or inertia
track, (2) a carefully selected working fluid (shown in blue color), (3) two rubber tube-forms (shown in green color), (4) ashaft (shown in black color) that connects the two inner members (shown in dark blue color) of the tube-forms, and (5) an
outer housing. The carefully selected fluid and designed orifice/inertia track is what makes this damper to work. The damper
needs to provide an adequate amount of damping to control the helicopter ground resonance. The lead-lag motion of the

HTFF 135-1

blades makes the working fluid of the hydraulic damper to move back and forth between the two fluid chambers, creating
oscillatory fluid flow phenomenon.
There is another passive technology, called hydraulic engine mounts (or fluid or fluidlastic mounts), shown in Figure 2,
that also experience oscillatory fluid flow phenomenon. As seen in Figure 2, a passive fluid (or hydraulic) engine mount [2,3, 4] comprises of a fluid (shown in blue color) that is housed in two elastomeric cavities (or fluid chambers) that are joinedby an inertia track.

Fig. 2: Hydraulic (fluid) engine mountThe pressure drop from one fluid chamber to the other can be shown as follows:̇∆𝑃𝑃 = 𝐼𝐼𝑓𝑓 𝑄𝑄 + ∆𝑃𝑃𝐿𝐿(1)
𝜌𝜌𝐿𝐿𝑡𝑡where fluid inertia is equal to 𝐼𝐼𝑓𝑓 = 𝐶𝐶
. In fluid inertia equation, 𝜌𝜌 is the working fluid density, 𝐿𝐿𝑡𝑡 is the actual length of𝐴𝐴𝑡𝑡the inertia track, 𝐶𝐶𝐿𝐿𝑡𝑡 is the effective length of the inertia track with the constant C being in the range of 1 to 4/3 [5], and 𝐴𝐴𝑡𝑡
is the inertia track cross-sectional area. In Equation (1), 𝑄𝑄 is the flow rate through the inertia track. The 2nd term in theequation (1) includes the effects of viscous shear, expansion/contraction, and other forces such as body and gravitational
forces. In this paper, we refer to the 2nd term of equation (1) as the flow losses. Accurate knowledge of the orifice/inertia
track flow losses and the effective length of the inertia track are the most important design parameters, when designing ahydraulic damper or a fluid mount. For the hydraulic dampers, since the length of the inertia track is in general low thus low
fluid inertia, the 2nd term in the equation (1), namely ∆𝑃𝑃𝐿𝐿 , is the most important design parameter. For fluid mounts, accurate
knowledge of both the effective inertia track length and the flow losses is very much needed. In order to accurately know theconstant C and ∆𝑃𝑃𝐿𝐿 , both CFD and experimental results are needed. Here in this paper, the CFD numerical results are only
shown.
Although real fluid mounts and hydraulic dampers can have very complicated geometries, the simplified geometry
shown in Figure 3a can be used to characterize the critical device parameters. In the physical model of Figure 3a, the wallsof the end chambers are assumed rigid. This allows an isolation of the fluid mechanics effects from the elastomer complianceeffects, which can be analysed separately. Here in this paper, we focus on fluid mounts rather than hydraulic dampers sinceboth the inertia track effective length as well as its flow losses will be important.

As we apply a sinusoidal displacement across the chambers of Figure 3a, the fluid is forced to move through the narrowinertia track causing a pressure drop across the mount. To determine the efficiency and improve the performance of themount, the prediction of this pressure drop is an important criterion. Therefore, the magnitude of the pressure drop, which isa function of inertia track geometry, fluid density and viscosity, and input displacement, is analysed by considering a Physicalmodel which is simulated in MATLAB and ANSYS FLUENT.

HTFF 135-2

2. Numerical Model of the Simplified Fluid Mount Model

The numerical modelling and analysis of the oscillating fluid flow in the simplified fluid mount model is being carriedout using CFD tool ANSYS Fluent. The preliminary objective of the approach is to model the oscillating fluid flow in a test
section and to initially identify and measure the pressure drop across the inertia track and validate the data with themathematical model developed.
The problem is solved in a two-dimensional CFD grid. It is initially assumed that there is no heat transfer or heat loss
across the walls of the simplified fluid mount model and there is no phase change accounted for during the simulation i.e.,
the liquid flow is single phase, and the fluid is incompressible.
2.1. Computational Domain and Mesh Generation
The simplified fluid mount model is defined with two fluid chambers and an inertia track between them, see Figure 3a.
At the end of the two chambers, inlet 1 and inlet 2 are defined, along with the wall and interior surface body. The fluid isallowed to oscillate inside the domain with the applied frequency and displacement. Since the model is axisymmetric, thegeometry is modelled across one side of the axis. In Fluent, the axis of symmetry is the x-axis.
In the ANSYS Meshing module, a computational mesh or grid is created. Using common "sizing" methods, a CFD meshor grid is built with an elemental size of 0.0005m of the surface body to improve performance quality. Figure 3b represents
the geometry of the simplified fluid mount model developed in Fluent and the CFD grid's construction. The grid's goodquality was demonstrated by the lack of elements with negative computational mesh areas, orthogonality in the range of
0.87-1, and skewness in the range of 0-0.35.

VP1y-axisV
P2

Inlet_2

Inlet_1
Inertia track

x-axisFig. 3: (a) Physical Model of the Simplified Fluid Mount Model (b) Domain of simulation with inlets and mesh2.2. Mathematical Description
The oscillating flow inside the flow bench is hydrodynamically fully developed in the inertia track with diameter 𝑑𝑑𝑡𝑡 .
The equation is considered by omitting the component of twisting flow, neglecting external forces, the flow direction isassumed to be parallel to the x-axis therefore the velocity has x-direction component only, which also indicated that thevelocity to be constant in the direction that is parallel to the centre line. The governing conservation equation for the massand momentum for the incompressible fully developed flow is expressed as;
𝜕𝜕𝜕𝜕
(2)
=0
𝜕𝜕𝜕𝜕

1 𝜕𝜕𝜕𝜕
𝜕𝜕 2 𝑢𝑢 1 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕=−+ 𝜗𝜗 � 2 +�𝜌𝜌 𝜕𝜕𝜕𝜕
𝑟𝑟 𝜕𝜕𝜕𝜕
𝜕𝜕𝑟𝑟
𝜕𝜕𝜕𝜕

(3)

𝑉𝑉1 = 𝑉𝑉2 = 0.08 cos(162.357t)(4)

Where the x, r, u, p, 𝜌𝜌 and 𝜗𝜗 represents the axial and radial coordinates, axial velocity, pressure, density, and kinematic
viscosity of the fluid respectively. At the inlet_1 and at inlet_2 the boundary condition is set with a piston motion amplitude
as 0.508 mm and at a frequency of 25.84Hz. Therefore, the velocity input at inlet_1 and inlet_2 is expressed as.

HTFF 135-3

A transient analysis of the oscillating fluid flow is simulated in the ANSYS Fluent, and the solver is pressure-based. The
SST k-omega model is used to solve the problem numerically. In this turbulence model, the k-𝜀𝜀 model is used to solve near
the wall and k-omega model is used in the free stream region, thereby combining the best features of both models. The model
uses a blending function to smoothly transition between the two models based on the distance from the wall, thus allowing
the model to accurately predict flows with wide range of turbulence levels and flow conditions.

3. Mathematical Model of the Simplified Fluid Mount Model

The physical model of the simplified fluid mount model was shown Figure 3a. This model is developed as a basis forfurther mathematical modelling and evaluation of design parameters. Since the concerned system is dealing with multiple
energy domains, mechanical and fluid systems, bond graph modelling technique [6] is employed for developing themathematical model of the simplified fluid mount model of Figure 3a.
In the physical model, shown in Figure 3a, an input velocity is applied to the left and right chamber ends in the same
direction. This causes the fluid to be displaced from chamber 1 to the inertia track and further to the chamber 2; respectively.
The fluid flow from the chamber 1 to chamber 2 through the inertia track is due to the pressure difference that is caused bythis moving chamber ends. As the fluid passes through the inertia track, the fluid undergoes flow losses, due to the friction,
viscous shear, and surface roughness, and this is one of the major concerns of oscillating flow simulation.
The bond graph model of the physical model of Figure 3a is shown in Figure 4. The mechanical domain and fluid domainof the simplified fluid mount model is incorporated while developing the bond graph model.
4

0P1
5

𝑆𝑆𝑓𝑓 : 𝑉𝑉𝑖𝑖𝑖𝑖

1

I

6

1

7

8
0P2

C

9

R
C

Fig. 4: Bond Graph of the System

In the above bond graph, the input velocity is considered as the source of flow which is represented by bond 1. The massof the chamber ends can be neglected since we are actuating the ends in a displacement mode. The fluid energy domain isthen introduced in the bond graph model when the chamber ends meet the fluid. A transformer is employed for the simplified
fluid mount model between the bonds 2 and 3. The employed transformer relation is Q =𝐴𝐴𝑝𝑝 𝑉𝑉𝑖𝑖𝑖𝑖 , which connects the input
velocity to the fluid's volumetric flow rate through the inertia track, which is equal to the chamber end’s surface area. Multiple
power dissipation factors are encountered as the power flows in the fluid energy domain. Specifically, the fluid's volumetricstiffnesses of each chamber (𝑘𝑘𝑣𝑣 ), inertia track flow losses (𝑅𝑅𝑜𝑜 ), and inertia track fluid inertia (𝐼𝐼𝑓𝑓 ). The compliance represented
by 𝐶𝐶 are for the compressibility of the working fluid. The flow losses and fluid inertia inside the inertia track is calculated
from the Fluent model. From the bond graph model, the state space equations in terms of displacement and momentum
variables are found as shown below,𝑃𝑃
𝑞𝑞4̇ = 𝐴𝐴𝑝𝑝 𝑉𝑉𝑖𝑖𝑖𝑖 − 𝐼𝐼 6
(5)
6

𝑃𝑃
𝑞𝑞
𝑞𝑞
𝑃𝑃6̇ = 𝐶𝐶4 − 𝑅𝑅7 𝐼𝐼 6 − 𝐶𝐶9
46𝑃𝑃𝑞𝑞̇ 9 = 𝐼𝐼 6 − 𝐴𝐴𝑃𝑃 𝑉𝑉𝑖𝑖𝑖𝑖
6

9

(6)

(7)

From these state space equations, the state space matrices of the system have been represented below, the output
variables are chamber 1 pressure (𝑃𝑃1 ), and chamber 2 pressure (𝑃𝑃2 ). MATLAB simulation is carried out with the above
represented state space matrices to plot the outputs with respect to time.
HTFF 135-4

⎡0
𝑞𝑞4
⎢1
�𝑃𝑃6 � = ⎢𝐶𝐶
4
𝑞𝑞9
⎢
⎣0
1

4. Results and Validations

𝑃𝑃𝐶𝐶� 1� = � 4
𝑃𝑃2
0

1
𝐼𝐼6
𝑅𝑅
− 𝐼𝐼 7
6
1
𝐼𝐼6

−

0
0

0 ⎤
𝐴𝐴𝑝𝑝
𝑞𝑞4
1⎥
− ⎥ �𝑃𝑃6 � + � 0 � 𝑉𝑉𝑖𝑖𝑖𝑖
𝐶𝐶9
−𝐴𝐴𝑝𝑝
⎥ 𝑞𝑞9
0 ⎦
0 𝑞𝑞4
0
𝑃𝑃
1 � � 6 � + �0� 𝑉𝑉𝑖𝑖𝑖𝑖
𝑞𝑞9
𝐶𝐶

(8)

(9)

9

4.1. Numerical Simulation Results
The pressure and velocity contour for the oscillating fluid flow inside the Fluent CFD model is represented in Figures 5
and 6. We can observe that inertia track is said to have a higher value of velocity compared to the chambers. At the sametime, considering the total pressure contour, since an input velocity is applied to both sides of the chamber ends which
compresses and expands the chambers simultaneously, the total pressure in the chambers is said to have a higher value than
the inertia track pressures, which can also be observed from Figure 5. This pressure drop across the chamber results in the
motion of fluid through the inertia track. Figure 6 shows the fluid jetting outside of the inertia track, which impacts the inertia
track effective.

Fig. 5: Pressure contour for flow simulation (fluid flowfrom right to left)

Fig. 6: Velocity contour for flow simulation (fluid flowfrom right to left)

4.2. Grid Independent Study (Mesh Sensitivity Analysis)

Refinement of mesh is conducted in both spatial and temporal domain. The sinusoidal pressure data is extracted at apoint in one of the chambers and the data for the respective refined mesh is represented in the Figure 7. For mesh refinement
and comparing the pressure data, we have taken the model parameters as 10 Hz, 0.508 mm, SST K-omega turbulence model,
with surface roughness as zero. The working fluid has a density of 1765 𝐾𝐾𝐾𝐾/𝑚𝑚3 and 1.8 cst viscosity. From Figure 7, we
can conclude that as we refine the mesh in both temporal and spatial domain, the error between each data reduces. Therefore,
we can conclude that for the case of 10 Hz, the mesh size of 0.001 and time step size 0.001 can be used for further calculation,as very fine mesh size and time step size may also have large truncation error.

HTFF 135-5

Element size=0.00025 & time step
size=0.0000625

1.50E+04

Pressure (Pa)

1.00E+04

Element size=0.0005 & Time step
size=0.000125

5.00E+03
0.00E+00
0.00E+005.00E-021.00E-011.50E-012.00E-012.50E-013.00E-013.50E-01-5.00E+03-1.00E+04-1.50E+04

Element size =0.001 & Time step
size= 0.000125
Element size 0.001 & Time step
size = 0.00025
Element size 0.001 & Time step
size = 0.0005
Element size=0.001 & Time step
size =0.001

Time (sec)Fig. 7: Pressure data in left chamber at different spatial
and temporal refined mesh.

4.3. Parametric Analysis on the Numerical Model

To calculate the pressure drop across the chambers, two points P1 and P2 have been considered shown in Figure 8. The
transient pressure data has been extracted at these two points in CFD post processing and the corresponding results ofsinusoidal pressure has been plotted in Figure 9. The frequency and displacement for this analysis is taken as 25.84Hz and
0.508mm; respectively. From Figure 9, we can observe that the results have converged as the cycles of pressure is repeating
at the same amplitudes. However, we can also observe a shift towards positive y-axis, and this shift could be explained dueto the pressure drop accounted by the flow losses. The Table 1 represents the comparison of the pressure data extracted at
the points P1 and P2, by changing the numerical model, and surface roughness and by keeping the geometry, boundary
conditions and the working fluid the same. From the Table, we can analyse that, the two CFD model SST K-omega andLaminar have a slightly different pressure amplitudes at the two chambers. Also, analysis, on 2D and 3D model has also
been done. From comparison of pressure data for 2D versus 3D, we can infer that, the pressure data doesn’t have significant
difference in its values thus no need to make 3D CFD runs. To see the effect of surface roughness of the chambers and theinertia track on the pressure data, a Fluent simulation was conducted. We can conclude that, as the surface roughness
increases the pressure amplitude in two chambers increases there by increasing the pressure drop inside the system. Analysis
on the pressure drop with respect to input frequency and displacement has been carried out with another working fluid withthe density of 935kg/𝑚𝑚3 and the viscosity of 10 cst.

Pressure (Pa)P1
6.00E+04

P2

1.00E+04
0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01-4.00E+04-9.00E+04

Fig. 8: Location of two points P1 & P2 considered in two
chambers.

HTFF 135-6

Time (sec)Fig. 9: Pressure data at the points P1 and P2

Table 1: Comparison of pressure data at different CFD model parameters
Reynolds
number𝑅𝑅𝑅𝑅
𝜌𝜌𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑
=
𝜇𝜇
999

StrouhalNumber𝜔𝜔𝜔𝜔
𝑆𝑆𝑆𝑆 =
𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚

2D VS 3D

CFD Model

SurfaceRoughness(m)

2D

SST k-omega

0

74400

Pressure
amplitude
rightchamber, P2
(Pa)
74900

3D

SST k-omega

0

75900

75800

2D

Laminar

0

74300

74800

999

44

2D

SST k-omega

0.0005

74700

75300

999

44

Pressure
amplitudeleft chamber,
P1 (Pa)

999

44
44

From Table 2, we can conclude that as the frequency increases, the pressure drop across the chambers also increases,
which represents the square dependence of pressure drop with frequency. If we assume a linear relationship between thepressure drop, ∆𝑃𝑃, and the flow rate, 𝑄𝑄, and the rate of flow rate, 𝑄𝑄̇ , as shown in equation (10):(10)∆𝑃𝑃 = 𝐼𝐼𝑓𝑓 𝑄𝑄̇ + 𝑅𝑅𝑜𝑜 𝑄𝑄
where 𝐼𝐼𝑓𝑓 , and 𝑅𝑅𝑜𝑜 represents inertia track fluid inertia, and the flow losses; respectively, then 𝐼𝐼𝑓𝑓 and 𝑅𝑅𝑜𝑜 can be estimated,
as shown in Table 2. Similarly, to analyse the influence of displacement amplitude on chamber pressures, numericalsimulations have been conducted by keeping all the model and geometry parameters the same as that of the above case, withdisplacement amplitudes changing from ±0.0381 mm to ±0.0762mm and to ±0.1524mm. The Figure 10 represents thecomparison of pressure amplitude at different frequency for the three different displacements amplitudes. Figure 10 showsthat the chamber pressures are related to square of the input displacement.
Table 2: Data at constant amplitude and varying frequency
Frequency(Hz)

Pressuredrop(Pa)±0.0381±0.0381±0.0381±0.0381±0.0381±0.0381±0.0381±0.0381±0.0381±0.0381

10
20
30
40
50
60
70
80
90
100

635.6
2416.15375.39529.814873.521503.9883
29364.9121
38310.4082
48977.3047
61107.4356

Fluid
Inertia
𝑰𝑰𝒇𝒇 = 𝑪𝑪𝝆𝝆𝑳𝑳𝑡𝑡𝑨𝑨𝑡𝑡

5927900
5937100
5950900
5997000
6020100
6043200
6052400
6089300
6135400
6158500

Approx
Flow
Loss,
𝑹𝑹𝟎𝟎

1.90E+08
2.30E+08
2.95E+08
3.45E+08
4.00E+08
4.88E+08
5.82E+08
6.23E+08
7.04E+08
8.72E+08

140000120000100000Pressure (Pa)

Displacement
Amplitude
(mm)

80000
60000
40000
20000
0

0

20

40

60

80

100

Frequency (Hz)0.0381mm

0.0762mm

0.1524mm

Fig. 10: Pressure amplitude at different displacements

Since in fluid mounts, the location of the notch frequency [2, 3, 4] is directly related to the inertia track fluid inertia andfluid inertia is directly related to the effective length, it is vital to accurately know the effective length of the intertia track
through CFD simulations and experimental data. As mentioned in literature, the effective length is equal to 1 to 1.33 times
the geometric length [5]. The increase in the effective length could be due to the jetting effect of the fluid from the intertia
track at high displacement amplitudes and frequencies. The velocity streamlines shown in Figures 11 and 12 for the case of±0.0381mm at frequencies of 10 and 100 Hz; respectively, validates the jetting effect theory. This shows the importance offlow visualization.

HTFF 135-7

0.1285 m

0.1335 m

Fig. 11: Effective length of the inertia track for ±0.0381mm@10Hz

Fig. 12: Effective length of the inertia track for ±0.0381mm@100Hz4.4. MATLAB Simulation Results

The pressure data obtained from MATLAB simulation is represented in Figure 13. CFD flow visualization, plus thebond graph model is used to find a value for 𝐼𝐼𝑓𝑓 and 𝑅𝑅𝑜𝑜 . The pressure drop for the cases of ±0.0381mm from 10Hz to 100Hz,
of CFD model and Mathematical Model, are tabulated below for comparing the accuracy of the pressure data and theeffectiveness of the bond graph model to estimate 𝐼𝐼𝑓𝑓 and 𝑅𝑅𝑜𝑜 . From the Table 3, we can infer that the pressure drop value
obtained from CFD Model and Mathematical model is approximately equal with a maximum error of 3.3%. This increase inpercentage error at higher frequencies could be due to the effective length not accurately approximated from the Fluent
model.
Table 3: Pressure drop comparison for CFD Model and Math Model

Frequency
(Hz)
10
20
30
40
50
60
70
80
90
100

Pressure drop(Pa)
CFD Model
635.546295
2416.05396
5375.32178
9529.78955
14873.54052
21503.9883
29364.9121
38310.4082
48977.3047
61107.4356

Pressure drop(Pa)
Mathematical Model
642.0342406.365377.06
9502.4614978.98
21795
29709.2
39226.2
50412.2
63198.8

Percentage
error %
1.01
0.4
0.03
0.28
0.7
1.33
1.15
2.33
2.84
3.3

Fig. 13: Pressure data for Mathematical Model

Conclusion

Flow visualization from the CFD Model has clearly indicated the importance of it when determining the effective lengthof the inertia track used to calculate the fluid inertance, 𝐼𝐼𝑓𝑓 . This provides insights on the importance of using CFD models todevelop a relation between the input displacement amplitude and frequency with the effective length of the inertia track, as
this length if not calculated correctly, can result in large errors in calculation of the flow losses. The exact values of fluid
inertia and flow loss is very important for calculating the exact location and depth of the notch frequency for fluid mounts.
Further validation of the CFD and Math models should be carried out using experimental results.

References[1] D. P. McGuire, “Hybrid Fluid and Elastomer Damper,” U.S. Patent 5 501 434, May 26, 1996.
[2] W. C. Flower, “Understanding hydraulic mounts for improved vehicle noise, vibration and ride qualities,” SAE Trans.,
pp. 832–841, 1985.
[3] M. Clark, “Hydraulic engine mount isolation,” SAE Technical Paper, 1985.
[4] N. Vahdati, “Variable volumetric stiffness fluid mount design,” Shock and Vibration, Vol. 11, issue 1, p. 21, 2004.
[5] E.O. Doebelin, System Dynamics: Modeling Analysis, Simulation, Design, Marcel Dekker, New York, p. 243-244, 1998.
[6] R.C. Rosenberg, D.C. Karnopp, Introduction to Physical Systems Dynamics, Mc Graw Hill Book Company, New York,1983.

HTFF 135-8