UNIVERSITY OF CINCINNATI
February 23, 2005
Date:___________________
Sergiy Bograd
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Masters of Science
in:
Mechanical Engineering
It is entitled:
Application of the Transfer Path Analysis to Hydraulic Systems
This work and its defense approved by:
Dr. Randall J. Allemang
Chair: _______________________________
Dr. David L. Brown
_______________________________
Dr. Allyn W. Phillips
_______________________________
_______________________________
_______________________________
Application of the Transfer Path Analysis to
Hydraulic Systems
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
in the Department of Mechanical Engineering of the College of
Engineering
2005
by
Sergey Bograd
B.S.M.E. Michigan Technological University 2002
Committee Chair: Dr. Randall J. Allemang
Abstract
Application of the traditional transfer path analysis (TPA) to hydraulic pipes, like the ones found
in the braking systems, can not be easily performed due to the fluid-structure interaction and rigid
coupling between the pipes and the piping unit. This thesis focused on studying how the
vibrational energy is transferred in hydraulic pipes with the fluid-structure coupling and how the
transfer path analysis can be applied to such systems.
A test bench for testing the propagation of vibration from a hydraulic pump to simplified
structure was constructed and tested. It consisted of the hydraulic exciter, generating pressure
pulses by two opposing piezo-stacks, a brake pipe filled with hydraulic oil and connected with
rigid mounts to the steel sheet. Transfer functions that could be used for the TPA were acquired.
Parallel to the experimental work, the FE model of the test bench was created in ANSYS.
Correlation with the experimental model was performed and transient and harmonic analysis data
was acquired. In addition, an initial analysis on a system with more complicated geometry was
performed.
Preface
“An engineer is one who passes as an exacting expert on the strength of being able to turn out,
with prolific fortitude, strings of incomprehensible formulas calculated with micrometric
precision from extremely vague assumptions which are based on debatable figures acquired from
inconclusive tests and incomplete experiments carried out with instruments of problematic
accuracy by persons of doubtful reliability and of rather dubious mentality with the particular
anticipation of disconcerting and annoying everyone outside their own profession”.
[Someone Said It! http://www.sdrl.uc.edu/newsletter/newsl_93_11.html]
This thesis was sponsored and performed at Robert Bosch GmbH. I would like to thank
Dr. Karl Bendel for his supervision and guidance during this project and in the scope of the work
performed in the last two years. Also, my thanks go to Wolfgan Rottler and Ilan Brauer for their
help in construction of the experimental model and technical support.
The simulation part of the project was implemented with the help of the Institute for
Mechanics A, University of Stuttgart. I would like to thank all the people in the institute and
particularly Mattias Maess for providing help and advice in model construction and simulation.
Many thanks go to Randy Allemang for bringing me into this program, which allowed me
to learn many new things and get to know many interesting people. Thanks for the support as a
teaching assistant and for pizza and drinks, it allowed me to survive the grad school. Thank you
Dave Brown for all the lessons and discussions in history, vibrations, and other things. I really
enjoyed my stay at UC-SDRL; thank you my fellow colleagues – students, faculty and staff for
making this place a great place to learn, work, and have fun.
Thank you Chuck Van Karsen for the introduction into the world of vibrations and help
with transition from Michigan Tech to UC.
My gratitude goes to my parents; without their help my education would not be possible.
And thanks to my family and all my friends who helped to make my education very long, but
interesting and fun.
Table of Contents
Chapter 1
Introduction............................................................................................................................................. 1
1.1 Motivation .............................................................................................................................................................. 1
1.2 Thesis structure ...................................................................................................................................................... 1
Chapter 2
Transfer Path Analysis........................................................................................................................... 3
2.1 Application and theory of TPA .............................................................................................................................. 3
2.2 Methods of predicting propagation of energy in hydraulic systems..................................................................... 6
Chapter 3
Fluid-Structure Coupling .................................................................................................................... 10
3.1 The acoustic wave equation ................................................................................................................................. 11
3.2 Fluid-structure interaction and its implementation in Finite Element analysis................................................. 14
Chapter 4
Experimental and Finite Element Models ......................................................................................... 17
4.1 Experimental system description ......................................................................................................................... 18
4.2 FE model description and construction............................................................................................................... 22
4.3 Model correlation................................................................................................................................................. 25
Chapter 5
Finite Element Results.......................................................................................................................... 31
5.1 Transient Simulation results ................................................................................................................................ 31
5.2 Harmonic Simulation results ............................................................................................................................... 35
Chapter 6
Finite Element Modifications / Results .............................................................................................. 36
6.1 Small model description....................................................................................................................................... 36
6.2 TPA using pressure – force transfer functions .................................................................................................... 38
6.3 TPA using strain transfer functions..................................................................................................................... 43
Chapter 7
Experimental Model Results ............................................................................................................... 47
7.1 Selection of transducers ....................................................................................................................................... 47
7.2 Testing conditions ................................................................................................................................................ 49
i
7.3 Linearity of the hydraulic system......................................................................................................................... 53
7.4 Application of reciprocity in hydro-mechanical systems .................................................................................... 54
7.5 Hydraulic pump results / modifications............................................................................................................... 56
Chapter 8
Discussion, Modifications, and the Future Work ............................................................................. 57
8.1 Discussion ............................................................................................................................................................ 57
8.2 Modifications / Future work ................................................................................................................................ 59
Chapter 9
Conclusions............................................................................................................................................ 63
References ...................................................................................................................................................................... 65
Appendix A:
Additional Diagrams.............................................................................................................. 67
Appendix B:
Data Tables ............................................................................................................................. 69
Appendix C:
Modeling code......................................................................................................................... 72
ii
Table of Figures
Figure 2.1 Schematic representation of the TPA. Excitation propagates from the source via the coupling
elements to the structure. Vibration or acoustic response is measured on the structure or at a different
points of interest............................................................................................................................................ 3
Figure 2.2 An example of TPA calculation. Propagation of vibration from the electric motor into car
compartment via three coupling elements (9 paths). Vector diagram at 1375 Hz [4]. ............................... 5
Figure 2.3 An example of the pipe system used for TMM ............................................................................ 8
Figure 3.1 A small volume of fluid in space, with mass flow going into and out of the volume. ............... 11
Figure 4.1 Experimental set up. ................................................................................................................. 18
Figure 4.2 Schematic drawing of the pipe and sheet of experimental set up with dimensions (cm). ......... 19
Figure 4.3 Hydraulic exciter in the disassembled form. ............................................................................ 20
Figure 4.4 Electrical circuit of the experimental set up connected to the data acquisition system. .......... 22
Figure 4.5 Finite element assembly of the test structure............................................................................ 24
Figure 4.6 Experimental set up of the modal analysis performed on a 30 cm section of the brake pipe... 25
Figure 4.7 An example of the convergence of the first bending mode of 30 cm pipe segment with an
increase in mesh resolution. ....................................................................................................................... 26
Figure 4.8 Experimental and simulation results. Left side: P/F frequency response function, with first
fluid mode occurring at 722 Hz. Right side: representation of the first fluid mode, FE analysis. ............. 28
Figure 4.9 Experimental set up for modal analysis of the complete structure. Inputs, provided by the
modal hammer and output measurement grid are shown. .......................................................................... 29
Figure 4.10 An example of the experimental and FE modal analysis correlation..................................... 30
Figure 5.1 Pressure excitation and response signal for transient simulation............................................ 32
Figure 5.2 Displacement of the point on the pipe, near the pump connection, caused by the application of
pressure impulse. ........................................................................................................................................ 33
Figure 5.3 Transient simulation; pressure pulse propagation through the system.................................... 34
Figure 5.4 Harmonic simulation; acceleration response on the sheet due to the pressure excitation
provided by the pump.................................................................................................................................. 35
Figure 6.1 Schematic of the small FE model; one and two pipe configurations........................................ 37
Figure 6.2 Mode comparison for small and full scale models. .................................................................. 38
Figure 6.3 TPA estimation for one pipe system using force and pressure transfer functions; excitation
case 1. ......................................................................................................................................................... 39
iii
Figure 6.4 TPA estimation for one pipe system using force and pressure transfer functions; excitation
case 2. ......................................................................................................................................................... 40
Figure 6.5 Schematic of indirect force estimation. .................................................................................... 41
Figure 6.6 Indirect force estimation; there was a 0 N force applied in x-direction................................... 42
Figure 6.7 Indirect force estimation; there was a 0 N force applied in y-direction................................... 42
Figure 6.8 Indirect force estimation; there was a 13 N force applied in z-direction................................. 43
Figure 6.9 TPA estimation for one pipe system using strain transfer function; excitation case 1............. 44
Figure 6.10 TPA estimation for one pipe system using strain transfer function; excitation case 2........... 44
Figure 6.11 TPA estimation for two pipe system using strain transfer functions. ..................................... 45
Figure 6.12 TPA calculation on a two pipe system; contribution of the paths to the total response......... 46
Figure 7.1 Pump, used for application of the static pressure, and analog pressure meter........................ 47
Figure 7.2 Effect of varying boundary conditions on pressure response of the system. ............................ 50
Figure 7.3 Effect of varying boundary conditions on acceleration response of the structure. .................. 50
Figure 7.4 Effect of varying static pressure on dynamic pressure response of the system. ....................... 52
Figure 7.5 Effect of the varying static pressure on the acceleration response of the system..................... 52
Figure 7.6 Check for the linearity in the system; pressure response. ........................................................ 53
Figure 7.7 Check for the linearity in the system; acceleration response. .................................................. 54
Figure 7.8 Application of the reciprocity principal to experimental system.............................................. 55
Figure 7.9 Modal analysis of the hydraulic pump. First deformation mode for the original and modified
design. ......................................................................................................................................................... 56
Figure 8.1 Properties of the hydraulic fluid [16]....................................................................................... 58
Figure A.1 Hydraulic pump mounted on a mass. Rigid boundary conditions testing. .............................. 67
Figure A.2 First draft of the hydraulic pump. Some dimensions were modified in the final model. ........ 67
Figure A.3 Steel joints. First draft, some dimensions were changed in final model.................................. 68
iv
Chapter 1 Introduction
1.1 Motivation
The research of the sound produced by an automobile is a significant part of the automotive
development. Even though some components of the sound might be desirable, like the low
frequency noise of the exhaust of sports cars, there is a general tendency to decrease the level of
the noise produced by an automobile. The noise reduction is particularly critical in the design of
the automotive components. The noise produced by a fuel pump or a windshield wiper motor
should not be louder than the engine operating at the idle speed.
Often the component itself does not produce the significant noise, but rather induces the vibration
of the car body where it is mounted. It is important to study how the vibration propagates from
the source and the way it excites the structure.
The goal of this project was to study the propagation of vibrational energy through the hydraulic
pipes from the source to the receiver structure and find the techniques for application of the
Transfer Path Analysis to these systems. Examples of such systems can be the propagation 7of
the vibrational energy from the ABS (Anti-Lock Brake Systems) or ESP (Electronic Stability
Program) hydraulic units through brake pipes to the car’s body.
1.2 Thesis structure
This thesis will focus on studying the propagation of vibrational energy through fluid and
structural paths. Then application of the TPA to hydraulic systems will be investigated. In order
to understand how TPA can be used in evaluation of hydraulic systems and what has to be
studied in fluid structure interaction, theory of the TPA and its applications will be discussed.
1
Further, the acoustic wave equation will be derived, and its discretization and coupling with the
solid structure will be shown.
The project was implemented using a test bench and a finite element model of it, so description
of both will be presented. This section will describe how and why the models were designed and
constructed and give the results of their correlation.
The results of the thesis are given in three subsequent chapters, starting with the FE results for
the modeled structure. Then results for the small model, which was created for increasing the
computing speed of the FE simulations, will be introduced. Finally, experimental results and
work that has been done on experimental structure will be presented. Analysis of the results and
ways to improve and modify the structure will be discussed in the end.
2
Chapter 2 Transfer Path Analysis
2.1 Application and theory of TPA
An important task in the evaluation of the noise and/or vibration performance of the mechanical
system is to define the sources and the forces they generate, paths through which the energy
propagates, and the response at the point of interest. The technique which is used for this purpose
is known as a transfer path analysis. It is a well established technique, which has applications in
the evaluation of multiple consumer products. It is most commonly used in the automotive
industry [1], [2], where it can be implemented to rank the individual components of the noise or
vibration propagation from the power-train or suspension into the car compartment. Another
area where TPA is used is in evaluation of home appliances [3].
Figure 2.1 Schematic representation of the TPA. Excitation propagates from the source via the coupling elements
to the structure. Vibration or acoustic response is measured on the structure or at a different points of interest.
Traditional TPA is based on the linear superposition principal, which is valid for linear, timeinvariant systems. The individual contribution of one path can be estimated in the frequency
domain as
3
x m (ω ) = H mn (ω ) ⋅ Fn (ω )
(2.1)
is response due to one path at a point of interest
where xm
Hmn is the frequency response function of the path
is the force spectrum
Fn
The total response at a point of interest, where response can be acceleration, displacement, sound
pressure level, or another quantity, can be estimated as the sum of the individual contributions.
N
N
n =1
n =1
X m (ω ) = ∑ xm (ω ) = ∑ H mn (ω ) ⋅ Fn (ω )
(2.2)
The schematic of the TPA can be seen in Figure 2.1. It is important to note that each junction
element consists of three DOFs, if only the translation is used, and six DOFs, if rotations are
included in the measurements. Each DOF constitutes a separate path and complex phasors are
summed at each frequency to get the total response. It is convenient to use a vector diagram to
visualize the contribution of the individual components and what effect they have on the total
response. The example of the TPA calculation can be seen in Figure 2.2.
4
Figure 2.2 An example of TPA calculation. Propagation of vibration from the electric motor into car compartment
via three coupling elements (9 paths). Vector diagram at 1375 Hz [4].
There are two methods that are usually used for the indirect estimation of the operating forces
acting on the structure. The first method is used, when there are elastic mounts between the
source and the structure. Operational displacements are measured on both sides of the mount,
and dynamic stiffness is estimated either experimentally, analytically, or using FE software.
Fn (ω ) = K n (ω ) ⋅ (u n1 (ω ) − u n 2 (ω ))
where Fn
(2.3)
is the force spectrum
Kn is the dynamic stiffness of the mount
un1 is the displacement on the source side
5
un2 is the displacement on the structure side
The second method is used when there is a rigid connection between the source and the structure,
or the mounts are too stiff in comparison to the structure’s stiffness and measuring relative
displacements is not feasible [5], [6]. The method involves an inversion of the FRF matrix
(Equation 2.1 in the matrix form), and using the operational response at the point of interest
⎧ F1 ⎫ ⎡ H 11
⎪F ⎪ ⎢H
⎪ 2 ⎪ ⎢ 21
⎨ ⎬=
⎪ M ⎪ ⎢ M
⎪⎩ FN ⎪⎭ ⎢⎣ H N 1
H 12 K H 1M ⎤
O
M ⎥⎥
O
M ⎥
⎥
L L H NM ⎦
−1
⎧ x1 ⎫
⎪x ⎪
⎪ 2⎪
⎨ ⎬
⎪ M ⎪
⎪⎩ x M ⎪⎭
(2.4)
The assumptions used in the inversion of the FRF matrix are linearity and reciprocity. In order to
reduce the noise and error during the inversion (force estimation) it is desirable to solve the
system in the least squared sense [3]. For this, the number of measurements of the response
vector should be larger than the force vector (M>N).
2.2 Methods of predicting propagation of energy in hydraulic systems
The goal of the project is application of the TPA to hydraulic systems. Even though, classical
TPA is a fairly well established technique, its application to hydraulic systems has not been
studied extensively. Before the method of the application of the TPA can be developed and used
experimentally, there is a need to perform the fundamental study of how the energy propagates
through the hydraulic systems. There are different methods which describe the dynamic behavior
of the fluid filled pipes and energy propagation through them.
The analytical method developed by Fuller and Fahy [7] describes the wave propagation and
energy distribution in cylindrical elastic shells filled with fluid. The method uses Donnell-
6
Mushtari shell equations to describe the dynamic behavior of the shell and fluid loading term to
describe the coupling of the shell with fluid. It can be used for prediction of dynamic behavior of
the elastic shells, but becomes impractical when the geometry of the piping system increases in
complexity.
Another applicable technique is the transfer matrix method (TMM) [8-10]. It can be used for the
description of the complex systems, with straight pipes, elbows, joints, valves, etc. Each
component is represented by a transfer matrix, which relates the excitation at one side of the
component to the response at the output
{Qo } = [Toi ]{Qi }
where Qo
(2.5)
is the state vector at the output
Toi is the transfer matrix
Qi is the state vector at the input
So for a simple system consisting of the multiple components, like in Figure 2.3, the total transfer
matrix can be calculated by multiplication of the transfer matrix of the components [8]
[T ] = [T ]⋅ [T
sys
pipe
43
elbow
32
]⋅ [T ]
pipe
21
(2.6)
Then using the boundary conditions the total equation of the system can be solved.
7
Figure 2.3 An example of the pipe system used for TMM
Finite element modeling can be used to characterize the behavior of the fluid filled piping
systems. The total system deformation is approximated as a combination of finite elements,
which are deformed according to predefined shape functions. The more detailed description of
the FEM can be found in the following chapter.
TMM and FEM each have their advantages and disadvantages. The advantage of the TMM is
that the size of the matrices does not increase with the increase of the physical dimensions of the
system. So if the complexity of the system remains the same, but the size of a certain segment,
such as pipe length, is increased, it is still represented by a single transfer matrix. However, the
number of elements, and thus computing time, for the FE method would increase. During the
implementation of the TMM numerical problems can arise, particularly at higher frequencies.
Also, the formulation of the transfer matrices including rotational matrices, models of joints,
valves, etc., requires significant background knowledge of the physical concepts and the software
for the method is not as widespread and easily obtainable as the FE software. The advantage of
the FEM is its flexibility in construction and definition of the systems, as well as the
8
representation of the output or the results of the modeling, which can be readily obtained in the
time or frequency domain. Since FEM was chosen for the analysis of the hydraulic systems in
this project, it is described more comprehensively and a more detailed description of TMM can
be found in references [8] and [11].
9
Chapter 3 Fluid-Structure Coupling
The FE modeling of the fluid-structure interaction (FSI) is based on two equations: the acoustic
wave equation and the structural dynamics equation. The acoustic wave equation will be derived
in the first part of the chapter; then it will be discretized into the finite element form and linked
with the structural dynamics equation. The following terminology and symbols will be used in
the chapter:
r
u , (ux, uy, uz) – particle velocity of an element
s = ( ρ- ρ0 )/ρ0 – condensation
k – bulk modulus of a fluid
ρ – instantaneous density
ρ0 – equilibrium density
P – instantaneous pressure
P0 – equilibrium pressure
p = P-P0 – acoustic pressure
c=
k / ρ 0 – thermodynamic speed of sound in the fluid
S – fluid-structure interface surface
{ } – vector notation
[ ] – matrix notation
n – unit normal to the interface surface
δp – virtual change in pressure
N – element shape function for pressure
N’ – element shape function for displacement
Pe – nodal pressure vector
10
Ue – nodal displacement vector
…F – fluid superscript
…S – structural superscript
…T – transpose
M – mass matrix
K – stiffness matrix
R – coupling matrix acting at the fluid structure interface
F – force vector
3.1 The acoustic wave equation
Assuming that the condensation is small, the acoustic pressure can be approximated by
p ≈ k ⋅s
(3.1)
Referring to Figure 3.1 it can be seen that the net influx of mass in an arbitrary small volume of
Figure 3.1 A small volume of fluid in space, with mass flow going into and out of the volume.
space resulting from the flow in the x direction is
∂ ( ρu x ) ⎞⎤
∂ ( ρu x )
⎡
⎛
⎢ ρu x − ⎜ ρu x + ∂x dx ⎟⎥ dy dz = − ∂x dV
⎝
⎠⎦
⎣
(3.2)
11
Summing the net influx from the y and z directions to get the total net influx
⎛ ∂ ( ρu x ) ∂ ( ρu y ) ∂ ( ρu z ) ⎞
r
⎟⎟dV = −∇ ⋅ ( ρu )dV
− ⎜⎜
+
+
∂y
∂z ⎠
⎝ ∂x
(3.3)
The rate of increase of the mass in the volume should be equal to the net influx of the fluid
r
∂ρ
+ ∇ ⋅ ( ρu ) = 0
∂t
(3.4)
r
In this exact continuity equation, instantaneous density (ρ) and particle velocity ( u ) are acoustic
variables, however acoustic density can be written as
ρ = ρ 0 (1 + s )
(3.5)
and if the equilibrium density is a weak function of time and s is very small, then Equation 3.4
becomes
ρ0
r
∂s
+ ∇ ⋅ ( ρ 0u ) = 0
∂t
(3.6)
Newton’s second law can be applied to the arbitrary small volume of the fluid in space
r r
f = a dm
(3.7)
The force acting on the volume in non-viscous fluid in x direction is
∂P
∂P ⎞⎤
⎡
⎛
df x = ⎢ P − ⎜ P +
dx ⎟⎥dy dz = −
dV
∂x
∂x ⎠⎦
⎝
⎣
(3.8)
Taking into account the force acting in y and z directions and vertical gravitational force acting
on the volume of fluid, the total force is
r
r
df = −∇P dV + gρ dV
(3.9)
The particle velocity is a function of both time and space, so the particle acceleration is
r
r
r
r
r ∂u
∂u
∂u
∂u
a=
+ ux
+ uy
+ uz
∂t
∂x
∂y
∂z
(3.10)
or written more concisely
12
r
r
r
r ∂u
a=
+ ( u ⋅ ∇ )u
∂t
(3.11)
Substituting Equations 3.9 and 3.11 into Equation 3.7 and taking into account that the mass of
the element is ρ dV
r
r
r
r⎞
⎛ ∂u
− ∇P + gρ = ρ ⎜ + (u ⋅ ∇)u ⎟
⎝ ∂t
⎠
(3.12)
If there is no acoustic excitation
r
∇P = ∇p + gρ 0
(3.13)
and substituting 3.13 and 3.5 into Equation 3.12
−
r
r
r
r⎞
⎛ ∂u
∇p + gs = (1 + s )⎜ + (u ⋅ ∇)u ⎟
ρ0
⎝ ∂t
⎠
1
(3.14)
Assuming the small amplitudes of the variables, which is valid for most acoustic processes the
r
r
r
r
following deductions can be made: gs << ∇p / ρ 0 , s << 1 , and (u ⋅ ∇)u << ∂u / ∂t , then
r
∂u
= −∇p
ρ0
∂t
(3.15)
which is linear Euler’s equation.
Equations 3.1, 3.6, and 3.15 can be combined together to yield the linear differential equation in
terms of the single variable. Taking the divergence of the Equation 3.15
r
⎛ ∂u ⎞
∇ ⋅ ⎜ ρ 0 ⎟ = −∇ 2 p
⎝ ∂t ⎠
(3.16)
Then taking the time derivative of the Equation 3.6 and noting that ρ 0 is a weak function of time
ρ0
r
∂2s
⎛ ∂u ⎞
+
∇
⋅
ρ
⎜
⎟=0
0
∂t 2
⎝ ∂t ⎠
(3.17)
Combining two previous equations and substituting condensation from Equation 3.1 and the
formula for the thermodynamic speed of sound in a fluid, results in the acoustic wave equation
13
1 ∂2 p
− ∇2 p = 0
2
2
c ∂t
(3.18)
This is linear, lossless wave equation for the propagation of sound in fluids with phase speed c
[12].
3.2 Fluid-structure interaction and its implementation in Finite Element analysis
In order to discretize the Equation 3.18, the following matrix operators for divergence and
gradient are used
⎡∂ ∂ ∂⎤
T
∇ ⋅ ( ) = {L} = ⎢
⎥
⎣ ∂x ∂y ∂z ⎦
(3.19)
∇( ) = {L}
(3.20)
For discretization of the wave equation the Galerkin procedure is used. At the fluid-structure
interface the relationship between the normal pressure gradient and the normal acceleration of the
structure is applied. Substituting Equations 3.19 and 3.20 into the wave equation, multiplying by
the virtual change in pressure and integrating over the volume of the domain [13] with some
manipulation results in
2
∂2 p
1
T
T ∂ {u}
{
}
{
}
{
}
+
=
−
δ
p
dV
(
L
δ
p
)(
L
p
)
dV
ρ
δ
p
n
dS
∫ c 2 ∂t 2
∫
∫S 0
∂t 2
V
V
(3.21)
The equations for the dependable variables P and u in terms of their shape functions and nodal
vectors are
p = {N } {Pe }
(3.22)
u = {N ′} {u e }
(3.23)
T
T
and denoting the application of the gradient in matrix form to a shape function as
[B ] = {L}{N }T
(3.24)
14
The terms in Equation 3.21 that do not change over the element can be taken out of the
integration sign. Since the virtual change of pressure was assumed as an arbitrary non-zero
variable, it can be factored out and substituting Equations 3.22 – 3.24 and applying described
manipulations to Equation 3.21.
{ }
1
{N }{N }T dV P&&e + ∫ [B ][B]T dV {Pe } + ρ 0 ∫ {N }{n}T {N ′}T dS {u&&e } = 0
c 2 V∫
V
S
(3.25)
or written in matrix notation
[M ]{P&& }+ [K ]{P }+ ρ [R ] {u&& } = 0
F
T
F
e
e
e
0
(3.26)
[M ] = c1 ∫ {N }{N } dV
T
F
where
2
V
[K ] = ∫ [B][B] dV
T
F
V
ρ 0 [R
]
T
= ρ 0 ∫ {N }{n} {N ′} dS
T
T
S
Fluid-structure interface can be incorporated in the dynamic equation of the structural part of the
system by adding the fluid pressure loading term.
[M ]{u&& }+ [K ]{u } = {F }+ {F }
S
S
e
S
pres
e
(3.27)
The fluid pressure load vector acting at the interface area S can be calculated by integrating the
pressure over the area. Applying the finite element discretization the expression for the loading
vector is
{F } = ∫ {N ′}{N } {n}dS {P }
T
pres
e
(3.28)
S
Examining the above expression, it can be seen that the load vector can be represented as
{F } = [R]{P }
pres
e
(3.29)
and substituting the acquired fluid loading term into Equation 3.27.
15
[M ]{u&& }+ [K ]{u }− [R]{P } = {F }
S
S
e
S
e
(3.30)
e
The combination of Equations 3.26 and 3.30 results in a complete description of the discretized
finite element equations of the fluid-structure interaction [14]
[ ]
⎡ MS
⎢
T
⎣ ρ 0 [R ]
[0] ⎤ ⎧{u&&e }⎫ ⎡[K ]S
[M F ]⎥⎦ ⎨⎩{P&&e }⎬⎭ + ⎢⎣ [0]
{ }
{ }
− [R ]⎤ ⎧{u e }⎫ ⎧ F S ⎫
⎬=⎨
⎬
⎥⎨
K F ⎦ ⎩{Pe }⎭ ⎩ F F ⎭
[ ]
(3.31)
Equation 3.31 does not include the dissipation due to damping at the boundary. It can be added
into the system by means of the symmetric damping matrix, however since damping at the
interface of the fluid and structure was not used during the simulation in the current project it will
not be discussed here.
16
Chapter 4 Experimental and Finite Element Models
This project encompassed both, the construction of the experimental test bench and the use of the
Finite Element for simulation. The need to use the experimental model was dictated by the fact
that TPA is an experimental technique, so the procedure, the measurements of the necessary
quantities (transfer functions, forces, accelerations, etc.), the selection of sensors should be tested
on a real structure. Also, it is still not possible to rely completely on simulation software and use
tabulated material properties during the FE model construction. Model correlation should be
performed, where the FE results should correspond well (according to the user requirements)
with the experimental model.
The use of the FEM in the project had three main functions:
- approximate model calculations before the experimental test bench construction
- study of the parameters which can be used for the TPA
- to study the effects of the change of the physical parameters, such as dimensions and
materials
Initially, the FE calculations were used to see if the experimental model would exhibit similar
frequencies of the natural modes as can be found in an automotive structure and to approximate
the computing times of the simulation. After the experimental model was constructed and the
correlation with the FE model was performed, it was possible to use the simulation for the TPA:
different parameters, such as different transfer functions, could be acquired and used in the TPA
calculations. The last reason for implementation of the FEM is the relatively easy ability to
change minor physical parameters and see what effect this would have on the overall dynamic
behavior of the system.
17
4.1 Experimental system description
The propagation of vibration and/or noise through the pipes can be found in systems of vastly
varying dimensions and configurations. Also, other parameters such as the type of fluid in the
system, velocity of the fluid in pipes, frequency range of interest, and etc., will have an effect on
choosing the method of the analysis and thus the final results of the study. This project is dealing
with automotive components and at the initial stage particularly with components related to
braking systems, so the experimental set up should be representative of the braking system of a
typical automobile.
There is not a lot of literature dealing with the topic of energy propagation in the small hydraulic
tubes, so the research had to be started with a simple fundamental model. The picture of the
experimental set up can be seen in Figure 4.1.
Pressure sensor 1
Accelerometer
Pressure sensor 2
Figure 4.1 Experimental set up.
18
The set up consists of a 1 m long steel brake pipe with a 90o bend approximately at the 70 cm
location. The drawing of the set up with dimension is shown in Figure 4.2. Table B.1 contains
information with the exact dimensions of components of the experimental set up. The pipe is
connected via two steel pillow blocks to the steel sheet, representing the body of an automobile.
Each pillow block consists of two parts held together with two bolts passing through the top part
and is attached to the plate with one bolt which screws into the bottom part (Figure A.3). The
diameter of the pillow blocks’ hole is slightly smaller than the pipe’s diameter, so that rigid
attachment is achieved without the use of glue or sealing materials. Such construction of the
attachment was intended for simplification of the simulation model.
The initial test bench consisted of one pipe and was intended for studying how the energy
propagates through the coupled hydro-mechanic system. However, the TPA is usually intended
for ranking the
Figure 4.2 Schematic drawing of the pipe and sheet of experimental set up with dimensions (cm).
19
propagation through multiple paths, so the experimental model could be updated by the addition
of one or more pipes. The pipes would originate from a source and be attached to the same sheet
(car body). However, due to the time constraints, only one pipe experimental set up was used in
this project.
Another important part of the test bench is the source of the vibration; a hydraulic exciter was
designed and constructed for this reason, a picture of which can be seen in Figure 4.3.
Membranes
Pressure sensor mounting hole
Pipe outlet
Figure 4.3 Hydraulic exciter in the disassembled form.
One of the requirements for the exciter was that it had to behave as a rigid body in the frequency
range of interest (0 to 3-4 kHz). The reason for this is that hydraulic units of an ABS or ESP can
be simply represented as a “black box” which behaves as a rigid body and has outputs for brake
pipes. Also, if the need for modeling of the exciter arises, it can be modeled simply as a
rectangular body of certain dimensions behaving as a rigid body. The pump is constructed from
steel and brass blocks which are held together with bolts. The parts of the pump behave as rigid
bodies in the frequency range of interest.
20
Another goal in the design of the exciter was to minimize the mechanical excitations that it
would introduce into the system. This would allow to study how the “pure” hydraulic energy
propagates without the initial coupling of the structural excitation. If there would be a need to
have either structural or both types of the excitation at once, a mechanical shaker could be
connected to the hydraulic unit. The pressure pulses are provided by two oppositely located
piezo-stacks, which push on the thin copper membranes.
In order to provide static pressure in the system, there was an opening made in the back plate of
the hydraulic unit. A short pipe segment with a valve was attached, through which the system
could be pressurized.
It was important to measure the pressure in the system at the outlet of the pump, so a hole above
the outlet was drilled which allowed the installation of the pressure transducer. Another sensor
was installed at the end of the pipe. For the measurement of the mechanical excitation induced
by the energy propagated from the hydraulic unit, an accelerometer was attached in an arbitrarily
chosen location on the sheet.
The electrical diagram of the system is shown in Figure 4.4. Signal source was provided by the
PC,
21
Figure 4.4 Electrical circuit of the experimental set up connected to the data acquisition system.
was amplified by an amplifier and sent to the piezo-actuators. Voltages across the system and
across the resistor were monitored so as not to exceed the voltage limit of the piezo-actuators and
stay under the 800 Watts limit of the power amplifier.
4.2 FE model description and construction
The simulation part of the project was performed using ANSYS 7.1. The FE model should
correlate well with the experimental model and should be able to be used for simulation.
However, creation of a good model does not necessarily mean copying every geometric feature
of the experimental model and making very fine resolution, because it will significantly raise
computational time and time spent on the development and modification of the model. The goal
is to create as simple a model as possible, which at the same time will give satisfactory results.
Three types of elements were used in the creation of the model. For the simulation of fluid
FLUID30 element was used. It is an 8-noded brick element used for the description of 3-D
acoustic fluid; it has an option of fluid-structure interaction. Fluid in the pipe was modeled as an
acoustic fluid, and on the boundary a solid fluid-structure interface was applied.
22
The pipe and joints were modeled with SOLID45 elements. It is an 8-noded brick element used
for the description of 3-D solids. Often fluid pipes are modeled as shell elements, however in
those cases the ratio of the pipe’s wall thickness to diameter is very small, which is not the case
for these hydraulic pipes with the ratio of 0.12-0.15 (depending on the pipe).
The thickness of the sheet is very small compared to the other dimensions, so it was modeled
with SHELL181 elements. It is a 4-node finite strain element. Because an accelerometer, used
for measurements on the sheet, provides significant mass loading, it was also incorporated in the
sheet model as a brick element.
The test structure was divided into substructures with simple geometric forms: pipe segments,
elbow, sheet, mounts. Each substructure was constructed mostly by using the “Top Down”
method: primitives were used for the creation of parts. Since the parts had to be joined together,
mapped meshing was performed. After the part was created, its geometry and load database
items were written to a file. In the final stage the assembly of the structure was performed by
reading in the saved geometry and load files and arranging them in the required pattern. This
approach permitted quickly changing material and geometric parameters and generating the new
model.
On the ends of the pipe there were couplings, which were not modeled for the simplicity of the
model. Also, on the termination end of the pipe, there was a pressure sensor mounted with a
coupling element. It was modeled as a solid segment of the pipe with a length and weight equal
to the length and weight of the coupling-pressure transducer assembly. Joints between the pipe
and the sheet were modeled as a single solid structure connected rigidly to both structures.
23
Modeling of the pump was not performed since the pump was assumed to behave as a rigid body
and experiments were performed with the pump rigidly clamped to a large mass. The boundary
conditions for the structure were modeled as a rigid connection at the pipe-pump interface and
free-free for the other parts of the structure.
Figure 4.5 Finite element assembly of the test structure.
Picture of the FE assembly of the test structure can be seen in Figure 4.5. The exact materials and
their parameters for the experimental model were unknown, so parameters for general materials
such as steel and hydraulic fluid were used. The initial parameters that were chosen for material
properties are shown in Table 4.1.
24
Modulus of
Poisson Ratio
Density (kg/m3)
Speed of Sound
Elasticity (N/m2)
(m/s)
Steel
2.06e11
0.33
7800
5000
Hydraulic Fluid
-
-
1100
1400
Table 4.1 Table of the initial material parameters used in the Finite Element model.
4.3 Model correlation
In order to make sure that FE model adequately describes the real system, model correlation has
to be performed. Modal analysis of the experimental system should be compared to simulated
modal analysis of the FE model. However, when performing correlation on the full model, there
are a lot of parameters that can influence the final results, and since the exact material properties
of the structure were unknown, it made sense to start with a simpler system.
The first system that was tested was a 30 cm section of an empty brake pipe. An experimental
modal analysis was performed with free-free boundary conditions (pipe was supported with weak
rubber strings). Set up of the test is shown in Figure 4.6. Since the structure is very light, and
the use of accelerometers would introduce significant mass loading effect, a scanning laser
vibrometer has been used for velocity measurements.
Figure 4.6 Experimental set up of the modal analysis performed on a 30 cm section of the brake pipe.
25
An impact hammer was used to provide the excitation. 8 modes were identified in the frequency
range of interest: 4 different bending modes, with each mode having a repeated root.
Material parameters and resolution were correlated in the test. Initial calculations were
performed on a structure with large mesh size. As mesh resolution was increased, the frequency
of the specific mode approached a constant value. An example of increasing mesh resolution and
convergence of the first bending mode can be seen in Figure 4.7.
Convergence of the FE model
385
frequency (hz)
380
375
370
365
experimental frequency
simulated frequency
360
1
2
3
4
5
6
case
Figure 4.7 An example of the convergence of the first bending mode of 30 cm pipe segment with an increase in
mesh resolution.
Once the mesh resolution was adjusted and model convergence was achieved, the material
parameters could be modified to achieve better correlation.
26
The influence of hydraulic oil on the structure needed to be studied. However, first in order to
encase the oil in the pipe, two thin (1 mm) aluminum disks were glued to the ends of the pipe.
Modal analysis on the structure was performed and the parameters were adjusted. Then the pipe
was filled with oil and correlation was performed again.
The first three tests permitted choosing an appropriate mesh resolution and adjusting material
parameters. The only parameter that was not extracted from measurements was the speed of
sound in the fluid. Standing waves in the fluid were not strong enough to excite the deformation
in the structure and had to be measured separately. For this purpose another set up was used. A
1 meter long pipe was closed with an aluminum disk at one end, and a pressure sensor was
installed at the other end. The pipe was excited with an impact hammer axially and the
frequency response function between the force and pressure was measured. Experimental and FE
results for the first fluid mode occurring at 723 Hz can be seen in Figure 4.8.
27
Figure 4.8 Experimental and simulation results. Left side: P/F frequency response function, with first fluid mode
occurring at 722 Hz. Right side: representation of the first fluid mode, FE analysis.
The last step was the correlation of the complete experimental and FE models. The experimental
model was tested with an attached pump, which was rigidly clamped to the 15 kg mass. The
structure was supported at two upper corners of the sheet and at the elbow by thin rubber strings.
Pressure and acceleration transducer were mounted, and the structure was filled with oil at
atmospheric pressure. An excitation was performed with an impulse hammer at two locations
and output was measured with laser vibrometer. A diagram of the measuring set up can be seen
in Figure 4.9.
28
Figure 4.9 Experimental set up for modal analysis of the complete structure. Inputs, provided by the modal hammer
and output measurement grid are shown.
The FE modal analysis was performed on a structure shown in Figure 4.5. The results of the
correlation agree within 9 %. Further improvement in correlation could be made, however the
results were sufficient for the initial analysis. An example of mode shapes can be seen in Figure
4.10.
The results were compared in the lower end of the frequency range of interest. The structure is
very flexible, both the pipe and the sheet produce multitude of modes by themselves and in
combination. There are about 33 modes in the 0-500 Hz range. Modes at the higher frequencies
should have been compared; another possibility was to generate and compare driving point FRFs
from the experimental and FE structures. Due to the time constraints and some technical
difficulties, this was not done.
29
Figure 4.10 An example of the experimental and FE modal analysis correlation.
30
Chapter 5 Finite Element Results
5.1 Transient Simulation results
After the construction of experimental model initial tests were performed. During the test,
pressure readings on the termination end of the pipe did not show logical results. It was attributed
due to lack of static pressure in the system, and inability of the pump to provide enough
excitation energy. The transient simulation was performed in order to check what magnitude of
pressure the transducer should register.
A transient simulation should satisfy the Courant-Friedrichs-Lewy (CFL) condition, which states
that minimum transition time for the element should be larger than the chosen step size. It means
that time step chosen for the simulation should be less than the time required for the sound wave
to travel through the length of one element. The speed of sound is higher in the structural
elements than in fluid, so it should be used for calculation. With the speed of sound in steel
approximately equal to 5000 m/s and the smallest length of the element 3 mm, the time step
should be lower then 6e-7 s.
Figure 5.1 shows pressure excitation signal applied at the pump-pipe interface boundary and
pressure response measured at the other end of the pipe. It can be seen that there an
approximately 20 % drop in pressure. Damping or viscosity was not used in the model, so
pressure drop would be larger, but not significantly.
In order to use TPA, transfer functions for different paths should be measured. Force
measurements on the experimental structure are difficult, however strain measurements are
31
possible and can be used in transfer function estimation. The simulation was performed in order
to
Transient Pressure History
10000
excitaion signal
response at the end
pressure (Pa)
5000
0
-5000
0
0.5
1
1.5
time (sec)
2
2.5
3
-3
x 10
Figure 5.1 Pressure excitation and response signal for transient simulation.
32
-7
2.5
Transient Displacement History
x 10
x
y
z
2
displacement (m)
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
time (sec)
2
2.5
3
-3
x 10
Figure 5.2 Displacement of the point on the pipe, near the pump connection, caused by the application of pressure
impulse.
see what strain levels the system would experience, so that proper transducers could be chosen.
Figure 5.2 shows the displacement of an arbitrarily chosen point on the pipe near the connection
to the pump. It can be seen that deformation in axial (z) direction is significantly larger than in
the other two directions, and possibly only axial deformation needs to be measured in order to
find the transfer function of acceptable reliability. However, experimental tests should be
performed to check this statement.
33
Figure 5.3 Transient simulation; pressure pulse propagation through the system.
Figure 5.3 shows overall deformation of the system caused by the pressure pulse propagation. A
transient simulation can be used for visualization of the deformation, which gives a better
understanding of the behavior of the system. It can show how the location of the sheet
connectors with respect to the elbow would affect overall dynamics of the system, which can be
valuable for system modification.
The current FE model contains 20545 elements, so for transient analysis, it takes approximately
10-15 hours for 1e-3 sec of simulation. In order to convert the transient data into the frequency
domain for further analysis, at least 0.2 seconds are needed. It can be seen that transient
simulation can not be used for transfer function estimates with the model of current size and
provided computing power.
34
5.2 Harmonic Simulation results
A harmonic simulation allows acquisition of structure’s frequency response functions. It has a
significant calculation time advantage over the transient simulation; it takes approximately 15
hours for the analysis of the system with 500 lines.
Harmonic Simulation
-9
10
-10
10
-11
magnitude
10
-12
10
-13
10
-14
10
-15
10
-16
10
0
500
1000
1500
frequency(Hz)
2000
2500
Figure 5.4 Harmonic simulation; acceleration response on the sheet due to the pressure excitation provided by the
pump.
With one simulation, it is possible to calculate frequency response functions for one input and
output condition of the complete structure. The result should be correlated with experimental
frequency response functions, to make sure that the FE model can be used for further predictions.
This is particularly important for higher frequency applications, since effects of the viscosity
have not been included. Damping caused by viscous fluid should play a greater role at the
propagation of waves at higher frequencies.
35
Chapter 6 Finite Element Modifications / Results
One of the main reasons for using FE model was the ability to find suitable transfer functions for
transfer path analysis. The type of function it should be: acceleration/strain, acceleration/force,
acceleration/pressure, etc. How and where (location wise) these quantities should be measured.
This goal required the large number of simulations, which was not feasible with full scale model
and available computing power. Some modifications should have been made, which would
allow reaching the objective.
The solution for the problem was to create a smaller physical model, with the smaller number of
elements, which would be used not for the correlation purposes, but just to study the phenomenal
effects of the system. Once the necessary parameters would be found, testing could be
performed on a full scale model and in the experiment.
6.1 Small model description
A smaller model with the same shape of geometry was created. All geometrical parameters were
reduced, but not in a uniform order. The length of the original pipe was approximately 1 m; it
was reduced to 25 cm. The pipe’s diameter was reduced by 1 mm, leaving the same wall
thickness. The dimensions of the sheet were reduced by 3 times. The model was not scaled
exactly, because this would make some parameters, such as pipe’s wall thickness, very small,
which would result in poor meshing conditions.
Even though the new model did not describe the experimental structure, it would have been good
for it to have a similar mode density and behavior. The material parameters were arbitrarily
36
adjusted, so that the modes of the small and experimental models would be in the same range.
The new model does not represent any physical structure, but the idea was that if TPA
parameters could be found and successfully applied to this “imaginary” structure, the same
procedure can be used for the full scale and experimental model.
The hydraulic pump was added in the small model, so that a free-free boundary conditions could
be checked. The new model has 3405 elements, where the full scale model has 20545 elements.
Also the model with two pipes was created, which has 5609 elements (Figure 6.1).
Figure 6.1 Schematic of the small FE model; one and two pipe configurations.
37
It can be seen in Figure 6.2 that the small model shows similar mode behavior, however the
model is stiffer, in particular the pipe segment adds stiffness to the sheet.
Figure 6.2 Mode comparison for small and full scale models.
The fluid structure interaction creates unsymmetrical matrices, because of that reciprocity
between the structural and fluid elements cannot be checked in the FE calculation. However, it is
not necessary in the FEM, while direct relationships can be calculated and the principles can be
tested. Then the procedure can be applied to experimental structure, where it is easier to acquire
the parameters using reciprocity.
6.2 TPA using pressure – force transfer functions
Two transfer functions can be measured: one caused by force acting on a structure
H1 ( f ) =
a
F
(6.1)
and another caused by pressure pulsations
H2( f ) =
a
P
(6.2)
38
During the operation of the pump, both the force and the pressure should be measured and using
the following equation
H1 × Pmeasured + H 2 × Fmeasured = a
(6.3)
the acceleration of the structure can be estimated. Applying the same technique to every pipe, it
is possible to rank contribution of each pipe to the overall vibration of the structure.
Vibration Propagation (excitation case 1)
-1
10
calculated
measured
-2
10
-3
magnitude
10
-4
10
-5
10
-6
10
-7
10
0
200
400
600
frequency (Hz)
800
1000
1200
Figure 6.3 TPA estimation for one pipe system using force and pressure transfer functions; excitation case 1.
Figure 6.3 and 6.4 show examples of a TPA calculation with this method for two different
excitation levels. The calculated curve is obtained by using Equations 6.1 - 6.3 and the measured
curve is obtained by directly getting the measurement of acceleration during the simulation of the
operational conditions.
39
The procedure is very simple to perform in simulation and it gives good results, however
transferring the same technique to experimental structure is not so easy. The operational pressure
can be measured directly with a pressure transducer, but there are no good transducers that would
measure the force applied to the structure and it has to be estimated in another way.
Vibration Propagation (excitation case 2)
-3
10
calculated
measured
-4
10
-5
magnitude
10
-6
10
-7
10
-8
10
-9
10
0
200
400
600
frequency (Hz)
800
1000
1200
Figure 6.4 TPA estimation for one pipe system using force and pressure transfer functions; excitation case 2.
The applied force can be estimated indirectly using reciprocity. A schematic of this principle is
shown in Figure 6.5. Force is applied in the spots marked with red dots, and response is
measured near the pipe. With these measurements, FRF matrix can be created.
⎡ H Ax , F 1x
⎢H
⎢ Ay , F 1x
H =⎢ .
⎢ .
⎢
⎣ H Az , Fnx
H Ax , F 1 y
H Ax , F 1z
H Ay , F 1 y
.
.
.
.
.
.
H Az , Fnz
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(6.4)
40
The measurement of a set of the FRF functions for one point would be sufficient, but using more
points gives a better estimation. The least squares approximation can be used to find the force by
measuring the operational accelerations and inverse of the FRF matrix
{F } = [H ]pinv × {a}
(6.5)
Figure 6.5 Schematic of indirect force estimation.
An estimation has been performed using two points on the pump and one operational simulation.
13 N operational force was applied in the z direction; forces in other directions were not applied.
Pressure was applied on the boundary at pipe-pump interface. Figure 6.6-6.8 show the indirect
force estimation in the directions x, y, and z respectively. It can be seen that the method does not
give good (any) predictions for a simple constant force spectrum, so its performance for
estimating real application force is doubtful, at least in the form that it has been used.
41
Indirect Force Estimation (x-direction)
300
250
200
force (N)
150
100
50
0
-50
-100
0
50
100
150
200
250
300
frequency (Hz)
350
400
450
500
Figure 6.6 Indirect force estimation; there was a 0 N force applied in x-direction.
Indirect Force Estimation (y-direction)
30
20
10
force (N)
0
-10
-20
-30
-40
-50
-60
0
50
100
150
200
250
300
frequency (Hz)
350
400
450
500
Figure 6.7 Indirect force estimation; there was a 0 N force applied in y-direction.
42
Indirect Force Estimation (z-direction)
700
force (N)
25
600
20
15
10
5
force (N)
500
50
100
150
200
250
frequency (Hz)
300
350
400
400
300
200
100
0
0
50
100
150
200
250
300
frequency (Hz)
350
400
450
500
Figure 6.8 Indirect force estimation; there was a 13 N force applied in z-direction.
6.3 TPA using strain transfer functions
Measuring operational force is not very easy, so instead of measuring the force, strain on the pipe
close to the pump can be measured. If the force and the pressure excitation are applied to the
structure, it is possible to measure strain and acceleration caused by them. The transfer function
for the acceleration due to the strain can be calculated.
⎛a a ⎞ ⎛ε ε ⎞ a
H ( f ) = ⎜ + ⎟ /⎜ + ⎟ =
⎝P F⎠ ⎝P F⎠ ε
(6.6)
Then measuring the operational strain, acceleration on the structure can be predicted.
H × ε measured = a
(6.7)
The results for two excitation cases for the one pipe structure can be seen in Figure 6.9 and 6.10.
The calculated curve was obtained by using Equation 6.7, and the measured curved is the
acceleration directly obtained from the FE simulation.
43
Vibration Propagation (excitation case 1)
-3
10
calculated
measured
-4
10
-5
magnitude
10
-6
10
-7
10
-8
10
-9
10
0
200
400
600
frequency (Hz)
800
1000
1200
Figure 6.9 TPA estimation for one pipe system using strain transfer function; excitation case 1.
Vibration Propagation (excitation case 2)
-1
10
calculated
measured
-2
10
-3
magnitude
10
-4
10
-5
10
-6
10
-7
10
0
200
400
600
frequency (Hz)
800
1000
1200
Figure 6.10 TPA estimation for one pipe system using strain transfer function; excitation case 2.
44
The same technique was applied to system with two pipes. The structure was excited by the
applied force and pressure at both pipes and the transfer function due to the strain in the first pipe
was calculated.
⎛a a
ε ε
ε ⎞ a
a
a ⎞ ⎛ε
H 1 ( f ) = ⎜⎜ + + + ⎟⎟ / ⎜⎜ 1 + 1 + 1 + 1 ⎟⎟ =
⎝ P1 F1 P2 F2 ⎠ ⎝ P1 F1 P2 F2 ⎠ ε 1
(6.8)
Similarly, the transfer function for the pipe two or for any number of pipes can be calculated.
Then the total response of the structure due to the different paths is
H 1 × ε operational 1 = a path1 ⎫
⎪
+ ⎪
⎪
H 2 × ε operational 2 = a path 2 ⎬ atotal
⎪
+ ⎪
M
H n × ε operational n = a path n ⎪⎭
(6.9)
Figure 6.11 shows the differences between the acceleration calculated with Equation 6.9 and
measured directly from the simulation.
Two Pipe System TPA (using Strain Transfer Functions (y))
-3
10
calculated
measured
-4
10
-5
magnitude
10
-6
10
-7
10
-8
10
0
50
100
150
200
250
frequency
300
350
400
450
500
Figure 6.11 TPA estimation for two pipe system using strain transfer functions.
45
The individual contributions of each path to the total measured response are shown in Figure
6.12. Path 1 was excited with 10 N force and 1x105 Pa pressure; path 2 was excited with 21 N
force and 2x104 Pa pressure.
Two Pipe System TPA
-3
10
-4
magnitude
10
-5
10
-6
10
-7
10
-8
10
0
50
100
150
200
250
300
350
400
450
500
path 1
path 2
total response
phase angle (rad)
4
2
0
-2
-4
0
50
100
150
200
250
300
frequency
350
400
450
500
Figure 6.12 TPA calculation on a two pipe system; contribution of the paths to the total response.
46
Chapter 7 Experimental Model Results
After the construction of the experimental model, there were still many parameters which could
influence the outcome of the results. What kind of transducers should be used for measuring
pressure? What testing conditions should be applied and implemented, and what should be
measured in the experiment that later can be applied to the TPA performed on a real life
structures? The answers to the questions could be partially estimated from the FE results;
however, most of them had to be acquired by testing.
7.1 Selection of transducers
Before the measurements could be started, it was necessary to choose proper pressure
transducers. Their selection depended on the type of test that had to be performed. One of the
requirements was ability to measure static pressure. There was an analog pressure meter
available (Figure 7.1), which was mounted on the pump used for the application of static
pressure.
Figure 7.1 Pump, used for application of the static pressure, and analog pressure meter.
47
The range of the pressure meter was 0 to 6 bars, which was sufficient for initial measurements.
However, automotive braking systems can apply pressures over 100 bars, and the behavior of the
system should be studied in the comparable range. Also, in this set up, the sensor was located
near the static pump. During the measurements the valve near the pump (Figure 4.1) was closed
and/or pipe leading to it was disconnected, so it was not possible to monitor pressure. In case of
a leakage in the system, the pressure data would not be accurate.
For measurement of the dynamic pressure in the system, there were available piezo-electric
(quartz) sensors with a measurement range of 0 to 250 bars and sensitivity of -16 pC/bar. These
sensors generally performed well, however such large dynamic range of the sensors was not
needed and higher sensitivity would be preferable. There is similar type of sensor which has five
times higher sensitivity, application of which would possibly improve the quality of acquired
data.
Piezo-resistive sensors provide the capability of measuring both static and dynamic pressure at
the same time and have high sensing range. This would give an ability to monitor the static
pressure in the system during the test and apply the static pressure comparable to automotive
systems’ magnitude. However, during the testing, selected pressure sensor had insufficient
sensitivity and did not register the applied signal. It was well suited for measuring static and
dynamic pressure of large magnitudes, but did not perform well for the dynamic measurements
of low magnitude. Since the excitation input provided by the hydraulic shaker was not very
large, it was not possible to measure it with this sensor.
48
It was necessary to measure the pressure at pump’s output, so piezo-electric sensor was used at
that location. Measurement of the dynamic pressure at pipe’s termination was not needed for the
experiments performed, so piezo-resistive sensor was used.
7.2 Testing conditions
Initial testing had to be done with rigidly clamped hydraulic pump, because it would significantly
simplify correlation with FE model. The pump did not have to be modeled and it was assumed
that pipe at pump’s interface was rigidly constrained. Another reason for clamping the pump was
application of force inputs into the structure. Even though pump was designed with the goal of
minimizing mechanical inputs into the system, there still would be some unbalances and
structural forces would be introduced. By rigidly constraining the pump, structural forces input
into the system are minimized and excitation comes mostly from hydraulic input.
Three cases of different boundary conditions were tested:
- rigidly constrained pump
- free-free pump resting on a foam with pipe for static pressure application connected to
the system
- free-free pump with static pressure pipe disconnected at the valve location
Measurements were done to quantify the effect of these boundary conditions on a pressure and an
acceleration responses of the system. Figure 7.2 and 7.3 show the frequency response functions
of the pressure and the acceleration due to voltage inputs for varying boundary conditions.
49
Pressure/excitation FRF comparison with different BCs
4
10
3
10
magnitude
2
10
1
10
0
10
pump rigidly clamped
pump F-F, pipe coulpled
pump F-F, pipe decoupled
-1
10
0
500
1000
1500
2000
2500
frequency (Hz)
3000
3500
4000
4500
Figure 7.2 Effect of varying boundary conditions on pressure response of the system.
Acceleration/excitation FRF comparison with different BCs
2
10
1
10
magnitude
0
10
-1
10
-2
10
pump rigidly clamped
pump F-F, pipe coulpled
pump F-F, pipe decoupled
-3
10
0
500
1000
1500
2000
2500
frequenca (Hz)
3000
3500
4000
4500
Figure 7.3 Effect of varying boundary conditions on acceleration response of the structure.
50
The influence of the varying static pressure in the system was investigated. Again, the frequency
response functions for the pressure (Figure 7.4) and the acceleration (Figure 7.5) of the system
due to the voltage inputs were acquired.
Separate FRFs for acceleration and pressure were acquired instead of using acceleration due to
pressure FRFs, so that it could be seen what and how quantities change with varying test
conditions. It can be seen from the plots that there is significant variation in pressure and
acceleration response due to eliminating rigid coupling of the hydraulic pump. Mostly there is
shift in amplitude of the response; however there is some frequency shift as well. Eliminating
the coupling between the static pressure pipe and pump does not have a significant effect on the
system. The only noticeable deviation between the curves occurs at 1300-1400 Hz range for both
pressure and acceleration figures.
The variation in static pressure has small effect on the change in magnitude of the dynamic
pressure response. However, there is significant shift in frequencies, which can be especially
noticeable for large pressure differences as it can be seen in Figure 7.4. There are slight changes
in acceleration response due to the varying static pressure, but they would be insignificant if the
acceleration/pressure frequency response function would be calculated.
51
Pressure/excitation FRF comparison with varying static pressure
4
10
3
10
magnitude
2
10
1
10
0
10
60 bar
100 bar
110 bar
-1
10
0
500
1000
1500
2000
2500
frequency (Hz)
3000
3500
4000
4500
Figure 7.4 Effect of varying static pressure on dynamic pressure response of the system.
Acceleration/excitation FRF comparison with varying static pressure
2
10
1
10
magnitude
0
10
-1
10
-2
10
60 bar
100 bar
110 bar
-3
10
0
500
1000
1500
2000
2500
frequency (Hz)
3000
3500
4000
4500
Figure 7.5 Effect of the varying static pressure on the acceleration response of the system.
52
7.3 Linearity of the hydraulic system
The linearity of the system was checked by applying an excitation signal of the varying
magnitudes. Cases for the different boundary conditions and static pressure levels were tested,
but they all gave similar result. Figure 7.6 shows the frequency response function for pressure
response due to the excitation voltage. Three arbitrarily chosen levels of the magnitude, with
approximate ratio of 1:2:3 were used. Figure 7.6 shows change in the magnitude of the FRF
curves as well as the shift in frequency of some peaks. Frequency shifts are particularly
noticeable at the fluid resonance modes, like for the second dominant peak around 700 Hz.
Pressure/Excitation FRF for different excitation levels, rigid pump BC
4
10
3
10
magnitude
2
10
1
10
high
0
10
intermidiate
low
-1
10
0
500
1000
1500
2000
2500
frequency (Hz)
3000
3500
4000
4500
Figure 7.6 Check for the linearity in the system; pressure response.
Figure 7.7 shows the frequency response function of acceleration due to the excitation voltage.
Levels of two magnitudes were tested. Comparable to the pressure response FRF, there is also
shift in magnitude of the curves, however there are no frequency shifts like in the previous case.
53
Acceleration/Excitation FRF for varying exciation levels
2
10
high
low
1
10
magnitude
0
10
-1
10
-2
10
-3
10
0
500
1000
1500
2000
2500
frequency (Hz)
3000
3500
4000
4500
Figure 7.7 Check for the linearity in the system; acceleration response.
7.4 Application of reciprocity in hydro-mechanical systems
In transfer path analysis transfer functions of the paths, through which noise propagates, are
measured. Often the only way of measuring the transfer functions requires the application of the
reciprocity principal. For a mechano-acoustical system the following equation can applied [15]
⎛ p 2′ ⎞
⎛ v′′ ⎞
⎟⎟
⎜⎜
= −⎜⎜ x1 ⎟⎟
all forces and
all forces and
⎝ U 2′′ ⎠ moments
⎝ Fx′1 ⎠ moments
at position 1
at position 1
zero , except Fx′1 ; U 2′ =0
where p2′
(7.1)
zero
is sound pressure, direct experiment
Fx′1
is force in direction x at position 1, direct experiment
v′x′1
is translational velocity in direction x at position 1, reciprocal experiment
U 2′′
is volume velocity, reciprocal experiment
54
Equation 7.1 can be applied to the experimental system in this project, as it is shown in Figure
7.8, but there are some changes that need to be applied. In this case application of force near the
accelerometer is not direct, but rather reciprocal experiment. A direct experiment would be the
p2′
Fx′1
v′x′1
U 2′′
Figure 7.8 Application of the reciprocity principal to experimental system.
application of a volumetric velocity, or in this case application of pressure (since it is a one
dimensional flow, volumetric velocity is proportional to the excitation pressure). Instead of
measuring the sound pressure level and velocity, there is measurement of the pressure and the
acceleration respectively. The acceleration can be easily transformed into the velocity by
integrating with respect to time. So the equation for this system looks like
H( f ) =
Ppump
Fexcitaion plate
∝
v plate
Pexcitaion pump
(7.2)
The first FRF in Equation 7.2 is acquired by exciting the structure at accelerometer location and
measuring the pressure response at the pump location. During the direct experiment operational
pressure is measured and velocity (acceleration) can be estimated by multiplying it with H(f).
Reciprocity is based on the linear network theory, so one of the necessary requirements for the
system is linearity, otherwise the approach will not work.
55
7.5 Hydraulic pump results / modifications
An experimental modal analysis has been performed on the hydraulic pump. The first mode,
torsional motion of the plates used for mounting pipe inlet and outlet around the center block,
occurred at 952 Hz. This frequency is significantly lower then was expected and required for the
first mode of the pump. There was a rubber gasket across the surface of the plates, so there was
no direct metal to metal contact between the parts. After the rubber gasket was substituted for a
machined groove and a small rubber ring just around the outlet and inlet, the frequency of the
first mode went up to 3036 Hz, which satisfied the design criteria. Figure 7.9 shows the first
mode of the original and modified systems.
Figure 7.9 Modal analysis of the hydraulic pump. First deformation mode for the original and modified design.
56
Chapter 8 Discussion, Modifications, and the Future Work
8.1 Discussion
Frequencies of the modes in the model correlation show up to 9 % differences in the simulation
and experiment in the 0 to 500 Hz range (correlation should be done for the higher frequencies).
The pipe used in the experiment has higher stiffness than the sheet, so if it is rigidly connected to
the sheet, there should be sheet stiffening in the area of the connection. This can be seen in FE
simulation; modes of the sheet are not symmetric, motion is constrained on the side of pipe
mounting. Same effect can be noticed in the experimental modal analysis of the structure,
however this effect is not nearly as pronounced as in the FE simulation. This means that the
mounting between the sheet and the pipe is not completely rigid, and should be adjusted in the
FE model. This difference is particularly noticeable at higher frequencies.
Simulation in the project was performed without the application of the static pressure, but most
of the experimental testing was done with some static pre-pressure. Figure 8.1 shows the
variance
57
Figure 8.1 Properties of the hydraulic fluid [16].
of the bulk modulus with pressure. In the testing range performed experimentally during the
project, bulk modulus changed in the range of 8 %, which results in less than 3 % change in the
speed of sound in the fluid. The static pressure should not have a large effect on the propagation
of speed in the structure. However, it can be seen from the measurements (Figure 7.4) that there
is 15 % shift of some peaks in the FRF acquired at different static pressures. Less than 2 % of
the shift can be attributed to the change in pressure according to the change in bulk modulus.
Figure 8.1 shows the data for a “perfect” fluid: without air bubbles, without water, and other
substances. However, in the experiment it was not possible to get rid of the bubbles in the fluid.
Also, during the refills of the test structure, each time there was a different amount of air mixed
into the fluid, and even the small amount of air bubbles in the fluid can cause a dramatic
reduction in the wave speed [17]. When the pressure was increased, some of the gas in the fluid
was compressed and dissolved, which caused large variation in the wave speed. Since in the
simulation none of these effects were included, the results between the experiment and FE model
do not correlate well.
58
The experimental structure shows large non-linearities (Figure 7.6), which can be also attributed
to the air bubbles in the fluid. Non-linear behavior of the structure means that the reciprocity can
not be applied, so the transfer function for pressure response due to force applied on the sheet can
not be measured. All of these factors hinder the application of TPA to the experimental structure.
The FE model did not incorporate the friction between the fluid and the structure in the interface.
Also, the viscosity of the fluid was not used nor checked in the experiments, which introduces
further discrepancies in the correlation of the simulated and experimental results.
The TPA performed on the small FE model showed good results when both the force and the
pressure transfer functions are measured and used in the analysis. However, the operational force
measurements are difficult to perform, and indirect force estimation gave poor results in the
modeling. Without the reliable method of accurately predicting the force transmitted into the
pipe, the TPA with the force and the pressure transfer functions can not be used. Using the
transfer functions with strain gives worse results than in the previous case, when performed in the
FE simulation; however they were still acceptable and could be used for the TPA. Application of
this technique to the experimental system can be more feasible than the force-pressure method.
8.2 Modifications / Future work
Modifications and further research should continue in both experimental and FE simulation.
When using the experimental model, attention should be paid to the fluid used in the experiment,
more importantly to the way the structure is filled with fluid. Since the pipes in the model are
narrow, it is difficult to fill them with the conventional method of just pouring the liquid in and
59
avoid mixture of the fluid with air. If the fluid would be in a closed loop, recirculation would be
possible, where the fluid would be pumped from a container with airless fluid.
More sensitive pressure sensors would give better results, particularly during the reciprocity
measurements. However, first the problem of air bubbles should be solved, because it has an
effect on the pressure sensors as well, and it may be that the sensors would perform better with
the airless fluid.
Experiments with the strain gages should be performed and analyzed. The simulation showed
that the strain can be used as a parameter in the application of the TPA, but there are some errors.
How much these errors will affect the result in the experiment needs to be checked. Also, other
experimental difficulties might arise, such as what strain should be used (torsion, tension, etc.)
and how it should be measured.
The indirect force estimation did not show good results in the simulation. Only forces were used
in approximation of structural inputs into the structure. However, moments also play a
significant role in the transmission of energy through the structure; particularly at higher
frequencies and when the source is close to structural discontinuities [18]. It is applicable to this
case when forces are estimated with the use of reciprocity. When the force is applied at the
corner of the pump (Figure 6.5) it generates a moment in the center point where the pipe is
mounted. For estimation of moments rotational degrees of freedom should be measured, or since
the pump behaves as a rigid body, rigid body dynamics can be used to calculate them from the
translational data.
60
There was a limited number of simulations run for the harmonic analysis on the full model as
well as on the small one pipe model. There were just two cases of the simulation of the two pipe
small model. More cases should be run for these set ups, with different excitation parameters and
boundary conditions. Also, once the basics are established, the results should be tested on more
complex systems with more than two pipes.
The viscosity of the liquid has not been taken into account during the modeling of the structure.
It can be seen in Figure 4.8 that the first fluid mode in the experiment is strongly excited,
however the higher harmonics of the standing fluid modes are not well distinguishable from the
experimental data. In the FE simulation higher harmonics of the fluid modes have a strong
participation. Damping can be added in ANSYS as a boundary absorption coefficient. Some
additional experiments should be done, where the decay of the fluid waves in the pipe would be
investigated and incorporated in the FE model.
Fine tuning of the FE model should provide good results and correlation with experimental data
for the simulation of the fluid-structure coupling up to a certain frequency, which is dependent on
the mesh size and accordingly to the CFL number for the time step. If the experimental model
can be modified, for example to exclude the joint mountings between the pipe and the sheet, or
use a different set up for them, then the minimum mesh size can increase and calculation time
can be improved by 2-3 times. However, FE simulation still would be time consuming,
particularly for larger systems. There are model reduction techniques suited to raising
computational efficiency; one of the reduced component models was developed at University of
Stuttgart (L.Gaul & M. Maess). It is a component mode synthesis with low number of DOFs.
61
The analysis procedure consists of two steps. First, model reduction by modal truncation up to
the user-specified limit frequency is performed. Then truncation of the modal states with low
contribution to the transfer function by use of the observability and controllability indices is
implemented.
In order to perform the model reduction, a fully coupled model with the fluid-structure
interaction is generated by the FE code. Then the model is imported into MatLab with system
matrices, element table, node table, and DOF definitions. The model reduction and the mode
synthesis are performed in MatLab with the Structural Dynamics Toolbox.
This procedure would not have significant effect on the system used in this project, because the
geometrical size of the system was not very large. The technique can be used efficiently for the
systems containing repeating element, or for the system design, when only one or a few elements
are changed, and overall system remains the same.
The Transfer Matrix Method should be also investigated in more detail to see if the application of
the method can be useful in the project. The optimal tool for a design engineer in the future
might be a combination of experimental and simulation techniques, where a system can be tested
on the design level previously to manufacturing.
62
Chapter 9 Conclusions
The propagation of vibrational energy in the fluid filled pipes was analyzed in the project. The
coupling of the structural and fluid domains was implemented and studied with the help of
experimental model and finite element simulation. The results of the study were applied toward
the application of transfer path analysis to hydraulic systems.
The finite element model of the experimental structure was created and correlation was
performed. The maximum difference in the frequency of the modes between the experimental
and FE models was 9 %. A small difference in the mode shapes was detected as well: pipe
connected to the sheet caused stiffening of the sheet in the FE model, where it did not produce
such noticeable effect in the experimental structure. This can be attributed to modeling the
connection of the joint with sheet and pipe as completely rigid, and not having a rigid connection
in the experimental structure.
The viscosity of the fluid has not been taken into account during the simulation. It should be
estimated from experimental data and introduced in terms of damping at the fluid-structure
interface. These changes should fine tune the model and bring the results into agreement with
the experimental tests.
A small FE model for studying application of the TPA to hydraulic systems has been created. It
allowed significant reduction in the calculation time and identification of the parameters suitable
for the implementation of the TPA on an experimental structure. The transfer function of the
acceleration due to the strain in the system showed a potential as one of the useful techniques in
TPA application. Another method is using two transfer functions: acceleration due to the applied
63
force and acceleration due to the applied pressure. This method gives better results than the
strain technique, however it is complicated by inability to easily measure the operational force.
The indirect force estimation had been simulated, but the results did not show successful
possibility of using this method. Experimental measurements should be performed to test both of
these techniques. For the indirect force estimation, calculation of moments applied to the system
might be crucial and possibly can improve the results.
The biggest difficulties in the experimental system arose from an inability to fill the set up with
pure airless fluid. Air bubbles in the system caused non-linearities and large frequency shifts due
to different static pre-pressures. This made the experimental model not useful for correlation of
the data, which had large dependence on the propagation of waves in the fluid.
There is no final answer given to the goal of the project – application of the TPA to hydraulic
systems, however research on the propagation of the energy in hydraulic pipes was performed
and direction in which the research toward the application of TPA should continue was found.
Further experimental work, simulations and new methods such as the model reduction should be
applied for the successful completion of the objective.
64
References
1. Wyckaert, K., Van der Auweraer, H., “Operational Analysis, Transfer Paths Analysis, Modal
Analysis: Tools to Understand Road Noise Problems in Cars”, Proc. SAE Noise and Vibr
Conf. 1995, pp 139-143.
2. Van der Auweraer, H., Wyckaert, K., Hendricx, W., Van der Linden, P., “Noise and
Vibration Transfer Path Analysis”, Technical Review, LMS International, 1995.
3. Van Karsen, C., Gwaltney, G., Blough, J., “Applying Transfer Path Analysis to Large Home
Appliances”, Shock and Vibration Digest, Vol.3, no 1, Jan 2000, pp 961-965.
4. Transfer Path Aanalysis software module, Robert Bosch GmbH.
5. Plunt, J., “Strategy for Transfer Path Analysis (TPA) Applied to Vibro-Acoustic Systems at
Medium and High Frequencies”, presented at ISMA 23, Leuven, Belgium
6. Yap, S., Gibbs, B., “Structure-Borne Sound Transmission from Machines in Buildings, Part
1: Indirect Measurement of Force at the Machine-Receiver Interface of a Single and MultiPoint Connected System by a Reciprocal Method”, Journal of Sound and Vibration 1999,
222(1), pp 85-98.
7. Fuller, C., Fahy, F., “Characteristics of Wave Propagation and Energy Distribution in
Cylindrical Elastic Shells Filled with Fluid”, Journal of Sound and Vibration 1982, 81(4), pp
501-518.
8. De Jong, C., “Analysis of Pulsations and Vibrations in Fluid-Filled Pipe Systems”, Ph.D.
dissertation, 1994, Eindhoven University of Technology, Eindhoven, the Netherlands.
9. Forbes, T., Tentarelli, S., “Dynamic Behavior of Complex Fluid-Filled Tubing Systems –
Part 1: Tubing Analysis”, Journal of Dynamic Systems, Measurement, and Control, March
2001, Vol. 123, pp 71-77.
10. Forbes, T., Tentarelli, S., “Dynamic Behavior of Complex Fluid-Filled Tubing Systems –
65
Part 2: System Analysis”, Journal of Dynamic Systems, Measurement, and Control, March
2001, Vol. 123, pp 78-84.
11. Tentarelli, S., 1989, “Propagation of Noise and Vibration in Complex Hydraulic Tubing
Systems”, Ph.D. dissertation, Lehigh University, Bethlehem, Pa.
12. Kinsler, L., Frey, A., Coppens, A., Sanders, J., “Fundamentals of Acoustics”, John Wiley &
Sons, Inc., 2000.
13. Zienkiewicz, O., Newton, R. E., “Coupled Vibrations of a Structure Submerged in a
Compressible Fluid”, Proceedings of the Symposium on Finite Element Techniques, June
1996, University of Stuttgart, Germany .
14. ANSYS, Inc. ANSYS Release 7.1 Documentation, 2003.
15. Wolde, T., “On the Validity and Application of Reciprocity in Acoustical, MechanoAcoustical and other Dynamical Systems”, Acustica, Vol. 28 (1973), pp 23-32.
16. Simulationsmodelle für Bremsflüssigkeiten, Robert Bosch GmbH, 2001.
17. Junger, M., Feit, D., “Sound, Structures, and Their Interaction”, MIT Press, Cambridge,
1986.
18. Yap, S., Gibbs, B., “Structure-Borne Sound Transmission from Machines in Buildings, Part
2: Indirect Measurement of Force and Moment at the Machine-Receiver Interface of a Single
Point Connected System by a Reciprocal Method”, Journal of Sound and Vibration 1999,
222(1), pp 99-113.
66
Appendix A: Additional Diagrams
Figure A.1 Hydraulic pump mounted on a mass. Rigid boundary conditions testing.
Figure A.2 First draft of the hydraulic pump. Some dimensions were modified in the final model.
67
Figure A.3 Steel joints. First draft, some dimensions were changed in final model.
68
Appendix B:
Pipe
Sheet
Joints
Data Tables
Length Thickness Width
Units - cm
97.7
30
0.1
30
2.2
1
2
Inner
Diameter
Outer
Diameter
0.46
0.6
0.59
Table B.1 Dimensions of the structure's components.
Modulus Poisson Density Speed
Ratio (kg/m3)
of
of
Sound
Elasticity
(m/s)
(N/m2)
Steel
2.15E+06
Hydraulic
Fluid
0.33
-
7800
1046
5000
1440
Table B.2 Final material parameters used in the simulation.
Modal Analysis results of a 30 cm brake pipe used for model correlation. Three cases: empty
pipe, closed empty pipe (with thin aluminum disks glued on the ends), and previous case filled
with hydraulic oil
1.Empty pipe
pipe30cm_m1 > 1
Frequency (hertz) : 383.6054 Damping (percent) : 0.521447
pipe30cm_m1 > 2
Frequency (hertz) : 384.0943 Damping (percent) : 0.827953
pipe30cm_m1 > 3
Frequency (hertz) : 1052.812 Damping (percent) : 0.234696
pipe30cm_m1 > 4
Frequency (hertz) : 1052.935 Damping (percent) : 0.260768
pipe30cm_m1 > 5
Frequency (hertz) : 2047.854 Damping (percent) : 0.147389
69
pipe30cm_m1 > 6
Frequency (hertz) : 2048.856 Damping (percent) : 0.141475
pipe30cm_m1 > 7
Frequency (hertz) : 3353.187 Damping (percent) : 0.098906
pipe30cm_m1 > 8
Frequency (hertz) : 3356.247 Damping (percent) : 0.084532
******************************************************************************
2. Pipe with end plates
endpla_m1 > 1
Frequency (hertz) : 371.3634
Damping (percent) : 0.533203
endpla_m1 > 2
Frequency (hertz) : 372.1432
Damping (percent) : 0.5995
endpla_m2 > 3
Frequency (hertz) : 1017.457
Damping (percent) : 0.227981
endpla_m2 > 4
Frequency (hertz) : 1019.524
Damping (percent) : 0.212655
endpla_m3 > 5
Frequency (hertz) : 1979.16
Damping (percent) : 0.126648
endpla_m3 > 6
Frequency (hertz) : 1983.086
Damping (percent) : 0.133028
endpla_m4 > 7
Frequency (hertz) : 3240.695
Damping (percent) : 0.080027
endpla_m4 > 8
Frequency (hertz) : 3248.573
Damping (percent) : 0.087008
******************************************************************************
70
3. Pipe with oil
pipe30_oil_m1 > 1
Frequency (hertz) : 342.6865 Damping (percent) : 0.53476
pipe30_oil_m1 > 2
Frequency (hertz) : 342.9354 Damping (percent) : 0.8428330
pipe30_oil_m2 > 3
Frequency (hertz) : 938.0328 Damping (percent) : 0.228703
pipe30_oil_m2 > 4
Frequency (hertz) : 939.787
pipe30_oil_m3 > 5
Frequency (hertz) : 1825.333 Damping (percent) : 0.119495
pipe30_oil_m3 > 6
Frequency (hertz) : 1829.018 Damping (percent) : 0.130254
pipe30_oil_m4 > 7
Frequency (hertz) : 2990.219 Damping (percent) : 0.075218
pipe30_oil_m4 > 8
Frequency (hertz) : 2996.453 Damping (percent) : 0.08555300
Damping (percent) : 0.238237
71
Appendix C:
Modeling code
!*********************************************************************
!*
Sheet model
*
!*********************************************************************
fini
/clear
!
/prep7
!
! ************************************************************
!**************************************************************
!
! Inputs
width=0.30 ! width of the joint
length=0.30
thickness=0.001
exs=2.06e11
! Young's modulus sheet
nuxs=0.33
! Poisson number sheet
denss=7800
! sheet density
!Divisions
tlw=150
!Divisions over width
tll=60
!Divisions over length
!tlt=1
!Divisions over thickn
! ********************************************************************
!*********************************************************************
!
ET,1,shell181
MAT,1
MP,EX,1,exs
MP,NUXY,1,nuxs
MP,DENS,1,denss
!
ET,2,solid45
!
MAT,2
MP,EX,2,exs
MP,NUXY,2,nuxs
!
72
MP,DENS,2,28800
block,0,width,0,length,0,thickness
!sheet creation
lsel,s,loc,x,width/2
lesize,all,,,tlw
lsel,s,loc,y,length/2
lesize,all,,,tll
!lsel,s,loc,z,thickness/2
!lesize,all,,,tlt
type,1
mat,1
asel,s,,,1
amesh,all
allsel
r,,0.001
!
allsel
block,.02,.02+.002,.25,.25+0.005,-0.005,0 ! accelerometer creation
type,2
mat,2
vsel,s,VOLU,,2
lsel,s,,,23
lsel,a,,,19
lsel,a,,,20
lesize,all,,,1
vmesh,all
allsel
clocal,11,1,,,,0,0,90,,,
csys,11
vgen,2,all,,,,90,,,,1
csys,0
vgen,2,all,,,,.015,,,,1 !y
vgen,2,all,,,-.2,,,,,1
!X
vgen,2,all,,,,,.05,,,1
!Z
! rotates the combination
nummrg,node
CDWRITE,comb,sheet
73
!*****************************************************************************
******************!
!******* This file reads the prepared models of different piping segments
with
**************!
!******* CDREAD command and joins them together as a complex piping system.
**************!
!******* First, the model of each segment should be present by running the
.txt file of it. ***!
!*****************************************************************************
******************!
finish
! Reset
/clear
!
/FILNAME,system_sm2
/CONFIG,nres,6001
/PREP7
!***************** pipe 1 *******************************************
CDREAD,comb, boundary_support
(BC)
vgen,2,all,,,,,-0.005,,,1
CDREAD,comb,pipe_l19sm
vgen,2,all,,,,,-0.19,,,1
!
clocal,11,1,,,,,-90,,,,
csys,11
vgen,2,all,,,,90,,,,1
csys,0
vgen,2,all,,,,.010,,,,1
!
CDREAD,comb,elbow_small
vgen,2,all,,,-0.010,,0,,,1
!
clocal,11,1,,,,0,0,90,,,
csys,11
vgen,2,all,,,0,90,,,,1
csys,0
! reads the support model
! reads the pipe model
! rotates the combination
! reads the elbow model
! rotates the combination
vgen,2,all,,,,,.005,,,1
CDREAD,comb,joint_block1_r3_10_10_5
vgen,2,all,,,0,,0.04,,,1
CDREAD,comb,pipe_l4_sm
vgen,2,all,,,0,,0.005,,,1
! reads the joint model
! reads the pipe model
CDREAD,comb,joint_block1_r3_10_10_5
vgen,2,all,,,0,,0.02,,,1
CDREAD,comb,pipe_cap_l2_sens_sm
CDREAD,comb,sheet10_10
vgen,2,all,,,0,,-0.08,,,1
vgen,2,all,,,0.205,,0,,,1
74
clocal,11,1,,,,0,90,0,,,
csys,11
vgen,2,all,,,0,90,,,,1
! rotates the combination
!***************** pump *******************************************
csys,0
CDREAD,comb,pump_cent
vgen,2,all,,,,.02,0,,,1
CDREAD,comb,pump_side
vgen,2,all,,,,-.04,0,,,1
CDREAD,comb,pump_cent
vgen,2,all,,,,-.02,0,,,1
CDREAD,comb,pump_side
! reads the pump model
! reads the pump model
! reads the pump model
! reads the pump model
vgen,2,all,,,,.02,0,,,1
!***************** pipe 2 *******************************************
CDREAD,comb, boundary_support
(BC)
vgen,2,all,,,,,-0.005,,,1
CDREAD,comb,pipe_l4_sm
vgen,2,all,,,,,-0.04,,,1
!
clocal,11,1,,,,,0,90,,,
csys,11
vgen,2,all,,,,180,,,,1
clocal,11,1,,,,0,90,0,,,
csys,11
vgen,2,all,,,,90,,,,1
csys,0
vgen,2,all,,,,.010,,,,1
!
CDREAD,comb,elbow_small
vgen,2,all,,,-0.010,,0,,,1
!
clocal,11,1,,,,0,90,0,,,
csys,11
vgen,2,all,,,0,180,,,,1
! reads the support model
! reads the pipe model
! rotates the combination
! rotates the combination
! reads the elbow model
! rotates the combination
csys,1
vgen,2,all,,,0,90,,,,1
csys,0
vgen,2,all,,,,.010,,,,1
75
!
CDREAD,comb,elbow_small
vgen,2,all,,,-0.010,,0,,,1
clocal,11,1,,,,0,0,90,,,
csys,11
vgen,2,all,,,0,90,,,,1
! reads the elbow model
! rotates the combination
csys,1
vgen,2,all,,,0,90,,,,1
clocal,11,1,,,,0,90,0,,,
csys,11
vgen,2,all,,,0,180,,,,1
csys,0
CDREAD,comb,pipe_l13sm
vgen,2,all,,,,,-0.13,,,1
clocal,11,1,,,,,-90,,,,
csys,11
vgen,2,all,,,,90,,,,1
clocal,11,1,,,,,0,-90,,,
csys,11
vgen,2,all,,,,90,,,,1
csys,0
vgen,2,all,,,,.010,,,,1
!
CDREAD,comb,elbow_small
vgen,2,all,,,-0.010,,0,,,1
!
clocal,11,1,,,,0,0,90,,,
csys,11
vgen,2,all,,,0,90,,,,1
csys,0
! rotates the combination
! reads the pipe model
! rotates the combination
! rotates the combination
! reads the elbow model
! rotates the combination
vgen,2,all,,,,,.005,,,1
CDREAD,comb,joint_block1_r3_10_10_5
vgen,2,all,,,0,,0.04,,,1
CDREAD,comb,pipe_l4_sm
vgen,2,all,,,0,,0.005,,,1
! reads the joint model
! reads the pipe model
CDREAD,comb,joint_block1_r3_10_10_5
vgen,2,all,,,0,,0.02,,,1
CDREAD,comb,pipe_cap_l2_sens_sm
!******
vgen,2,all,,,0,0,-.08,,,1
vgen,2,all,,,.205,0,0,,,1
vgen,2,all,,,0,.02,0,,,1
clocal,11,1,,,,0,90,0,,,
! rotates the combination
76
csys,11
vgen,2,all,,,0,90,,,,1
csys,0
nummrg,node
!csys,1
!asel,s,loc,z
!asel,r,loc,x,0.0013,0.002
!nsla,s,1
!d,all,ux,,,,,uy,uz
!allsel
!modmsh,deta
!ematwrite,yes
/SOLU
ANTYPE,2
MODOPT,unsym,20,5,100
MXPAND,20
Allsel
!*********** TRANSIENT ***********!
!Transient simulation of the one pipe system
!***********************************************
!
/solu
ANTYPE,TRANS
TRNOPT,FULL
alphad,2
!damping parameters
betad,10e-6
!damping parameters
!TINTP,0
!
csys,1
!load step 1
asel,s,loc,z,
asel,r,loc,x,0,0.0023
da,all,pres,1e4
!pressure application
csys,0
!
deltime,6e-7
time,.1e-3
kbc,0
outres,nsol,10
!
solve
csys,1
!load step 2; “unloading”
asel,s,loc,z,
77
asel,r,loc,x,0,0.0023
da,all,pres,0
! zero (no) pressure applied
csys,0
!
deltime,6e-7
time,.2e-3
kbc,0
outres,nsol,10
!
solve
csys,1
!load step 3; system response due to initial pressure pulse
asel,s,loc,z,
asel,r,loc,x,0,0.0023
da,all,pres,0
csys,0
!
deltime,6e-7
time,3e-3
kbc,0
outres,nsol,10
!
solve
78