HABILITATION THESIS
Numerical and experimental simulation of
waves and pulses propagation through
inhomogeneous elastic medium
ABSTRACT
Prof.univ.dr.ing. Ioan Călin ROŞCA
Transilvania University of Braşov
- 2015 –
Content
Chapter 1 - Waves in industrial applications
1.1. General considerations
1.2. Technological applications of the waves
1.3. Conclusions
Chapter 2 - Elements of the waves and pulse propagation in solid one-dimensional environment
2.1. Introduction
2.2. Theoretical considerations
2.2.1. Generalities
2.2.2. Waves propagation equation
2.2.3. Longitudinal wave energy
2.2.4. Characteristic phenomenon of wave propagation
2.3. Waves’ amplitude magnification
2.3.1. Horn with rapid change of cross section (stepped horns)
2.3.2. Horn with continuous cross section variation
2.4. The ultrasonic horns with continuous cross section variation design
Chapter 3 - Numerical methods used in wave propagation analysis
3.1. Finite-difference method used in wave propagation through horns with variable cross-section
variation analysis
3.2. The analyse of the wave propagation in horns with harmonic cross-section variation
3.3. The transfer matrix method
Chapter 4 - Optimisation algorithms of cross-section horn
4.1. General considerations
4.2. Optimisation algorithm
4.2.1. The functional calculus
4.2.2. Solution of the equation
4.3. Horns testing
4.4. Conclusions
Chapter 5 - Studies and personal contributions in the field of waves propagation in inhomogeneous
solid media
5.1. Generalities
5.2. Simulation of Gaussian pulse propagation in elastic medium with periodical inhomogenity
5.2.1. Mathematical model
5.2.2. Solution of the wave equation
5.2.3. Physical meaning of the solution u~ (x ,)
5.2.4. Pulse propagation in medium with periodically variable properties
5.2.5. Results
5.3. Wave propagation in optimised horns
5.3.1 General considerations
5.3.2. Transfer matrix method
5.3.3 Computer simulation using coupled-oscillators model
5.3.4 Computer simulation using finite element method
5.3.5 Ultrasonic horns with input and output couplings
Chapter 6 - Studies and future developments
References
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Chapter 1
Waves in industrial applications
1.1. General considerations
Present day industrial and non-industrial technologies based on wave emission and
propagation are originated in the six-decade of the XXth Century. At that time, the technical progress,
on all its aspects, permitted that the theoretical considerations about the waves’ generation and
propagation, in different media, took shape in practical applications. There were developed new and
high performance technologies but, in the same time, were developed new research area as Non
Destructive Testing (NDT) and Non Destructive Evaluation (NDE).
The use of elastic waves in different domains is strong tied of their frequency. In usual
applications there are used waves with a frequency that is in the range of minimum 20 kHz and
maximum few GHz. The highest values are limited of the technological facilities of generating.
In the case of industrial applications, one of the most important components is the ultrasonic
horn. The technical role of these devices is to amplify the signal amplitude that is emitted by a
transducer and to focalise the ultrasonic wave in the working area or to the cutting tool.
The ultrasonic horns are characterized by some important properties, such as:
a) The amplify the amplitude of the tool vibrations;
b) Concentrate and focalise the ultrasonic energy in the manufacturing area;
c) Allow the use of ultrasounds in a very large gamut of manufacturing processes;
d) Allow the ultrasonic energy transfer from the generator to the point of manufacturing;
e) Introducing a new vibrating motion in the working area it is created the possibility of
the manufacturing capability increasing;
f) The ultrasonic waves create the support of manufacturing improvement in the
meaning of decreasing energetically consumption of the cutting process.
1.2. Technological applications of the waves
Now, in the manufacturing area, and not only there, there is a large and diversified gamut of
high frequency waves applications. These applications can be classified according with the
technological support:
a) Applications in solid medium that can be split in:
 Machine working (e.g. turning, milling, boring, broaching, threading, etc.);
 Plastic deformations (e.g. die forging, throttling, moulding, sheet-metal stamping,
forging, etc.);
b) Applications in liquid medium which can be grouping in two categories:
 Technological applications (e.g. welding, soldering, casting, metal coating, etc.);
 Passive applications where there is not mechanical processing (e.g. emulsion obtaining,
pieces cleaning, diffusion acceleration, crystallization, filtering, dissolution, reaction
acceleration, etc.);
c) Applications in gas medium are used for fume sagging, drying, etc.
1.3. Conclusions
a) High frequency waves propagation have a large applicability, their use can be found in both
industrial and non-industrial domains;
3
b) In case of industrial applications ultrasonic waves are used in manufacturing in all types of
environments (solid, liquid and gas);
c) The use of high frequency in technological applications has advantages and disadvantages;
d) A special branch of the high frequency waves applications is connected with the material
testing and can be split in Non Destructive Testing (NDT) and Non Destructive Evaluation
(NDE).
Chapter 2
Elements of the waves and pulse propagation in solid one-dimensional
environment
2.1. Introduction
In the present work there are analysed the elastic waves propagation considered as physical
element useful in technical applications. The aim of the paper is to describe some research objectives
in the field of waves and pulses propagation in homogeneous and inhomogeneous one-dimensional
media. The inhomogeneous characteristics of the media are related to the geometry and structure.
The geometrical inhomogenity is related to the cross section variation along the longitudinal
axis of an ultrasonic horn and the structural inhomogenity is refers to the Young’s modulus, and
density variation along the same axis. In the same time, the considered horns are axis symmetrical.
2.2. Theoretical considerations
2.2.1. Generalities
As it is known the waves developed in solid continuous media are generated by a source.
These sources can be classified according with their geometry as:
a) Point like source - that has the negligible dimensions compared with the observation
point of the wave propagation;
b) Linear sources – that have a continuous distribution of point like sources that are
situated on a curve;
c) Surface sources – that are made of point like sources continuous distributed on a
surface.
Considering the nature of the waves that propagate in a media and are used in technological
applications there are defined the following types:
a) Elastic waves that consists in a local mechanical deformation (perturbation) which
propagates in solid, liquid or gas media but they cannot developed in emptiness media;
b) Electromagnetic waves that are developed in solid or emptiness media and propagates
electromagnetic fields;
c) Magnetohidrodynamic waves that are presented in plasma media and have as source
complex perturbations of plasma having both mechanical and electromagnetically
characteristics;
d) Heat waves that are generated by a temperature variation;
e) Broglie waves which are associated with the micro-particles.
4
From media nature point of view as propagating media there considered the following types:
homogeneous, isotropic, conservatives, dissipative and linear.
The waves can be classified according with different criteria:
a) Frequency criteria: infra-sound waves, sound waves, ultrasonic waves, hypersonic waves;
b) Particle trajectory: longitudinal waves, quasi-longitudinal waves, transversal (bending)
waves, torque waves, surface waves (Rayleigh), plate waves (Lamb;
c) Direction propagation: progressive waves, regressive waves.
2.2.2. Waves propagation equation
Based on the literature, in case of small source oscillations, in an ideal media the wave
propagation is described by the following differential equation:
 2  2  2 1  2
 2  2  2 2 ,
x 2
y
z
c t
(2.3)
where c is the wave velocity in the solid considered media being dependent on the media mechanical
characteristics and the wave type,  is wave function, t is time and x , y , z are the Cartesian
coordinates
In the case of 3-D study, the propagation equation (2.3), in Cartesian coordinates, can be
written in different shapes according with the function:
a) Acoustical pressure p p(x , y , z ,t)
2 p 2 p 2 p 1 2 p
;



x 2 y 2 z 2 c 2 t 2
(2.11)
b) Velocity wave potential    (x , y , z ,t)
2 2 2 1 2

 2  2 2 ;
x 2
y 2
z
c t
(2.12)
c) Wave elongation u u(x , y , z ,t)
 2u  2u  2u 1  2u
.



x 2 y 2 z 2 c 2 t 2
(2.13)
2.2.3. Longitudinal wave energy
In practical applications there are used the following physical quantities: wave total density
energy, average total density energy, instantaneous energy density, energy flux, wave intensity,
average wave intensity, instantaneous intensity, specific impedance, mechanical impedance and wave
impedance.
2.2.4. Characteristic phenomenon of wave propagation
The main phenomenons that are developed in longitudinal wave propagation are: reflexion
and refraction, interference, diffraction, diffusion, and absorption.
2.3. Waves’ amplitude magnification
Based on amplitude magnification principle there were done some special devices called
ultrasonic horns (ultrasonic amplifiers).
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2.3.1. Horn with rapid change of cross section (stepped horns)
The horn is made of two (or more)
different sectors of beams each one with
constant cross section S1 and S2 . The cross
sections are chose as the first one is larger than
the second S1 S2 and the reference point is
considered the point of section jump (figure 2.1).
Based on physics laws the pressure of the
three waves, incidental, transmitted and
Figure 2.1 Stepped horn
refracted can be written as:
j(t kx )
a) Incident wave
Pi P1e
b) Reflected wave
Pr P2 e
j(t kx )
,
(2.46)
,
(2.47)
j(t kx )
c) Transmitted wave
.
(2.48)
Pt P3e
The level of transmitted energy through the jump plane can be evaluated by the reflection and
transmitted coefficients R and TR :
 S1I2 P22 (S1  S2 )2 (S12 1)2
 2
R 
2
2,
 S1I1 P1 (S1  S2 ) (S12 1)

2
T  S2I3  S2P2  4 S1 S2  4 S12 .
 R S I S P 2 (S  S )2 (S 1)2
11
1 1
1
2
12

(2.50)
2.3.2. Horn with continuous cross section variation
The horns with continuous cross section variation it is presented in many books and articles
such as 6, 10, 56, 64, 80, etc.
Figure 2.4 Ultrasonic horn with continuous cross section variation: 1) magnetostrictive transducer; 2
ultrasonic horn; 3 tool
The ultrasonic horns with continuous cross section variation have are very useful in practical
application as a result of the following advantages:
a) The acoustical energy can be transmitted in the desired direction;
b) They can balance the source impedance with the impedance of the radiating media;
c) They multiply the transmitted energy.
The wave propagation equation is given by:
6
 2  
1  2
lnSx  2 2 ,

x 2 x x
c t
(2.52)
where S x is the cross-section area at a distance x from the origin of the reference system (figure
2.4).
2.4. The ultrasonic horns with continuous cross section variation design
The ultrasonic horns design it is described in many technical books and papers. The main
design principles can be found in 6, 8, 9, 10, 16, 17, 32, 53, 56, 58, 59, 61, 64,
74, 80, 85, 91, 93, 99, 133, 134, 148, 149, 157, etc.
Based on all description and taking into consideration the practical aspects of industrial
applications one can say that the use of these devices can be considered as part of the nonconventional technologies.
Taking into consideration the mathematical functions of cross-section variation in practice
there are used the following horns type:
a) With stepped variable cross-section;
b) With linear cross-section variation;
c) With exponential cross-section variation;
d) With different cross-section function variations, as catenoidal, parabolic, Gaussian,
hyperbolic, and/or their combinations.
Chapter 3
Numerical methods used in wave propagation analysis
3.1. Finite-difference method used in wave propagation through horns with variable
cross-section variation analysis
Longitudinal waves propagation through horns with variable cross-section can be modelled in
a good shape using the method of finite-difference method. This method is very useful taking into

l nS x  from the motion wave equation (2.57).
consideration the component
x
Based on this method one can found the variation of amplitude, velocity and acceleration of
the waves in a large number of points. Details about this method can be found in many books or
papers such as: 52, 82, 83, 86, 93, 100, 141, 146, etc.
3.2. The analyse of the wave propagation in horns with harmonic cross-section
variation
It is considered a horn with a harmonic cross-section variation given by the following equation:
Sx S0 S  coskp x  ,
(3.1)
where for S there were considered some values equal with initial cross-section S 0 fractions, and k p
is the wave number that corresponds to the cross section periodicity:
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kp 
2
p
,
(3.2)
where  p represents the length of the periodicity.
Based on relationship (3.1), the wave propagation equation (2.57) is, in terms of spatial
components:
u"
k p Ss in(k p x)
S0  Scos (k p x)
u' k 2u  0 .
(3.3)
In figures 3.3 and 3.4 there are presented the variation shapes for two different horns where
were considered in equation (3.1) S  0,2 S0 and S  0,5 S0 . As method it was considered the centred
finite difference method.
g
g
1.
5
2
1.
5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1.5
-1
0
0.2
0.4
0.6
0.8
1
1.2
Lungimea concentratorului (adimensional)
Figure 3.3 Signal magnitude g for the case
S  0,2 S0
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
Lungimea concentratorului (adimensional)
Figure 3.4 Signal magnitude g for the case
S  0,5 S0
Based on finite-differenced method simulation with harmonic cross-section variation one can
find the both the magnification of the ultrasound amplitude and the points where these are achieved.
3.3. The transfer matrix method
The transfer matrix method is used for its main advantage that consists in the fact that its
components o not depends on both elastic properties of the propagation media and wave type [B6].
Elastic constant that are in the transfer matrix there are introduced by the boundary conditions at the
level of boundary layers which are dependent on interface type between layers [36], [102], [155].
The advantages of the method are useful in finding the velocities of the longitudinal waves in
solid media as:
a) Small cylindrical samples;
b) The method permits to realise a study of dispersion for a given material sample. The
experimental set-up can be sized in a such way as the phase velocity can be found in a
large range of frequencies;
c) The method permits to generalise the attenuation influence;
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d) The method can offer important information if it is applied to binary periodically
systems with extrapolation in the area of acoustic-optical properties of solid media;
e) The method is useful to be applied for special materials as ceramics used in extreme
conditions and with special mechanical properties.
Chapter 4
Optimisation algorithms of cross-section horn
4.1. General considerations
In papers 119 and 120 it is presented a synthesis of the main considerations about the
design of ultrasonic horns. This synthesis refers to:
I. Assumptions concerning the ultrasound propagation;
II. Boundary conditions;
III. The horns’ shape;
IV. Dimensions restrictions;
V. Shape variation influence on wave propagation;
VI. Phase velocity variation;
VII. Increasing the design criteria of ultrasonic horns.
4.2. Optimisation algorithm
In papers 120 and 121 it is presented an optimisation method for the ultrasonic horns. In
the case of this model there are considered the maximum value of the cross-section Sm , the imposed
magnification coefficient g, and the stiffness value of the contact between the horn and the tool.
4.2.1. The functional calculus
In the ultrasonic waves propagation theory it was defined the average energy density in a
point of the elastic media 6 that can be found using the relationship:
Emed 
2
pmax
,
2 c2
(4.1)
where pmax is the maximum acoustic pressure,  is the horn’s material density, and c is the velocity
of the longitudinal wave in considered media.
Considering the wave propagation through a horn with variable cross-section, written based on
Bernoulli-Fourier method, on space coordinates, results:
d 2u d
du
 (lnS)  k 2u  0 .
2
dx dx
dx
(4.6)
The considered functional includes two element tied on the horn structure and work: cross-section
S  S(x) and the spatial signal component u  u(x) . Based on 112 and 149 one can write the
following functional:
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L
L
  2  u' 
   Sdx   x Su" S' u' k 2uS dx  1   (L) (L)  ,
2 
0
0
 2E
(4.7)
d
l nS  S' .
dx
S
Based on the functional (4.7) it is obtained the following equation:
where 1 and x  (x) are Lagrange multipliers and
u'2 k 2u2 C 0
.
(4.37)
4.2.2. Solution of the equation (4.37)
The solution of the equation (4.37) is:
u(x)u0ekx C2 ekx ekx  ,

(4.38)

where from results:
(4.39)
u'(x)k u0ekx C2 ekx ekx  .
A condition that can be imposed at the limits (contact between horn and tool) is that of the
contact is given in 27, 28:
E S( L )
u
x
k S u(L) 0 ,
(4.40)
x L
where k s is the contact stiffness between the horn and the tool.
Based on the previous condition it is obtained:
C2 
 2 S( L ) 
1 
kL
.
u
e

0
e kL e kL 
gk S ku0 E 
(4.45)
Introducing (4.45) in (4.38) and considering that at the end of the horn (x  L) it is obtained the
maximum value of the amplitude (a magnification with the coefficient g of the signal) it is obtained:
 2 S(L) 
e kL e kL 
kL
,
gu0 u0 e  kL kL  u0 e 
e e 
gkS ku0 E 
kL
(4.46)
where the signal ( - ) is introduced to obtain a nodal point along the longitudinal axis of the horn.
From (4.46), after some calculations, it is obtained:
 2


gk S ku02 E ge kL e kL 2
.
S(L) e kL e kL 
(4.47)
Introducing (4.38) in (4.40), it is obtained for the coefficient C 2 the following relationship:
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An important geometrical element in horns functionality is the nodal point that can be found
based on relation (4.38):
xnod 
1  C2
ln
2k  C2 u0

 .

(4.49)
4.3. Horns testing
Considering the optimisation procedure described above, it was designed and tested an
ultrasonic horn made of alloy steel having the following characteristics: Young’s modulus
E  2  1011 N m2 , density   7850 kg m3 , longitudinal wave velocity in finite space cl  5050 m s ,
length L  126.9 mm and the end cross-section diameter dL  8,4 mm .
The ultrasonic signal was generated
with a magnetostrictive transducer having a
magnification coefficient g  5 at a frequency
of fr 19900 Hz .

The data acquisition system was
made of a Brüel & Kjær Pulse platform
12(type 3039 with control module 7539),

two accelerometers 4517-002 and dedicated
soft for time and frequency domains
analysis.

Figure 4.6. The set-up for experimental testing of the
horns: 1 – amplifier; 2 – magnetostrictive transducer;
3 – ultrasonic horn; 4 – Pulse 12 platforme
[m/s²]
[m/s²]
Time(Input) - Input
Wo rking : Input : Input : FFT A nalyzer
800
500
700
450
600
400
500
350
400
300
300
250
200
200
100
150
0
100
-100
50
-200
Fo urier Spectrum(Input) - Input (M agnit ude)
Wo rking : Input : Input : FFT A nalyzer
0
0
1m
2m
3m
4m
5m
[s]
6m
7m
8m
9m
10m
18k
18.4k
18.8k
19.2k
19.6k
20k
[H z]
20.4k
20.8k
21.2k
21.6k
22k
a)
b)
Figure 4.8 Input signal: a) time domain representation; b) Fourier spectrum
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[m/s²]
[m/s²]
Time(Output) - Input
Wo rking : Input : Input : FFT A nalyzer
800
Fo urier Spectrum(Output) - Input (M agnitude)
Wo rking : Input : Input : FFT A nalyzer
500
450
600
400
400
350
200
300
250
0
200
-200
150
-400
100
-600
50
-800
0
0
1m
2m
3m
4m
5m
[s]
6m
7m
8m
9m
10m
18k
18.4k
18.8k
19.2k
19.6k
20k
[H z]
20.4k
20.8k
21.2k
21.6k
22k
a)
b)
Figure 4.9 Output signal: a) time domain representation; b) Fourier spectrum
4.4. Conclusions
The ultrasonic horns design presents some special particularities as follows:
a) One way to optimise their shape can be focused on variational principles;
b) The general type of horns have decreasing cross-section diameter. As a result of this it
is considered as objective function the volume minimisation;
c) The parameters that were chose for optimisation were: the value of maximum crosssection diameter that is equal with a quarter of wave length, the magnification
coefficient restricted to the value 5 to have only longitudinal waves,;
d) The boundary conditions refer to the attached mass and the “connection” stiffness k s
between the horn and the tool;
e) In case of using the variation principle for optimisation, the length of the horn can be
limited only to a half of the wave length;
f) This type of concentrators comparing with all known horns in the literature present a
design particularity, the fact that the nodal point is in the same point was is the largest
cross-section.
Chapter 5
Studies and personal contributions in the field of waves propagation in
inhomogeneous solid media
5.1. Generalities
The subject of waves of high frequency propagation, in homogeneous and inhomogeneous
media is a challenge subject from both point of view theoretical and experimental approaches.
5.2. Simulation of Gaussian pulse propagation in elastic medium with periodical
inhomogenity
5.2.1. Mathematical model
In papers [120] and [134] it is treated, exhaustively, a mathematical method, useful in
computer simulation of pulse propagation in an elastic medium with periodical inhomogeneities. It
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was considered the general case and, in the end it was took into the consideration the particular case
of an elastic semi-finite rod that has a periodical variation of Young’s modulus space dependent.
It is considered the wave propagation of a longitudinally acoustic wave in an elastic inhomogeneous
medium:
 2u  
u 
f x  2  g(x)  ,
t x 
x 
(5.1)
where f (x) and g(x) there are two functions that describe the inhomogenity of the propagation
medium:
 f (x)  (x)S(x) ,
(5.2)

g(x)E (x)S(x).
From all three inhomogeneities, density, Young’s modulus and cross-section variations along
the longitudinal axis, the first two have, for the moment, only theoretical interest. The cross-section
variation has a practical application being possible the experimental cheeking. For the study it was
considered a periodicity of 500 mm.
1. The solution of the wave propagation equation (5.1) has a Fourier and one can write:

1
u(x ,t )  u~(x ,)e  jt d ,
2 
(5.3)
2. The medium inhomogeneities have small variations, comparing with homogeneous
medium, and there can be written the following relations:
 f (x) f0 11 p(x)  ,

g(x)g0 12 q(x) .
(5.4)
Introducing (5.3) in equation (5.1) it is obtained:
or in space components u x :

u~(x ,)  2
g
(
x
)
 f (x)u~(x ,) 0 ,
x 
x 
(5.5)
g(x)uxg'(x)ux  2 f (x)ux 0 .
(5.6)
The functions f (x) and g(x) there considered that have the necessary conditions to be written
in a Taylor expansion C4, C5, C6, S4, etc. in a point x 0 :



f ( n) ( x 0 )
n
f
(
x
)

(
x

x
)

an (x  x 0 )n ,


0

n
!
n 0
n0


( n)


g(x) g (x 0 )(x x )n  b (x x )n .
 n!
 n 0
0

n0
n0
(5.7)
13
where:
 (n)
n f
f
(
x
)


0
x n


n
g (n) (x )   g
0

x n

,
x  x0
(5.8)
.
x  x0
5.2.2. Solution of the wave equation
The solutions of the wave propagation are:

 ()
()
jt
u
(
x
,
t
)

 0
 u~0 (x ,)e d




 u () (x ,t )  u~ () (x ,)e  jt d
 0
 0


(5.28)
that correspond to the progressive and regressive pulses in medium characterised by the functions
f (x0 ) şi g(x0 ) .
The solutions (5.28) can be developed in a Taylor expansion:

u~0() (x ,)  C n   (x  x0 )n ,
(5.29)
n0
with the following coefficients:

    
C 0  c0 ,

    
  
C1   j c0 ,
 v0 


2

1
      
C     


1


p
(
x
)
 C n .

1
0 
 n2


(
n

1
)
(
n

2
)
1


q
(
x
)
c




2
0


where v 0 is the wave velocity in the medium with periodical inhomogeneities.
(5.30)
5.2.3. Physical meaning of the solution u~ (x ,)
The solutions of the equation (5.5) for different values of the frequency  represent the
modeshape of medium oscillation. The function u(x ,t ) represents the superposition of all possible
modeshapes. The superposition calculation can be realised considering the initial conditions that are
useful for finding the phase and the amplitude. Thus, the solution u(x ,t ) can be written in the shape
(5.3).
To the function u~(x ,)A(x ,)e j ( x , ) one can attach different quantities as wave number or
phase velocity. These two quantities are dependent on coordinates and they can have the following
shapes:
14
  ( x , )

k(x ,)   x ,

v (x ,)   .
 
k
(5.35)
Practically, the quantities (5.35) can be found after there are found the complex solutions of
~
u(x ,) for the studied medium.
The use of the previous methodology has the advantage that the solution u(x ,t ) at the time
t  0 can offer the possibility to find out the temporary evolution of each modeshape using a simple
variation of the temporal phase.
5.2.4. Pulse propagation in medium with periodically variable properties
In paper [35] it is considered another shape variation for the functions f (x) and g(x) . For
there is considered a periodical variation of p :
 f (x  np )  f (x),

g(x  np )  g(x),
g'(x )  0.
0

(5.36)
Solution of the equation represents a problem of eigenvalues problem for the studied
medium. The discrete character of the solutions spectrum is given by the medium characteristics. For
the studied case, the conditions (5.36) impose to be used a supplementary restriction given by the
derivative of the function g(x) in point x 0 .
The propagation study of a wave in a medium with periodical variable properties imposes a
previous analysis of the medium particularities. So:
1. Making the variable substitution xx  p , the equation (5.5) becomes:
 u~(x  p ,) 
 
g
(
x


)
  2 f (x  p ) u~(x  p ,)  0 ,
p


x 
x

(5.37)
and between solutions of the equations (5.5) and (5.37) it is the dependency:
~
~

u(x  p ,)  Cu(x ,),


 C  1.
(5.38)
2. Condition C  1 is necessary for two reasons: the first one - to be developed in considered
medium bounded oscillations, and the second one – considering the absence of a dissipative force
then the amplitude of the oscillation has to satisfy the conservative condition and so the constant C
has to be a real number equal with the unit.
15
5.2.5. Results
In the present paragraph there are presented some results concerning a Gaussian pulse
propagation simulation in a medium cu periodical cross-section variations. In simulation there were
considered the following periodic functions that satisfy condition (5.24) in point x0  0 :


 2 
 f (x) f0 11 cos x  ,


 p 


 2 

 x .
g
(
x
)

g
1


cos

0
2

 

 p 

(5.42)
There were considered all three possible:
Case 1: 1 0 şi 2 0 - Young’s modulus variation E

 2
E  E0 1  cos

 p

x  ,

 2
1  0;2  ; q(x)  cos
 p

x 

Case 2: 1 0 şi 2 0 - density variation 

 2 
x 

p


  0 1  cos

 2 
x 

p


2  0;1  ; p(x)  cos
Case 3: 1 2  0 - Variaţia secţiunii S

 2
S  S0 1  cos
 p


x 

 2 
x 

 p 
1  2  ;q(x)  p(x)  cos
The obtained simulations there are presented in the following graphs.
x/p
b)
a)
Figure 5.1 Graphical representation of the solution u~(x ,) for the case 1 2  : a) real
component; b) imaginary component
16
k(x,)
x/p
b)
a)
Figure 5.2 Variation of: a) pseudo-wave number k(x ,) ; b) amplitude A(x) and phase velocity V(x)
for case 1 2 
u(x,t)
x/p
b)
a)
Figure 5.3 Gaussian pulse propagation in time domain with initial conditions 1 2 0,2 : a) at the
beginning of the propagation; b) at the end of horn.
a)
b)
Figure 5.4 Gaussian pulse with the initial conditions 1 2 0,5 : a) at the beginning of the
propagation; b) at the end of horn.
5.3. Wave propagation in optimised horns
5.3.1 General considerations
Computer simulations are used to characterise the USH obtained in chapter 4 in terms of
resonance frequencies and standing wave amplitude. Two 1D methods are employed in order to
obtain the resonance spectrum:
17
a) The Transfer Matrix Method (TMM);
b) The Coupled-Oscillators (CO) model. Both have some notable characteristics.
Both methods have some distinguished particularities:
a) The TMM has a simple and compact theoretical formulation and allows fast and accurate
solutions to be obtained computationally;
b) The CO model reflects the physical phenomena involved in elastic wave propagation,
which is its main advantage; nevertheless CO simulations require a lot of computational
time and generally result in low accuracy in the obtained spectra.
For these reasons the TMM was used both for spectral characterisation and for the design of
the couplings that are part of the working setup, while the CO model was used to check that the
resonance spectrum returned by the TMM was complete for a certain interval of frequencies.
Moreover, the 1D simulation results were further verified with a Finite-Element Method (FEM) run for
3D models of the USH and working setup.
5.3.2. Transfer matrix method
In the transfer matrix method (TMM) the sound wave propagation along the horn is
characterized by a wave transfer matrix [39], [40], [41], [102], [154]. If we consider a progressive wave
of spectral amplitude A  A f  and frequency f moving from an input end to the output end of the
USH, and a regressive wave of spectral amplitude B  B f  moving in the opposite direction, then the
total output wave is given by:
 Aout   Ain 

  T 
 Bout   Bin 
(5.45)
Then the transfer matrix from input to output could be expressed as the right-product of the
propagation and discontinuity matrices along the path of propagation:
Tin,out  PN DN 1, N PN 1...P3D2,3P2D1, 2P1 .
(5.49)
The propagation in the opposite direction is characterized by the matrix:
Tout,in  P1D2,1P2 ...PN 1DN , N 1PN .
(5.50)
The propagation from input to output and back again is characterized by the total propagation
matrix:
Ttotal  Tout,in RTin,out .
(5.51)
5.3.3 Computer simulation using coupled-oscillators model
In many cases, for 1D propagation, the USH can be modelled as a linear arrangement of
coupled oscillators (CO) [65]. If an initial perturbation is applied to one end, the displacement u j t  of
the oscillators can be obtained by numerically integrating of the equations of motion:
mi ui  ki*1, i ui 1  ui   ki*, i 1 ui 1  ui  .
(5.53)
18
A number of N osc oscillators of index j , each with mass m j , were set at equal distances xosc
and connected with springs of elastic constants k *j , j 1 were considered. The parameters were
obtained from the local Hooke’s law:
xosc 
L
N osc  1
, m j   S j xosc , k
*
j , j 1
 c 

 m j 
 xosc 
2
(5.54)
Equation (5.53) was solved by a 4th-order Runge-Kutta integration method [114].
5.3.4 Computer simulation using finite element method
For the 3D simulation a Finite Element Method (FEM) was applied [114] using the Elmer FEM
software [60]. The USH was modeled using Gmsh [70], as a mesh with 3820 nodes and 17624
tetrahedral volume elements as shown in figure 5.7. Next the Linear Elasticity module of the Elmer
Solver was used in order to determine the USH eigenmodes illustrated in figure 5.8.
Figura 5.7. The 3D mesh of the USH
Figure 5.8 Longitudinal eigenmodes in the USH based on FEM: (a) 12091.67 Hz, (b) 27841.47 Hz,
(c) 44802.37 Hz, (d) 62225.28 Hz.
19
5.3.5 Ultrasonic horns with input and output couplings
The next step is to consider the coupling elements added to the surfaces of the horn. For
practical reasons it was chose to equip the USH with two rods with constant section at the two ends
of the horn. This configuration simulates the transducer and the tool connected at the USH ends [92],
[67] (figure 5.9). In case the generator and tool are significantly long, they are considered to be part of
the system; otherwise they may be complemented with the couplings in order to ensure the desired
resonance frequency, g factor and nodal point position.
The coupling affects the values of eigenfrequencies of the system, which will be different from
those of the isolated USH. It was considered that the two couplings are made of the same material as
the USH body.
Choosing an input coupling length L1  38.9 mm and an output coupling length L2  32.1mm ,
numerical simulations give a resonance frequency of the whole system at fres  19.9 kHz . The position
of the nodal point was derived and a theoretical value of xn  34.7 mm was found, measured from the
input surface of the horn.
Figura 5.9. Radius vs. x of the USH with input and output couplings.
Figures 5.10, 5.11 and 5.13 illustrate the changes produced by taking into consideration the
whole ultrasonic system including the added couplings. The resonance spectrum contains the working
frequency and superior resonance frequencies that may be used too as working frequencies with
proper adjustments in the couplings and support point for the USH.
The Transfer Matrix Method gives the waveform at 19900 Hz with the prescribed node
position (Figure 5.10).
The spectrum in figure 5.11 gives the highest amplitude at 25788 Hz (second resonance
frequency). This is most probably a computation artifact: since the frequency error in the Fourier
transform employed in the Coupled Oscillators method is of 153 Hz, the resonance maximum in the
spectrum are not well determined.
The 3 methods employed in simulations give similar values for the resonance frequencies,
validating the designed working set-up.
The 3D FEM simulations recover longitudinal eigenmodes at these frequencies. It can be
concluded that the plane-wave assumption used in the optimisation and the 1D simulations are
adequate for the design of the USH and working set-up.
20
b)
a)
Figure 5.10. The frequency response of the USH with end couplings from Figure 5.9, simulated with
156 TMM: a) the condition of resonance; the resonance frequency values are 19900, 25802, 38144,
51955 Hz, with a tolerance of  1 Hz; b) the wave amplitude along the USH system, described by the
real part of A+B at f  19900 Hz . The node is at a distance xn  34.72 mm from the input into the USH
body
Figure 5.11. The frequency response of the USH with end couplings, obtained with the coupledoscillators model with 3119 oscillators.
Figura 5.12 The 3D mesh of the USH with end couplings.
21
Figure 5.13. Longitudinal wave eigenmodes in the USH with end couplings obtained at frequencies:
(a) 19729 Hz, (b) 25937 Hz, (c) 37875 Hz, and (d) 51326 Hz.
a)
b)
Figure 5.14. Amplification and first node position: a) as functions of frequency for the isolated USH
and the whole working set-up; b) the node position x n measured from the input into the USH body.
Chapter 6
Studies and future developments
The study of waves and pulses propagation in different media with technological applications
is a boundary domain between applied physics and mechanical engineering.
The use of the undulatory phenomenon during the technological processes and development studies
in this area implies create mixed teams of engineers and physicians.
At this time, in the frame of TRANSILVANIA University of Brasov there were developed two
laboratories, one of Modal Analysis, at the Faculty of mechanical Engineering, and the other one of
22
Technical Acoustics, in the Faculty of Electronics and Computer Science. The whole work (theoretical
and experimental) presented by the author in the present paper was developed in both laboratories.
The equipment infrastructure is a modern one and consist of Brüel & Kjæer – platforms PULSE 12 with
specific soft, excitation systems with particular soft of vibration generating and control produced by
Vibration View company, accelerometers with the frequency range up to 25 kHz, excitation systems,
both impact hammers and shakers - TiraVib and Brüel & Kjæer – for high level forces (up to 440 N)
and small level forces (up to 10 N) with a frequency of 20 kHz. There are also separate data
acquisition systems like National Instruments, two Doppler lasers (for displacements and velocities
measurements), Fourier analyser Stanford SR 760, specialised soft for acquisition and processing data
(Labview 8.2, 2013).
Based on existing equipment the following works that can be developed are focused on:
1. a continuity of studies concerning waves and pulses propagation through mediums
with periodically and non-periodically variable parameters (Young’s modulus, density
and cross-section - E  E(x),    (x), S  S(x) );
s i nx
2. follow-on the studies of pulses propagation considering Gaussian, rectangular,
,
x
etc. in inhomogeneous medium;
3. development the studies on transfer matrix method and modal analysis based on this
method;
4. the use of transfer matrix method for the systems with high level of natural
frequencies (multi-layer medium, sonic crystals, periodically media with permitted and
non-permitted frequency ranges in sonic and ultrasonic field);
5. statistical analysis of data information based on Skewness and/or Kurtosis methods
with applications in complex systems as buildings, bridges, pipes subjected to
undulatory loads;
6. an interesting subject that can developed is the design of ultrasonic horns made of new
inhomogeneous materials;
7. it is proposed to be developed the algorithm of modelling and design of ultrasonic
horns taking into consideration real boundary conditions and the fatigue of the
materials used;
8. based on previous studies about the pulse and ultrasonic waves propagation there will
be extended studies for composite materials behaviour analysis subjected at this type
of signals;
9. mechanical characteristics identification for composite materials based on vibroacoustic methods, materials that are used for designing different pieces as vertical
and/or horizontal walls with wholes, ribs and other structural inhomogeneities, plates
with variable thick, etc.;
10. development of algorithms for designing structures with pre-defined behaviour at
waves and pulses propagation;
11. creating an interdisciplinary group that can develop complex research subjects for
future doctoral studies and projects.
23
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