theoretical and experimental contributions concerning the projection ...

SISOM 2007 and Homagial Session of the Commission of Acoustics, Bucharest 29-31 May
THEORETICAL AND EXPERIMENTAL CONTRIBUTIONS CONCERNING THE
PROJECTION OF SHAPED ULTRASOUND ENERGETIC CONCENTRATOR
PROFILED TYPE USED FOR THE ULTRASOUND WELDING OF MIXED MATERIALS
Cornelia LUCHIAN, Gheorghe AMZA, Florea DUMITRACHE
University “Politehnica” of bucharest, e-mail: cornelia_lucky@yahoo.com
In the paper it is being presented the results obtained in the case of bell-type profiled
concentratrors used for the ultrasound welding of compound materials used in the automobile
construction.
Keywords: ultrasound, acoustic concentrators
1. THEORETICAL CONTRIBUTIONS
The ultrasound energetic concentrators must be calculated so that they should work with each
component and fulfill the following functions: to conduct ultrasound energy from the transductor to the place
where the welding takes place, to concentrate and easy the ultrasound energy in the area, to increase the
particle’s speed value watching the enlargement of the work tool and at the same time the acoustic intensity,
through their form they lead to a very diverse scale of welded constructions and to obtaining different kinds
of acoustic waves ( longitudinal, transversal, of surface, of torsion, radial or combinations), to allow
sustaining or fixing the entire acoustic system within the whole welding installation in the crucial plans.
To work with good efficiency the concentrator must be tuned with an approximation of a few
frequencies with which the ultrasound transductor is calculated, so the entire moving system must work in
terms of resonance so that the particle's speed amplitude at the top of the sonotrode was even bigger and, so,
also the acoustic intensity [1], [2].
For the calculus of acoustic transformers it can be applied the theory of elastic waves spreading
through variable section rods, being taken into consideration the unmatched vibrations case, where the
waves’ equation has the following expression:
2
∂ Φ
− c
2
∂t
2
2
∂Φ∂
2 ∂ Φ
(ln S
x
) − c = 0
( 1 )
2
∂x∂x
∂x
in which: Φ is the speed potential; Sx – the surface area of the bar’s section at a distance x from its
origin; c – the speed of the acoustic waves spreading waves through the material from which the rod is made
of.
It is obvious that the waves’ equation will take different shapes related to the variation type of the
rod’s section in length: linear, in scales, exponential, catenarial, the different order (parabola) and
combinations of the forms mentioned above.
In case of acoustic transformers that have the following section’s form:
−αx
S S e
( 2 )
x
= 0
where α is the characteristic coefficient of the section’s variation type, the equation (1) becomes:
2
2
∂ Φ ∂Φ 1 ∂ Φ
−α = .
( 3 )
2
2 2
∂x
∂x
c ∂t

Considering as the equation’s (3) general solution is:
j(
ω t+γx)
( t,
x) = Ce
Φ ( 4 )
and by replacing in the equation Φ(t,x) and its derivatives you obtain the general solution of the equation (3):
Φ
⎛
⎞
α
2 α
x
2 2
jx k −
2 jx k α
jωt
t x = e
⎜ − − / 4
4
( , ) Ae + Be
⎟
e
( 5 )
⎜
⎝
where k = ω / c, and A and B are two invariables that can be determined from the limit-conditions.
From (5) we realize that there are two waves in the rod: the first one spreads from the bottom to the
top and the other one spreads from the top to the bottom coming from the reflection (regressive wave).
If you know the speeds potential you can determine the acoustic pressure’s expression in a point of
distance x from the origin with the following relation:
p
x
2
⎟
⎠
α ω
ω
∂Φ
x ⎛ − j x j x ⎞
2 c
c jωt
ρ jρωe
⎜ '
'
= − = − Ae Be ⎟e
t
+
( 6 )
∂
⎝
⎠
The VX particle’s speed expression can be determined in the same way with the expression:
Where
v
x
=
c’ = c/
∂Φ
∂x
= e
α
x
2
⎡ ⎛α
ω ⎞
⎢A⎜
− j ⎟e
⎣ ⎝ 2 c'
⎠
1 ω
ω
− j x
c'
⎛α
+ B⎜
+
⎝ 2
ω ⎞
j ⎟e
c'
⎠
ω
j x
c'
⎤
⎥
⎦
( 7 )
2 2 2
− α c / 4
( 8 )
and ρ – the density of the material the rod is made from.
For the use of acoustic transformers it is necessary that we know what is the particle’s speed at the
initial and final sections.
⎡ ⎛α
ω ⎞ ⎛α
ω ⎞⎤
jωt
x⇒
0 → v = ⎢A⎜
− j ⎟ + B⎜
+ j ⎟ e
0
c
c
⎥
⎣ ⎝ 2 ' ⎠ ⎝ 2 ' ⎠⎦
( 9 )
x⇒
L → v
L
= e
α
L
2
⎡ ⎛α
⎢A⎜
−
⎣ ⎝ 2
ω ⎞ ⎛α
j ⎟ + B⎜
+
c'
⎠ ⎝ 2
ω ⎞
j ⎟e
c'
⎠
ω
j L
c'
⎤
⎥e
⎦
jωt
( 10)
In most cases it is preferable that the length of the acoustic transformers L was equal with a whole
number of wave length values, meaning:
λ' nc'
L = n =
2 2 f
A very important measure that characterizes the acoustic transformer is the N amplifying coefficient,
defined as the particle’s speed on the final section against the particle’s speed on the initial section.
( 11 )
N
v
=
v
L
0
=
α
L
n 2
( −1)
=
e
S
S
0
f
( 12 )
If you know the N amplifying coefficient (as an initial condition for the transformer) you can
determine that coefficient α with the following relation:
385

2ln N
α = ( 13 )
L
The most important element of the transformer remains, still, the length - L – upon which must be
applied certain corrections depending on the variation type of the sections and the way in which the elements
are connected. In addition, using the relations (10), (12) and (13) you can determine the length of the
acoustic transformer with the following expression:
in which:
f e
nc
L = . 2
f e
1 π
2
+ _(ln N / n )
( 14 )
is the calculus frequency in Hz, c – the longitudinal waves’ spreading in a material from which
the rod is made in m/s, n – full number.
Another important parameter for the calculus of a moving system, but also for the concentrator is the
knowing of the nodal plans, necessary for fixing the entire moving system in the installation that he is part
of. For determining the positions of the nodal plans it is imposed the canceling of the Vx = 0 speed, and for
the relation (7) you obtain:
⎡α
⎛ ω ⎞ ω ⎛ ω ⎞⎤
(A+B) ⎢ cos⎜
x⎟
− sin⎜
x⎟
= 0
2 ' ' '
⎥
⎣ ⎝ c ⎠ c ⎝ c ⎠⎦
Table 1: The values of the ‘N’ multiplication coefficient
Crr. The section’s form and the concentrator’s type
Nr.
The multiplication
coefficient in functioning
estate
Observations and
notes
0 1 2 3
1
( 15 )
N
=
D
0
D f
cylindrical,
exponential
2
N
=
a
0 .
b
0
a f
b f
rectangular,
exponential
3
N
=
D
2
D
− d
2
circular, exponential
4
N r
=
cos Kl
N
cos Kl
1
2
cylindrical,
exponential
K =
ω
c
l = 0 3
386

that results in:
c'
⎛α
c'
⎞
x nodal
= arctg⎜
+ n'
π ⎟
( 16 )
ω ⎝ 2 ω ⎠
Taking in consideration that L is a full number with semi-lengths of waves, expression (16) becomes:
L ⎛ 1 ⎞
x nodal
= arctg⎜
ln N + n'
π ⎟
( 17 )
nπ
⎝ nπ
⎠
in which n and n’ are full numbers, n’ = 0 for the first nodal point and n’ = 1 for the second nodal point etc.
In table 1 are given the values for the ‘N’ multiplication coefficient for some types of sections, in
table 2 are indicated the parameters of some types of acoustic transformers [2], [5] .
Table 2: The parameters of some types of acoustic transformers
The type of
the
concentrator
The
section’s
variation
law
D
x
=
N =
D
D
D
0
f
0
e
α
− x
2
ω ln N
α =
c π +
( ln N ) 2
D x
= D 0
for
x
D f
0 ≤ x ≤
1
2
D = for 1≤ x ≤1
D
x
α =
=
ω
c
D
0
e
α
− x 4
The
theoretical
coefficient
of
amplifying
of the
amplitude
The length
of the
concentrators
The nodal
point’s
coordinates
(nodal x)
N
D
D
r
= 0 =
r
N
2
N
r
2
⎛ D ⎞
= = N
⎜
D ⎟
⎝ f ⎠
0 2
c ⎛ln
N ⎞
L = 1+ ⎜ ⎟
L
2 f ⎝ π ⎠ f
x
nod
L ln N
1 c
= arctg
x
π π
nod = =
2 4f
Nr
= 1+
4lnN
= λ c
2 =
λ ⎡ 2 ln N
L = ⎢ +
1
2 ⎣ π 2
2
λ arctg(2 ln N )
+ ⋅
2 π
x
nod
L 2ln N
arctg
⎤
⎥ +
⎦
= π π
387

2. EXPERIMENTAL CONTRIBUTIONS
Using the methodic projection of the transductors and ultra-acoustic concentrators it has
been created the ultrasound concentrator and transductor presented in the figures from 1 to 7 [3], [4].
Fig. 1: the geometrical dimensions of the ultra-acoustic system with a bell-type concentrator used for experiments: 1 –
piezoceramic group; 2 – movement amplifier; 3 – bell-type concentrator;
Fig. 2: the geometrical dimensions of the ultra-acoustic system with a prismatic-type concentrator used for experiments: 1 –
piezoceramic group; 2 – movement amplifier; 3 – prismatic-type concentrator;
Fig.3: general view on the ultra-acoustic system with a bell-type concentrator:
1 – piezoceramic group; 2 – movement amplifier; 3 - concentrator
388

Fig 4: general view on the ultra-acoustic system with a a prismatic-type concentrator:
1 - piezoceramic group; 2 - movement amplifier; 3 – concentrator
Fig. 5: The geometrical dimensions of the movement amplifier
Fig. 6: the geometrical dimensions of the bell-type acoustic concentrator
Fig. 7: the geometrical dimensions of the prismatic-type acoustic concentrator
389

3. CONCLUSIONS
The acoustic intensity obtained through an ultrasound transductor has a limited value determined by the
nature of the material that it is made of, this is why it is necessary to create in a relatively reduced volume
much bigger densities of acoustic energy which should allow the ultrasound welding of composed materials.
The ultrasound energy concentrators which represent the ultra-acoustic brain of the system must be
calculated in such way that they should fulfill the following functions: they conduct the ultrasound energy
from the transductor to the place of the welding; they concentrate and focus the ultrasound energy in the
combining zone; they amplify the particle’s speed value by increasing the amplitude of the work tool and
also the acoustic intensity; through their shape they allow a very diverse scale of welded combinations and
the obtaining of different types of acoustic waves (longitudinal, transversal, of surface, of torsion, radial or
combinations); they allow the sustaining or fixing of the entire acoustic system in the whole of the
installation that must be welded in the nodal plans.
To work with better efficiency, the ultrasound system should work in terms of resonance so that the
particle’s speed amplitude and so the acoustic intensity are as big as in the welding zone.
For the calculus of ultrasound energy concentrators it is being applied the elastic waves spreading
theory in rods of variable sections while the primary calculus elements are: the concentrator’s length, the
multiplication coefficient, the coefficient of the variation section, the diameter in each section, the nodal
points’ coordinates, the weight and the relative disagreement of frequency.
For a quick projection of the concentrators there has been build calculus nomograms, the problem of
executing them through different methods remaining open. The functioning of the concentrators at the
established efficiency and their fiability in functioning depend on the way they are executed, on the surfaces
qualities, on the dimensional precision and on the nature of the material that they are made of. The surface of
the concentrators must have a roughness Ra