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77
CHAPTER 9
DESIGN OF HORNS
9.1 INTRODUCTION TO HORN DESIGN
Ultrasonic horns are tuned components designed to vibrate in a
longitudinal mode at ultrasonic frequencies for thermoplastic welding.
Reliable performance of such horns is normally decided by the uniformity of
amplitude of vibration at the working surface and the stresses developed
during loading conditions. This chapter discusses horn configurations which
satisfy these criteria and investigates the design requirements of ultrasonic
horns in ultrasonic system. Design requirement includes amplitude required at
the tool end and minimal stress distribution throughout the horn while
subjected to loading. The mathematical equations are developed for analyzing
vibration system to determine the displacement and stresses. Cylindrical,
conical and exponential horns are analysed for their behaviour. This provides
a basis for design of horns. There are two approaches attempted here in
modeling the vibrations of horns, (a) Use of classical mathematical model and
(b) Use of ANSYS software. Approach (a) provides the basic understanding
of the horn vibration and (b) provides a solution that industries can use.
Computer Aided Design (CAD), Computer Aided Engineering (CAE) and
Computer Aided Manufacturing (CAM) are the procedures adopted by
industries in arriving at the best design of tools that can be manufactured with
reduced time. A CAE based approach is useful as it gives user-friendly and
quick method for manufacturing of horns. The horn gets heated as it transmits
energy and there are different temperature zones in the horn during welding.
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The areas with higher temperature if known can help the designer in
improving the life and performance of the horn by altering the design of the
horn. Therefore determining the temperature distribution in the horn is useful
for the industry.
9.2 MATHEMATICAL MODEL
The horn or the mechanical resonator is designed on the basis of
axial vibration of an elastic member with varying cross-section. It is
considered to be free-free vibration (Seah et al 1993) of a non- uniform bar.
Plane wave propagation in the rod is assumed to be only in axial direction and
propagation along lateral directions is neglected.
The generalized wave equation that is applicable is
0.12
2
2
2
U
CxS
xU
SxU (9.1)
which can be solved for different boundary conditions.
9.2.1 Solution for the Plane Wave Equation for Uniform Horn
tqxU . (9.2)
x =function of spatial coordinate ‘x’
tq = function of time ‘t’
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Since for uniform rod (Figure 9.1) change in area of cross section
xS =0 and )(2
2
tU
ttU
=
tUU
t
)( = U 2 then the equation (9.1)
becomes
2
2
2
2
xUE
tU
(9.3)
Figure 9.1 Cylindrical horn
The solution for the partial differential equation 9.1 is
x
CDx
CCtBtAU sincossincos (9.4)
Boundary conditions are
At x = 0 and x = L, 0
xU
At x = 0, U=Uo and V=Vo
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tC
xcosBCADtC
xcosBCAD
tC
xsinBDACtC
xsinBDAC
C2xU
(9.5)
Applying boundary condition at 0,0
xUx
tC
xUU coscosmax
(Final displacement equation) (9.6)
To find out stresses in vibrating uniform bar
Stress = ExU
tC
xC
EUcal cossinmax
(Final stress equation) (9.7)
9.2.2 Solution for the Plane Wave Equation for Conical Horn
The taper of a conical horn (Figure 9.2)
)( 121
11
ddxldld
dd
x (9.8)
where 1d is the diameter at the small end 2d is the diameter at larger end and
xd is the diameter at distance x from the large end.
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Figure 9.2 Conical horn
121
1221ddxld
ddxS
S
(9.9)
Substituting in wave equation (9.1)
02
2
2
121
122
2
U
CxU
ddxlddd
xU
(9.10)
The solution for the partial differential equation is
12
121
12
121
121
12 sincosdd
ddxldC
Bdd
ddxldC
Addxld
ddU
(9.11)
The constants A and B can be determined by applying appropriate
boundary condition. At x=0, U=U1 and at x=L, U=U2. At x=0, L, xU =0.
Finding the value of A and B and substituting in displacement equation (9.11)
we get
Final displacement equation
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lx
Cclx
Cddld
xddldUddU
sincos
12
2
121
212 (9.12)
To find out stresses in vibrating conical bar
Stress = ExU
lx
Clxdd
xddldUdd
Ecal cos122
121
212
lx
Clxdddc
ddddl
C
sin2122
2
12
212
(9.13)
9.2.3 Solution for the Plane Wave Equation for Exponential Horn
The exponential horn (Figure 9.3) mathematically described by
mxeSS 0 (9.14)
Figure 9.3 Exponential horn
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mxmeSxS
0 where m= taper factor and 0S =area at x=0
Substituting in wave equation (9.1)
0112
2
200
2
2
tU
CmeS
xU
eSxU mx
mx (9.15)
The solution for the partial differential equation is
xMtBDtxMDA
xMtBCxMtACeU xm
11
11)2/(
sin*sincos*sincos*sincos*cos
(9.16)
The constants can be determined by applying appropriate boundary
condition. At x=0, U=U1 and at x=L, U=U2. At x=0, L, xU =0. Finding the
value of constants and substituting in displacement equation (9.16) we get
xM
MmxMetUU
xm
11
1max sin2
cos*sin)2/(
(9.17)
To find out stresses in vibrating exponential bar
Stress = ExU
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xmxm
xmxm
cal
emxMsinxMcosM*eMm
xMsineMe*xMcosm
U*E2
1112
1
12
12
1
0
22
2
(9.18)
These equations are useful in obtaining classical solutions. But the
manufacturing of horn is by machining the stock. Therefore CAD/CAE
approach is useful. So ANSYS is used for simulating the vibrations. A tuned
horn can be validated in ANSYS and NC code can be generated for
manufacturing.
9.3 ANSYS ANALYSIS
The horn or the mechanical resonator is designed on the basis of
axial vibration of an elastic member with varying cross-section. It is
considered to be free-free vibration of a non uniform-bar. The system is
allowed to vibrate in the lowest available frequency of 20 KHz. In the present
study Aluminum has been used as horn material. Since sound wave
propagates through the resonator, generalized wave equation is applicable
here. To obtain maximum vibration at the extreme end of the rod, the length
should be of half wavelength, i.e. /2. There must be a gain for the
amplification of the amplitude of vibration from the transducer end to the
required level at the tool end. After selecting the suitable material for horn the
next step is to calculate the wavelength using the relationship among wave
length( ), frequency( f )and velocity of sound (C ) given by fC
and
EC . For axial-mode sonotrodes of certain shapes, the length of the horn
should be half of the wavelength. The diameter and shape of the horn is
85
decided by considering required value of amplitude at the tip of horn.
Vibration amplitude of the system at the booster end is 20m. This value is
taken as input amplitude to the horn and this vibration takes place only along
axial direction. Tetrahedral element is selected for modeling horn in ANSYS.
For vibration analysis this element will provide better results. The analysis is
carried out for plane strain condition. The boundary condition is applied by
restricting the displacement in tangential direction, and by allowing vibrations
in axial and radial directions.
9.3.1 Preprocessing for Cylindrical Horn Using ANSYS The cylindrical horn is modeled using ANSYS based on the
theoretical dimensions calculated and meshed using tetrahedral element. The
analysis is carried out as shown in Figure 9.4. The boundary conditions are
applied by restricting the displacement in tangential direction and by allowing
vibrations in axial and radial directions.
Figure 9.5 shows the transient loading with different time step. One
cycle of vibration can be divided in to as many number of time steps required.
The time at end of load steps and number of sub steps are mentioned as
shown in Figure 9.5.
Loading of cylindrical horn is as shown in Figure 9.6 Horn is
connected to the booster at one end. The output from booster is taken as input
to the horn. This amplitude can be converted into acoustic pressure by using
the equation
Acoustic pressure = ρ×ω×C×U (9.19)
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Figure 9.4 Applying plane strain condition-Cylindrical horn
Figure 9.5 Transient loading-cylindrical horn
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Figure 9.6 Applying pressure at booster end-Cylindrical horn 9.3.2 Preprocessing for Conical Horn Using ANSYS
Preprocessing steps for conical horn are similar to those steps stated
above in the case of cylindrical horn. The conical horn is modeled using
ANSYS based on the theoretical dimensions, which are obtained depending
on the vibration amplitude required at the other end.
9.3.3 Preprocessing for Exponential Horn Using ANSYS
Preprocessing steps for exponential horn are similar to those steps
stated above in the case of cylindrical horn. The exponential horn is modeled
using ANSYS based on the theoretical dimensions, which are obtained
depending on the vibration amplitude required at the other end.
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9.4 ANALYSIS OF HORN FOR TEMPERATURE DISTRIBUTION
USING ANSYS
Ultrasonic welding horn is a waveguide focusing device with cross
sectional area which decreases from the input end to output end. It amplifies
the input amplitude of vibration so that at the output end the amplitude is
sufficiently large for welding. Stress inhomogeneties in a vibrating body give
rise to fluctuation in temperature and hence to local heat currents. These heat
current increase the temperature of the vibrating horn.
ANSYS analysis was done on vibrating horn to find out the internal
temperature rise due to vibration. A stepped horn was used for the analysis.
Harmonic analysis was done in ANSYS environment. The type of element
used for the analysis is PLANE 223 because it is having four degrees of
freedom and the fourth one is temperature. The frequency range of
15,000-25,000 Hz was used. Vibration amplitude of the system at the booster
end is 20 µm. This value is taken as input amplitude to the horn. The
boundary condition is applied by restricting the displacement in tangential
directions and by allowing only in axial directions.
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