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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

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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.