1

A Planar Bézier Profiled Horn for Reducing Penetration Force in Ultrasonic

Cutting

Dung-An Wang*, Hai-Dang Nguyen

Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung40227, Taiwan, ROC

February 23, 2013

Abstract

An ultrasonic planar horn with a Bézier profile is developed. The first longitudinal

displacement mode of the horn is exploited for high displacement amplification in order

to reduce the penetration force required to enter and cut materials. The displacement

amplification and stress distribution characteristics of the Bézier horn and the commonly

used catenary horn are examined. The penetration force by the Bézier horn is nearly 75%

of that by the catenary horn with a penetration speed of 0.25 mm/s during cutting a tissue

stimulant. At a penetration speed of 0.5 mm/s, the penetration force by the Bézier horn is

nearly 85% of that by the catenary horn for cutting a polymethylmethacrylate (PMMA)

material. The decrease in the penetration force by the Bézier horn is attributed to the fact

that the displacement amplification of the Bézier horn is 30% higher than that of the

traditional catenary horn with the same length and end surface widths.

PACS: 87.55.de; 06.30.Bp

Keywords: Bézier horn; Ultrasonic cutting; Penetration force

____________* Corresponding author. Tel.:+886-4-22840531; fax:+886-4-22858362E-mail address: daw@dragon.nchu.edu.tw (Dung-An Wang).

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1. Introduction

Recent development in material technologies requires the use of high strength and

hard materials which are difficult to machine using conventional methods. The need for

less tissue damage during surgical applications demands the development of medical

instruments with small penetration force. Pandey and Shan [1] indicated that among the

non-conventional methods to cut materials, ultrasonic cutting has been used extensively

in the machining of composites, ceramics, quartz, and graphite. Charles et al. [2] used

ultrasonic cutting in ophthalmic surgery. Iula et al. [3] utilized ultrasonic bistoury in

surgical cutting procedures. An ultrasonic machine usually includes an ultrasonic

generator, a transducer, an ultrasonic horn and a tool. Kremer et al. [4] reported that the

effective ultrasonic cutting demands a high vibration amplitude at the tool tip in order to

achieve high cutting rates and low penetration force. Hence, the oscillation amplitude of

the transducer must be amplified using an ultrasonic horn to provide high displacement

output to the cutting tool.

Amin et al. [5] suggested that the performance of ultrasonic cutting depends on

the design of the ultrasonic horn to facilitate an increase in tool-tip vibration, allowing a

significant amount of material to be removed. Ultrasonic horn of different profiles have

been proposed and investigated by many researchers. Lee and Lal [6] designed a

Gaussian horn for lifetime testing of thin films. Eisner [7] deduced a horn with a Fourier

profile. Bangviwat [8] used a stepped horn as a particle velocity amplifier.

Sindayihebura and Bolle [9] developed a stepped horn for fine atomization of liquids.

Nagarkar and Finch [10] designed a sinusoidal horn for impedance transformation. Graff

[11] described a design procedure for catenoidal, conical, and exponential horns. Salmon

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[12] synthesized a new family of horns in which the exponential horn forms a central

member. Abramov [13] pointed out that conical, exponential, catenoidal, stepped and

Gaussian are the most commonly used horns, and the displacement amplification of

catenoidal horns is greater than that of exponential or conical ones and less than that of

stepped horns.

Rogers and Adams [14] indicated that the parametric curve based geometry is

flexible enough to give a much better control over the profile of horns for design purpose.

In parametric form each coordinate of a point on a curve is represented as a function of a

single parameter. Therefore, it has more potential to find higher displacement

amplification while keeping the stress in the horns low. Because the parametric curve

has more freedom to define the horn profile, it is a more difficult problem to optimize the

performance of the horn. Woo et al. [15] have used finite element method (FEM) to

study and analyze behaviors of horns. Using FEM, detailed stress and displacement

distributions can be obtained. Fu et al. [17] discussed the design of a piezoelectric

transducer with a stepped horn via multiobjective optimization. They formulated the

optimization problem using Pareto-based multiobjective genetic algorithms. In order to

design horns with conflicting design objectives, the genetic algorithms capable of finding

multiple optimal solutions in a single optimization run may be used.

In this investigation, design and analysis of a planar horn for high displacement

amplification are presented. The profile of the horn is based on a cubic Bézier curve.

The design is based on a procedure described by Wang et al. [18]. Prototypes of horns

are fabricated and tested. The experimental results are in good agreement with those

based on the design procedure. Penetration force during ultrasonic cutting is measured to

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show the advantage of the developed Bézier horn. The present work is an extension of

the previously proposed cylindrical Bézier horn by the authors [18], and is geared

towards the cutting applications of the planar Bézier horn.

2. Design

Fig. 1 shows a schematic of an ultrasonic system for cutting. An ultrasonic horn

is driven by a Langevin transducer. The horn is a displacement amplifier designed to

work in a longitudinal mode. The front end of the plate-shaped horn can function as a

tool for cutting materials. The Langevin transducer is composed of a couple of

piezoelectric disks clamped between the back metal block and the front metal block. The

flange positioned at the longitudinal nodes the Langevin transducer is mounted to a

support structure. The designated frequency of the Langevin transducer is 28.0 kHz. The

first longitudinal frequency of the horn is set to be 28.0 kHz accordingly. A typical

displacement distribution curve is also shown in the figure. By proper design of the horn

structure, the longitudinal mode of vibration of the horn can be excited, and a large

displacement amplification can be obtained. In this investigation, the performance of the

Bézier horn is compared with the commonly used catenary horn. Fig. 2(a) and (b) shows

the profiles of the Bézier horn and the catenary horn, respectively. A Cartesian

coordinate system is also shown in the figure.

The design of the Bézier horn for high displacement magnification is based on an

optimization procedure where the profile of the horn is optimized via the parameters of a

cubic Bézier curve to meet the requirement of displacement amplification. The cubic

Bézier curve is determined by a four-point Bézier polygon Q0 Q1 Q2 Q3 as shown in Fig.

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2(a). Rogers and Adams [14] described that the first and last points, Q0 and Q3,

respectively, on the curve are coincident with the first and last points of the defining

polygon. The tangent vectors at the ends of the curve have the same directions as the first

and last polygon spans, respectively. Rogers and Adams [14] gave the parametric cubic

Bézier curve as

10)1(3)1(3)1()(

3

2

1

0

3223

t

PPPP

tttttttP

Q

Q

Q

Q

(1)

where t is the parameter, andiQP is the position vector of the point iQ .

The profile of the horn is optimized by allowing points Q1 and Q2 to move in the

design space enclosed by the dashed rectangle in Fig. 2(a). The positions of the points Q0

and Q3 are fixed by the specified widths of the back and front end of the horn, W1 and W2,

respectively, and the length of the horn, L. The thickness of the horn, t , is specified as

1.2 mm. The optimization procedure has been developed and described in detail by

Wang et al. [18]. The procedure is based on the non-dominated sorting genetic algorithm,

which has been applied by Deb et al. [19] for the optimization of horn profiles. The

algorithm is suitable for solving constrained multiobjective problems.

In the optimization process, the working frequency f and the geometry parameters

W1, W2 and L are specified. The objective functions of the optimization problem are

0

3

0

Q

Q

u

uMMax

ffMin

(2)

where f0 is the first longitudinal modal frequency of the population of each generation of

the horn. M is the amplification of the displacement defined by the ratio of the

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longitudinal displacement at the front end to that of the back end of the horn. The

proposed horn is designed to have the same working frequency as the Langevin

transducer. Due to the geometry complexity, the modal frequency f0 and the

displacement amplification M of the Bézier horn cannot be calculated analytically. Finite

element analysis by a commercial software ANSYS is utilized to obtain f0 and M of the

horn.

3. Analyses

3.1 Finite element model

In order to obtain accurate modal frequency and displacement solutions for the

developed horn, finite element analyses are carried out. Fig. 3(a) shows a schematic of

the Bézier horn with the length L, the thickness t and the widths W1 and W2 of the back

and front ends, respectively. A Cartesian coordinate system is also shown in the figure.

Fig. 3(b) shows a mesh for a finite element model. Modal analysis and harmonic analysis

are carried out to obtain the modal frequencies and the amplitude-frequency

characteristics, respectively, of the horn. In this investigation, the material of the horn, a

stainless steel (SS41), is assumed to be linearly elastic. The Young’s modulus and

Poisson’s ratio are taken as 210 GPa and 0.3, respectively. The density is taken as 7800

Kg/m3. The commercial finite element programs ANSYS is employed to perform the

computations. Two-dimensional 8-node quadratic element PLANE183 is used to model

the horn. In order to compare the performances of the Bézier horn with classical horns, a

catenary horn, a stepped horn, and a linear horn are also modelled. Fig. 4(a) and (b) is a

schematic and a mesh, respectively, for the catenary horn. Fig. 4(c) and (d) shows a

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schematic and a mesh, respectively, for the stepped horn. A schematic and a mesh for the

linear horn are shown in Fig. 4(e) and (f), respectively.

3.2 Numerical analysis

In the optimization process, the length L and the widths W1 and W2 of the back

and front end of the Bézier horn are specified as 94 mm, 20 mm and 1.5 mm, respectively.

The working frequency f is set to be 28.0 kHz. The Bézier horn is designed to have its

first modal frequency equal to the working frequency of the Langevin transducer.

Following the optimization procedures reported by the authors [18], the optimal design of

the planar Bézier horn is obtained. Fig. 5 shows the profile of an optimized Bézier horn.

The position of its control points are also shown in the figure. Table 1 lists the

coordinates of the control points of the optimum design.

Modal analysis is carried out to determine vibration modes of the optimized horn.

Fig. 6 shows the results of the modal analysis in the frequency ranging from 0 kHz to

30.0 kHz. In total, one longitudinal mode (20th mode), four torsional modes (5th, 10th,

12th and 16th mode) and fifteen bending modes are observed in the frequency range of 0-

30 kHz. The modal frequency of the first longitudinal mode, the 20th mode, is 28.0 kHz

which is the specified working frequency f . Fig. 7(a) shows the vibration amplitude of

the front end of the Bézier horn as a function of the driving frequency during a harmonic

analysis. A harmonic displacement in the y direction is applied to the nodes at the back

end surface of the horn. The vibration amplitude of the front end of the horn is

normalized by the maximum amplitude at the resonance frequency. As shown in Fig.

7(a), the first longitudinal frequency of the Bézier horn is 28.0 kHz which is in a good

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agreement with the specified working frequency. Cardoni et al. [20] pointed out that

when the system is driven in the operating mode and a combination resonance occurs,

then a large amount of energy flows from the operating mode to the internal modes. In

the experiments, the system is excited close to its first longitudinal mode. The flexural

displacement of the front end of the horn at the driving frequency of 28.0 kHz is nearly

zero based on the harmonic analysis. This result indicates that the intermodal responses

may not be possible.

A catenary horn, a stepped horn and a linear horn are also designed in order to

verify the effectiveness of the Bézier horn. For fair comparison, the catenary horn has

the same back and front end widths and length as those of the proposed horn. The

working frequency of the catenary horn obtained by a finite element analysis is 27.8 kHz

(see Fig. 7(b)). The back and front end widths of the stepped horn and the linear horn are

the same as those of the proposed horn. In order to have the same working frequency of

28.0 kHz as the proposed horn, the length of the stepped horn can be obtained by finite

element analyses. The computed length of the stepped horn and the linear horn are 89

mm and 106 mm, respectively. Fig. 7(c) and (d) shows the results of the harmonic

response of the stepped horn and the linear horn, respectively. The profiles of the

catenary horn, the stepped horn and the linear horn are also shown in Fig. 5.

Fig. 8(a) is a plot of the normalized displacements along the normalized length of

the Bézier horn, the catenary horn, the stepped horn and the linear horn based on finite

element computations. The displacements are normalized by the displacement at the

back end of the horns. Therefore, the normalized displacement at the normalized length

of 1 represents the displacement amplification. The stepped horn has the largest

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displacement amplification among the four types of the horns. The displacement

amplification of the Bézier horn is much larger than that of the catenary horn and the

linear horn. As shown in Fig. 5, the narrower width near the neck (the center of the horns)

of the Bézier horn compared to the neck width of the catenary horn and the linear horn

may explain its higher displacement amplification. Fig. 8(b) shows the Mises stress

along the normalized length of the horns based on the finite element computations. The

ordinate axis in Fig. 8(b) is given in the log scale due to the large stress of the stepped

horn. For the stepped horn, high stress occurs near the abruptly changing section. Stress

concentration of the Bézier horn is significantly less than the stepped horn, and is less

than the catenary horn. The lower Mises stress of the Bézier horn can be attributed to its

bell-shaped profile (see Fig. 5). The very smooth stress distribution of the linear horn is

due to its linearly changing contour contrasted to the more severely changing contours of

the other three types of horns.

Fig. 9(a), (b), (c) and (d) shows the contour plot of the Mises stress distribution of

the Bézier horn, the catenary horn, the stepped horn and the linear horn, respectively. It

is observed that a very high stress concentration occurs at the abruptly changing section

of the stepped horn. The stress of the Bézier horn, the catenary horn and the linear horn

increases gradually along their longitudinal axis from the back end and reaches relatively

high values near the center of the longitudinal axis. The maximum stress of the Bézier

horn is lower than that of the catenary horn due to the much larger cross-section area of

the Bézier horn near its back end. The linear horn has the smoothest stress distribution.

The values of the displacement amplification M and maximum Mises stress max of the

four types of the horns are listed in Table 2. Although the stepped horn gives the highest

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displacement amplification, its high stress concentration at the step discontinuity makes it

prone to failure. The catenary horn has smaller displacement amplification and a

smoother stress distribution than the stepped horn. The Bézier horn has larger

displacement amplification than the catenary horn, and lower stress concentration than

the catenary horn and the stepped horn. Although, the smoothest stress distribution is

observed in the linear horn, but its displacement amplification is the lowest among the

four types of horns considered here.

For successful applications of the ultrasonic horns, displacement uniformity at the

front end of the horns is generally required. The uniformity of the horns is examined by

moving the probe tip of the laser displacement sensor along the width direction of the

front end surface of the horns. Fig. 10 shows the simulated vibration amplitude at the

front end surface of the horns along the width direction. The Bézier horn, the catenary

horn, the stepped horn and the linear horn all show high displacement uniformity.

4. Fabrication and experiments

In order to verify the effectiveness of the proposed planar Bézier horn, prototypes

of a Bézier horn and a catenary horn are fabricated by a laser cutting process from a

stainless steel (SS41). Dimensions of the prototypes are based on the finite element

analyses. Fig. 11 is a photo of the fabricated horns and a Langevin transducer. The horns

were driven by the Langevin transducer. Fig. 12(a) is a schematic of the experimental

apparatus for measurement of the displacement/vibration of the horns. The horns were

mounted on an optical table. The ultrasonic frequency was provided by a sinusoidal

wave from a function generator (WW5062, Tabor Electronics Ltd., Israel). This

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sinusoidal wave was then amplified by a linear voltage amplifier (LA75, Cedrat

Technologies, France) and fed into the ultrasonic horn. The amplifier can perform

amplifying operations in the -20/150 V range and has a high current capability. The

output of the Langevin transducer was monitored by an oscilloscope (DSO3062A,

Agilent Technologies, US). The vibration amplitude of the horns was measured by a

laser displacement sensor (LK-G5001, KEYENCE Corporation, Japan). The laser head

LK-H020 was held by a micro manipulator. The measurement was recorded and

analyzed by a KEYENCE software (LK-Navigator2). Fig. 12(b) is a photo of the

experimental setup.

Fig. 13 is the harmonic response of the designed and fabricated Bézier horns near

the working frequency, 28.0 kHz. The experimental working frequency of the Bézier

horn, 28.2 kHz, is in a good agreement with that predicted by the finite element analyses.

The discrepancy can be attributed to the machining error of the threaded connection

between the horn and the front metal block, as shown in Fig. 11. Fig. 14 shows the

measured vibration amplitude of the Bézier horn and the catenary horn as functions of the

applied voltage. The Bézier horn and the catenary horn are driven at 28.2 kHz and 28.1

kHz, respectively. The vibration amplitudes of both horns increase as the driving voltage

increases. The average displacement of the Bézier horn is approximately 30% greater

than that of the catenary horn for the driving voltages considered. The simulation results

listed in Table 2 shows that the Bézier horn has the displacement amplification 61%

larger than the catenary horn while the experiment shows a nearly 30% of improvement.

The manufacturing error and the misalignment due to assembly process may contribute to

the discrepancy. It should be noted that high power ultrasonic components are prone to

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non-linear response with cubic softening or hardening. The discrepancy may also be

attributed to the assumption that the finite element analyses carried out in this

investigation are based on linear elastic response.

Ultrasonic cutting is a promising technology to enhance cutting processes of soft

and hard materials. Sinn et al. [21] suggested that one effect of ultrasonic cutting is to

reduce penetration force compared to conventional cutting. The penetration force is the

force required to enter the workpiece. We investigate the effects of ultrasound and horn

types on the penetration force of ultrasonic cutting. Fig. 15 is a photo of the experimental

apparatus. The entire experimental apparatus is mounted on top of an optical table. The

horn is driven by the Langevin transducer, which is fixed to the guideway of a linear

motor system (KK5002C-200A1-F2CS2, Hiwin Technologies Co., Taiwan). The

penetration speed of the horn is controlled by an AC servo motor controller (D Series

Servo Drive, Hiwin Technologies Co., Taiwan). The specimen is attached to the top of a

digital scale (GM-5000, Lutron Electronic Enterprise Co., Taiwan), which is used to

measure the penetration force. The scale has a maximum capacity of 5000 gram and a

measurement resolution of 1 gram. Acquired data are transmitted to a personal computer

through the serial RS232 interface, and the measurement output is analyzed by the

LabVIEW software (National Instruments Co., US). Two different specimens are used in

this investigation: a tissue stimulant, raw potato, as representative for soft material, and a

PMMA material as representative for hard material. The length, width and thickness of

the specimen are 70 mm, 35 mm and 5 mm, respectively. Orthogonal cutting tests in the

longitudinal direction of the horns are performed. The penetration speed set by the servo

motor controller is held constant in each test run.

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Fig. 16 is a typical penetration force history when cutting a tissue stimulant (raw

potato) by the fabricated Bézier horn. The penetration speed is 0.2 mm/s. The plot

shows two cycles of entering and exiting the potato. First cycle and second cycle show

the force required to enter the potato with and without the ultrasound, respectively. The

penetration force with the ultrasonic excitation is approximately 30% of that without

ultrasound. Sinn et al. [21] also observed a reduction in penetration force for cutting

wood with ultrasound. They interpreted their results in terms of reduced friction forces

caused by ultrasonic vibration of the cutting tool. Zhou et al. [22] attributed the lower

penetration force of ultrasonic cutting to dynamic friction and aerodynamic lubrication

between the tool and the workpiece. They pointed out that the aerodynamic lubrication

promoted by ultrasonic excitation decreases the friction. Due to the high frequency of the

alternating contact and separation between the tool and workpiece, cooling and

lubrication is greatly increased during ultrasonic cutting, and that will cause a remarkable

reduction in the penetration force.

Fig. 17 shows the penetration force of cutting the tissue stimulant by the Bézier

horn and the catenary horn for five repeated trials. The penetration speed is 0.25 mm/s.

The averaged penetration force by the Bézier horn is nearly 75% of that by the catenary

horn. Fig. 18(a) and (b) shows the penetration force as a function of the penetration

speed for cutting the tissue stimulant and the PMMA material, respectively. The Bézier

horn and catenary horns are used in these tests. The penetration depths for cutting the

tissue stimulant and the PMMA material are kept constant at 2 mm and 3 mm,

respectively. The error bars in Fig. 18 indicate the scatter of the penetration force in five

consecutive measurements. As shown in Fig. 18(a) and (b), the penetration forces

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increases with the penetration speed for both materials. For the tests carried out, the

penetration force by the Bézier horn is significantly lower than that by the catenary horn.

At a penetration speed of 0.5 mm/s, the penetration force by the Bézier horn is nearly

85% of that by the catenary horn for cutting the PMMA material.

The results presented above show that the horn types and the penetration speed

significantly affect the penetration force in ultrasonic cutting. Regarding the need for less

tissue damage during surgical applications, the Bézier horn may require less applied

penetration force to cut fresh tissue than the traditional catenary horn based on the

experimental results. O'Daly et al. [23] suggested that for successful tissue cutting, the

reduced penetration force in ultrasonic systems is favorable to avoid excessive residual

damage to the local tissue structures. Zhou et al. [22] described that from the mechanics

point of view, ductile-regime machining of brittle and hard materials can be realized by

applying ultrasonic cutting, and therefore the material can be removed plastically. The

penetration force can be an important factor to decide the size of the stress field and the

scale of deformation in the material in front of the cutting edge of the tool. A small

penetration force makes the plastic deformation occur more easily. Zhou et al. [22]

measured the penetration force by a stepped type tool with and without ultrasonic

vibration, and found that with the ultrasonic vibration smaller penetration force is

required to enter the workpiece. They reported that the reduced penetration force is an

effective measure for increasing the critical cutting depth in ultrasonic cutting, and below

the critical cutting depth, material can be removed plastically. When the plastic

deformation is the predominant mode of deformation, a very smooth and fine surface can

be obtained. Zhou et al. [22] indicated that a cleavage process characterized by a larger

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penetration force causes a cloudy surface which is covered with residual cracks. Usage

of Bézier horns in ultrasonic cutting can reduce the penetration force, and may achieve

high quality mirror surface when machining brittle materials.

Cyclic stresses may damage the mechanical components of the ultrasonic cutting

system due to material fatigue. The low and smooth stress distribution of the Bézier horn

is favored to prevent fatigue damage during ultrasonic cutting. In order to achieve

effective ultrasonic cutting, the vibration amplitude of the ultrasonic horn with the cutting

tool attached should be sufficiently large. The developed Bézier horn with low stress

concentration, smooth stress distribution and high displacement amplification may have a

great potential in ultrasonic cutting applications.

5. Conclusions

A planar horn with high displacement amplification and low stress concentration

is developed. The profile of the horn is a cubic Bézier curve. The contour of the

designed Bézier profiled horn is a bit closer to that of the stepped horn than the

commonly used catenary horn whilst exhibiting a much smooth stress distribution than

the stepped horn. Based on the finite element analyses, maximum Mises stress of the

proposed horn is 8% lower than that of the catenary horn. Prototypes of the horn have

been fabricated and tested. The displacement amplification of the proposed horn is 30%

higher than that of the traditional catenary horn with the same length and end surface

widths based on the experiments.

Penetration forces required to enter the workpieces during ultrasonic cutting of a

tissue stimulant and a PMMA material using the Bézier horn are much lower than those

using the catenary horn. The penetration force by the Bézier horn is nearly 75% of that

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by the catenary horn with a penetration speed of 0.25 mm/s during cutting of the tissue

stimulant. The penetration force increases as the penetration speed increases based on the

experiments. At a penetration speed of 0.5 mm/s, the penetration force by the Bézier

horn is nearly 85% of that by the catenary horn for cutting the PMMA material. The

developed Bézier horn with low stress concentration, smooth stress distribution and high

displacement amplification has a great potential in ultrasonic cutting applications.

Acknowledgement

Support of this work by a grant from National Science Council, Taiwan, ROC

(Grant Number: NSC 101-2221-E-005-003) is greatly appreciated. The authors would

like to express their appreciation to the National Center for High-Performance

Computing (NCHC), Taiwan for their assistance. Helpful discussions with Mr. Yao-

Tang Lin of Precision Machinery Research and Development Center are greatly

appreciated.

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Gordon and Breach Science Publishers, The Netherlands, 1998.

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Table 1. The coordinates of the control points of the optimum design

Variable Values (mm)

),( xy of the point 0Q (0, 10.00)

),( xy of the point 1Q (60.96, 5.24)

),( xy of the point 2Q (0, 0.75)

),( xy of the point 3Q (94.00, 0.75)

Table 2. Comparison of the four types of the horns

21 /WW Length (mm) M max (MPa)

Catenary 13.3 94 4.6 151.1

Linear 13.3 106 2.3 52.7

Stepped 13.3 89 13.2 6926.0

Bézier 13.3 94 7.4 140.2

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List of figures

Fig. 1 Schematic of a horn and a Langevin transducer.

Fig. 2 (a) A profile of a Bézier horn and its four control points Q0, Q1, Q2, and Q3. (b)

A profile of a catenary horn.

Fig. 3 (a) Schematic of the horn with the length L and the widths W1 and W2 of the

back and front ends. (b) A mesh for a finite element model of a Bézier horn.

Fig. 4. (a) Schematic; (b) a mesh of a catenary horn. (c) Schematic; (d) a mesh of a

stepped horn. (e) Schematic; (f) a mesh of a linear horn.

Fig. 5. Profiles of the horns.

Fig. 6. Mode shapes and modal frequencies of the Bézier horn. In total, one

longitudinal mode (20th mode), four torsional modes (5th, 10th, 12th and 16th mode) and

fifteen bending modes are observed in the frequency range of 0-30 kHz.

Fig. 7. Harmonic response of (a) the Bézier horn; (b) the catenary horn (c) the stepped

horn and (d) the linear horn based on the finite element analyses.

Fig. 8. (a) Normalized displacements along the normalized length of the horns. (b) von

Mises stress along the normalized length of the horns.

Fig. 9. Stress distribution of the (a) the Bézier horn; (b) the catenary horn, (c) the

stepped horn, and (d) the linear horn based on the finite element analyses.

Fig. 10. Simulated vibration amplitude of the front end of the horns.

Fig. 11. Fabricated prototypes and a Langevin transducer.

Fig. 12. (a) Schematic of the experimental apparatus. (b) A photo of the experimental

setup.

Fig. 13. Harmonic response of the designed and fabricated Bézier horns.

Fig. 14. Measured vibration amplitude of the horns as functions of the driving voltage.

Fig. 15. Photo of the experimental apparatus for ultrasonic cutting.

Fig. 16. A typical penetration force history when cutting a tissue stimulant.

Fig. 17. Penetration force of ultrasonic cutting by the Bézier horn and the catenary horn.

Fig. 18. The penetration force as a function of the penetration speed for (a) the tissue

stimulant; (b) the PMMA material.

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Fig. 1. Schematic of a horn and a Langevin transducer.

22

Fig. 2. (a) A profile of a Bézier horn and its four control points Q0, Q1, Q2, and Q3. (b) A

profile of a catenary horn.

23

Fig. 3. (a) Schematic of the horn with the length L and the widths W1 and W2 of the back

and front ends. (b) A mesh for a finite element model of a Bézier horn.

24

Fig. 4. (a) Schematic; (b) a mesh of a catenary horn. (c) Schematic; (d) a mesh of a

stepped horn. (e) Schematic; (f) a mesh of a linear horn.

25

Fig. 5. Profiles of the horns.

26

Fig. 6. Mode shapes and modal frequencies of the Bézier horn. In total, one longitudinal

mode (20th mode), four torsional modes (5th, 10th, 12th and 16th mode) and fifteen

bending modes are observed in the frequency range of 0-30 kHz.

27

Fig. 7. Harmonic response of (a) the Bézier horn; (b) the catenary horn (c) the stepped

horn and (d) the linear horn based on the finite element analyses.

28

Fig. 8. (a) Normalized displacements along the normalized length of the horns. (b) von

Mises stress along the normalized length of the horns.

29

Fig. 9. Stress distribution of the (a) the Bézier horn; (b) the catenary horn, (c) the stepped

horn, and (d) the linear horn based on the finite element analyses.

30

Fig. 10. Simulated vibration amplitude of the front end of the horns.

31

Fig. 11. Fabricated prototypes and a Langevin transducer.

32

Fig. 12. (a) Schematic of the experimental apparatus. (b) A photo of the experimental

setup.

33

Fig. 13. Harmonic response of the designed and fabricated Bézier horns.

34

Fig. 14. Measured vibration amplitude of the horns as functions of the driving voltage.

35

Fig. 15. Photo of the experimental apparatus for ultrasonic cutting.

36

Fig. 16. A typical penetration force history when cutting a tissue stimulant.

37

Fig. 17. Penetration force of ultrasonic cutting by the Bézier horn and the catenary horn.

38

Fig. 18. The penetration force as a function of the penetration speed for (a) the tissue

stimulant; (b) the PMMA material.