Theoretical and Experimental Research on Design of Non ...

THEORETICAL AND EXPERIMENTAL RESEARCH ON DESIGN OF NON-RESONANT GEAR TRANSFORMER

Wang Shiying College of Mechanical Engineering of Taiyuan

University of Technology Taiyuan City, Shanxi Province, China

Lv Ming College of Mechanical Engineering of Taiyuan

University of Technology Taiyuan City, Shanxi Province, China

Ya Gang College of Mechanical Engineering of Taiyuan

University of Technology Taiyuan City, Shanxi Province,

China

Liu Jiancheng School of Engineering and

Computer Science University of the Pacific

3601 Pacific Ave., Stockton, CA 95211-0197

Liang Guoxing College of Mechanical Engineering of Taiyuan

University of Technology Taiyuan City, Shanxi Province,

China

ABSTRACT Honing process is widely used to improve the surface finish

of hardened gears. The existing challenging for this

manufacturing process includes its low efficiency and excessive

forces exerted on both of engaged gears and tools. Ultrasonic

assisted honing has a number of superior advantages over the

traditional methods. This paper presents a new theoretical

method for the parameter determination of non-resonant gear

transformer, which is the key component of ultrasonic assisted

honing systems. A mathematical model for the ultrasonic

assisted honing system is first established and solved with

different design parameters by using the numerical method. The

numerated results then are verified by the FEM analysis and

experiments. It is found that the proposed method is effective

and useful. The details will be addressed in the paper.

1 INTRODUCTION

Carburized and quenched gears have been widely employed

in automobiles, tractors and machine tools due to their high

bearing capacity, longer operational life, compact size, and low

volume to weight ratio[1,2]

. A number of manufacturing

processes including honing are used to fabricate gears. Honing

operation for gear fabrication is mainly used to improve the

surface finish of the gear teeth. However the existing gear

honing process is of excessive machining forces, low efficiency

and frequent jam of honing wheels[3]

. It is known that

ultrasound assisted machining has many superior advantages for

the hard and brittle materials[4]

. These advantages include: (1)

higher material removal rates due to the strong impacting

acceleration of abrasives that is thousands times higher than the

acceleration of gravity; (2) the explosion machining action

produced by ultrasonic cavitations effect on machining fluids;

(3) the better cleaning action on honing wheels created by

cavitation effect; (4) the lower machining force due to

lubrication effect of ultrasonic vibration. Therefore, the

ultrasound assisted gear honing has the potential to make up the

shortcomings of conventional gear honing.

In this research, a carburized and quenched gear is the

object to be machined. It is attached to the ultrasonic vibration

system. Conventional gears are simplified as a thin annular

plate because of its relatively large diameter compared to its

thickness. In an ultrasound-assisted honing process, since a gear

is to be machined, its size is not determined by the design

frequency of ultrasonic vibration system, but decided by the

application requirements of the gear. It is known that the

adjustable frequency of actual ultrasonic vibration system

consisting of a transducer and an ultrasonic generator etc. is

limited to a certain range, so it is difficult to assure that the

resonant frequency of arbitrary size gear matches the range of

ultrasonic vibration system. Therefore, the gear is considered to

be a non-resonant load, the transformer consisting of the gear

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-39714

1 Copyright © 2014 by ASME

and a horn cannot be designed by applying the whole resonant

theory[5]

. It has been theoretically and experimentally proved

that the resonant frequency of a complex structure has a

relationship to its combined units[6]

. According to this principle,

a complex ultrasonic vibration system consisting of a gear with

a fixed size and a horn with adjustable size can vibrate at the

resonant frequency in the permitted range of transducer and

ultrasonic power; therefore ultrasonic gear machining becomes

feasible. As the resonant frequencies of gear and horn are

different from the frequency of ultrasonic generator, this

composite unit of a gear and horn is defined as a non-resonant

vibration system. Therefore, it is clear that how to design a

transformer in light of the structural size of machined gear and

realize the resonance vibration of the transformer in the range

of specified frequency is a technical issue that needs to be

solved when designing an ultrasound assisted gear honing

system.

At present, there are two approaches to design an ultrasonic

vibration system: one is that the cutting tool head is ignored in

the design of transformer when the tool size is small; another

one is that the tool head with a large size is included. Its

vibration is assumed to be free vibration. Normally, the honing

wheel and lapping wheel are typical plate-shape parts and their

sizes and masses are much larger than small drill or milling

cutters. They cannot be omitted. Lots of researches have been

conducted about the vibration characteristics of a single plate or

an annular plate. Zhang Chuan et al. studied the frequency

equation of an axisymmetric flexural vibrating free-edged thin

plate and its solving process[7]

. Liu Shi-qing investigated the

equivalent circuit of a thin annular plate with a tapered section

plane and derived its frequency equations[8]

. Wang D S et al.

studied the acoustics characteristics of a bending vibration disc

and the reduction methods of its transverse vibration equations,

and also presented the method of resonant frequency[9]

. The

sizes of honing wheel and lapping wheel can arbitrarily be

decided because they are the tool of ultrasonic machining

system. Therefore, the sizes of an annular plate or a circular

plate can be designed according to the resonant frequency of

ultrasonic machining system by means of the above mentioned

methods. Through assembling the plate and the horn with

same resonant frequency together, the formed transformer

vibrates around the designed frequency. The above mentioned

research results solved the design problem of the transformer

which consists of an annular plate and a horn in a particular

situation where the annular plate size can be determined

arbitrarily.

The essence of the above mentioned design method for

transformer is that the stress of the joint between the annular

plate and the horn of transformer is smaller as they are in free

resonant vibration. It is considered that the stress acting on the

joint will be significant if the annual plate and the horn are of

different resonant frequencies[10,11,12,13,14]

. The excessive stress

results in a shorter life of the system. Therefore, the objective of

this research is to introduce a new design algorithm for

transformer. This method takes the coupling stress of the joint

between the annular plate and the horn as one of the boundary

conditions. Resonant frequency equation of transformer is

derived by use of the established vibration equations of the

annular plate and the horn. The resonant frequency is obtained

by solving the above mentioned equations to obtain the

theoretical solutions of design parameters. The obtained

theoretical results are verified through the finite element method

and experimental study for the fabricated gear transformer

system. The consistency of theoretical and experimental results

shows that the proposed theoretical design method is valid and

efficient for the design of transformer consisting of a non-

resonance annular plate and a horn, which provides a new

design method of the ultrasonic assisted gear honing vibration

system. The details of this method are addressed as follows.

2 CONFIGURATION AND MATHEMATICAL MODEL OF ULTRASONIC GEAR HONING VIBRATION SYSTEM.

2.1 Structural configuration of ultrasonic gear honing

vibration system

Gear honing operation resembles a pair of gear’s meshing.

One gear is as the cutting tool whose surface is made of

abrasives; another one is the gear to be machined. The

machined gear is mounted on the end of a mandrel. The

mandrel is held between the headstock and the footstock on the

table of the gear honing machine. The honing tool is attached in

a cutter shaft, which rotates at a determined speed. When the

machined gear and the honing tool are engaged each other

under a certain level of pressure, the gear is honed. While

engaging, the machined gear vibrates, which materializes the

ultrasound assisted gear honing. The headstock side of the

ultrasonic gear honing system is shown in Fig.1.

The ultrasonic vibration of machined gear is excited by a

transducer, a rod and a horn. The rod is attached to a sleeve

with bolts at the vibration nodal circle; the sleeve is mounted in

the headstock supported by two bearings; the headstock is

attached on the machine tool table. Note the machined gear is

attached to the horn and tightened through a nut. It is rotating

during machining, thereby the power of the transducer is

required to be provided by a pair of brushes, which are

connected with the ultrasonic generator.

Fig. 1 Structural schematics of vibration system


of ultrasonic gear honing

2.2 Mathematical model for vibration system of ultrasonic

gear honing.

The ultrasonic vibration system is designed separately for

each component. The summation of the lengths of the

transducer, rod and transformer is chosen to be equal to the

half-wavelength based on the whole resonant theory. Fig. 2(a)

shows the structure of the transformer consisting of a rod, a gear

and a nut. Although the gear structure is comparatively

complicated and the horn is a stepped shaft with an exposed

surface, the transformer is treated as a whole because the gear,

the horn and the nut are made of the same material. They have

tight junctions where the sound media such as Vaseline is laid

on. The mathematical model for the vibration system of

ultrasonic gear honing is shown in Fig. 2(b). The machined gear

is simplified as an annular plate with an outer diameter 2R2 of

its reference circle and the inner one 2R1 of its hole (a little bit

large than the inner diameter of the gear). The horn is a

composite with one cone and two cylinders, it can be divided

into three parts of the cone, cylinder and cylinder with a single

generatrix function respectively according to the requirement

for building different boundary conditions. The calculating

inner bore diameter of annular plate is 2R1, the length of the

second and third cylinders are L2, L3, and the thickness of gear

is omitted here. The chosen vibration modes in gear honing are

that the gear vibrates transversely with only circular nodal lines,

without linear ones, and the horn vibrates longitudinally. The

boundary conditions of the mathematical model can be gained

as following:

The annular plate vibrates transversely and its outer edge is

in free vibration, so the external shearing force is zero. We have

02

RrrQ (1)

(a) Structure

(b) Model

Fig. 2 Structure and model of the transformer

The bending moment is also zero at the outer boundary. So,

02

RrrM (2)

The slope of the deflection is zero because of the rigid

connection at the inner edge of machined gear with the horn:

0)( 1 Rw (3)

Because the displacement is continuous at the common part

between the annular plate and horn, so the boundary condition

is:

)()( 1212 RwLL (4)

The force at the right end of the first cylinder horn equals

the sum of one at the inner perimeter of the annular plate[11]

and

one of the left end of the second cylinder horn:

2121123 LLzLLzRrr FFQ

(5)

The displacement of the left end of conical horn reaches its

maximum, so the boundary condition is set as:

0)0(1 (6)

Because of the continuity of the displacement and the

acting force of the junction between the cone and the first

cylinder, the boundary conditions at the junction are expressed

by Equations (7) and (8):


)()( 1211 LL (7)

)()( 1111 LFLF (8)

Because of the continuity of the displacement at the

junction between the first and the second cylinder, the boundary

condition at the junction is as follows:

)()( 213212 LLLL (9)

Because the right end of the second cylinder is free, its

boundary condition is stated as follows:

0)( 3213 LLL (10)

3 FREQUENCY EQUATIONS OF ULTRASONIC GEAR HONING TRANSFORMER

The gear is simplified as a thin annular plate because the

thickness of gear is constant and much smaller compared to its

diameter. The cylindrical coordinates (r, θ , z) of transformer

is used as shown in Fig. 2(a) for convenience. When the annular

plate vibrates transversely with only circular nodal lines and

without linear ones, its displacement (the time factor tje is

neglected and so is in the following equations) is represented by

the following equation[6]

:

)()(

)()()(

0403

0201

rKCrIC

rYCrJCrw

(11)

Where J0, Y0 are zero-order Bessel’s functions of the first

and second kind, respectively. I0 and K0 are modified zero-order

Bessel’s functions, respectively. ω is the angle frequency, γ is

the flexural rigidity of the annular plate,

)]1(12/[ 22 Eh

Where, E, ρ, σ are the elastic modular, the density and the

Poisson’s ratio respectively.

The displacement equation of the conic horn is shown as

follows[15]

:

zBzA

z

z

sincos1

1)( 111

(0≤z<L1) (12)

Where NL

N 1 ，

1

3

R

RN ，

c

，L is the

length of the conic horn, c is the light speed of longitudinal

wave:

Ec .

The displacement equation of the first cylindrical horn is:

zBzAz sincos)( 222 （13）

The displacement equation of the second cylindrical horn

is:

zBzAz sincos)( 333 （14）

In order to decrease computing quantity, two integrated

constants A1 and B3 can be eliminated by means of the boundary

conditions in equation (6) and (10).

According to Eq. (6) 0)0(1 : 011 BA

Substituting the above equation into equation (8) and then

taking the first order derivative yields the following equations:

zzz

Bz

sincos

1)( 1

1

(15)

zzz

zzz

Bz

sin11

cos11

)(

2

2

2

211

(16)

From equation (10) 0)( 3213 LLL , we have

0)(cos

)(sin

3213

3213

LLLB

LLLA

zLLLzAz sin)(tancos)( 32133

(17)

zLLLzAz cos)(tansin)( 32133 (18)

Substituting equations(11)、(13)、(15)-(18) into boundary

conditions equations (1)-(5) 、 (7)-(9) yields the following

equations(19)-(26)：

From equation (1) 02

RrrQ ：

01

2

22

2

Rr

Rrrr

w

rr

w

rDQ

Where: 2hD

Taking the derivate for equation (11) and substituting it into

above equation will yield:


0)(K)(I

)(Y)(J

421321

221121

CRCR

CRCR

(19)

From equation (2): 02

RrrM :

0

2

22

2

Rr

Rrrr

w

rr

wDM

Taking the derivative for equation (11) with respect to r and

substituting it into the above equation yields:

0)(K1

)(K

)(I1

)(I

)(Y1

)(Y

)(J1

)(J

421

2

20

321

2

20

221

2

20

121

2

20

CRR

R

CRR

R

CRR

R

CRR

R

（20）

Taking the derivate for equation (11) with respect to r and

substituting it into the equation（3） 0)( 1 Rw yields:

0)(K)(I

)(Y)(J

114113

112111

RCRC

RCRC

（21）

From equation（4） )()( 1212 RwLL :

0)(sin)(cos

)(K)(I

)(Y)(J

212212

104103

102101

LLBLLA

RCRC

RCRC

（22）

The shearing force per unit arc at the inner perimeter of the

annular plate is[11]

:

1

1

2

2

3

3

22

2

3

3

1

1

11)(

Rr

Rrr

dr

wd

rdr

wdD

dr

dw

rdr

wd

rdr

wdDRq

The resultant shearing force transmitted to the main system

is then estimated by:

411311

211111

3

1

11

)(K)(I

)(Y)(J2

)(2

CRCR

CRCRDR

RqRQ rr

The force of the right end of the conical horn is[15]

:

zSEzF

)(

)(cos)(tan

)(sin

)(

21321

2123

21323 21

LLLLL

LLESA

LLESF LLz

)(cos

)(sin

)(

2122

2122

21222 21

LLESB

LLESA

LLESF LLz

As equation（5）21211

23 LLzLLzRrr FFQ :

0)(cos)(tan

)(sin

)(cos)(sin

)(K2)(I2

)(Y2)(J2

21321

2123

21222122

41113111

21111111

LLLLL

LLESA

LLESBLLESA

CRDRCRDR

CRDRCRDR

（23）

From equation（7） )()( 1211 LL :

0sincos

sincos1

1212

11

1

1

LBLA

LLL

B

（24）


From equation（8） )()( 1111 LFLF and considering

of z

SEzF

)( :

0cossin

sin11

cos11

12212

1

1

2

2

1

2

1

1

2

1

1

LzBBLA

LLL

LLL

B

（25）

From equation（9） )()( 213212 LLLL :

0)(sin)(tan

)(cos

)(sin)(cos

21321

213

212212

LLLLL

LLA

LLBLLA

（26）

To simplify the expression of equations （19）-（26）,

their coefficients are revised with Cij (i,j=1,2,3,4,5,6,7,8),

Where i is the sequential number of 1C 、 2C 、 3C 、 4C 、

1B 、 2A 、 2B 、 3A , j is the sequential number of equations

（19）-（26）:

0414313212111 CCCCCCCC

0424323222121 CCCCCCCC

0434333232131 CCCCCCCC

0247246

444343242141

BCAC

CCCCCCCC

0358257256

454353252151

ACBCAC

CCCCCCCC

0266165 ACBC

0277175 BCBC

0378277276 ACBCAC

If non-zero solutions are obtained, the determinant of the

above equations must be equal to zero because the coefficients

Cij are not all zeroes:

0

00000

000000

000000

0

00

0000

0000

0000

888786

7775

6665

58575654535251

474644434241

34333231

24232221

14131211

CCC

CC

CC

CCCCCCC

CCCCCC

CCCC

CCCC

CCCC

（27）

Equation (27) is the frequency estimation equation of the

transformer. It is a transcendental equation and can not be

solved analytically, therefore it must be solved by using of the

numerical method.

4 DESIGN OF DIMENSIONS OF ULTRASONIC TRANSFORMER

The material properties of the transformer are:

E=2.092×1011

N/m2, σ =0.29, and ρ=7810kg/m

3 respectively.

The inner and outer radii of the annular plate are R1=22.25mm

and R2=66mm, respectively. Note that the machined gear is of

teeth number z=44, mode number m=3, the reference circle

diameter 132 mm. The radius R3 is set to be 29mm and its

thickness h to be 20mm. As previously indicated, the equation

(27) is a transcendental equation; its analytical results cannot be

obtained. Therefore, considering the frequency of the

transformer and the length of the horn as variables, the value

Δ of frequency equation is calculated, wherein the frequency

varies between 13~33kHz, the length of the horn varies between

65~265mm. The relationships among the value, the vibration

frequency and the length of the first cylinder are obtained and

shown in Fig. 3.

Fig.3 Relationships among the value,

vibration frequency and the length of cylinder


http://www.iciba.com/number/

In Fig. 3, it can be found that there are a number of peaks

above the zero plane and a number of valleys under the zero

plane. This result means that there are many zeros at the curved

surface of valueΔ , these frequencies and lengths at the zero are

all solutions of the frequency equation (27). Fig.3 also shows

that there is only a solution at lower frequency among the

analytical lengths, whereas more than two solutions at higher

frequency. This means that the number of solutions will

increases as the design frequency increases. Therefore, it is can

be decided that the designed transformer is feasible.

In order to accurately determine the design parameters,

select 20kHz as the design frequency according to the central

frequency of used ultrasonic machining devices, and allow the

variation range of the length of horn L2 be between 0~200mm,

the solved results are plotted in Fig. 4. The length of the horn

L2=38.9mm can be obtained as the valueΔ of frequency

equation equals only -2.765×10-6

.

The curved surface in Fig. 3 can be analyzed further by

means of the curve in Fig. 4. The following calculation is made

as L2=19.6mm:

499.0568488.1

)256.1920(02428.0)( 321

LLL

As L2=19.7mm:

50004.0570916.1

)257.1920(02428.0)( 321

LLL

Fig. 4 Relationships between the value and

the length of cylinder at f=20000Hz

From the above calculations, the two lengths of the horns is

a quarter of wavelength, and 2

)( 321

LLL is an

asymptote of )(tan 321 LLL , it means that the plus pole

at L2=19.6mm and the minus pole at L2=19.7mm are caused by

)(tan 321 LLL , and the intersection point of the linking

line between the two poles and the line of Δ =0 is not a

solution of frequency equation. In a similar way, the intersection

point between L2=149mm and L2=149.1mm is also not a

solution. The solution of frequency equation should fall in the

range of L2=19.7mm and L2=149mm. Therefore, it can be

concluded that the L2=38.9mm is the solution of the frequency

equation of Fig.4.

To demonstrate that the transformer consists of the non-

resonant annular plate and horn, the resonant frequencies of

their free and lone vibration are calculated. The first and second

order resonant frequencies[6]

of the transverse vibration of the

annular plate are 9594.98Hz and 62976.79Hz respectively and

the first order longitudinal vibration frequency[16]

have reached

29118Hz, they are so far from the design frequency of 20000Hz

that it can be fully proved that the transformer consists of the

non-resonant annular plate and horn.

5 SUBSTITUTABILITY OF THE WHOLE-RESONANCE DESIGN THEORY BY THE NON-RESONANCE ONE

The whole-resonant design theory of a transformer requires

that the dimensions of the plate and horn can be designed

individually according to the selected resonance frequency of

the ultrasonic vibration system and then the transformer

assembled vibrates around this frequency. In order to explain

the substitutability of the whole-resonance design theory by the

new non-resonance one presented in the paper, let the free

resonant frequency of an annular plate and a conical horn be

20000Hz. Their resonant structural dimensions are chosen to be

R3=29 mm，R1=12.5 mm，L=137.85mm for the conical horn

and R1=12.5mm，R2=75 .8mm，h=12mm for the annular plate,

therefore a new transformer can be obtained which consists of a

resonant horn and a resonant annular plate. Fig. 5 shows the

value Δ of the frequency equation of the transformer with the

frequencies. It can be found that the value Δ at around

20000Hz is not equal zero; there is also an extreme value. This

means that there is a resonant frequency for transformer

vibration near the 20000Hz. In fact, the value of the frequency

equation is only -0.00038. It is small and may be caused due to

the computational errors. Therefore, the new design method in

the paper can substitute the existing whole-resonant design

theory.

Fig. 5 Value of frequency equation with frequencies


6 MODE ANALYSES AND DYNAMIC EXPERIMENT OF TRANSFORMER

To verify the validity of the above mentioned numerical

solution, the FEM is used to analyze the designed results [17,18]

.

The length, the frequency and the material parameters of the

FEM are as the same as those used in numerical calculation.

The meshes are generated automatically with solid 95 elements

by the ANSYS software as showed in Fig.6, the Block Lanczos

method is used to extract the mode and 30-order modes are

extracted. The resonant frequency solved by FEM is 23504Hz

which is near to the resonant frequency of 20000Hz, and the

theoretic and FEM results are in better agreement.

Fig. 6 Vibration mode of ultrasonic honing gear system

(wherein: vibration frequency=23504Hz)

A transformer as shown in Fig. 7 can be build after

connecting the cone, cylinder and nut, which are fabricated

based on the theoretical design structural dimensions. The

transformer is connected with transducer showed in Fig. 1. The

main technical specifications of a used transducer YP-5520-4Z

are: the resonant frequency: 20±1KHz, the power: 1.2KW, the

diameter: Φ55mm.

Fig. 7 Experimental mode of the plate

In order to study the transverse vibration orders of the

annular plate, the 120# SiC abrasives are scattered uniformly on

it. After the transformer vibrates for a time-interval, the

abrasives is getting together quickly into a circular nodal line.

This phenomenon shows that the annular plate is really

undergoing the first order transverse vibration with a simple

circular node line.

The gear substituting for the plate was assembled with the

horn, the gear transformer was obtained as shown in Fig. 8.

There is solely a circular nodal line produced on the gear after

doing the same experiment as the above. Hence, it is proper to

simplify a gear to an annular plate with the outer diameter of the

gear reference circle.

Fig. 8 Experimental mode of the gear

7 CONCLUSIONS

A new design theory for ultrasonic vibration systems with

non-resonant units is proposed in this paper. It is found that the

machined gear can be simplified as an annular plate with the

outer diameter of its reference circle and the inner diameter of

its assembling hole, and the dynamical equations of the annular

plate are established based on displacement equations and

boundary conditions, and one of boundary conditions is the

force coupling at the joint between the annular plate and the

horn. The frequency equation is derived analytically and solved

by use of the numerical method; the designed parameters can be

obtained. The finite element method and dynamic experiments

are employed to validate the theoretical method. It is improved

that the structural parameters and vibration modes decided

theoretically are consistent with design requirements. The

dynamical experiment of corresponding gear showed that it is

feasible to simplify a gear to an annular plate; the design

precision meets the demand of industrial application.

Consequently, the non-resonant design method is an efficient

method for the transformer design. The new design method

described in the paper can also substitute for the existing whole-

resonance method. The non-resonant theory discussed in this


paper provides the theoretical foundation for the design of

ultrasonic vibration system with arbitrary size gears.

ACKNOWLEDGMENTS The author gratefully acknowledges the support of K.C.

Wang education foundation, Hong Kong; and the support of the

National Science Foundation of China under Grant No.

50975191.

REFERENCES

[1] B. Karpuschewski, H.-J. Knoche, M. Hipke. Gear finishing by abrasive processes[J] . CIRP Annals - Manufacturing Technology 57 (2008) 621–640

[2] LV Ming，MA Hongmin，XU Zeling. Study on new manufacturing process of gear-honing-tool used for hardened gear[J] .Key Engineering Materials, 2004，259-260：10-13.

[3] ZHANG Yundian ， LI Jianlin ， YU Jiaying ， et al. Research Development of ultrasonic honing mechanism of ductile materials[J].Electromachining,1998(1) ： 31-34.(in Chinese)

[4] T. B. THOE，D. K. ASPINWALL，M. L. H. WISE. REVIEW ON ULTRASONIC MACHINING. Int. J. Mach. Tools Manufact. Vol. 38, No. 4：239–255, 1998

[5] Cao Fengguo, Zhang Qinjian. Research Situation and Development Trends of the Ultrasonic Machining Technology.Electromachining and mold supplement, 25-31, 2005, (in Chinese)

[6] He Zhayong, Zhao Yufang. Theoretical Foundation of Acoustics[M].Beijing: Defence Industrial Press, 1981, (in Chinese)

[7] ZHANG Chuan，YAN Yuhun. Analysis of vibration of free-edged thin plate with multi-nodal circles applied to compound flexural ultrasonic transducer[J] .Technical Acoustics，1998，17（1）：38-40 (in Chinese).

[8] LIU Shiqing, LIN Shuyu, WANG Chenghui. Radial vibration equivalent circuit of annular plate concentrator with tapered section plane[J].Journal of Shaanxi Normal University (Natural Science Edition) .2005，33（3）：

31-33.(in Chinese) [9] WANG D S，ZHOU A P，LIU C S, et al. Study of

acoustics characteristics of bending vibration disc theoretical analysis[J].Key Engineering Materials，2001，202：359-363.

[10] WANG Shi Ying, LÜ Ming, YA Gang. Dynamical characteristics of exponential transformer in gear honing[J].Chinese Journal of Mechanical Engineering，2007，43（6）：190-193.(in Chinese)

[11] AMABILI M ， PIERANDREI R ， FROSALI G. Analysis of vibrating circular plates having non-uniform constraints using the modal properties of free-edge plates: application to bolted plates[J] .Journal of Sound and Vibration 1997,206（1）：23-38.

[12] HYEONGILL L, RAJENDRA S. Acoustic radiation from out-of-plane modes of an annular disk using thin and thick plate theories[J].Journal of Sound and Vibration 2005，282：313–339

[13] PARKER R G，SATHE P J. Exact solutions for the free and forced vibration of a rotating disk spindle system [J].Journal of Sound and Vibration ，1999，223(3)：445-465.

[14] LV Ming, WANG Shiying, YA Gang. Displacement Characteristics of Transverse Vibratory Disc Transformer in Ultrasonic Gear Honing. Chinese Journal of Mechanical Engineering，2008，44（7）：106-111.(in Chinese)

[15] LIN Zhongmao. Principle and Design of Ultrasonic Horn[M].Beijing: Science and Technology Press, 1987

[16] Wang Shiying, Lv Ming, Ya Gang. Research on system design of ultrasonic-assisted honing of gears. Advanced Materials Research, Vols. 53-54 in September 2008

[17] WANG Shiying，YA Gang，ZHAO Li. The design and experimental research on the horn in ultrasonic machining [C]// Dalian，China.Beijing：Beijing World Publishing Corporation. ISTM/2005：64-67.

AMIN G，AHMED M H M，YOUSSEF H A. Computer aided design of acoustic horn for ultrasonic machining using finite-element analysis [J] .Journal of Material Processing Technology，1995，55：254-260


Theoretical and Experimental Research on Design of Non ...

Documents