theoretical and experimental research on design of non
TRANSCRIPT
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
2 Copyright © 2014 by ASME
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):
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)()( 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:
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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)
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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
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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
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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
8 Copyright © 2014 by ASME
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
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