Draft - University of Toronto T-Space · design theory of ultrasonic honing vibration system. Spiral bevel gear includes spiral bevel gears and hypoid gear during ultrasonic lapping

Draft

Influence of Gear Parameters on Dynamic Characteristics of

Ultrasonic Vibration System

Journal: Transactions of the Canadian Society for Mechanical Engineering

Manuscript ID TCSME-2017-0036.R1

Manuscript Type: Article

Date Submitted by the Author: 05-Dec-2017

Complete List of Authors: Zhu, Derong; School of Mechatronic Engineering,Northwestern Polytechnical University Yang, Jianjun; School of Mechatronic Engineering, Henan University of Science and Technology Deng, Xiaozhong; Henan Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing, Henan University of Science and

Technology Jiang, Chuang; School of Mechatronic Engineering, Henan University of Science and Technology Li, Jubo; School of Mechatronic Engineering, Henan University of Science and Technology

Is the invited manuscript for consideration in a Special

Issue? : N/A

Keywords: spiral bevel gear, ultrasonic vibration system, dynamic characteristics, gear characteristic parameter, tooth flank error measurement

https://mc06.manuscriptcentral.com/tcsme-pubs

Transactions of the Canadian Society for Mechanical Engineering

Draft

INFLUENCE OF GEAR PARAMETERS ON DYNAMIC CHARACTERISTICS OF

ULTRASONIC VIBRATION SYSTEM

De-Rong Zhu1 Jian-Jun Yang2* Xiao-Zhong Deng3 Chuang Jiang1 and Ju-Bo Li2

1 School of Mechatronic Engineering,Northwestern Polytechnical University,127 Youyi West Rd,Xi’an 710072,China

2 School of Mechatronic Engineering, Henan University of Science and Technology, 48 Xiyuan Rd,Luoyang 471003,China

3 Henan Collaborative Innovation Center of Machinery Equipment Advanced Manufacturing, Henan University of Science

and Technology, 48 Xiyuan Rd,Luoyang 471003,China

*Corresponding author: [email protected]

ABSTRACT: Spiral bevel gear belongs to a special load in ultrasonic lapping, which is not only processing object but also

processing tool, it is necessary to study dynamic characteristics of ultrasonic vibration system. Firstly, the spiral bevel gear is

reasonably simplified to a frustum, then which is combined with horn to form a new type of composite horn. Based on the

theory of plane longitudinal wave propagation, the resonance mathematical model of the gear ultrasonic vibration system is

established, the frequency equation of vibration system with gear characteristic parameters is obtained. Secondly, the

frequency and displacement characteristics of ultrasonic vibration system are analyzed by means of design examples,

especially, the influence law of gear characteristic parameters on dynamic characteristics of vibration system is studied.

Finally, the ultrasonic lapping and vibration measurement experiment are carried out using two pairs of different hypoid

gears. The results show that the change of dynamic characteristics of ultrasonic vibration system has a large influence on the

finish machining quality of the gear. Therefore it is necessary to consider the gear characteristic parameters when the

ultrasonic vibration system is designed. The research results provide an accurate theoretical basis for the detailed design of

ultrasonic lapping vibration system.

Keyword: spiral bevel gear; ultrasonic vibration system; dynamic characteristics; gear characteristic parameter; tooth flank

error measurement

1. INTRODUCTION

Ultrasonic assisted machining technique is a non-conventional material removal methods, which not only does not

thermally damage the work-piece, but also not introduce significant residual stress, and been successfully used in machining

both metallic materials and non-metallic materials, such as ultrasonic lapping, ultrasonic honing, ultrasonic grinding,

ultrasonic polishing, ultrasonic drilling and ultrasonic turning, as reported by T.B.Thoe et al. [1998]. Some study results,

which were presented by H.Hocheng et al.[2002] and B.Y.Wei et al.[2007], showed ultrasonic assisted machining can reduce

machining time, cutting power and cutter wear, and improve the material’s machinability, work-piece surface quality and

tool life compared to the conventional machining. Therefore, further research on ultrasonic-assisted machining has a good

theoretical value and application prospect.

One of the key technologies of ultrasonic assisted machining is the design of ultrasonic vibration system. Both the

resonant frequency and the displacement amplitude are analyzed when the ultrasonic vibration system is designed.

Especially the displacement amplitude, which has a very important influence on the quality of ultrasonic machining and tool

life, because it is an important performance index to represent the output power of the vibration system. At present, the study

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on the dynamic characteristics of ultrasonic vibration system can be divided into three types [Wang, S.Y. et al. 2012].

The first type is the small size tool head used in ultrasonic assisted machining, the tool head can be ignored because of

small size, the study on the dynamic characteristics can be transformed to the research on ultrasonic horn. A kind of

ultrasonic horn for grinding was designed by using finite element analysis, as reported by Y.J.Choi et al.[2013]. The abrasive

materials was electroplated at the end of the horn. The vibration experiment was completed by changing the power of the

ultrasonic generator. The output amplitude and frequency of the horn can be measured by laser vibrometer measurement and

Tektronix oscilloscope. The output end face of ultrasonic horn was designed with convex shape used in the sheet forming,

which was presented by Hojin Bae and Keun Park [2016]. The optimal vibration mode and corresponding natural frequency

of ultrasonic horn can be obtained by modal analysis. The horn’s vibration characteristics at input end and output end were

measured by using fiberoptic displacement sensors.

The size and quality of the tool head for second type is much larger than the first type in ultrasonic combined machining,

which cann’t be ignored because of big head size and quality when the dynamic characteristics was studied. But the tool

head can be designed according to the resonance requirements, at this time the ultrasonic horn and the processing tool need

be designed according to the full resonance theory. The overall design method was adopted by Jiangtao Che et al.[2016] and

Ioan-Calin Rosca et al. [2015], the workpiece and ultrasonic horn were connected with a stud. The effect of workpiece

quality on the resonance oscillations of ultrasonic horn was ignored because of small material removal in theoretical

calculation, then the theoretical characteristic values and the size of the ultrasonic system, including the length, nodal

displacement and amplification coefficient can be achieved。

For the third type, the processing tool is a special kind of load. Compared to the conventional small tool head, the

quality of the tool is larger and the shape is special, whose parameters are determined by operating requirements. The tool

head has great influence on the vibration characteristics of the ultrasonic vibration system, so not be ignored when the horn

is designed. A comprehensive consideration of the horn and gear was researched when studying the ultrasonic gear honing

by LÜ Ming, et al.[2008a, 2008b, 2008c, 2012, 2013]. The gear was simplified to a circular plate, whose outer diameter was

equal to the pitch diameter of gear. According to the coupling relation between the gear and the horn, the dynamic model of

the combined system of the horn and the gear was established. The dynamic characteristics of ultrasonic vibration system,

such as resonance frequency, amplitude distribution, node position and amplification coefficient, were studied. Experiments

proved that the method can solve the design problem of circular plate type transformer, and finally formed a non-resonant

design theory of ultrasonic honing vibration system.

Spiral bevel gear includes spiral bevel gears and hypoid gear during ultrasonic lapping process. Their contour are

conical, and the shape of the conical surface are very complex, which belongs to the third type, and is a more specific loads.

Spiral bevel gear is not only the processing object of ultrasonic lapping, but also the processing tool. Compared to the

conventional design of vibration system, the ultrasonic lapping vibration system design of spiral bevel gear is more difficult.

In recent years, many scholars have done a lot of researches on some new special machining methods of spiral bevel

gear, and a lot of researches have been contributed to the finishing of spiral bevel gear. Deng Xiaozhong et al. [2007]

established the lapping dynamic model of spiral bevel gear in the case of considering the ultrasonic excitation and tooth

surface geometric transmission error excitation. The dynamic model had two degrees of freedom with the clearance and

vibrated along peripheral direction. Then the calculating formula of the dynamic lapping force is obtained, which the effect

of ultrasonic excitation on the machining of spiral bevel gear was revealed. The ultrasonic lapping method was adopted in

finish machining of spiral bevel gear [Wei, B.Y. et al. 2007], and the experimental research on ultrasonic lapping of small

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module spiral bevel gear was carried out. The results showed that the tooth surface roughness Ra can reach 0.2 µm, the

material removal rate is three times as the ordinary lapping, and the noise of gear pair is reduced by 3~8dB after ultrasonic

lapping. An ultrasonic lapping system for middle module hypoid gear was established by Yang Jianjun et al.[2010, 2013],

the resonance frequency was calculated by means of finite element model, axial vibration and stress of the horn were

analyzed. The experiment results showed that the vibration of gear pair is weakened, and the tooth surface quality is

improved, meanwhile, the noise and vibration energy is decreased obviously.

Compared to the conventional lapping gear method, ultrasonic lapping gear was able to obtain better surface quality

and higher lapping efficiency. But there were a few researches on the design methods of ultrasonic lapping vibration system,

the analysis on frequency characteristics and displacement characteristics are no further. In addition, the studies about the

influence of the gear characteristics change on the resonant frequency, displacement amplitude, amplification coefficient of

the vibration system are also not enough. There is still no an effective design method of ultrasonic gear lapping system in

theory.

In this paper, the dynamics of the vibration system is studied. First, the spiral bevel gear is simplified as a cone, and the

ultrasonic lapping vibration system is built by combining the cone and the horn. Then, the mathematical model of ultrasonic

vibration system was established, the frequency characteristics and displacement characteristics of ultrasonic vibration

system are analyzed. The focus of the study is the influence of gear characteristic parameters on the dynamic characteristics

of vibration system. Finally, the ultrasonic lapping experiments and the vibration test of the gear pair are carried out.

2. MACHINING PRINCIPLE AND SETUP OF GEAR ULTRASONIC LAPPING

The principle of ultrasonic lapping of spiral bevel gear is shown in Fig. 1(a). In Fig. 1(a), V and H represent the offset

direction and axial direction of the pinion, respectively. J is the axial direction of the gear. The realization of ultrasonic

lapping between the pinion and gear shall meet following requirements. First, in order to control the location of tooth flank

in lapping region and the gear backlash of gear pair, the relative direction of V, H and J are adjusted. Then, the brake torque

is emerged by the eddy current brake and the gear, which provides the pressure of tooth flank during lapping. Meanwhile,

the lapping liquid which contains some grains is injected around the meshing region by spray nozzle. Last, the pinion

endures the ultrasonic longitudinal vibration along axial direction with a certain frequency(f = 10～20kHz ) and a certain

amplitude(A = 5～10µm), the grains will constantly suffer from hammering, impact, friction and cavitation gear flank, the

materials of gear flank are peeled off, and then the error of gear flank is amended, the carrying capacity is enhanced.

The ultrasonic lapping vibration system of the pinion is the key equipment to realize the ultrasonic lapping processing

of spiral bevel gear, as shown in Fig. 1(b). The system consists of ultrasonic power supply, transducer, the horn and the

pinion with shaft. The pinion with shaft is fixed on the small end face of horn through the tail thread. Alternating current

(220V or 380V) is transformed to the ultrasonic-frequency drive signal by ultrasonic power supply, which excites the

transducer to produce axial mechanical vibration. The vibration displacement is magnified by the horn. The energy is

concentrated on the big end face of pinion, which drives the pinion to vibrate with amplifying ultrasonic frequency.

The whole vibration system should be simplified before design. The effect of the transducer on the resonance frequency

of vibration system can be neglected because the transducer is only a drive system. As show in Fig 1(b), ① is a flange plate,

it is located at the displacement nodal position of ultrasonic vibration, which can be ignored. ② is a small plane, in order to

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make the disassembly of the pinion convenient, which also can be ignored. The axle of pinion is regarded as a part of the

horn, whose gear teeth can be simplified to a circular truncated cone with pitch cone line as the generatrix. Then a new

four-part composite horn is formed through the combination of simplified horn and the pinion, which is signed as Ⅰ, Ⅱ,

Ⅲ and Ⅳ . The center line of rectangular coordinate system is set as x axis, y axis is at the interface of Ⅰ and Ⅱ,as

shown in Fig. 1(b).

3. MATHEMATICAL MODEL OF THE VIBRATION SYSTEM FOR GEAR ULTRASONIC LAPPING

Ultrasonic composite horn is a composite structure with four sections, including two cylinders and two cones, and the

cross-section of each component is circular. It is assumed that the composite horn is composed of homogeneous and

isotropic material, and mechanical loss and damping vibration aren’t considered. When the cross-section size of the

composite horn is much smaller than the wavelength, it can be considered that the particle displacement distribution in the

cross-section is uniform when the plane longitudinal wave propagates along axial direction. Under the precondition of

one-dimensional sinusoidal vibration, wave equation of single-stage horn is shown in equation (1) [Lin, Z.M. 1987]:

22

2

10n n n

n n

n n n n

Sk

x S x x

ξ ξξ

∂ ∂ ∂+ + =

∂ ∂ ∂ （1）

In Eq. (1): n represents four sections of composite horn, n=1, 2, 3 and 4, corresponding to Ⅰ, Ⅱ, Ⅲ and ;Ⅳ ( )n n n

xξ ξ= is

the displacement function of the particle located to the cross-section of single-stage horn;

( )n n nS S x= is a cross-sectional

area function of single-stage horn; nk is a circular wavenumber; and 2n n nk c f cω π= = ; is the circular

frequency; /n n nEc ρ= is propagation velocity of the longitudinal wave, nE is the elasticity modulus of the material;

nρ

is the density of the material.

The section Ⅰ and ofⅢ composite horn are cylindrical, ( )in nS x S= （n =1 or 3，i = 1 or 2）, which is a constant. The

displacement function and stress function of the section Ⅰ and Ⅲ can be obtained when i

S is put into equation (1), as

shown in equation (2).

( )= cos( ) sin( )

( )sin( ) cos( )

n n n n n n n n

n nn n n n n n n n

n

x k x A k x B

xk k x A k k x B

x

ξξ

⋅ + ⋅∂ = − ⋅ + ⋅ ∂

（2）

Among them: the coefficients nA 、 nB ( n=1 or 3) are integral constant.

The section Ⅱ and section Ⅳ of composite horn are conical, and the change law of conical section is shown in

equation (3):

21( ) (1 )n n j j nS x S xα += −

（3）

Among them: j = 1 or 3; jS represents cross-sectional area of the big end of the cone; 1 1 11 ( 1)

j j jj N N lα + + ++ −=

w

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represents taper coefficient, and 1 1j j jN S S+ += represents area coefficient.

The displacement function and stress function of section Ⅱand Ⅳ can be achieved by solving simultaneous equations of

(3) and (1), as shown in equation (4).

2 2

cos( ) sin( )( )

1 1

( ) sin( ) cos( ) cos( ) sin( )

1 ( 1 ) 1 ( 1 )

n n n nn n n n

n n

n n n n n n n n n n n nn n

n n n n n n n n n

k x k xx A B

x x

x k k x k x k k x k xA B

x x x x x

ξα α

ξα α α α

= ⋅ + ⋅ − −

∂ = − − ⋅ + − ⋅ ∂ − − − −

（4）

Where the coefficient nA and nB ( n = 2 or 4) are integral constant.

When the composite horn does one dimensional longitudinal vibration along axial direction, the displacement and

internal force on each cross section are continuous [Liang, X. 2015]. The displacement and internal force must meet the

requirements of continuity, as shown in equation (5) and (6), respectively.

1 2

2 2 3 2

3 2 3 4 2 3

1 1 2 20 0

2 2 3 3

3 3 4 4

( ) ( ) 0

( ) ( ) 0

( ) ( ) 0

x x

x l x l

x l l x l l

x x

x x

x x

ξ ξ

ξ ξ

ξ ξ

= =

= =

= + = +

− =

− =

− =

（5）

1 2

2 2 3 2

4 2 33 2 3

1 1 2 21 1 1 2 2 2

1 20 0

3 32 22 2 2 3 3 3

2 3

3 3 4 43 3 3 4 4 4

3 4

( ) ( )( ) ( ) 0

( )( )( ) ( ) 0

( ) ( )( ) ( ) 0

x x

x l x l

x l lx l l

x xE S x E S x

x x

xxE S x E S x

x x

x xE S x E S x

x x

ξ ξ

ξξ

ξ ξ

= =

= =

= += +

∂ ∂ − =

∂ ∂

∂∂− =

∂ ∂

∂ ∂ − = ∂ ∂

（6）

In addition, the left end of composite horn is connected to the transducer, in where the displacement is the largest but

the strain is zero. The right end of composite horn is under the state of free vibration, whose strain is also zero. As a result,

the boundary condition of the left end and right end is shown as eq. (7).

1 1

4 2 3 4

1 1

1

4 4

4

( )0

( )0

x l

x l l l

x

x

x

x

ξ

ξ

= −

= + +

∂=

∂∂ = ∂

（7）

By means of Eqs. (5) ~ (7), the mathematical model of the vibration system for ultrasonic lapping gear is established

under the resonance, which is a complex transcendental equation group.

4. DYNAMIC CHARACTERISTICS OF ULTRASONIC LAPPING VIBRATION SYSTEM

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4.1. Frequency Equation of Ultrasonic Lapping Vibration System

The displacement function and stress function (Eqs. (2) ~ (4)) are put into the Eqs. (5) ~ (7), respectively. The

homogeneous equation set, including the size parameters of the pinion and the natural frequency of longitudinal vibration,

can be obtained, as shown in matrix Eq. (8).

12

21 23

32 33 34

43 44 45 46

53 54 55 56

65 66 67 68

75 76 77 78

87 88

11 1

1

2

2

3

3

4

4

C

0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0

C

C C

C C C

C C C C

C C C C

C C C C

C C C C

C C

A

B

A

B

A

B

A

B

=

（8）

Where ij

C ( i , j =1～8) are some coefficients of variables (n

A ,n

B ( n=1, 2, 3 and 4 )) determined by Eqs. (5) ~ (7).

In order to solve the equations, and make the coefficients (n

A ,n

B ( n =1、2、3、4 )) not all zero, the value of the

determinant consisted of the coefficient ijC in Eq. (8) must be equal to zero. As shown in the following Eq. (9). Equation (9)

is the frequency equation of the ultrasonic lapping vibration system for bevel gear. If the characteristic parameters of the

horn and the pinion are known, the resonant frequency of the vibration system can be solved by the Eq. (9). On the contrary,

if the frequency of the vibration system is determined, the unknown parameters of the horn and the pinion with shaft can be

solved.

11 12

21 22 23

32 33 34

43 44 45 46

53 54 55 56

65 66 67 68

75 76 77 78

87 88

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 00

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0

C

C C C

C C D

C C C C

C C C C

C C C C

C C C C

C C

C

∆ = = （9）

4.2. Displacement Characteristic of Ultrasonic Lapping Vibration System

The left end of composite horn is connected to the transducer, and the displacement at the connecting surface is

continuous. The maximum output displacement of the transducer is equal to the maximum input displacement (0ξ ) of the

composite horn’s left end. And then, Eq. (10) can be obtained the displacement function of the composite horn among Eq (2).

The result is shown in eq (10):

1 11 1 1 1 1 1 1 1 0= -( ) = cos( ) sin( )

x lx k l A k l Bξ ξ⋅ − ⋅ = （10）

An inhomogeneous equation set is composed by Eq. (10) and (8). A set of special solutions of the undetermined coefficient

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nA ,

nB （ n =1, 2, 3 and 4）can be obtained by Eq. (11).

12

22

32 33 34

43 44 45 46

53 54 55 56

65 66 67 68

75 76 77 78

87 88

1

111

231 0

2

2

3

3

4

4

C

0

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0

C

C C

C C C

C C C C

C C C C

C C C C

C C C C

C C

A

B

A

B

A

B

A

B

ξ

−

′ ′

− = ⋅

（11）

In Eq. (11): ( )22 1 1sinC k l′ = , ( )23 2 1 1cosC k lα′ = , Other coefficients ij

C（ i，j =1～8）are the same as Eq. (8). Then

the results are put into the Eq. (2) and the Eq. (4). The displacement distribution along axial direction can be obtained when

the vibration system of ultrasonic lapping gear produces resonance.

4.3. Design Example of Ultrasonic Lapping Vibration System

The designed parameters of the horn and the pinion with shaft in vibration system of ultrasonic lapping gear, are shown

in Table 1. The resonant frequency f is independent variables, and the value∆of determinant is dependent variable, the

search range of resonant frequency is set to f = 16～20kHz. The value ∆ of determinant is solved by Eq.(9). The

relationship between f and ∆ can be acquired, as shown in Fig.2.

Fig. 2 shows that the error ∆ of which is solved by the frequency equation is a smooth curve. Frequency equation has a

unique solution when ∆ =0, namely resonant frequency f is 15.993 kHz. The error from design frequency is 0.4375‰,

which can meet the requirement of engineering applications.

Assuming the maximum output displacement of the transducer is 0ξ = 5µm, the displacement distribution along axial

direction of ultrasonic vibration system can be obtained by solving Eq. (2), (4) and (11), as shown in Fig. 3. It can be known

that there are two displacement nodes in the ultrasonic vibration system. The distance of the nodes from the coordinate

origin are 22.89 mm and 185.3 mm, respectively. In this paper, the flange plate is set at 79.89 mm from the left end of the

composite horn. The displacement amplitude is 8.156µm at the end of the vibration system (namely the small end of pinion).

The amplification coefficient of amplitude (Mp) can be obtained by Eq (12), which is 1.6312.

4 2 3 4

1 1

4 4

1 1

( )=

( )

x l l l

P

x l

xM

x

ξ

ξ= + +

=−

（12）

4.4. Characteristic Analysis of Ultrasonic Lapping Vibration System

40, 41, 42 …… 50 mm are chosen as the diameter of gear ( 3d ) while other parameters unchanged, the corresponding

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resonance frequency can be obtained using Eq 8. In a similar way, the resonance frequency also can be achieved when the

gear length 4l =26、27、28……36 mm are decided. The results are shown in Table 2. Table 3 shows gear diameter, gear length

and the resonance frequency after relativization processing with Eq 13.

3 3

4 4

3 3

4 4

45

31

15.993

15.993

d d

l l

d d

l l

f f

f f

∆ = − ∆ = −∆ = − ∆ = −

（13）

The relationship between the relative value of resonance frequency ( f∆ ) and the relative value of gear parameters

( 3d∆ or 4l∆ ) is illustrated in Fig. 4. It can be known from Fig. 4 that the resonance frequency of ultrasonic lapping vibration

decreases linearly with the increase of gear parameters, the effect of gear length on the resonance frequency is more obvious

than gear diameter, the average effect value of gear length is 5.6Hz bigger than gear diameter. On the contrary, the resonance

frequency increases linearly when the gear parameters declines, the effect value is about 7.8Hz. In other words, the average

value of resonance frequency decreases 74Hz or increases 82.2Hz when the gear length increases or decreases 1 mm. The

resonance frequency decreases 72Hz or increases 79.4Hz on average when gear diameter increases or decreases 1 mm.

The relationship between the amplification coefficient of ultrasonic vibration system (P

M ) and the relative value of

gear parameters ( 3d∆ or 4l∆ ）) is illustrated in Fig.5. It can be seen from Fig.5 that the amplification coefficient (PM )

decreases linearly with the increase of gear parameters. On the contrary, the amplification coefficient increases linearly with

the decline of gear parameters. The influence of gear diameter on amplification coefficient (P

M ) is more significant than

gear length. From analysis above, it can be known that the effect of ultrasonic vibration on bevel gear is different.

It also can be seen from Fig. 4 and 5, the higher amplification coefficient corresponds to higher resonance frequency,

lower amplification coefficient is produced with lower resonance frequency. The effect of gear diameter and gear length on

resonance frequency is same basically. As far as the amplification coefficient, the effect of gear diameter is more obvious

than gear length.

The design of gear is on basis of operating requirement in the actual production, the actual structure parameter of gear

is completely different, which leads to the change of resonance frequency of ultrasonic vibration system. Simple harmonic

vibration will not be generated by the primary ultrasonic vibration system, so ultrasonic vibration system need to be

designed again. For new ultrasonic vibration system, in order to ensure that simple harmonic vibration is produced by new

ultrasonic vibration system under primary resonant frequency, the length of composite horn in primary ultrasonic vibration

system should be adjusted if ultrasonic power supply and the transducer are still used. In this paper, the length 3l of

composite horn can be changed, in this way, ultrasonic lapping processing can be carried out under a same resonant

frequency for different gear.

The length of horn ( l3 ) is set as 25, 26, 27…… 35 mm, the corresponding resonant frequency (3lf ) and amplification

coefficient (3l

Mp ) can be obtained using Eq (9), (11) and (12), the results are shown in Table 2. The length of composite

horn and resonant frequency of the vibration system are processed with Eq (14), the results are shown in Table 3. The curve

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of relative resonant frequency and amplification coefficient are illustrated in Fig. 4 and Fig. 5, respectively.

3 3

3 3 30

15.993l l

l l

f f

∆ = −∆ = −

（14）

It can be seen from Fig. 4, the resonant frequency of ultrasonic vibration system decreases with the increase of the

length of composite horn 3l . On the contrary, the resonant frequency increases with the decrease of 3l . As a result, the

characteristic of composite horn can be used to adjust the change of resonant frequency caused by the change of gear

parameter, the resonant frequency of ultrasonic vibration system keeps constant. Known form Fig. 5, the amplification

coefficient of ultrasonic vibration system has a little change when the length of composite horn ( 3l ) increases or decreases.

This shows that the amplification coefficient is insensitive to the change of the length of composite horn when the length of

composite horn ( 3l ) is used to adjust the resonant frequency of vibration system, so the characteristic of primary system

keeps unchanged.

5. EXPERIMENT AND ANALYSIS

5.1. Machining of Experimental Gear

In the experiment, two pairs of hypoid gear used in automotive rear bridge are chosen, whose tooth are 7/37 and 8/41

respectively. The structure of pinion with shaft is shown in Fig 6, the shaft design parameters of two pairs of pinion are same.

The parts of spiral gear tooth are designed according to actual operating requirements. The basic parameters of two gear

pairs are shown in Table 4.

The tooth flank processing of the gear and pinion contains two process: rough-milling and finish-milling. Gear milling

machine Y2250 is used to rough-milling, gear cutting machine Y2950 is used to finish-milling of the gear, gear cutting

machine GH35 is used to the finish-milling of pinion. The processing parameters of two gear pairs are shown in Table 5 and

6. The gear pair processed is shown in Fig. 7.

5.2. Manufacture of Ultrasonic Vibration System

The length of the composite horn ( 3theoryl ) and the amplification coefficient of vibration system ( theoryMp ) can be

obtained through Eq. (9), (11) and (12) when the resonant frequency is 16kHz, according to the characteristic parameters of

the pinion, the results are shown in Table 7.

The length value of 3theoryl is accurate to four decimal places, which is against the machining of composite horn. As a

result, some work need to be done before machining. First, it is necessary to revise the data downward, the data are rounded

to 33 and 27 mm respectively. Then, the revised resonant frequency and amplification coefficient can be get through Eq. (9),

(11) and (12), the resonant frequency are 16.006kHz and 16.013kHz after revising respectively. Compared to the theoretical

resonant frequency, the difference are 0.375‰ and 0.8125‰ respectively, which shows a higher precision. It can be

considered that the resonant frequency of the two vibration systems is equal to 16 kHz.

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At the same time, the corresponding amplification coefficient can also be achieved, 1.7122 and 1.5562 respectively.

The value is essentially unchanged compared to the rounding before, which verifies the previous analysis on the effect of

composite horn length 3l . The rounded results are shown in Table 7.

Ultrasonic lapping gear vibration system is set up by using processed the horn, the pinion with shaft and outsourcing

transducer, as shown in Fig.8. The large end surface of horn is connected half wavelength cylindrical transducer, model

YP-6016-4Z, power of 2000W, resonance frequency of 16 kHz. The appropriate amount of Vaseline is daubed at the small

end surface of the horn, which can enhance the efficiency of ultrasonic transmission. The pinion with shaft is positioned by

the inner hole Φ30 at small end surface of horn, and it is fixed with the horn via thread M18 in tail end of the pinion with

shaft.

5.3. Experiment of Gear Ultrasonic Lapping

In order to achieve the uniform lapping of whole tooth flank, the checkout on the tooth contact zone should be

conducted before gear ultrasonic lapping. The lapping points (large end, small end, tooth top and tooth root) are selected as

check points respectively, as shown in Fig 9. The rolling tester Y9550 is used in the experiment. Red lead powder is painted

on the tooth flank of the gear, the value of V, H and J are adjusted according to the actual position and shape of tooth flank

contact trace. The results are shown in Table 8.

In the experiment, the gear ultrasonic setup is shown in Fig 10. The nut screw pair is drived along linear guide by motor

② and ③, to realize the adjustment of lapping point H and J. The deflexion of pinion’s spindle system is achieved by

synchronous belt drived by motor , the adjustment of lapping point V can be realized. The loading (2Nm) on the gear

tooth flank is produced by brake . Lapping liquid is composed of special lapping oil and white corundum W40#, whose

concentration is 25%. The main rotation motion of pinion is provided by motor , the speed is 480r/min, the rotation

direction is convex flank of the gear and concave flank of pinion. Meanwhile, the longitudinal ultrasonic vibration with 16

kHz is loaded on the pinion by ultrasonic-frequency power supply. In order to realize uniform lapping, the method of

multipoint cycle motion is adopted, a circulation cycle is A-B-O-D-C-B-O-D-A, as shown in Fig 9. Repeat this cycle until

the ultrasonic lapping is finished, but the lapping time is 3 min at most.

5.4. Measurement and Analysis of Tooth Flank Error

In order to examine the actual effect of gear pair after the ultrasonic lapping, the tester of spiral bevel gear (JD45+) is

used. In the experiment, the tooth flank error of the gear before and after ultrasonic lapping are measured, as shown in Fig 11.

The head diameter of measuring pin is 2 mm in JD45+ tester. The convex tooth flank and concave tooth flank of random

tooth can be chosen as measurement object. The number of measurement point for one tooth flank is 45, the distribution of

points are shown in Fig 12.

The convex and concave flank measurement error of the gear for gear pair Ⅰare shown in Table 9 and 10, respectively.

Positive number shows that measurement error is bigger than theoretical error, the minus means that measurement error is

littler than theoretical error. It can be seen from Table 9 and 10, the sum of the square error of convex tooth flank are about

0.6409mm2 and 0.1861mm2 before and after ultrasonic lapping, respectively. The deduction of error is about 0.4548mm2.

The sum of square error of concave tooth flank are about 1.4969mm2 and 1.2137mm2 respectively. The deduction of error is

about 0.2832mm2, as shown in Fig 13. Known from analysis above, the error of convex and concave tooth flank reduce

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obviously, so ultrasonic vibration has a positive effect to improve tooth flank error. In addition, the deduction of convex

tooth flank error is bigger than concave tooth flank, this phenomenon can be explained by that the ultrasonic vibration

directly impacts on the convex tooth flank of the gear when the spiral bevel gear is processed with the method of ultrasonic

lapping.

The measurement error of convex and concave tooth flank is plotted with grids, as shown in Fig 14 and 15. Seen from

Fig 14 and 15, the measurement error of each point decreases in different extent, the deduction of convex tooth flank is more

obvious.

The convex and concave flank measurement error of the gear for gear pair Ⅱ are shown in Table 11 and 12, respectively.

It can be seen from Table 11 and 12, the sum of the square error of convex tooth flank is about 0.2602mm2 and 0.1087mm2

before and after ultrasonic lapping, respectively. The deduction of error is 0.1515mm2. The sum of the error’s absolute value

of concave tooth flank is 1.0984mm2 and 0.8874mm2 before and after ultrasonic lapping, respectively. The deduction of

error is 0.2110mm2, as shown in Fig 13. It can be known that the results are consist with gear pair Ⅱ. The cause of results is

same as gear pair Ⅱ. In the same manner as gear pair Ⅱ, the measurement error of convex and concave tooth flank is plotted

with grids, as shown in Fig 16 and 17. Observation from Fig 16 and 17, the errors all decrease in different extent, the

decrement of convex tooth flank is obvious than the concave tooth flank.

Known from the comparison of Fig. 14~17, the cusp and inflection points of grid curves are removed for gear pair Ⅱ

and Ⅱ, the convex and concave tooth flank become smoother. Analysis from the error variation along tooth length direction

before and after ultrasonic lapping, the error of gear helical angle is improved, the contact zone of tooth flank tends to the

midpoint of tooth length direction. And then the early wear of gear can be avoided, the noise is reduced. Analysis from the

error variation along tooth height direction before and after ultrasonic lapping, the error of gear pressure angle is also

improved, the contact zone of tooth flank tends to the midpoint of tooth height direction. And then these phenomenon of

tooth breakage and tooth beat are avoided.

Seen from Fig. 13, the error variation of convex tooth flank of gear pair Ⅱ is the biggest, which is 1.6 times as the

concave tooth flank of gear pair Ⅱ, 3 times as the convex tooth flank of gear pair Ⅱ. The conclusion can be obtained that

gear vibration system with same resonant frequency has different effect on different gear pair. The ultrasonic effect of

convex tooth flank of the gear is most obvious for gear pair Ⅱ. It can be explained by that the amplification coefficient of

displacement amplitude in ultrasonic vibration system is 1.7122, which is bigger than the amplification coefficient (1.5562)

of displacement amplitude in ultrasonic vibration system for gear pair Ⅱ. The results is consistent with the theoretical

analysis. The ultrasonic effect of concave tooth flank for gear pair Ⅱ and Ⅱ has no big difference, which may be caused by

the rotation direction of ultrasonic vibration system, only with convex surface of the gear and concave surface of pinion.

5.5. Vibration Test and Analysis of Gear Pair

The vibration comparison experiment of two bevel gear pairs is conducted on rolling inspection machine Y9550 after

ultrasonic lapping. Arrangement of vibration test equipment is shown in Fig. 18. The vibration acceleration sensor is

adsorbed on the front end face of pinion’s bearing block, the location is 100 mm far away from the spindle center of pinion.

The rotation speed of pinion is 710 r/min. The load torque of the gear is 65 Nm.

The vibration picking signal is magnified by the amplifier, get by data acquisition instrument (MPS-140801), recorded

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and analyzed with computer. A vibration comparison of test results after ultrasonic lapping for gear pair Ⅱ and Ⅱ are shown in

Fig 19 and 20.

Seen from Fig. 19 and 20, the shock is obvious when two gear pairs mesh after ultrasonic lapping, which is caused by

the error of tooth flank. The range of instantaneous acceleration of gear pair Ⅱ is 0.6m/s2. For gear pair Ⅱ, the range is

higher than 0.6m/s2. It is illustrated that there are many convex points and concave points on tooth flank of gear pair Ⅱ, the

tooth quality after ultrasonic lapping is worse than gear pair Ⅱ. The existence of convex and concave points have adverse

effect on smooth scrolling of tooth flank. As a result, the vibration of gear pair Ⅱ is inevitably bigger than gear pair Ⅱ during

transmission.

Further testimony from Fig. 19 and 20, the tooth flank accuracy of gear pair Ⅱ is higher, the error amplitude of gear

transmission is weakened, the smooth scrolling feature of tooth flank is better, the dynamic meshing performance of tooth

flank pair is good. It is verified that the ultrasonic effect produced by the vibration system with a same resonant frequency is

best for gear pair Ⅱ.

6. CONCLUSIONS

Ultrasonic lapping is a new technology, which makes the ultrasonic vibration cutting technology applied in precision

machining of the spiral bevel gear. In this paper, the design method was proposed based on the theory of wave propagation,

which is a good solution to the resonant design of the ultrasonic vibration system for spiral bevel and hypoid gears.

Experimental results showed that the design method has reached the production requirements. For the system with a similar

combined structures, the method has a significant theory and application value for the vibration system design and dynamics

study.

(1) In the ultrasonic lapping gear vibration system, the transducer can only produce the driving system with high

frequency vibration. The resonant frequency of the vibration system is mainly determined by the horn and the spiral bevel

gear. Compared to the general ultrasonic machining tool, the spiral bevel gear has a special structure, complex shape and

larger quality, so resulting in a large effect on the resonant frequency of the vibration system. It is necessary to combine the

horn and the gear into a complete system while design the ultrasonic vibration system.

(2) Owing to the different design requirements and the difference of the processing technology, the characteristic

parameters of the gear are not constant, which must affect the dynamic characteristics of the vibration system during the

process of ultrasonic lapping. Gear ultrasonic lapping experiment and gear vibration tests have confirmed that the tooth

surface quality and meshing characteristics are obviously different. Thus, the change of gear characteristic parameters must

be considered when the ultrasonic vibration system is designed.

(3) The resonant frequency changes of the vibration system can be amended by designing the reasonable size of the

horn. The combination system of the horn and gear generates the harmonic vibration at the design frequency, thereby, it is

not necessary to design the vibration system with the inherent frequency of the horn and gear.

(4) Different gears can be ultrasonic lapped at the same resonant frequency, but the cumulative effect and the steady

characteristics generated by the ultrasonic vibration system have large differences. In the ultrasonic vibration system with

larger amplitude magnification, the machined gear can obtain better tooth surface error, higher meshing accuracy and higher

lapping efficiency.

(5) The ultrasonic vibration effect on the convex and concave surfaces is different after the ultrasonic lapping. This may

be related to the rotation direction of the ultrasonic vibration system and the structural characteristics of the tooth surface,

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which needs further study.

ACKNOWLEDGEMENTS

Financial assistance was provided by the National Natural Science Foundation of China (No.51475141 and No.

51375144 and No. 51405135), and Science and Technique Plan Projects of Henan Province (No.102102310372), and the

Program for Science&Technology Innovation Talents in Universities of Henan Province (No.15HASTIT025).

REFERENCES

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Journal of Materials Processing Technology. 231: 312-318.

Deng, X.Z., Yang, J.J. and Wei, B.Y. 2007. Dynamic Lapping Force Analysis of Spiral Bevel Gear Lapping System

with Ultrasonic Excitation. China Mechanical Engineering. 18(19): 2359-2361.

Hocheng H. and Kuo K. 2002. Fundamental study of ultrasonic polishing of mold steel. International Journal of

Machine Tools & Manufacture. 42: 7-13.

Hojin, B. and Keun, P. 2016. Design and Analysis of Ultrasonic Horn for Polymer Sheet Forming. International Journal

of Precision Engineering and Manufacturing-Green Technology. 3: 49-54.

Ioan,C.R., Mihail, I.P. and Nicolae, C. 2015. Experimental and numerical study on an ultrasonic horn with shape

designed with an optimization algorithm. Applied Acoustics. 95: 60-69.

Liang, X,. LÜ, M. and Wang, S.Y. 2015. Design of longitudinal-flexural coupling vibration transformer with large

amplitude used in gear ultrasonic machining. Chinese Journal of Engineering Design. 22(2): 172-177.

Lin, Z.M. 1987. Principle and design of ultrasonic horn. Beijing: Science Press.

LÜ, M., She, Y.Z. and Qin, H.B. 2013. Design method for a vibration system of ultrasonic gear honing and its dynamic

characteristics. Journal of Vibration and Shock. 32(2): 147-152.

LÜ, M., Wang, S.Y. and Ya, G. 2008a. Displacement Characteristics of Transverse Vibratory Disc Transformer in

Ultrasonic Gear Honing. Chinese Journal of Mechanical Engineering. 44(7): 106-111.

She, Y., LÜ, M. and Wang, S.Y. 2012. Design of transformer with non-resonance structure and its dynamic study.

Journal of Mechanical Engineering. 48(7): 49-55.

Thoe, T., Aspinwall, D. and Wise, M. 1998. Review on ultrasonic machining. International Journal of Machine Tools

and Manufacture. 38: 239-255.

Wang, S.Y., Li, X.P. and Zhang, C.H. 2012. Dynamics characteristics of cone transformer in ultrasonic gear honing.

Journal of Vibration Engineering. 25: 294-301.

Wang, S.Y., LÜ, M. and Ya, G. 2008b. Research on system design of ultrasonic-assisted honing of gears. Advanced

Materials Research. (53/54): 191-196.

Wang, S.Y., LÜ, M. and Ya, G. 2008c. Study on dynamical characteristics of catenary transformer in gear honing. Key

Engineering Materials. 359-360: 414-419.

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Wei, B.Y., Deng, X.Z. and Fang, Z.D. 2007. Study on ultrasonic-assisted lapping of gears. International Journal of

Machine Tools & Manufacture. 47: 2051-2056.

Wei, B.Y., Yang, J.J. and Deng, X.Z. 2007. Insensitive vibration cutting mechanism and experiments on ultrasonic

lapping of gears. Mechanical Science and Technology. 26(4): 491-494.

Yang, J.J., Deng, X.Z. and Wei, B.Y. 2010. Study and experiments of dynamic mechanism of excited gear lapping of

spiral bevel gear. Journal of Aerospace Power. 25(12): 2839-2845.

Yang, J.J., Zhang, H. and Deng, X.Z. 2013. Ultrasonic lapping of hypoid gear: system design and experiments.

Mechanism and Machine Theory. 65(3): 71-78.

Young, J.C., Kyung, H.P. and Yun, H.H. 2013. Effect of Ultrasonic Vibration in Grinding; Horn Design and Experiment.

International Journal of Precision Engineering and Manufacturing. 14: 1873-1879.

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TABLE 1: Designed parameters of ultrasonic vibration system

Parameter Value

Material 45

Designed frequency ( kHz ) 16

Density ρ (3/ mkg ) 7800

Elasticity modulus Ε (Gpa ) 210

Poisson ratio 0.3

Segment Ⅱ 1l ( mm ) 57

1d ( mm ) 75

Segment Ⅱ 2l

( mm ) 193

Segment Ⅱ 3l ( mm ) 30

2d ( mm ) 42

Segment Ⅱ

4l ( mm ) 31

3d ( mm ) 45

δ ( º ) 10

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TABLE 2: Resonance frequency and amplification coefficient under different characteristic parameters

3d (mm) 40 41 42 43 44 45 46 47 48 49 50

df(kHz) 16.390 16.308 16.227 16.148 16.07 15.993 15.918 15.845 15.773 15.702 15.633

3dMp 1.8872 1.8362 1.7850 1.7338 1.6824 1.6312 1.5802 1.5294 1.4790 1.4288 1.3792

4l (mm) 26 27 28 29 30 31 32 33 34 35 36

lf(kHz) 16.404 16.318 16.235 16.153 16.072 15.993 15.916 15.841 15.767 15.695 15.624

4lMp 1.7522 1.7290 1.705 1.6808 1.6562 1.6312 1.6062 1.5808 1.5556 1.5302 1.5048

3l (mm) 25 26 27 28 29 30 31 32 33 34 35

lf(kHz) 16.245 16.194 16.143 16.093 16.043 15.993 15.944 15.896 15.848 15.800 15.753

3lMp 1.6302 1.6306 1.6310 1.6312 1.6314 1.6312 1.6310 1.6308 1.6302 1.6296 1.6290

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TABLE 3: Relative values of resonant frequency under different characteristic parameters

3 4 3/ / ( )d l l mm∆ ∆ ∆ -5 -4 -3 -2 -1 0 1 2 3 4 5

3( )df kHz∆

0.397 0.315 0.234 0.155 0.077 0 -0.075 -0.148 -0.220 -0.291 -0.360

4( )

lf kHz∆

0.411 0.325 0.242 0.160 0.079 0 -0.077 -0.152 -0.226 -0.298 -0.369

3( )

lf kHz∆

0.252 0.201 0.150 0.100 0.050 0 -0.049 -0.097 -0.145 -0.193 -0.240

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TABLE 4: Essential parameters of hypoid gear pairs

Serial Number of gear pairs Ⅱ Ⅱ

Parameter Pinion Gear Pinion Gear

Tooth number 7 37 8 41

Modulus /mm 4.611 4.162

Offset /mm 25 30

Width of tooth surface /mm 29.78 25 30.99 25

Height of outer end tooth top 6.56 0.98 6.36 1.12

Height of outer end tooth root 2.16 7.53 2.31 7.32

Height of outer end total tooth 8.72 8.51 8.67 8.44

Diameter of outer end /mm 55.77 171 58.35 171.09

Diameter of the gear pitch 170.62 170.64

Helix angle 50˚22' 31˚4' 50˚27' 27˚15'

Pitch cone angle 10˚36'28 78˚46' 10˚45' 78˚19'

Face cone angle 16˚1' 79˚37' 15˚57' 79˚18'

Root cone angle 9˚48' 73˚5' 9˚51' 72˚44'

Spiral direction left right left right

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TABLE 5: Processing parameters of hypoid gear pair Ⅱ (7/37)

Parameter Gear Pinion

Concave Flank

Pinion

Convex Flank

Radius of cutter head /mm 95.25 97.60 90.58

Pressure angle of blade 22˚30' 14˚0' -35˚0'

Installation angle 70˚10' -2˚46' -5˚1'

Vertical cutter position /mm 85.58

Horizontal cutter position /mm 9.37

Horizontal gear position /mm +0.59 -0.65 +2.15

Bed position /mm 0 2.06 7.82

Vertical gear position /mm 21.75 26.83

Radial cutter position /mm 161.55 85.38

Angle cutter position 95˚57' 86˚50'

Roll ratio 5.04 5.28

Cutter angle -0' 333˚59'

Cutter inclination angle 57˚20' 55˚22'

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TABLE 6: Processing parameters of hypoid gear pair Ⅱ (8/41)

Parameter Gear

Pinion

Concave Flank

Pinion

Convex Flank

Radius of cutter head /mm 95.25 97.60 90.58

Pressure angle of blade 22˚30' 14˚0' -35˚0'

Installation angle 69˚1' -2˚57' -4˚53'

Vertical cutter position /mm 90.61

Horizontal cutter position /mm 8.77

Horizontal gear position /mm +0.64 -0.46 +1.99

Bed position /mm 0 1.90 6.71

Vertical gear position /mm 26.62 31.55

Radial cutter position /mm 211.36 91.05

Angle cutter position 96˚8' 87˚55'

Roll ratio 4.89 5.07

Cutter angle -0' 333˚55'

Cutter inclination angle 63˚13' 58˚35'

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TABLE 7:Characteristic and Dynamic Parameters of ultrasonic lapping gear vibration system

Serial Number

of gear pairs

Characteristic Parameters Dynamic Parameters

3d 4l δ °

theoryf /kHz 3theoryl theoryMp 3revisel revisef /kHz

reviseMp

Ⅱ 43.9843 30.29 10.6078 16 33.1307 1.7120 33 16.006 1.7122

Ⅱ 46.5995 31.73 10.7597 27.2671 1.5563 27 16.013 1.5562

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TABLE 8: Relationship between lapping gear position and V/H/J

Gear pair Ⅱ Ⅱ

Position V H J V H J

Midpoint O 0 0 0 0 0 0

Big end C -0.50 +0.60 -0.15 -0.35 +0.40 -0.10

Small end A +0.40 -0.50 +0.13 +0.25 -0.40 +0.09

Gear top B -0.20 +0.20 -0.05 -0.20 +0.20 -0.05

Gear root D +0.20 -0.20 +0.05 +0.10 -0.20 +0.04

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TABLE 9:Convex flank error of gear in gear pair Ⅱ (unit:mm)

Location 1 2 3 4 5 6 7 8 9 ( )2error∑

Before

ultrasonic

lapping

1 0.1566 0.1429 0.1037 0.0696 0.0369 0.0007 -0.0330 -0.0712 -0.1159 0.64085057

2 0.1362 0.1104 0.0789 0.0393 0.0037 -0.0417 -0.0810 -0.1149 -0.1511

3 0.1127 0.0812 0.0482 0.0078 0.0000 -0.0704 -0.1156 -0.1606 -0.2018

4 0.0952 0.0582 0.0221 0.0412 0.0014 -0.1087 -0.1511 -0.2029 -0.2421

5 0.0734 0.0345 -0.0090 -0.0544 -0.0994 -0.1445 -0.1940 -0.2487 -0.2847

After

ultrasonic

lapping

1 0.0881 0.0865 0.0777 0.0669 0.0544 0.0412 0.0170 0.0006 -0.0168 0.18612056

2 0.0756 0.0679 0.0626 0.0415 0.0249 0.0133 -0.0127 -0.0320 -0.0594

3 0.0588 0.0461 0.0399 0.0167 0.0000 -0.0171 -0.0451 -0.0665 -0.0965

4 0.0411 0.0244 0.0148 -0.0055 -0.0319 0.0487 -0.0792 -0.0998 -0.1313

5 0.0215 0.0034 -0.0108 -0.0376 -0.0555 -0.0800 -0.1100 -0.1405 -0.1656

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TABLE 10:Concave flank error of gear in gear pair Ⅱ (unit:mm)

Location 1 2 3 4 5 6 7 8 9 ( )2error∑

Before

ultrasonic

lapping

1 -0.2496 -0.1830 -0.1019 -0.0342 0.0396 0.1588 0.2260 0.2753 0.3035 1.49686842

2 -0.2638 -0.2017 -0.1223 -0.0588 0.0106 0.1239 0.1893 0.2473 0.2606

3 -0.2829 -0.2228 -0.1490 -0.0886 0.0000 0.0939 0.1502 0.2146 0.2281

4 -0.2997 -0.2463 -0.1742 -0.1140 -0.0065 0.0600 0.1142 0.1768 0.1804

5 -0.3173 -0.2589 -0.1996 -0.1378 -0.0392 0.0248 0.0719 0.1384 0.1349

After

ultrasonic

lapping

1 -0.2276 -0.1768 -0.0922 -0.0276 0.0418 0.1212 0.2049 0.2517 0.2835 1.21365009

2 -0.2418 -0.1829 -0.1015 -0.0275 0.0105 0.1047 0.1678 0.2236 0.2413

3 -0.2514 -0.2015 -0.1345 -0.0673 0.0000 0.0747 0.1310 0.1796 0.2014

4 -0.2714 -0.2263 -0.1546 -0.0829 -0.0142 0.0254 0.0945 0.1526 0.1676

5 -0.2918 -0.2257 -0.1841 -0.1236 -0.0310 0.0254 0.0741 0.1211 0.1515

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TABLE 11:Convex flank error of gear in gear pair Ⅱ (unit: mm)

Location 1 2 3 4 5 6 7 8 9 ( )2error∑

Before

ultrasonic

lapping

1 -0.0321 -0.0361 -0.0457 -0.0576 -0.0652 -0.0752 -0.0954 -0.1085 -0.1272 0.26018819

2 0.0178 0.0176 -0.0030 -0.0238 -0.0347 -0.0443 -0.0546 -0.0578 -0.0587

3 0.0636 0.0211 0.0287 0.0005 0.0000 0.0076 0.0210 0.0325 0.0185

4 0.1034 0.0743 0.0578 0.0453 0.0614 0.0684 0.0785 0.0621 0.0682

5 0.1421 0.1139 0.1059 0.1174 0.1278 0.1230 0.1254 0.1324 0.1286

After

ultrasonic

lapping

1 -0.0248 -0.0261 -0.0373 -0.0455 -0.0497 -0.0626 -0.0755 -0.0849 -0.1034 0.10867547

2 -0.0047 -0.0157 -0.0181 -0.0318 -0.0338 -0.0395 -0.0496 -0.0419 -0.0656

3 0.0204 0.0179 0.0094 -0.0005 0.0000 -0.0024 -0.0101 0.0224 -0.0217

4 0.0410 0.0377 0.0388 0.0352 0.0357 0.0378 0.0346 0.0351 0.0230

5 0.0652 0.0623 0.0692 0.0666 0.0758 0.0798 0.0816 0.0776 0.0744

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TABLE 12:Concave flank error of gear in gear pair Ⅱ (unit: mm)

Location 1 2 3 4 5 6 7 8 9 ( )2error∑

Before

ultrasonic

lapping

1 -0.2022 -0.1407 -0.0752 0.0404 0.0959 0.1630 0.2097 0.2663 0.3314 1.09843839

2 -0.2063 -0.1378 -0.0708 -0.0479 0.0412 0.1131 0.1673 0.2174 0.2712

3 -0.2286 -0.1479 -0.0968 -0.0730 0.0000 0.0602 0.1117 0.1631 0.2113

4 -0.2435 -0.1653 -0.1201 -0.1016 -0.0734 -0.0321 0.0715 0.1142 0.1573

5 -0.2671 -0.2221 -0.1864 -0.1301 -0.0912 -0.0536 0.0255 0.0912 0.0921

After

ultrasonic

lapping

1 -0.1622 -0.0987 -0.0452 0.0204 0.0759 0.1324 0.1897 0.2463 0.3014 0.88735497

2 -0.1863 -0.1288 -0.0742 -0.0179 0.0395 0.0931 0.1457 0.1984 0.2506

3 -0.2099 -0.1542 -0.1059 -0.0540 0.0000 0.0504 0.1026 0.1474 0.1935

4 -0.2253 -0.1753 -0.1303 -0.0817 -0.0401 0.0044 0.0517 0.0968 0.1364

5 -0.2490 -0.2007 -0.1570 -0.1101 -0.0706 -0.0272 0.0177 0.0532 0.0882

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Figure Captions

Fig. 1 Machining principle and vibration system for ultrasonic lapping gear

Fig. 2. Relationship between ∆ and f Fig. 3 Displacement amplitude of vibration system in the axial direction

Fig. 4. Curve of relative resonant frequency f∆

Fig. 5. Curve of amplification coefficient PM

Fig. 6. The hypoid pinion shaft of a certain automotive rear bridge

Fig. 7. Two pairs of gear pairs processed

Fig. 8. Revised vibration system of ultrasonic lapping gear

Fig. 9. Test position and motion mode of gear ultrasonic lapping in contact zone

Fig. 10. Experiment setup of gear ultrasonic lapping

Fig. 11. Measurement scene graph of tooth flank error

Fig. 12. Position of Measuring point of tooth flank error

Fig.13. Comparison of the square error sum of tooth flank error

Fig.14. Tooth flank error grid of the gear before ultrasonic lapping (Gear Pair Ⅰ)

Fig.15. Tooth flank error grid of the gear after ultrasonic lapping (Gear Pair Ⅰ)

Fig.16. Tooth flank error grid of the gear before ultrasonic lapping (Gear Pair Ⅱ)

Fig.17. Tooth flank error grid of the gear after ultrasonic lapping (Gear Pair Ⅱ)

Fig.18. Arrangement of vibration test equipment

Fig.19. Vibration test results of gear pair Ⅰ

Fig.20. Vibration test results of gear pair Ⅱ

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1l 2l 3l 4l

2d x

y1s 2s 3s 4s

3d

4d

1d

δ

510

Amµ

=�

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4l

3d

δ

30

Φ

18

M

28

152

100

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20

~5

=

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δ

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0.15151272

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Documents

Draft - University of Toronto T-Space · design theory of ultrasonic honing vibration system. Spiral bevel gear includes spiral bevel gears and hypoid gear during ultrasonic lapping