Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications

Mechanical Systems and Signal Processing 91 (2017) 167–182

Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications Tengjiao Lin a,⇑, Zeyin He a,b a b

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, PR China School of Mechatronics and Vehicle Engineering, Chongqing Jiaotong University, Chongqing 400074, PR China

a r t i c l e

i n f o

Article history: Received 23 May 2016 Received in revised form 23 December 2016 Accepted 2 January 2017

Keywords: Gear system Transmission errors Machining errors Assembly errors Tooth modifications

a b s t r a c t We present a method for analyzing the transmission error of helical gear system with errors. First a finite element method is used for modeling gear transmission system with machining errors, assembly errors, modifications and the static transmission error is obtained. Then the bending-torsional-axial coupling dynamic model of the transmission system based on the lumped mass method is established and the dynamic transmission error of gear transmission system is calculated, which provides error excitation data for the analysis and control of vibration and noise of gear system. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction As a complicated elastic machinery system, gear system will produce vibration and noise under dynamic excitation. So it is a primary question for the dynamic characteristics of gear system to research the dynamic excitation in the gear meshing process and determine the type and features of the dynamic excitation. As is well known to all, gear transmission transmits power by the mesh force along the mesh line. The dynamic excitation along the mesh line will then be produced in the transmission process, which adds the displacement difference of pinion and gear in the direction of the meshing line. This is what is called transmission error, which can be divided into static and dynamic transmission error. The displacement excitation depends greatly on the gear profile design, machining method and assembly error, and it is an important excitation source of the gear vibration and noise, which has an important influence on the dynamic characteristics. Therefore, the research about reduction of vibration and noise of gear transmission usually focuses on the reduction of displacement excitation along the mesh line [1], corresponding representatives are Velex and Kahraman. They studied the influence of modification, assembly error and profile error on the transmission error [2–5], summed up each factor’s influence linearly to get the static transmission error [6,7], analyzed the correlation between the tooth load and transmission error [8], and verified their result by experiment [9,10]. At the same time, there are lots of researchers who took the single level gear pair, multi level gear pair and the planetary transmission system as study objects, and carried out many research works about the calculating method of error excitation. The method uses the machining error given by the gear accuracy

⇑ Corresponding author. E-mail address: [email protected] (T. Lin). http://dx.doi.org/10.1016/j.ymssp.2017.01.005 0888-3270/Ó 2017 Elsevier Ltd. All rights reserved.

168

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

grade, synthesizes the contact ratio and base pitch error to calculate the simplified tooth error excitation while assuming that the tooth error follows the half sine distribution from tooth root to tooth tip [11–15]. In conclusion, many researchers carried out lots of research works about the error excitation. However, there are many simplifications when simply linearly summing up each error’s (machining error, assembly error and modification) influence on the static error, because the effect of interaction of each characteristic error on dynamic error excitation is hardly considered. Therefore, this work proposes a calculation method of static transmission error (STE) which considers the coupling of multi factors such as machining error, assembly error and modification. The bending-torsional-axial coupling dynamic model of transmission system is also established, and the dynamic transmission error (DTE) of gear transmission system is studied, which will provide error excitation data for the analysis and control of vibration and noise of gear system. 2. Model method for gear pair with error 2.1. Mathematical model of machining error Because of the factors such as gear machining and machine tool accuracy, there is always a certain error between the actual tooth surface and theoretical tooth surface. The actual surface error is a comprehensive error, and can be divided into several single errors, such as profile error (involute incline error f Ha and profile shape error f f a ), helix error (helix incline error f Hb and helix shape error f fb ), and pitch error f pt . As is shown in Fig. 1, the pitch error is 0 order error, involute incline error and helix incline error is 1 order error, and the profile shape error and helix shape error is high order error [16]. The gear machining error has a great effect on the transmission error of gear transmission system, and researchers have done a lot of research work about the machining error. However, most of them sum up each error linearly, which does not well reflect the combined effects on gear transmission error of the interaction of the individual errors. Therefore, we want to build a model of a gear pair with error based on the gear meshing theory to study the effect of the machining error on the static transmission error. While deriving the equation of tooth surface with machining error, the machining coordinate system of helical gear with error is Opxpyp in the cutter coordinate systems Sp, as is shown in Fig. 2, which the coordinate origin Op locates at the initial point of the straight line portion for cutter rack, xp is along the vertical direction to the straight line portion for cutter rack, yp is along the straight line portion for cutter rack. The coordinate system and the modeling process were similar to our group’s paper [17], which has the detailed explanations. The cutter error is assumed to be the sine function with the wave period of 1/W. The equation of cutter profile with error is

xp ¼ A sin Wyp

ð1Þ

where A is the amplitude of profile error function; W is the wave frequency of error. The machining coordinate system of helical gear is shown in Fig. 3. In Fig. 3, S0(O0  x0y0z0) is fixed to the cross section of the generating rack, y0, z0 locates on the pitch plane, x0 locates on the symmetric plane of the cross profile of the generating rack and is perpendicular to the pitch plane, and the coordinate

Fig. 1. The actual tooth surface with machining errors.

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

169

Fig. 2. The manufacturing system of actual tooth surface with error.

Fig. 3. Coordinate systems.

origin O0 locates at the center of the tooth breadth; S1(O1  x1y1z1) is fixed to the end plane of the generating rack, y1, z1 locates on the pitch plane, x1 locates on the symmetric plane of the end profile of the generating rack and is perpendicular to the pitch plane; S2(O2  x2y2z2) is fixed to the end face of the helical gear; S3(O3  x3y3z3) is fixed to the ground; r is the pitch radius of the gear, and u is the rotation angle of S2 along z3 axis. In the Sp coordinate system, the profile equation of the generating rack with error is R1sp , and the tooth root transition curve equation is R2sp which is assumed with no error; in the coordinate system S0, the profile equation of generating rack is Rs0 while the profile part and tip ellipse part are respectively R1s0 and R2s0 . Through the coordinate conversion from Sp to S0, the profile equation R1s0 and root transition curve R2s0 of generating rack with error in coordinate system S0 is

(

R1s0 ¼ M 0p R1sp

ð2Þ

R2s0 ¼ M 0p R2sp where M 0p is the coordinate conversion matrix from Sp to S0. In the coordinate system S0, tooth profile equation R1s0 and root transit curve R2s0 can be expressed as

8 < R1 ¼  R1x s0 s0 : R2 ¼  R2x s0

s0

R1y s0

R1z s0

1

R2y s0

R2z s0

1

T T

ð3aÞ

170

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

which is to say,

8 1 0  A sin at ðsin WlÞ þ l cos at  ðhan mn  xn mn þ sÞ > > > > B A cos a ðsin WlÞ  ðl sin a  ðh m  x m þ sÞ tan a  ðpm =4 þ x m tana ÞÞ C > > t t n n t n n n t C B 1 > an n > C > Rs0 ¼ B > A @ > 0 > > > > < 1    1 0  han mn  xn mn þ s þ q sin an  q sin h > > >   >     C B  > q cos an pmn > C B > > R2 ¼ B   han mn  xn mn þ s tan at  4 þ xn mn tanat  cos b0 þ q cos h C > > s0 C B > > A @ 0 > > > : 1

ð3bÞ

where l is the length of BD on the cutter; s is the distance along xn axis between the projections of point B and C; han⁄ is the tooth addendum coefficient; c⁄n is the tip clearance coefficient; at1 is the transverse pressure angle; mn is the normal modulus; xn is the addendum modification coefficient; q is the corner radius; ±respectively stands for the left and right flank of cutter. Using the coordinate transformation from S0 to S1, the tooth surface equation R1s1 and root transit surface R2s1 of the rack cutter with error in transverse coordinate system S1 can be obtained

(

R1s1 ¼ M 10 R1s0

ð4Þ

R2s1 ¼ M 10 R2s0 where M 10 is the transformational matrix from S0 to S1.

0

1 0 0

0

0

1

1

B 0 1 0 l sin b0 C z C B M 10 ¼ B C @ 0 0 1 lz cos b0 A 0

0

0

where b is the helical angle of the gear with helix machining error, which is assume to be a sine function. The equation R1s1 and R2s1 of tooth surface and root surface in transverse coordinate system S1 is respectively

8 < R1 ¼  R1x s1 s1 : R2 ¼  R2x s1

s1

R1y s1

R1z s1

1

R2y s1

R2z s1

1

T T

ð5Þ

The unit normal vector n1s1 and n2s1 of any point on the tooth surface and root surface can be expressed as

8 @R1 @R1 s1  s1 > > @l @lz  1 >  n ¼ > s1 > @R1s1 @R1s1  > <  @l  @lz 

ð6Þ

@R2 @R2 > s1  s1 > > @h @lz  2 >  n ¼ > > : s1 @R2s1 @R2s1  @h @lz

The relative velocity vector

v 1s1 and v 2s1 of any point on the tooth surface and root surface is respectively

1 8 0  > x R1y > s1  r u1 > > B C > > C v 1s1 ¼ B > > @ A xR1x > s1 > > < 0 1 0  > > x R2y > s1  r u2 > > 2 C B > > v ¼ B xR2x C > > A > s1 @ s1 > : 0

ð7Þ

Through the coordinate transformation from S1 to S2, the tooth surface equation R1s2 and root transit surface R2s2 of helical gear in coordinate system S2 is

(

R1s2 ¼ M 21 R1s1 R2s2 ¼ M 21 R2s1

Then the mathematical model of tooth profile of helical gear with profile shape error and helix shape error is

ð8Þ

171

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

8 3 2 1x > ðRs1 þ rÞ cos u1 þ ðr u1  R1y > s1 Þ sin u1 >   > 7 6 > > 6 ðR1x þ rÞ sin u  ru  R1y cos u 7 > > 6 s1 1 1 1 17 s1 > > R ¼ 7 6 > s2 > 7 6 1z0 > > 5 4 R > s1 > > > < 1 3 2    2y > R2x > s1 þ r cos u2 þ r u2  Rs1 sin u2 > 7 6 > >    7 6  2x > > 6 R þ r sin u  r u  R2y cos u 7 > > 2 7 6 > 2 2 2 s1 s1 R ¼ > s2 6 7 > > 7 6 > 2z > Rs1 5 4 > > > : 1

ð9Þ

where M 21 is the transformational matrix from S1 to S2; u1 and u2 are the rotation angle of coordinate system S2 while generating the gear surface. Table 1 shows the main parameter of the gear pair of one marine gearbox, and the parameter of gear II and gear I are the same. Table 1 The main parameters of gear pairs. Parameters

Value

Tooth number z1 (z2) of gear I (II) Tooth number z3 of gear III Normal modulus mn/mm Reference circle pressure angle a/° Helical angle b/° Tooth width b/mm Center distance a/mm Addendum coefficient h*an Tip clearance coefficient cn

41 (left-handed) 161 (right-handed) 12 20 12 185/180 1250 1 0.4

Based on the equations of the tooth profile with error derived above, the parametric modeling program of helical gear is developed via Matlab code to generate exact three-dimensional (3-D) coordinates of discrete points on the tooth surface of helical gears. In order to directly show the helix shape error and profile shape error, the deviation of helical line and profile is amplified, and the discrete points on the tooth surface is shown is Fig. 4. The amplitude of deviation of helical line and profile is respectively 25 lm and 35 lm in this figure, and the wave period number is assumed to be f ¼ 2. 2.2. Model of gear transmission system with error

Tooth depth x/mm

Tooth depth x/mm

There is always assembly error in the assembly process of gearbox, and this error has great influence on the transmission error of gear system. Fig. 5 shows the schematic diagram of assembly error of gear pair, where O1O2 and O3O4 are the shaft axis of a gear pair without assembly error, O5O6 is the shaft axis with assembly error. There are two angle deviations between O1O2 and O5O6, which are respectively angle g in the vertical plane and angle f in the horizontal plane.

To oth wi dth z

helix deviation

/m m

Tooth

(a) helix shape error

t

ess hickn

y/mm

To oth wi dth z

profile deviation

/m m

ness

To

(b) profile shape error Fig. 4. Tooth profiles with deviations.

ick oth th

y/mm

172

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

Fig. 5. The definition of assembly errors.

In addition, there are profile modifications (tip and root modification) and crowning in the gear pair of the marine gearbox, the equation derivation of tooth surface is similar to the equation derivation of tooth surface with machining error. The modification process and modeling method is integrated into software GEMTE, and the software copyright has been applied. Synthesizing the machining error, assembly error and modification, the calculation model of static transmission error of the gear transmission system with multi-factors coupling of error and modification is established, and the comprehensive influence of such factors on the static transmission error of gear transmission system is studied. Fig. 6 shows the static contact finite element analysis model of the gear transmission system, there are two input gear shafts (gear I and gear II), one output gear and one output shaft, amounting to 409770 nodes. The different mesh models with the intermediate node are established to calculate the stress of the gear pairs. Our group finds that the stress of gear pairs changes small when the number of nodes is 409770. Therefore, our group supposes that the mesh is fine enough to properly represent local Hertzian deformation in the tooth contact zone. 3. Calculation of static transmission error for the gear system 3.1. The influence of tooth modification on static transmission error Adopting GEMTE software, the geometric model of the gear pair with modification is established. The static transmission error of the helical gear pair without modification is shown in Fig. 7.

Fig. 6. The finite element mesh of gear transmission system.

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

173

1.47

STE/μm

14

13

127 12

11 0

5

10

15

20

25

rotational angle θ /° Fig. 7. The static transmission error of ideal gear pairs.

From the result, while without modification, the average value of static transmission error is 12.7 lm, the peak-to-peak value of the static transmission error is 1.47 lm. The static transmission error of gear pair with profile modification is shown in Fig. 8. It displays that the average value of static transmission error is 12.3 lm, the peak-to-peak value of static transmission error is 0.8 lm. Compared to the result without modification, the average static transmission changes little, but peak-peak value decreases. On the basis of profile modification, the static transmission error of gear pair with crowning is calculated, as is shown in Fig. 9. It shows that the average value of static transmission error linearly increases with the increase in the crowning amount, the peak-to-peak value changes little while crowning is small, and when the crowning surpasses 30 lm, the peak-to-peak value increases greatly. 3.2. The influence of assembly error on static transmission error In order to study the influence of assembly error in vertical plane (V plane) and horizontal plane (H plane) on the transmission error of gear pair, the static transmission error with different g and f are calculated, as is shown in Figs. 10 and 11, and the average and peak-to-peak values of transmission error is listed in Table 2. Looking at the calculations in Table 2, we can see that the average transmission error caused by assembly error in the horizontal plane will be 1–2 times greater than the transmission error caused by assembly error in vertical plane, and the peak-peak value is 3–5 times. This may be mainly from that the contact condition with assembly error in horizontal plane is better than that with assembly error in vertical plane. 3.3. Influence of machining error on static transmission error Fig. 12 shows the static transmission error of the gear pair with machining error (f fb = 25 lm and f f a = 35 lm). We note that the average value of static transmission error will decrease gradually, and the peak-to-peak value will increase gradually as the alignment wave period number f increases. Compared to the gear pair without machining error, the average and peak-to-peak value of the static transmission error are both much greater, and the curve regulation has also changed, which are mainly from that the tooth surface contact condition has changed.

13

12.3

0.8

STE/μm

14

12

11 0

5

10

15

20

25

rotational angle θ /° Fig. 8. The static transmission error of gear pairs with profile modification.

174

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

1.01 0.83

16.2

0.82

STE/μm

16

17.7

14.5

0.80

19

12.3

13

C=0 μm

C=15 μm

C=30 μm

C=45 μm

10 0

5

10

15

20

25

rotational angle θ /° Fig. 9. The static transmission error of gear pairs with profile modification and crowning.

24

STE/μm

18

12

η=0° η=0.02°

η=0.01° η=0.03°

6 0

5

10

15

20

25

rotational angle θ /° Fig. 10. The effects of assembly errors in the V plane on static transmission error.

36

STE/μm

30 24 18 12 6

ζ=0°

ζ=0.01°

ζ=0.02°

ζ=0.03°

0 0

5

10

15

20

25

rotational angle θ /° Fig. 11. The effects of assembly errors in the H plane on static transmission error.

175

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182 Table 2 The effects of assembly errors on static transmission error.

g/°

0 12.3 0.8 0 12.3 0.8

Average values of transmission error/lm Peak-to-peak values of transmission error/lm f/° Average values of transmission error/lm Peak-to-peak value of transmission error/lm

0.01 15.8 1.1 0.01 21.2 4.9

0.02 19.0 2.0 0.02 27.5 7.6

0.03 21.7 2.6 0.03 32.0 9.8

30

STE/μm

24 18 12 6

f=1

f=2

f=3

0 0

5

10

15

20

25

rotational angle θ/° Fig. 12. The static transmission error of helical gears with machining errors.

3.4. Synthesizing the effect of all errors on static transmission error In order to research the comprehensive influence of all factors on the static transmission error, Fig. 13 shows the static transmission error of the modified gear pairs with machining error (f fb = 25 lm and f f a = 35 lm) and assembly error (error g = 0.02°) in the vertical plane (V plane). Compared to the result while considering the machining error and assembly error separately, the average value of static transmission error is 22 lm, which is smaller than the gear pair only with machining error, and bigger than the gear pair only with assembly error in the vertical plane; the peak-to-peak value is 9 lm, which is not only bigger than the gear pair only with machining error, but also bigger than the gear pair only with assembly error. From the analysis of the frequency spectrum, the frequency domain of the static transmission error has more high frequency components at multiples of the mesh frequency. So we would not simply linearly summing up each error’s (machining error, assembly error and modification) influence on the static error. Fig. 14 shows the static transmission error of modified gear pairs with machining error and assembly error in the horizontal plane (H plane). It shows that the average value of the static transmission error is 26.8 lm, and the peak-to-peak value is 10.8 lm, which is greater than both the gear pair only with machining error and the gear pair only with assembly error in horizontal plane. From the analysis of the frequency spectrum, we can note that the static transmission error has more high frequency components at the multiplication of mesh frequency, but those high frequency components are smaller compared to the gear pair with machining error and assembly error in vertical plane. 4. Calculation of dynamic transmission error of gear system 4.1. The discrete dynamic model of the gear transmission system In order to study the dynamic transmission error of the gear system, the bending-torsional-axial discrete dynamic model of the gear transmission system is established with lumped mass method on the basis of the static transmission error, shown in Fig. 15. The gear I and gear II are input gears, the gear III is output gear, and the helical angle of the pinions is b which are left handed. The dynamic model with 12 degree of freedom of the gear system is established, and the generalized displacement array of the system can be expressed as

fxp1 yp1 zp1 hp1 xp2 yp2 zp2 hp2 xg yg zg hg gT

ð10Þ

where xpi ; ypi ; zpi ; hpi (i = 1,2) are respectively the linear and rotational displacements of input gear I and II along x, y, z, directions; xg ; yg ; zg ; hg are respectively the linear and rotational displacements of output gear III along x, y, z, directions. The relative displacements dni (i = 1,2) along the normal direction of the mesh point of gear pair I and II are

dni ¼ ðxpi þ xg Þ sin a þ ðypi þ yg þ r pi hpi  rg hg Þ cos a cos b þ ðzpi þ zg Þ cos a sin b  ei ðtÞ

ð11Þ

176

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

30

STE/μm

25 20 15

assembly error in V plane

10 machining error 5 assembly error in V plane and machining error 0 0

5

10 15 rotational angle θ /°

20

25

4000

5000

(a) time domain 3 assembly error in V planeV machining error

STE/μm

2

assembly error in V plane

1

0 0

1000

2000 3000 frequency f /Hz

(b) frequency domain Fig. 13. The transmission error of modified gear pairs with machining error and assembly errors in V plane.

where b – the helical angle of the helical gear; rp1 , r p2 , r g -the base circle radius of helical gear 1, 2, 3; hp1 , hp2 , hg – the rotation angle of helical gear 1, 2, 3; e1(t), e2(t)-the static transmission error of the gear pair I and II which takes profile modification, crowning, machining error, assembly error and mesh phase difference into consideration. The tooth dynamic mesh force and its components along each axial direction are

8 _ > > > F ni ¼ kni ðtÞdni þ cni dni > < F ¼ sin a½k ðtÞd þ c d_  xi ni ni ni ni > F yi ¼ cos a cos b½kni ðtÞdni þ cni d_ ni  > > > : F zi ¼ cos a sin b½kni ðtÞdni þ cni d_ ni 

ði ¼ 1; 2Þ

ð12Þ

where kni, cni are respectively the normal mesh stiffness and damping of gear pair I (gear I and gear III) and II (gear II and gear III), which the cni is 154257.74, and the normal mesh stiffness has taken the profile modification, crowning, machining error, assembly error (horizontal plane) and mesh phase difference into consideration. The time-varying mesh stiffness and static transmission error are respectively shown in Figs. 16 and 17. The motion direction of each component under the effect of input torque is defined as the positive direction of each component’s rotation displacement and the positive direction of the equal linear displacement along the mesh line. The mesh line direction when the tooth surface is under press is defined as the positive direction of the relative displacement. Focusing on the transmission system of a marine gearbox, the bending-torsional-axial coupling dynamic model with time-varying mesh stiffness, profile modification, crowning, assembly error and machining error is established

177

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

35

STE/μm

28 21 assembly error in H plane

14

machining error

7

assembly error in H plane and machining error 0 0

5

10 15 rotational angle θ /°

20

25

(a) time domain 3

assembly error in H plane

STE/μm

machining error assembly error in H plane and machining error

2

1

0 0

1000

2000

3000 frequency f /Hz

4000

5000

(b) frequency domain Fig. 14. The transmission error of modified gear pairs with machining error and assembly errors in H plane.

8 m €x þ cp1x x_ p1 þ kp1x xp1 ¼ F x1 > > > p1 p1 > > > m €x þ cp2x x_ p2 þ kp2x xp2 ¼ F x2 > > p2 p2 > > > mg €xg þ cgx x_ g þ kgx xg ¼ F x1  F x2 > > > > €p1 þ cp1y y_ p1 þ kp1y yp1 ¼ F y1 > mp1 y > > > > > €p2 þ cp2y y_ p2 þ kp2y yp2 ¼ F y2 y m > p2 > > > < mg y €g þ cgy y_ g þ kgy yg ¼ F y1  F y2 mp1 €zp1 þ cp1z z_ p1 þ kp1z zp1 ¼ F z1 > > > > > mp2 €zp2 þ cp2z z_ p2 þ kp2z zp2 ¼ F z2 > > > > > > mg €zg þ cgz z_ g þ kgz zg ¼ F z1  F z2 > > > > > Ip1 €hp1 ¼ T p1  F y1 r p1 > > > > > > Ip2 €hp2 ¼ T p2  F y2 r p2 > > > : € Ig hg ¼ T g þ F y1 r g þ F y2 r g

ð13Þ

where kpij , cpij – the bearing stiffness and damping of the input shaft, and i = 1, 2, j = x, y, z; kgj , cgj – the bearing stiffness and damping of the output shaft, and j = x, y, z. The dynamic characteristics of the gear transmission system with input rotation rate n1 = n2 = 800 r/min and input torque Tp1 = Tp2 = T = 2387.2 N m is calculated. We would convert the static transmission error eðtÞ and the time-varying stiffness kðtÞ into Fourier expansion, take them into the above equation and adopt the four order variable step Runge-Kutta numerical integration method to solve the motion differential Eq. (13).

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

Fig. 15. The discrete parameter dynamic model of gear transmission system.

35

k /N•μm -1 •mm-1

178

gear pair I

gear pair II

25

15

5 0

5

10

15

20

rotational angle θ /° Fig. 16. The time-varying meshing stiffness.

25

179

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

50

STE/μm

40

30

20

gear pair I

gear pair II

10 0

5

10

15

20

25

rotational angle θ /° Fig. 17. The static transmission error.

4.2. Analysis of the dynamic transmission error Figs. 18 and 19 shows the dynamic transmission error (DTE) and dynamic mesh force (DMF) of the transmission system with input rotation rate n1 = n2 = 800r/min and input torque Tp1 = Tp2 = T = 2387.2 N m. From the result, there is difference between the transmission error of gear pair I and II because there exists interaction and phase difference between the mesh forces. Figs. 20 and 21 shows the peak-to-peak valueof the dynamic transmission error (PPTE) and the root mean square (RSM) value of the dynamic mesh force of the gear transmission system while input torques are Tp1 = Tp2 = T and input speed are n1 = n2 = n (n = 50 r/min, 100 r/min, 150 r/min, . . ., 2000 r/min). In the figures, the outer envelope lines of the dynamic transmission errors of gear pair I and gear pair II are used to represent the transmission error of the whole transmission system, and the outer envelope lines of the dynamic mesh force of gear pair I and gear pair II are used to represent the dynamic mesh force of the whole transmission system. It shows that the peak-peak value of the dynamic transmission error and the root mean square value of the dynamic mesh force of the gear system both increase firstly and then decrease as the rotation speed increases, and the peak values appear at the speed of 550 r/min and 1100 r/min (corresponding frequencies 375 Hz and 750 Hz are the natural frequencies of the gear 66

50

DTE/μm

DTE/μm

65

35 20 5 30

31

32

33

34

44 22 0 30

35

31

32

33

time t /ms

time t /ms

(a) gear pair I

(b) gear pair II

34

35

Fig. 18. The dynamic transmission error of helical gear transmission system.

20

15

DMF2 /kN

DMF1 /kN

20

10 5 0 90

15 10 5

91

92

93

time t /ms

(a) gear pair I

94

95

0 90

91

92

93

time t /ms

(b) gear pair II

Fig. 19. The dynamic meshing force of helical gear transmission system.

94

95

180

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

80

PPTE/μm

60 40

gear pair I 20

gear pair II outer envelope line

0

0

500

1000

1500

2000

speed n /r∙min-1

RSM dynamic meshing force /kN

Fig. 20. The peak-to-peak of transmission error (PPTE) of helical gear pair.

16

12

gear pair I 8

gear pair II Outer overlope line

4 0

500

1000

1500

2000

speed n /r∙min-1 Fig. 21. The RSM dynamic meshing force of helical gear pair changing with the rotational speed.

Gear I

O3 G3

Gear III

θ3

Gear II

G1

θ1 O1

Fig. 22. The finite element model of helical gear transmission system.

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

181

90

DTE/μm

60 30 0 lumped mass method -30 -60 10

finite element method 12

14

16

18

time t /ms

(a) gear pair I 90

DTE/μm

60 30 0

lumped mass method -30

finite element method

-60 10

12

14

16

18

time t /ms

(b) gear pair II Fig. 23. The dynamic transmission error for gear system.

system), which means changes of the parameter of gear should be taken to avoid the occurrence of those two peak frequencies.

4.3. Dynamic transmission error calculation based on the finite element method In this section we will adopt the explicit dynamic contact algorithm based on finite element method to calculate the dynamic transmission error of the gear system, so as to verify the rationality of the transmission error calculated by the discrete parameter model. Fig. 22 shows the dynamic contact finite element model of the transmission system. In the figure, G1, G2, G3, are respectively an arbitrary node on the inner surface of gear I, II and III. At any moment t in the mesh process, those three nodes’ n o n o n o corresponding displacements are xGt 1 ; yGt 1 ; zGt 1 , xGt 2 ; yGt 2 ; zGt 2 and xGt 3 ; yGt 3 ; zGt 3 ; at the next computation moment t þ Dt, n o n o n o those three nodes’ corresponding displacements are xGtþ1 Dt ; yGtþ1Dt ; zGtþ1Dt , xGtþ2 Dt ; yGtþ2 Dt ; zGtþ2 Dt and xGtþ3 Dt ; yGtþ3 Dt ; zGtþ3 Dt . Then the rotation angle of gear I can be expressed as

1

h1 ¼ sin

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2ffi xGt 1  xGtþ1Dt þ yGt 1  yGtþ1 Dt þ zGt 1  zGtþ1Dt r1

ð14Þ

Likewise, the rotation angle h2 and h3 of gear II and gear III can be solved. Fig. 23 shows the dynamic transmission error of the gear transmission system which is calculated by finite element method with input rotation rate of n1 = n2 = 800 r/min, and input torque of Tp1 = Tp2 = T = 2387.2 N m, the dynamic transmission error calculated by discrete parameter method is also displayed to make a comparison. It describes that the average values of the dynamic transmission error calculated by finite element method and discrete parameter method respectively are close and have similar change law, while the peak-to-peak values calculated by finite element method is a little smaller mainly because the number of the nodes of the dynamic model is too large to include the input and output shafts. There are some higher frequency contents of the FE method because the nonlinear factors exist in meshing process of gear pairs, such as the variable stiffness, the transmission error and the load sharing performance, which would affect the fatigue of gear pairs.

182

T. Lin, Z. He / Mechanical Systems and Signal Processing 91 (2017) 167–182

5. Conclusion (1) Synthesizing the profile modification and crowning, machining errors and assembly errors, the calculation model of the static transmission error of the gear transmission system with the coupling of multi factors such as errors and modification is proposed. (2) Reasonable profile modification has little effect on the average value of the static transmission error, but it can reduce the peak-to-peak value of the static transmission error. Along with the increase of the crowning amount, the average value of the static transmission error increases linearly, but the peak-to-peak value changes little when the crowning is small. The transmission error caused by assembly error in horizontal plane is greater than the transmission error caused by the same assembly error in vertical plane. (3) Based on the discrete parameter method, the bending-torsional-axial coupling dynamic model of the transmission system with time-varying mesh stiffness, profile modification and crowning, assembly error and machining error is established. The dynamic transmission error of the gear transmission system is calculated, which can provide error excitation data for the analysis and control of vibration and noise of gear system.

Acknowledgements The authors are grateful for the financial support provided by the National Science & Technology Support Project of PR China under Contract No. 2015BAF06B02, Chongqing Research Program of Basic Research and Frontier Technology No. cstc2016jcyjA0514 and Project Supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission No. KJ1600503. References [1] J.D. Smith, Gear Noise and Vibration, 2nd ed., revised., Marcel Dekker, New York, 2004. [2] P. Velex, M. Maatar, A mathematical model for analyzing the influence of shape deviations and mounting errors, J. Sound Vib. 5 (1996) 629–660. [3] M. Kubur, A. Kahraman, D.M. Zini, Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment, J. Vib. Acoust., Trans. ASME 3 (2004) 398–406. [4] S.T. Li, Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications, Mech. Mach. Theory 42 (2007) 88–114. [5] S.T. Li, Effects of machining errors, assembly errors and tooth modification on loading capacity, load-sharing ratio and transmission error of a pair of spur gears, Mech. Mach. Theory 42 (2007) 698–726. [6] P. Velex, M. Ajmi, On the modelling of excitations in geared systems by transmissions errors, J. Sound Vib. 3–5 (2006) 882–909. [7] P. Velex, J. Bruyère, D.R. Houser, Some analytical results on transmission errors in narrow-faced spur and helical gears: influence of profile modifications, J. Mech. Des., Trans. ASME 3 (2011) 1–11. [8] P. Velex, M. Ajmi, Dynamic tooth loads and quasi-static transmission errors in helical gears – approximate dynamic factor formulae, Mech. Mach. Theory 11 (2007) 1512–1526. [9] V.K. Tamminana, A. Kahraman, S. Vijayakar, A study of the relationship between the dynamic factors and dynamic transmission error of spur gear pairs, J. Mech. Des., Trans. ASME 1 (2007) 75–84. [10] M.A. Hotait, A. Kahraman, Experiments on the relationship between the dynamic transmission error and the dynamic stress factor of spur gear pairs, Mech. Mach. Theory 70 (2013) 116–128. [11] J.B. Han, G. Liu, L.Y. Wu, Synthetic method of error excitation of gear transmission system, Chin. J. Mech. Transmiss. 5 (2009) 24–26. [12] T.J. Lin, Y.J. Liao, R.F. Li, Numerical simulation and experimental study on radiation noise of double-ring gear reducer, Chin. J. Vib. Noise 3 (2010) 43–47. [13] T.J. Lin, Z.Y. He, F.Y. Geng, Prediction and experimental study on structure and radiation noise of subway gearbox, J. Vibroeng. 4 (2013) 1838–1846. [14] J.X. Zhou, G. Liu, S.J. Ma, Vibration and noise analysis of gear transmission system, Chin. J. Vib. Noise 6 (2011) 234–238. [15] C.C. Zhu, B. Lu, C.S. Song, Dynamic analysis of a heavy duty marine gearbox with gear mesh coupling, Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 11 (2009) 2531–2547. [16] Z.Y. Shi, H. Lin, Multi-degrees of freedom theory for gear deviation, Chin. J. Mech. Eng. 1 (2014) 55–60. [17] Z.Y. He, T.J. Lin, T.H. Luo, Parametric modeling and contact analysis of helical gears with modifications, J. Mech. Sci. Technol. 30 (2016) 4859–4867.