Improved analytical models for mesh stiffness and load sharing ratio of spur gears considering structure coupling effect

Mechanical Systems and Signal Processing 111 (2018) 331–347

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

Improved analytical models for mesh stiffness and load sharing ratio of spur gears considering structure coupling effect Chongyang Xie a,b,c,⇑, Lin Hua b,c,⇑, Jian Lan a,b,c, Xinghui Han a,b,c, Xiaojin Wan a,b,c,d, Xiaoshuang Xiong a,b,c a

Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan 430070, PR China School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, PR China c Hubei Collaborative Innovation Center for Automotive Components Technology, Wuhan University of Technology, Wuhan 430070, PR China d State Key Laboratory of Digital Manufacturing Equipment and Technology of China, Huazhong University of Science and Technology, Wuhan 430074, PR China b

a r t i c l e

i n f o

Article history: Received 6 November 2017 Received in revised form 15 February 2018 Accepted 15 March 2018

Keywords: Spur gears Mesh stiffness Load sharing ratio Fillet foundation stiffness The minimum potential energy principle

a b s t r a c t Due to the lack of efficient formulas, fillet foundation stiffness under double tooth engagement has long been a challenging problem for mesh stiffness calculation of spur gears by analytical method. Direct summation of the mesh stiffness of each meshing tooth pair overestimates the total mesh stiffness, which may lead to a large calculation error. An improved fillet foundation stiffness calculation method is proposed considering the structure coupling effect, namely, one gear body is shared by two meshing teeth simultaneously. On the basis of the proposed method, a mesh stiffness model and a load sharing model are analytically established. In the mesh stiffness model, fillet foundation stiffness correction factor is introduced to improve the calculation accuracy. Based on the minimum potential energy principle (MPEP), the load sharing ratio is determined by the proposed load sharing model. The accuracy of the proposed models is validated by comparing with finite element method (FEM). Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction As a key component in various mechanical transmissions, spur gear is an important research object for many researchers. For spur gear system, Time-varying mesh stiffness (TVMS) is considered as an important internal excitation source, which has a key influence on gear dynamics. Generally, TVMS can be evaluated either by rectilinear mesh stiffness or torsional mesh stiffness. Rectilinear mesh stiffness is defined as the meshing force needed for unit deformation along the line of action [13]. Torsional mesh stiffness is defined as the input torque needed for unit rotation of driving gear hub (driven gear hub being fixed) [11]. The relationship between the two kinds of mesh stiffness can be expressed by the following equation [33]:

ktorsional ¼ R2b1 krectilinear

ð1:1Þ

where ktorsional represents the torsional mesh stiffness; krectilinear represents the rectilinear mesh stiffness; Rb1 is the radius of driving gear hub. ⇑ Corresponding authors at: Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan 430070, PR China. E-mail address: [email protected] (C. Xie). https://doi.org/10.1016/j.ymssp.2018.03.037 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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Many calculation methods are developed to study the TVMS of spur gear pairs, i.e., the experimental methods [1–4], the FEM [5–11], the analytical method [12–25] and the analytical FEM [26–29].There is no doubt that the experimental methods are accurate and reliable. However, special devices and measurement methods are needed for all the reported experimental methods [1–4]. Hence, more finite element (FE) models, analytical models and analytical FE models are established to calculate the TVMS of spur gears. FEM is widely accepted and used by many researchers for its convenience and accuracy. For example, Wang et al. [5] studied the torsional mesh stiffness of spur gear pairs using FEM. Rectilinear mesh stiffness of spur gear pair with tip relief was calculated by Ma et al. [6] with a FE model. Using FEM, Lin et al. [7] investigated the influence of backlash on dynamic properties of gear drives. Based on FEM, the rectilinear mesh stiffness of modified gears with tooth root crack is determined by Pandya et al. [8]. In general, FEM is powerful and efficient. However, it is time-consuming and computationally expensive. In order to reduce the computational cost, analytical FEM was also adopted by some researchers. Using a 2D FE model, the combined torsional mesh stiffness of spur gears was determined by Timo et al. [26], and some empirical formulas were derived based on the FE results. A FEM based procedure was presented by Fernandez del Rincon et al. [27] to investigate the loaded transmission error (LTE) as well as the rectilinear mesh stiffness and load sharing ratio of spur gear transmissions. Chang et al. [28] developed a robust analytical FE model to calculate the rectilinear mesh stiffness of cylindrical gears. Pedersen et al. [29] proposed a rectilinear mesh stiffness calculation model for spur gears, where gear rim size and contact zone size were found to have obvious influence on mesh stiffness. Different from FEM and analytical FEM, analytical method studied the TVMS of gear pairs based on Timoshenko beam theory [34]. The analytical method, which is also known as the potential energy method, was first proposed by Yang and Lin [13] with the assumption that the total energy stored in a meshing tooth consists of three parts: Hertzian contact energy, bending energy and axial compressive energy. Later, shear energy was also considered by Tian et al. [14]. In order to calculate the fillet foundation deflections, Sainsot et al. [15] derived an analytical formula based on the theory of Muskhelishvili. Chen and Shao et al. [16] established a general analytical rectilinear mesh stiffness model, where tooth deviations and spacing errors were considered. Chaari et al. [17] studied the effect of tooth spalling and breakage on the rectilinear mesh stiffness of spur gear pairs. Liang et al. [18] analytically evaluated the rectilinear mesh stiffness of a planet gear set with tooth crack. Using the analytical method, Saxena et al. [19] investigated the influence of tooth crack on the rectilinear mesh stiffness of geared rotor system. Pedrero et al. [20] proposed a load sharing model based on the potential energy method. In Pedrero’s model, MPEP was adopted. Later, Pedrero’s [20] model was further improved by Miryam et al. [21] with the Hertzian contact energy being considered. Actually, traditional analytical method may lead to a large calculation error for evaluating the TVMS in double tooth engagement region, i.e., the TVMS is directly calculated by the summation of the mesh stiffness of each meshing gear pair [16–23]. However, it should be noted that the analytical formula in Ref. [15] is derived under single tooth engagement situation. For double tooth engagement situation, the rectilinear fillet foundation stiffness must be reconsidered. Unfortunately, little work has been done to analytically evaluate the rectilinear fillet foundation stiffness in double tooth engagement region. Having noted the deficiency, many researchers proposed various revised models using the FEM and the analytical method [5,25,26]. Such as Wang et al. [5] treated the gear body stiffness (torsional) as a constant value in double tooth engagement using FEM. Timo et al. [26] proposed a revised mesh stiffness model assuming that the gear body stiffness (torsional) in double tooth engagement region is proportional to that in single tooth engagement region with a fixed ratio using FEM. Ma et al. [25] also proposed a revised mesh stiffness model with similar assumption using analytical method. In Ma’s [25] model, a constant fillet foundation stiffness (rectilinear) correction factor was introduced, which was determined by a specially designed FE model. Generally, methods proposed in Ref. [5,25,26] try to revise the gear body stiffness (or fillet foundation stiffness for rectilinear stiffness situation) in double tooth engagement region by a mathematical way, and are surely effective to some extent. However, some important points are neglected and may need further investigation: 1. The gear body stiffness (or fillet foundation stiffness for rectilinear stiffness situation) during double tooth engagement is considered as load-independent for a given mesh position, which is only determined by geometry parameters. This means the meshing force of one gear pair has no influence on the other gear pair theoretically, i.e., the structure coupling effect is ignored. 2. The analytical load sharing model is still based on the traditional mesh stiffness model, which may lead to some calculation errors inevitably. The remaining part of this paper is organized as follows. In Section 2, a revised fillet foundation stiffness calculation method and an improved mesh stiffness model are proposed. Section 3 presents an improved load sharing ratio model based on MPEP. Finally, general FE models are established to prove the correctness of the proposed method in Section 4.

2. Analytical mesh stiffness calculation model Traditional mesh stiffness model (rectilinear) together with some revised models (rectilinear and torsional) proposed by other researchers are briefly introduced in Section 2.1. An improved mesh stiffness model (rectilinear) is proposed in

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Section 2.2. It should be noted that the proposed mesh stiffness model belongs to the rectilinear mesh stiffness model, which is derived by: k ¼ F=d. 2.1. Problem description According to the Timoshenko beam theory [34], the potential energy U beam stored in a cantilever beam consists of three parts: shear energy U s , axial compressive energy Ua , and bending energy U b , which can be expressed as:

U beam ¼ U s þ U a þ U b

ð2:1Þ

As shown in Fig. 1, the total energy Utooth stored in a single tooth pair with the meshing force being F can be further expressed as follow: p w U tooth ¼ U h þ U ptooth þ U w tooth þ U body þ U body ¼

F2 2k

ð2:2Þ

where U h is the Hertzian contact energy; U itooth ði ¼ p; wÞ is the energy considering the gear tooth as a cantilever beam with variable cross-section; U ibody ði ¼ p; wÞ is the energy caused by the elastic deformation of the gear body; superscripts p and w denote the pinion and wheel, respectively. In Eq. (2.2), U h , U itooth ði ¼ p; wÞ, U ibody ði ¼ p; wÞ can be further written as[24]:

Uh ¼

F2 F2 F2 F2 F2 ; U itooth ¼ i þ i þ i ; U ibody ¼ i 2kh 2ks 2ka 2kb 2kf

ð2:3Þ

i

i

where kh is the Hertzian contact stiffness; ks ði ¼ p;wÞ is the shear stiffness; ka ði ¼ p;wÞ is the axial compressive stiffness; i kb ði

i kf ði

¼ p;wÞ is the bending stiffness; ¼ p;wÞ is the fillet foundation stiffness. Substituting Eq. (2.3) into Eq. (2.2), one can obtain the following equation:

k¼

1 kh

1 þ k1p þ k1w þ k1p þ k1w þ k1p þ k1w þ k1p þ k1w s

s

a

a

b

b

f

ð2:4Þ

f

In Eq. (2.4), the Hertzian contact stiffness kh is nonlinear and load-dependent, which has been proved by experimental method and analytical method. An empirical formula has been proposed to conveniently calculate the Hertzian contact stiffness expressed as follow [32]:

p U body

p U tooth w U tooth

Uh

w U body

Fig. 1. Total potential energy stored in a meshing gear pair.

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C. Xie et al. / Mechanical Systems and Signal Processing 111 (2018) 331–347 0:8

kh ¼

E0:9 b F 0:1 1:275

ð2:5Þ

where E is Young’s modulus; b is the tooth width. It is worth noting that similar formula was also adopted in Refs. [21,24], of which the Hertzian contact stiffness was assumed to be load-independent as:

kh ¼

pEb 4ð1  m2 Þ

ð2:6Þ

where m is the Poisson’s ratio. In this paper, Eq. (2.5) is adopted for its higher accuracy. Based on cantilever beam theory (see Fig. 2), ks , ka and kb can be expressed as follows [13,14]:

1 ¼ ks 1 ¼ ka 1 ¼ kb

Z

d

0

Z

d

0

Z

d

0

1:2 cos2 am dx GAx

ð2:7Þ

2

sin am dx EAx

ð2:8Þ 2

½cos am ðd  xÞ  sin am h dx EIx

ð2:9Þ

where G is shear modulus; Ax is the area of cross section; Ix is the area moment of inertia of the cross section; For the calculation of fillet foundation stiffness (see Fig. 3), Sainsot et al. [15] proposed an analytical formula expressed as:

1 cos2 am ¼ kf Eb

(   )   2   uf  uf  2 þ P ð1 þ Q tan am Þ þM L Sf Sf

ð2:10Þ

where the coefficients L ; M  ; P  ; Q  are constants for a given mesh point One can find more details about the variables in Eq. (2.10) in Ref. [15]. Based on Eqs. (2.4), (2.5) (2.7), (2.8), (2.9) and (2.10), the mesh stiffness of single gear pair can be analytically calculated. For double tooth engagement situation, the mesh stiffness is calculated as springs connected in parallel (see Fig. 4), which means:

kdouble ¼

2 X 1 i¼1 khi

1 þ k1p þ k1w þ k1p þ k1w þ k1p þ k1w þ k1p þ k1w si

si

ai

ai

bi

bi

fi

ð2:11Þ

fi

where i denotes the gear pair number.

y

Fy

F

d m

Fx

h O

x

hx

dx

Fig. 2. Cantilever beam model of meshing tooth.

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F m

uf

-

f

f

sf rint

rf

Fig. 3. Fillet foundation stiffness calculation parameters [15].

k fp1

ksp1, kap1, kbp1

kh1

ksw1 , kaw1, kbw1

k wf1

pinion

wheel

k fp2

ksp2 , kap2 , kbp2

kh 2

ksw2 , kaw2 , kbw2

k wf2

Fig. 4. Traditional mesh stiffness model.

However, Eq. (2.11) overestimates the mesh stiffness in double tooth engagement region by simply summing the mesh stiffness of each gear pair. The main reason for this problem is that the two meshing tooth pairs share one gear body simultaneously [5,25], while Eq. (2.10) in Ref. [15] is derived under single tooth engagement situation. Hence, traditional mesh stiffness model must be revised to solve this problem. Wang et al. [5] proposed an improved torsional mesh stiffness model (shown in Fig. 5) for double tooth engagement situation. In Wang’s [5] model, the tooth stiffness(torsional) is considered as springs connected in parallel, while the gear body stiffness(torsional) kB was assumed to be constant. The calculation accuracy for mesh stiffness (torsional) was improved by Wang’s [5] model in double tooth engagement region. However, the gear body stiffness kB and the tooth part stiffness kT in Wang’s [5] model are calculated by FEM, which needs special mesh refinements and thus is time-consuming. Timo et al.[26] also proposed a torsional mesh stiffness model that is similar to Wang’s[5] model. In Timo’s[26] model, the torsional mesh stiffness in single tooth engagement region (ksingle ) is calculated by the following equations:

ksingle ¼

1

kp body

1 þ l2 k1w þ l2 k1w þ l2 kw1 contact body tooth contact

þ kp1 þ kp 1 tooth

ð2:12Þ

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kT1 kBp

kBw

pinion

wheel kT2

Fig. 5. Wang’s [5] mesh stiffness model.

with i

ð2:13Þ

i

ð2:14Þ

i

ð2:15Þ

kbody ¼ cB Ebi ln ðr fi  r si Þ1:6 r 1:6 si ; ði ¼ p; wÞ kcontact ¼ cC Ebi m1:85 z2i T 1:85 ; ði ¼ p; wÞ i ktooth ¼ cT Ebi m2 z2:2 i ; ði ¼ p; wÞ

l¼

zp zw

ð2:16Þ

where cB ¼ 9:555e  4; cC ¼ 7:937e  5; cT ¼ 3:2e  5; r f is the dedendum radius; r s is the hub bore radius; m is the module of the gear pair; z is the tooth number. For double tooth engagement region, the total mesh stiffness is calculated by the following equation:

kdouble ¼

1 p 1:1kbody

1 þ 1:1l21kw þ 2l2 k1w þ l2 kw1 contact body tooth contact

ð2:17Þ

þ 2kp1 þ kp 1 tooth

with i

kcontact ¼ c0C Ebi m1:85 z2p T 0:068 ; ði ¼ p; wÞ p

ð2:18Þ

where c0C = 1.1905e4. According to Eqs. (2.12-2.18), curve-fitted formulas were derived in Timo’s[26] model, which may be more efficient and convenient compared with Wang’s [5] model. However, three points should be noted in Timo’s [26] model: 1. The curve-fitted formula itself may lead to some calculation errors, too. 2. Too many simplifications are adopted for double tooth engagement situation. For example, the tooth stiffness is assumed to be twice as high as that in single tooth engagement region, which may be not so reasonable.

ksp1 , kap1, kbp1 12

kh1

ksw1 , kaw1, kbw1

k fp

22

pinion

k wf

wheel

ksp2 , kap2 , kbp2

kh 2

ksw2 , kaw2 , kbw2

Fig. 6. Ma’s mesh stiffness model [25].

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3. The physical model for mesh stiffness is actually the same as Wang’s [5] model. Based on Wang’s [5] model, Ma et al. [25] assumed the fillet foundation stiffness in double tooth engagement region is proportional to that in single tooth engagement region with constant coefficients (k12 ; k22 ¼ 1:1) (see Fig. 6). There is no doubt that the above listed revised models [5,25,26] are effective to some extent. However, the problem of traditional mesh stiffness model is mainly approached in a mathematical way. The load sharing model is still not improved. Actually, some physical factors should also be noted, which have an important influence on the mesh stiffness and load sharing ratio of gear pairs. 2.2. Improved analytical mesh stiffness calculation model A special designed FE model is established in this section to calculate the fillet foundation stiffness correction factors. Before the calculation process, some basic assumptions used in this paper must be explained first: 1. The material behavior is assumed to be linear and isotropic, which means the Young’s modulus, Poisson’s ratio and density are constants. 2. In order to calculate the gear body induced displacement (along the line of action) at the meshing point, elements of the tooth part are assigned to a material with Young’s modulus being 10,000 times higher than the gear body (see Fig. 7). Hence, the tooth part can be seen as a relative rigid region compared with the gear body under this condition. Then, the displacements contributed by local Hertzian contact effect and the elasticity deformation of the tooth part can be neglected. 3. A local coordinate system is established, of which the x axis is along the line of action. The displacement of the meshing point is measured along the x axis of the local coordinate system. As shown in Fig. 7, two 2D FE models are established in FEM software Abaqus CAE 6.12Ò [30] using plane stress elements (CPS3 and CPS4R). In Fig. 7(a), the inner ring nodes are coupled with the master node located at the center of the gear, and the master node is constrained in all directions. A unit force is applied at O1 , and the displacements of O1 and O2 are measured along the line of action. Similar as Fig. 7(a), in Fig. 7(b), a unit force is applied at O2 , and again the displacements of O1 and O2 are measured along the line of action. In order to make the calculation method easier to understand, we also define the following symbols listed in Table 1. According to the FEM results, one can obtain the following equations:

kf 1 ¼

1

ð2:19Þ

d11

Etooth =10000E

o x Local coordinate

O2 O1

Etooth =10000E

y

O2

1N

y

o x Local coordinate

1N O1

E

slave node

RP−1

master node

O

E

slave node

RP−1

master node

O

(a)Unit force applied on gear pair 1

(b)Unit force applied on gear pair 2

Fig. 7. Fillet foundation stiffness considering structure coupling effect.

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C. Xie et al. / Mechanical Systems and Signal Processing 111 (2018) 331–347 Table 1 Symbol definition.

kf 2 ¼

Symbol

Meaning

F i ði ¼ 1; 2Þ kfi ði ¼ 1; 2Þ K fi ði ¼ 1; 2Þ d1i ði ¼ 1; 2Þ

Actual meshing force applied at Oi The fillet foundation stiffness of gear pair i when Fi is applied at Oi The fillet foundation stiffness of gear pair i whenF 1 is applied at O1 and F 2 is applied at O2 The displacement of Oi when a unit force is applied at O1

d2i ði ¼ 1; 2Þ di ði ¼ 1; 2Þ

The displacement of Oi when F 1 is applied at O1 and F 2 is applied at O2

The displacement of Oi when a unit force is applied at O2

1

ð2:20Þ

d22

Based on linear superposition assumption, d1 and d2 can be calculated by the following equations:

d1 ¼ F 1 d11 þ F 2 d21

ð2:21Þ

d2 ¼ F 1 d12 þ F 2 d22

ð2:22Þ

Hence, K f1 and K f2 can be written as follows:

Kf1 ¼

F1 d1

ð2:23Þ

Kf2 ¼

F2 d2

ð2:24Þ

Further, the load sharing ratios can be defined as:

Lsr1 ¼

F1 F1 þ F2

ð2:25Þ

Lsr2 ¼

F2 F1 þ F2

ð2:26Þ

Substituting Eqs. (2.21), (2.22), (2.25) and (2.26) to Eqs. (2.23) and (2.24), one can obtain the following equations:

Kf1 ¼

Kf2 ¼

Lsr 1 Lsr 1 d11

ð2:27Þ

þ ð1  Lsr 1 Þd21 Lsr 2

ð2:28Þ

Lsr 2 d22 þ ð1  Lsr 2 Þd12

Based on Eqs. (2.19), (2.20), (2.27) and (2.28), we can define the fillet foundation stiffness correction factors ki ði ¼ 1; 2Þ as follows:

k1 ¼

Kf1 Lsr 1 d11 ¼ kf 1 Lsr 1 d11 þ ð1  Lsr 1 Þd21

p p 1 kf1

ð2:29Þ

ksp1, kap1, kbp1

kh1

ksw1 , kaw1, kbw1

w w 1 kf1

pinion

wheel p p 2 kf 2

ksp2 , kap2 , kbp2

kh 2

ksw2 , kaw2 , kbw2

w w 2 kf 2

Fig. 8. Revised mesh stiffness calculation model proposed in this paper.

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k2 ¼

Kf 2 Lsr 2 d22 ¼ 2 kf 2 Lsr 2 d2 þ ð1  Lsr 2 Þd12

ð2:30Þ

Finally, a revised mesh stiffness model is established as shown in Fig. 8. 3. Load sharing model In this section, a load sharing model will be proposed based on the mesh stiffness model proposed in Section 2. 3.1. Traditional load sharing model Traditional load sharing model is shown in Fig. 9. In Fig. 9, ki ði ¼ 1; 2; . . . ; nÞ is the mesh stiffness of gear pair i, and it is assumed to be completely load-independent. Hence, the load sharing ratio of gear pair i can be calculated by the following equation:

Lsr i ¼

ki Dx k i Dx ki ¼ n ¼ n X X F ki Dx ki i¼1

ð3:1Þ

i¼1

It is worth noticing that the traditional load sharing model is based on the mesh stiffness model depicted in Fig. 4. Hence, it may lead to some calculation errors. 3.2. The MPEP based load sharing model MPEP is based on the assumption that the potential energy contained in a balance system is the minimum. It was used to calculate the load sharing ratio of spur gears by Miryam et al. [21]. However, the load sharing model can be further improved in the following points: 1. The Hertzian contact stiffness is calculated through Eq. (2.6), which means the Hertzian contact stiffness is linear and load-independent. Actually, Eq. (2.5) shows better accuracy than Eq. (2.6). 2. The fillet foundation stiffness is neglected, and this may lead to some calculation errors. 3. Due to the lack of efficient mesh stiffness model, traditional mesh stiffness model was adopted. Hence, same results will be obtained if the load sharing ratio is calculated by Eq. (3.1) with the fillet foundation stiffness in ki being neglected. However, Ref. [21] offers an important load sharing ratio calculation method from the point of view of energy. Based on the MPEP theory, an improved load sharing model is proposed in this section considering the following points: 1. The Hertzian contact stiffness is calculated by Eq. (2.5). 2. Fillet foundation stiffness is considered, and the fillet foundation stiffness correction factor is calculated by Eq. (2.29) and (2.30). 3. The revised mesh stiffness model proposed in Section 2 is adopted (see Fig. 8).

F F

pinion

x

x

x

x

pinion k1

wheel

k2

k3

...

kn

k1

wheel Fig. 9. Traditional load sharing model.

k2

k3

...

kn

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Based on the above points, the total energy stored in two meshing gear pairs can be expressed by the following equation:

U¼

F 21 F2 þ 2 2k1 2k2

ð3:2Þ

where ki ði ¼ 1; 2Þ is the mesh stiffness of gear pair i expressed as follow:

ki ¼

1 khi

1 þ k1p þ k1w þ k1p þ k1w þ k1p þ k1w þ kp1kp þ kw1kw si

si

bi

bi

ai

ai

i

fi

i

ð3:3Þ

fi

Considering the constraint:

F1 þ F2 ¼

T ¼F Rb1

ð3:4Þ

where T is the input torque; Rb1 is the radius of base circle of the pinion. Then the load sharing model can be transformed into an optimization problem, namely:

8 min UðF 1 ; F 2 Þ > > > < s:t: F þ F ¼ F 1 2 > 0 6 F 6 F 1 > > : 0 6 F2 6 F

ð3:5Þ

Here, Genetic Algorithm Toolbox in Matlab R2013bÒ [31] is used to get the optimal solution. 4. Model verification and discussion 4.1. Mesh stiffness calculation model verification In this section, the mesh stiffness calculation model is verified by comparing with the FEM results. As shown in Fig. 10, a 2D FE model is established in Abaqus CAE 6.12Ò using plane stress elements (CPS3 and CPS4R) [30]. The inner ring nodes of pinion and wheel are coupled with the master nodes located at the center of pinion and wheel, respectively. The master node located at the center of the wheel is constrained in all directions, while the master node located at the center of the pinion is only constrained in x axis and y axis with an input torque T being applied. The entire FE model contains 73,486 elements and

Fig. 10. FE model used in Section.4.

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C. Xie et al. / Mechanical Systems and Signal Processing 111 (2018) 331–347 Table 2 Gear parameters. Parameters

Pinion

Wheel

Number of teeth z1 =z2 Module m (mm) Pressure angle a (°) Pitch diameter dw (mm)  The addendum coefficient ha The tip clearance coefficient c Young’s modulus E (Pa) Poisson’s ratio v Tooth width b (mm) Center distance a (mm) Hub bore radius r int (mm)

50 3.0 20 150 1 0.25 2.068  1011 0.3 20 150 5; 15;25

50 3.0 20 150 1 0.25 2.068  1011 0.3 20

44,880 nodes (r int ¼ 25mm), and such an accuracy is generally considered to be adequate to obtain a result close to the real physical model. The gear parameters are listed in Table 2. The entire FE model is solved as a quasi-static problem, and the mesh stiffness is calculated as follows [11]:

k¼

T hb R2b1

ð4:1Þ

where hb is the rotation angle of the master node located at the center of the pinion. 4.1.1. Influence of input torque on mesh stiffness In this section, the TVMS obtained by the proposed method is compared with those by FEM, Miryam’s [21] method, Chen’s [16] method and Timo’s [26] method(shown in Fig. 11). It is worth noticing that Timo’s [26] mesh stiffness model was derived in terms of torsional mesh stiffness, and was represented by rectilinear mesh stiffness using Eq. (1.1) for comparison purpose in this paper. According to Fig. 11, the following phenomena can be observed: (1) The mesh stiffness increases with the increase of input torque, as can be observed from the results obtained by FEM and the proposed method. (2) The results obtained by the proposed method agree well with those by FEM both in single tooth engagement region and double tooth engagement region. (3) Miryam’s [21] method ignores the fillet foundation stiffness, and thus overestimates the mesh stiffness both in single and double tooth engagement region. Moreover, the mesh stiffness obtained by Miryam’s [21] method is completely load-independent. (4) Chen’s [16] method agrees well with the FEM results in single tooth engagement region. However, it results large calculation errors in double tooth engagement region. Same as Miryam’s [21] method, the mesh stiffness is also loadindependent. (5) Timo’s [26] method belongs to a fast calculation method with many simplifications being adopted, which may inevitably lead to some calculation errors. Besides, there are no variations for mesh stiffness both in double and single tooth engagement regions, as can be observed from the other four methods. The calculation errors of the proposed mesh stiffness model relative to FE model with different input torques are listed in Table 3. 4.1.2. Influence of hub bore radius on mesh stiffness In this section, different hub bore radii are studied with a constant input torque T ¼ 50N  m being applied. The results are shown in Fig. 12. According to Fig. 12, the following phenomena can be observed: (1) Hub bore radius has a big influence on the mesh stiffness of gear pair both in double and single tooth engagement region, as can be seen from Fig. 12(a). (2) The proposed method agrees well with the FEM both in single and double tooth engagement regions. (3) Chen’s [16] method overestimates the mesh stiffness in double tooth engagement region, and this phenomenon is especially obvious for gear pair with smaller hub bore radius. This can be explained by the fact that fillet foundation stiffness decreases with the decrease of hub bore radius, and the contribution of the fillet foundation stiffness to the overall mesh stiffness is getting larger. Hence, the calculation error is amplified under this situation. (4) According to Miryam’s [21] method, hub bore radius has little influence on the mesh stiffness. This can be explained by the fact that gear body elasticity is ignored in Miryam’s [21] mesh stiffness model.

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Fig. 11. Mesh stiffness of gear pair in Table 2 with different input torques (r int ¼ 5 mm).

Table 3 Mesh stiffness calculation errors relative to FE model with different input torques. T = 50 Nm

Maximum error (%)

T = 100 Nm

T = 200 Nm

T = 400 Nm

Double

Single

Double

Single

Double

Single

Double

Single

1.26

1.57

0.83

1.37

0.70

1.12

0.64

0.87

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Enlarged

Enlarged

Enlarged

Fig. 12. Mesh stiffness of gear pair in Table 2 with different hub bore radii (T ¼ 50 N  m).

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(5) The gear body elasticity was considered in Timo’s [26] method, and the mesh stiffness increases with the increase of hub bore radius, which can also be seen from Fig. 12(a),(b) and (c). The calculation errors of the proposed mesh stiffness model relative to FE model with different hub bore radii are listed in Table 4. 4.2. Load sharing model verification In this section, the load sharing model is validated by comparing with the FE results and the results obtained by other researchers’ models. 4.2.1. Influence of input torque on load sharing ratio Different input torques are studied to investigate the influence of input torque on the load sharing ratio of spur gear pairs. As shown in Fig. 13, two contact pairs are established at gear pair 1 and 2, respectively. The contact forces, namely F 1 and F 2 , are obtained to calculate the load sharing factors using Eqs. (2.25) and (2.26). The results obtained by different methods are shown in Fig. 14. According to Fig. 14, the following phenomena can be observed: (1) Input torque has no obvious influence on the load sharing ratio of gear pair, as can be seen from the results obtained by FEM method and the other methods. (2) The proposed load sharing model is in better agreement with the FE model. (3) The largest calculation error occurs at the starting point of double tooth engagement region and the hand-over point for all the three methods.

Table 4 Mesh stiffness calculation errors relative to FE model with different hub bore radii. Rint = 5 mm

Maximum error (%)

Rint = 15 mm

Rint = 25 mm

Double

Single

Double

Single

Double

Single

1.26

1.57

2.84

0.90

4.15

0.52

slave node

pinion

T

I

contact pair 2 master node

F1 contact pair 1

Enlarged view

F2 slave node

wheel

II

master node

Fig. 13. FE model used to calculate the load sharing ratio.

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Fig. 14. Load sharing ratios of gear pair in Table 2 with different input torques (r int ¼ 5 mm).

Table 5 Load sharing ratio calculation errors relative to FE model with different input torques.

Maximum error (%)

T ¼ 50 N  m

T ¼ 100 N  m

T ¼ 200 N  m

T ¼ 400 N  m

3.23

2.99

3.61

3.91

(4) The load was assumed to be equally shared by both teeth pairs in Timo’s[26] model, which results Lsr1 ¼ 0:5 in double tooth engagement region. The calculation errors of the proposed load sharing model relative to FE model with different input torques are listed in Table 5. 4.2.2. Influence of hub bore radius on load sharing ratio Similar as Section 4.1.2, different hub bore radii are studied and a constant input torque T ¼ 50 N  m is applied to the pinion. The results obtained by different methods are shown in Fig. 15. According to Fig. 15, the following phenomena can be observed: (1) Hub bore radius has no obvious influence on the load sharing ratio of gear pair, as can be seen from the results obtained by FEM method, the proposed method and Miryam’s [21] method. (2) Chen’s [16] method shows sensitivity to the hub bore radius, as the calculation error increases with the decrease of hub bore radius. (3) According to the results obtained by Miryam’s [21] load sharing model, hub bore radius has no influence on the load sharing ratio. This can be explained by the fact that the fillet foundation stiffness is ignored. (4) The proposed load sharing model is in better agreement with FE model.

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Fig. 15. Load sharing ratioof gear pair in Table 2 with different hub bore radii (T ¼ 50 N  m).

Table 6 Load sharing ratio calculation errors relative to FE model with different hub bore radii.

Maximum error (%)

r int =5 mm

r int =15 mm

r int =25 mm

3.23

3.67

2.10

The calculation errors of the proposed load sharing model relative to FE model with different hub bore radii are listed in Table 6. 5. Conclusion In this paper, an improved analytical mesh stiffness model and load sharing model are proposed to study the mesh stiffness and load sharing ratio of spur gears. Main conclusions are drawn as follows: (1) For double tooth engagement situation, the meshing force on one gear pair has an obvious influence on the other gear pair. Thus, the two meshing gear pairs are actually coupled with each other. Hence, from this viewpoint, the fillet foundation stiffness is also load-dependent and nonlinear in double tooth engagement region. (2) Input torque and hub bore radius have obvious influences on the mesh stiffness of gear pair, whereas have limited influences on the load sharing ratio. The proposed models accurately reflect these phenomena and thus agree well with the FE results. Admittedly, the proposed method still needs further improvements. For example, the fillet foundation stiffness correction factor in Section 2.2 is obtained from a specially designed FE model, and this may be not so convenient and efficient to some extent. However, it is difficult to derive an accurate expression analytically. Hence, in this paper, the main purpose is to

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reveal the influence of structure coupling effect on mesh stiffness and load sharing ratio of spur gears, which is rarely considered in the past papers. In our future work, more efforts will be taken to improve the calculation efficiency and accuracy of the proposed method. Acknowledgment This research is supported by the National Natural Science Foundation (Grant no. 51575417), the Fundamental Research Funds for the Central Universities of China (Grant nos. 2014-zy-091) and State Key Laboratory of Digital Manufacturing Equipment and Technology of China Open Project Fund (Grant no. DMETKF2013010). References [1] R.G. Munro, D. Palmer, L. Morrish, An experimental method to measure gear tooth stiffness throughout and beyond the path of contact, Proc. Inst. Mech. Eng. C J.Mech. Eng. Sci. 215 (7) (2001) 793–803. [2] Y. Pandya, A. Parey, Experimental investigation of spur gear tooth mesh stiffness in the presence of crack using photo elasticity technique, Eng. Fail. 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