Study on teeth profile modification of cycloid reducer based on non-Hertz elastic contact analysis

Mechanics Research Communications 48 (2013) 87–92

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Study on teeth profile modification of cycloid reducer based on non-Hertz elastic contact analysis Yu Hong-Liu ∗ , Yi Jin-hua, Hu Xin, Shi Ping Institute of Biomechanics and Rehabilitation Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

a r t i c l e

i n f o

Article history: Available online 3 January 2013 Keywords: Heavy-duty cycloid reducer Pin teeth Profile modification Non-Hertz contact

a b s t r a c t To solve the problem of contact failure for heavy-duty cycloid reducers, the teeth mesh was originally studied as a Non-Hertz elastic contact problem. An analysis method called non-Hertz flexibility matrix method (NHFMM) for the teeth contact was developed for the teeth profile modification of pin gear. The NHFMM analysis shows that the edge concentrated pressure calculated by non-Hertz method is 2.11 times of that by Hertz method on the pin teeth without profile modification. In the end, the analysis results of NHFMM were verified by a specially designed photo-elastic experiment. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Contact damage of gear tooth is one of the critical technical problems for heavy-duty cycloid power transmission. High transmission torque can bring high contact stress between the tooth of cycloid gear and pin gear, which results in the pressure feedback on tooth profile, even may cause pitting damage and agglutinative destruction. In theory, the axial modification of pin gear can avoid the edge concentration problem for tooth load and improve the status of deformation. Hertz put forward the theory of contact problem under positive pressure condition for the first time as early as in 1881. However, Hertz contact theory cannot always get a solution with enough accuracy needed in some cases, especially for the contact problems of bodies with complex surfaces. So studies on non-Hertz contact problems, most of which focused on searching for numerical solutions of non-Hertz contact problems developed very fast from 1970s (Francavilla and Zienkiewicz, 1975; Krishna and Burton, 1974; Nayak, 1982). Conry and Seireg first studied the contact problems by computer programming with the mathematic model deduced according to the principle of minimum potential energy in 1971 (Conry and Seireg, 1971). Then a linear equation set associated with the unknown contact pressure was established by Singh and Paul through using flexibility analysis to solve the non-Hertz contact problem (Singh and Paul, 1974). Kannel and Hartnett proved through theoretical and experimental studies that the results

calculated with non-Hertz method were more accurate than that calculated with Hertz theory when solving the contact problem between two short cylinders with unsmooth surface generatrix (Kannel and Hartnett, 1983). Researchers tried studying the contact between rollers and track surfaces in bearings as a non-Hertz contact problem, and achieved more accurate results (Oh and Trachmann, 1976; Heydari and Gohar, 1979). This was one of the earliest cases applying non-Hertz method in dealing with the engineering problems. In recent years, more studies on non-Hertz contact problems were reported, which shows that non-Hertz method was attached more importance in resolving the contact problems in practice (Rahmejat and Gohar, 1979; Oh and Trachmann, 1976; Kogut and Etsion, 2002). In this paper, an equation based on tooth elastic deformation and load distribution flexibility was established by a non-Hertz elastic contact method. By solving this equation, the load distribution of tooth profile can be easily and accurately analyzed, and the axial pin tooth modification shape can also be analyzed satisfying the technical conditions and improving contact conditions. In addition, an ideal profile modification curve for even load distribution can also be found by solving this equation. This method was proved as an effective way to improve the situation of heavy-duty tooth contact models.

2. Materials and methods 2.1. Single tooth meshing mathematical model

∗ Corresponding author at: Institute of Biomechanics and Rehabilitation Engineering, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China. Tel.: +86 21 55271205; fax: +86 21 55271057. E-mail address: [email protected] (Y. Hong-Liu). 0093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechrescom.2012.12.007

Under certain assumptions, we can establish mathematical model of the single tooth meshing under considered contact area shape and deformation transfer effect of discrete contact

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Fig. 1. Contact status between cycloid wheel and pin gears.

points. The single tooth meshing status under loads is shown in Fig. 1. As shown in Fig. 2, the pressure on single tooth meshing under loads can be shown as below:

⎫ ⎪ ⎪ (Aij + Bij )pj + Ei ⎪ −f (xi , yi ) + ı = ⎪ ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ k ⎬ 

Fig. 3. Pressure distribution on pressure unit.

k 

F(Pj ) = W



⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

j=1

Pj ≥ 0, (xj , yj ) ∈ ˝ Pj = 0, (xj , yj ) ∈ /˝

Suppose the total displacement of elastic deformation for unit i is Wi . So, (1)



e(xi , yi ) = f (xi , yi ) + f  (xi , yi ) − ı

(3)

Suppose the dimensionless quantity d¯ ij (0, y¯ ij ) =

Ed(0,y¯ ij ) (1−2 )pj ai

,

dimensionless mathematical model can be obtained by

where F is unit center pressure, Pj is unit pressure function of independent variable, W is the normal total load of pin tooth, and it relates to the force of pendulum wheel, Aij is a flexibility component of contact deformation for unit j under unit i tooth profile, Bij is a flexibility component of pin teeth deflection for unit j under unit i tooth profile, Ei is processing and assembly errors of center normal tooth for unit i, ı is normal contact deformation between teeth, and f(xi , yi ) is an imaginary shape at any point i of pin teeth under-load without considering contact deformation. Moreover,



Wi = f (xi , yi ) − e(xi , yi ) − Ei

(2)

where f(xi , yi ) is the shape at pin point i under pre-load, f ’ (xi , yi ) is the deformation under load at pin point i without considering contact deformation, and e(xi , yi ) is the shape under load at pin point i. Stiffness of the supporting structure is usually very large. To simplify calculation, the above model can be further simplified by ignoring the support deformation and processing and assembly errors.

Wi = k 

k 

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

(Iij + Jij )p¯ j

j=1

F(Pj ) = W

(4)

⎪ ⎪ ⎪ ⎪  ⎪ P¯ j ≥ 0, (xj , yj ) ∈ ˝ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ j=1

/˝ P¯ j = 0, (xj , yj ) ∈

where Iij =

E1 A , a0 (1−2 ) ij

and Iij is a dimensionless flexibility com-

ponents of elastic contact deformation. Jij =

E1 B , a0 (1−2 ) ij

and Jij is a P

dimensionless flexibility components of deflection. P¯ j = P j , and P0 0 is the center pressure of contact area. a0 is the width of the central contact area. E and  are tooth elastic modulus and tooth Poisson’s ratio respectively. 2.2. Non-Hertz deformation calculation Numerical analysis of non-Hertzian method is from Boussinesq formula which is the elastic semi-infinite solutions on the profile of a concentrated load (Jackson and Green, 2005; Jackson et al., 2005). It divides unit grids on contact area. Based on numerical integration, displacement flexibility coefficient among each unit can be obtained, and pressure displacement equations can be established and solved. So, the physical model of single tooth meshing can be simplified as a contact model between cylinder and semi-infinite body (Bryant and Keer, 1982; Johnson, 1985; Galina et al., 2007). 2.2.1. Lateral load distribution assumptions and the selection of computing units Generally the lateral pressure distribution on cylindrical contact area is an ellipse (Jin et al., 2007). The pressure distribution on pressure unit is shown in Fig. 3. A pressure can be expressed as

P(x, y) = Pm Fig. 2. Contact status of single pair of gear teeth in cycloid reducer.

2 1− 2 a

1/2 (5)

Y. Hong-Liu et al. / Mechanics Research Communications 48 (2013) 87–92

where Pm is pressure distribution on the whole contact area, it is a pressure intensity located on arbitrary cross section. 2.2.2. Calculating elastic contact deformation component Suppose that: p is the element local center of pressure peak and || pm = pj 1 − c . j

The pressure on unit j can be expressed as

 pj (, ) = pj

1−

  

1−

cj

2

1/2 (6)

a2j

Based on distributed load displacement formula of elastic semiinfinite body, the deformation of unit i under the effect of unit j can be expressed as

d(x, y) =

(1 − 2i )pj Ei



 

aj cj

1−

|| cj

 −aj −cj



2

1−

2 a2

1/2

j

+ yij − 2

dd

(7)

For non-dimensional calculation the unit bottom can be equiva¯  = cj , ¯ ai = a¯ i ci , and lent to square unit. Suppose ytj = cj y¯ tj ,  = aj , aj = a¯ j cj . With dimensionless quantities, Eq. (7) can be written as d¯ ij (0, y¯ ij ) =



E1



1 − 21 pj ai

dij (0, yij )

(8)

So, the dimensionless displacement coefficient of the unit i under the effect of unit j can be obtained as

d¯ ij (0, y¯ ij ) =

aj ai

1 1

  1/2 (1 − )(1 −  2 ) ¯ ¯ dd 

(9)

−1 −1

aj

(10)

a0

Here we can consider the all supports for gears as rigid constraint, and intentionally set unit number and unit locations as the discrete units in contact area in the effective contact length of the pin gear (Li et al., 2010; Rajendrakumar and Biswas, 1996). An unit concentrated load was applied on each node j in the contact area respectively, and the line displacement cij at node j in contact area can be computed. Transferring the distribution pressure in each unit into a concentrated pressure at unit center, we can obtain the following equation.



 

aj cj

aj cj ∗ pj −aj −cj

1−

    cj

×

1−

2 a2j

1/2 dd =

 acp 2 j j j (11)

So, flexibility components of dimensionless deflection can be written as Jij =

3.1. Analysis for elastic deformation and profile modification of pin teeth 3.1.1. Selection of pin teeth profile modification With analyzing the contact between short cylinder roller and elastic semi-infinite body by Reusner, and considering the meshing characteristics and process requirements of cycloid gear tooth, different profile shapes can be adopted six pin tooth profile modification shape under pressure or partial load (Rahmejat and Gohar, 1979). (1) (2) (3) (4) (5)

Straight line slope at both ends. Smooth arc connection at both ends. Non-smooth arc connection at both ends. Integral arc. Integral arc including small curvature arc trimming at both ends. (6) Integral arc including straight line slope at both ends.

3.1.2. Selecting initial deformations According to elastic deformation formula for finite length steel cylinder under pressure among elastic semi-infinite body by Palmgren, the profile modification deformations of pin teeth can be calculated by ı =

A × 10−10 Q 0.9 0.8 L (m)

(13)

where A is deformation coefficient, Q is designed load, L is effective contact length. According to the load in this paper, the value A is from 0.7 to 1.0. 3.2. Methods used for numerical calculation

2.3. Finite element analysis of pin tooth deflection flexibility component

WX =

3. Calculation

2

¯ + (y¯ ij − ) (a¯ j ) ¯ 2

So, the flexibility component of elastic contact deformation in mathematical model of Eq. (4) can be expressed as Iij = d¯ ij (0, y¯ ij )

89

2 Ei aj cj 2(1 − 21 )

cij

(12)

3.2.1. Methods for solving the problem of singular integral The problem of singular integral occurs when calculating the flexibility component of contacting deformation with the above Eq. (3). When x =  and y = , the integrand function is boundless. Many scholars paid attention to this problem to try to achieve a high accuracy for the flexibility matrix. Though different methods have been used to solve the problem, most of the methods basically focused on changing the integral limits which led to a result with high error (Kovalev et al., 2009; Braghin et al., 2002). For these traditional methods are neither good nor convenient, a method of Gauss–Legendre algorithm with minor error was used for integration here on basis of separating the singular unit in advance, which improved the accuracy of the whole flexibility matrix. The calculation results show that the new method has very good stability and convergence. Furthermore, double precision is also advised in calculation to improve the accuracy of Iij . 3.2.2. Methods for solving the problem of ill-conditioned matrix In the numerical analysis of non-Hertz elastic contact problem, the condition number of flexibility matrix Iij + Jij is generally very big which will bring the calculation result with a high error if no actions were taken. Therefore, the iteration method was used to improve the accuracy of result. Calculation results show that the method is of more convenience, higher accuracy and faster convergence. 3.2.3. Methods for solving the problem of edge singular unit According to the viewpoint of Oh and Trachmann (Oh and Trachmann, 1976), singular units must be employed for the contact problem where the pressure fields interrupt at two ends when a shot cylinder without modification contacts with an infinite body.

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Fig. 4. Load distribution curves for different profile modifications. Fig. 6. Specially designed experimental instrument.

The program was designed to automatically treat the edge pressure units as half tri-angular units with a good result. 4. Results 4.1. Theoretical results As an example, the flexibility matrix Eq. (4) was solved for the cycloid reducer of model VMS306-9398 with 4 profile modification methods, which has different connection curve at two ends of a pin gear including two types of straight line, smooth arc and integral arc. Simulation results show that the pressure distribution is improved under profile modification method. And the edge concentrated pressure of pin teeth without profile modification by non-Hertz method is 2.11 times of that by Hertz method. The load distribution curves under different profile modification ways are shown in Fig. 4. The calculation results show that there is a best pressure distribution under the smooth arc connection for pin teeth profile modification. 4.2. Photo-elastic experimental verification 4.2.1. Experiment scheme The stress was analyzed by three-dimensional photo-elastic model with the “freezing” sections on single tooth meshing model. Two tests were done under line-slope profile modification in Fig. 5 and no profile modification for pin gear tooth. The geometry similar rate of original model is 1:1. Maleic anhydride was used for

Fig. 5. Pin gear tooth with line-slope profile modification used for experiment.

curing agent of model material. Modeling samples were made using precision casting method. Load was placed through shelves lever. Those experiment tools were to assure similar boundary conditions and equivalent static load. A special experimental instrument was designed to simulating the load exerted on the pin gear tooth as shown in Fig. 6. The load for the models of pin gear tooth with and without modification is respectively 6.64 kg and 7.42 kg. The whole experimental instrument after load exerting was put into an oven for “freezing”. 4.2.2. Stress observation After a period of “freezing” in the oven controlled according to a required curve in temperature, the model pin gear tooth was successfully frozen and was moved out for being cut in pieces. Ten positive cut pieces are obtained as shown in Fig. 7. The iso-chromatic curves for all of 10 cut pieces of pin gear tooth model can be observed in dark field. Two typical cut pieces for the pin gear teeth model with and without profile medication are respectively shown in Fig. 8. A high concentration stress can be observed in No. 10 cut piece at the edge of pin gear tooth without profile modification, while a low stress appears at the edge of pin gear tooth with profile modification. The contacting stress for all of the ten positive cut pieces is calculated by, according to the iso-chromatic curves observed, as shown in Table 1.

Fig. 7. Cutting position and calculation model for pin gear tooth.

Y. Hong-Liu et al. / Mechanics Research Communications 48 (2013) 87–92

91

Fig. 8. Iso-chromatic curves for typical cut pieces of experimental model of pin gear tooth in dark field. (a) No. 7 cut piece of pin gear tooth with modification, (b) No. 10 cut piece of pin gear tooth with modification, (c) No. 7 cut piece of pin gear tooth without modification, and (d) No. 10 cut piece of pin gear tooth without modification.

4.2.3. Stress conversion Theoretically, the photo-elastic experiment should follow the rule of Poisson’s similarity for similar simulation (Guagliano and Vergani, 2005), that is,  = m

(14)

where  and m is respectively the Poisson’s ratio of material for model and prototype. Then we can have the conversion coefficient of model stress and prototype stress as follow,



C =

Cq CE Cl

(15)

where Cq = ppm LLm = ppm (p and pm is respectively the load exerted on the prototype and experiment model). CE = EEm (E and Em is respectively the elastic modulus of material for the prototype and experiment model).

Cl = LLm = 1 (L and Lm is respectively the contact length of pin gear teeth for the prototype and experiment model). Steel GCr 15 was used for the material of prototype and E = 2.10 × 1011 (N/m2 ), while epoxy resin was adopted for the material of experiment model and Em = 2.50 × 104 (N/m2 ). With the ab × e formula (2), we can conveniently do the conversion between the stresses on the model in experiment and on the prototype. The maximum meshing force in prototype was calculated as below:

Fb = p = 41,115.98(N) where Fb is also the rated load for prototype design. In the above photo-elastic experiment, the simulated load exerted on the pin gear tooth with modification is pm = 7.42 kg, so

Table 1 Contacting stress  ym at each cut piece of experimental model of pin gear tooth (×105 N/m2 ). No.

1

2

3

4

5

6

7

8

9

10

 ym

With modification Without modification

8.08 2.11

6.67 7.33

4.44 6.04

4.32 6.05

4.48 6.06

4.39 6.14

4.47 6.08

4.34 6.10

4.56 8.12

8.64 3.27

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Table 2 Converted contacting stress  y at each cut piece of experimental model of pin gear tooth with profile modification (×109 N/m2 ). No.

1

2

3

4

5

6

7

8

9

10

y

0.46

1.61

1.33

1.33

1.34

1.35

1.40

1.34

1.79

0.72

gear tooth without profile modification. The stress concentration effect becomes significantly worse when considering the deflection of pin tooth. (3) Pin tooth profile modification which meets the technical requirements and avoid stress concentration can improve the contact status, and greatly enhance the load capacity of pin teeth. (4) Photo-elastic experimental results with three-dimensional “frozen” sections show that the proposed non-Hertz elastic contact method is feasible and accurate for analyzing elastic deformation and tooth profile modification of gear tooth. Acknowledgements Fig. 9. Contact load distribution comparison between theoretical calculation and photo-elastic experiment.

we can calculate the conversion coefficient as below by formula (2), CE =

E = 8571.4 Em

Cq =

p Lm p = 565.43 = pm L pm

 Cy =

 Cq CE = 565.43 × 8571.4 = 2201.49 Cl

Therefore, the contacting stress at the center of each cut piece of prototype with profile modification can be calculated by the following formula (16), y = 2201.49ym

(16)

The experimental results corresponding to Table 1 for the pin gear tooth with profile modification are calculated as shown in Table 2, and the comparison result curves of contact load distribution of theoretical computation and photo-elastic experiments are shown in Fig. 9. Experimental results show that there is a sharp concentration at the contact edge for the model without profile modification. However, there only appears local stress concentration at the slope connections for the model with profile modification. Furthermore, the stress concentration the model with profile modification was significantly improved compared to the model without profile modification. Experimental results show a good consistence with the theoretical curves. 5. Conclusions (1) The maximum contact stress computed by the proposed nonHertz flexibility matrix method is 2–3 times of that computed by classical Hertz method. (2) A sharp stress concentration effect will occur at both ends of edge when straight pin gear tooth meshes with the cycloid

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