Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set

Mechanism and Machine Theory 76 (2014) 20–38

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Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set Xihui Liang, Ming J. Zuo ⁎, Mayank Pandey Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada

a r t i c l e

i n f o

Article history: Received 19 February 2013 Received in revised form 3 February 2014 Accepted 3 February 2014 Available online 2 March 2014 Keywords: Mesh stiffness Potential energy method Planetary gear set Crack modeling

a b s t r a c t Time-varying mesh stiffness, caused by the change of tooth contact number and contact position, is one of the main sources of vibration of a gear transmission system. In order to comprehensively understand the vibration properties of a planetary gear set, it is necessary to evaluate the mesh stiffness effectively. When a crack happens in one gear, the mesh stiffness will decrease and consequently the vibration properties of the gear system will change. This change of vibration can be characterized through dynamic simulation of a gearbox and processed further to detect the crack severity and location. In this paper, the potential energy method is used to analytically evaluate the mesh stiffness of a planetary gear set. A modified cantilever beam model is used to represent the external gear tooth and derive the analytical equations of the bending, shear and axial compressive stiffness. A crack propagation model is developed and the mesh stiffness reduction is quantified when a crack occurs in the sun gear or the planet gear. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction When a pair of spur gear meshes, the tooth contact number and the tooth mesh position change during meshing. It leads to a periodic variation in the gear mesh stiffness. The mesh stiffness variation is one of the main sources of vibration in a gear transmission system [1]. Crack may occur in gears due to excessive service load, inappropriate operating conditions or simply fatigue [2]. When a crack takes place, the gear mesh stiffness will reduce and consequently the vibration characteristics of the gear system will change. If the stiffness reduction can be quantified for different crack levels, the corresponding vibration signal can be obtained through dynamic simulation. The vibration signal can be processed further for crack detection and prognosis. Both the finite element method (FEM) and the analytical method (AM) have been used to evaluate the gear mesh stiffness [2–4]. But, FEM is complicated and time consuming. While AM can offer a simple and effective way to evaluate the time-varying mesh stiffness. In this paper, we use external–external gears to denote a meshing gear pair which contains two external gears. Similarly, we define external–internal gears as a meshing gear pair which contains one external gear and one internal gear. Many researchers have applied the analytical method to evaluate the mesh stiffness of a pair of fixed shaft external–external gears. Yang and Lin [6] proposed the potential energy method to calculate the mesh stiffness of a pair of external–external spur gears by considering Hertzian contact stiffness, bending stiffness and axial compressive stiffness. Later, Tian et al. [7] introduced an additional term called the shear stiffness in the potential energy method. Recently, Zhou et al. [5] took the deformation of the gear body into consideration. The gear tooth was modeled as a cantilever beam which started from the base circle [5–7]. Also, Zhou et al. [5], Tian et al. [7], and Pandya and Parey [8] considered that the gear crack follows a linear path starting from the point of intersection of the base circle and the involute curve as shown in Fig. 1. Actually, the gear tooth starts from the root circle rather than the base circle as given in Fig. 1. Thus, their models ignored the gear tooth part between the root circle and the base circle. The tooth profile of this part (tooth fillet area) is not an involute curve and it is basically determined by cutting tool tip trajectory. Using a different cutting tool tip trajectory, the generated curve will be different and there is not a uniform function to depict it [10]. However, the ⁎ Corresponding author. Tel.: +1 780 492 4466; fax: +1 780 492 2200. E-mail address: [email protected] (M.J. Zuo). 0094-114X/$ – see front matter © 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2014.02.001

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

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ν

Fig. 1. Crack model from Refs. [5,7].

ignorance of this part will change the stiffness of the meshing gears, especially when the distance between the base circle and root circle is bigger. Chaari et al. [11] presented an analytical method to evaluate the mesh stiffness of a pair of external–external gears and modeled the crack as a straight line starting from the root circle. They mentioned that the gear mesh stiffness can be calculated by taking into account the tooth thickness reduction. However, they did not provide the analytical equations of the mesh stiffness for a crack gear. Chen and Shao [12] proposed an analytical mesh stiffness model with tooth root crack propagating along both the tooth width and the crack depth. Further, they [13] investigated the effect of the tooth profile modification on the mesh stiffness. Their model is more realistic as compared to other models. However, their analytical equations are not convenient to use. Their method was based on single point estimation of the mesh stiffness. In this case, for time-varying stiffness measurement, it is required that information at every point on the gear are known beforehand which is repeated in nature for different gears and time consuming. In this paper, a model is proposed which solves this problem and does not need variable information at every point. Rather, a model is developed which can calculate all these information by itself and use it to determine the time-varying stiffness. The analytical equations proposed in this paper are much easier to use. Given the gear geometries and the material properties, the mesh stiffness can be expressed as a function of the angular displacement of the gear. For a planetary gear set, there are pairs of sun–planet gears (external–external gears) and pairs of ring–planet gears (external–internal gears) meshing simultaneously. Chaari et al. [14] and Walha et al. [15] used a square waveform to approximate the time-varying mesh stiffness of a planetary gear set. In their method, the amplitudes of the sun–planet mesh stiffness and the ring-planet mesh stiffness were assumed without a specific qualification method. Besides, the square waveform reflects only the effect of the change in the tooth contact number, but ignores the effect of the change in the tooth contact position. Liang et al. [16] evaluated the mesh stiffness of a perfect planetary gear set using the potential energy method. They also treated the sun gear and the planet gear as a cantilever beam starting from the base circle. It is mentioned in Ref. [14] that the amplitude modulation can be used to obtain the mesh stiffness of a planetary gear set with crack. Fig. 2 [14] illustrated the amplitude loss of 50% due to a crack in the sun gear. But, they modeled the stiffness reduction only in the double tooth contact duration while ignored the stiffness decrease in the single tooth contact duration. Also, physical meaning of this loss was not described, like how much crack propagation will lead to the amplitude loss by 50%. In this study, we propose an approach to overcome these shortcomings. In this paper, the mesh stiffness of a planetary gear set is analytically evaluated using the potential energy method. To model the external gear, i.e. the sun gear and the planet gear, a modified beam model is proposed by considering the gear tooth starting from the root circle. Further, a crack propagation model is developed and the mesh stiffness equations are derived when a crack takes place in the sun gear or the planet gear. This crack model can be used for the detection of the crack severity and the crack

Fig. 2. Amplitude loss of 50% due to a crack in the sun gear [14].

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ψ

Fig. 3. Beam model of an external gear tooth when root circle is smaller than base circle.

position. The crack propagation of the ring gear will be modeled in the future research. A shorter version of this journal paper is included in the conference proceedings of QR2MSE 2013 [17]. 2. Mesh stiffness of fixed-shaft external–external gears In earlier studies [5–7], the gear tooth is assumed to start from the base circle. In this section, we model the gear teeth more rigorously by considering the gear tooth starting from the root circle when all teeth are perfect. The impact of the differences in these two modeling methods on the mesh stiffness will be illustrated. The potential energy method is used to evaluate the mesh stiffness of a pair of fixed-shaft external–external gears with the consideration of Hertzian energy, bending energy, shear energy and axial compressive energy. The new analytical equations for the bending stiffness, shear stiffness and axial compressive stiffness are derived. The overall mesh stiffness is represented as a function of the rotation angular displacement of the driven

ψ

Fig. 4. Beam model for an external gear tooth when root circle is bigger than base circle.

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Table 1 Physical parameters of a planetary gear set [16]. Parameters

Sun gear

Planet gear

Ring gear

Number of teeth Module (mm) Pressure angle Face width (m) Young's modulus (Pa) Poisson's ratio Base circle radius (mm) Root circle radius (mm) Reduction ratio

19 3.2 20° 0.0381 2.068 × 1011 0.3 28.3 26.2 5.263

31 3.2 20° 0.0381 2.068 × 1011 0.3 46.2 45.2

81 3.2 20° 0.0381 2.068 × 1011 0.3 120.8 132.6

gear. In Refs. [7–9], the gear system is assumed to be perfect without friction and transmission error and the gear body is treated as solid. The same assumptions will be applied in this paper. 2.1. Bending, shear and axial compressive stiffness For an external gear, the gear root circle may be bigger or smaller than the base circle according to the geometry of gears. If the gear is a standard spur gear with the pressure angle of 20°, the root circle is bigger if the tooth number is more than 41. It is smaller if the tooth number is less than 41. In the industrial applications, both types of gears are commonly used. In this paper, these two cases will be discussed separately. 2.1.1. Case 1: the gear root circle is smaller than the base circle If the root circle is smaller than the base circle, the beam model of the gear tooth is shown in Fig. 3. Gear tooth profile follows an involute curve up to the base circle (curves IN' and JD'). The tooth profile between the base circle and the root circle is not an involute curve and hard to describe analytically [10]. Therefore, straight lines NN' and DD' are used to simplify the curve. According to the properties of the involute curve, the action line of two meshing gears is always tangent to the gear base circle and normal to the tooth involute profile. The action force F which is along the action line, can be decomposed into two orthogonal forces Fa and Fb, as shown in Fig. 3. F a ¼ F sinα 1

ð2:1Þ

F b ¼ F cosα 1

ð2:2Þ

Applying the beam theory, the bending, shear and axial compressive energies stored in a tooth can be expressed as follows [6,7]: Ub ¼

F2 ¼ 2kb

Us ¼

F2 ¼ 2ks

Ua ¼

F ¼ 2ka

2

Z

Z Z

½ F b ðd−xÞ−F a h2 dx 2EIx 0

ð2:3Þ

1:2F b 2 dx 0 2GAx

ð2:4Þ

d

d

d

2

Fa dx 0 2EAx

ð2:5Þ

where kb, ks and ka denote the bending, shear and axial compressive stiffness, respectively, E and G represent Young's modulus and shear modulus, respectively, h shows the distance between the gear contact point and the tooth central line, d is the distance from the contact point to the gear root, and Ax and Ix indicate the area and the area moment of inertia of the section where the distance to the tooth root is x (see Fig. 3). According to the characteristics of the involute curve, h, hx, d, Ix and Ax can be expressed as follows: h ¼ Rb ½ðα 1 þ α 2 Þ cosα 1 − sinα 1   Rb sinα 2 ; if 0≤x ≤d1 hx ¼ Rb ½ðα 2 −α Þ cosα þ sinα ; if d1 bx≤d d ¼ Rb ½ðα 1 þ α 2 Þ sinα 1 þ cosα 1 −Rr cosα 3 Ix ¼

1 2 3 3 ð2hx Þ L ¼ hx L 12 3

Ax ¼ 2hx L

ð2:6Þ ð2:7Þ ð2:8Þ ð2:9Þ ð2:10Þ

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

t

24

Fig. 5. Mesh stiffness of a fixed shaft external–external gear pair. * Model comes from Tian et al. [7].

where Rb, Rr and L denote the base circle radius, root circle radius and tooth width of the external gear, respectively, hx is the height of the section where the distance to the tooth root is x, α2 represents the half tooth angle on the base circle [18] while α3 describes the approximated half tooth angle on the root circle (see Fig. 3). α2 ¼

π þ tanα 0 −α 0 2N

ð2:11Þ

  R sinα 2 α 3 ¼ arcsin b Rr

ð2:12Þ

where N is the tooth number of the external gear and α0 is the pressure angle. Substituting Eqs. (2.1), (2.2) and (2.6) to (2.9) into Eq. (2.3), the bending stiffness of the external gear can be expressed as: h i cosα 1 cosα 3 3 1− ðN−2:5NÞcosα −ð1− cosα 1 cosα 2 Þ3 1 0 ¼ kb 2EL cosα 1 sin3 α 2 Z α 2 2 3f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα þ dα: 2EL½ sinα þ ðα 2 −α Þ cosα 3 −α 1

ð2:13Þ

Substituting Eqs. (2.2), (2.7) and (2.10) into Eq. (2.5), the shear stiffness of the external gear is given as:   N−2:5 1:2ð1 þ ν Þ cos2 α 1 cosα 2 − cosα 3 1 N cosα 0 ¼ ks EL sinα 2 Z α 2 1:2ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1 dα: þ EL½ sinα þ ðα 2 −α Þ cosα  −α 1

ð2:14Þ

Substituting Eqs. (2.1), (2.7) and (2.10) into Eq. (2.4), the axial compressive stiffness of the external gear can be obtained:   N−2:5 2 sin α 1 cosα 2 − cosα 3 1 N cosα 0 ¼ ka 2EL sinα 2 Z α 2 ðα 2 −α Þ cosα sin2 α 1 dα: þ −α 1 2EL½ sinα þ ðα 2 −α Þ cosα 

t

9

Fig. 6. Mesh stiffness of a fixed shaft external–internal gear pair. * Model from Liang et al. [16].

ð2:15Þ

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

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p c

s

r Fig. 7. Structure of a planetary gear set with the ring gear fixed.

2.1.2. Case 2: the gear root circle is bigger than the base circle If the root circle is bigger than the base circle, the beam model of the gear tooth is considered starting from the root circle as shown in Fig. 4. The whole gear tooth profile (curves IN and JD) follows the involute curve. As compared to Case 1 when the root circle is smaller than the base circle, the expressions for the tooth effective length d and tooth section width hx change as follows: d ¼ Rb ½ðα 1 þ α 2 Þ sinα 1 þ cosα 1 −Rr cosα 4

ð2:16Þ

hx ¼ Rb ½ðα 2 −α Þ cosα þ sinα 

ð2:17Þ

where α4 is the half tooth angle on the root circle of Case 2. Applying the similar derivation procedures as Case 1, the bending, shear and axial compressive stiffness of Case 2 can be expressed as follows: 1 ¼ kb 1 ¼ ks 1 ¼ ka

Z

Z

Z

3f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g2 ðα 2 −α Þ cosα dα 2EL½ sinα þ ðα 2 −α Þ cosα 3 −α 1 α5

ð2:18Þ

2

α5

1:2ð1 þ νÞðα 2 −α Þ cosα cos α 1 dα EL½ sinα þ ðα 2 −α Þ cosα  −α 1

ð2:19Þ

ðα 2 −α Þ cosα sin2 α 1 dα: −α 1 2EL½ sinα þ ðα 2 −α Þ cosα 

ð2:20Þ

α5

The symbol α5 represents the angle between the action force F and the decomposed force Fb when the distance between the meshing point and the root circle is 0. The value of α5 can be obtained through the following equation set. 

Rr sinα 4 ¼ Rb ½ðα 2 −α 5 Þ cosα 5 − sinα 5  Rr cosα 4 −Rb cosα 2 ¼ Rb ½ cosα 5 −ðα 2 −α 5 Þ sinα 5 − cosα 5 

ð2:21Þ

2.2. Hertzian contact stiffness From the result derived by Yang and Sun [19], the Hertzian contact stiffness, kh, for a pair of external–external gears, is linearized to a constant along the entire line of action independent of both the position of contact and the depth of interpenetration. πEL  kh ¼  4 1−ν2

ð2:22Þ

where E, L, and ν denote Young's modulus, tooth width and Poisson's ratio, respectively. Table 2 Relative phases of a planetary gear set [16]. γs1

γs2

γs3

γs4

γr1

γr2

γr3

γr4

γrs

0

0.75

0.5

0.25

0

−0.25

−0.5

−0.75

0

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

θ

θ

θ

θ

θ

θ

Fig. 8. Mesh stiffness of a planetary gear set when the ring gear is fixed.

Fig. 9. Gear crack propagation path [22]. (a) Experiment and (b) finite element method.

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γ

Fig. 10. Cracked tooth model when root circle is smaller than base circle.

2.3. Overall mesh stiffness For a pair of spur gears with contact ratio between 1 and 2, one pair and two pairs of tooth contact take place alternatively. For the single-tooth-pair meshing duration, the total effective mesh stiffness can be calculated as [7]: kt ¼

1 1 1 1 1 1 1 1 þ þ þ þ þ þ kh kb1 ks1 ka1 kb2 ks2 ka2

where subscripts 1 and 2 represent the driving gear and the driven gear, respectively.

γ

Fig. 11. Cracked tooth model when root circle is bigger than base circle.

ð2:23Þ

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38 Table 3 Crack levels and the corresponding crack length in the sun gear. Crack levels

Crack length

10% 25% 50% 75% 100%

q1 q1 q1 q1 q1

= = = = =

0.78 1.95 3.90 3.90 3.90

mm mm mm mm, q2 = 1.95 mm mm, q2 = 3.90 mm

For the double-tooth-pair meshing duration, there are two pairs of gears meshing at the same time. The total effective mesh stiffness can be obtained as [7]: kt ¼ kt1 þ kt2 ¼

2 X i¼1

1 1 1 1 1 1 1 1 þ þ þ þ þ þ kh;i kb1;i ks1;i ka1;i kb2:i ks2;i ka2;i

ð2:24Þ

where i = 1 for the first pair and i = 2 for the second pair of meshing teeth. Table 1 gives the parameters of a planetary gear set [16]. Suppose the parameters of the driving gear and the driven gear of a pair of external–external gears are the same to the sun gear and the planet gear, respectively. The mesh stiffness of this external–external gear pair can be evaluated using the method developed in this paper. The proposed model is compared with the model given in Ref. [7]. The results are shown in Fig. 5. Fig. 5 illustrates the mesh stiffness (15.5 times of meshes) in half revolution of the driven gear. In Ref. [7], the gear tooth is considered starting from the base circle, while in this paper it is modeled starting from the root circle. The difference in the radius of the base circle and the root circle is 2.1 mm (7.4% of the base circle radius) for the sun gear and 1 mm (2.2% of the base circle radius) for the planet gear as given in Table 1. It is found that the mesh stiffness decreases about 40%, if the gear tooth is considered staring from the root circle. This demonstrates that substantial difference in the mesh stiffness of a pair of external–external gears is likely when the origin of the gear tooth is changed from the base circle to the root circle, especially when the difference between the base circle and the root circle is bigger. It is important to consider this difference because the dynamic response of the gear system will be affected due to the stiffness change. 3. Mesh stiffness of fixed-shaft external–internal gears The external–internal gear pair includes two types of gears, namely an external gear and an internal gear. For an external gear, the root circle may be bigger or smaller than the base circle. However, for an internal gear, the root circle is always bigger than the base circle. Liang et al. [16] derived analytical equations of the mesh stiffness for a pair of external–internal gears. But, they modeled the external gear tooth starting from the base circle. In this study, we will improve the results of Liang et al. [16] by modeling the gear tooth starting from the tooth root circle. The stiffness equations of the external gear are derived in Section 2.

Fig. 12. Mesh stiffness of a sun–planet pair of five crack levels in sun gear.

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

+γ

29

θ

+γ

θ

+γ

θ

Fig. 13. Mesh stiffness of sun–planet gear pairs with 25% crack in sun gear.

For the internal gear, the equations of bending, shear, and axial compressive stiffness reported in Ref. [16] will be used directly in this research. Thus, the difference of the mesh stiffness for a pair of external–internal gears between our model and the model from Ref. [16] comes from the difference in the modeling of the external gear teeth. We evaluate the mesh stiffness of a pair of fixed-shaft external–internal gears of which the external gear and the internal gear have the same parameters to the planet gear and the ring gear (listed in Table 1), respectively. The overall mesh stiffness is shown in Fig. 6. The comparison with Liang et al. [16] is also presented in Fig. 6. It can be seen that even the radius difference of the base circle and the root circle is only 1 mm for the planet gear; the mesh stiffness decreased about 17% in the double-tooth-pair meshing duration and about 23% in the single-tooth-pair meshing duration. This percentage change is smaller than the external–external gear case because the same stiffness equations are used for the internal gear in these two models. Though relatively small, this result indicates the importance of modeling the gear tooth of the external gear starting from the root circle as it is more realistic. 4. Mesh stiffness of a planetary gear set Fig. 7 shows the structure of a planetary gear set which will be considered in this study. It is comprised of a sun gear (s), a ring gear (r) and four equally-spaced planets (p) which are held by a common carrier (c). In this structure, the ring gear is fixed. The parameters of this planetary gear set are listed in Table 1. A planetary gear set comprises several pairs of sun–planet gears (external–external gears) and ring–planet gears (external–internal gears) meshing simultaneously. While each of the sun–planet meshes (or ring–planet meshes) has the same shape of mesh stiffness variation, they are not in phase with each other [20]. The phasing relationships of multiple meshing gears are required to be considered. The mesh phasing of the planetary gear set was calculated in Ref. [16] and shown in Table 2. The value of γsn (n = 1, 2, 3, 4) is the relative phase between the nth sun–planet pair with respect to the 1st sun–planet pair. The value of γrn is the relative phase between the nth ring–planet pair with respect to the 1st ring–planet pair. The value of γrs is the relative phase between the nth ring–planet mesh with respect to the nth sun–planet mesh. The γrs = 0 shows that the sun–planet gears and the ring–planet gears mesh at the pitch point simultaneously [16]. Once the mesh stiffness equations for a pair of fixed-shaft external–external gears and a pair of fixed-shaft external–internal gears are derived, the mesh stiffness of a planetary gear set can be obtained using the approach reported in Ref. [16]. The time-varying mesh stiffness of the planetary gear set is plotted in Fig. 8. There are 12.56 times of gear meshing in half revolution of the planet gear. The points P' and P illustrate that both the sun–planet pair and the ring–planet pair mesh at the pitch point at t = 0. The curves ksp1, ksp2, ksp3 and ksp4 represent the 1st, 2nd, 3rd and 4th pairs of the sun–planet mesh stiffness, respectively. Similarly, the curves krp1 krp2, krp3 and krp4 denote the 1st, 2nd, 3rd and 4th pairs of

Table 4 Crack levels and the corresponding crack length in the planet gear. Crack levels

Crack length

10% 25% 50% 75% 100%

q1 q1 q1 q1 q1

= = = = =

0.86 2.15 4.30 4.30 4.30

mm mm mm mm, q2 = 2.15 mm mm, q2 = 4.30 mm

30

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

Fig. 14. Mesh stiffness of a sun–planet pair of five crack levels in planet gear.

the ring–planet mesh stiffness, respectively. The symbol θm is the rotation angular displacement of the planet gear in one mesh period, θm = 2πZr/Zp/(Zs + Zr) [16]. Zs, Zp, and Zr are the tooth number of the sun gear, planet gear and ring gear, respectively.

5. Crack modeling of the external gear Lewichi [21] found that several factors such as rim and web thicknesses, initial crack location and backup ratio (rim thickness divided by tooth height) decided the gear crack propagation. Belsak and Flasker [22] investigated the propagation path of the crack both experimentally and computationally. The results (see Fig. 9) indicated that the crack propagation paths were smooth, continuous, and in most cases, rather straight with only a slight curvature; similar to the findings in Ref. [21]. It is pointed out by both Kramberger et al. [23] and Belsak and Flasker [22] that the crack mostly initiated at the point of the maximum principle stress in the tensile side of a gear tooth (critical area in Fig. 9). In this paper, the crack is modeled as a straight line staring from the critical area of the tooth and two cases will be analyzed. The first case is when the gear root circle is smaller than the base circle and the second case is when the root circle is bigger than the base circle. When the crack happens, the gear mesh stiffness reduction is modeled by taking into account the thickness reduction of the tooth section and the effective length changing of the beam model.

θ

Fig. 15. Mesh stiffness of a sun–planet pair with 25% crack in planet gear.

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

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Fig. 16. Mesh stiffness of a ring–planet pair of five crack levels in planet gear.

5.1. Case 1: the gear root circle is smaller than the base circle The point M (see Fig. 10), located within the critical area of the tooth, is considered as the starting point of the crack. The crack is modeled as a straight line which passes through point N which is the intersection of the straight line NN′ and the root circle. The crack propagates along the straight line until reaching the tooth central line at point B. Then, it changes the propagation direction towards point D where the tooth breaks. The line segment MN can be interpreted as an initial notch which stimulates higher stress concentration which then leads to fully developed crack. It is similar to what Lewicki [24] described as an initial notch in his crack experiment. This notch will not be considered in the mesh stiffness derivation in this paper. The fillet curve is difficult to express analytically [10], therefore, in the proposed beam model, we simplify it and represent it using a straight line NN′ as shown in Fig. 10. The angle between the crack line and the tooth central line is defined as υ. Though we have modeled that the crack gradually grows until it reaches the breaking point D, sometimes, sudden tooth breakage may also take place especially when the crack goes over the central line. Similar to the perfect situation, the overall mesh stiffness is considered as the summation of the Hertzian stiffness, bending stiffness, shear stiffness and axial compressive stiffness. The Hertzian stiffness and axial compressive stiffness will not be affected by the crack propagation [7,9]. Only the bending stiffness and the shear stiffness will be affected due to the change in the tooth length and the tooth height caused by the crack. In order to derive the bending and shear stiffness with the propagation of the crack, we need to consider four conditions. Condition 1 and condition 2 may happen when the crack is below the tooth central line. Condition 3 and condition 4 may happen when the crack goes over the tooth central line.

θ

Fig. 17. Mesh stiffness of ring–planet pairs with 25% crack in planet gear.

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38 Table 5 Major parameters of the spur gears used in this model [26]. Gear type

Standard involute, full-depth teeth

Material Modulus of elasticity Poisson's ratio Face width Module Number of teeth Pressure angle Theoretical contact ratio Theoretical angle of meshing cycle Addendum Dedendum

Aluminum 69 GPa 0.33 0.015 m 6 mm 23 20° 1.59 24.912° 1.00 m 1.25 m

Condition 1: When ha ≥ ho & α1 N αa The symbol ha represents the distance from the crack end point A to the tooth central line when the crack has not reached the tooth central line. Half of the roof chordal tooth thickness is denoted by ho. The angle αa corresponds to the force action point K (see Fig. 10). For a tooth with crack, the tooth section area and the area moment of inertia can be expressed as follows:  Ax ¼

ðha þ hx ÞL 2hx L

if x ≤da if xNda

8 1 > < ðha þ hx Þ3 L 12 Ix ¼ > : 1 ð2h Þ3 L x 12

ð5:1Þ

if x≤da

ð5:2Þ

if xNda

where ha and hx have the following expressions: ha ¼ Rb sinα 2 −q1 sinυ

ð5:3Þ

8 < Rr sinγ; hx ¼ Rb sinα 2 ; : Rb ½ðα 2 −α Þ cosα þ sinα ;

if 0≤x ≤d1 if d1 ≤x ≤d2 if d2 bx ≤d:

ð5:4Þ

Substituting Eq. (5.2) into Eq. (2.3) and substituting Eq. (5.1) into Eq. (2.4), the bending and shear stiffness of the cracked tooth can be obtained respectively: 1 ¼ kb

i2   − cosα þ cosα 3 − cosα r − qR1 cosυ cosα 1 r dα  3 α3 EL sinα 3 þ sinα− qR1 sinυ r h i ðN−2:5Þ cosα 1 cosα 3 3 4 1− −4ð1− cosα 1 cosα 2 Þ3 N cosα 0 þ  3 EL cosα 1 2 sinα 2 − Rq1 sinυ b Z α 2 2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα þ dα h i3 −α a EL sinα 2 − Rq1 sinυ þ sinα þ ðα 2 −α Þ cosα b Z −α 2 a 3f1 þ cosα ½ðα −α Þ sinα− cosα g ðα −α Þ cosα 1 2 2 dα þ 2EL sinα þ α ½ ð −α Þ cosα 3 −α 1 2

Z

αr

12 sinα

h

N cosα 0 N−2:5

  N−2:5 2 2:4 ð 1 þ ν Þ cos α cosα − cosα 1 2 3 αr 1 2:4ð1 þ ν Þ cos α 1 sinα N cosα 0   ¼  3 dα þ q1 ks α 3 EL sinα − q1 sinυ þ sinα EL 2 sinα − sinυ 3 Rr 2 Rb Z α 2 2 2:4ð1 þ νÞðα 2 −α Þ cosα cos α 1

dα þ q −α a EL sinα 2 − 1 sinυ þ sinα þ ðα 2 −α Þ cosα Rb Z −α 2 a 1:2ð1 þ ν Þðα −α Þ cosα cos α 2 1 þ dα: EL½ sinα þ ðα 2 −α Þ cosα  −α 1 Z

ð5:5Þ

2

ð5:6Þ

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33

Table 6 Mesh stiffness comparison. Gear tooth condition

Maximum difference (%)

Perfect 4.7 mm crack

Single mesh period

Double mesh periods

5.9 6.7

6.3 2.9

Condition 2: When ha b ho or when ha ≥ ho & α1 ≤ αa In this condition, the expressions of the tooth section area and the area moment of inertia are as follows: Ix ¼

1 3 ðh þ hx Þ L 12 a

ð5:7Þ

Ax ¼ ðha þ hx ÞL:

ð5:8Þ

Similar to condition 1, the bending and the shear stiffness of the cracked tooth are obtained: 1 ¼ kb

i2   − cosα þ cosα 3 − cosα r − qR1 cosυ cosα 1 r dα  3 α3 EL sinα 3 þ sinα− qR1 sinυ r h i cosα 1 cosα 3 3 4 1− ðN−2:5NÞcosα −4ð1− cosα 1 cosα 2 Þ3 0 þ  3 EL cosα 1 2 sinα 2 − Rq1 sinυ b Z α 2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g2 ðα 2 −α Þ cosα þ dα h i3 −α 1 EL sinα 2 − Rq1 sinυ þ sinα þ ðα 2 −α Þ cosα Z

αr

12 sinα

h

N cosα 0 N−2:5

ð5:9Þ

b

  N−2:5 2:4ð1 þ ν Þ cos2 α 1 cosα 2 − cosα 3 1 2:4ð1 þ ν Þ cos α 1 sinα N cosα 0   ¼  3 dα þ q1 ks α 3 EL sinα − q1 sinυ þ sinα EL 2 sinα 2 − sinυ 3 Rr Rb Z α 2 2 2:4ð1 þ ν Þðα 2 −α Þ cosα cos α 1

dα: þ q −α 1 EL sinα 2 − 1 sinυ þ sinα þ ðα 2 −α Þ cosα Rb Z

2

αr

ð5:10Þ

Condition 3: When hc b ho or when hc ≥ ho & α1 ≤ αc The angle αc corresponds to the force action point E which is the mirror of the point E'. The point E' is relative to the crack point C in Fig. 10. The tooth section area and the area moment of inertia are given by: Ix ¼

1 3 ðh −hc Þ L 12 x

ð5:11Þ

Ax ¼ ðhx −hc ÞL

ð5:12Þ

where hc = q2 sin υ. The bending and shear stiffness of the cracked tooth are derived as:

1 ¼ kb

    i2 q2 3 − cosα þ cosα 3 − cosα r − sinα sinυ − Rr cosυ cosα 1 dα  3 α3 EL sinα− qR2 sinυ r h i 3 cosα 1 cosα 3 3 4 1− ðN−2:5NÞcosα −4ð1− cosα 1 cosα 2 Þ 0 þ  3 EL cosα 1 sinα 2 − Rq2 sinυ b Z α 2 2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα þ dα h i3 −α 1 EL sinα þ ðα 2 −α Þ cosα− Rq2 sinυ Z

αr

12 sinα

h

N cosα 0 N−2:5

b

ð5:13Þ

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

  N−2:5 2 ð Þ 2:4 1 þ ν cos α cosα − cosα 1 2 3 α r 2:4ð1 þ ν Þ cos α sinα 1 N cosα 0 1   ¼  3 dα þ q2 ks α 3 EL sinα− q2 sinυ sinυ EL sinα 2 − Rr Rb Z α 2 2:4ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1

dα: þ q −α 1 EL sinα þ ðα 2 −α Þ cosα− 2 sinυ Rb Z

2

ð5:14Þ

Condition 4: When hc ≥ ho & α1 N αc In this condition, the tooth is considered as the constitution of the tooth sections whose height hx is bigger than hc [9]. If the section height hx is smaller than hc, the corresponding tooth part can be ignored. If the section height hx is bigger than hc, the expressions of the tooth section area and the area moment of inertia are the same as Eqs. (5.11) and (5.12). The bending and shear stiffness of the cracked tooth are expressed as follows:     i2 q2 3 − cosα þ cosα 3 − cosα r − sinα sinυ − Rr cosυ cosα 1 dα  3 α3 EL sinα− qR2 sinυ r h i cosα 1 cosα 3 3 4 1− ðN−2:5NÞcosα −4ð1− cosα 1 cosα 2 Þ3 0 þ  3 EL cosα 1 sinα 2 − Rq2 sinυ b Z α 2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g2 ðα 2 −α Þ cosα þ dα h i3 −α c EL sinα þ ðα 2 −α Þ cosα− Rq2 sinυ

ð5:15Þ

  N−2:5 2 2:4ð1 þ ν Þ cos α 1 cosα 2 − cosα 3 1 2:4ð1 þ ν Þ cos α 1 sinα N cosα 0   ¼  3 da þ q2 ks α 3 EL sinα− q2 sinυ sinυ EL sinα 2 − Rr Rb Z α 2 2:4ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1

dα: þ q −α c EL sinα þ ðα 2 −α Þ cosα− 2 sinυ Rb

ð5:16Þ

1 ¼ kb

Z

12 sinα

αr

h

N cosα 0 N−2:5

b

Z

2

αr

5.2. Case 2: the gear root circle is bigger than the base circle Fig. 11 shows the tooth crack model when the root circle is bigger than the base circle. The crack is modeled as a straight line starting from the intersection (point M) of the involute curve and the root circle. The point M is within the critical area of the gear tooth. Similar to Case 1, four conditions are considered in the derivation of the bending and shear stiffness. In each condition, the expressions of the tooth section area and the area moment of inertia are the same as derived for the condition when the root circle is smaller than the base circle. But, the expression of the tooth height, hx, changes as:  hx ¼

if 0≤x ≤d1 if d1 bx ≤d:

Rr sinγ; Rb ½ðα 2 −α Þ cosα þ sinα ;

ð5:17Þ

Applying the same procedure as in Case 1, the bending stiffness kb, and shear stiffness ks for the four conditions are derived and listed as follows: Condition 1: When ha ≥ ho & α1 N αa

1 ¼ kb

Z

12 sinα

αr

h

N cosα 0 N−2:5

α3

Z þ

r

12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g2 ðα 2 −α Þ cosα h i3 dα −α a EL N−2:5 sinα − q1 sinυ þ sinα þ ðα −α Þ cosα 4 2 N cosα R α5

0

Z þ

i2   − cosα þ cosα 4 − cosα r − qR1 cosυ cosα 1 r dα  3 EL sinα 4 þ sinα− qR1 sinυ

b

3f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g2 ðα 2 −α Þ cosα dα 2EL½ sinα þ ðα 2 −α Þ cosα 3 −α 1 −α a

ð5:18Þ

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

1 ¼ ks

Z

Z −α 2 a 1:2ð1 þ ν Þðα −α Þ cosα cos α 2:4ð1 þ ν Þ cos2 α 1 sinα 2 1 dα  3 dα þ EL½ sinα þ ðα 2 −α Þ cosα  α 3 EL sinα þ sinα− q1 sinυ −α 1 4 Rr Z α 5 2:4ð1 þ νÞðα 2 −α Þ cosα cos2 α 1

dα: þ N−2:5 q −α a EL sinα 4 − 1 sinυ þ sinα þ ðα 2 −α Þ cosα N cosα 0 Rb

35

αr

ð5:19Þ

Condition 2: When ha b ho or when ha ≥ ho & α1 ≤ αa 1 ¼ kb

Z

αr

12 sinα

h

N cosα 0 N−2:5

α3

Z þ

r

2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα h i3 dα −α 1 EL N−2:5 sinα − q1 sinυ þ sinα þ ðα −α Þ cosα 4 2 R N cosα

Z

ð5:20Þ

α5

0

1 ¼ ks

i2   − cosα þ cosα 4 − cosα r − qR1 cosυ cosα 1 r dα  3 EL sinα 4 þ sinα− qR1 sinυ

b

2:4ð1 þ ν Þ cos2 α 1 sinα  3 dα α 3 EL sinα þ sinα− q1 sinυ 4 Rr Z α 5 2:4ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1

dα: þ N−2:5 q −α 1 EL sinα 4 − 1 sinυ þ sinα þ ðα 2 −α Þ cosα N cosα 0 Rb αr

Fig. 18. Time-varying mesh stiffness (a) FEM result from Ref. [26] and (b) obtained from the method developed in this study.

ð5:21Þ

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

Condition 3: When hc b ho or when hc ≥ ho & α1 ≤ αc 1 ¼ kb

Z

12 sinα

αr

h

N cosα 0 N−2:5

α3

Z þ

    i2 q2 4 − cosα þ cosα 4 − cosα r − sinα sinυ − Rr cosυ cosα 1 dα  3 EL sinα− qR2 sinυ r

2 12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα dα h i3 −α 1 EL sinα þ ðα 2 −α Þ cosα− Rq2 sinυ

ð5:22Þ

α5

b

1 ¼ ks

Z

Z

2

αr

2:4ð1 þ ν Þ cos α 1 sinα  3 dα α 3 EL sinα− q2 sinυ R

þ

r

2:4ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1

dα: q −α 1 EL sinα þ ðα 2 −α Þ cosα− 2 sinυ Rb α5

ð5:23Þ

Condition 4: When hc ≥ ho & α1 N αc 1 ¼ kb

Z

12 sinα

αr

h

N cosα 0 N−2:5

α3

Z þ

    i2 q2 4 − cosα þ cosα 4 − cosα r − sinα sinυ − Rr cosυ cosα 1 dα  3 EL sinα− qR2 sinυ r

2

α5

ð5:24Þ

12f1 þ cosα 1 ½ðα 2 −α Þ sinα− cosα g ðα 2 −α Þ cosα dα h i3 −α c EL sinα þ ðα 2 −α Þ cosα− Rq2 sinυ b

1 ¼ ks

Z

2:4ð1 þ ν Þ cos2 α 1 sinα  3 dα α 3 EL sinα− q2 sinυ R αr

r

Z þ

2:4ð1 þ ν Þðα 2 −α Þ cosα cos2 α 1

dα: q −α c EL sinα þ ðα 2 −α Þ cosα− 2 sinυ Rb α5

ð5:25Þ

6. Crack effect on the mesh stiffness In this section, the crack effect on the mesh stiffness of a planetary gear set (see Fig. 7) is investigated when a crack takes place in the sun gear or the planet gear. Two cases exist when a crack occurs in the planet gear (sun gear side or ring gear side). If it is in the sun gear side, the ring–planet mesh stiffness is assumed not to be affected as the crack part can still bear the compressive stiffness as if no crack exists. Similarly, if it is in the ring gear side, the sun–planet mesh stiffness will not be affected. Overall, three cases are considered in this study; crack in the sun gear, crack in the planet gear (sun side) and crack in the planet gear (ring side). For each case, the mesh stiffness at five crack levels will be evaluated. These levels are 10%, 25%, 50%, 75% and 100% cracks (tooth missing). Fifty percent (50%) crack occurs when the crack line q1 reaches the tooth central line (see Fig. 10). Twenty five percent (25%) crack means the crack length is half of 50% crack length. One hundred percent (100%) crack indicates that the crack line q2 reaches the tooth root circle when the tooth breaks. Seventy five percent (75%) crack happens when the crack line q2 has the same length of q1 with 25% crack. The angle, υ, between the crack line and the tooth central line is assumed to be a constant in the calculation and set as 45°. 6.1. Crack in the sun gear Table 3 shows the five crack levels in the sun gear and the corresponding crack length. According to the parameters of the planetary gear set (see Table 1), the root circle of the sun gear is smaller than the base circle. Applying the equations derived in Sections 2 and 5, the mesh stiffness of a pair of sun–planet gears (ksp1) is calculated and shown in Fig. 12 for the five crack levels. It shows the period (1.6 mesh periods) that was experienced from the crack tooth starting to mesh to the ending of mesh. As the size of the crack grows, the mesh stiffness reduces correspondingly. Correct stiffness measurement is important for fault diagnosis of the gears. For each crack level, corresponding vibration signal can be obtained through dynamic simulation. By analyzing these vibration signals, fault severity indicators can be generated, which can be then used to detect the fault severity of the gear system. If the sun gear has a crack, the crack tooth will mesh with each of the planet gears successively. Therefore, the mesh stiffness of all sun–planet pairs will be affected. The time intervals of the crack tooth in meshing are calculated analytically and labeled in Fig. 13. Fig. 13 shows one crack mesh cycle of the sun–planet mesh stiffness, which is 19 times of gear meshes corresponding to a 272.4 degree revolution of the planet gear. The symbol θm represents the angular displacement of the planet gear in one mesh period, which expression was given in Section 4. The point P' is the pitch point. The mesh stiffness of the ring–planet gears is assumed to be not affected by the crack just like what is shown in Fig. 8. 6.2. Crack in the planet gear (sun gear side) Table 4 gives the information about the five crack levels in the planet gear. For the planet gear given in Table 1, the root circle is also smaller than the base circle. The mesh stiffness will decrease gradually along with the crack growth as shown in Fig. 14.

X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

37

In this case, one crack mesh cycle includes 31 times of the gear meshing corresponding to a 444.4 degree revolution of the planet gear. The ring–planet gear pairs are treated as the perfect condition (see Fig. 15 for the mesh stiffness of the perfect case). One big difference from Case 1 is that the mesh stiffness of only one pair of sun–planet gears will be affected by the crack (see Fig. 15). This difference in the mesh stiffness will cause the corresponding difference in the vibration signal which can be obtained through dynamic simulation. Signal processing methods can be applied to distinguish the crack location in the sun gear or the planet gear. 6.3. Crack in the planet gear (ring gear side) In this case, the crack lengths of the five crack levels are the same as Case 2 (see Table 4). Fig. 16 illustrates the mesh stiffness reduction trend for a pair of ring–planet gears at five crack levels. For each crack level, the mesh stiffness decrease is quantified. 1.94 mesh periods are experienced from the crack tooth starting to mesh to the ending of mesh. This information is essential to obtain the dynamic response of a planetary gear set when crack is present. Fig. 17 presents the mesh stiffness of a pair of ring–planet gears when the planet gear has the crack level of 25%. Gear mesh takes place 31 times during one crack mesh cycle which corresponds to a 444.4 degree revolution of the planet gear. The point P corresponds to the pitch point. The sun–planet gear pairs are regarded as the perfect condition (see Fig. 15 for the mesh stiffness of the perfect case). In Case 1 and Case 2, when the crack happens, the sun–planet mesh stiffness decreases and the ring–planet mesh stiffness keeps the same as the perfect situation. This difference in the mesh stiffness can be used in the dynamic simulation to observe the vibration response when crack happens in different positions. By analyzing these vibration signals, fault position indicators can be generated to track the crack position of a planetary gear set. 6.4. Summary Figs. 12, 14 and 16 describe the crack effect on a planetary gear set for three different situations, namely, crack in the sun gear, crack in the planet gear (sun gear side) and crack in the planet gear (ring gear side), respectively. To better illustrate the differences, these three figures are drawn by focusing on the time periods when the crack tooth is in meshing. When the crack is present in the sun gear or the planet gear (sun gear side), the sun–planet mesh stiffness will reduce as shown in Figs. 12 and 14, respectively. Comparing these two figures with each other, they seem similar. However, if one examines them carefully, he/she will see some differences, for example, the stiffness values of a 50% crack are different (one is 1.64 × 109 N/m and the other one is 1.58 × 109 N/m) at the initial mesh point (time 0 of the x-axis). Unless the sun gear and the planet gear have the same gear geometry and material properties, the stiffness reduction for these two situations will not be the same. In addition, as shown in Figs. 13 and 15, if the crack is in the sun gear, the stiffness of four pairs of sun–planet gears are affected, while, if the crack is in the planet gear (sun gear side), only the stiffness of one pair of sun–planet gears is affected. If the crack is in the planet gear (ring gear side), only the stiffness of one pair of ring–planet gears will decrease as shown in Figs. 16 and 17, while the stiffness of the sun–planet gear pairs will not be affected. However, the gear body deformation is not considered in this study. In Ref. [25], an analytical formula for gear body-induced tooth deflection was derived for perfect gears by assuming linear and constant stress variations at root circle. However, when there is a crack in the tooth root, the assumption of linear and constant stress variations at root circle fails. Until now, there are no available analytical equations available based on our literature survey to consider the gear body-induced tooth deflection when there is a crack in the tooth root. 2. The stiffness component due to the gear body deformation when there is a crack in the tooth root will be investigated in our future research. 7. Verification A planetary gear set comprises several pairs of sun–planet gears (external–external gear pairs) and several pairs of ring–planet gears (external–internal gear pairs). In this study, the stiffness equations are analytically derived for the external gear tooth in the perfect and the crack conditions. For the internal gear, the stiffness equations reported in Ref. [16] are used directly. The stiffness equations derived in this study for the external gear tooth have the same expressions, no matter the external gear belongs to an external–external gear pair or an external–internal gear pair. Therefore, verification of the stiffness value of a pair of external–external gear pair can demonstrate the correctness of the derived equations for the external gear tooth. In Ref. [26], a two-dimensional (2-D) finite element model was developed to evaluate the torsional stiffness of a pair of external–external gears with 1:1 transmission ratio. The gears were modeled using quadratic 2-D plane strain elements and the contact effect was modeled using 2-D line-to line general contact elements which included elastic Coulomb frictional effects. The relationship between linear and torsional mesh stiffness can be expressed as follows [26]: 2

ktb ¼ kt  Rb

ð7:1Þ

where ktb and kt represent the torsional mesh stiffness and linear mesh stiffness, respectively, and Rb is the radius of the gear base circle. The major parameters of this gear pair used in Ref. [26] are listed in Table 5. Table 6 and Fig. 18 give the comparisons between the mesh stiffness results from Ref. [26] and that obtained from the analytical model developed from this study. Two health conditions are analyzed: health condition with no crack and crack condition with a 4.7 mm crack at the root of one gear tooth. The

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X. Liang et al. / Mechanism and Machine Theory 76 (2014) 20–38

biggest difference in the mesh stiffness between the FEM result and the analytical result from this study is within 7%, which verifies the stiffness theory and equations developed in this study. 8. Conclusion In this paper, the potential energy method is applied to analytically evaluate the time-varying mesh stiffness of a planetary gear set. The time-varying mesh stiffness is represented as a function of the angular displacement of the gear, which is easy to use. Hertzian contact stiffness, bending stiffness, shear stiffness and axial compressive stiffness are considered. A modified cantilever beam model is proposed for the external gear tooth and the analytical equations of the bending, shear and axial compressive stiffness are derived. The results show that it is important to model the gear tooth starting from the root circle rather than the base circle. A crack propagation model is developed for the external gear and the analytical equations for the mesh stiffness are derived. The mesh stiffness reduction is quantified when a crack happens in the sun gear, the planet gear (sun gear side) or the planet gear (ring gear side). In the future, the vibration signals of a planetary gear set will be obtained for different crack levels and crack positions through dynamic simulation. Crack severity and the crack position indicators can be generated by analyzing the vibration signals. Acknowledgment This research is supported by the Natural Science and Engineering Research Council of Canada (NSERC) and China Scholarship Council (CSC). Comments and suggestions from reviewers and the Editor are very much appreciated. References [1] T. Eritenel, R.G. Parker, An investigation of tooth mesh nonlinearity and partial contact loss in gear pairs using a lumped-parameter model, Mech. Mach. Theory 56 (2012) 28–51. [2] F. Chaari, W. Baccar, M.S. Abbes, M. 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