Analytical modeling of spur gear corner contact effects

Mechanism and Machine Theory 96 (2016) 146–164

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Analytical modeling of spur gear corner contact effects Wennian Yu ⁎, Chris K. Mechefske Department of Mechanical and Materials Engineering, Queen's University, Kingston, ON K7M 3N6, Canada

a r t i c l e

i n f o

Article history: Received 3 May 2015 Received in revised form 29 August 2015 Accepted 1 October 2015 Available online xxxx Keywords: Corner contact Loaded static transmission error Mesh stiffness model Linear tip relief

a b s t r a c t Corner contact, which happens due to the tooth flexibility when subjected to a heavy load, is normally neglected in the dynamic analysis of gear transmission systems with profile modifications when using analytical methods. To fill this gap, an analytical method to study corner contact effects on the mesh stiffness, static transmission error and the dynamic response of a spur gear pair with tip relief is introduced. In order to stress the significance of corner contact effects on gear dynamic analysis two types of gear mesh stiffness model are differentiated, and three types of commonly used single degree of freedom (SDOF) models are generalized. Comparisons with experimentally measured results presented in the literature shows that corner contact effects should not be neglected in the gear dynamic analysis when no or an insufficient amount of profile modification is applied. Detailed comparisons of the steady state responses predicted by the proposed models are made by introducing different amounts of linear tip relief into the analysis, and the advantages and disadvantages of each model are stated. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Due to the tight restrictions on noise and vibration amplitudes generated by gear transmission systems during operation and the increasingly intense competition in the market for industrial machinery, there have been numerous studies conducted focused on the reduction of gear noise and vibration [1,2]. Research on gear noise and vibration has revealed that the fluctuation of the number of gear tooth pairs in the meshing zone is one of the most significant factors exciting large amplitude vibrations and high levels of noise. Gear unloaded static transmission error (i.e. gear manufacturing errors and tooth profile modifications) as well as the backlash, which inevitably exist between mating pairs of gear teeth, are also considered as sources of major internal excitation and gear vibration [2–4]. In addition to these, external excitation such as torque fluctuation, mass unbalance of the shaft, gear and shaft misalignment and the common occurrence of gear tooth faults and bearing faults will all contribute to the overall vibration and noise level of gear transmission systems [5–17]. The literature reports on a large number of investigations dealing with the effect of multiple types of excitations on gear dynamic behavior. The finite element (FE) method is widely used to calculate mesh stiffness, static transmission error, and even the dynamic response of gear transmission systems [1–8]. It offers significant advantages in its representation of crucial tooth contact and modeling fidelity. However, FE models are normally computationally expensive since they need sufficient mesh refinements as well as necessary re-meshing for every contact position. On the other hand, analytical approaches for the gear mesh stiffness calculation and discrete parameter models (i.e. lumped parameter models) for gear dynamic analysis have significant advantages in their computation speed and modeling efficiency. Even though a number of simplifying assumptions are usually employed, analytical models built in previous studies still yield consistent results with the experimental measured results [9–17]. Detailed reviews of the discrete parameter models used in the literature are given in [18,19]. ⁎ Corresponding author at: Room 321, McLaughlin Hall, Queen's University, Kingston, Ontario K7M 3N6, Canada. Tel.: +1 613 449 6390. E-mail addresses: [email protected] (W. Yu), [email protected] (C.K. Mechefske).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.10.001 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

147

Although theoretical models used in the past differ in some aspects, the analytical studies made by Lin [20–24] and Kahraman [9–12], together with the experimental results provided by Munro [25] and Kahraman [14–17], clearly demonstrate that the unavoidable fluctuation of the number of meshing gear tooth pairs and the gear backlash must be included in gear dynamic analysis so that an accurate prediction of gear dynamic behavior can be obtained. Gear damping due to Hertzian contact [26], lubrication and friction forces developed between meshing gear tooth faces [27], is usually simplified by using an energy-equivalent constant damping ratio obtained from experimental measurements [28,29] or an empirical formula [30]. Although some recent research has studied the effects of the friction forces [29,30] and lubrication damping [31,32] separately and included them in the model acting as further nonlinear elements, it was found that an equivalent constant damping ratio is sufficient to account for frictional loses and lubrication factors [9–12]. The fluctuation of the number of meshing gear tooth pairs is universally treated as a time-varying gear mesh stiffness (GMS) in most models, whereas backlash is usually modeled as a piecewise linear function. However, gear models established by different researchers are still slightly different from each other. The main discrepancies include the treatment of gear profile errors and corner contact (i.e. tip interference). In some literature, both the undesirable gear manufacturing errors and the intentional profile modifications are treated as displacement excitation [3,5,6,9–12], while in other papers, instead of being treated as a displacement excitation the smoothing effect coming from the profile modifications is implicitly reflected in the mesh stiffness [1,2,18–21,33]. In addition, the premature or delayed engagement of a mating tooth pair due to the corner contact are completely neglected in some studies [5,6], whereas in some other papers this effect is incorporated by slightly modifying the mesh stiffness function [28,34]. These confusions were first noticed by Kasuba and Evans [35]. In their paper, the GMS that is totally independent of the gear error and the load is defined as the fixed-variable mesh stiffness (FVMS), whereas the GMS that is influenced by the gear error and load is defined as the variable–variable mesh stiffness (VVMS) as opposed to the FVMS. Based on the two different definitions of mesh stiffness, two different types of gear model can be established. In the FVMS model: 1) gear tooth errors have negligible effect or no effect on mesh stiffness; 2) the contact ratio and/or mesh stiffness are not affected by transmitted load, premature or delayed engagement (i.e. corner contact); 3) dynamic simulations are based on uninterrupted periodic mesh stiffness functions and error displacement strips. In the VVMS model, gear tooth errors and corner contact effects are reflected in the calculation of the VVMS in the static analysis, and therefore eliminated from the governing equation but reflected in the VVMS. The differences between these two models have been noticed for a while, but they are both widely used. Systematic and direct comparisons between these two models have not yet been made. Liu and Parker [5] have compared the dynamic response predicted by some existing discrete parameter models against a FE benchmark. However, the VVMS model was not included in their analysis. Which model is the most effective and efficient for gear dynamic analysis remains an open question. This constitutes one of the primary motivations of this study. Analytical methods to determine the effect of corner contact on the STE and GMS of heavily-loaded spur gears has been initiated to some degree. Several kinds of stiffness functions have been discussed by Umezawa et al. [28] to accommodate tip interference, and an experimental test was conducted to see which stiffness function better depicts the behavior of transmission error under static load and dynamic meshing conditions. Lin et al. [24] has derived an exact version of analytical formulae for the calculation of the transmission error outside the normal path of contact (also referred as the gear separation distance), whereas Munro et al. [36] and Seager [37] developed approximate formulae that are simple to use and highly accurate. However, in their discussions, the effect of gear tooth profile errors on corner contact is not included. It is commonly believed that corner contact can be avoided if sufficient tip and/or root relief on the teeth is used, which is the main reason that this effect is neglected in gear dynamics studies [5,6]. However, when insufficient tip and/or root relief is applied, especially for the gears working under heavily-loaded condition, the corner contact might still exist. Analytical studies for this case are limited. In this study, an analytical method to determine corner contact effects on gear static transmission error and mesh stiffness with the introduction of gear tooth profile errors will be presented. Based on this new method, two types of mesh stiffness model are differentiated and three distinct types of SDOF models of gear dynamics are generalized. They are the FVMS model, the VVMS model and a time-invariant approximate model [38,39], using the loaded static transmission error (LSTE) as the input. In order to determine which model is most effective and efficient for gear dynamics prediction, the simulated results based on these three models will be directly compared against the experimentally measured results that are provided in the literature. Finally, a parameter study will be conducted to define the relationships among these models. The ultimate purpose of this study is to fill the gap in previous analytical research findings regarding the interaction of gear tooth profile errors and corner contact effects, clarify remaining confusion about gear time-varying mesh stiffness, differentiate several SDOF models developed and used in previous research, and generalize the advantages and disadvantages of each model. 2. Gear mesh stiffness model 2.1. Gear tooth pair compliance The compliance of a single tooth pair C i (i = 1,2,⋯,N) can be acquired by both computational (finite element based) and analytical methods. In this paper, an analytical method based on the potential energy principle is used. According to Weber [40] and Cornell [41], the analytical calculation of Ci can be divided into three separate factors which can be summarized as: 1) the local deformation of each tooth caused by the Hertzian contact between mating teeth; 2) the tooth beam deflection in the direction of the load when the gear tooth is considered as a non-uniform cantilever while the tooth

148

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

foundation is assumed as perfectly rigid; and 3) the tooth deflection in the direction of the load caused by the flexibility of the foundation when considered as an elastic plane while the tooth is assumed as perfectly rigid. Therefore, by superposition, the total tooth deflection at the point j in the direction of load Fj of a single tooth pair is:




 δ j ¼ δh j þ δt j þ δt j þ δ f j þ δ f j d

p

g

p

g

ð1Þ

where δhj is the local Hertzain contact deformation between mating teeth; (δtj)p and (δtj)g are the tooth beam deflections of the driving gear tooth and the driven gear tooth respectively; (δfj)p and (δfj)g are the fillet-induced deflections of the driving gear tooth and the driven gear tooth respectively. Various analytical equations have been proposed to evaluate each deflection based on Weber's work. Thorough discussions about it can be found in [40–44]. Hence, the compliance of the ith tooth pair at the contact point j is defined as: i Cj

¼


 δj

:

d

Fj

ð2Þ

Analytical studies show that gear tooth pair compliance is independent of the load, and determined only by the gear tooth macro-geometries (i.e. tooth width and contact ratio, etc.) and material properties (i.e. Young's modulus and Poisson's ratio, etc.). 2.2. Gear static transmission error and load sharing ratio The transmission error, x, which is defined as the difference between the actual and ideal positions of the driven gear, is usually expressed as the linear displacement along the line of action [45]: x ¼ R1 θ1 −R2 θ2 :

ð3  a; bÞ

The sign convention used for the transmission error is positive behind the ideal position of the driven gear. Considering a spur gear pair with a normal contact ratio (1 b CR b 2, hence i = 1,2) running at a low speed subjected to the load F, the transmission error of the gear pair working under such circumstances is usually called the loaded static transmission error (LSTE), (xj)s, which normally includes the tooth deflections (δij)d, gear manufacturing errors (δij)m and profile modifications (δij)p at the contact position j, and should be equal for every tooth pair in mesh at j: 8



 1 1 1 > < ^x j ¼ δ j þ δ j þ δ j s d p



m : 2 2 2 > : ^x j ¼ δ j þ δ j þ δ j s

d

p

ð4  a; bÞ

m

In these equations, corner contact effects are not included and will be considered in the next section (hence a ‘hat’ symbol is added on LSTE for differentiation). Substituting Eq. (1) into the above equations and noting that the sum of the load shared by every tooth pair in mesh equals the total static load F, gives: 8
 1 1 1 > > >
^x j s ¼ C j F j þ e j < ^x j ¼ C 2j F 2j þ e2j > s > > 1 2 : F¼F þF j

ð5  a; b; cÞ

j

where Fij is the static load shared by the ith tooth pair at the contact position j; eij = (δij)p + (δij)m is the equivalent gear profile error regarding the ith tooth pair in mesh, which includes the gear manufacturing errors and tooth profile modifications. It is interesting to note that the static transmission error can be experimentally measured if the driving gear runs at a low speed. When experimentally measuring unloaded static transmission error, the gear pair should be worked under a light load so that the gear tooth deflection will be negligible. In Eq. (5-a, b, c), F, C1j , C2j , e1j and e2j can be either calculated or measured directly. F1j , F2j and ð^x j Þs are the three unknowns. Solving Eq. (5-a, b, c) simultaneously gives: 8
 > C 1 C 2 F þ C 1j e2j þ C 2j e1j > > > ^x j ¼ j j > > s C 1j þ C 2j > > > > 2 21 < C 1 j F þ ej Fj ¼ 1 > C j þ C 2j > > > > > C 1j F−e21 > 2 j > > Fj ¼ 1 > : C j þ C 2j

ð6  a; b; cÞ

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

149

p q where epq j = ej − ej (p, q = 1, 2). The load sharing ratio (LSR) is defined as the ratio of the load shared by one tooth pair in mesh to the total static load. According to the Eq. (6-a, b, c):

8 F 1j C 2j F þ e21 C 2j þ e21 > 1 j j =F > > ¼ ¼ ¼ LSR > j < F C 1j þ C 2j C 1j F þ C 2j F : > F 2j C 1j F−e21 C 1j −e21 > 2 j j =F > > ¼ ¼ LSR ¼ : j F C 1j F þ C 2j F C 1j þ C 2j

ð7  a; bÞ

As to the high contact ratio (CR N 2) gear pair, the loaded static transmission error, the static load shared by a tooth pair and the corresponding load sharing ratio can be obtained by following the same procedure described above: 8
   Xn

n n q >
 epj ∏p¼1 C pj F þ ∏ C > q¼1; q≠p > j > > ^
p¼1  ¼ x X > j > n n s > ∏q¼1;q≠p C qj > > >
 p¼1   > Xn

n > n q > < ∏q¼1;q≠i C j F− p¼1 ∏q¼1;q≠i;p C qj eipj i  Fj ¼ Xn
n q > > ∏ C > q¼1;q≠p j p¼1 >

X
  > > n n n > > ∏q¼1;q≠i C qj − ∏q¼1;q≠i;p C qj eipj =F > > i p¼1 > >  Xn
n > LSR j ¼ > : ∏q¼1;q≠p C qj p¼1

ð8  a; b; cÞ

where n is the nearest integer that is larger than CR. 2.3. Corner contact Corner contact, or contact outside the normal path of contact, can occur in spur as well as helical gear transmission systems owing to the elastic deflection of the loaded teeth, which will lead to premature and delayed engagement. This can be easily explained by observing Fig. 1. The load-carrying tooth pair #2 deforms elastically, which causes the incoming tooth pair #3 to enter contact earlier than the theoretical start of contact B. Similarly, the loaded outgoing tooth pair #1 will leave contact later than the theoretical end of contact E. This extends the tooth contact zone and increases the contact ratio. Obviously, the degree of this increased contact is dependent on the torque applied, since the higher the transmitted load, the greater the increase in elastic deflection due to the loaded teeth. The analysis by Lin et al. [24] shows that neglecting corner contact effects results in underestimating resonant speed and overestimating the maximum dynamic load. Lin et al. used the concept of separation distance to analytically analyze the influence of corner contact on transmission error and produce the final LSTE curve. 2.3.1. Gear teeth separation distance Gear teeth separation distance is defined as the distance between a pair of teeth just out of contact, during approach or recess, if there is no elastic deformation [24]. This distance, expressed along the line of action (Sa and Sr as shown in Fig. 1), will be

Fig. 1. Spur gear tooth pairs in mesh at the beginning (B) and end (E) of a meshing cycle and the separation distance in: (a) approach (b) recess.

150

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

compared with the LSTE to determine the contact condition. Lin et al. [24] have derived an exact version of analytical formulae for calculation of gear teeth separation distance, whereas Munro et al. [36] and Seager [37] developed several approximate formulae that are simple to use and highly accurate. 2.3.2. LSTE including corner contact effects The influence of corner contact on transmission error has been thoroughly discussed by Lin et al. [24] with gear tooth profile errors neglected, which may be a reasonable assumption for high-quality, heavily-loaded gears. However, in this study, a general form of LSTE including corner contact effects will be derived. Considering a spur gear pair with a normal contact ratio, corner contact mainly happens when an incoming tooth pair is just coming into the theoretical starting point of the engagement and an outgoing tooth pair is just leaving the theoretical end point of the engagement. In this study, we assume: 1) corner contact affects the gear mesh only at the start and end of each single tooth pair mesh zone, and do not influence the double mesh zone; and 2) in the single mesh zone where tooth pair #2 is theoretically the only tooth pair in mesh as shown in Fig. 1, profile errors on this tooth pair are not sufficient to lead to loss of contact. The first assumption is reasonable since according to FEA results [7], the hand over region between single and double mesh zones will move slightly with increasing load so that the single zone reduces when corner contact happens whereas the double zone is relatively stable. The second assumption is introduced for the purpose of disregarding profile irregularities due to local spalls, pits or even tooth breakage. As a result, there will be three distinct cases when corner contact happens. ① Tooth pair #3 is in corner contact, and tooth pair #1 is not in corner contact. In this case, tooth pair #2 carries the most load. Tooth pair #3 comes into contact earlier and gradually increases its load share of the total transmitted load. Tooth pair #1 is out of contact and shares no load. Since the transmission error for each tooth pair should be equal, thus: 8
 3 3 3 3 > ¼ C F þ Sa j þ e j x > > < j
s  j j 2 2 2 xj ¼ C j F j þ ej > s > > 2 3 : F¼F þF j

ð9  a; b; cÞ

j

where Sa3j is the separation distance of tooth pair #3 during approach when tooth pair #2 contacts at point j. Solving these equations will give:
 8 >
 C 2j C 3j F þ Sa3j þ C 2j e3j þ C 3j e2j > > > xj ¼ > > > s C 2j þ C 3j > > > < 2 C j F þ Sa3j þ e32 2 j : Fj ¼ > > C 2j þ C 3j > > > > > C 2j F−Sa3j −e32 > 3 j > > Fj ¼ : C 2j þ C 3j

ð10  a; b; cÞ

However, under some circumstances, the introduction of tooth profile errors will prevent the corner contact of tooth pair #3 during approach and only the tooth pair #2 carries the load. Therefore, whenever a non-positive value is yielded for the load carried by the tooth pair #3 (i.e. F3j ≤ 0): 8
 2 2 > > < xj s ¼ C j F þ ej 2 Fj ¼ F > > : 3 Fj ¼ 0

ð11  a; b; cÞ

② Tooth pair #3 is not in corner contact, and tooth pair #1 is in corner contact. In this case, elastic deflection causes the tooth pair #1 to remain in contact even after the theoretical end point of engagement. Its load share gradually decreases to zero. Meanwhile, tooth pair #3 does not come into contact. Thus: 8
 1 1 1 1 > ¼ C F þ Sr j þ e j x > > < j
s  j j 2 2 2 xj ¼ C j F j þ ej > s > > 1 2 : F¼F þF j

j

ð12  a; b; cÞ

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

151

where Sr1j is the separation distance of tooth pair #1 during recessing when tooth pair #2 contacts at point j. Solving these equations will give:
 8 2 1 1 >
 C C F þ Sr þ C 1j e2j þ C 2j e1j > j j j > > xj ¼ > > 1 > s C j þ C 2j > > > < 2 C j F−Sr1j −e12 1 j : F ¼ j > > C 1j þ C 2j > > > 1 1 12 > > C j F þ Sr j þ e j > 2 > > Fj ¼ : 1 2 Cj þ Cj

ð13  a; b; cÞ

Similarly, whenever a non-positive value is yielded for the load carried by the tooth pair #1 (i.e. F1j ≤ 0): 8
 2 2 > > < xj s ¼ C j F þ ej 1 Fj ¼ 0 > > : 2 Fj ¼ F

ð14  a; b; cÞ

③ Tooth pair #3 and tooth pair #1 are both in corner contact. In this case, elastic deflection causes tooth pair #1 to remain in contact and tooth pair #3 to come into contact earlier. This is a triple contact zone where tooth pair #2 carries most of the load but tooth pair #1 and #3 are both in contact. Thus 8
 1 1 1 1 > ¼ C F þ Sr j þ e j x > > > j
s  j j > > 2 2 2 < xj ¼ C j F j þ ej :
 s 3 3 3 3 > > x j ¼ C j F j þ Sa j þ e j > > s > > : 1 2 3 F ¼ Fj þ Fj þ Fj

ð15  a; b; c; dÞ

Solving these equations will give:
 8 > C 2j  C 1j C 3j F þ C 3j Sr 1j þ C 1j Sa3j þ C 1j C 2j e3j þ C 2j C 3j e1j þ C 1j C 3j e2j > >
 > > xj ¼ > > s C 1j C 2j þ C 2j C 3j þ C 1j C 3j > >
 > > 2 3 3 12 2 3 1 2 3 > > C j C j F−C j e j −C 2j e13 > j − C j þ C j Sr j þ C j Sa j > 1 > > Fj ¼ < C 1j C 2j þ C 2j C 3j þ C 1j C 3j : 1 3 1 23 3 1 1 3 > > C j C j F−C 3j e21 > 2 j −C j e j þ C j Sr j þ C j Sa j > F ¼ > j > > C 1j C 2j þ C 2j C 3j þ C 1j C 3j > >
 > > 1 2 2 31 2 1 1 2 3 > > C j C j F−C j e j −C 1j e32 > j þ C j Sr j − C j þ C j Sa j > 3 > F ¼ > j : C 1j C 2j þ C 2j C 3j þ C 1j C 3j

ð16  a; b; c; dÞ

Similarly, whenever a non-positive value is yielded for the load carried by the tooth pair #1 (i.e. F1j ≤ 0), meaning tooth pair #1 is not in corner contact, then return to Case 1. Whenever a non-positive value is yielded for the load carried by the tooth pair #3

Table 1 Static transmission error with corner contact effect for a NCR spur gear pair. Tooth pair #1

Tooth pair #3

In corner contact

Not in corner contact

In corner contact

Not in corner contact

fC 2j

C 2j ðC 3j FþSa3j ÞþC 2j e3j þC 3j e2j

 ðC 1j C 3j F þ C 3j Sr 1j þ C 1j Sa3j Þ þC 1j C 2j e3j þ C 2j C 3j e1j þ C 1j C 3j e2j g C 1j C 2j þC 2j C 3j þC 1j C 3j

C 2j ðC 1j FþSr 1j ÞþC 1j e2j þC 2j e1j C 1j þC 2j

C 2j þC 3j

C2j F + e2j

152

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

(i.e. F3j ≤ 0), meaning tooth pair #3 is not in corner contact, then return to Case 2. However, when non-positive values are yielded for both tooth pair #1 and tooth pair #3 (i.e. F1j ≤ 0 and F3j ≤ 0): 8
 2 2 > xj ¼ C j F þ ej > > > s > < 1 Fj ¼ 0 : 2 > > Fj ¼ F > > > 3 : Fj ¼ 0

ð17  a; b; c; dÞ

Table 1 shows the formulae of the LSTE with corner contact effect (i.e. (xj)s) under various conditions. The unique way to determine whether a tooth pair is in corner contact is through its shared load calculated based on the corresponding equations mentioned above. Regarding the high contact ratio (CR N 2) gear pair, the loaded static transmission error and the static load shared by a tooth pair can be obtained by following the same procedure described above but will not be detailed here. Table 2 shows the formulae of static transmission error with the corner contact effect for a high contact ratio spur gear pair. The corner contact happens when the tooth pair #(n + 1) is just about to enter the mesh zone and the tooth pair #1 just leaves the mesh zone, where n is nearest integer that is larger than CR. 2.4. FVMS and VVMS The FVMS of the ith tooth pair KiFj at a contact position j, is directly defined as the reciprocal of its equivalent compliance: i

KFj ¼

1 C ij

ð18Þ

Therefore, as with the compliance, FVMS is determined by the gear macro-geometries and the material properties, and independent on the gear error and the static load applied. In other words, FVMS is “fixed” for a given gear tooth pair and totally independent of the gear micro-geometries and the load. The FVMS of the gear pair is defined as the sum of the FVMS of every tooth pair in mesh: KFj ¼

Xn i¼1

i

KFj :

ð19Þ

The VVMS of the gear KVj and the ith tooth pair KiVj at a contact position j are defined as:
 8 < K iV j ¼ F ij = x j
s : K ¼ F= x Vj j

ð20  a; bÞ

s

where (xj)S is the loaded static transmission error calculated in the previous section considering corner contact effects. Therefore, unlike FVMS, VVMS is determined not only by the gear macro-geometries and the material properties, but also by the gear microgeometries and the transmitted load. It should be noted that KFj, KVj and (xj)s are all position-dependent, and normally not constant along the contact positions. This indicates their time-varying characteristics. Hence, they can be also written as KF(t), KV(t) and xs(t).

Table 2 Static transmission error with the corner contact effect for a HCR spur gear pair. Tooth pair #1 In corner contact Tooth pair # (n + 1)

In corner contact

Not in corner contact nþ1

nþ1

n

fð∏p¼1 C pj ÞF þ ∑p¼1 ðð∏q¼1;q≠p C qj Þepj Þ nþ1 n þð∏q¼2 C qj ÞSr 1j þ ð∏q¼1 C qj ÞSanþ1 g j nþ1 nþ1 ∑p¼1 ð∏q¼1;q≠p C qj Þ

Not in corner contact

n

n

n

fð∏p¼1 C pj Þ F þ ∑p¼1 ðð∏q¼1;1≠p C qj Þepj Þ n þð∏q¼2 C qj ÞSr 1j g n

n

∑p¼1 ð∏q¼1;q≠p C qj Þ

nþ1

nþ1

nþ1

fð∏p¼2 C pj Þ F þ ∑p¼2 ðð∏q¼2;q≠p C qj Þepj Þ

n g þð∏q¼2 C qj ÞSanþ1 j nþ1 nþ1 ∑p¼2 ð∏q¼2;q≠p C qj Þ n n n ð∏p¼2 C pj Þ Fþ∑p¼2 ðð∏q¼2;q≠p C qj Þepj Þ n n ∑p¼2 ð∏q¼2;q≠p C qj Þ

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

153

2.5. Linear tip relief Intentionally modifying the gear tooth profiles has been proved theoretically and also experimentally to significantly affect the STE, dynamic transmission error (DTE), dynamic load and the gear operation life [1,2,5–8,16,20–23]. Proper profile modifications will not only reduce the occurrence of tip interference, but also greatly minimize the gear vibration and corresponding noise generated during operation and also increase the working life. Among all kinds of profile modification methods, gear tooth tip reliefs and root reliefs are the most commonly used modification strategies. They are basically an intentional removal of the material from the perfect involute profile, as shown in Fig. 2. Normally, the same amount and length of profile modification are applied to the tooth tip of both gear and pinion. Since modifying the root of a tooth is equivalent to modifying the tip of its counterpart, only tip relief will be considered in this study. In practice, modifying the root of a gear tooth will be much more difficult than modifying the tip especially for some extremely low contact ratio gears, making it preferable to apply only tip modification [20,21]. The conventional amount of tip relief, as stated in [20], is equal to the sum of the combined deflection evaluated at the HPSTC (This is for low contact ratio gear pairs with a contact ratio between 1 and 2. For high contact ratio gear pairs with contact ratio between 2 and 3, it should be evaluated at the highest point of the second double tooth contact region, HP2DTC) and twice the maximum spacing error, which is assumed to be zero in this study. The conventional amount of tip relief is chosen as a reference value in this study. The non-dimensioned modification amount is designed as Δ, and equals 1 for the conventional amount of tip relief. The conventional length of tip relief starts from the HPSTC (For HCR gear pairs, it should start at the HP2DTC) and ends at the tooth tip point. This value is set as the reference value in this study. The non-dimensioned modification length is designed as Ln. As a result, by varying Δ and Ln, one can define any amount and length of tip relief. The last key point that needs to be ascertained for tip relief is how it is going to apply at the tooth tip. In [22,23], three different types of profile modification are introduced. They are described as linear, parabolic I and parabolic II, as shown in Fig. 3. The linear modification has a linear tip relief trace on a profile chart. The parabolic I modification has a parabolic trace with zero slope at the start of the modification (tangent to the involute profile) in the profile chart, whereas the parabolic II modification has an infinite slope (vertical) at the end of the modification, i.e. the tooth tip point. In the following simulation, only linear tip relief will be considered. 2.6. Effects of toque and profile modifications on LSTE and VVMS Fig. 4 shows the variation of the VVMS, LSR and LSTE of a spur gear pair (as described in Table 3) in one theoretical mesh cycle under different torques when there are no gear profile errors (i.e. eij = 0). It should be noted that there will be no corner contact when the applied torque T = 0. In this case (T = 0 and eij = 0), the FVMS is same with VVMS. Compared with the FVMS (when T = 0), VVMS is load-dependent, and the abrupt change of mesh stiffness in the transition regions between single and double mesh zone is smoothed. Since the fluctuation of mesh stiffness can considerably affect the dynamics of gear transmission systems, one can expect that the corner contact effects may play a significant role in gear dynamics, especially for heavily-loaded gear transmission systems. The LSR of the three meshing tooth pairs (i.e. the middle subfigure of Fig. 4) shows, in the theoretical single mesh zone, there may be double tooth pairs in mesh, or even triple tooth pairs in mesh (when T = 340 Nm), since the corner contact admits the early engagement of the incoming tooth pair and delayed contact of the outgoing tooth pair. As a result, the actual tooth contact zone is extended, meaning the contact ratio is increased compared with the theoretical value. The variation of CR when the load increases from 0 to 340 Nm is shown in Fig. 5. It is obvious that the larger the transmitted load, the more the contact ratio increases. This means an increasing transmitted load will increase the average mesh stiffness (in terms of VVMS), and therefore, increase the natural frequency. This is another significant difference between the FVMS and VVMS models. Amount of tip reliefΔ Length of tip relief Ln HPSTC Pitch point Length of root relief Ln

LPSTC LPT Amount of root reliefΔ Fig. 2. Tip and root relief.

154

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

End of modification Modification

Parabolic II Parabolic I Amount of Modification Δ

Linear Start of modification

Roll angle Fig. 3. Three different types of tip relief.

Fig. 6 shows the variation of the VVMS, LSR (only for the middle tooth pair #2) and LSTE of a spur gear pair (as described in Table 3) in one theoretical mesh cycle with different amounts of Δ, but the same length (Ln = 1) of linear tip relief when the applied torques are 0 and 340 Nm respectively. By comparing Fig. 6(a) and (b), one can find that as the amount of tip relief increases, the corner contact effect is gradually reduced. When the tip relief amount exceeds a certain value (in this case Δ = 1), the corner contact effect can be completely avoided. Chen and Shao [31] point out that the existence of corner contact can be estimated by observing whether there is an abrupt ‘jump’ in the transmission regions of mesh stiffness. In most cases, gears are working under varying loads, which means a certain amount of tip relief may be sufficient to smooth the abrupt change of mesh stiffness in the transition region to avoid corner contact for a specific load, but may be insufficient for anther load. Therefore, corner contact effects should not be neglected even when a certain amount of tip relief is applied, and the dynamic analysis based on the load-independent mesh stiffness (i.e. FVMS) model may not be accurate to predict the gear dynamic behavior.

3. Three types of SDOF model Some important effects that should be included in the dynamic model for a gear pair are the time-varying mesh stiffness, the clearance (i.e. backlash) and the excitation due to the gear error. In this study, the friction force developed between the gear tooth

Outgoing tooth pair # 1

Middle tooth pair # 2

Incoming tooth pair # 3

Fig. 4. Variation of VVMS, LSR and LSTE under different torques without gear errors.

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

155

Table 3 Parameters of gear pairs from [15–17].

Teeth number Module (mm) Teeth width W (mm) Pressure angle α0 Young modulus E (N/mm2) Pitch diameter (mm) Root diameter (mm) Poisson's ratio μ Theoretical contact ratio Backlash (2b) on line of action (mm)

Gear (driven gear)

Pinion (driving gear)

50 3 20 20° 2.06 × 105 150 140.68 0.3 1.75 0.136

50 3 20 20° 2.06 × 105 150 140.68 0.3

mesh faces is not considered. However, to account for the frictionless model additional damping is introduced into the analysis [9–12], as shown in Fig. 7. In this situation the dynamic equations can be written as [46]: (

J 1 €θ1 ðt Þ ¼ T 1 −ð F ðt Þ þ Gðt ÞÞR1

ð21  a; bÞ

J2 € θ2 ðt Þ ¼ −T 2 þ ð F ðt Þ þ Gðt ÞÞR2

where F(t) and G(t) represent the elastic and damping forces during the contact, respectively; Ji, θi, Ti, Ri are the moment of inertia, angular displacement from the nominal position, the applied toque, and the radius of the base circle of gear i as shown in Fig. 7. 3.1. FVMS model In this model: 1) the gear error e(t) has no effect on mesh stiffness, and is treated as a displacement excitation, 2) corner contact effects are not considered and have no effect on mesh stiffness. Therefore, the elastic and damping forces can be represented as: 8 <

F ðt Þ ¼ hk Fðt Þðxðt Þ−eðtÞÞ Gðt Þ ¼ hc x ðt Þ− e ðt Þ : h ¼ ½ sgnðxðt Þ−eðt Þ−bÞ þ sgnðxðt Þ−eðt Þ þ bÞ=2 



ð22  a; b; cÞ

where x(t) = R1θ1(t) − R2θ2(t) is the dynamic transmission error; e(t) is the tooth profile error; h ∈ {1, 0, − 1} is the tooth contact function that determines drive-side contact (1), contact loss (0), or back-side contact (− 1); and b is the tooth backlash. Substituting Eq. (22-a, b, c) into Eq. (21-a, b) can reduce Eq. (21-a, b) to a single degree of freedom model: m€xðt Þ þ c x ðt Þ þ hk F ðt Þðxðt Þ−eðt ÞÞ ¼ f 0

ð23Þ



Fig. 5. Variation of contact ratio versus toque.

156

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

Fig. 6. Variation of VVMS, LSR and LSTE under different amounts of tip relief: (a) T = 0, (b) T = 340 Nm.

where the equivalent mass m and the static load f0 are defined as: m¼

J1 J2 T T ;f ¼ 1¼ 2 : J 1 R2 þ J 2 R1 0 R1 R2

ð24  a; bÞ

Fig. 7. Rotary model of a meshing spur gear pair.

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

157

The influence of gear error on the damping term is neglected as they have minimum effect on the response [9–12,17]. The time-varying mesh stiffness kF(t) and the gear error e(t) can both be considered as periodic and therefore can be expanded in Fourier series: k F ðt Þ ¼ k F0 þ

∞ X

k Fi cosði2π f m t þ φ Fi Þ;

eðt Þ ¼ e0 þ

i¼1

∞ X

ei cosði2π f m t þ θi Þ

ð25  a; bÞ

i¼1

where t is the time [s]; fm is the fundamental excitation frequency (mesh frequency) [Hz]; kF0 and e0 are the mean value of kF [N/m] and e [m]; kFi and φFi are the amplitude and phase of the ith component of kF(t); ei and θi are the amplitude and phase of the ith component of e(t). Eq. (23) can be rewritten in a non-dimensional form [1,5]: € ~ ~k ðτ Þð~xðτ Þ−~eðτ ÞÞ ¼ ~f ~xðτ Þ þ 2ζ ~x ðτÞ þ h F 0 

ð26Þ

~ ðτÞ, ẽ(τ), ~f and h ~ are the non-dimensional dynamic transmission where τ is the non-dimensional time; ζ is the damping ratio; ~xðτÞ, k F 0 error, mesh stiffness, gear error, load and contact functions. Moreover: 8 xðτ Þ eðτ Þ f > > ~xðτ Þ ¼ ; ~eðτ Þ ¼ ; ~f 0 ¼ 0 ; τ ¼ ωn t; > > k b b > F0 b > > > ~ ¼ ½ sgnð~xðτ Þ−~eðτ Þ−1Þ þ sgnð~xðτÞ−~eðτÞ þ 1Þ=2 > < h X∞ ~ ðτÞ ¼ k F ðτÞ ¼ 1 þ ~ cosðiΛτ þ φ Þ k k > F Fi > i¼1 Fi > k F0 > rffiffiffiffiffiffiffiffi > > > 2π f m > ~ ¼ k Fi ; ω ¼ k F0 ; ζ ¼ c ; > :Λ ¼ ; k Fi n ωn k F0 m 2mωn

ð27Þ

where Λ is the non-dimensional excitation frequency; ωn is the natural frequency of the system. FVMS model has been widely used in the literature [3,5,6,9–12]. Since FVMS is fixed for a given gear tooth pair, some researchers built the FVMS model based on the individual loads of each tooth pair in order to consider the partial contact loss [5,6]:
 i xðt Þ−e ðt Þ  i Gðt Þ ¼ ðt Þ−e ðt Þ > h

i > > : hi ¼ sgn xðt Þ−ei ðt Þ−b þ sgn xðt Þ−ei ðt Þ þ b =2 8 > > > <

F ðt Þ ¼

Xn

i i h k F ðt Þ i¼1
X n i hc x i¼1 



ð28  a; b; cÞ

where kiF(t), ei(t) and hi are the individual FVMS, tooth error and contact function for the ith tooth pair in mesh respectively. n is the maximum number of tooth pair in mesh. 3.2. VVMS model Some researchers have used the VVMS of gear pair to analyze gear dynamics, especially for some FE models [1,2,20–23,33] used for dynamic analysis. The gear error e(t), instead of being treated as displacement excitation, was used to evaluate the mesh stiffness of the gear pair under the static analysis. Besides, corner contact effects are normally incorporated in the calculation of mesh stiffness. According to Eq. (20-a, b), the elastic and damping forces can be represented as: 8 <

F ðt Þ ¼ hkV ðt Þxðt Þ : Gðt Þ ¼ hc x ðt Þ : h ¼ ½sgnðxðt Þ−eðt ÞÞ þ sgnðxðt Þ−eðt ÞÞ=2 

ð29  a; b; cÞ

Still, if we follow the same procedure described in the previous section, we can get the non-dimensional form: € ~ ~k ðτ Þ~xðτ Þ ¼ ~f : ~xðτ Þ þ 2ζ ~x ðτÞ þ h V 0 

ð30Þ

The VVMS model has been recognized or initiated to some degree in [1,2,20–23,33]. Kasuba and Evan [35] developed a large scale digitized extending gear modeling including the VVMS to analyze spur gearing dynamics. Compared with their approach, the proposed VVMS model is much more concise and easy to solve analytically since it averts the coupling of parametric excitation and displacement excitation. In fact, a generalized solution methodology was introduced in [10] based on the harmonic balance method and Newton–Raphson procedure to analytically solve the differential equation with combined parametric excitation and clearance non-linearity. Another significant merit of the VVMS model over the FMVS model is that the corner contact effects can be easily incorporated in the dynamic analysis through the VVMS. However, since the VVMS is dependent on gear microgeometries and transmitted load, the dynamic tooth load division between the individual tooth pair in mesh is neglected, meaning partial contact loss cannot be simulated through this model.

158

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

3.3. LSTE model Both FVMS equations and VVMS equations incorporate the time-varying meshing stiffness acting as a parametric excitation to the gear system. Ozguven [38] and Cai [39] both introduced a linear approximated equation, in which a constant mesh stiffness is assumed. However, the self-excitation effect of mesh stiffness variation is indirectly included into this approximate equation by using the loaded static transmission error as the input. The non-dimensional form of this model is: €xðτ Þ þ 2ζ x ðτÞ þ hxðτ Þ ¼ hxs ðτÞ : 

ð31Þ

In the LSTE model, a constant mesh stiffness assumption with a displacement excitation is used to represent the basic characteristic of actual mesh stiffness. Although they are derived based on the deletion of the alternating component of mesh stiffness, the self-excitation effect of time-varying mesh stiffness is included in the analysis with approximate terms. It has been proved that, under some circumstances, this approximate method based on LSTE can show a good agreement with that including time-varying mesh stiffness [38]. Besides, corner contact effects can be easily incorporated (see Section 2.3.2). However, the LSTE model includes the fluctuating mesh stiffness only indirectly and also neglects the dynamic tooth load division between the individual tooth pair and partial contact loss [5]. 4. Verification with the experimental results Verification of the analytical method to determine corner contact effects with profile modifications introduced in Section 2, and the proposed SDOF models in Section 3 is given in this section by comparing the simulation results against experimental results provided in the literature. Kahraman and Blankenship have conducted a series of experiments to investigate the dynamics of a gear pair with backlash clearance, parametric and internal displacement excitation due to gear manufacturing errors [15–17]. Their experimental studies guided many modeling efforts and have been used extensively for modal verification. Different sets of spur test gear pair are considered in the experimental study representing different modification parameters and involute contact ratios. However, some common parameters of the spur gear sets are described in Table 3. Their tests were conducted over a speed range from 600 to 4100 rpm which corresponds to a gear mesh frequency fm from 500 to 3400 Hz. Applied torque T was varied from 0 to 340 Nm. The measured DTE values are given in terms of the RMS (root mean qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 square) Arms ¼ ∑i¼1 A1 , which include only the first three gear mesh harmonic amplitudes. 4.1. Corner contact effects In this section, we will research whether corner contact effects affect gear dynamics by comparing the dynamic response predicted by the proposed models to that of the measured value for an unmodified spur gear pair. In Fig. 8, the experimental measured Arms for a gear pair with no modification (e = 0) at T = 340 Nm [17], are compared to predictions of the three proposed models. For the VVMS and LSTE model, a damping ratio of 0.02 is used as indicated in [17] through the experimental investigation. However, for the FVMS model, such a low damping ratio will yield such an intense

Fig. 8. Comparison of measured [17] and predicted Arms versus speed for an unmodified gear pair at 340 Nm.

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

159

response that the double-sided impact (DSI) solution will be obtained in the primary resonance region, which did not happen in the experiment. In order to be consistent with the measured value, Liu and Parker [4] suggested a damping ratio of 0.07 for the load-independent mesh stiffness model (i.e. FVMS model). The primary resonance, super-harmonic resonances, softening non-linearity and the classical jump phenomena near resonance regions that appeared in the experimental measured response are also accurately predicted by the proposed models, as shown in Fig. 8. However, it seems that dynamic response results predicted by the VVMS and LSTE model are more consistent with the experimental measured results compared with those of the FVMS model. This means that the load-dependent mesh stiffness model appears more representative of the physical system, and the corner contact effects should not be neglected in the gear dynamic analysis. As a further comparison, Fig. 9(a) from [14] presents the experimental measured Arms versus fm for an unmodified gear pair under three different torques T = 100, 200, and 300 Nm. The amplitude of the resonance response is noticeably larger for increasing torque. In addition, it appears that the jump frequencies are load-dependent. There is a nearly 20% difference in natural frequency when the toque changes from 100 Nm to 300 Nm in the experiment. Fig. 9(b), (c) and (d) show the predicted responses from VVMS, LSTE and FVMS model respectively. Compared with the FVMS model, the LSTE and VVMS models agree well with the experimental results in terms of not only the response amplitude but also the load-dependent jump frequencies. One exception is that the difference in natural frequencies between the 100 Nm and 300 Nm torque is not as obvious as the experimental result. However, the load-dependent mesh stiffness model is clearly more appropriate to capture the gear dynamic behavior than the load-independent mesh stiffness model, which again shows that corner contact effects should not be neglected in the dynamic analysis. This is contrary to the conclusion drawn in [4] which stated that the expectedly more accurate load-dependent stiffness model yields conflicting results with the experiments. The reason is because Liu and Parker [4] used a universal damping ratio of 0.07 in their discrete SDOF models which was too large for the load-independent mesh stiffness model and that the non-linearity behavior of the gear pair working at high loads was suppressed.

(a) Experiment

(b) VVMS model

(c) LSTE model

(d) FVMS model

Fig. 9. Experimentally measured [14] and predicted Arms versus fm for an unmodified gear under three different torques: (+) 100, (◊) 200 and (•) 300 Nm.

160

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

4.2. The effect of tip relief Fig. 10 shows the comparison of the experimentally measured A1 (the first harmonic response) from [16] and the predicted A1 over the non-dimensional frequency range 0.6 b Λ b 1.2 (this is the frequency range in which A1 dominates the dynamic response) for two tip relieved gear pairs with the same tip relief amount δ = 10 μm (Δ = 0.62) but different tip relief starting angles, α = 22.2° and 20.9° (Ln = 0.92 and 1.15). The reason that the comparisons are made based on the non-dimensional frequency Λ is because the natural frequencies of the gear pairs are dependent on the tip relief and applied torque. Besides, in order to provide valid comparisons among the three SDOF models, the same damping ratio of 0.02 is used for all the discrete SDOF models. Compared with the response for the unmodified gear pair (Fig. 9), the introduction of tip relief can substantially decrease the dynamic response amplitude especially in the primary resonance region when α = 20.9°. The VVMS and LSTE model can relatively accurately capture the experimentally measured results, whereas the FVMS model presents a much more intense softening nonlinearity than that of the experimental results. A number of computational studies have been done and it was found that the corner contact effect is the main factor leading to this discrepancy. This proves that the corner contact effects are still in play, and therefore should not be neglected even though a certain amount of the tip relief is applied. Besides, the proposed analytical method to calculate the LSTE and VVMS considering corner contact effects with tip relief is valid and can provide relatively consistent results compared with the experimental results. In addition, it seems that the dynamic response predicted by the VVMS model is very close to that of the LSTE model in most cases. A detailed analysis will be provided in the next section.

(a) α = 22.2 degrees

(b) α = 20.9 degrees Fig. 10. Experimentally measured [16] and predicted A1 versus Λ for two modified gear pairs with δ = 10 μm and α = 22° and 20.9° at 340 Nm.

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

161

5. Comparisons and discussion In this section, detailed comparisons of the simulated steady-state responses of the gear pair described in Table 3 with varying amounts of linear tip relief by using the proposed 3 SDOF models will be provided. The advantages and disadvantages of each model will be discussed.

Fig. 11. Simulated Arms versus Λ for various amounts of tip relief with Ln = 1 using 3 models: (a) Δ = 0 (b) Δ = 0.2 (c) Δ = 0.5 (d) Δ = 0.8(e) Δ = 1 (f) Δ = 1.2.

162

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

5.1. Comparisons of the 3 SDOF models In order to make valid dynamic performance comparisons among these models, all the response curves in this section are based on non-dimensionless frequency Λ. Fig. 11 shows the steady-state response in terms of Arms of the DTE versus Λ with Δ = 0, 0.2, 0.5, 0.8, 1 and 1.2 (i.e δ = 0, 3.2, 8.0, 12.9, 16 and 19.2 μm) when Ln = 1 (i.e. α = 21.7°) at T = 340 Nm. The corresponding mesh stiffness curves for each modification amount have already been shown in Fig. 6(a). Three interesting phenomena can be noticed. 1) The introduction of linear tip relief can significantly affect the dynamic behavior of the gear pair. In fact, a proper modification can significantly minimize the DTE fluctuation of the gear pair working at a specific design load, and this point has been proved theoretically and also experimentally in many previous published papers. 2) As the amount of relief increases, the corner contact effect is gradually alleviated (as shown in Fig. 6(b)) resulting in the closer and closer agreement between the load-independent mesh stiffness model and the load-dependent mesh stiffness model especially in the non-resonance region. When Δ = 1, there will be no corner contact effect. However, in Fig. 11(e), there still exist some discrepancies between the FVMS model and VVMS model in the resonance region, which is mainly due to different treatments of tip relief in these two models. 3) Steady-state responses of the VVMS model become closer and closer to those of the LSTE model. Especially when Δ = 1, the amplitudes of the responses given by these two models are entirely the same at each non-dimensioned excitation frequency Λ . In fact, a simple analytical calculation can prove that the VVMS equation (Eq. (30)) is approximately equivalent to the LSTE equation (Eq. (31)) as long as the fluctuation of LSTE (or VVMS) is comparatively small. 5.2. Discussion From the comparisons made above, it is obvious that corner contact is still in play when an insufficient amount of tip relief is applied. Therefore, the FVMS model that uses a load-independent mesh stiffness curve may yield inconsistent results compared with those of the VVMS model and LSTE model, especially in the non-resonance region. As the amount of tip relief increases, the corner contact effect is gradually reduced, and the dynamic response predicted by the load-independent mesh stiffness model moves closer to that of the load-dependent mesh stiffness model, which justifies the employing of a loadindependent mesh stiff model in the dynamic analysis for the multi-mesh gear set [5] and the planetary gear set [6] with sufficient amount of profile modifications. However, due to the different treatments of the tooth profile errors, there are still some discrepancies between the FVMS model and VVMS model especially in the resonance regions. The FVMS model presents a much softer nonlinearity in these regions than the VVMS model. One of the biggest advantages of the FVMS model is that it allows for the partial contact loss by dividing the dynamic tooth load into individual loads for each tooth pair in mesh as shown in Eq. (28-a, b, c), which can be called the ITMS (individual tooth mesh stiffness) model. Liu and Parker [5] found that this model best agrees with the FE benchmark for dynamic predictions regardless of different loads, profile modifications and bearings. It has been shown in this paper that the LSTE model, which is initially proposed by Ozguven and Houser [36], is approximately equivalent to the VVMS model proposed in this paper as long as the fluctuation of the LSTE (or VVMS) is comparatively small. The LSTE model can easily incorporate corner contact effects into analysis based on the analytical method proposed in this paper, which simplifies the problem and provides reasonable estimates of the gear dynamics under certain conditions [5,38]. However, it neglects the dynamic tooth load division between individual tooth pair in mesh. Besides, since a constant mesh stiffness is used in this model, the Mathieu-Hill type stability characteristic of the system cannot be studied. The VVMS model proposed in this paper can include corner contact effects into the analysis, and shows a good agreement with the experimental results. Unlike the FVMS model that treats the tooth profile error as a displacement excitation to the system, the VVMS model includes the tooth profile error in the VVMS during the static analysis. However, like the LSTE model, the dynamic tooth load division between the individual tooth pair in mesh is neglected. Probably, those are the reasons why the dynamic response predicted by the VVMS model still shows some discrepancies with the experimental measured results. 6. Conclusions This paper proposes an analytical method to calculate the LSTE considering corner contact effects for the spur gear pair with tip relief, based on which the effect of the corner contact on the dynamic response of the spur gear pair has been studied. Two types of the gear mesh stiffness model used in the literature have been differentiated, and three types of commonly used SDOF models (the FVMS model, the VVMS model and the LSTE model) are generalized and their corresponding non-dimensional governing equations are given. Comparisons of the predicted response from the proposed three models with experimentally measured results provided in literature were made. It was found that the load-dependent mesh stiffness model yields more consistent results, which proves that corner contact effects should not be neglected in the gear dynamic analysis when no or an insufficient amount of profile modification is applied. Besides, the proposed analytical method used to calculate the LSTE and VVMS considering corner contact effects with tip relief is valid and can provide relatively consistent results compared with experimental results. Detailed comparisons of the steady state responses predicted by these three types of SDOF models were made by

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

163

introducing different amounts of linear tip relief into the analysis, and the advantages and disadvantages of each model were generalized which can be concluded as: 1) The FVMS model disregards the corner contact effect. However, this model can yield consistent results when a sufficient amount of tip relief is applied. Besides, this model allows for load division between the individual tooth pair in mesh. Therefore, partial contact loss can be simulated through this analysis. 2) The LSTE model is approximately equivalent to the VVMS model as long as the fluctuation of the LSTE (or VVMS) is comparatively small. Plus, this model can easily incorporate corner contact effects into the analysis. However, partial contact loss and Mathieu-Hill type stability analysis cannot be studied using this model. 3) The VVMS model proposed in this paper can include corner contact effects into the analysis, and shows good agreement with experimental results. However, the dynamic tooth load division between individual tooth pair in mesh is neglected, which may explain some discrepancies between the predicted results from this model and experimental results. It should be noted that even though this study mainly focuses on the spur gear pair, the conclusions also apply to the other types of gears. The corner contact effect should not be neglected in the gear dynamic analysis when no or insufficient tip relief is applied especially under heavily-loaded conditions. Acknowledgments The authors acknowledge the financial support of the China Scholarship Council (201306050004) and are grateful for the facility resources and support provided by the Natural Sciences and Engineering Research Council of Canada (203023-06). References [1] M. Faggioni, F.S. Samani, G. Bertacchi, F. Pellicano, Dynamic optimization of spur gears, Gear Technol. 46 (2011) 544–557. [2] G. Bonori, F. Pellicano, Non-smooth dynamics of spur gears with manufacturing errors, J. Sound Vib. 306 (2007) 271–283. [3] V.K. Tamminana, A. Kahraman, S. Vijayakar, A study of the relationship between the dynamic factors and the dynamic transmission error of spur gear pairs, J. Mech. Des. 129 (2007) 75–84. [4] R.G. Parker, S.M. Vijayakar, T. Imajo, Non-linear dynamic response of a spur gear pair: modeling and experimental comparisons, J. Sound Vib. 237 (2000) 435–455. [5] G. Liu, R.G. Parker, Dynamic modeling and analysis of tooth profile modification of multi-mesh gear vibration, J. Mech. Des. 130 (2008) 1214021–12140213. [6] C. Bahk, R.G. Parker, Analytical investigation of tooth profile modification effects on planetary gear dynamics, Mech. Mach. Theory 70 (2013) 298–319. [7] J.D. Wang, Numerical and experimental analysis of spur gears in mesh(Ph. D thesis) Curtin's University of technology, Perth, Australia, 2003. [8] J.D. Wang, I. Howard, Finite element analysis of high contact ratio spur gears in mesh, J. Tribol. 127 (2005) 469–483. [9] A. Kahraman, R. Singh, Interactions between time-varying mesh stiffness and clearance non-linarites in a geared system, J. Sound Vib. 146 (1991) 135–156. [10] G.W. Blankenship, A. Kahraman, Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity, J. Sound Vib. 185 (1995) 743–765. [11] A. Kahraman, G.W. Blankenship, Interactions between commensurate parametric and forcing excitations in a system with clearance, J. Sound Vib. 194 (1996) 317–336. [12] A. Kahraman, R. Singh, Nonlinear dynamics of a spur gear pair, J. Sound Vib. 142 (1990) 49–75. [13] R.J. Comparin, R. Singh, Non-linear frequency response characteristics of an impact pair, J. Sound Vib. 134 (1989) 259–290. [14] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, Part-I: characterization of forced response, ASME Power Transmission and Gearing Conference, San Diego, 1996. [15] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, Part-II: effect of involute contact ratio, ASME Power Transmission and Gearing Conference, San Diego, 1996. [16] A. Kahraman, G.W. Blankenship, Gear dynamics experiments, Part-III: effect of involute tip relief, ASME Power Transmission and Gearing Conference, San Diego, 1996. [17] A. Kahraman, G.W. Blankenship, Experiments on nonlinear dynamic behavior of an oscillator with clearance and periodically time-varying parameters, J. Appl. Mech. 64 (1997) 217–226. [18] H.N. Ozguven, D.R. Houser, Mathematical models used in gear dynamics—a review, J. Sound Vib. 121 (1987) 383–411. [19] J.J. Wang, R.F. Li, X.H. Peng, Survey of nonlinear vibration of gear transmission systems, Appl. Mech. Rev. 56 (2003) 309–329. [20] H.H. Lin, D.P. Townsend, F.B. Oswald, Profile modification to minimize spur gear dynamic loading, NASA Technical Memorandum1988. [21] H.H. Lin, F.B. Oswald, D.P. Townsend, Dynamic loading of spur gears with linear and parabolic tooth profile modifications, Mech. Mach. Theory 29 (1994) 1115–1129. [22] G. Padmasolala, H.H. Lin, F.B. Oswald, Influence of tooth spacing error on gears with and without profile modifications, NASA Technical Memorandum2000. [23] H.H. Lin, J. Liu, Effect of tooth profile modifications on the dynamics of high contact ratio gears with different tooth spacing errors, Proceeding of 2006 ASME International Mechanical Engineering Congress and Exposition, IMECE2006 – Design Engineering, 2006. [24] H.H. Lin, J.F. Wang, F.B. Oswald, J. Coy, Effect of extended tooth contact on the modeling of spur gear transmissions, Gear Technol. 11 (1994) 18–25. [25] R.W. Gregory, S.L. Harris, R.G. Munro, Dynamic behavior of spur gears, Inst. Mech. Eng. Proc. 178 (1963) 207–226. [26] D.C.H. Yang, J.Y. Lin, Hertzain damping, tooth friction and bending elasticity in gear impact dynamics, J. Mech. Transm. Autom. Des. 109 (1987) 189–196. [27] K.J. Huang, M.R. Wu, J.T. Tseng, Dynamic analyses of gear pairs incorporating the effect of time-varying lubrication damping, J. Vib. Control. 17 (2011) 353–363. [28] K. Umezawa, T. Sato, J. Ishikawa, Simulation of rotational vibration of spur gears, Bull. JSME 27 (1984) 102–109. [29] B. Rebbechi, F.B. Oswald, D.P. Townsend, Measurement of gear tooth dynamic friction, NASA Technical Memorandum1996. [30] S. He, S.M. Cho, R. Singh, Prediction of dynamic friction forces in spur gears using alternate sliding friction formulations, J. Sound Vib. 309 (2007) 843–851. [31] R. Guilbault, S. Lalonde, M. Thomas, Nonlinear damping calculation in cylindrical gear dynamic modeling, J. Sound Vib. 331 (2012) 2110–2128. [32] K. Huang, M. Wu, J. Tseng, Dynamic analyses of gear pairs incorporating the effect of time-varying lubrication damping, J. Vib. Control. 17 (3) (2010) 355–363. [33] Z. Chen, Y. Shao, Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack, Mech. Mach. Theory 62 (2013) 63–74. [34] L. Vedmar, B. Henriksson, General approach for determining dynamic forces in spur gears, J. Mech. Des. 120 (1998) 593–598. [35] R. Kasuba, J.W. Evans, An extended model for determining dynamic loads in spur gearing, J. Mech. Des. 103 (1981) 398–409. [36] R.G. Munro, L. Morrish, D. Palmer, Gear transmission error outside the normal path of contact due to corner and top contact, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 213 (4) (1999) 319–400. [37] D.L. Seager, Separation of gear teeth in approach and recess, and the likelihood of corner contact, ASLE Trans. 19 (2) (1976) 164–170. [38] H.N. Ozguven, D.R. Houser, Dynamic analysis of high speed gears by using loaded static transmission error, J. Sound Vib. 125 (1988) 71–88.

164

W. Yu, C.K. Mechefske / Mechanism and Machine Theory 96 (2016) 146–164

[39] Y. Cai, T. Hayashi, The linear approximated equation of vibration of a pair of spur gears (theory and experiment), J. Mech. Des. 116 (1994) 558–564. [40] C. Weber, The deformation of loaded gears and the effect on their load carrying capacity, sponsored research (Germany), British Dept. of Sci. and Ind. Res. Report No. 3, 1949. [41] R.W. Cornell, Compliance and stress sensitivity of spur gear teeth, ASME J. Mech. Des. 103 (1981) 447–459. [42] Z. Chen, Y. Shao, Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth, Eng. Fail. Anal. 18 (2011) 2149–2164. [43] P. Sainsot, P. Velex, Contribution of gear body to tooth deflections—a new bi-dimensional analytical formula, ASME J. Mech. Des. 126 (2004) 748–752. [44] W. Yu, Y. Shao, C.K. Mechefske, The effects of spur gear tooth spatial crack propagation on gear mesh stiffness, Eng. Fail. Anal. 54 (2015) 103–119. [45] M. Amabili, A. Rivola, Dynamic analysis of spur gear pairs: steady-state response and stability of the SDOF model with time-varying meshing damping, Mech. Syst. Signal Process. 11 (1997) 375–390. [46] D.C.H. Yang, Z.S. Sun, Rotary model for spur gear dynamics ASME (Paper), 107, 1985 529–535.