Improved analytical calculation model of spur gear mesh excitations with tooth profile deviations

Mechanism and Machine Theory 149 (2020) 103838

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Research paper

Improved analytical calculation model of spur gear mesh excitations with tooth profile deviations Zaigang Chen∗, Ziwei Zhou, Wanming Zhai, Kaiyun Wang State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, 610031, PR China

a r t i c l e

i n f o

Article history: Received 28 November 2019 Revised 31 January 2020 Accepted 4 February 2020

Keywords: Mesh stiffness Coupling effect Gear transmission Teeth contact Tooth error Tooth profile modification

a b s t r a c t Time-varying mesh stiffness and gear transmission errors are the major excitations to gear dynamics system. How to calculate them more accurately and efficiently has attracted much attention worldwide for a long period. Especially consideration of the gear body structure coupling effect and the tooth profile error will make interactions between the tooth pairs in mesh more complicated so that traditional analytical methods could not applicable any more. In this paper, a comprehensive and general analytical gear mesh model is proposed by considering all the deformations including the teeth deformation, teeth contact deformation, fillet-foundation deformation, and gear body structure coupling effect, as well as tooth profile deviations. The calculation formulas for the mesh stiffness, load sharing ratio and static transmission error are derived, and the corresponding efficient solving method is also proposed. Based on the proposed model, the influencing mechanism of the gear body structure coupling effect on the total mesh stiffness is revealed. The proposed mesh stiffness calculation model is proven to be more general and comprehensive since the traditional analytical models are the special cases of the proposed model. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Gear transmission is a key component for delivery of motions and powers in mechanical systems and has been widely applied in different areas such as railway locomotive, automotive, ship, wind turbine, helicopter [1,2]. As the promptly increasing concerns on the improvement of gear product quality in recent decades, the vibration and noise problem of precision gear transmission systems have drawn great attentions, especially for the high-speed and/or heavy load conditions. As the two major dynamic excitations to gear dynamic system, time-varying mesh stiffness and gear transmission error are well-known and have been investigated by numerous researchers since the middle of the last century after the first systematic studies in gear dynamics performed in 1920s [3]. To predict the time-varying mesh stiffness excitation more accurately is the eternal theme that the researchers have been seeking for since it is the basic foundation for a reasonable gear dynamic performance analysis. Hence, there are many methods to obtain the gear mesh stiffness, such as experimental method [4,5], finite element method (FEM) [6–11], analytical-FEM hybrid method [12,13], and analytical method [14–30]. The experimental method and the FEM are usually regarded as the most accurate techniques, and the obtained results were used to verify and validate the results calculated by the analytical and/or the analytical-FEM hybrid methods. However, high cost and low efficiency are the major limitations to these methods, namely the experimental method and the FEM, although they have the inherent capability to cover as ∗

Corresponding author: E-mail address: [email protected] (Z. Chen).

https://doi.org/10.1016/j.mechmachtheory.2020.103838 0094-114X/© 2020 Elsevier Ltd. All rights reserved.

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Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

Nomenclature m

module of gear pressure angle of gear pair contact ratio dc operating center distance rint radius of hub bore z number of Teeth W teeth width ha ∗ addendum coefficient c∗ tip clearance coefficient x profile shift coefficient Dp pitch diameter ha addendum height hf dedendum height E Young modulus of the material G shear modulus of the material ρ density of the material ν Poisson’s ratio  gap between the gear teeth to be in mesh δ gear tooth deformation Tp1 /Tp2 torque applied to the driving gear Tg1 /Tg2 torque applied to the driven gear T0 torque applied to the gear pair in the case study F1 /F2 teeth contact force of 1st/2nd tooth pair F total gear mesh force h( t) switch between single- (h(t) = 0) and double-tooth (h(t) = 1) mesh regions Kf ji the equivalent stiffness due to the gear body structure coupling effect Kpt / Kgt tooth stiffness of pinion/gear Kpf / Kgf tooth fillet-foundation stiffness of pinion/gear Kh contact stiffness of tooth pair e tooth profile deviations Kb /Ks /Ka bending/shear/axial compressive stiffness of a gear tooth d effective tooth length for integration h half of the tooth thickness at the position that the mesh force is applied to α1 angle between the mesh force and the direction perpendicular to the tooth center line Ix /Ax the area moment of inertia and the area of the section that has a distance x away from the acting point of the applied force along the tooth center line hx half of the tooth thickness Ch /κ gear teeth contact coefficient L/M/P/Q coefficients used for tooth fillet-foundation stiffness calculation in Eq. (5) Mi /Qi /Ti /Ui /Li /Pi /Ri /Si /Vi coefficients used for equivalent stiffness calculation of gear body structure coupling effect in Eqs. (8) and (9) Ai /Bi /Ci /Di /Ei /Fi /Gi /Hi /Ii coefficients for calculating tooth fillet-foundation stiffness or equivalent stiffness of gear body structure coupling effect whose values can be found in Tables A.1–A.3 in the Appendix δ t /δ f /δ h /δ ij deformation of tooth/fillet-foundation/tooth-contact/gear body structure coupling effect Z the integer set N number of tooth pairs in contact Rbp radius of pinion base circle Rf radius of gear dedendum circle Lsr load sharing ratio of gear pair E˜12 general tooth error function Np total number of the mesh excitation calculation steps θ rotational angle of the pinion θ total total rotational angle of the pinion for mesh excitation calculation θ rotational angle interval for mesh excitation calculation n counter for mesh excitation calculation Ca amplitude of tooth tip relief L a length of tooth tip relief

α0 ε

Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

Cn L n Aj

ϕj

Ne

θ max LOA NSTE LSTE floor(x) mod(x,y) FEA FEM TR SCE TPE

3

normalized amplitude of tooth tip relief normalized length of tooth tip relief amplitude of the jth harmonic of the tooth profile error phase angle of the jth harmonic of the tooth profile error total number of the harmonics of the tooth profile error gear rotational angle corresponding to the mesh point moving in the theoretical segment along LOA line of action non-loaded static transmission error loaded static transmission error function for round x toward its negative infinity function for calculating modulus after division, it equals to x-floor(x/y)∗ y finite element analysis finite element method tip relief structure coupling effect tooth profile error

many influencing factors as possible. Besides, FEM needs extremely high mesh refinement so as to capture the nonlinear teeth contact property and approximate the tiny tooth profile errors at the grade of micrometer, thus resulting in a very huge model scale that is very computationally expensive. As an alternative approach, the analytical method has been widely concerned worldwide since the middle of the last century due to its high computational efficiency with acceptable agreement with the results of FEM in the calculation of gear mesh stiffness. In 1949, Weber [14] proposed a complicated analytical method to calculate the spur gear tooth deflections by regarding the tooth as a non-uniform cantilever beam, where the nonlinear teeth contact, tooth deformation, and deformation of gear body represented by a semi-infinite elastic plate were considered. And then, this method was widely used and extended by other scholars. For example, Sainsot et al. [15] proposed a new bi-dimensional analytical formula for gear body deformation calculation by accounting for actual solid disk wheels rather than the assumption of semi-infinite elastic plate, based on which Chen et al. [16] further extended this method to cover the effect of tooth root crack on the tooth deflection due to gear body deformation. And then Chen et al. [17] applied this improved fillet-foundation stiffness to the dynamic investigation of locomotive gear transmission with tooth root crack. Chaari et al. [18] proposed a method to calculate the time-varying mesh stiffness in presence of gear tooth root crack based on the Weber’s method [14], and the method was validated by the results from the FEM. And in later 1980s, Yang and Lin [19] proposed an analytical method for gear mesh stiffness calculation based on the potential energy principle. And this method was extended by Wu et al. [20] by considering the tooth shear deflection. Based on this method, Chen and Shao [21] further proposed an analytical model for gear mesh stiffness in presence of a tooth root crack that was non-uniformly distributed along tooth width, where the tooth was fixed to the dedendum circle and the gear body deformation was considered. Then, this method has been widely used in the gear mesh stiffness calculation, such as Brethee et al. [22], Mohammed et al. [23,24], Ma et al. [25], Wang et al. [26]. And Yu et al. [27] also used this theory to calculate the time-varying mesh stiffness of helical gears by slicing the helical gear into many slices, of which each sliced thin gear was regarded as a spur gear. Then, Liang et al. [28] derived the gear tooth stiffness components in the analytical form which was convenient for the mesh stiffness calculation in presence of tooth crack fault. And later, Chen and Shao [29] proposed a general analytical mesh stiffness model by considering the gear tooth errors, where the analytical relationship between the gear tooth errors and the gear mesh parameters were derived. This method was also applied by Ma et al. [30] in calculating the mesh stiffness of the profile shifted gears, and validated by comparing with FEM. Zhang et al. [31], He et al. [32], Jiang et al. [33] applied this method to calculate the time-varying gear mesh stiffness for dynamic analysis of railway locomotives. And Wei et al. [34] extended this method to the mesh stiffness calculation of helical gear transmission. The similar idea was also used by Cao et al. [35] in the gear dynamic simulation with eccentricity. Recently, Sánchez et al. [36] studied the influence of tooth profile modification on the mesh stiffness, load sharing and transmission error based on a previously developed approximate equation, and the desired tooth profile modification could be obtained. Yi et al. [37] performed nonlinear gear dynamic investigation considering the effect of time-varying pressure angle and gear backlash and the mesh stiffness was calculated using the potential energy principle. In these works including both the analytical methods and the FEM with single-tooth model, the tooth deflection due to the neighboring loaded tooth was not considered for a long period, which will make a considerable overestimation to the multi-tooth mesh stiffness and the practical contact ratio. This may be caused by the intuitional thought that the total mesh stiffness is the direct superposition of the individual tooth pairs in mesh. Actually, as early as in 1960s, Attia [38] derived the calculation formula for the spur gear tooth deflection when its neighboring tooth was loaded, where only the thin gear rim was considered. In recent years, the considerable reduction of the double-tooth mesh stiffness due to the extra deflection of the tooth caused by the loaded neighboring tooth through the gear body elastic deformation was observed in some

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Fig. 1. Multi-tooth contact pattern of a gear pair in presence of tooth deviation.

literatures [6–13]. This is mainly attributed to the methods or models that could automatically include this coupling effect from the neighboring loaded tooth, such as the experimental test [6], FEM [6–11], and analytical-FEM [12,13]. However, the efficient analytical method is still very limited due to the complexity of the gear body deformation. Based on the traditional analytical methods with potential energy principle used in many literatures [21,23–25,29], Ma et al. [7] introduced a so-called “correction coefficient” to revise the fillet-foundation stiffness so as to solve the problem that the traditional methods with direct summation of the mesh stiffness from the individual tooth pairs will overestimate the double-tooth mesh stiffness due to the repeated consideration of the gear body stiffness. However, this “correction coefficient” needs to be obtained by FEM due to the lack of existing analytical method to calculate it. The similar ideal had been also used by Wang [10] and Kiekbusch et al. [11]. And recently, Xie et al. [39] also used the FEM to calculate the “correction coefficient” from the perspective of gear body structure coupling effect, and improved the load sharing ratio calculation based on the minimum potential energy principle, based on which they [40] further derived the analytical stiffness calculation formula to reveal the tooth posture variations due to the gear body flexible deformation based on the work done by Sainsot et al. [15]. These works mentioned above are effective in improving the calculation accuracy of double-tooth mesh stiffness by considering the effect of gear body flexibility. Especially, this analytical method proposed in [40] is a good method to calculate the double-tooth mesh stiffness due to its accuracy and efficiency, however, the gear tooth profile deviations were not considered in their work. As is known that gear tooth profile deviations are inevitable due to the intentional tooth profile modification, the undesired manufacturing or assembly errors, and/or the tooth profile defect during operation. Presence of the gear tooth profile deviation will complicate the gear teeth interaction, especially when the gear body structure coupling effect and nonlinear teeth contact are considered. For this condition, the existing analytical models could not applicable directly any more. To solve this problem, an improved analytical mesh stiffness model is proposed and the efficient solving method is also established. By using this proposed method validated by FEM results, effects of the tooth profile modification, tooth contact nonlinearity, loaded forces, as well as tooth profile errors on the gear mesh stiffness, static transmission error, and load sharing ratio are investigated. 2. Analytical gear mesh excitation calculation model for spur gear pair 2.1. Problem description Time- or angular-varying mesh stiffness and tooth deviations from the theoretical position generated in design, manufacturing, or assembly process are the two major dynamic excitations in gear transmission systems. In the single-tooth mesh region, the tooth deviations will excite the gear dynamics system by the means of making the two teeth get contact with each other ahead or delaying, while, the mesh stiffness will begin to take part in the effect on gear motions after the teeth get contact. It can be seen that the mesh stiffness and the tooth deviation are independent with each other to excite the gear vibrations. However, this situation will be changed completely for the multiple tooth mesh region. The tooth pair with the least tooth deviations will firstly get contact, and then more tooth pairs will get contact in sequence as the increasing of the load which results in a greater gear tooth deformation. For example, Fig. 1 displays the possible contact patterns in the double-tooth mesh region. When the relative displacement of the gear pair is just equal to the smaller tooth pair error, this tooth pair is just getting contact but with no force generated (see tooth pair 2 in Fig. 1a). At this time, there is also a gap  between the two teeth of tooth pair 1. Then, the tooth pair 2 starts to deform in the amplitude of δ which is smaller than  when a light external torque is applied to the gear pair. In this case, only one tooth pair, namely the tooth pair 2, carries the load due to the presence of the tooth errors even if the gear pair is essentially in the theoretical double-tooth mesh region, in other words, the total mesh stiffness of the gear pair in the double-tooth mesh region is equal to the single-tooth mesh stiffness. When a larger external torque is applied to the gear pair so that the deformation of the tooth pair 2 is

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Fig. 2. Improved analytical model for gear mesh stiffness calculation.

larger than the gap , both the two tooth pairs are in contact to carry the load. For this case, it is obvious that the relative displacement of the gear pair with tooth deviations is larger than that without tooth deviations. This indicates that existence of tooth deviations contributes not only to the displacement excitation but also to the reduction of the total mesh stiffness for the multi-tooth mesh conditions. It should be noted that the tooth deviation mentioned in this paper is assumed too small to change the gear tooth stiffness and the teeth contact stiffness. Actually, a gear tooth deformation under a force applied to it will cause the deformations of its neighboring teeth through the gear body, and vice versa. This gear body structure coupling effect had also been pointed out by Xie et al. [39] and the corresponding analytical calculation equations were also derived by them in their work [40]. It is evident that the coupling effect between the deformations of the neighboring teeth will make the contact forces of the tooth pairs in mesh interact more complicatedly, especially when the nonlinear characteristics of teeth contact are considered, and this coupling effect was seldom considered in the analytical methods for gear mesh stiffness calculation in previous work. Besides, it can be seen from above analysis that the tooth deviations are likely to considerably change the contact forces of the tooth pairs in mesh, namely, the load sharing ratio is likely to be changed by the tooth deviations. Existence of the tooth deviation together with the aforementioned coupling effect will further complicate the interaction among the gear teeth contact forces in multi-tooth mesh region, which has not been analyzed in literatures. To make up this gap, a comprehensive gear mesh stiffness analytical calculation model is proposed and shown in Fig. 2, where the action mechanism of the coupling effect between neighboring teeth is fully considered and the tooth deviations are also included. Although only a double-tooth mesh model is presented here, it can be easily extended to more tooth pairs in mesh using the same idea. The springs representing the tooth deformation due to coupling effect, the tooth filletfoundation deformation, the tooth flexible deformation, and the teeth contact deformation are connected in series according to their action mechanism. It can be seen that the tooth deviations represented by ei will change the teeth contact force Fi , which will further bring extra deformation (Fi /Kpf ji + Fi /Kgf ji ) to its neighboring teeth, namely the tooth pair j, through the gear body structure coupling effect, where the subscripts i,j = 1,2, and i = j. The symbol h(t) is a switch between the single(h(t) = 0) and double-tooth (h(t) = 1) mesh regions. Both the total deformations of the two tooth pairs are calculated based on the reference of the two hub bores of the gear pair. In the following subsections, the detailed mesh stiffness calculation procedure will be introduced. 2.2. Stiffness calculation of gear tooth As shown in Fig. 2, deformation of a gear under an external load usually consists of tooth deformation (Ki t ), tooth filletfoundation deformation (Ki f ), tooth deformation due to coupling effects from the neighboring teeth (Ki f jk ) which was concerned in recent years, and the contact deformation of the tooth pair (Kh ). Here, the subscripts, i = p, g, denoting the pinion and the gear, respectively, and j and k are the tooth pair number in mesh. 2.2.1. Gear tooth deformation For the gear tooth deformation, it is usually regarded as a non-uniform cantilever beam which could facilitate the analytical calculation of the gear tooth deformation. The model for the tooth deformation calculation is shown in Fig. 3. The tooth deformation is contributed by the bending, shear, and axial compressive deformations, and their corresponding stiffness are calculated as [20,21],

1 = Kb 1 = Ks 1 = Ka



d

(x cos α1 − h sin α1 )2 E Ix

0



d

0



d 0

dx

(1)

1.2 cos2 α1 dx GAx

(2)

sin α1 dx E Ax

(3)

2

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Fig. 3. Modelling a spur gear tooth as non-uniform cantilever beam [21].

where, the symbols, Kb , Ks , Ka , denote the bending stiffness, shear stiffness, and axial compressive stiffness, respectively; d is the effective tooth length for integration; h represents half of the tooth thickness at the position that the mesh force is applied to; α 1 is the angle between the mesh force and the direction perpendicular to the tooth center line; E and G are the Young’s modulus and shear modulus of the gear material, respectively. Ix and Ax represent the area moment of inertia and the area of the section that has a distance x away from the acting point of the applied force along the tooth center line, and they are calculated as Ix = 2hx 3 W/3, Ax = 2hx W. The symbol hx is half of the tooth thickness, and W is the tooth width. 2.2.2. Gear teeth contact deformation The contact area will deform when two gear teeth get in contact to carry the external load. There have been many research work related to the calculation of the contact deformations, including the Hertz contact theory, finite element method, and empirical formulas etc. The calculation formula of the teeth contact deformation δ h is summarized here as,

δh = Ch F κ

(4)

where, the symbol F is the contact force, and the parameters Ch and κ are the contact coefficients. When the teeth contact

−ν ) deformation is regarded as linear by ignoring the nonlinear component, the two parameters are given as Ch = 4(π1W E , κ = 1, respectively, and this has been also adopted by many scholars in their research work, such as [21,23-24,30]; while when the nonlinear characteristics of the teeth contact is considered, the two parameters are calculated as Ch = 0.9E10.275 .9 W 0.8 , κ = 0. 9, respectively, and this is derived from the gear teeth contact stiffness calculation equation that has been used in many works such as [39,41]. Usually, the second case is regarded as the more accurate calculation method for the gear teeth contact deformations due to the truth that the gear teeth contact is nonlinear. 2

2.2.3. Gear tooth fillet-foundation deformation The gear tooth fillet-foundation deflection is another factor that softs the stiffness of a gear tooth. The analytical calculation of this deflection was derived in [15] based on the theory proposed by Muskhelishvili [42] who assumed a linear distribution of normal stress and a constant tangential stress at the gear dedendum circle. And the derived equation was then firstly adopted by Chen and Shao [21] to calculate the gear mesh stiffness together with the potential energy principle, which has been used widely by other scholars. The gear tooth fillet-foundation deflection is calculated as [15,21,40],

  2

1 1 u = cos2 α1 L Kf WE S



  u + M + P 1 + Q tan2 α1 S

(5)

where, u and S are given in Fig. 4; and the coefficients L, M, P, Q can be numerically calculated and curve-fitted by polynomial functions for 2≤hf ≤5 and 0.04 ≤ θ f ≤ 0.08 with the following forms [40]:

Xi (hf , θf ) = Ai θf3 + Bi /h3f + Ci θf2 + Di /h2f + Ei θf /h2f + Fi θf /hf + Gi θf2 /hf + Hi /hf + Ii

(6)

Xi (hf , θf ) = Ai /θf3 + Bi h3f + Ci /θf2 + Di h2f + Ei h2f /θf + Fi hf /θf + Gi hf /θf2 + Hi hf + Ii

(7)

where Xi denotes the coefficients L, M, P and Q. And when calculating Q, Eq. (6) is used, while calculating L, M, P, Eq. (7) should be used. The symbol hf = Rf /rint . The symbols Rf , rint and θ f are defined in Fig. 4, and the values of Ai , Bi , Ci , Di , Ei , Fi , Gi , Hi , and Ii are given in Table A.1 in the Appendix which is extracted from Ref. [40].

Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

7

Fig. 4. Deformation calculation of a gear tooth fillet-foundation [15,21].

Fig. 5. Gear tooth deformation due to the coupling effect of gear foundation structure.

2.2.4. Gear tooth deformation due to the nearby loaded teeth It has been found that it will not only make the tooth itself deformed when a gear tooth is loaded but also make the posture change of the neighboring teeth through the gear foundation structure. How to calculate this gear body structure coupling effect between the loaded tooth and its neighboring teeth has attracted the attention of the researchers. Some researchers, such as Ma et al. [7] and Xie et al. [39], considered this coupling effect in the form of a correction factor to the tooth fillet-foundation stiffness by using the finite element method which is of course capable of calculating this effect automatically. And later, Xie et al. [40] derived the analytical formulas to calculate the gear body structure coupling effect based on the elastic circular ring theory of Muskhelishvili [42] by further assuming a cubic distribution of normal stress and parabolic distribution of tangential stress which is different from the assumption made in the work by Sainsot et al. [15], namely a parabolic distribution of normal stress and a linear distribution of tangential stress. In this paper, the analytical formulas derived by Xie et al. [40] are used to calculate the gear body structure coupling effect. The gear body structure coupling deformation 1/Kf ij is defined as the displacement of the intersection point of the ith tooth profile and the line of action when a unit force is applied to the intersection point of the jth tooth profile and the line of action. The direction for both the displacement and the unit force is along the line of action. For example, when the gear pair is in the double-tooth mesh region shown in Fig. 5, the 1/Kf12 denotes the displacement of the point where F1 (F1 = 0) is applied to due to the gear foundation deflection when F2 = 1, and the 1/Kf21 denotes the displacement of the point where F2 (F2 = 0) is applied to due to the gear foundation deflection when F1 = 1. They can be calculated by [40],

1 1 = cos Kf21 WE

   u1 u2 2

α1 cos α2 L1

S2

+ (tan α2 M1 + P1 )

u1 S

u2 + (tan α1 Q1 + R1 ) + (tan α1 S1 + T1 ) tan α2 + U1 tan α1 + V1 S 1 1 = cos Kf12 WE

   u1 u2 2

α1 cos α2 L2

S2

+ (tan

α1 M2 + P2 )

u2 S

u1 + (tan α2 Q2 + R2 ) + (tan α2 S2 + T2 ) tan α1 + U2 tan α2 + V2 S

 (8)

 (9)

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Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

where, the symbols ui and α i (i = 1,2 denoting the tooth number) has the same meaning as the symbol u and α 1 in Eq. (5). The Mi, Qi, Ti, Ui (i = 1,2) are calculated by Eq. (6), and Li, Pi, Ri, Si, Vi (i = 1,2) are calculated by Eq. (7). The values of the coefficients Ai , Bi , Ci , Di , Ei , Fi , Gi , Hi , and Ii for calculating Kf21 using Eq. (8) and Kf12 using Eq. (9) are given in Tables A.2 and A.3, respectively, which are also extracted from Ref. [40]. It should be noted that the values of the coefficients in the last line of Table A.3 has been corrected by multiplying the corresponding values in [40] with −1. 2.3. Mesh stiffness calculation of gear pair The major advantage of the proposed gear dynamic excitation (e.g. time-varying mesh stiffness and the tooth error excitation) calculation model shown in Fig. 2 is the involvement of the tooth deviations, the tooth deformation due to the gear body structure coupling effect, and the nonlinear teeth contact. Consideration of these three factors simultaneously will eventually complicate the interactions among the contact forces of the tooth pairs in mesh. And of course, the traditional calculation methods are not suitable to solving such a complicated problem. In this subsection, a relative simple solving method is proposed, and the corresponding solving flowchart is giving out for better illustration. It can be seen from proposed mesh stiffness calculation model shown in Fig. 2 that all the deformations of the gear tooth pairs are calculated by referencing to the hub bores of the gear pair. When calculating the total mesh stiffness and the tooth error excitation for a certain mesh position, the driven gear hub bore is fixed firstly and then the driving gear begins to rotate from its theoretical position under the external applied torque T0 at the hub bore. It will rotate freely until at least one tooth pair is just in contact, in which case, the error excitation δ NSTE , namely the non-loaded static transmission error (NSTE), can be obtained as,

 

δNSTE = min ep1 + eg1 ,...,epi + egi , ..., epN + egN

(10)

And then, the driving gear is continuing to rotate so as to compress the tooth pairs in mesh until the total force generated by the springs representing the gear deformations in Fig. 2 is equal to that from the external torque. At this time, the gear transmission system is in a statically balanced status, and the total deformation of each tooth pair is equal to the equivalent displacement on the line of action due to the driving gear rotation. In other words, summation of the teeth deformations δ t , tooth fillet-foundation deformations δ f , teeth deformations due to gear body structure coupling effect δ ij , teeth contact deformation δ h , and teeth deviations e of each tooth pair in mesh is equal to the loaded static transmission error (LSTE) of the gear pair when an external torque T0 is applied. It can be represented mathematically as, N 

δti + δfi +

δi j + δhi + epi + egi = δLSTE , (i ∈ Z, 1 ≤ i ≤ N )

(11)

j=1, j=i

where, the symbol Z denotes the integer set, and N is the number of tooth pairs in contact. And Eq. (11) can be rewritten in the form of the stiffness introduced in the Section 2.2, which yields,



Fi

1 1 1 1 1 1 + + + + + Kapi Kbpi Kspi Kagi Kbgi Ksgi N 

+



Fj

j=1, j=i

1 1 + Kpi j Kgi j





+ Fi

1 1 + Kfpi Kfgi

+ Ch Fiκ + epi + egi = δLSTE ,

(i ∈ Z, 1 ≤ i ≤ N )

(12)

where Fi is mesh force of the ith tooth pair in contact, and the symbols p and g in the subscripts denote the pinion and gear, respectively. And the total mesh force of all the tooth pairs in contact is just used to resist the external torque to keep the gear transmission system in a static equilibrium, namely, N 

Fi = T0 /Rbp , Fi ≥ 0

(13)

i=1

where Rbp is the radius of the pinion base circle. It can be seen from the Eqs. (12) and (13) that there are totally N + 1 unknown numbers (F1 , F2 ,…,FN , δ LSTE ) while the number of equations is also N + 1, indicating all these unknown numbers can be obtained directly and uniquely by solving these equations. Finally, the total mesh stiffness and the load sharing ratio can be calculated as,

Kmesh =

T0 Rbp (δLSTE − δNSTE )

Lsri = Fi Rbp /T0

(14)

(15)

For the widely used spur gear pair with a contact ratio between 1 and 2, namely the number of tooth pairs in mesh is alternately changed between 1 and 2. For double-tooth mesh region in this case, the set of equations denoted in Eqs. (12) and

Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

(13) will be in a simpler form shown as follows,

⎧  1 ⎪ F1 Kap1 + ⎪ ⎪ ⎪ ⎨  F2 ⎪ ⎪ ⎪ ⎪ ⎩

1 Kap2

+

1 Kbp1

+

1 Kbp2

+

1 Ksp1

+

1 Ksp2

+

1 Kag1

+

1 Kag2

+

1 Kbg1

+

1 Ksg1

1 Kbg2

+

1 Ksg2

 

+ F1 + F2

 

1 Kfp1

+

1 Kfg1

1 Kfp2

+

1 Kfg2

 

+ F2 + F1

 

1

1

Kp12

+

Kg12

1 Kp21

+

1 Kg21

 

9

+ Ch F1κ + ep1 + eg1 = δLSTE + Ch F2κ + ep2 + eg2 = δLSTE

(16)

F1 + F2 = T0 /Rbp , F1 ≥ 0, F2 ≥ 0

If the gear teeth contact is regarded as linear, namely κ = 1, the exact solution to the Eq. (16) can be obtained as,

F1 =

F

1

K2

1 K1

+

− 1 K2

1 K12

−



+ E˜12

1 K12

−

1 K21

, F2 =

F 1 K1

1

K1

+

− 1 K2

1 K21

−



1 K12

− E˜12 −

1 K21

(17)

where, F is the total mesh force (F = T0 / Rbp ); and the general tooth error function is defined as E˜12 = ep2 + eg2 − ep1 − eg1 ; the mesh stiffness of the single tooth pair is calculated as,

1 1 1 1 1 1 1 1 1 1 = + + + + + + + + , i = 1, 2 Ki Kapi Kbpi Kspi Kagi Kbgi Ksgi Kfpi Kfgi Khi

(18)

Without loss of generality, it is assumed that the tooth deviation of the tooth pair 1 is less than that of the tooth pair 2, namely, the total deformation of tooth pair 1 due to the load is greater than that of tooth pair 2. Substitution of Eqs. (16) and (17) into Eqs. (14) and (15) yields,

Kmesh = Lsr1 =

K1 + K2 − K1 K2 (1/K12 + 1/K21 ) 1 + K2 E˜12 (1 − K1 /K12 )/F − K1 K2 /(K12 K21 )

1/K2 − 1/K12 + E˜12 /F 1/K1 + 1/K2 − 1/K12 − 1/K21

(19)

(20)

When the deformations of the teeth due to the gear foundation coupling effect are not considered, namely 1/K12 = 0 and 1/K21 = 0, Eqs. (19) and (20) will degenerate into,

Kmesh =

K1 + K2 1 + K2 E˜12 /F

K1 Lsr1 = K1 + K2



K2 E˜12 1+ F

(21)

(22)

Eqs. (21) and (22) are the same as that in Ref. [29] where the gear body structure coupling effect was not considered. That is to say the proposed mesh stiffness calculation model is more general and comprehensive than the traditional ones, and in reverse, the traditional models are special cases of the proposed model in this paper. Comparing Eq. (19) with (21) reveals the influencing mechanism of the gear body structure coupling effect on the total mesh stiffness, namely, the tooth deformation due to the coupling effect plays a role in the total mesh stiffness calculation through interacting with both the single-tooth mesh stiffness and the errors assembled from the individual tooth deviations. The calculation flowchart for the gear time-varying mesh stiffness and the transmission error excitation with using the proposed model is shown in Fig. 6. By using this proposed calculation method, the angular-varying mesh stiffness and the non-loaded static transmission error can be obtained. These excitations can be further imported into the gear dynamics system for dynamic performance analysis. 3. Results and discussions In order to reveal the effectiveness of the proposed gear excitation calculation model and method, this section is going to carry out some calculations for case studies. Although the proposed analytical method could be applicable to spur gear pairs having any contact ratio, this section gives out only the case studies with contact ratio between 1 and 2 due to the lack of existing analytical methods for tooth deformation calculation due to gear body structure coupling effect with the contact ratio greater than 2. The detailed design parameters of the spur gear pair is shown in Table 1 which has the same parameters as the gear pair 1 in [40]. In this section, the effectiveness of the proposed method is firstly verified by comparison with the results by FEA in the published literatures, and then the effects from the tooth profile modifications, the applied torques, the teeth linear or nonlinear contacts, the gear foundation structure coupling deformation, as well as the tooth profile errors, are analyzed to highlight the advantages of the proposed method. The time- or angular-varying mesh stiffness results of the spur gear pairs with different radius of hub bore calculated by the proposed method are shown in Fig. 7. It is noted that the nonlinear teeth contact is considered in all the following calculated results. For comparison, the results by FEA are also extracted out from Ref. [40] and displayed in this figure. It can be seen that the results from the two methods agree well with each other for all the three cases with different radius

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Fig. 6. Calculation flowchart of gear mesh excitations.

of the hub bore. In addition, the gear mesh stiffness of the other two gear pairs, namely gear pairs 2 and 3 in Ref. [40], are also calculated by the proposed method and compared with their FEA results although they are not given out here due to the similar trend to results in Fig. 7. The compared results indicate that the maximum error between the calculated mesh stiffness results by the proposed method and by the FEA is less than 6.9% for all the three gear pairs. Finally, the mesh stiffness of the gear pair investigated by Ma et al. [43] with using a correction coefficient obtained by FEM to revise the fillet-foundation stiffness are also extracted out and compared with the results calculated by the proposed method, and the trend and amplitude of the mesh stiffness for both single- and double-tooth regions indicate also a good agreement. Consequently, the effectiveness of the proposed method has been verified and it has a good accuracy and efficiency in calculating the gear mesh stiffness. The intentional tooth profile modification is an effective approach that has been widely used for vibration and noise reduction of gear transmissions. However, it will inevitably complicate the dynamic interactions between the teeth and the tooth pairs in mesh. The gear mesh results, namely the mesh stiffness, loaded static transmission error, and load sharing ratio of the spur gear pair with different tooth tip relief are calculated by using the proposed analytical method in

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Table 1 Design parameters of the spur gear pair. Pinion Tooth Shape Module m (mm) Pressure angle α 0 (deg.) Contact ratio ε Operating center distance dc (mm) Radius of hub bore rint (mm) Number of Teeth z Teeth width W (mm) Addendum coefficient ha∗ Tip clearance coefficient c∗ Profile shift coefficient x Pitch diameter Dp (mm) Addendum height ha (mm) Dedendum height hf (mm) Young modulus E (N/mm2 ) Density ρ (kg/ m3 ) Poisson’s ratio ν

Gear

Standard involute 1 20 1.71 40 4; 6; 8 40 40 20 20 1 1 0.25 0.25 0 0 40 40 1 1 1.25 1.25 2.068 × 105 2.068 × 105 7.8 × 103 7.8 × 103 0.3 0.3

Fig. 7. Time-varying mesh stiffness in one mesh cycle with: (a) rint = 4 mm, (b) rint = 6 mm, (c) rint = 8 mm.

Section 2 and shown in Figs. 8 and 9. The definitions of the tooth tip relief are the same as that in [29] and shown in Fig. 5, where the normalized relief parameters are calculated as: Cn = Ca /0.02 m, Ln = La /0.6 m. Here, the symbol m represents the module of the gear. It can be seen that the double-tooth region will reduce and the slope of the transition region between single- and double-tooth mesh will become greater as the amplitude increase of the tooth tip relief with a fixed length of tooth profile modification. The loaded static transmission error curves have the similar variation trend but with an inverse shape compared with the mesh stiffness curves. The tooth load sharing ratio results also indicate that this tiny modification at the level of micrometer will cause a considerable variation between the mesh forces of the multiple tooth pairs. The similar phenomenon could be observed in Fig. 9 by varying tooth relief length with fixed modification amplitude. It should be noted that the amplitude variations of the loaded static transmission error will decrease first and then increase as increasing of the tooth tip relief length, namely a potential optimum tooth tip relief exists at the view point of minimum loaded static transmission error variations, and an excessive tooth tip relief will bring adverse effects to the gear dynamic performance. The external applied load is also an important factor that influences the gear teeth mesh characteristics. The calculated results of the spur gear pair loaded with different torques are displayed in Fig. 10. It can be seen that the proportion of the double-tooth mesh region increase with the loaded torque. Amplitude and its variation of the loaded static transmission error shown in Fig. 10(b) are also becoming greater as the increasing of the torque. And the mesh stiffness amplitudes, as shown in Fig. 10(a), increase with the increasing loaded torque for both single- and double-tooth mesh, which is due to the nonlinear teeth contact that highly depends on the mesh forces, namely, the nonlinear teeth contact stiffness is greater with a larger acting force. However, the linear teeth contact stiffness that ignored the nonlinear characteristics has also been adopted by many scholars in their research work, such as in Refs. [20–21,23–24,29]. Here, both the nonlinear and linear teeth contact stiffness calculation methods are used for gear mesh calculation and compared in Fig. 11. It can be seen that the gear mesh stiffness considering the linear teeth contact that is independent on the acting force has the similar shape and amplitude variations as that when the nonlinear teeth contact is considered, but has a larger mean value of the mesh

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Fig. 8. Results for the gear mesh with different tooth tip relief amplitudes (Ln = 0.5, T0 = 40 Nm): (a) mesh stiffness, (b) loaded static transmission error, (c) load sharing ratio of tooth pair 1, and (d) load sharing ratio of tooth pair 2.

Fig. 9. Results for the gear mesh with different tooth tip lengths (Cn = 0.6, T0 = 40 Nm): (a) mesh stiffness, (b) loaded static transmission error, (c) load sharing ratio of tooth pair 1, and (d) load sharing ratio of tooth pair 2.

stiffness. The linear teeth contact method will overestimate the gear mesh stiffness by about 12% when the loaded torque is 120 Nm and by about 16% with the torque 20 Nm for the calculated gear case. The time-varying mesh stiffness with using linear teeth contact calculation method could capture most of the dynamic excitation characteristics, so it is reasonable to some extent for many previous works to use this method in calculating gear mesh stiffness for a preliminary analysis. However, for a more accurate gear dynamic simulation, the more practical nonlinear teeth contact calculation method is recommended to be used. As stressed in Refs. [7,39] and in this paper, the gear body structure coupling effect has considerable effect on the gear mesh stiffness. Especially, this influencing factor will complicate the gear mesh process and the corresponding solving method when the tooth profile deviations are taken into consideration. The time-varying gear mesh stiffness results shown in Fig. 12 are calculated by four analytical methods: (1) the traditional method in [21] that could not consider effects of the tooth profile deviation and the gear foundation structure coupling, (2) the traditional method in [29] that could not consider effects of the gear foundation structure coupling, (3) the traditional method in [39-40] that could not consider effects of the tooth profile deviation, and (4) the proposed method that enables involvement of the effects of both the tooth profile deviation and the gear foundation structure coupling. It can be seen from Fig. 12(a) that the traditional analyt-

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Fig. 10. Results for the gear mesh with different loaded torques (Cn = 0.6, Ln = 0.5): (a) mesh stiffness, (b) loaded static transmission error, (c) load sharing ratio of tooth pair 1, and (d) load sharing ratio of tooth pair 2.

Fig. 11. Results for the gear mesh with using different teeth contact deformation calculation methods (Cn = 0.6, Ln = 0.5, T0 = 120 Nm): (a) mesh stiffness, (b) loaded static transmission error, (c) load sharing ratio of tooth pair 1, and (d) load sharing ratio of tooth pair 2.

ical calculation methods that could not consider the gear body structure coupling effect will overestimate the amplitude of double-tooth mesh stiffness by about 43.9% when compared with the methods that include the coupling effect. In addition, the gear body structure coupling effect could also reduce the proportion of the double-tooth mesh region when the gear teeth profiles are modified. While in the order spectrum shown in Fig. 12(b), it can be seen that the gear body structure coupling effect mainly reduces the amplitudes of the low orders, e.g. the orders 40, 80, and 160, while the tooth tip relief mainly reduces the amplitudes of the higher orders. Gear teeth deviations are usually generated in the gear machining or assembly process and they are usually undesired due to their adverse effect on the product quality and dynamic performance. Here, the tooth profile errors are considered by using the proposed analytical method to calculate the gear mesh indicators. The profile errors are assumed identical for all the teeth in this study due to the fact that they appear a similar pattern in the same flank of successive teeth in the manufacturing process [12]. The tooth profile error curve can be expressed by summation of a series of sinusoidal curves

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Fig. 12. Time-varying mesh stiffness with using different calculation methods (Cn = 0.6, Ln = 0.5, T0 = 40 Nm): (a) mesh stiffness, and (b) the corresponding order spectrum. (TR: tip relief; SCE: structure coupling effect).

Fig. 13. Results for the gear mesh with tooth profile errors (Cn = 0.6, Ln = 0.5, T0 = 40 Nm): (a) mesh stiffness, (b) tooth errors and transmission errors, and (c) load sharing ratio. (TR: tip relief; TPE: tooth profile error; NSTE: non-loaded static transmission error; LSTE: loaded static transmission error).

versus the gear rotation, and it is given as,

e (θ ) =

Ne 

A j sin(2π jθ /θmax + ϕ j ), θ ∈ [0

θmax ]

(23)

j=0

where, the symbol e is the tooth profile error; Aj denotes the amplitude of the jth harmonic; θ max is the gear rotational angle corresponding to the mesh point moving in the theoretical segment along the line of action; ϕ j is the phase angle of the jth harmonic; and Ne is the total number of the harmonics. In this study, the values of the detailed tooth profile error coefficients are given as: a) the amplitudes of the harmonics A2 /A4 /A8 are 3/2/1.5 μm, respectively, and b) the phases of the harmonics ϕ 2 /ϕ 4 /ϕ 8 are given as 0.25π /0.5π /π , respectively. Thus, the profile error curve of one tooth pair can be obtained and shown as the green thin solid curve in Fig. 13(b), where the tooth tip reliefs are also included. With the tooth profile deviations, the mesh stiffness, the transmission errors, and the tooth load sharing ratio could be obtained by using the proposed method and shown in Fig. 13. It can be seen that the tiny tooth errors at the grade of micrometer will make drastic oscillations to the gear mesh stiffness in the double-tooth regions in Fig. 13(a), and thus the tooth load sharing ratio also varies greatly in Fig. 13(c). The corresponding transmission errors are presented in Fig. 13(b). It can be found that all the tooth profile errors are contributed to the non-loaded static transmission error and they have no effect on the mesh stiffness for the single-tooth mesh regions (e.g. CD and FG regions). However, this situation will be completely different for the double-tooth mesh regions, namely the AC and DF regions. When the contact point of the tooth pair 2 lies at the region AB, the tooth pair 1 is in region DE. For this case, the amplitude of the tooth profile error of tooth pair 2 (see the green thin solid curve in region AB) is larger than that of tooth pair 1 (see the green

Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838

15

thin solid curve in region DE), thus the tooth profile error of tooth pair 1 is copied to the region AB as the non-loaded static transmission error (see the blue dotted thick curve) and the difference in tooth profile errors between the two tooth pairs are contributed to the mesh stiffness reduction and variation. The same analysis could also be suitable to the transmission errors between the regions EF and BC. It can be concluded that reduction of the double-tooth mesh stiffness will be greater with the increase in the tooth profile error difference between the two tooth pairs in mesh. In addition, the loaded static transmission error curve with tooth profile errors have the same shape as the non-loaded static transmission error, but with larger amplitude due to contribution of the deformations from the gear teeth and gear body. 4. Conclusions An improved mesh stiffness analytical calculation model of spur gear pairs is proposed in this paper with especially considering both the gear body structure coupling effect and the tooth profile deviations. The influencing mechanism of the gear body structure coupling effect and the tooth profile deviations on the gear mesh stiffness and load sharing ratio are revealed analytically. And the detailed calculation flowchart by using the proposed model is also given out. The proposed mesh stiffness calculation model and the corresponding solving method are verified by comparison with the FEA results extracted from previously published papers, and a good agreement is presented. Then, the case studies are performed to reveal the effects of some influencing factors on the gear mesh indicators. Some conclusions can be obtained as follows. Tooth profile modification has apparent effect on the gear mesh stiffness and the load sharing ratio. It could make the transition between the single- and double-tooth mesh region more smooth and obtain an optimum profile modification for minimizing the loaded static transmission error. The linear teeth contact deformation calculation method will overestimate both the single- and double-tooth mesh stiffness amplitudes when compared with the nonlinear method. And ignoring of the gear body structure coupling effect in traditional analytical methods will make a considerable overestimation of the double-tooth mesh stiffness, and the gear body structure coupling effect mainly reduces the amplitudes of the low order harmonics of the gear mesh stiffness while the tooth tip relief mainly reduces the amplitudes of the higher order harmonics. Finally, the tiny tooth errors at the grade of micrometer will make drastic oscillations to the gear mesh stiffness, load sharing ratio, and static transmission errors in the double-tooth regions. Consequently, for a more accurate gear dynamic simulation, the nonlinear teeth contact method is recommended to be used due to the time-varying teeth acting forces due to the complicated dynamic excitations, such as the time-varying mesh stiffness and the gear tooth profile deviations. And the gear body structure coupling effect should be considered due to that it will soft the gear mesh stiffness considerably and complicate the mesh force interactions between the tooth pairs in mesh. Thus, the proposed comprehensive analytical calculation model for gear mesh stiffness and transmission error excitations is recommended for gear dynamic analysis due to its accuracy and efficiency. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China [grant numbers 51775453, 51735012]; the Fundamental Research Funds for the State Key Laboratory of Traction Power of Southwest Jiaotong University [grant number 2019TPL-T09]; and the Fundamental Research Funds for the Central Universities [grant number 2682019YQ04]. Appendix Values of the coefficients Ai /Bi /Ci /Di /Ei /Fi /Gi /Hi /Ii in Eqs. (6) and (7) for determinations of the coefficients L, M, P, Q in Eq. (5) to calculate the tooth fillet-foundation stiffness and the coefficients Li , Mi , Pi , Qi , Ri , Si , Ti , Ui , Vi (i=1,2) in Eqs. (8) and (9) to calculate the equivalent stiffness of gear body structure coupling effect were obtained by Xie et al. [40], and these values are also given out in the following three tables for a better readability. It should be noted that the values of the coefficients in the last line of Table A.3 have been corrected by multiplying the corresponding values in [40] with −1. Table A.1 Coefficients for L, M, P and Q for plane strain condition. Ai L M P Q

−3.785e−5 −8.177e−5 −4.415e−5 50.15

Bi 1.931e−3 4.312 e−3 1.669 e−2 2.190

Ci 1.293e−3 2.547 e−3 2.053e−3 −4.765

Di −2.254e−2 3.4e−2 −8.121e−3 −4.636

Ei 9.922e−5 −2.184e−3 −2.115e−4 5.705

Fi −1.306e−2 −4.419e−2 6.338e−3 −7.071

Gi 2.6e−4 1.095e−3 −1.318e−4 11.26

Hi 0.2451 0.6473 0.9658 3.4580

Ii 3.5260 0.5298 0.4352 −0.1916

16

Z. Chen, Z. Zhou and W. Zhai et al. / Mechanism and Machine Theory 149 (2020) 103838 Table A.2 Coefficients for L1 , M1 , P1 , Q1 , R1 , S1 , T1 , U1 , V1 for plane strain condition. Ai L1 M1 P1 Q1 R1 S1 T1 U1 V1

−2.42e−5 40.21 −3.620e−5 −39.58 −3.610e−5 −3.570e−5 3.597e02 −3.564e02 −4.210e−5

Bi 1.607e−3 −0.3763 1.967e−3 0.3751 1.960e−3 1.099e−2 9.839e−2 −9.638e−2 1.638e−2

Ci 8.513e−4 −8.284 1.074e−3 8.117 1.070e−3 1.689e−3 −61.51e 60.86e 1.962e−3

Di −1.614e−2 0.5267 1.940e−2 −0.5241 1.952e−2 −0.1434 −0.1819 0.1778 −3.132e−3

Ei −4.55e−5 −5.404 −1.100e−3 5.376 −1.101e−3 −3.802e−4 −2.452 2.503 −2.771e−4

Fi −1.001e−2 3.073 −2.254e−2 −3.019 −2.252e−2 8.386e−3 0.8327 −0.8984 7.486e−3

Gi 2.076e−4 −12.77 5.622e−4 12.47 5.621e−4 −1.440e−4 22.93 −22.53 −1.463e−4

Hi 0.1931 −0.2074 0.3177 0.2050 0.3170 0.6640 0.1655 −0.1627 0.9336

Ii −0.8056 0.5662 −0.2317 −0.5785 −0.2305 −0.7143 0.1135 −0.1146 −0.5324

Table A.3 Coefficients for L2 , M2 , P2 , Q2 , R2 , S2 , T2 , U2 , V2 for plane strain condition. Ai L2 M2 P2 Q2 R2 S2 T2 U2 V2

−2.42e−5 −40.21 −3.620e−5 39.58 −3.610e−5 −3.570e−5 −3.597e02 3.564e02 −4.210e−5

Bi 1.607e−3 0.3763 1.967e−3 −0.3751 1.960e−3 1.099e−2 −9.839e−2 9.638e−2 1.638e−2

Ci 8.513e−4 8.284 1.074e−3 −8.117 1.070e−3 1.689e−3 61.51e −60.86e 1.962e−3

Di −1.614e−2 −0.5267 1.940e−2 0.5241 1.952e−2 −0.1434 0.1819 −0.1778 −3.132e−3

Ei −4.55e−5 5.404 −1.100e−3 −5.376 −1.101e−3 −3.802e−4 2.452 −2.503 −2.771e−4

Fi −1.001e−2 −3.073 −2.254e−2 3.019 −2.252e−2 8.386e−3 −0.8327 0.8984 7.486e−3

Gi 2.076e−4 12.77 5.622e−4 −12.47 5.621e−4 −1.440e−4 −22.93 22.53 −1.463e−4

Hi 0.1931 0.2074 0.3177 −0.2050 0.3170 0.6640 −0.1655 0.1627 0.9336

Ii −0.8056 −0.5662 −0.2317 0.5785 −0.2305 −0.7143 −0.1135 0.1146 −0.5324

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