ACCURATE BENDING STRENGTH ANALYSIS OF ASYMMETRIC GEARS USING THE NOVEL ES-PIM WITH TRIANGULAR MESH

International Journal of Automotive an Mechanical Engineering (IJAME) ISSN: 2229-8649 (Print); ISSN: 2180-1606 (Online); Volume 4, pp. 373-396, July-December 2011 Universiti Malaysia Pahang DOI: http://x.oi.org/10.15282/ijame.4.2011.1.0031 ACCURATE BENDING STRENGTH ANALYSIS OF ASYMMETRIC GEARS USING THE NOVEL ES-PIM WITH TRIANGULAR MESH S. Wang 1, G. R. Liu 1, 2, G. Y. Zhang 2 an L. Chen 1 1 Centre for Avance Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore (NUS), 9 Engineering Drive 1, 117576, Singapore. Tel: +0065-65164797 Email: g0800348@nus.eu.sg 2 Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, 117576, Singapore ABSTRACT This paper extens the ege-base smoothe point interpolation metho (ES-PIM) to bening strength analysis of asymmetric gears with complex outlines. Five sets of asymmetric gears with pressure angles of 20 /20, 25 /20, 30 /20, 35 /20 an 40 /20 were generate by a specially esigne rac cutter. Four ey factors, e.g. accuracy, convergence, the convergence rate an the computational efficiency of the present ES-PIM were chece in great etail on these five moels, an the istributions of bening stresses at the fillet of the rive sie were carefully investigate. The finite element metho (FEM) was also use to calculate the abovementione factors to stress the avantages of ES-PIM. The numerical results inicate that ES-PIM can provie more efficient an accurate solutions in the stress fiel than the FEM, an is very suitable for stress analysis of complicate asymmetric gears. Keywors: Asymmetric gear; Bening strength; Accurate stress; Numerical metho; Point interpolation metho (PIM); Smoothing operation. INTRODUCTION Gear transmission is one of the most important mechanical transmissions in engineering systems, so its reliability is essential. However, fracture always occurs in gear teeth ue to their low bening strength, which leas to invaliation an thus failure of the entire transmission (Bible et al., 1994). To avoi this failure, several methos have been use to improve the bening strength of gear teeth. Among these, moifying the shape of the tooth is an effective an wiely use metho (Bible et al., 1994; Cavar et al., 2005). To ate, several ins of asymmetric gears have been evelope. Deng et al. (2003) mae changes to the pressure angle on the coast sie to increase the bening stiffness of the tooth. Kapelevich (2000) an Kapelevich an Kleiss (2002) have suggeste using a larger pressure angle on the rive sie an a fixe pressure angle on the coast sie. With a larger pressure angle on the rive sie, Litvin 373

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 et al. (2000) propose a moifie geometry of an asymmetric spur gear rive esigne as a combination of an involute gear an a ouble crowne pinion. Muni et al. (2007) an Kumar et al. (2008) use a irect gear esign metho for the optimisation of bening strength of asymmetric spur gear rives. Xiao et al. (2006), Xiao et al. (2008) an Xiao (2008) have use a non-strane asymmetric rac cutter to generate the gear pair via a series of analyses on bening stresses, vibration, thermal conuction, etc. All these researches have put forwar a series of solutions to improve the loa carrying ability of asymmetric gears. During gear transmission, the teeth withstan tensile stresses on the loae sie (rive sie) an compressive stresses on the opposite sie (coast sie). Tooth fracture always occurs at the fillet of the tensile-stress sie (rive sie), where there is also a stress concentration. So, it is essential to carry out accurate bening stress analysis at the fillet of the tensile-stress sie (rive sie) in orer to chec the strength of the teeth (Wang, 2003). Among these stresses at the fillet, the maximum bening stress of the tooth has a significant impact on fracturing an shoul be carefully chece. Examination of run-in teeth reveals that when a single tooth carries the full loa an the loa is applie at the highest point of single tooth contact (HPSTC), the maximum bening stress occurs (Wang, 2003). Therefore, when performing bening stress analysis, the loa is always applie at the HPSTC. In the literature, the bening strength analysis of the gear is always ealt with as a 2-D problem, an the finite element metho (FEM) is utilise to perform the analysis (Bible et al., 1994; Cavar et al., 2005; Deng et al., 2003). Since the linear triangular element can be generate efficiently without manual operation, even for complicate omains, it is quite suitable for the gear problems with complex outlines. However, because of the well-nown overly-stiff behaviour of the fully compatible isplacement of the FEM moel base on the Galerin wea form, the FEM using linear triangular elements usually gives poor solutions, especially for stress components. Recently, a generalise graient smoothing technique, employing a efinition of G space which inclues both continuous an iscontinuous functions an the notion of a weaene wea (W 2 ) formulation, also calle the Generalise Smoothe Galerin (GS-Galerin) wea form, has been evelope by Liu (2008, 2010). The W 2 formulation is the founation of a series of novel an effective numerical methos which can effectively overcome the problems iscusse previously (Liu an Zhang, 2008). Compare to the compatible FEM moel, these methos behave more softly an possess a number of attractive properties accoring to the formation of smoothing omains (Liu an Zhang, 2008; Liu, 2008, 2010). The ege-base smoothe point interpolation metho (ES-PIM) is one of these novel methos, in which the strains are smoothe over the smoothing omains associate with the eges of bacgroun triangles (Liu an Zhang, 2008; Liu et al., 2008). The polynomial an/or raial PIM shape functions, which possess the Kronecer elta property an hence facilitate the treatment of essential bounary conitions (Liu, 2002), are use to construct isplacement fiels in this metho. Base 374

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh on the triangular elements, four schemes of selecting support noes for creating PIM (polynomial or raial) shape functions have been propose, an four moels have been evelope, which are the ES-PIM of T3-scheme (ES-PIM (T3)), the ES-PIM of T6/3-scheme (ES-PIM (T6/3)), the ES-RPIM of T6-scheme (ES-RPIM (T6)) an the ES-RPIM of T2L-scheme (ES-RPIM (T2L)). The T-scheme can solve not only the singularity problem encountere in constructing polynomial PIM shape functions, but also improves the efficiency of the numerical methos (Liu an Zhang, 2008). Numerical results have emonstrate that the ES-PIM moels have a close-to-exact stiffness an prouce super-convergence an ultra-accurate solutions (Liu an Zhang, 2008; Liu et al., 2008; Liu, 2002). Most importantly, the ES-PIM (T3) moel has been foun to achieve the highest computational efficiency compare to the other ES-PIM moels an the linear FEM moel. However, the conclusions escribe above were rawn base on problems with simple geometries, e.g. beam an plate. The efficiency of this metho in terms of problems with complex geometries has not been carefully chece yet. This wor just extens the ES-PIM (T3) metho to the bening strength analysis of an asymmetric gear which possesses very complex outlines. First, the basic principles of the ES-PIM (T3) metho are introuce briefly. Then, base on a specially esigne rac cutter, five ifferent asymmetric gear moels were generate, with a greater pressure angle (20, 25, 30, 35, 40 ) on the rive sie an a normal pressure angle (20 ) on the coast sie. Thir, some important ES-PIM (T3) properties, e.g. computational accuracy in the stress fiel, convergence of strain energy, the convergence rate of the strain energy norm an computational efficiency were chece in great etail an compare with the FEM-T3 metho. Finally, the bening stress istributions at the fillet of the rive sie were stuie. From this analysis, it can be seen that the ES-PIM (T3) metho is one of super-convergence, an provies more efficient an accurate solutions in the stress fiel than the FEM with the same coarse triangular mesh. The ES-PIM (T3) metho is quite suitable for bening strength analysis of complicate asymmetric gear problems with complex outlines an is capable of ealing with stress concentration problems. THE ES-PIM (T3) IN BRIEF The main ifferences in numerical implementation between the ES-PIM (T3) an FEM are the calculation of the strain fiel an the proceure of numerical integration. A brief illumination of the ifferences is presente in this section. Displacement Fiel Approximation Using PIM The point interpolation metho (PIM) obtains an approximation by letting the interpolation function pass through the function values at each scattere noe within the local supporting omain. In total, there are two types of PIM that have been 375

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 evelope using ifferent basis functions, i.e., polynomial PIM using polynomial basis functions (Liu an Gu, 2001) an raial PIM (RPIM) using raial basis functions (Wang an Liu, 2002). For polynomial PIM, the formulations start with the following assumption: n T p a u x x p x a (1) i1 i where u(x) is a fiel variable function efine in the Cartesian coorinate space; p i (x)is the basis function of monomials which is usually built utilising the Pascal s triangles; a i is the corresponing coefficient an n is the number of noes in the local support omain. The complete polynomial basis of orers 1 an 2 can be written as: 1 x y i T p x Basis of complete first orer (2) 1 x y x xy y p T x 2 2 Basis of complete secon orer (3) For raial PIM, using raial basis functions augmente with polynomials, the fiel function can be approximate as follows: n T T r a p b i i j j i1 j1 m u x x x r x a p x b (4) where r i (x) an p j (x) are the raial basis functions an polynomial basis functions, respectively; a i an b j are the corresponing coefficients; n is the number of noes in the local support omain an m is the number of polynomial terms. In the ES-PIM (T3), the linear polynomial PIM is use to approximate the isplacement fiel an form the global stiffness matrix. The coefficients in Eq. (4) can be etermine by enforcing the fiel function to be satisfie at the n noes within the local support omain. 1 a P u s (5) where P is the moment matrix; for a 2-D linear integration problem, P can be expresse as: 1 x1 y1 P 1 x2 y 2 (6) 1 x3 y 333 u s is the isplacement component vector at all the n noes of the support omain: u T s u1x u1y unx u ny (7) 1 Note that P may not exist. Approaches ealing with this problem can be foun in the literature (Liu, 2002). Therefore, we just consier that the moment matrix is invertible. 12 n 376

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh Substituting Eq. (6) into Eq. (5), we can obtain: n T u u x x Φ xu (8) i1 i i s where i x is the shape function which processes the Kronecer elta property (Wang an Liu, 2002) an is efine by (Liu an Que, 2003): i x p x P (9) T 1 ij ji In the above formula, it is note that we nee to properly select n noes for interpolation, thus ensuring a non-singular moment matrix (Liu, 2002). The next subsection will focus on the principles of selecting noes. Cell-base T-scheme for Noe Selection The problem omain is first iscretise with three-noe triangular cells (elements). In the frame wor of ES-PIM (T3), a linear isplacement fiel is constructe within each triangular cell, which is exactly the same as that in the stanar FEM. As illustrate in Figure 1a, no matter the point of interest x locate in an interior element (element i) or a bounary element (element j), only the three noes of the corresponing cell (i1-i3 or j1-j3) are selecte. (a) Noe selection (b) Construction of smoothing omains Ege-base Smoothe Strains Figure 1. Illustration of the T3-scheme In the framewor of the W 2 formula, the graient of the fiel function (strains) will be obtaine using the following generalise smoothing operation which consiers both continuous an iscontinuous isplacement functions (Liu an Zhang, 2008; Liu et al., 2005). 377

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 ε 1 1 A ε x A 1 n A L u x L u x n when when where ε is the compatible strain obtaine by: u x is continuous in u x is iscontinuous in 0 x ε Lux where L, 0 (2) x y y y x where L n is the matrix of unit outwar normal; ε is the smoothe strain over the smoothing omain an 3 2 A is the area of. To perform generalise strain smoothing, the problem omain is first iscretise using three-noe triangular elements an then the stationary an non-overlapping smoothing omains are constructe base on these triangles, such that an i j, i j in which N s is the number of 1 2 Ns smoothing omains. For the ES-PIM (T3), smoothing omains are constructe with respect to the eges of the triangular elements by connecting two ens of an ege to the centrois of two ajacent cells, as illustrate in Figure 1b. Thus, the number of smoothing omains (N s ) equals the number of eges of triangles (N ege ). Substituting Eq. (2) into Eq. (1), the ege-base smoothe strain, ε, can now be written in the following matrix form of noal isplacements: (1) 1 ε L Φ xu B x u (3) T n s i si A in infl where Φ is the matrix of PIM shape functions; N infl is the number of fiel noes involve in constructing the smoothe strain fiels within an B x is terme as i the smoothe strain matrix that can be expresse as: x 0 0 b x b x 3 2 bix B x x b i iy iy ix (4) 378

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh where 1 bil x i nl l x, y A x x (5) Using a Gauss integration scheme, the above integration can be further expresse as follows: Nseg Ngau 1 bil x wn i xmn ni x m l x, y (6) A m1 n1 where N seg is the number of segments of the bounary Γ ; N gau is the number of Gauss points locate in each segment on Γ ; w n is the corresponing weight number of Gauss integration scheme. As linear shape functions are use in this wor, N gau =1 is aopte to perform the integration. Discretise System Equations For the ES-PIM moels, the following generalise smoothe Galerin (GS-Galerin) wea form is use to erive the iscretise system equations (Liu, 2008, 2010): T T T ε u D ε u u b u t 0 (7) which has exactly the same form as the stanar Galerin wea form. Thus, the formulation proceure is exactly the same as that in the stanar FEM, an all we nee to o is to use the ege-base smoothe strain ε in place of the compatible strain ε. The overall proceure of the presente methos is as follows. First, the isplacement fiel will be constructe by using PIM. Following this, the smoothe strains ε will be obtaine using Eq. (3). Finally, by substituting the assume isplacements an the smoothe strains into the generalise Galerin wea form (Eq. (7)), an invoing the arbitrary nature of the variation operations, a set of iscretise algebraic system equations can be obtaine in the following matrix form: Ku f (8) where f is the force vector that can be obtaine as: T T f Φ b Φ t (9) an the stiffness matrix K is assemble from the sub-stiffness matrix for all the integration cells, which are exactly the ege-base smoothing omains for the present metho. Ns Ns T ij ij A i j 1 1 K K B DB (10) 379

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 where the smoothe strain matrixes are obtaine using Eq. (4). Substituting the stiffness matrix K into the static system of equations: KU F (11) where F is the equivalent force vector. The vector U of the isplacements at all the noes thus can be calculate by Eq. (11) base on any methos, e.g. Gauss elimination, LU ecompositions an iterative methos (Liu an Que, 2003). Finally, the intereste variables, e.g. stresses an strains in the fiel, can be calculate base on the vector U following stanar proceures use as in the FEM. ESTABLISHMENT OF ASYMMETRIC GEAR MODELS Formulation of the Asymmetric Gear In this section, five asymmetric gear moels, each of which possesses a greater pressure angle on the rive sie an a normal pressure angle on the coast sie, were generate by a specially esigne rac cutter. Figure 2 presents the schematic of the specially esigne rac cutter which uses a larger pressure angle on one sie an a normal pressure angle on the opposite sie. Only one roun corner where there is only one raius (ρ) is esigne at the top of the rac cutter as the connection of two straight outlines of the rac to ensure that the fillet of the generate asymmetric gears is trochoial. The esign parameters of the rac cutter are liste in Table 1.. Figure 2. Profile of the specially esigne rac cutter with one fillet at the tip 380

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh Table 1. Parameters of the specially esigne rac cutter Parameter Symbol Quantitative value Moule m 5 mm Pressure angle on the coast sie Pressure angles on the rive sie Aenum coefficient on the coast sie c 20 20 /25 /30 /35 /40 h c 1 #1 Tip raius #2 Bottom clearance coefficient on the coast sie #3 Bottom clearance coefficient on the rive sie c c c ep hc m(tanc tan ) cos sec sin tan c c (1 sin c) m (1 sin ) m c c c #4 Aenum coefficient on the rive sie h h h c c c c #1,2,3,4 For ifferent, the quantitative values of, c, c, h are ifferent, an are liste in Table 2. c Table 2. Quantitative values of, c, c, h associate with special c Unit: mm 20 25 30 35 40 2.2424 1.9657 1.6592 1.3186 0.9390 c c 0.2951 0.2587 0.2183 0.1735 0.1236 c 0.2951 0.2270 0.1659 0.1125 0.0671 h 1 1.0317 1.0524 1.0611 1.0565 Base on envelope principles (Fetvaci an Imra, 2008; Su an Houser, 2000), the outlines of five asymmetric gear moels can be rawn as shown in Figure 3. The parameters of these five gear moels are list in Table 3. Note that the maximum bening stress occurs at the time when a single tooth carries the full loa. Therefore, 381

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 the selection of one single tooth contact is enough for an investigation into the maximum bening stress (Wang, 2003). a) Moel 1 with pressure angles 20 /20 b) Moel 2 with pressure angles 25 /20 c) Moel 3 with pressure angles 30 /20 ) Moel 4 with pressure angles 35 /20 e) Moel 5 with pressure angles 40 /20 Figure 3. Five asymmetric gear moels with moifie pressure angles (20, 25, 30, 35 an 40 ) on the rive sie an a normal pressure angle (20 ) on the coast sie Table 3. Basic parameters of the five asymmetric gears Moule Number of teeth Pressure angle Aenum coefficient Tooth with m =5 mm z 20 c 20 20 / 25 / 30 / 35 / 40 h ac 1 B 20 mm 382

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh Loa Application Figure 4 shows the loa F applie at HPSTC M which is locate on the rive sie of the asymmetric tooth. Here, the loa F is ivie into the tangent loa F t an the normal loa F n. As the transferre power is assume to be P=50 W an the rotation spee of the gear is assume to be n=1000 r/min in this analysis, the transferre tangent loa F t an the normal loa F n can then be calculate base on Eq. (14); thus, the exact values are liste in Table 4. The torque applie at the tooth is: T 9550 P (12) n The whole force applie at the tooth is: T T F B r cos B r M M b (13) an the tangent force an normal force applie at the tooth are: Ft Fcos Fn Fsin where B is the with of the gear; z is the tooth number of the pinion; FM FM (14) M is the pressure angle at point M, M r 2π 1 b arctan tan arccos r hac m ; z an is the z FM is the loa angle at point M, FM inv M M inv contact ratio on the rive sie. Figure 4. Force applie at the HPSTC on the rive sie of the asymmetric tooth 383

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 Table 4. Forces applie at the HPSTC with ifferent asymmetric gear moels Unit: N/mm Pressure angle Total force F Tangent force F t Normal force F n 20 /20 508144.89 481551.68 162231.99 25 /20 526862.96 476574.41 224636.17 30 /20 551369.51 471014.73 286624.24 35 /20 582919.87 465552.06 350794.59 40 /20 623331.98 460744.50 419830.04 COMPUTATIONAL INVESTIGATION OF THE ES-PIM (T3) METHOD In this section, the gear moel with the pressure angle of 20 /20, as shown in Figure, was investigate to test the ey properties of the present ES-PIM (T3), e.g. the accuracy of stresses, the convergence properties of the strain energy, the convergence rate of the strain energy norm an the computational efficiency. Figure 5. Gear moel subjecte to Dirichlet an Neumann bounary conitions An ES-PIM (T3) coe evelope in MATLAB was use to perform the bening stress analysis. For comparison, the FEM in-house coe was also evelope here. The properties of the material of the gear were E=2.16 10 11 Pa an v=0.3. A mesh with 521 irregularly istribute noes, as shown in Figure 6a, was generate for the analysis. As the exact stress was not available for the gear problem, a refine mesh with 10,233 irregular noes (see Figure 6b) was also generate to calculate the reference stress. Note that there as a higher noe ensity at the fillet in orer to pay special attention to the stress istribution at that location. The plane strain triangular element was use in this analysis. 384

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh a) Mesh with 521 noes b) Mesh with 10,233 noes Figure 4. Meshes generate for analysis a) Particular mesh near the loa point b) Concentrate loa applie at the HPSTC c) Distribute loa applie at the nine points associate with the HPSTC Figure 5. Loa istribution from a concentrate loa to a istribute loa The Dirichlet an Neumann bounary conitions were applie at the gear, as shown in Figure 5. For the Dirichlet bounary, the isplacements of relevant noes 385

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 were constraine in both the X an Y irections. For the Neumann bounary conition, the loas of F t =481551.67 N an F n =162231.99 N were applie at the HPSTC. In orer to avoi the infinite value because of the concentrate loa, the region near the HPSTC was particularly meshe (Figure 5a); the concentrate loa was linearly iscretise to nine extremely ajacent noes on a line associate with the HPSTC using the Gauss integration metho, as shown in Figure 5b an Figure 5c. Accuracy of the Stress Fiel It has been note that gear teeth usually fail by cracing at the base of the tooth on the tensile-stress sie (Wang, 2003). Therefore, it is meaningful to focus on the stress istribution at the fillet of the rive sie. Figure 6 presents the istribution of the von Mises stress at the fillet of the rive sie using both the present ES-PIM (T3) an FEM. From these three figures, it can be seen that, at the region near the loa point, the stress istribution of the ES-PIM (T3) was much closer to the reference istribution, which means that the ES-PIM (T3) can eal with a concentrate loa problem better than the FEM. The istribution of von Mises stresses from the solutions of the present ES-PIM (T3), the FEM an the reference mesh are shown in Figure 7. Figure 8 presents the curves of the stress istributions. From a comparison of the curves in these two figures, it can be seen that the present ES-PIM (T3) solution was much closer to the reference solution compare to the FEM, which means that the ES-PIM (T3) solution was more accurate than that obtaine from the FEM computation. Consiering the maximum von Mises stress, a 23.67% reuction in the error was foun with the ES-PIM (T3) compare to the FEM. Convergence of Strain Energy The strain energy inicator of a whole omain is efine as follows: 1 1 2σ ε 2σ D σ (15) T T 1 e exact = exact exact exact exact where σ exact is the exact stress vector of an arbitrary point of the omain; ε exact is the exact strain vector corresponing to the arbitrary point an D is the material constant matrix. Five sets of mesh (321 noes, 521 noes, 1102 noes, 1984 noes an 4225 noes) were generate in this subsection for analysis. As the exact strain energy cannot be irectly obtaine for the gear problem with complex outlines, a reference strain energy base on a refine mesh of 10,233 noes, which will replace e exact in Eq. (24), was also calculate. Table 5 lists the strain energies of the present ES-PIM (T3), the FEM metho an the reference solution. Figure 9 shows the convergence process of the solutions with an increase in the DOFs. From this figure, it can be seen that i) the strain energies of both the ES-PIM (T3) an FEM converge to the reference solution with an 386

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh increase in the DOFs; ii) the strain energy of the ES-PIM (T3) is much closer to the reference solution than that of FEM. It is well-nown that the linear FEM moel shows overly stiff behaviour an hence gives poor stress solutions. A softening effect has been introuce into the ES-PIM (T3) owing to the implementation of the strain smoothing operation, which maes it behave more softly an hence obtains more accurate results than the FEM moel (Liu an Zhang, 2008). a) ES-PIM (T3) solutions with 521irregular noes b) FEM solutions with 521irregular noes c) Reference solutions with 10,233 irregular noes Figure 6. Rainbow figures of the von Mises stress from ES-PIM (T3), FEM an reference solutions 387

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 Figure 7. Setch map of the von Mises stress at the fillet base on the present ES-PIM (T3) an FEM 240 220 FEM ES-PIM(T3) Reference Solu. 200 Mises stress: Mpa 180 160 140 120 100 80-7 -6.5-6 -5.5-5 -4.5-4 X coorinate: mm Figure 8. Comparisons of the von Mises stress at the fillet base on the present ES-PIM (T3) an FEM Table 5. Strain energy with ifferent DOFs Case DOF Strain energy of ES-PIM (T3) Strain energy of FEM 1 642 4.1680 3.9507 2 1042 4.2037 4.0429 3 2204 4.2945 4.1925 4 3968 4.3157 4.2385 5 8450 4.3459 4.2920 Reference 4.3591 (20466 DOFs) 388

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh 4.4 4.35 4.3 Strain energy 4.25 4.2 4.15 4.1 4.05 4 FEM 3.95 ES-PIM(T3) Reference solu. 3.9 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 Figure 9. Solutions (in strain energy) converging to the exact solution for the gear using both ES-PIM (T3) an FEM Convergence Rate of Strain Energy Norm The strain energy error norm inicator is efine as follows: e e e exact e e exact num DOF where e exact is the exact strain energy of the gear moel obtaine by Eq. (16) an e num is the numerical strain energy of the gear moel. Here, the e exact is also replace by the reference strain energy from a refine mesh as in section 4.2. The convergence rate in energy error norm, converging with the reucing average noal spacing (h), is plotte in Figure 10. From this figure, it can be foun that the convergence rate of the present ES-PIM (T3) of 0.99 is 1.46 times higher than that of the FEM of 0.68. 12 (16) -0.5-0.6-0.7 Log 10 (e e ) -0.8-0.9-1 -1.1-1.2 FEM(r=0.68) ES-PIM(T3)(r=0.99) -1.3-1.8-1.7-1.6-1.5-1.4-1.3-1.2-1.1 Log 10 (h) Figure 10. Convergence rate of the numerical results in the energy norm for the gear using both ES-PIM (T3) an FEM 389

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 Computational Efficiency A comparison of computational efficiency was investigate on the same computer (Dell PC with an Inter Pentium(R) 2.80GHz CPU, 1 GB of RAM) using moels with the same DOFs. In orer to carry out this stuy in an efficient manner an to test the computation time with reuce measurement error, a moel with four types of irregularly istribute noes (3010 noes, 10461 noes, 30286 noes an 44407 noes) was evelope an the MFree2D was use in a test where a well-coe banwith solver with an on-column storage technique was available (Liu an Liu, 2003; Wu et al., 2008; Chen et al., 2010). As is nown, the computation costs mainly come from two parts: the cost of solving the system equations an the cost of interpolation which mainly solves the inverse of moment matrix to form the shape functions. Table 6 an Table 7 list the CPU time require for these two parts (Chen et al., 2009), respectively. From these two tables, it can be seen that i) the CPU time of ES-PIM (T3) an FEM of interpolation was shorter than that of solving the system equations (especially as the number of noes became larger) an coul be neglecte for the whole computation, especially when the moel was huge with more DOFs; ii) the CPU time of ES-PIM (T3) for solving the system equations was 2-3.5 times longer than that of FEM. This was mainly because more local noes were selecte by ES-PIM (T3) to approximate the isplacement fiel, which mae the banwith of the global stiffness matrix two times greater than that of FEM (Wu et al., 2008; Chen et al., 2009). Actually, the CPU time of ES-PIM (T3) for solving the system equations shoul be four times greater than that of FEM base on banwith theory (Chen et al., 2009). Therefore, the test results agree very well with theoretical consierations. Although more CPU time (2-3.5 times more) was use by ES-PIM (T3), the ES-PIM (T3) metho stans out if we consier the computational efficiency (strain energy error norm/cpu time). Tests (Liu an Zhang, 2008; Liu et al., 2008; Chen et al., 2009) have inicate that the solution accuracy of ES-PIM (T3) is much better (by about 10 times) than the linear FEM with the same triangular mesh. Thus, ES-PIM (T3) is a much better choice for the calculation of mechanical problems. Table 6. Comparison of the CPU time (s) require to solve the system equations by ES-PIM (T3) an FEM with triangular bacgroun elements Unit: secons Case DOF ES-PIM (T3) FEM 1 7020 0.359 0.156 2 20922 3.807 1.331 3 60572 23.309 9.314 4 88814 63.391 19.22 390

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh Table 7. Comparison of the CPU time (s) of interpolation by ES-PIM (T3) an FEM with triangular bacgroun elements Unit: secons Case DOF ES-PIM (T3) FEM 1 7020 0.281 0.267 2 20922 1.227 0.767 3 60572 4.228 2.353 4 88814 4.478 3.622 INVESTIGATION INTO THE STRESS DISTRIBUTION The fillet stress istribution on the rive sie of an asymmetric gear was investigate in this section. The five asymmetric gear moels establishe in section 3 were utilise. The respective moels with their esign parameters can be foun in Figure 3 an Table 3. The materials assigne at the moels were all linear elastic, with a Young s moulus of E=2.16 10 11 Pa an a Poisson s ratio of v=0.3. During the analysis, the five gear moels were iscretise with approximately irregularly istribute noes an the equal noe space was set on the outline; the noe ensity was higher at the fillet with a noe spacing of 0.0003 mm an was normal in the other parts of the outline with a noe spacing 0.0007 mm. Five sets of refine meshes were generate, respectively, for their reference solutions. The noe information is liste in Table 8. The element type, Dirichlet an Neumann bounary conitions are the same as those use in section 4. Figure 11 shows the istribution of von Mises stress, an Figure 12 an Figure 13 present the comparisons of these stresses at the fillet. From these figures, it can be seen that i) the solutions from both ES-PIM(T3) an FEM gave the same trens of stress istribution; ii) the maximum bening stress ecrease with an increase in the rive sie pressure angle; iii) the fillet move to the negative irection of the X-axis an the length of fillet ecrease with an increase in the rive sie pressure angle an iv) the location of the maximum stress move to the top of the fillet with an increase in the pressure angle on the rive sie. Figure 14 presents a comparison of the maximum bening stress with ifferent pressure angles on the rive sie. It was again note that the maximum bening stress ecrease with an increase in the rive sie pressure angle. It can also be seen that the stresses calculate by ES-PIM (T3) were much closer to the reference values than those of the FEM, which means that the ES-PIM (T3) shoul give more accurate solutions than the FEM. The reuce errors were 23.67%, 26.31%, 25.90%, 25.41% an 29.73%, respectively, for the present ES-PIM (T3). This conclusion can verify the conclusions foun in section 4. Another conclusion rawn from Figure 14 is that the maximum bening stresses ecrease approximately linearly with an increase in the rive sie pressure angle. This curve can be synthesise in a straight line with the expression in Eq. (17): 391

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 a1) ES-PIM (T3) solutions for the 20 /20 moel a2) FEM solutions for the 20 /20 moel b1) ES-PIM (T3) solutions for the 25 /20 moel b2) FEM solutions for the 25 /20 moel c1) ES-PIM (T3) solutions for the 30 /20 moel c2) FEM solutions for the 30 /20 moel 1) ES-PIM (T3) solutions for the 35 /20 moel 2) FEM solutions for the 35 /20 moel e1) ES-PIM (T3) solutions for 40 /20 moel e2) FEM solutions for 40 /20 moel Figure 11. von Mises stress rainbow of ES-PIM (T3), FEM an reference solutions base on the five asymmetric gear moels with pressure angles 20 /20, 25 /20, 30 /20, 35 /20 an 40 /20 392

Accurate bening strength analysis of the asymmetric gear using the novel ES-PIM with triangular mesh Max 0.877 245.22 (17) where α is the pressure angle on the rive sie an σ Max is the maximum bening stress corresponing to α in MPa. Table 8. Noe information of the five asymmetric gear moels Gear moel Noes numbers for comparison Noe number for reference 20 /20 521 10233 25 /20 524 11100 30 /20 519 11061 35 /20 510 10919 40 /20 500 10928 Mises stress: Mpa 220 200 180 160 140 120 Stress comparision with ifferent pressure angles ---ES-PIM(T3) 20/20 25/20 30/20 35/20 40/20 100 80-8.5-8 -7.5-7 -6.5-6 -5.5-5 -4.5-4 X coorinate: mm Figure 12. Stress istributions at the fillet of the five asymmetric gear moels base on ES-PIM (T3) Mises stress: Mpa 220 200 180 160 140 120 Stress comparision with ifferent pressure angles ---FEM 20/20 25/20 30/20 35/20 40/20 100 80-8.5-8 -7.5-7 -6.5-6 -5.5-5 -4.5-4 X coorinate: mm Figure 13. Stress istributions at the fillet of the five asymmetric gear moels base on FEM 393

Wang et al. / International Journal of Automotive an Mechanical Engineering 4(2011) 373-396 Max mises stress: Mpa 235 230 225 220 215 210 205 200 195 Max stress comparision with ifferent pressure angles FEM ES-PIM(T3) Reference Solu. 190 185 20 25 30 35 40 Pressure angle at riven sie: Figure 14. Maximum bening stress comparison base on the present ES-PIM (T3) an FEM using the same coarse triangular elements CONCLUSIONS In this wor, newly esigne asymmetric gears which use moifie pressure angles on the rive sie an a stanar pressure angle on the coast sie were propose, an the ES-PIM (T3) metho using the bacgroun mesh of triangular cells was use to conuct asymmetric gear bening stress analysis. Through the formulation an numerical examples, it can be conclue that ES-PIM (T3) possesses significant avantages over the FEM in problem analysis with very complex geometries. More accurate solutions in the stress fiel were rawn by the ES-PIM (T3) than by FEM. Higher convergence, a higher convergence rate of the strain energy error an a higher computational efficiency were also obtaine by the ES-PIM (T3) in this analysis. The problem with stress concentration can be better solve by the present metho. With the attractive properties of goo accuracy, super-convergence an high computational efficiency, the ES-PIM (T3) is very convenient for bening strength analysis of complicate asymmetric gears, which involves complex outlines, stress concentration at the fillet an concentration loa at the HPSTC. ACKNOWLEDGEMENTS This wor is partially supporte by AStar, Singapore. It was also partially supporte by the Open Research Fun Programme of the State Key Laboratory of Avance Technology of Design an Manufacturing for Vehicle Boies, Hunan University, P. R. China uner the grant number 40915001. 394

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