ISSN 10526188, Journal of Machinery Manufacture and Reliability, 2015, Vol. 44, No. 3, pp. 227–231. © Allerton Press, Inc., 2015. Original Russian Text © F.I. Plekhanov, 2015, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2015, No. 3, pp. 43–49.
MECHANICS OF MACHINES
Deformability of Units of a Planetary Gear and its Effect on Load Distribution in Gear Meshes F. I. Plekhanov Glasov Engineering and Economical Institute, Branch of Izhevsk Kalashnikov State Technical University, ul. Kirova 36, Glasov, Udmurt Republic, 427622 Russia email:
[email protected] Received June 18, 2014
Abstract—A method is considered to determine the axis compliance of a planetary pinion subject to the deformation of the carrier web. It is revealed that the deformability of the axis, its mating parts, and sun gear affects the load distribution over power flows and rims of a tworow pinion. Variation factors for the load distribution in gear meshes are determined by solving of sets of displacement compatibility equations including the compliance of gear units and by its preparation error. DOI: 10.3103/S1052618815030164
Multipinion planetary gears with external and internal gears meshes are widely applied in engineering because of the high load capability, good weight–size parameters, and small power losses for friction. One of simplest and manufacturable structures of this gear contains axes cantilevered in the carrier with spher ical pinion bearings and two central gears [1, 2]. When the gear has a limited radial size, pinions are mounted in two to three rows (Fig. 1). The presence of redundant constraints in such structures leads to a nonuniform load distribution in gears meshes and a decrease in the multiengine effect. The implementa tion of the sun gear as a “floating pinion” allows for eliminating the nonuniformity of load distribution over pinions where their number is no greater than three. At the same time, the compliance of links of the planetary train (especially cantilever axes) promotes load balancing in meshes and, for corresponding parameters, may provide its distribution closed to uniform even for the comparatively small accuracy of the drive manufacture and in the absence of complex mechanisms for selfadjustment of units of the plan etary gear. In this connection it is significant to determine the compliance of the axis and its mating parts and reveal the degree of its effect on the variation factor of the load distribution in gear meshes on which the load capability depends generally. With allowance made for bending of the pinion axis under the applied forces and the increased com pliance of the internal race of the bearing at its faces, the axle load from the bearing can be represented by the equation (Fig. 2) πx πP πx q ( x ) = q m sin ⎛ ⎞ = sin ⎛ ⎞ , ⎝ l⎠ ⎝ 2l l⎠ where qm is maximum linear load (in the middle of a section with length l). Then, there are the bending moment M(x) = 0.5P l sin ⎛ πx ⎞ – x in an arbitrary section, transverse π ⎝ l⎠ force Q(x) = 0.5P 1 – cos ⎛ πx ⎞ , and axle deflection ⎝ l⎠ 2
3
2 Pl ⎛ 5 – 1.11Pl 0.5 l y 0 = 0.5π 3– 1⎞ + ⎛ 0.25 + ⎞ = 0.87q ( 1.65 + l ), ⎝ ⎠ ⎝ ⎠ π SG IE 96 E 2π
which are conditioned by them in the middle of the area of contact with the bearing race and determined via material resistance formulas using the Moore’s integral, where l = l/d, d is the diameter of the pinion axis; and q = P/l is the average linear load. 227
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PLEKHANOV
P y
ν
q(x)
ω(z)
ξ Fig. 1. Planetary gear with tworow pinions and self adjusting sun gear.
x z
The equation for the deformed axis in the zone of its conjugation with the carrier web has the fol lowing form:
b
l
Fig. 2. Schematic of loading of the pinion axis and the 2
2
carrier web. w ( z) = d y ( z) = M ( z) + 1.11 w ( z) , 1 d (1) 2 IE SG C dz 2 dz where C is the specific contact rigidity of the conjugation, which is found by the experimental method (C ≅ E/1.2 [3]); I is the axial moment of inertia; S is the area of the cross section of the axis; E and G are the
∫
z
modulus of elongation and the rigidity modulus, respectively; and M(z) = – w ( ν ) ( z – ν ) dν . 0
After the double differentiation of equality (1) there is 4
2
d w ( z ) 1.11C d w ( z ) C – + w ( z ) = 0, 4 SG dz 2 IE dz from this we have w ( z ) = C 1 sinh ⎛ αz ⎞ sin ⎛ βz ⎞ + C 2 cosh ⎛ αz ⎞ sin ⎛ βz ⎞ + C 3 sinh ⎛ αz ⎞ cos ⎛ βz ⎞ + C 4 cosh ⎛ αz ⎞ cos ⎛ βz ⎞ , (2) ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ ⎝ b⎠ where C 1.11 IEC C 1.11 IEC α = b 4 cos 0.5 arccos ⎛ ⎞ , β = b 4 sin 0.5 arccos ⎛ ⎞ . ⎝ 2FG ⎠ ⎝ 2FG ⎠ IE IE To determine integration constants C1–C4, the following boundary conditions and static equations are used: b
b
∫ w ( z ) dz = –P = –2F cos α , ∫ w ( z ) ( b – z ) dz = 0.5Pl;
−− W(z)
w
0
0
8
2
w ( z) = 1.11C at z = 0 d w ( z ); 2 SG dz – 0.5CPl + 1.11C w ( z ). IE SG
2
and z = b
d w ( z) = 2 dz
2
1
1
−6
Here, F is the normal force in opinion meshes with the frozen gear and the sun gear; and αw is the angle of gear meshes.
−13
Figure 3 represents the graph of changing of relative linear load 1 z W ( z ) = W ( z ) over the thickness of the carrier web at z = and q b
−20
4
0
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0.4
0.6
3
0.8
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−z
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CP (N/m−2) 169974144 84987072 0
Fig. 4. Computer model of the pinion axis.
various b = b/d at l = l/d = 1: (1) b = 2; (2) b = 1.5; (3) b = 1; and (4) b = 0.6. The axis displacement in the zone of the bearing mounting, which is caused by the compliance of the axis–carrier web conjugation, from equation (2) is 1.2 l dw ( z ) y H = w ( z ) + E 2 dz
z=b
⎧ = 1.2 ⎨ sinh α sin β C 1 + 0.5 l ( C 2 α – C 3 β ) E⎩ b
l l + cosh α cos β C 4 + 0.5 ( C 2 β + C 3 α ) + sinh α cos β C 3 + 0.5 ( C 1 β + α ) b b ⎫ l + cosh α sin β C 2 + 0.5 ( C 1 α – C 4 β ) ⎬. b ⎭ The validity of the analytical determination of the axis compliance subject to the deformability of the carrier web is supported by the finiteelement analysis of the stress–strain state of the pinion axis–carrier web conjugation in a Solid Works environment (Fig. 4). The difference in computation results by these methods is no greater than 10%. To determine the variation factor for the load distribution over opinions, displacement compatibility equations [2] are used, including in them, apart from errors of making and mounting of the gear, the axis deformation and its mating parts (carrier webs, rolling bearing): q1 l F 1 = = b w C w [ ε – δ 1 – ( y 1 + y n1 ) cos α w ], 2 cos α w ……………, qi l = b w C w [ ε – δ i – ( y i + y ni ) cos α w ], F i = 2 cos α w (3)
……………, qN l = b w C w [ ε – δ N – ( y N + y nN ) cos α w ], F N = 2 cos α w N
∑
i=1
l F i = 2 cos α w
N
∑q
i=1
i
Nl = q c , 2 cos α w
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KW(Δ) 1
1.38 1.25
2
3 4
1.13 1.00
0
10
20
30
40
Δ
Fig. 5. Variation factor of the load distribution over four pinions as a function of relative initial teeth gapping: (1) b = 2; (2) b = 1.5; (3)
where Fi is the normal force in the gear mesh with the ith pinion; qi is average linear load acting on the ith axis from the bearing race; qc is the average linear load acting on the pinion axis with the uniform load distribution over power flows; N is the opinion number; αw is the mesh angle (given set of equations corresponds to equality of gears mesh angles: αw = αwa = αwb); Cw is the specific mesh rigidity (this value is less changed with changing the mesh phase [4], espe cially for selfadjusting pinions, and the nearuniform load distribu tion over the tooth length therefore can be taken to be equal to approximately 0.075E [2]); δi is the initial teeth gapping (clearance) in meshes of the ith pinion with gears, which is conditioned by errors in the circumferential arrangement of axes and the teeth pitch (clearance in the internal mesh is equal to the clearance in the external mesh); bw is the width of the pinion rim; yi is the displace ment of the ith axis in the place of bearing mounting, which is caused its deformation and the deformation of the carrier web (yi =
y0i + yHi = qi f( l , b )/E); ε = const; and yni is the displacement of the ith pinion, which is caused by the compliance of the rolling bearing and the conjugation of its race with the axis (with allowance made for changing the bearing compliance in wide ranges depending on the load [5], its averaged magnitude is πq 5q taken, and then is yni ≅ i + i ). 2C E b = 1; and (4) b = 0.6.
Initial teeth gapping in gears meshes with the most loaded pinion is absent (δ1 = 0). Then, knowing δi corresponding to the degree of the accuracy of the gear manufacture, from equations (3), one can found qi, ε, and the variation factor of the load distribution KW = qmax/qc = ωnmax/ωn (ωn, ωmax are the average and maximum linear normal load in pinion meshes with gears). Figure 5 gives the graph of the dependence of the variation factor of the load distribution over pinions δE 2δE cos α of the fourpinion gear (N = 4) (Fig. 1) on Δ = = w at αw = 20°, l = 1, b w = bw/l = 1, δ1 = ωn qc δ2 = 0, δ3 = δ4 = δ (the most favorable case with respect to the load distribution), and various b . From the graphs it follows that at b = b/d = 0.6 the load distribution irregularity over pinion is small even for significant error in gear manufacture. Using tooth gears with compliant rims in planetary mech anism [6] makes it possible to reduce still more the negative effect of manufacture errors on the load dis tribution in meshes; however, in this case the bending strength of the rim may limit the load capability of the gear and to effect its lifetime negatively. A gear with pinions arranged in two rows has the load distribution irregularity over rims of the multip iece pinion, which is caused both by manufacture errors of the planetary mechanism and by the torsion of the sun gear under action of the moment applied to it. The variation factor of the load distribution over rims of the tworim pinion (Fig. 1) is determined from solving of the set of equations q1 l = b w c w [ ε – ( y 1 + y n1 ) cos α w ], 2 cos α w (4)
q2 l = b w c w [ ε – δ – y ϕ – ( y 2 + y n2 ) cos α w ], 2 cos α w q 1 + q 2 = 2q c .
Here, δ is the clearance between teeth of the second rim of pinion g and central gears a, b (δ = δgb = δga) with the close contact of teeth in meshes of the first rim of the pinion with central gears; yϕ is the difference JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY
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half of the teeth displacement of the sun gear in its cross sections by symmetry planes of pinion rims as a result of torsion, 2
0.5r ba b w y ϕ = 0.5ϕr ba = ( 0.875t 2 + 0.125t 1 ), Ip G where the linear torsion moment of the gear at the area of its conjugation with the pinion rim is ti = 0.5Nqirba/cosαw; rba is the radius of the main gear circumference; Ip is the polar inertia moment of the gear section; and ϕ is the difference of section torsion angles. Equations (4) are described for a case when the most loaded (first) pinion rim is located from the direc tion of the moment feed to the sun gear (unfavorable case), and the load is uniformly distributed over the width of single selfadjusting pinion rims and over pinions (“floating” sun gear, N = 3). Computations show that in the case of compliance (cantilever) pinion axes and relative sun gear width b a = 1–3, its torsion does not have a substantial effect on the load distribution over single pinion rim, and at b = b/d = 0.6 and Δ ≤ 30 the variation factor is no greater than 1.2. The given dependences and research results based on them can be used in designing both gears of the considered type (2KH) and other planetary mechanisms (including effective structures of type KHV gears with the roller mechanism for motion recovery from pinions [7, 8]), which makes it possible to choose rational parameters of the mechanical drive in accordance with the degree of the manufacture accuracy and to enhance its technical and economical characteristics. REFERENCES 1. Airapetov, E.L. and Genkin, M.D., Deformativnost’ planetarnykh mekhanizmov (Deformation Ability of Plane tary Mechanisms), Moscow: Nauka, 1973. 2. Kudryavtsev, V.N. and Kirdyashev, Yu.N., Planetarnye peredachi: spravochnik (Planetary Gearings. Handbook), Moscow: Mashinostroenie, 1977. 3. Plekhanov, F.I., Ovsyannikov, A.V., and Kazakov, I.A., Experimental research of deformation ability for plane tary gearing elements, Sb. Tr. n.tekhnich. konf. “Nauchnotekhnicheskie i sotsial’noekonomicheskie problemy regional’nogo razvitiya” (Proc. Sci.Tech. Conf. “ScientificTechnical and SocialEconomical Problems of Regional Development”), Glazov, 2010, pp. 76–78. 4. Kosarev, O.I., Bednyi, I.A., and Mamonova, M.G., The way to reduce vibration of tooth herringbone gearing, Vestn. Mashinostr., 2011, no. 11, pp. 19–24. 5. Nakhatakyan, F.G., Numerical determination of the elastic compliance of roller bearings using the Hertz the ory, J. Mach. Manuf. Reliab., 2011, vol. 40, no. 1, p. 23. 6. Kahraman, A., Ligata, H., and Singh, A., Influence of ring gear rim thickness on planetary gear set behavior, J. Mech. Design, 2010, vol. 132, p. 021002. 7. Plekhanov, F.I. and Gol’dfarb, V.I., RF Patent 2460916 IPC F16H 1/32, Byull. Izobret., 2012, no. 25. 8. Plekhanov, F.I., RF Patent 2492376 IPC F16H 1/32, Byull. Izobret., 2013, no. 25.
Translated by S. Ordzhonikidze
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