The gear contact stress formula has a history of more than 100 years, and it is widely used in many aspects such as gear transmission and tooth surface elastic fluid dynamic pressure lubrication. In the derivation of the contact stress, the two tooth faces are replaced by an equivalent cylinder and then expanded into a 2nd order paraboloid. If the contact area is small, the error is not large; but if the contact area is very deformed, especially when the two involutes are meshed, the contact area is large, and it is not enough to expand to the 2nd order. In this paper, the involute is expanded to the third order, and the corresponding contact stress formula is derived. At present, the commonly used method is to regard the vicinity of the gear contact point as a cylinder whose radius is equal to the radius of curvature, and then directly use the Hertz formula.
The 3rd order approximation of the involute is defined by the Law of Dau, and any curve can be expanded at a certain point: r(u)=r(u0) dduru 12dduru2 13dduru3r (1) for the involute equation: r=[rb(sin -cos)rb(cos sin)](2)(2) where rb is the base circle, is the involute angle, and r is the radial diameter. Derivation: r=rb[sincos](3), then the tangential and normal vectors are: =[sincos]n=[-cossin](4) and then derivation: r=rb[sin coscos-sin]3 derivations :r=[2cos sin-(2sin cos)] The distance from the curve to its tangent: MM0=r(u)-r(u0)=dduru 12dduru2r 13dduru3 (5) Since: rn=0rn=-rrn=-2rb :=-rb2/2-(2rb)3/6(6) establishes the x coordinate along the tangent and establishes the y coordinate along the normal. Arc length: s=rr=rb=x, then =x/rb=-x22rb-x36r2b3(7) Relative inferior surface of the two involutes at the meshing point: r=-121rb11-1rb22x2-131r2b131-1r2b232x3=-k2x2-k3x3 (8) k2, k3 are coefficients of the 2nd and 3rd terms.
2 Using the Chebyshev polynomial to find the contact stress to introduce the relative difference surface and the comprehensive elastic abbreviation mark = 2 (1-21) E1 2 (1-2) E2, the contact problem is transformed into the rigid body press in the formula (8) Contact stress in an infinite half plane. Due to the asymmetry of equation (8), the contact stress and the contact area are also asymmetrical. Let the contact zone occur in the segment -bxa, where a, -b are the two roots of equation (8) closest to the origin.
The gear asymmetric contact introduces a transformation: x = lc, y = l (9) where l = (ab) / 2, c = (ab) / 2, when -bxa, -11 is substituted into (8): = y / l=k2l(lt c)2 k3l(lt c)3=(k2c2 k3c3)/l (2k2c 3c2k3)t (k2l 3lck3)t2 k3l2t3(10) The displacement is now expanded with Chebyshev polynomial:=y/l =Nn=0bnTn(t)=b0T0 b1T1(t) b2T2(t) b3T3(t)=b0 b1t b2(2t2-1) b3(4t3-3t)(11) By comparing the corresponding coefficients, coefficients such as b0b3 can be determined.
According to the research by GLGladwell and Popov et al. [3], when the contact surface displacement is expressed by the Chebyshev polynomial, the pressure distribution of the contact surface can be expressed in the form of closed integral and also by Chebyshev polynomial. When w=-d Nn=0bnTn(), the contact stress p(t)=(1-t2)Nn=0anTn(t)(11)(11) where: an=-1nbn, (11) can be rewritten :(1-t2)-1/2p(t)=anTn(t)(12) Obviously, when t=1,-1, the right side of (12) is zero, and the formula (12) is rewritten as (1-t2). -1/2p(t)=anTn(t)-anTn(1)=a0[T0(t)-T0(1)] a1[T1(t)-T1(1)] a2[T2(t)-T2 (1)] a3[T3(t)-T3(1)]=a1(t-1) 2a2(t2-1) 4a3t(t2-1)p(t)=-1(a1 a3)t-1t 11 /2 2a(t2-t)1/2 4a3t(1-t2)1/2 The total load is equal to F, and the relationship between a, b and F is determined. If the 3 term is ignored, the Hertz stress formula is re-returned. Using the formula and conclusions of this paper, many conclusions and formulas such as gear lubrication mechanics, gear elastic fluid dynamic pressure lubrication, etc. can be re-examined. Therefore, I will not go into details.
3 Example: An object with a parabolic bottom surface with a radius of curvature R, the shape of which: -ya=x22aR=at22R=a(2t2-1)4R a4R=a4R[T0(t) T2(t)] B2=-a4R, p(t)=2P0(1-t2)1/2a, P0=a22R, a=2RP0 is the same as the Hertz formula. Pmax=p(0)=2P0a=2P0R Set a pair of gears d1=70mm, d2=230mm, b=70mm, i=3.28, elastic coefficient 1=189.8/MPa, and the comprehensive radius of curvature is R=d1isin2(i 1)= 9.178mm, the load per unit line length P0=64.17N/mm, the maximum contact stress pmax=499.75N/mm, and the maximum contact stress calculated by the third-order expansion is equal to 52974N/mm, and the stress is increased by about 6.
4 Conclusions The improvement of gear contact stress is multi-faceted. The main point of this paper is that the tooth surface should not be simply viewed as a cylinder, but should be regarded as a third-order parabola, which can improve the calculation accuracy. The results and methods herein can be applied to other aspects.
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