The effect of making the offset difference on the transmission performance of the face gear

1.1 Considering the influence of machining offset error The ideal coordinate line contact orthogonal face gear transmission is difficult to achieve due to machining error, assembly error and load deformation. Once the idealized line contact state is broken, the orthogonal face gear drive will produce edge contact, resulting in the occurrence of eccentricity of the orthogonal face gear drive. In order to avoid the above situation, the eccentric gear of the local point contact is realized by selecting the spur gear with the number of teeth less than 1 to 3 teeth of the plunging tool to mesh with the orthogonal face gear.
The machining offset error εP is the main error in the machining error of the orthogonal face gear. It refers to the shortest distance between the two axes when the tool rotation axis OPZP and the workpiece rotation axis OFZF are perpendicularly intersected in space. Based on the workpiece rotation axis OFZF, the tool rotation axis OPZP is “ ” when it is biased to the right side of the reference, and “-” when it is biased to the left side. The machining contact offset error takes into account the influence of the machining offset error on the point contact orthogonal face gear transmission coordinate system as shown. Among them, SM-XMYMZM is the medium gear static coordinate system, SP-XPYPZP is the machining offset error coordinate system, SM'-XM'YM'ZM' is the medium gear following coordinate system, and the medium gear rotates around the axis OM'ZM'. The angular velocity is ωM, the rotation angle is φM; SG-XGYGZG is the cylindrical gear static coordinate system, SG'-XG'YG'ZG' is the cylindrical gear following coordinate system, the cylindrical gear rotates around the axis OGZG, the angular velocity is ωG, and the rotation angle is φG; SF-XFYFZF is the orthogonal coordinate gear coordinate system, SF'-XF'YF'ZF' is the orthogonal face gear following coordinate system, the orthogonal face gear rotates around the axis OFZF, the angular velocity is ωF, the rotation angle is φF; The error coordinate system is established by the medium gear static coordinate system along the XM axis translation εP distance; the X, Y and Z axes of the cylindrical gear static coordinate system are in the same direction as the X, Y and Z axes of the medium gear static coordinate system, and the cylindrical gear is static. The OGXGYG plane of the coordinate system is coplanar with the OMXMYM plane of the static coordinate system of the medium gear, and the origin of the two coordinate system is a, a is the center distance of the spur gear and the medium gear; the motion relationship between the medium gear and the orthogonal gear determines the medium gear OFXFZ in OMXMZM plane and orthogonal face gear static coordinate system in static coordinate system The F planes are parallel and the height is the media gear tip circle radius ra, the plane OMXMYM and the plane OFYFZF are parallel to each other and the distance is d, d is the positioning reference of the media gear tooth width center plane, and d follows the medium gear relative to the orthogonal face gear The initial position changes and changes.
Media gear spur gear orthogonal gear machining tool actual position Brown F gear transmission coordinate system from the coordinate system SM' to SP, the homogeneous transformation matrix is ​​MPM' = cosφM-sinφM00sinφMcosφM00010001, the homogeneous transformation matrix from the coordinate system SP to SM is MMP=100-εP010000100001, the homogeneous transformation matrix from coordinate system SM to SF is MFM=00-1-d-100-10ra01, and the homogeneous transformation matrix from coordinate system SF to SF' is MF'F=cosφF-sinφF00sinφFcosφF00010001 .
Therefore, considering the influence of the machining offset error, the homogeneous gear transformation matrix of the orthogonal gear coordinate system is MF'M'=MF'FMFMMMPMPM'.
The homogeneous transformation matrix from the coordinate system SG' to SG is MGG'=cosφG-sinφG00sinφGcosφG00010001, and the homogeneous transformation matrix from the coordinate system SG to SM is MMG=1000010a00100001. Since the assembly relationship does not consider the influence of error, the spur gear is converted to the medium gear. When the coordinate systems SP and SM are completely coincident, the homogeneous transformation matrix from the coordinate system SM to SM' is M'M=cosφMsinφM00-sinφMcosφM00010001, so the homogeneous transformation matrix of the spur gear to the medium gear is M'G'= MM'MMGMGG'.
1.2 The contact point equation in the transmission considering the influence of machining offset error is the involute spur gear, and its tooth profile surface coordinate system is as shown. Where rb is the base circle radius of the medium gear, rk is the radial diameter at any point k on the tooth profile, and θk is the angle between rk and the y axis.
yMrkz k purple MrbxM Therefore, the media gear tooth profile equation is xM=±rksinθk, yM=rkcosθk, zM=uk.
Where: θk=π/2z-invα invαk, invα=tanα-α, invαk=tanαk-αk, α is the medium gear index circle pressure angle, αk is the pressure angle at k point, uk is the medium gear tooth width variable, " " in xM
For the left tooth profile, the "-" in xM is the right tooth profile. Therefore, the homogeneous matrix of the medium gear tooth profile is RM=[xMyMzM1]T.
Since the medium gear and the orthogonal face gear satisfy the envelope relationship, the transformation of the medium gear through the coordinate system SM' to SF' results in a family equation of RZM=MF'M'RM.
According to the envelope principle, the envelope condition is RZMαk×RZMukRZMφM=0.
(2) The equations (1) and (2) are connected in series, that is, the contact line equation of the medium gear and the orthogonal face gear in the transmission considering the influence of the machining offset error is obtained.
Since the spur gear and the medium gear also satisfy the envelope relationship, the cylindrical gear is transformed by the coordinate system of SG' to SM', and the equation of the family is RZG=MM'G'RG. (3) According to the envelope principle, The envelope condition is RZGαk×RZGukRZGφM=0.
(4) where: RG = [xMyMzM1] T. The equations (3) and (4) are connected in series, that is, the contact line equation of the spur gear and the medium gear in the transmission considering the influence of the machining offset error is obtained.
The contact line equation of the medium gear and the orthogonal face gear in the transmission is connected with the contact line equation of the spur gear and the medium gear, so that the contact point equation in the transmission considering the influence of the machining offset error is obtained RZM=MF'M'RM, RZMαk ×RZMukRZMφM=0, RZG=MM'G'RG, RZGαk×RZGukRZGφM=0.
1.3 Machining offset error The offset error of the orthogonal face gear in the transmission causes the orthogonal gear tooth profile to shift circumferentially. When the machining offset error is positive, the orthogonal gear tooth profile is offset in a clockwise direction based on its axis of rotation; when the machining offset error is negative, the tooth profile is offset in a counterclockwise direction; The offset increases as the machining offset error increases. The machining offset error has little effect on the geometry of the orthogonal face gear profile. At the same time, it has a symmetrical relationship with the corresponding contact points on the left and right tooth profiles of the orthogonal face gear in the transmission and the contact point with respect to the tooth profile of the orthogonal face gear. The impact of location is not obvious.
2 Influence of machining offset error on the principal curvature and contact characteristics at the contact point in the transmission 2.1 Influence of machining offset error on the principal curvature at the contact point The spur gear and the orthogonal face gear cylinder are used in the point-to-point contact orthogonal face gear transmission. The first basic amount and the second basic amount of the tooth profile surface of the gear and the orthogonal face gear.
According to the surface method curvature solution method, the main directions of the contact points of the spur gear and the orthogonal face gear are substituted respectively, that is, the principal curvature kGI, kGII at the contact point of the spur gear and the principal curvature kFI, kFII at the contact point of the orthogonal face gear are obtained. .
According to the calculation method of the principal curvature at the contact point, the influence of the machining offset error on the contact characteristics at the contact point of the spur gear and the orthogonal face gear 2.2 when the point is in contact with the orthogonal surface gear transmission due to the machining offset error The contact analysis is based on the solution of the Bushinesk problem and establishes the basic equation for the elastic contact problem. According to the analysis of the geometric relationship of the surface of the contact object, the points with the same distance on the surface of the object form an elliptical region on the common cut surface, so the distribution of the load Fm in the contact area satisfies z=3Fm2πρxρy1-x2ρ2x-y2ρ2y. In the formula: ρx and Ρy is the long radius and short radius of the contact ellipse.
From the relationship between the principal curvature of the contact point and the elastic coefficient of the two elastic bodies and the contact area, the ellipse length and short radius of the contact area should satisfy ρx=u33Fm(θG θF)8(kGI kGII kFI kFII), ρy=v33Fm(θG θF) 8 (kGI kGII kFI kFII).
Where: u and v coefficients are elliptic integral coefficients, which can be obtained by looking up the elliptic integral coefficient table [8]; θi is calculated by equation (6): θi=4(s2i-1)/(Eis2i).
Where: i is G and F, respectively, si is the coefficient of the ratio of the longitudinal extension of the material to the transverse compression, and Ei is the elastic modulus of the material.
The maximum compressive stress equation in the contact region is σmax=0.92uv3Fm(kGI kGII kFI kFII)2(θG θF)2, and the maximum elastic deformation amount of the contact region is δmax=3JFm(θG θF)/(8πρx). Where: J is the elliptic integral coefficient, and the solution method is the same as the coefficients u and v.
According to the contact characteristic equation, for the orthogonal gears in the transmission considering the influence of the machining offset error, it is assumed that the force at the contact point is 3000N; the longitudinal extension and the lateral compression ratio of the cylindrical gear and the orthogonal gear are 3; The elastic modulus of the spur gear and the orthogonal gear is 210GPa. It is known from the calculation results that the machining offset error has little influence on the contact characteristics of the point contact orthogonal face gear transmission. When the machining offset error is " " and gradually increases, the maximum compressive stress on the left tooth profile of the orthogonal face gear will increase slightly, but the maximum compressive stress at the tip of the tooth will decrease slightly; the maximum compressive stress on the right tooth profile will Slightly reduced, but the maximum compressive stress at the top of the tooth is slightly increased. Considering the influence of machining offset error, the maximum compressive stress on the left and right tooth profiles of the orthogonal face gear in the transmission is compared.
3 Conclusions 1) The machining offset error has a significant effect on the circumferential tooth distribution of the orthogonal face gear, but has no significant effect on the geometric tooth profile of the orthogonal face gear and the position of the contact point in the transmission relative to the orthogonal face gear.
2) Machining offset error to the orthogonal face gear 3) When the machining offset error is " " and gradually increases, the maximum compressive stress on the left tooth profile of the orthogonal face gear will increase slightly, but the rate of change will be slightly reduced. The maximum compressive stress on the right tooth profile will be slightly reduced, but the rate of change will increase slightly.
In summary, the contact characteristics of the point contact orthogonal face gear drive are not sensitive to the machining offset error of the orthogonal face gear.

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