CN114048560A - Zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method - Google Patents

Zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method Download PDF

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CN114048560A
CN114048560A CN202111264708.4A CN202111264708A CN114048560A CN 114048560 A CN114048560 A CN 114048560A CN 202111264708 A CN202111264708 A CN 202111264708A CN 114048560 A CN114048560 A CN 114048560A
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蒋进科
杨创战
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Changan University
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Abstract

The invention provides a zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method, and relates to a tooth surface vibration reduction and noise reduction method. The new method compensates the initial contact clearance of different meshing positions of the modified gear teeth, so that the deformation in the meshing period is equal to the maximum deformation before uncompensation, namely ALTE of the meshing position is zero, and then the compensation clearance is reversely solved by improving the original LTCA solving method. The size of the compensation gap is constrained by the meshing position of the tooth surface, the compensation gap is superposed on the preset easy-off tooth surface to form a zero ALTE modification tooth surface, and the zero ALTE modification tooth surface has good fusion and is beneficial to processing. The method ensures that the spiral bevel gear not only meets the requirement of high strength under a small proper amount, but also eliminates rigidity excitation, fundamentally improves the dynamic performance of the gear pair, and provides a method for designing and analyzing a high-performance tooth surface; meanwhile, relevant software is compiled, and the product design efficiency can be improved.

Description

Zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method
Technical Field
The invention relates to the technical field of tooth surface design, in particular to a zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method.
Background
The spiral bevel gear has the advantages of high contact ratio, high bearing capacity and the like, is widely applied to crossed or staggered shaft gear transmission systems of aviation, navigation, vehicles and the like, and has complex geometry and high processing difficulty. In the design of the Tooth surface, Tooth surface Contact quality and Transmission dynamic performance are controlled mainly by Tooth surface static Analysis including Contact Analysis (TCA) and Tooth surface bearing Contact Analysis (LTCA) to obtain Tooth geometry Transmission Error, Tooth surface impression, bearing Transmission Error (LTE) and load. The LTE reflects the deviation condition of actual meshing and ideal meshing of the gear, and the size of the amplitude of transmission error (ALTE) borne by the gear is the direct excitation of vibration in the working process and is an important factor for generating vibration and noise. Although ALTE may reduce vibration, ALTE reduction also tends to result in increased high frequency components in the transmission error curve, creating new vibrations. To reduce vibration and noise more effectively, it is necessary to further eliminate transmission error excitations (stiffness excitations) to reduce ALTE to zero.
The size of the gear LTE is greatly related to the actual contact ratio and the geometric parameters of the loaded gear teeth, the actual contact ratio of the gear teeth can be changed by gear tooth modification, and the method is an effective way for reducing the gear ALTE. The traditional tooth surface design and processing method focuses on correcting parabolic second-order, fourth-order and high-order transmission error tooth surfaces of motion parameters of the shaking table type machine tool, ALTE is obtained through calculation of TCA and LTCA, and tooth surface processing parameters are obtained by optimizing the minimum ALTE. The Ease-off tooth surface modification technology is based on the Ease-off tooth surface modification technology, the Ease-off modification curved surface is obtained by mainly modifying the machining parameters of a shaking table type machine tool, and then the contact area matching verification is carried out by combining a TCA method under light load, so that the design problem of the zero ALTE tooth surface is not involved. In addition, the conventional LTCA numerical method cannot reversely design the modified tooth surface by designing LTE of the meshing position as a constant value. Therefore, a new technical scheme is particularly provided.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a zero-bearing transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method, which is expected to meet the high-strength requirement under a small proper amount, simultaneously eliminate rigidity excitation fundamentally, further improve the dynamic characteristic of a gear pair and provide a method for high-performance tooth surface design and analysis.
The invention is realized by the following technical scheme:
a zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method comprises the following steps:
step 1, determining a small wheel tooth surface completely conjugated with a large wheel according to a gear meshing principle;
step 2, a normal trimming curved surface is designed, and the normal trimming curved surface is superposed on the conjugate pinion tooth surface to obtain a pinion preset Ease-off trimming tooth surface;
step 3, determining the maximum bearing deformation when the preset Ease-off modified tooth surface is meshed with the big gear tooth surface by adopting a gear bearing contact analysis method;
step 4, reconstructing a gear bearing contact analysis equation to enable the bearing deformation in the meshing period to be equal to the maximum bearing deformation, and obtaining an additional tooth space gap of a preset Ease-off modification tooth surface by reversely solving the bearing contact analysis equation;
step 5, converting the additional inter-tooth gap into an additional geometric transmission error of the gear pair, and determining the geometric transmission error of the zero ALTE gear pair according to the additional geometric transmission error of the gear pair;
and 6, according to the relation of the geometric transmission error of the zero ALTE gear pair, generating small wheels by using a large wheel to obtain the zero ALTE small wheel modified tooth surface.
Preferably, the bit vector R of the tooth flanks of the small wheel which is completely conjugated in step 110And vector N10The following were used:
Figure BDA0003326606260000021
wherein the tooth surface position vector r2、n2The original tooth surface phase vector and the normal vector of the big wheel are respectively; u and beta are parameters of any point on the tooth surface; m1p、Mpq、Mqr、Mrs、Ms2For a coordinate transformation matrix, L1p、Lpq、Lqr、Lrs、Ls2Is the corresponding 3 × 3 sub-matrix.
Preferably, in step 2, in combination with the interdental space and the contact line normal space, a modified curved surface reflecting the contact line normal space is superimposed on the small wheel tooth surface only containing the transmission error, so as to obtain the bit vector R of the small wheel preset Ease-off tooth surface1rVector N1rComprises the following steps:
Figure BDA0003326606260000022
wherein R is1、N1The gear surface phase vector and the normal vector of the small wheel only contain geometric transmission errors; deltacIs a contact line normal clearance curved surface; u and beta are tooth surface parameters.
Preferably, in step 3, the compliance coefficient of the gear pair is calculated by a finite element numerical method, the initial contact gap of the gear pair is calculated according to a tooth surface geometric contact analysis method, the compliance coefficient of the gear pair and the initial contact gap of the gear pair are used as input data of the LTCA, the forced contact of the gear pair is converted into solving of the mechanical balance problem of a finite number of discrete contact points of the tooth surface according to the initial gap and the compliance coefficient of the multi-tooth pair at the meshing position, under the action of a load P, instantaneous 2 pairs of tooth contacts are set, the gear teeth generate elastic deformation, the small wheel is fixed, the large wheel moves along the normal direction under the action of the load to displace Z, the friction of the tooth surface is ignored, and the linear contact is generated along the major axis of an instantaneous contact ellipse, and the bearing contact equation of the tooth surface is as follows:
Figure BDA0003326606260000031
wherein p isjk(j ═ 1,2, … n) is the normal load at discrete point j of the instantaneous contact circle major axis of tooth pair k; djk(j ═ 1,2, … n) is the tooth flank clearance after deformation at the discrete point j of the major axis of the instantaneous contact ellipse of tooth pair k, dkIs djkThe set of vectors; z is the normal displacement vector of the gear teeth, namely the bearing deformation;Fkis a normal compliance matrix, w, of tooth pairs kkAnd for the initial contact clearance vector of the tooth pair, converting Z into displacement on the meshing line after solving the tooth surface bearing contact equation to obtain the maximum bearing deformation of one meshing period.
Preferably, the coordination equation of the displacement in the new gear load contact analysis equation set in step 4 is as follows:
Fkpk+wk+ek＝Z+dk k＝Ι,ΙΙ
the new LTCA equation system is a nonlinear programming formed by the known parameter F, P, W, Z and the unknown parameters p, d and e, and the compensation initial contact clearance e of the gear tooth meshing position can be obtained by inverse solution.
Preferably, in the step 5, the additional inter-tooth space is converted into angular displacement on a meshing line to obtain additional geometric transmission errors of different meshing positions of the tooth surface, and the additional geometric transmission errors of n meshing positions of the tooth surface are fitted by adopting a cubic uniform B spline to further obtain a geometric transmission error of a zero ALTE gear pair.
Preferably, the expression of the zero ALTE gear pair geometric transmission error relationship is as follows:
θ2＝z1/z2(θ1-θ10)+θ20+ψe(θ1)
preferably, the data of n meshing positions of the tooth surface are fitted by adopting cubic uniform B splines, and the geometric transmission error psi is addedeThe expression of the curve is as follows:
Figure BDA0003326606260000032
in the formula, xiIs a small wheel corner, ViControl point column vector, i ═ 1 … n-1; m is a constant matrix.
Preferably, the zero ALTE small-wheel modified tooth surface position vector R1zAnd vector N1zIs determined by superimposing the backlash caused by the drive error on the preset modified tooth surface R1rTo obtainZero ALTE modification tooth surface position vector and normal vector R1z、N1zAfter simplification, it can be expressed as:
Figure BDA0003326606260000041
in the formula R1Ψ、N1ΨFor small wheel tooth surface position vector and normal vector including preset geometric transmission error and additional geometric transmission error, R1Ψ、N1ΨThe expression of (a) is as follows:
Figure BDA0003326606260000042
compared with the prior art, the invention has the following beneficial technical effects:
the invention provides a zero-bearing transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method which comprises the steps of firstly deducing a small gear tooth surface completely conjugated with a large gear according to an engagement principle, designing a free Ease-off curved surface according to a tooth space gap and tooth surface normal gap generation principle, and superposing the free Ease-off curved surface and the conjugated tooth surface to represent a preset Ease-off tooth surface of the small gear. Secondly, a novel low temperature co-fired ceramic (LTCA) numerical method based on zero ALTE is provided by combining a tooth surface bearing contact analysis (LTCA) numerical method; the novel LTCA principle is that tooth surface initial clearance of a contact position is obtained according to a certain step length by combining tooth surface contact analysis (TCA), then after tooth surface maximum deformation is obtained according to an original LTCA method, bearing deformation of other meshing positions is set to be equal to the maximum deformation through a mechanical balance equation, the flexibility coefficient and the initial contact clearance of the meshing positions are used as known quantities, and additional tooth space clearance and load of the meshing positions are reversely solved through improvement of the original LTCA solving method. The size of the gap between the additional teeth is restricted by the meshing position of the tooth surface, the additional teeth are superposed on the preset easy-off tooth surface to form a zero ALTE modified tooth surface, and the additional teeth have good fusion and are convenient to process. The invention ensures that the spiral bevel gear not only meets the requirement of high strength under a smaller proper amount, but also can eliminate the rigidity excitation of specific working conditions, fundamentally improves the dynamic performance of the gear pair, and provides a method for designing and analyzing the high-performance tooth surface.
The method carries out free Ease-off modification tooth surface active design according to the principle of generating the tooth space gap and the tooth surface normal gap, ensures that the spiral bevel gear meets the high-strength requirement under a small proper amount, can eliminate rigidity excitation, fundamentally improves the dynamic characteristic of a gear pair, and provides a theoretical method for high-performance tooth surface design and analysis. And (3) the zero ALTE-based trimming spiral bevel gear tooth surface LTCA numerical simulation software is developed, and the product design efficiency is improved.
Drawings
FIG. 1 is a hypoid gear mesh coordinate system of the present invention.
FIG. 2 is a preset second-order transmission error curve, a preset contact line modification curve and a curved surface mapped by the curve.
FIG. 3 is a LTCA computational model of the present invention.
FIG. 4 is a flow chart of the tooth surface active Ease-off modification method of the spiral bevel gear with zero bearing transmission error amplitude of the invention.
FIG. 5a1 shows an Ease-off curve for a pre-set profile tooth surface according to the present invention.
FIG. 5a2 is a TCA simulation of a pre-contoured tooth surface in accordance with the present invention.
FIG. 5b1 shows the additional backlash and additional geometric drive error for a zero ALTE tooth surface 1 according to the present invention.
FIG. 5b2 shows additional tooth space for a zero ALTE tooth surface 1 in accordance with the present invention.
FIG. 5c1 shows the additional backlash and additional geometric drive error for a zero ALTE tooth surface 2 according to the present invention.
Fig. 5c2 shows additional tooth space for a zero ALTE tooth surface 2 of the present invention.
FIG. 5d1 shows the additional backlash and additional geometric drive error for the zero ALTE tooth surface 3 of the present invention.
FIG. 5d2 shows additional tooth space for a zero ALTE tooth surface 3 according to the present invention.
FIG. 5e1 shows the additional backlash and additional geometric drive error for a zero ALTE tooth surface 4 according to the present invention.
Fig. 5e2 shows additional tooth space for a zero ALTE tooth surface 4 in accordance with the present invention.
FIG. 5f1 shows the Ease-off curve for a zero ALTE tooth surface 2 of the present invention.
FIG. 5f2 is a TCA simulation of a zero ALTE tooth surface 2 of the present invention.
Fig. 5g1 shows the multi-load condition LTE curve of each preset modified tooth surface according to the present invention.
Fig. 5g2 is a multi-load condition LTE curve for each zero ALTE tooth face 2 of the present invention.
FIG. 5h is a comparison of the ALTE curves for multiple load conditions for each zero ALTE tooth surface of the present invention.
Detailed Description
The following takes a hypoid gear processed by an HFT method as an example, and the specific implementation of the zero-bearing transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method is further described by combining the accompanying drawings, and the method comprises the following steps:
step 1: according to the gear meshing principle, the small wheel tooth surface completely conjugated with the large wheel is determined.
According to the geometric parameters of the gear pair, the large tooth surface is obtained by initially calculating the large wheel machining parameters through a local synthesis method. When the small gear tooth surface is meshed with the large gear tooth surface, the small gear tooth surface and the large gear tooth surface are completely conjugated, the transmission ratio is equal to the nominal transmission ratio of the gear pair, and then the large gear is meshed at a rotating angle theta2Angle of rotation theta of engagement with small wheel1The relationship of (1):
θ2＝z1/z2(θ1-θ10)+θ20 (1)
in the formula [ theta ]10，θ20Designing the meshing rotation angles of the large wheel and the small wheel at the reference point; z is a radical of1And z2The number of teeth of the small gear and the large gear is respectively. The coordinate system of the big wheel and the small wheel is shown in figure 1, and the coordinate system Ss、SrAnd SqAs a reference coordinate system, S2A large-wheel moving coordinate system; v is the offset, H1、H2Distances from the vertex of the small and large wheel cone to the intersection point are respectively, and sigma is an axis intersection angle; the big gear surface is taken as an imaginary gear cutter, and is converted into a small gear surface coordinate system S based on the space meshing theory and coordinate transformation1Middle, fully conjugated small wheel tooth surface potential vector R10Vector N10Respectively as follows:
Figure BDA0003326606260000061
wherein the tooth surface potential vector r2、n2The original tooth surface phase vector and the normal vector of the big wheel are respectively; u and beta are parameters of any point on the tooth surface; m1p、Mpq、Mqr、Mrs、Ms2For a coordinate transformation matrix, L1p、Lpq、Lqr、Lrs、Ls2Is the corresponding 3 × 3 sub-matrix.
Step 2: the small wheel preset Ease-off tooth surface is obtained by designing a normal trimming curved surface and superposing the normal trimming curved surface on the basis of a conjugate tooth surface.
By designing the normal modification curved surface and superposing the normal modification curved surface on the basis of the conjugate tooth surface, the position vector and the normal vector of the preset Ease-off tooth surface of the small wheel are obtained. The normal shape-modifying curved surface is the initial contact clearance of the gear pair. The initial contact clearance of the tooth pair can be subdivided into the sum of the tooth clearance of the tooth and the normal clearance of the contact line, the actual contact ratio of the loaded tooth has great influence, and the initial contact clearance of the conjugate tooth surface (the initial clearance of the conjugate tooth surface is zero) can be changed by the modification. The geometric transmission error reflects the size of the gap between teeth, and can change the tooth surface load distribution and the bearing deformation among different meshing positions, and the influence on the vibration is larger.
The length and the contact path of the contact line can be changed by the normal clearance of the contact line, so that certain edge stress concentration is avoided; both of them have influence on the sensitivity of installation errors and reflect different aspects of transmission performance, so that when designing small wheel Ease-off tooth surfaces, the design of comprehensive tooth space and contact line normal space is needed. The modification curved surface reflecting the normal clearance of the contact line is superposed on the small wheel tooth surface only containing the transmission error, and the bit vector R of the small wheel preset Ease-off tooth surface capable of improving the comprehensive transmission performance is obtained1rVector N1rComprises the following steps:
Figure BDA0003326606260000062
r in the formula 31、N1The gear surface phase vector and the normal vector of the small wheel only contain geometric transmission errors; deltacIs a contact line normal clearance curved surface; u and beta are tooth surface parameters. Small wheel tooth surface phase vector R only containing preset transmission error1Vector N1The expression can be determined with reference to (1-2), except that the large wheel steering angle is expressed as:
θ2＝z1/z2(θ1-θ10)+θ20+ψ(θ1) (4)
Figure BDA0003326606260000071
the transmission error curve with psi as the predetermined geometry in equation 5 is shown in FIG. 2a, where λ1For the range of the rotation angle of the small wheel with the conjugate tooth surface, the preset parameter (epsilon)1、λ0) Can determine a0～a2Are curve parameters.
δcThe design of the gear needs to consider that the tooth root and the tooth top have certain tooth profile modification to avoid the edge stress concentration, and the contact trace also needs to avoid the edge contact of the tooth top and the two tooth sides, so the modification quantity of the engaging-in end and the engaging-out end should have certain distortion as shown in figure 2b
Figure BDA0003326606260000072
Theta in formula 6aFor the rotation transformation angle, ζ is the expression for the tooth profile modification amount, e0、e1、d1、d2For the sectional parabolic curve modification parameters, the parameters (d) can be preset1、d2、q1、q2And thetaa) And (6) determining. Presetting the parameter of the interdental contact gap (epsilon)1、λ0) Normal contact clearance parameter (d) with tooth surface1、d2、q1、q2And thetaa) A small round resolvable Ease-off modified tooth surface can be obtained.
And step 3: determining the maximum bearing deformation of the meshing of a preset Ease-off modified tooth surface and a big wheel by adopting a gear bearing contact analysis (LTCA) method;
the bearing deformation of different positions of the meshing period when the preset Ease-off tooth surface of the small wheel is meshed with the large wheel can be obtained through an LTCA method, and the maximum bearing deformation is further obtained. It should be noted that when the LTCA calculation is performed, the flexibility coefficient of the gear pair needs to be calculated by a finite element method, and the initial contact gap of the gear pair is calculated by a TCA method to provide input data for the LTCA calculation. The gear LTCA technical principle is that TCA and a finite element method are combined, and according to the initial gap of a multi-tooth pair at a meshing position, the forced contact of a gear pair is converted into the mechanical balance problem of solving a finite number of discrete contact points on the tooth surface. The mathematical model is shown in fig. 3, under the action of a load P, 2 pairs of teeth contact at a certain moment, the gear teeth elastically deform, a small wheel is arranged for fixation, and a large wheel moves along the normal direction under the action of the load to displace to Z. It is assumed that tooth flank rubbing is neglected and line contact occurs along the instantaneous contact ellipse major axis. The tooth flank contact can be described by:
Figure BDA0003326606260000073
in the formula pjk(j ═ 1,2, … n) is the normal load at discrete point j of the instantaneous contact circle major axis of tooth pair k; djk(j ═ 1,2, … n) is the tooth flank clearance after deformation at the discrete point j of the major axis of the instantaneous contact ellipse of tooth pair k, dkIs djkThe set of vectors; z is the normal displacement vector of the gear teeth, namely the bearing deformation; fkThe matrix is a normal flexibility matrix of the tooth pair k and is obtained by a finite element method; w is akIs the tooth pair initial contact gap vector. The above formula is a nonlinear programming composed of known parameters F, P, W and unknown parameters p, Z and d, the unknown parameters can be obtained after solving, Z is converted into displacement on a meshing line, and the displacement is expressed by a rotation angle, namely the bearing transmission error of a meshing period.
And 4, step 4: constructing a new gear bearing contact analysis LTCA equation, enabling the bearing deformation in the meshing period to be equal to the maximum bearing deformation, and obtaining an additional tooth space gap of a preset Ease-off modification tooth surface, namely a geometric transmission error, by reversely solving the new LTCA equation;
the normal deformation Z obtained in equation (7) for one meshing cycle is not equal, and in order to make it equal, it is necessary to compensate for the clearance of the meshing position, that is, by compensating for the initial contact clearance when the preset Ease-off tooth surface is meshed with the large wheel, the load-bearing deformation after compensation in the meshing cycle is equal to the maximum deformation amount before uncompensation, that is, ALTE of the meshing position is zero. Assuming that the additional initial gap at the contact location is greater than 0, the compensated load bearing deformation will always be greater than the maximum load bearing deformation before compensation, where the interdental gap is mainly compensated, the equation of coordination of the displacements of the LTCA model is modified as follows:
Fkpk+wk+ek＝Z+dk k＝Ι,ΙΙ (8)
the formula (8) is a nonlinear programming composed of a known parameter F, P, W, Z and unknown parameters p, d and e, and the compensation initial contact clearance e of the gear tooth meshing position can be obtained by improving the original LTCA solving method and reversely solving the method. It should be noted that in order to facilitate the machining of the modified tooth surface, the size of the compensation gap needs to be constrained by the meshing position of the tooth surface, so as to ensure that the curve of the compensation gap between the meshing position and the meshing position has no larger concave.
And 5: converting the additional tooth space gap into a gear pair additional geometric transmission error, and determining a zero ALTE gear pair geometric transmission error according to the gear pair additional geometric transmission error;
the compensated tooth space e is converted into angular displacement on the meshing line, and additional geometric transmission errors of different meshing positions of the tooth surface can be obtained.
ψe＝3600×180/πZ(r2×e·n2) (9)
Fitting the data of n meshing positions of the tooth surface by adopting a cubic uniform B spline, and adding a geometric transmission error psieThe expression of the curve is as follows:
Figure BDA0003326606260000081
in the formula, xiIs a small wheel corner, ViControl point column vector, i ═ 1 … n-1; m is a constant matrix. The additional geometric transmission error and the original geometric transmission error are synthesized into a geometric transmission error of a zero ALTE tooth surface, and the geometric transmission error relation of the zero bearing transmission error gear pair is determined and expressed as follows:
θ2＝z1/z2(θ1-θ10)+θ20+ψe(θ1) (11)
step 6: according to the relation of geometric transmission errors of the zero ALTE gear pair, the zero ALTE small wheel modified tooth surface is obtained by using a large wheel to generate a small wheel.
Superimposing the backlash caused by the transmission error in equation 9 on the preset modified tooth surface R in equation 31rObtaining the zero ALTE modification tooth surface position vector and the normal vector R1z、N1zAfter simplification, it can be expressed as:
Figure BDA0003326606260000091
r in the formula 121Ψ、N1ΨThe small wheel tooth surface position vector and the normal vector containing the preset geometric transmission error and the additional geometric transmission error can be determined by referring to (1-2), except that the large wheel rotation angle is determined by an equation 11, R1Ψ、N1ΨThe expression is as follows:
Figure BDA0003326606260000092
and 7: and calculating the zero ALTE modification gear pair TCA and LTCA.
In order to verify whether the fitted additional interdental space is reasonable or not and whether the fitted additional interdental space is better fused with the preset Ease-off modified tooth surface and the sensitivity of ALTE to the fitting error or not, TCA and LTCA simulation needs to be carried out on the zero ALTE modified tooth surface. The zero ALTE modified tooth surface in the invention has a determined analytic expression, and the tooth-to-tooth initial contact gap can be obtained by adopting a traditional TCA method, wherein the TCA expression is as follows:
Figure BDA0003326606260000093
in the formula (I), the compound is shown in the specification,
Figure BDA0003326606260000094
the angles of the driving wheel and the driven wheel are turned in the meshing process of the gear pair. The total number of the simplified equations is 5, and the equations are taken
Figure BDA0003326606260000095
As input quantity, solve for u1、β1、u2、β2、
Figure BDA0003326606260000096
A deterministic solution can be obtained for the system of unknowns. The initial contact clearance of the gear pair is obtained through TCA analysis, and then the initial contact clearance is used as input data of the original LTCA method for simulation analysis, so that the effectiveness of the novel LTCA method based on zero ALTE can be further verified.
And 8: the Ease-off profile amount of the zero ALTE profile tooth surface is determined.
The Ease-off relief surface reflects the degree of mismatch between the flank of the small wheel and the flank of the large wheel in mesh, the value of which is equal to the deviation between the flank of the small wheel fully conjugate with the flank of the large wheel and the relief flank of the small wheel. A flow chart of the zero ALTE spiral bevel gear tooth surface active Ease-off modification method is shown in figure 4. The Ease-off modification amount of the zero ALTE modification tooth surface is as follows:
δe(u,β)＝(R1z(u,β)-R10(u,β))gN10(u,β) (15)
zero ALTE tooth surface simulation and analysis examples.
In order to verify the effectiveness of the method, the geometric parameters and theoretical tooth surface processing parameters of the hypoid gear pair shown in tables 1 and 2 are taken as examples, the load rated torque of a bull wheel is 800N.m, and simulation analysis is as follows.
TABLE 1 hypoid gear set geometry parameters
Figure BDA0003326606260000101
TABLE 2 hypoid Gear Process parameters
Figure BDA0003326606260000102
TABLE 3 Preset Ease-off surface parameters
Figure BDA0003326606260000103
The parameters of the preset Ease-off shape-modifying curved surface under the condition that the contact trace is inclined at 25 degrees are shown in a table 3, and the parameters of the three-dimensional Ease-off curved surface are shown in a table 5a1The simulation of the preset modified tooth surface TCA is shown in FIG. 5a2The light-load contact impression is in the middle of the tooth width, and a larger geometric transmission error exists when the tooth top and the tooth root are in contact, so that the sensitivity of installation errors can be reduced; the Ease-off curve matches the contact area.
The shape of the additional tooth space influences the difficulty of processing, the ideal additional tooth space is in a parabolic shape from tooth surface engagement to tooth surface engagement, is aperiodic, and is not easy to be too small, so that the manufacturing and processing are convenient, the roll ratio can be controlled on a cradle type machine tool to realize, and in practice, in order to realize the equal bearing deformation in the engagement period, the compensated initial contact space of each engagement position of the tooth surface can be combined in multiple ways. One meshing period is divided into 8 equal parts, in the embodiment, the total number of one tooth from meshing to meshing contact positions is 22, and different additional interdental spaces can be obtained by the LTCA new method based on zero ALTE through the compensation position constraints of 1 st to 8 th type value points close to the meshing end, the middle 9 th to 16 th type value points, the 17 th to 22 th type value points close to the meshing end and the 8 type value point compensation quantity constraints of one meshing period. The following are three-dimensional plots of the additional backlash, additional geometric drive error, and compensation clearance for the 4 cases:
FIG. 5b1、5b2In order to compensate mainly for the zero ALTE tooth flanks 1 obtained by the tooth-engaging and tooth-disengaging end shaping,the compensation clearance is changed periodically, the compensation quantity close to the engaging end and the engaging end is large, the maximum compensation clearance is 3.5 mu m, and the corresponding maximum additional transmission error is 11 arc seconds; FIG. 5c1、5c2The zero ALTE tooth surface 2 obtained by mainly compensating the tooth engaging end modification amount of the gear teeth has non-periodic variation of the compensation clearance, the maximum compensation clearance is 18 mu m, and the corresponding additional transmission error is 56 arc seconds; FIG. 5d1、5d2The zero ALTE tooth surface 3 obtained by mainly compensating the tooth engaging end modification amount has non-periodic variation of the compensation clearance, the maximum compensation clearance is 5.4 mu m, and the corresponding additional transmission error is 22 arc seconds; FIG. 5e1、5e2The zero ALTE tooth surface 4 obtained by the modification amount of the middle part of the main compensation tooth surface has non-periodic variation of the compensation clearance, the maximum compensation clearance is 4.4 mu m, and the corresponding additional transmission error is 16 arc seconds;
③ zero ALTE tooth surface 2 as an example, FIG. 5c2The compensation clearance is superposed to a preset Ease-off curved surface, namely an Ease-off curved surface of the zero ALTE tooth surface 2 (see figure 5 f)1) The Ease-off curved surface has no concave points on the whole, namely the original Ease-off modified curved surface can well fuse additional gaps, and the processing is facilitated. TCA simulation of zero ALTE tooth surface 2 see FIG. 5f2The tooth surface impression of the gear tooth profile is basically unchanged, the geometric transmission error shape is not obviously concave on the whole, and the size of the geometric transmission error is the sum of the additional geometric transmission error and the geometric transmission error of the preset shape-modifying tooth surface. The multi-load comprehensive bearing deformation of the preset modified tooth surface and the zero ALTE tooth surface 2 is basically consistent, the deformation of the three-tooth meshing area is smaller than that of the two-tooth meshing area, and the deformation of the two-tooth meshing area and the three-tooth meshing area of the zero ALTE tooth surface 2 are nearly equal under the rated working condition (see figure 5 g)1,5g2)。
And fourthly, the graph 5h is an ALTE curve under the zero ALTE tooth surface multi-load working condition under 4 compensation quantities, and the amplitude of the load transmission error after compensation is basically zero and is greatly reduced compared with a conjugate tooth surface and a preset shape modification tooth surface.
A zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method relates to a tooth surface vibration reduction and noise reduction method. According to the method, Ease-off shape modification tooth surface design is carried out according to the generation principle of a contact tooth to an interdental gap and a contact line normal gap, and a new gear bearing contact analysis (LTCA) numerical method based on a zero bearing transmission error Amplitude (ALTE) is provided. The new method compensates the initial contact clearance of different meshing positions of the modified gear teeth, so that the deformation in the meshing period is equal to the maximum deformation before uncompensation, namely ALTE of the meshing position is zero, and then the compensation clearance is reversely solved by improving the original LTCA solving method. The size of the compensation gap is constrained by the meshing position of the tooth surface, the compensation gap is superposed on the preset easy-off tooth surface to form a zero ALTE modification tooth surface, and the zero ALTE modification tooth surface has good fusion and is beneficial to processing. The method ensures that the spiral bevel gear not only meets the requirement of high strength under a small proper amount, but also eliminates rigidity excitation, fundamentally improves the dynamic performance of the gear pair, and provides a method for designing and analyzing a high-performance tooth surface; meanwhile, relevant software is compiled, and the product design efficiency can be improved.
The above-mentioned contents are only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited thereby, and any modification made on the basis of the technical idea of the present invention falls within the protection scope of the claims of the present invention.

Claims (9)

1. A zero-bearing transmission error amplitude spiral bevel gear tooth surface active easy-off modification method is characterized by comprising the following steps:
step 1, determining a small wheel tooth surface completely conjugated with a large wheel according to a gear meshing principle;
step 2, a normal trimming curved surface is designed, and the normal trimming curved surface is superposed on the conjugate pinion tooth surface to obtain a pinion preset Ease-off trimming tooth surface;
step 3, determining the maximum bearing deformation when the preset Ease-off modified tooth surface is meshed with the big gear tooth surface by adopting a gear bearing contact analysis method;
step 4, reconstructing a gear bearing contact analysis equation to enable the bearing deformation in the meshing period to be equal to the maximum bearing deformation, and obtaining an additional tooth space gap of a preset Ease-off modification tooth surface by reversely solving the bearing contact analysis equation;
step 5, converting the additional inter-tooth gap into an additional geometric transmission error of the gear pair, and determining the geometric transmission error of the zero ALTE gear pair according to the additional geometric transmission error of the gear pair;
and 6, according to the relation of the geometric transmission error of the zero ALTE gear pair, generating small wheels by using a large wheel to obtain the zero ALTE small wheel modified tooth surface.
2. The zero-load transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method according to claim 1, characterized in that a bit vector R of a small wheel tooth surface which is completely conjugated in the step 110And vector N10The following were used:
Figure FDA0003326606250000011
wherein the tooth surface position vector r2、n2The original tooth surface phase vector and the normal vector of the big wheel are respectively; u and beta are parameters of any point on the tooth surface; m1p、Mpq、Mqr、Mrs、Ms2For a coordinate transformation matrix, L1p、Lpq、Lqr、Lrs、Ls2Is the corresponding 3 × 3 sub-matrix.
3. The active easy-off modification method for the tooth surface of the spiral bevel gear with zero bearing transmission error amplitude according to claim 1, wherein in the step 2, in combination with the inter-tooth gap and the normal gap of the contact line, a modification curved surface reflecting the normal gap of the contact line is superimposed on the tooth surface of the small gear only containing the transmission error to obtain the bit vector R of the preset easy-off tooth surface of the small gear1rVector N1rComprises the following steps:
Figure FDA0003326606250000012
wherein R is1、N1The gear surface phase vector and the normal vector of the small wheel only contain geometric transmission errors; deltacIs a contact line normal clearance curved surface; u and beta are tooth surface parameters.
4. The tooth surface active Ease-off modification method of the spiral bevel gear with zero bearing transmission error amplitude according to claim 1, characterized in that in step 3, a compliance coefficient of a gear pair is calculated through a finite element numerical method, an initial contact gap of the gear pair is calculated according to a tooth surface geometric contact analysis method, the compliance coefficient of the gear pair and the initial contact gap of the gear pair are used as input data of the LTCA, forced contact of the gear pair is converted into solving of a mechanical balance problem of a finite number of discrete contact points of the tooth surface according to a multi-tooth pair initial gap and a compliance coefficient of a meshing position, under the action of a load P, instantaneous 2 pairs of tooth contact are set, elastic deformation occurs in a gear tooth, a small gear is fixed, a large gear moves along a normal direction under the action of the load to displace Z, tooth surface friction is ignored, linear contact occurs along a major axis of an instantaneous contact ellipse, and a tooth surface bearing contact equation is as follows:
Figure FDA0003326606250000021
wherein p isjk(j ═ 1,2, … n) is the normal load at discrete point j of the instantaneous contact circle major axis of tooth pair k; djk(j ═ 1,2, … n) is the tooth flank clearance after deformation at the discrete point j of the major axis of the instantaneous contact ellipse of tooth pair k, dkIs djkThe set of vectors; z is the normal displacement vector of the gear teeth, namely the bearing deformation; fkIs a normal compliance matrix, w, of tooth pairs kkAnd for the initial contact clearance vector of the tooth pair, converting Z into displacement on the meshing line after solving the tooth surface bearing contact equation to obtain the maximum bearing deformation of one meshing period.
5. The zero-load transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method according to claim 4, characterized in that the displacement coordination equation in the new gear load contact analysis equation set in step 4 is as follows:
Fkpk+wk+ek＝Z+dk k＝Ι,ΙΙ
the new LTCA equation system is a nonlinear programming formed by the known parameter F, P, W, Z and the unknown parameters p, d and e, and the compensation initial contact clearance e of the gear tooth meshing position can be obtained by inverse solution.
6. The zero-bearing transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method according to claim 1, characterized in that in step 5, the additional inter-tooth gap is converted into an angular displacement on a meshing line to obtain additional geometric transmission errors at different meshing positions of the tooth surface, and the additional geometric transmission errors at n meshing positions of the tooth surface are fitted by adopting a cubic uniform B spline to further obtain a zero ALTE gear pair geometric transmission error.
7. The zero-bearing transmission error amplitude helical bevel gear tooth surface active Ease-off modification method according to claim 6, characterized in that the expression of the zero-ALTE gear pair geometric transmission error relationship is as follows:
θ2＝z1/z2(θ1-θ10)+θ20+ψe(θ1)
8. the zero-bearing transmission error amplitude spiral bevel gear tooth surface active Ease-off modification method according to claim 7, characterized in that three times of uniform B splines are adopted to fit data of n meshing positions on a tooth surface, and a geometric transmission error psi is addedeThe expression of the curve is as follows:
Figure FDA0003326606250000031
in the formula, xiIs a small wheel corner, ViControl point column vector, i ═ 1 … n-1; m is a constant matrix.
9. The zero-bearing drive error amplitude spiral bevel gear tooth surface active Ease-off modification method according to claim 8, characterized in that the zero-ALTE small wheel modificationPotential vector R of tooth surface1zAnd vector N1zIs determined by superimposing the backlash caused by the drive error on the preset modified tooth surface R1rObtaining the zero ALTE modification tooth surface position vector and the normal vector R1z、N1zAfter simplification, it can be expressed as:
Figure FDA0003326606250000032
in the formula R1Ψ、N1ΨFor small wheel tooth surface position vector and normal vector including preset geometric transmission error and additional geometric transmission error, R1Ψ、N1ΨThe expression of (a) is as follows:
Figure FDA0003326606250000033
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