CN107798200A - A kind of helical gears time-variant mesh stiffness computational methods for considering axial deformation - Google Patents
A kind of helical gears time-variant mesh stiffness computational methods for considering axial deformation Download PDFInfo
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Abstract
The present invention proposes a kind of helical gears time-variant mesh stiffness computational methods for considering axial deformation, it is intended to improves the computational accuracy of helical gears mesh stiffness.Realize that step is：Calculate end face bending stiffness, end face shearing rigidity, radial compression rigidity and the end-tooth base rigidity of helical gears；Calculate contact stiffness；Calculate the end face mesh stiffness of monodentate pair；Derive and calculate axial bending rigidity, axial shearing rigidity and axial tooth base rigidity；Monodentate is calculated to mesh stiffness；Calculate time-variant mesh stiffness.The present invention considers influence of the axial engagement force to helical gears time-variant mesh stiffness, helical gears axial bending rigidity is derived, the calculation expression that axial shearing rigidity and axial tooth base rigidity quantitatively calculate, and each rigidity for combining end face direction calculates the time-variant mesh stiffness of helical gears jointly, computational accuracy is improved, Dynamic Performance Analysis and optimization design available for helical gears.
Description
Technical Field
The invention belongs to the field of gear precision transmission technology and mechanical analysis, and relates to a method for calculating time-varying meshing stiffness of a helical gear by considering axial deformation, which can be used for dynamic performance analysis and optimal design of the helical gear.
Background
In a traditional gear system, the gear transmission mainly comprises various transmission modes such as straight gear transmission, helical gear transmission, worm and gear transmission and the like, wherein the helical gear transmission can be further divided into forms such as helical cylindrical gear transmission, helical conical gear transmission, helical section gear transmission and the like, and the helical cylindrical gear transmission is one of the most widely applied transmission forms in the transmission field.
In recent years, high-end numerical control equipment is continuously developing towards high speed, high precision and long service life, and higher requirements are put on the dynamic characteristics of a helical gear which is a key component. The periodic variation of the meshing stiffness is one of the main excitation forms of the transmission system, and is called as time-varying meshing stiffness, which directly influences the dynamic characteristics in the transmission process. The time-varying meshing rigidity is mainly caused by the fact that the contact ratio of the helical gear is generally not an integer, namely the number of teeth of the helical gear which simultaneously participate in meshing is periodically changed along with time. Therefore, whether the time-varying meshing stiffness of the helical gear can be accurately solved is a prerequisite for researching the dynamic characteristics of a helical gear transmission system, carrying out dynamic performance analysis and optimizing design, and therefore, a calculation method for researching the time-varying meshing stiffness of the helical gear is necessary.
From the currently published data, only the end face bending stiffness, the end face shearing stiffness, the radial compression stiffness and the end face tooth base stiffness of the helical gear are considered when calculating the time-varying meshing stiffness of the helical gear, for example, the chinese patent application with the application publication number CN 104573196A, entitled "an analytic calculation method of time-varying meshing stiffness of helical gear", discloses an analytic calculation method of time-varying meshing stiffness of helical gear. According to the method, the end face bending rigidity, the end face shearing rigidity, the radial compression rigidity and the end face tooth base rigidity of the helical gear are calculated by adopting an energy method, and then the time-varying meshing rigidity of the helical gear is calculated through the rigidity series-parallel connection relation. However, the analytical calculation method does not consider the influence of axial deformation of the helical gear on the time-varying meshing stiffness under the actual working condition, so that the method has the defect of low calculation accuracy caused by incomplete stiffness consideration.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, provides a helical gear time-varying meshing stiffness calculation method considering axial deformation, and aims to improve the calculation accuracy of the helical gear meshing stiffness.
In order to achieve the purpose, the technical scheme adopted by the invention comprises the following steps:
(1) calculating the gear tooth cutting of helical cylindrical gearEnd face bending stiffness value dk of sheettbEnd face shear stiffness value dktsRadial compression stiffness value dktaAnd face tooth base stiffness value dktf；
(1.1) based on the slicing theory, dividing the gear teeth into a plurality of slices with the width dl along the tooth width direction of the helical cylindrical gear, and decomposing the engaging force F vertical to any slice to obtain the end face engaging force F of the slicetAnd axial engagement force FaThen the end face engaging force F is appliedtDecomposing to obtain axial force F on the slicetaAnd a radial force Ftb；
(1.2) use of axial force F on the slicetaAnd a radial force FtbThe generated end face bending, end face shearing and radial compression are used for calculating the corresponding end face bending rigidity value dk of the slicetbEnd face shear stiffness value dktsAnd radial compression stiffness value dkta；
(1.3) calculating the face tooth base stiffness value dk of the slice based on Sainsit theorytf；
(2) Calculating the contact rigidity value dk of the slice based on Hertz contact theoryh；
(3) Based on the rigidity series-parallel connection theory, the end face bending rigidity value dk of the slice is utilizedtbEnd face shear stiffness value dktsRadial compression stiffness value dktaEnd face tooth base stiffness value dktfAnd value of contact stiffness dkhCalculating the end face meshing stiffness value of the single tooth pair
(4) Based on a beam deformation theory, respectively deducing calculation formulas of axial bending rigidity, axial shearing rigidity and axial tooth base rigidity of the helical gear teeth by adopting an energy method;
(5) calculating the axial bending rigidity value k of the helical gear tooth by using the calculation formula deduced in the step (4)abAxial shear stiffness valuekasAnd axial tooth base stiffness value kaf；
(6) Based on the rigidity series-parallel connection theory, the end face meshing rigidity value of a single tooth pair is utilizedAxial bending stiffness value kabAnd axial shear stiffness value kasCalculating the meshing stiffness value of a single tooth pair
(7) Based on the rigidity series-parallel connection theory, the end face tooth base rigidity value dk calculated in the step (1.3) is utilizedtfAnd (5) calculating the axial tooth base stiffness value kafAnd (6) calculating the meshing rigidity value of the single tooth pairAnd calculating the time-varying meshing rigidity value K of the helical cylindrical gear.
Compared with the prior art, the invention has the following advantages:
when the axial meshing rigidity of the helical gear is calculated, calculation expressions of the axial bending rigidity, the axial shearing rigidity and the axial tooth base rigidity of the helical gear are deduced based on a beam deformation theory and an energy method in Material mechanics, quantitative calculation of the axial bending rigidity, the axial shearing rigidity and the axial tooth base rigidity of the helical gear is realized, a rigidity series-parallel connection theory is adopted, and the calculated axial bending rigidity value, axial shearing rigidity value, axial tooth base rigidity value and rigidity values in the end face direction of the helical gear are used for calculating to obtain a time-varying meshing rigidity value of the helical gear, and the result shows that the time-varying meshing rigidity precision of the helical gear considering the axial meshing force is remarkably improved.
Drawings
FIG. 1 is a flow chart of an implementation of the present invention;
FIG. 2 is a schematic diagram of the slicing division of the helical gear according to the embodiment of the invention;
FIG. 3 is a schematic diagram of a helical gear according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of the force applied to the end face of the helical gear according to the embodiment of the present invention;
FIG. 5 is a schematic diagram of a time varying meshing stiffness model of a helical gear according to an embodiment of the invention;
FIG. 6 is a schematic view of the helical gear according to the embodiment of the present invention;
FIG. 7 is a schematic view of the axial tooth base stress of the helical gear according to the embodiment of the invention.
Detailed Description
The present invention will be described in further detail below with reference to the accompanying drawings and taking the parameters of the helical gear structure shown in table 1 as an example.
TABLE 1
Referring to fig. 1, a method for calculating time-varying meshing stiffness of a helical gear with consideration of axial deformation includes the following steps:
step 1) calculating the end face bending rigidity value dk of the helical gear tooth slicetbEnd face shear stiffness value dktsRadial compression stiffness value dktaAnd face tooth base stiffness value dktf；
Step 1.1) based on a slicing theory, dividing the gear teeth into a plurality of slices with the width dl along the tooth width direction of the helical cylindrical gear, and decomposing the engaging force F vertical to any slice to obtain the end face engaging force F of the slicetAnd axial engagement force FaThen the end face engaging force F is appliedtDecomposing to obtain axial force F on the slicetaAnd a radial force Ftb；
Referring to fig. 2, based on the slicing theory, the gear teeth are divided into a plurality of slices with the width dl along the tooth width direction B of the helical gear, and a plurality of slices are obtained;
because the stress on each slice is the same, for any slice, the meshing force F is orthogonally decomposed in the meshing plane, refer to fig. 3, wherein β is the helical angle of the helical gear, and the end face meshing force F of the slice is obtained by adopting orthogonal decompositiontAnd axial engagement force FaZ is the axial direction and y is the radial direction;
referring to FIG. 4, a schematic of a tooth is shown for applying an end-face engagement force FtOrthogonal decomposition is carried out to obtain the axial force F on the slicetaAnd a radial force FtbWhereinis the pressure angle at the point of contact, rbThe radius of the base circle is, the CD section is a transition curve part, and the BC section is an involute curve part.
Step 1.2) use of the axial force F on the slicetaAnd a radial force FtbThe generated end face bending, end face shearing and radial compression are used for calculating the corresponding end face bending rigidity value dk of the slicetbEnd face shear stiffness value dktsAnd radial compression stiffness value dkta；
Since the forces have the same effect on the driving and driven gears, only one of the gears is taken for analysis, see fig. 4, for radial force FtbWill exert bending and shearing action on the slice, axial force FtaWill generate a sliceRadial compression, and calculating the end face bending rigidity value dk corresponding to the slice by using an energy method in Material mechanicstbEnd face shear stiffness value dktsAnd radial compression stiffness value dktaThe specific formula expression is as follows:
wherein dktbFor the end face bending stiffness of the slice, dktsShear stiffness for slicing, dktaThe radial compression stiffness of the cut piece, E is the modulus of elasticity, G is the shear modulus, β is the helix angle of the helical gear,τc＝αc-(θb-invαc) in order to index the circular pressure angle,is the pressure angle at the point of contact, rbIs the base radius, θbIs half of the base tooth angle,the coefficient of the tooth crest height is,mis the modulus, gamma and tau are the integration range, y1For horizontal sitting at any point on the transition curveLogo, y2Is the horizontal coordinate at any point on the involute,is the section moment of inertia at any position on the transition curve,is the section moment of inertia at any position on the involute,is the cross-sectional area at any position on the transition curve,is the cross-sectional area at any position on the involute.
Step 1.3) calculating the end face tooth base stiffness value dk of the slice based on Sainsot theorytf；
Since the helical gear can also generate flexible deformation in the gear meshing process, and the flexible deformation of the tooth base can also influence the calculation of the whole rigidity, it is necessary to calculate the end face tooth base rigidity value dk of the slicetfThe formula disclosed in Sainot's article "distribution of gear body to tooth definitions-A new binary systematic for use" ("Transactions-American Society of Mechanical Engineers journal of Mechanical Design" 2004,126(4):748-754) is adopted, and the expression is:
in the formula SfIs the root width of the section, ufThe coefficients L, M, P, Q in the formula for the distance from the force to the tooth root to which the slice is subjected can be found by the following polynomial:
Xi(h,θf)＝Ai/θf 2+Bihr 2+Cihr/θf+Di/θf+Eihr+Fi
Ai,Bi,Ci,Di,Ei,Fithe values of (b) are shown in Table 2, hr＝Rf2/Ri，Rf2Denotes the root circle radius, RiRepresenting the inner bore radius, θfIs half of the angle corresponding to the curve of the whole tooth profile of the worm wheel.
TABLE 2Ai,Bi,Ci,Di,Ei,FiCoefficient value of
Step 2) calculating the contact rigidity value dk of the slice based on the Hertz contact theoryh；
Because the driving helical gear and the driven helical gear are in contact with each other in the transmission process, the contact rigidity exists, the Hertz formula and the energy method are adopted for calculation, and the expression of the contact rigidity is as follows:
wherein dkhFor the contact stiffness between slices, E is the modulus of elasticity, dl is the width of the slice, and v is the Poisson's ratio of the slice.
Step 3) based on the rigidity series-parallel connection theory, utilizing the end face bending rigidity value dk of the slicetbEnd face shear stiffness value dktsRadial compression stiffness value dktaEnd face tooth base stiffness value dktfAnd value of contact stiffness dkhCalculating the end face meshing stiffness value of the single tooth pair
Referring to fig. 5, when calculating the end face meshing stiffness of a single tooth pair, it is necessary to calculate the end face bending stiffness of the driving gear slice respectivelyEnd face shear stiffnessRadial compression stiffnessAnd end face bending stiffness of the driven gear segmentEnd face shear stiffnessRadial compression stiffnessAnd contact stiffness between slicesThe superscript i denotes a tooth pair, p denotes a drive gear, and g denotes a driven gear, so that the end face mesh stiffness of each segment of a pair of meshing gearsIs the rigidity of the end face sliced by the driving gearEnd face stiffness of driven gear segmentAnd rigidity of contact between the cut piecesFormed in series, but with a single toothThe end face meshing rigidity of the pair is equal to that of each pair of slicesAccording to the series-parallel connection relation of the rigidity, the end face meshing rigidity of each slice isIntegrating along the direction of contact line to obtain the end face meshing rigidity of single tooth pairThe calculation expression is as follows:
wherein,the end face meshing rigidity of the single tooth pair is achieved,the bending rigidity of the end face of the driving wheel slice is achieved,the shear rigidity of the end face of the driving wheel slice is achieved,the radial compression rigidity of the section of the driving wheel,the bending rigidity of the end face of the driven wheel slice is achieved,the shear rigidity of the end face of the driven wheel slice is achieved,for the radial compressive stiffness of the driven wheel slice,for the contact stiffness between the slices, L is the contact line length.
Step 4) based on a beam deformation theory, respectively deducing calculation formulas of axial bending rigidity, axial shearing rigidity and axial tooth base rigidity of the helical gear teeth by adopting an energy method;
referring to FIG. 6, the axial meshing force F received at the line of contact of a helical gear is shownaZ is the axial direction, y is the radial direction, the axial engagement force FaAxial bending and axial shearing action can be generated on the helical cylindrical gear;
referring to fig. 7, in analyzing the axial tooth base stiffness, the helical cylindrical gear is simplified into a semi-circular cantilever for analysis, and the axial meshing force F is analyzedaProduces bending action on the tooth base of the helical gear rfIs the root circle radius;
based on a cantilever beam theory and an energy method in Material mechanics, calculating formulas of axial bending rigidity, axial shearing rigidity and axial tooth base rigidity of the helical cylindrical gear tooth are respectively deduced, and the deduction process is as follows:
(i) deducing an axial bending rigidity calculation formula of the gear teeth, wherein the realization mode is as follows:
based on the theory of beam deformation, the bending moment M of the gear teeth causing the bending deformation of the gear teeth on the transition curve is obtaineda1And bending moment M on the involute line causing bending deformation of gear teetha2Then using an energy method, by Ma1And Ma2Giving axial bending energy U of gear teethabExpression of (2), finally synthesize Ma1、Ma2And UabDeducing an axial bending rigidity calculation formula of the gear teeth:
wherein,
(ii) deducing an axial shear stiffness calculation formula of the gear teeth, wherein the realization mode is as follows:
obtaining the torque M causing the gear tooth shear deformation based on the beam deformation theorytThen using an energy method, by MtGiving axial shear energy U of gear teethasExpression of (2), finally synthesize MtAnd UasDeducing an axial shear stiffness calculation formula of the gear teeth:
wherein,
(iii) deducing an axial tooth base rigidity calculation formula of the gear teeth, wherein the realization mode is as follows:
obtaining the torque M on the axial direction of the gear matrix based on the beam deformation theoryafThen using an energy method, by MafGiving axial tooth basis energy U of the toothafExpression of (2), finally synthesize MafAnd UafDeducing an axial tooth base stiffness calculation formula of the gear teeth:
wherein,
wherein k isabIs the axial bending stiffness, k, of helical gearsasAxial shear for helical gearShear stiffness, kafThe axial tooth base rigidity of the helical gear, E is the elastic modulus, G is the shear modulus,τc＝αc-(θb-invαc)，β is the helix angle of the helical gear,in order to index the circular pressure angle,is the pressure angle at the point of contact, rbIs the base radius, rfRoot circle radius, θbIs half of the base tooth angle,is the addendum coefficient, m is the modulus, gamma and tau are the integral range, y is the distance from the center of the helical gear, y1Is the horizontal coordinate, y, at any point on the transition curve2Is the horizontal coordinate at any point on the involute,is the axial section inertia moment at any position on the transition curve,is the axial section moment of inertia, I, at any position on the involutep1Is the polar moment of inertia, I, of a cross-section at any position on the transition curvep2Is the polar moment of inertia, I, of a cross-section at any position on the involuteafThe axial section inertia moment is the axial section inertia moment at any position away from the center of the helical gear.
Step 5) calculating the axial bending rigidity value k of the helical gear tooth by using the calculation formula deduced in the step 4)abShaft, shaftShear stiffness value kasAnd axial tooth base stiffness value kaf；
Combining with the specific embodiment, respectively substituting the parameters of the driving helical gear and the driven helical gear into the calculation formula deduced in the step 4), so as to calculate and obtain the axial bending rigidity value of the driving helical gearAxial shear stiffness valueAnd axial tooth base stiffness valueAnd axial bending stiffness value of driven helical cylindrical gearAxial shear stiffness valueAnd axial tooth base stiffness valueThe superscript i denotes a tooth pair, p in the subscript denotes a drive gear, and g in the subscript denotes a driven gear.
Step 6) based on the rigidity series-parallel connection theory, utilizing the end surface meshing rigidity value of the single tooth pairAxial bending stiffness value kabAnd axial shear stiffness value kasCalculating the meshing stiffness value of a single tooth pair
Referring to FIG. 5, mesh stiffness of a single tooth pairEnd face mesh stiffness including single tooth pairAxial bending stiffness kabAnd axial shear stiffness kasThree major parts, among them the axial bending stiffness kabAxial bending stiffness value including active helical cylindrical gearAnd axial bending stiffness value of driven helical cylindrical gearTwo parts, the axial shear stiffness includes the axial shear stiffness value of the driving helical gearAnd axial shear stiffness value of driven helical cylindrical gearTwo parts, the superscript i denoting the tooth pair, the subscript p denoting the drive gear, and the subscript g denoting the driven gear, because of the mesh stiffness of the single tooth pairIs the end face meshing rigidity of a single tooth pairAxial bending rigidity value of driving helical gearAxial bending rigidity value of driven helical cylindrical gearAxial shear stiffness value of driving helical cylindrical gearAnd axial shear stiffness value of driven helical cylindrical gearFive parts are connected in series, so that the meshing rigidity value of a single tooth pair is calculated according to the series-parallel connection relation of the rigidityThe calculation expression is as follows:
wherein,in order to provide the meshing rigidity of the tooth pair,the end face meshing rigidity of the single tooth pair is achieved,in order to provide the axial bending rigidity of the driving wheel,the axial shear rigidity of the driving wheel is provided,for the axial bending stiffness of the driven wheel,is the axial shear stiffness of the driven wheel.
Step 7) based on the rigidity series-parallel connection theory, the end face tooth base rigidity value dk calculated in the step 1.3) is utilizedtfAnd 5) calculating the axial tooth base stiffness value kafAnd step 6) calculating the meshing rigidity value of the single tooth pairAnd calculating the time-varying meshing rigidity value K of the helical cylindrical gear.
The time-varying meshing stiffness of the helical cylindrical gear is mainly caused by the fact that the contact ratio of the helical cylindrical gear is not an integer generally, namely the number of pairs of teeth which are simultaneously engaged with the helical cylindrical gear is periodically changed along with time, the number of pairs of teeth which are engaged with the helical cylindrical gear at any moment can be obtained by integrating the contact ratio epsilon, referring to fig. 5, the situation that the number of pairs of teeth which are simultaneously engaged with the helical cylindrical gear is 3 is given, and the meshing stiffness value of a single tooth pair exists in each tooth pair iAnd because the time-varying meshing stiffness value K at any one time is the meshing stiffness value of a single tooth pair from each tooth pair iAfter being connected in parallel, the rigidity k of the end face tooth base is further connected withtfAxial tooth base stiffness value k of driving helical gearafpAnd axial tooth base stiffness value k of driven helical gearafgThe time-varying meshing rigidity value K of the helical gear is calculated according to the series-parallel relation of rigidity, and the calculation expression is as follows:
wherein K is the time-varying meshing stiffness of the helical gear,for the meshing stiffness of the tooth pairs, ktfIs the end face tooth base rigidity, k, of the helical gearafpIs the axial gear base rigidity, k, of the driving helical gearafgThe axial tooth base rigidity of the driven helical gear is shown, epsilon is the contact ratio at different moments, and Ceil is an integral function.
Based on the steps and by combining with specific parameters of the embodiment of the invention, the maximum relative error of the time-varying meshing stiffness obtained by the embodiment of the invention is 5.9%, compared with the prior art only considering the end face bending stiffness, the end face shearing stiffness, the radial compression stiffness and the end face tooth base stiffness of the helical gear, the relative error is reduced by 21.7%, and the calculation precision is obviously improved.
Claims (8)
1. A time-varying meshing stiffness calculation method of a helical cylindrical gear considering axial deformation is characterized by comprising the following steps:
(1) calculating the end face bending rigidity value dk of the helical gear tooth slicetbEnd face shear stiffness value dktsRadial compression stiffness value dktaAnd face tooth base stiffness value dktf；
(1.1) based on the slicing theory, dividing the gear tooth into a plurality of slices with the width dl along the tooth width direction of the helical cylindrical gear, and carrying out engagement force F vertical to any sliceDecomposing to obtain the end surface engaging force F of the slicetAnd axial engagement force FaThen the end face engaging force F is appliedtDecomposing to obtain axial force F on the slicetaAnd a radial force Ftb；
(1.2) use of axial force F on the slicetaAnd a radial force FtbThe generated end face bending, end face shearing and radial compression are used for calculating the corresponding end face bending rigidity value dk of the slicetbEnd face shear stiffness value dktsAnd radial compression stiffness value dkta；
(1.3) calculating the face tooth base stiffness value dk of the slice based on Sainsit theorytf；
(2) Calculating the contact rigidity value dk of the slice based on Hertz contact theoryh；
(3) Based on the rigidity series-parallel connection theory, the end face bending rigidity value dk of the slice is utilizedtbEnd face shear stiffness value dktsRadial compression stiffness value dktaEnd face tooth base stiffness value dktfAnd value of contact stiffness dkhCalculating the end face meshing stiffness value of the single tooth pair
(4) Based on a beam deformation theory, respectively deducing calculation formulas of axial bending rigidity, axial shearing rigidity and axial tooth base rigidity of the helical gear teeth by adopting an energy method;
(5) calculating the axial bending rigidity value k of the helical gear tooth by using the calculation formula deduced in the step (4)abAxial shear stiffness value kasAnd axial tooth base stiffness value kaf；
(6) Based on the rigidity series-parallel connection theory, the end face meshing rigidity value of a single tooth pair is utilizedAxial bending stiffness value kabAnd axial shear stiffness value kasCalculating the meshing stiffness value of a single tooth pair
(7) Based on the rigidity series-parallel connection theory, the end face tooth base rigidity value dk calculated in the step (1.3) is utilizedtfAnd (5) calculating the axial tooth base stiffness value kafAnd (6) calculating the meshing rigidity value of the single tooth pairAnd calculating the time-varying meshing rigidity value K of the helical cylindrical gear.
2. A method for calculating the time-varying meshing stiffness of a helical gear considering axial deformation according to claim 1, wherein the calculating of the end face bending stiffness value dk corresponding to the slice in the step (1.2)tbEnd face shear stiffness value dktsAnd radial compression stiffness value dktaThe calculation expressions are respectively:
wherein dktbFor the end face bending stiffness of the slice, dktsShear stiffness for slicing, dktaThe radial compression stiffness of the cut piece, E is the modulus of elasticity, G is the shear modulus, β is the helix angle of the helical gear, in order to index the circular pressure angle,is the pressure angle at the point of contact, rbIs the base radius, θbIs half of the base tooth angle,is the crest coefficient, m is the modulus, γ and τ are the integration ranges, y1Is the horizontal coordinate, y, at any point on the transition curve2Is the horizontal coordinate at any point on the involute,is the section moment of inertia at any position on the transition curve,is the section moment of inertia at any position on the involute,is the cross-sectional area at any position on the transition curve,is the cross-sectional area at any position on the involute.
3. A method for calculating the time-varying meshing stiffness of a helical gear with axial deformation taken into account as set forth in claim 1, wherein the end face tooth stiffness value dk of the slice is calculated in the step (1.3)tfThe calculation expression is as follows:
wherein dktfFor the face-tooth base stiffness of the slice, E is the modulus of elasticity, β is the skewHelix angle, S, of the toothed spur gearfThe width of the root of the tooth is taken as the width of the root of the tooth,is the pressure angle at the contact point, ufDistance of force to tooth root for slicing, L*,M*,P*,Q*Are the coefficients of the formula.
4. The method for calculating time-varying meshing stiffness of a helical gear according to claim 1, wherein the step (2) of calculating the contact stiffness value dk of the sliced piecehThe calculation expression is as follows:
<mrow> <msub> <mi>dk</mi> <mi>h</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mi>&pi;</mi> <mi>E</mi> <mi>d</mi> <mi>l</mi> </mrow> <mrow> <mn>4</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow>
wherein dkhFor the contact stiffness between slices, E is the modulus of elasticity, dl is the width of the slice, and v is the Poisson's ratio of the slice.
5. The method for calculating the time-varying meshing stiffness of the helical gear considering the axial deformation according to claim 1, wherein the calculating of the end face meshing stiffness value of the single-tooth pair in the step (3)The calculation expression is as follows:
<mrow> <msubsup> <mi>k</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> <mi>i</mi> </msubsup> <mo>=</mo> <msub> <mo>&Integral;</mo> <mi>L</mi> </msub> <mfrac> <mn>1</mn> <mrow> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>b</mi> <mi>p</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>s</mi> <mi>p</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>a</mi> <mi>p</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>b</mi> <mi>g</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>s</mi> <mi>g</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mrow> <mi>t</mi> <mi>a</mi> <mi>g</mi> </mrow> <mi>i</mi> </msubsup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>dk</mi> <mi>h</mi> <mi>i</mi> </msubsup> </mrow> </mfrac> </mrow> </mfrac> </mrow>
wherein,the end face meshing rigidity of the single tooth pair is achieved,the bending rigidity of the end face of the driving wheel slice is achieved,the shear rigidity of the end face of the driving wheel slice is achieved,the radial compression rigidity of the section of the driving wheel,the bending rigidity of the end face of the driven wheel slice is achieved,the shear rigidity of the end face of the driven wheel slice is achieved,for the radial compressive stiffness of the driven wheel slice,for the contact stiffness between the slices, L is the contact line length.
6. The method for calculating the time-varying meshing stiffness of the helical gear considering the axial deformation according to claim 1, wherein the calculation formulas for deriving the axial bending stiffness, the axial shearing stiffness and the axial tooth base stiffness of the teeth of the helical gear in the step (4) are respectively as follows:
(i) deducing an axial bending rigidity calculation formula of the gear teeth:
based on the theory of beam deformation, the bending moment M of the gear teeth causing the bending deformation of the gear teeth on the transition curve is obtaineda1And bending moment M on the involute line causing bending deformation of gear teetha2Then using an energy method, by Ma1And Ma2Giving axial bending energy U of gear teethabExpression of (2), finally synthesize Ma1、Ma2And UabDeducing an axial bending rigidity calculation formula of the gear teeth:
wherein,
(ii) deducing an axial shear stiffness calculation formula of the gear teeth, wherein the realization mode is as follows:
obtaining the torque M causing the gear tooth shear deformation based on the beam deformation theorytThen using an energy method, by MtGiving axial shear energy U of gear teethasExpression of (2), finally synthesize MtAnd UasDeducing an axial shear stiffness calculation formula of the gear teeth:
wherein,
(iii) deducing an axial tooth base rigidity calculation formula of the gear teeth, wherein the realization mode is as follows:
obtaining the torque M on the axial direction of the gear matrix based on the beam deformation theoryafThen using an energy method, by MafGiving axial tooth basis energy U of the toothafExpression of (2), finally synthesize MafAnd UafDeducing an axial tooth base stiffness calculation formula of the gear teeth:
wherein,
wherein k isabIs the axial bending stiffness, k, of helical gearsasIs the axial shear stiffness, k, of helical gearsafThe axial tooth base rigidity of the helical gear, E is the elastic modulus, G is the shear modulus,τc＝αc-(θb-invαc)，β is the helix angle of the helical gear,in order to index the circular pressure angle,is the pressure angle at the point of contact, rbIs the base radius, rfRoot circle radius, θbIs half of the base tooth angle,is the addendum coefficient, m is the modulus, gamma and tau are the integral range, y is the distance from the center of the helical gear, y1Is the horizontal coordinate, y, at any point on the transition curve2Is the horizontal coordinate at any point on the involute,is the axial section inertia moment at any position on the transition curve,is the axial section moment of inertia, I, at any position on the involutep1Is the polar moment of inertia, I, of a cross-section at any position on the transition curvep2Is the polar moment of inertia, I, of a cross-section at any position on the involuteafThe axial section inertia moment is the axial section inertia moment at any position away from the center of the helical gear.
7. The method for calculating the time-varying meshing stiffness of the helical gear considering the axial deformation according to claim 1, wherein the step (6) of calculating the meshing stiffness value of the single-tooth pairThe calculation expression is as follows:
<mrow> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mi>t</mi> <mi>i</mi> </msubsup> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mrow> <mi>t</mi> <mi>t</mi> </mrow> <mi>i</mi> </msubsup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>p</mi> </mrow> <mi>i</mi> </msubsup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>p</mi> </mrow> <mi>i</mi> </msubsup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mrow> <mi>a</mi> <mi>b</mi> <mi>g</mi> </mrow> <mi>i</mi> </msubsup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mrow> <mi>a</mi> <mi>s</mi> <mi>g</mi> </mrow> <mi>i</mi> </msubsup> </mfrac> </mrow>
wherein,in order to provide the meshing rigidity of the tooth pair,the end face meshing rigidity of the single tooth pair is achieved,in order to provide the axial bending rigidity of the driving wheel,the axial shear rigidity of the driving wheel is provided,for the axial bending stiffness of the driven wheel,is axial of the driven wheelShear stiffness.
8. The method for calculating the time-varying meshing stiffness of the helical gear considering the axial deformation according to claim 1, wherein the time-varying meshing stiffness value K of the helical gear is calculated in the step (7) by the calculation expression:
<mrow> <mi>K</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>C</mi> <mi>e</mi> <mi>i</mi> <mi>l</mi> <mrow> <mo>(</mo> <mi>&epsiv;</mi> <mo>)</mo> </mrow> </mrow> </munderover> <mfrac> <mn>1</mn> <msubsup> <mi>k</mi> <mi>t</mi> <mi>i</mi> </msubsup> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>t</mi> <mi>f</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>f</mi> <mi>p</mi> </mrow> </msub> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>a</mi> <mi>f</mi> <mi>g</mi> </mrow> </msub> </mfrac> </mrow> </mfrac> </mrow>
wherein K is the time-varying meshing stiffness of the helical gear,for the meshing stiffness of the tooth pairs, ktfIs the end face tooth base rigidity, k, of the helical gearafpIs the axial gear base rigidity, k, of the driving helical gearafgThe axial tooth base rigidity of the driven helical gear is shown, epsilon is the contact ratio at different moments, and Ceil is an integral function.
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