1. A method for calculating dynamic meshing force of an external meshing straight-tooth cylindrical gear pair is characterized by comprising the following steps:

s1, acquiring basic parameters of a driving gear and a driven gear in an externally-meshed straight-tooth cylindrical gear pair;

s2, establishing an experimental model of an external meshing straight-tooth cylindrical gear pair;

s3, judging a theoretical meshing area where a current theoretical meshing point of the gear pair is located according to the experimental model;

s4, selecting three pairs of teeth in the gear pair to form a set to be examined according to a theoretical meshing area where a current theoretical meshing point of the gear pair is located, sequentially naming the set to be examined as a No. 1 pair of teeth, a No. 2 pair of teeth and a No. 3 pair of teeth along the rotation direction of the gear, and simultaneously establishing a three-tooth dynamic contact state matrix model: q₃F₃= E₃Wherein: q₃Representing a compliance coefficient matrix of three pairs of teeth, F₃Representing the meshing force vectors of three pairs of teeth, E₃Representing the transmission error vector of three pairs of teeth;

s5, solving the three-tooth dynamic contact state matrix model, obtaining each element value in the meshing force vector of three pairs of teeth, judging the actual meshing state of the gear pair, and outputting the calculated value of the dynamic meshing force of the gear pair.

2. The method for calculating the dynamic meshing force of an externally meshed spur gear pair according to claim 1, wherein: in the step S4, three pairsCompliance coefficient matrix Q of teeth₃Three pairs of teeth engagement force vector F₃And a transmission error vector E of three pairs of teeth₃Respectively as follows:

wherein the content of the first and second substances,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling compliance created at the point of tooth meshing.

3. The method for calculating the dynamic meshing force of an externally meshed spur gear pair according to claim 1, wherein: in the step S4, if the current theoretical meshing point of the gear pair is located in the theoretical double-tooth meshing area, calculating a tooth profile error value of a pair of teeth of the line outer meshing area along the gear rotation direction and a tooth profile error value of a pair of teeth of the line outer meshing area against the gear rotation direction, and taking a pair of teeth of the line outer meshing area with a smaller tooth profile error value and two pairs of teeth of the line inner meshing area to form a three-pair tooth inspection set; if the current theoretical meshing point of the gear pair is located in the theoretical single-tooth meshing area, the tooth form error value of one pair of teeth of the line external meshing area along the rotation direction of the gear and the tooth form error value of one pair of teeth opposite to the rotation direction of the gear are calculated, and two pairs of teeth of the line external meshing area and one pair of teeth of the line internal meshing area form a set of three pairs of teeth to be examined.

4. The method for calculating the dynamic meshing force of an externally meshed spur gear pair according to claim 1, wherein: the specific process of step S5 is as follows:

s5-1: solving a three-tooth dynamic contact state matrix model to obtain each element value in meshing force vectors of three pairs of teeth;

s5-2: if all elements in the meshing force vectors of the three pairs of teeth are positive values, judging that the current actual meshing state of the gear pair is a three-tooth meshing state, outputting a calculated value of the dynamic meshing force of the gear pair, and finishing the calculation; if all elements in the meshing force vectors of the three pairs of teeth have non-positive values, carrying out the next step;

s5-3: if the current theoretical meshing point of the gear pair is located in a theoretical double-tooth meshing area, one pair of teeth of the off-line meshing area is removed from the three pairs of tooth to-be-examined sets to form double-pair tooth to-be-examined sets, the double-pair tooth to-be-examined sets are sequentially named as No. 1 pair of teeth and No. 2 pair of teeth along the rotation direction of the gear, and meanwhile, a double-tooth dynamic contact state matrix model is established; if the current theoretical meshing point of the gear pair is located in a theoretical single-tooth meshing area, one pair of teeth with larger tooth shape error values in two pairs of teeth in the off-line meshing area are removed from the three pairs of tooth to-be-examined sets to form a double-pair tooth to-be-examined set, the double-pair tooth to-be-examined set is sequentially named as a No. 1 pair of teeth and a No. 2 pair of teeth along the rotation direction of the gear, and meanwhile, a double-tooth dynamic contact state matrix model is established;

s5-4: solving a double-tooth dynamic contact state matrix model, obtaining each element value in the meshing force vector of double pairs of teeth, if all the elements are positive values, judging that the current actual meshing state of the gear pair is a double-tooth meshing state, and outputting a calculation value of the dynamic meshing force of the gear pair; otherwise, the single-tooth meshing state is judged, and meanwhile, the calculated value of the dynamic meshing force of the gear pair is also output.

5. The method for calculating the dynamic meshing force of an externally meshed spur gear pair according to claim 4, wherein: the double-tooth dynamic contact state matrix model is as follows: q₂F₂=E₂Wherein Q is₂Representing a compliance coefficient matrix of the pairs of teeth, F₂Representing the vector of the meshing forces of the pairs of teeth, E₂Represents the transmission error vector of double pairs of teeth and has:

wherein the content of the first and second substances,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling compliance created at the point of tooth meshing.

Background

The external meshing straight-tooth cylindrical gear pair is a relatively basic and simple gear transmission form and can be seen in numerous mechanical devices, and accurate calculation of dynamic meshing force of the external meshing straight-tooth cylindrical gear pair is an important ring for researching the dynamics of a gear system and directly determines the accuracy of a dynamics model. At present, the calculation method of the dynamic meshing force of the straight-tooth cylindrical gear pair mainly comprises 3 methods: (1) finite element method: the method can accurately calculate the dynamic meshing force of the gear by means of a finite element method, and is a widely adopted method. However, the method is poor in computational economy, a fine finite element mesh model must be established, and the solution is time-consuming; (2) the traditional analytic method comprises the following steps: the method has the advantages that the relation of a plurality of pairs of loaded gear teeth is simplified into spring models which are connected in parallel, the rigidity of the gear pairs participating in meshing is calculated by using the assumption of a material mechanics cantilever beam, and the dynamic meshing force of the gear is calculated on the basis, but the result calculated by adopting a traditional analytic method has larger error because the structural coupling effect is ignored; (3) the experimental method comprises the following steps: the method is a method for calculating the load distribution rate by means of a strain sensor, a data acquisition instrument and a visualization device, has high measurement precision, but has high requirements on hardware and complex operation process.

Disclosure of Invention

The invention aims to solve the technical problem of providing a method for calculating the dynamic meshing force of an external-meshing straight-tooth cylindrical gear pair, which has the advantages of high accuracy, good calculation economy, relatively simple operation process and low requirement on hardware.

The technical scheme adopted by the invention for solving the technical problems is as follows: a method for calculating dynamic meshing force of an external meshing straight-tooth cylindrical gear pair comprises the following steps:

s1, acquiring basic parameters of a driving gear and a driven gear in an externally-meshed straight-tooth cylindrical gear pair;

s2, establishing an experimental model of an external meshing straight-tooth cylindrical gear pair;

s3, judging a theoretical meshing area where a current theoretical meshing point of the gear pair is located according to the experimental model;

s4, selecting three pairs of teeth in the gear pair to form a set to be examined according to a theoretical meshing area where a current theoretical meshing point of the gear pair is located, sequentially naming the set to be examined as a No. 1 pair of teeth, a No. 2 pair of teeth and a No. 3 pair of teeth along the rotation direction of the gear, and simultaneously establishing a three-tooth dynamic contact state matrix model: q₃F₃= E₃Wherein: q₃Representing a compliance coefficient matrix of three pairs of teeth, F₃Representing the meshing force vectors of three pairs of teeth, E₃Representing the transmission error vector of three pairs of teeth;

s5, solving the three-tooth dynamic contact state matrix model, obtaining each element value in the meshing force vector of three pairs of teeth, judging the actual meshing state of the gear pair, and outputting the calculated value of the dynamic meshing force of the gear pair.

Preferably, in step S4, the compliance coefficient matrix Q of three pairs of teeth₃Three pairs of teeth engagement force vector F₃And a transmission error vector E of three pairs of teeth₃Respectively as follows:

wherein the content of the first and second substances,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling compliance created at the point of tooth meshing.

Preferably, in step S4, if the current theoretical meshing point of the gear pair is located in the theoretical double-tooth meshing area, the tooth profile error value of one pair of teeth of the out-of-line meshing area along the gear rotation direction and the tooth profile error value of one pair of teeth opposite to the gear rotation direction are calculated, and the pair of teeth of the out-of-line meshing area with the smaller tooth profile error value and the pair of teeth of the in-line meshing area form a three-pair-of-teeth examined set; if the current theoretical meshing point of the gear pair is located in the theoretical single-tooth meshing area, the tooth form error value of one pair of teeth of the line external meshing area along the rotation direction of the gear and the tooth form error value of one pair of teeth opposite to the rotation direction of the gear are calculated, and two pairs of teeth of the line external meshing area and one pair of teeth of the line internal meshing area form a set of three pairs of teeth to be examined.

Preferably, the specific process of step S5 is as follows:

s5-1: solving a three-tooth dynamic contact state matrix model to obtain each element value in meshing force vectors of three pairs of teeth;

s5-2: if all elements in the meshing force vectors of the three pairs of teeth are positive values, judging that the current actual meshing state of the gear pair is a three-tooth meshing state, outputting a calculated value of the dynamic meshing force of the gear pair, and finishing the calculation; if all elements in the meshing force vectors of the three pairs of teeth have non-positive values, carrying out the next step;

s5-3: if the current theoretical meshing point of the gear pair is located in a theoretical double-tooth meshing area, one pair of teeth of the off-line meshing area is removed from the three pairs of tooth to-be-examined sets to form double-pair tooth to-be-examined sets, the double-pair tooth to-be-examined sets are sequentially named as No. 1 pair of teeth and No. 2 pair of teeth along the rotation direction of the gear, and meanwhile, a double-tooth dynamic contact state matrix model is established; if the current theoretical meshing point of the gear pair is located in a theoretical single-tooth meshing area, one pair of teeth with larger tooth shape error values in two pairs of teeth in the off-line meshing area are removed from the three pairs of tooth to-be-examined sets to form a double-pair tooth to-be-examined set, the double-pair tooth to-be-examined set is sequentially named as a No. 1 pair of teeth and a No. 2 pair of teeth along the rotation direction of the gear, and meanwhile, a double-tooth dynamic contact state matrix model is established;

s5-4: solving a double-tooth dynamic contact state matrix model, obtaining each element value in the meshing force vector of double pairs of teeth, if all the elements are positive values, judging that the current actual meshing state of the gear pair is a double-tooth meshing state, and outputting a calculation value of the dynamic meshing force of the gear pair; otherwise, the single-tooth meshing state is judged, and meanwhile, the calculated value of the dynamic meshing force of the gear pair is also output.

Preferably, the bidentate dynamic contact state matrix model is: q₂F₂=E₂Wherein Q is₂Representing a compliance coefficient matrix of the pairs of teeth, F₂Representing the vector of the meshing forces of the pairs of teeth, E₂Represents the transmission error vector of double pairs of teeth and has:

wherein the content of the first and second substances,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling compliance created at the point of tooth meshing.

Compared with the prior art, the method has the advantages that the dynamic meshing force of the external meshing straight-tooth cylindrical gear pair is calculated by establishing the three-tooth dynamic contact state matrix model, the operation process is simple, the hardware requirement is not high, and the method can be widely used in the mechanical field; compared with the traditional finite element method and the experimental method, the calculation result is more accurate, and the efficiency is more efficient.

Drawings

FIG. 1 is a schematic diagram of a theoretical single tooth mesh region of a current theoretical mesh point of an experimental model of a gear pair of the present invention;

FIG. 2 is a schematic diagram of a theoretical double-tooth meshing area with a current theoretical meshing point of an experimental model of a gear pair of the present invention;

FIG. 3 is a force analysis diagram of a spur gear of the present invention;

fig. 4 is a schematic view of a tooth base structure of the spur gear of the present invention.

Detailed Description

The invention is described in further detail below with reference to the accompanying examples.

As shown in fig. 1-4, a method for calculating the dynamic meshing force of an external-meshing straight-tooth cylindrical gear pair comprises the following steps:

s1, obtaining basic parameters of a driving gear 1 and a driven gear 2 in the externally-meshed straight-tooth cylindrical gear pair, wherein the basic parameters comprise basic geometric parameters (modulus, tooth number, pressure angle, addendum coefficient, tip clearance coefficient and excircle radius of a base body 4), gear material parameters (elastic modulus, Poisson ratio and density) and gear load parameters (input torque) of the gears;

s2, establishing an experimental model of an external meshing straight-tooth cylindrical gear pair;

s3, judging whether the current theoretical meshing point of the gear pair is located in a theoretical double-tooth meshing area or a theoretical single-tooth meshing area according to the experimental model;

s4, if the current theoretical meshing point of the gear pair is located in a theoretical double-tooth meshing area, calculating a tooth form error value of a pair of teeth of an off-line meshing area of a theoretical meshing line 3 along the rotation direction of the gear and a tooth form error value of a pair of teeth opposite to the rotation direction of the gear, taking a pair of teeth of the two pairs of teeth of the off-line meshing area of the theoretical meshing line 3 with smaller tooth form error values and two pairs of teeth of an in-line meshing area of the theoretical meshing line 3 to form a three-pair tooth inspection set, sequentially naming the three-pair teeth as a No. 1 pair, a No. 2 pair and a No. 3 pair along the rotation direction of the gear, and simultaneously establishing a three-tooth dynamic contact state matrix model: q₃F₃=E₃Wherein: q₃Representing a compliance coefficient matrix of three pairs of teeth, F₃Representing the meshing force vectors of three pairs of teeth, E₃Representing the transmission error vector of three pairs of teeth; if the current theoretical meshing point of the gear pair is positioned in the theoretical single-tooth meshing area, calculating the tooth profile error of a pair of teeth of the line-out meshing area of the theoretical meshing line 3 along the rotation direction of the gearThe numerical value and the tooth shape error numerical value of a pair of teeth against the rotation direction of the gear form a set to be examined, wherein three pairs of teeth to be examined are formed by two pairs of teeth in the line outside meshing area of the theoretical meshing line 3 and one pair of teeth in the line inside meshing area of the theoretical meshing line 3, the three pairs of teeth are sequentially named as a No. 1 pair of teeth, a No. 2 pair of teeth and a No. 3 pair of teeth along the rotation direction of the gear, and a three-tooth dynamic contact state matrix model is established at the same time: q₃F₃=E₃Wherein: q₃Representing a compliance coefficient matrix of three pairs of teeth, F₃Representing the meshing force vectors of three pairs of teeth, E₃Represents the transmission error vector of three pairs of teeth and has:

（1）

in the formula (1), the reaction mixture is,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling flexibility generated at the meshing point of the pair of teeth is as follows:

（2）

in the formula (2), the reaction mixture is,vrepresenting the poisson's ratio of the gear material,Erepresents the young's modulus of the gear material,brepresenting tooth width, superscript, of gearpRepresents a driving gear 1, superscriptgRepresents the driven gear 2 and has:

（3）

in the formula (3), the reaction mixture is,、、respectively representiThe axial compression flexibility, bending flexibility and shearing flexibility of the teeth belonging to the driving gear 1 in the number pair of teeth,、、respectively representiAxial compression compliance, bending compliance and shear compliance of the teeth belonging to the driven gear 2 in the pair of teeth,

in the formula (3), the reaction mixture is,andthe calculation formula is consistent with the flow and is different from the flowThe similar situation exists only in the condition that the parameter values of the driving gear 1 and the driven gear 2 are differentAnd、andin the following, a standard spur gear is taken as an example, and the first embodiment is shown in detailiAxial compression compliance of teethFlexibility of bendingAnd shear complianceThe calculation formula of (2) is as follows:

（4）

in the formula (4), the reaction mixture is,Grepresentative of the shear modulus of the material,representing the angle of action of the meshing force,drepresents the mesh point toyThe distance of the axis is such that,hrepresents the mesh point toxThe distance of the axis is such that,represents the abscissaxThe area of the cross section of the gear,represents the abscissaxThe moment of inertia of the cross section of the gear,representing points of engagementBThe value of the abscissa of (a) is,xrepresents the variation of the integral of the product,

in the formula (2), the reaction mixture is,andthe calculation formula is consistent with the flow, the difference is only that the parameter values of the driving gear 1 and the driven gear 2 are different, a standard straight-tooth cylindrical gear is taken as an example below, and the first calculation method is shown in detailiMatrix induced local compliance of teethThe calculation formula of (2):

（5）

in the formula (5), the reaction mixture is,、、、representing the polynomial coefficient, and the specific calculation method is as follows:

（6）

in the formula (6), the reaction mixture is,、、represent andn、coefficient of interest, whereinnThe value range of (a) is a positive integer,representing the ratio of the radius of the root circle of the gear to the radius of the central bore of the hub, i.e.，Representing the radius of the root circle,represents the radius of the central hole of the hub and has:

（7）

wherein、、、Represents a group represented by formula (I) and n,、The related coefficient is calculated by the following formula:

（8）

in the formula (8), the reaction mixture is,A ₁represents the first lame constant of the gear material,A ₂representing the second lame constant of the gear material,representing the first Lame constant of the gear materialA ₁And second Lame constant of gear materialA ₂The coefficients of relevance are such that,

in the formula (2), the reaction mixture is,andthe calculation formula is consistent with the flow, the difference is only that the parameter values of the driving gear 1 and the driven gear 2 are different, a standard straight-tooth cylindrical gear is taken as an example below, and the first calculation method is shown in detailiBase induced coupling compliance of teethThe calculation formula of (2):

（9）

in the formula (9), the reaction mixture is,、、、、、、、、representing the polynomial coefficient, and the specific calculation method is as follows:

（10）

（11）

in the formulae (6), (10), (11), Λ,、、、Represent andthe related coefficient, the calculation formula is as follows:

（12）

in the formulae 5 to 12, the compound represented by the formula,、、respectively representing the engagement force、、The angle of action of (a) is,、、respectively representing the engagement force、、The distance from the intersection point of the line of force action of (2) with the tooth center line 6 of the tooth body 5 to the intersection point of the root circle with the tooth center line 6 of the tooth body 5,represents the arc length of the tooth root arc,represents the arc center angle of the tooth root circular arc,representing the radius of the root circle,an integral infinitesimal representing a calculus,represents an integral variable;

s5, solving a three-tooth dynamic contact state matrix model to obtain meshing force vectors F of three pairs of teeth₃And determining the actual meshing state of the gear pair, and outputting the calculated value of the dynamic meshing force of the gear pair.

In the above embodiment, the specific process of step S5 is as follows:

s5-1: solving a three-tooth dynamic contact state matrix model to obtain meshing force vectors F of three pairs of teeth₃The value of each element in (1);

s5-2: if the engaging force vector F of three pairs of teeth₃If all elements in the gear pair are positive values, the current actual meshing state of the gear pair is judged to be a three-tooth meshing state, and the dynamic meshing force of the gear pair is outputDMF：

（13）

Finishing the calculation; if the engaging force vector F of three pairs of teeth₃If all elements in the solution have non-positive values, the next step is carried out;

s5-3: if the current theoretical meshing point of the gear pair is located in the theoretical double-tooth meshing area, a pair of teeth in the off-line meshing area of the theoretical meshing line 3 is removed from the three pairs of tooth to-be-examined sets to form a double-pair tooth to-be-examined set, the double-pair tooth to-be-examined set is sequentially named as a No. 1 pair of teeth and a No. 2 pair of teeth along the rotation direction of the gear, and a double-tooth dynamic contact state matrix model is established: q₂F₂= E₂Wherein: q₂Representing a compliance coefficient matrix of the pairs of teeth, F₂Representing the vector of the meshing forces of the pairs of teeth, E₂Representing a transmission error vector of the double pairs of teeth; if the current theoretical meshing point of the gear pair is located in a theoretical single-tooth meshing area, one pair of teeth with larger tooth shape error values in two pairs of teeth in an off-line meshing area of a theoretical meshing line 3 is removed from three pairs of tooth to-be-inspected sets to form a double-pair tooth to-be-inspected set, the double-pair tooth to-be-inspected sets are sequentially named as No. 1 pair of teeth and No. 2 pair of teeth along the rotation direction of the gear, and a double-tooth dynamic contact state matrix model is established at the same time: q₂F₂= E₂Wherein Q is₂Representing a compliance coefficient matrix of the pairs of teeth, F₂Representing the vector of the meshing forces of the pairs of teeth, E₂Represents the transmission error vector of double pairs of teeth and has:

（14）

wherein the content of the first and second substances,represents the firstiTooth profile error of the number pair of teeth;represents the firstiThe meshing force of the horn pair teeth;DTErepresenting the dynamic transmission error of the currently given gear pair,represents the firstiThe hertzian contact compliance of the horn pair of teeth,represents the firstiThe flexibility of the tooth body of the number pair of teeth,represents the firstiThe matrix of the pair of teeth induces local compliance,represents the firstjMeshing force on horn pair teethF _jIn the first placeiThe matrix induced coupling flexibility generated at the meshing point of the number pair teeth;

s5-4: solving the double-tooth dynamic contact state matrix model, wherein the solving process of the double-tooth dynamic contact state matrix model is the same as that of the three-tooth dynamic contact state matrix model, and the difference is only that: subscript of each parameter in solving process of three-tooth dynamic contact state matrix modeli ∈（) Andi, j∈（) And isi≠j(ii) a In the solving process of the double-tooth dynamic contact state matrix model, subscript of each parameteri∈（) Andi, j∈（) And isi≠jObtaining the meshing force vector F of the double pairs of teeth₂The value of each element in (1), if the meshing force vector F of the double pairs of teeth₂If all elements in the gear pair are positive values, the current actual meshing state of the gear pair is judged to be a double-gear meshing state, and the dynamic meshing force of the gear pair is outputDMF：

（15）

Otherwise, the current actual meshing state of the gear pair is judgedIs in a single-tooth meshing state and outputs the dynamic meshing force of the gear pairDMF：

（16）。

The scope of the present invention includes, but is not limited to, the above embodiments, and the present invention is defined by the appended claims, and any alterations, modifications, and improvements that may occur to those skilled in the art are all within the scope of the present invention.