CN104679941B - A kind of computational methods of tooth surface bending deformation quantity - Google Patents

Abstract

The present invention relates to a kind of computational methods of tooth surface bending deformation quantity, and gear teeth face flexural deformation softness factor matrix is extracted with reference to FInite Element and differential technique；Then, gear teeth face bending deformation quantity is established by coordinate transform and mapping and accurately solves computation model, a complicated space curved surface problem on deformation is reduced to a simple cantilever beam deformed problem；Finally, the bending deformation quantity of optional position on gear teeth face has been obtained by interpolation calculation.To realize that carrying out dynamic load contact analysis (LTCA) to gear by deformation compatibility condition lays a good foundation, and has also laid good place mat for the quasi-statics analysis method of gear.

Description

A Calculation Method of Bending Deformation of Gear Tooth Surface

technical field

The invention belongs to the field of mechanical analysis of gears, and in particular relates to a calculation method for the amount of bending deformation of the tooth surface when the gear teeth are loaded during the gear transmission process.

Background technique

In order to realize the dynamic loading contact analysis (LTCA) of gears through deformation coordination conditions, it is necessary to obtain the tooth surface bending deformation and tooth surface contact deformation of the gear pair at each contact moment.

In current engineering, the most commonly used method for calculating the force of the gear transmission process is to perform calculation and analysis through finite element software such as (ANSYS, ABAQUS, etc.). This method establishes the finite element model of the gear and assembles it. By applying loads and constraints, the static force of the gear can be obtained, and the deformation of the gear under force can be obtained. At this time, the deformation of the tooth surface includes contact deformation. and bending deformation. However, this method requires a lot of computer time, and each contact position needs not only adjustment and calculation, but also the deformation of each position on the tooth surface is not easy to extract, and is not accurate.

However, the contact deformation of the tooth surface can be easily calculated by Hertz theory, but the calculation model of the tooth surface bending deformation has not been deeply studied. In the early years, it was mentioned that the gear tooth is regarded as a cantilever beam to calculate the bending deformation, but due to the complex shape of the gear tooth, this calculation model is not accurate.

Contents of the invention

technical problem to be solved

In order to avoid the deficiencies of the prior art, the present invention proposes a calculation method for the bending deformation of the gear tooth surface, and extracts the gear tooth surface bending deformation compliance coefficient matrix in combination with the finite element method and the difference method; then, through coordinate transformation The calculation model for the accurate calculation of the bending deformation of the gear tooth surface is established with the mapping. This method simplifies a complex space surface deformation problem into a simple cantilever beam bending deformation problem; finally, an arbitrary position on the gear tooth surface is obtained through interpolation calculation. amount of bending deformation.

Technical solutions

A method for calculating the amount of bending deformation on the surface of a gear tooth, characterized in that the steps are as follows:

Step 1. Extract the tooth surface bending deformation compliance coefficient matrix:

Establish a single-tooth finite element model, obtain the node coordinates and node numbers on the working tooth surface, and at the same time obtain the normal vector of each node position on the tooth surface through the method of differential geometry;

Decompose a unit force onto the normal vector of each node to form a unit load vector on each node;

Import the single-tooth finite element model into the finite element software, and impose full-degree-of-freedom constraints on all nodes on the bottom and both sides of the single-tooth finite element model;

According to the node number, the unit load vector is applied to each node on the working tooth surface in turn, and the deformation of all nodes on the working tooth surface is extracted to obtain a comprehensive compliance coefficient matrix [C _z ] _n×n×3 , where: n Indicates that there are a total of n nodes on the working tooth surface,

The first dimension of the matrix represents the unit load vector applied to the i-th node; the second dimension represents the amount of deformation generated on the j-th node; the first column of the third dimension represents the amount of deformation of the j-th node in the X direction, The second column of the third dimension represents the deformation of the jth node in the Y direction, and the third column of the third dimension represents the deformation of the jth node in the Z direction;

Then apply full-degree-of-freedom constraints to the nodes on the bottom surface and both sides of the single-tooth finite element model, and at the same time, apply full-degree-of-freedom constraints to all nodes on the non-working tooth surface;

According to the node number, the unit load vector is applied to each node on the working tooth surface in turn, and the deformation of all nodes on the working tooth surface is extracted to obtain a contact deformation compliance coefficient matrix [C _C ] _n×n×3 ;

Subtract the two coefficient matrices to obtain the bending deformation flexibility coefficient matrix of the working tooth surface of a single gear tooth:

[C _f ] _n×n×3 = [C _z ] _n×n×3 −[C _c ] _n×n×3 ;

Step 2. Establish the calculation model of tooth surface bending deformation: obtain the parameter representation r _i (u _i , v _i ) of any node on the tooth surface of the single-tooth finite element model through the gear generation method, and use the coordinate projection and mapping to convert the finite element The nodes on the working tooth surface of the model are transformed into the plane coordinate system. The two axes of the plane coordinate system are u and v respectively. The flexibility coefficient matrix is used as the mechanical properties of the planar cantilever thin plate model, and the calculation model of the bending deformation of the tooth surface is obtained;

Step 3. Calculate the bending deformation of the tooth surface: According to the tooth surface contact analysis method, the distribution of contact position and contact pressure on the working tooth surface at each contact instant is obtained, and then the contact pressure of each contact area is equivalently distributed to all nodes in the area On the above, the load matrix of the calculation model of the bending deformation of the entire tooth surface is obtained, and the load matrix is multiplied by the bending flexibility coefficient matrix to obtain the bending deformation at any node;

For the non-node position on the working tooth surface, it is obtained by linear interpolation of the nodes near the position point.

Beneficial effect

A method for calculating the amount of bending deformation of the gear tooth surface proposed by the present invention combines the finite element method and the difference method to extract the flexibility coefficient matrix of the gear tooth surface bending deformation; then, establishes the bending deformation of the gear tooth surface through coordinate transformation and mapping Quantitatively and accurately solve the calculation model, simplifying a complex space surface deformation problem into a simple cantilever beam bending deformation problem; finally, the bending deformation at any position on the gear tooth surface is obtained through interpolation calculation. It lays a good foundation for the dynamic loading contact analysis (LTCA) of gears through deformation coordination conditions, and also lays a good foundation for the quasi-static analysis method of gears.

Description of drawings

Fig. 1(a) Finite element calculation model of non-working tooth surface without displacement constraints;

A-working surface

Figure 1(b) Finite element calculation model with displacement constraints on the non-working tooth surface;

B-non-working surface

Figure 2 Calculation model of tooth surface bending deformation;

Figure 3 Equivalent treatment of tooth surface contact pressure

Detailed ways

Now in conjunction with embodiment, accompanying drawing, the present invention will be further described:

Step 1, extract the tooth surface bending deformation compliance coefficient matrix:

Establish a finite element model of a single gear tooth, determine the node coordinates and node numbers on the working tooth surface, and at the same time obtain the normal vector of each node position on the tooth surface by means of differential geometry.

Decompose a unit force to the normal vector of each node of the working tooth surface to form a unit load vector on each node.

As shown in Fig. 1(a), in the finite element software (such as ANSYS), the degrees of freedom in all directions of all nodes on the bottom surface and both sides of the single-tooth finite element model are all constrained, and then, the first Apply the corresponding unit load vector to the node, extract the deformation of all nodes under a single unit load vector, then delete the load, apply the unit load vector corresponding to the next node, and extract the deformation of all nodes under a single unit load vector, then... ..., assuming that there are n nodes on the working tooth surface, it is repeated n times in sequence. From this, a comprehensive flexibility coefficient matrix [C _z ] _n×n×3 can be obtained. Among them, the first dimension of the matrix represents the unit load vector applied to the i-th node; the second dimension represents the deformation generated on the j-th node; the first column of the third dimension represents the deformation of the j-th node in the X direction, The second column of the third dimension represents the deformation amount of the jth node in the Y direction, and the third column of the third dimension represents the deformation amount of the jth node in the Z direction.

As shown in Figure 1(b), in the finite element software (such as ANSYS), the degrees of freedom in all directions of all nodes on the bottom surface and both sides of the single-tooth finite element model are all constrained, and at the same time, the non-working tooth surface of the gear tooth The degrees of freedom in all directions of all nodes on the node are all constrained. Then, apply the corresponding unit load vector to the first node on the working tooth surface, extract the deformation of all nodes under the single unit load vector, then delete the load, apply the unit load vector corresponding to the next node, and extract the single unit load vector Under the deformation of all nodes, next..., repeat n times in turn. From this, a contact deformation compliance coefficient matrix [C _C ] _n×n×3 can be obtained.

Therefore, the bending deformation flexibility coefficient matrix of the working tooth surface of a single gear tooth can be obtained by subtracting the two:

[C _f ] _n×n×3 = [C _z ] _n×n×3 −[C _c ] _n×n×3

(1) Establishment of calculation model for tooth surface bending deformation

The tooth surface of a gear is usually a space-complex surface, and the model for calculating the bending deformation by directly applying force on the tooth surface of the gear is relatively complicated. Usually, the space surface can be expressed in a parametric form, and r(u,v) below represents the parametric equation of the tooth surface studied in this paper. Then the parameters of any node on the tooth surface of the single-tooth finite element model can be expressed as r _i (u _i , v _i ). As shown in Figure 2, below, establish a plane coordinate system whose two axes are u and v respectively, and then transform all nodes on the tooth surface to the plane coordinate system through coordinate projection and mapping, and then obtain a plane grid Model, if a fixed constraint is imposed on one end of the grid, the solid model of the single-tooth finite element model under the constraints can be expressed as a planar cantilever thin plate bending calculation model. At the same time, the bending compliance coefficients of a series of discrete nodes on the planar cantilever thin plate are known. So far, the calculation model of the bending deformation of the entire gear tooth surface has been completely established. It is a planar network of the bending deformation of the tooth surface grid calculation model.

(2) Calculation of tooth surface bending deformation

According to the existing tooth surface contact analysis (TCA) technology, each contact instant of a pair of gear pairs, the contact position and the distribution area of the contact pressure on the working tooth surface can be obtained.

It is known that when the contact is made locally, the contact pressure on the tooth surface is similar to the Hertz pressure, and its pressure distribution can be expressed as:

p=p ₀ {1-(r/a) ² } ^1/2

Therefore, the pressure distribution curve of any section passing through the central axis can be shown by the curve in Figure 3. In "Material Mechanics", the pressure and deflection have a linear relationship. The following method can be used to simplify the calculation by evenly distributing the contact pressure to the nodes near the center of the contact ellipse. As shown in Figure 3, when the area of the square is equal to the curve and the axis contains When the area is , the calculation of the bending deformation at each point on the tooth surface is roughly accurate.

With this simplified method, the contact pressure of each contact area is equivalently distributed to each node in this area, so that the force of the calculation model of the bending deformation of the entire tooth surface can be obtained, combined with the bending flexibility coefficient matrix, it can be obtained The amount of bending deformation at any node. For the non-node position on the working tooth surface, it can be obtained by linear interpolation of the nodes near the position point. According to "Material Mechanics", in the case of small deformation, linear interpolation has higher calculation accuracy.

Claims (1)

1. a kind of computational methods of tooth surface bending deformation quantity, it is characterised in that step is as follows：

Step 1, extraction flank of tooth flexural deformation softness factor matrix：

Monodentate finite element model is established, obtains node coordinate, the node serial number on working flank, while the side for passing through Differential Geometry Method obtains the normal vector of each node location on the flank of tooth；

One unit force is decomposed on the normal vector of each node, forms the specific loading vector on each node；

Monodentate finite element model is imported in finite element software, to all sections on the bottom surface and two sides of monodentate finite element model Point applies full free degree constraint；

According to node serial number, apply specific loading vector to each node on working flank successively, and extract on working flank The deflection of all nodes, obtains a comprehensive softness factor matrix [C_z]_n×n×3, wherein：N represents a shared n on working flank A node,

First dimension of the matrix is represented applies specific loading vector in i-th of node；Second dimension is represented produces on j-th of node Raw deflection；Third dimension first row represent j-th of node X to deflection, third dimension secondary series represents j-th of node and exists The deflection of Y-direction, the row of the third dimension the 3rd represent deflection of j-th of node in Z-direction；

Apply the full free degree to node on monodentate finite element model bottom surface and two sides again to constrain, meanwhile, on non-working flank All nodes apply full free degree constraint；

According to node serial number, apply specific loading vector to each node on working flank successively, and extract on working flank The deflection of all nodes, obtains a juxtaposition metamorphose softness factor matrix [C_C]_n×n×3；

Two coefficient matrixes are done into additive operation and obtain the flexural deformation softness factor matrix of single gear teeth working flank：

[C_f]_n×n×3=[C_z]_n×n×3-[C_c]_n×n×3；

Step 2, establish flank of tooth bending deformation quantity computation model：Obtained by gear generating method on the monodentate finite element model flank of tooth The parameter of any node represents r_i(u_i,v_i), by coordinate projection and mapping, the node on finite element model working flank is become Change under plane coordinate system, two axis of plane coordinate system are respectively u, v, by entity of the monodentate finite element model under constraints Model is expressed as a plane cantilever sheet model, the power using flexural deformation softness factor matrix as plane cantilever sheet model Characteristic is learned, obtains flank of tooth bending deformation quantity computation model；

Step 3, calculate flank of tooth bending deformation quantity：Each contact is obtained instantaneously according to Tooth Contact Analysis method, on working flank The distribution of contact position and contact, then by all sections of the contact Equivalent Distributed of each contact area to the region On point, the loading matrix of whole flank of tooth bending deformation quantity computation model is obtained, by loading matrix with bending softness factor matrix phase Multiply, obtain the bending deformation quantity on arbitrary node；

For not a node position on working flank, the method that linear interpolation is carried out to node near the not a node is tried to achieve.