A method for calculating the meshing misalignment of herringbone gear pairs considering system flexibility

Abstract

The purpose of the present invention is to provide a method for calculating the meshing misalignment of a herringbone gear pair considering the flexibility of the system. Based on the basic idea of ​​axial multi-point meshing, the herringbone gear is uniformly discretized into a series of thin-sheet meshing units along the axial direction, and then the Euler-Bernoulli beam theory and the finite substructure technology are combined. By assembling the overall stiffness matrix of the system and solving the static equilibrium equation of the system, the meshing misalignment of the herringbone gear pair considering the flexibility of the system can be obtained. The present invention can calculate the meshing misalignment of the herringbone gear pair under different system structural parameters and various layout forms, providing an effective means for the quality control of tooth surface meshing and subsequent dynamic analysis and calculation.

Description

Technical Field

[0001] The present invention relates to a method for calculating a gear pair, in particular to a method for calculating the misalignment of a gear pair. Background Art

[0002] The herringbone gear pair consists of two pairs of large tooth width helical gear pairs with equal helix angles and opposite rotation directions. The single-side tooth width is generally more than 10 times the module. Since herringbone gears usually have a high degree of overlap and a large number of meshing gear pairs, the impact of manufacturing / assembly errors on the actual three-dimensional contact state of the herringbone gear tooth surface is more complicated than that of conventional straight / helical gear pairs. Due to the large radial size and unique and complex structural form, it is also more difficult to control manufacturing / assembly errors, tooth surface modification design, and vibration and noise control. In addition, the basic characteristics of large tooth width make the actual three-dimensional contact state of the tooth surface significantly affected by system deformation (shaft system deformation, bearing deformation, and housing deformation), which leads to a large difference in the actual meshing state of the left and right meshing tooth surfaces.

[0003] As a key parameter to measure the influence of system flexibility on the actual meshing state of the tooth surface, there are very few methods to calculate the meshing misalignment, and it is difficult to balance the calculation accuracy and solution efficiency. The calculation method of the meshing misalignment of herringbone gear pairs considering system flexibility has not been reported. Summary of the invention

[0004] The purpose of the present invention is to provide a method for calculating the meshing misalignment of a herringbone gear pair taking into account system flexibility, which organically combines beam unit theory and finite element substructure technology and has high calculation accuracy and high solution efficiency.

[0005] The object of the present invention is achieved in that:

[0006] The present invention provides a method for calculating the meshing misalignment of a herringbone gear pair taking into account system flexibility, which is characterized by:

[0007] (1) Discretize each axis into axis system units, and use Euler-Bernoull beam theory to establish the stiffness matrix of each axis system unit;

[0008] (2) The left and right meshing pairs of the herringbone gears are discretized into thin-slice gear meshing units along the axial direction, and the stiffness matrix of the thin-slice gear meshing unit is established according to the basic parameters of the gears and the meshing principle;

[0009] (3) Discretize the four bearings along the axial direction, establish the bearing-housing unit, and use a set of parallel springs to simulate their stiffness;

[0010] (4) The box is regarded as a basic unit, a three-dimensional finite element model of the box is established, the inner holes of the four bearing seats are coupled as four concentrated mass points, and the finite element substructure technology is used to extract its stiffness matrix;

[0011] (5) Based on the structural finite element method, the system stiffness matrix is ​​obtained by assembling the unit stiffness matrix, and then the system static equilibrium equations are established;

[0012] (6) By solving the static equilibrium equations of the system, the meshing misalignment of the herringbone gear pair considering the flexibility of the system is obtained according to the relative displacement of each slice gear meshing unit.

[0013] The present invention may also include:

[0014] 1. The flexibility matrix method is used to calculate the stiffness of the thin-plate gear meshing unit. The calculation formula of the flexibility matrix method is:

[0015]

[0016] Where [λ] is the elastic deformation compliance matrix of the contact point, {F} is the load borne by each contact point, C is the deformation of the contact point, and P is the total load.

[0017] 2. The static equilibrium equation of the bearing-housing unit is

[0018]

[0019] In the formula, K b is the bearing stiffness matrix, q bi is the displacement vector of the bearing node, q hi is the displacement vector of the equivalent node of the box.

[0020] 3. The static equilibrium equation of the box unit is

[0021] K h q h =0

[0022] In the formula, K h is the box stiffness matrix obtained by the finite element substructure method, q h is the displacement vector of the equivalent node of the box.

[0023] 4. The meshing misalignment of the herringbone gear pair is:

[0024]

[0025] In the formula, It represents the meshing misalignment of the i-th thin-plate gear unit in the left meshing pair, represents the meshing misalignment of the i-th thin-plate gear unit in the right meshing pair, It represents the relative displacement of the i-th thin-slice gear pair in the left meshing pair along the normal meshing line direction, It represents the relative displacement of the i-th thin-plate gear pair in the right meshing pair along the normal meshing line.

[0026] 5. The stiffness calculation formula of the thin-film meshing unit is written as:

[0027]

[0028] Wherein, N is the number of contact points of the sheet meshing unit.

[0029] The advantages of the present invention are that the present invention can accurately predict the tooth surface meshing quality under system flexibility, thereby effectively guiding the structural parameter design and selection of the gear transmission system, and can further lay the foundation for the tooth surface modification design considering system flexibility. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 It is a schematic diagram of the discretization of the gear pair along the axial direction;

[0031] Figure 2 It is a schematic diagram of a thin-plate gear meshing unit;

[0032] Figure 3 It is the schematic diagram of the finite element model of the box;

[0033] Figure 4 This is a schematic diagram of the meshing misalignment of a herringbone gear pair. DETAILED DESCRIPTION

[0034] The present invention is described in more detail below with reference to the accompanying drawings:

[0035] Combination Figure 1-4 The present invention provides a method for calculating the meshing misalignment of a herringbone gear pair considering system flexibility, and the steps of the method are as follows:

[0036] (1) According to the structural and layout characteristics of the gear-shaft system-bearing-box, the system is discretized and the Euler-Bernoull beam element is used to simulate the shaft system element.

[0037] (2) The left and right sides of the herringbone helical gear pair are discretized into a series of thin slices along the tooth width. The meshing process of the left meshing pair after axial discretization is as follows: Figure 1 As shown, where r p and r g The radius of the base circle of the driving and driven gears, ω p and ω g is the speed of the driving and driven gears, O p and O g It is the rotation center of the driving and driven gears. 1 N 2 is the theoretical meshing line, B 1 B 2 B 3 B 4 is the meshing plane, βb is the helix angle of the base circle. The model of the i-th nonlinear contact element is as follows Figure 2 As shown, where ψ and α are the installation angle and the meshing angle, and is the rotation center of the ith nonlinear contact element. is the meshing stiffness of the i-th nonlinear contact element of the left gear pair, Indicates meshing clearance.

[0038] The stiffness of the i-th thin-plate gear meshing unit of the meshing pair on the left and right sides and The calculation formulas are

[0039]

[0040] Where M and N are the number of contact points on the i-th thin-slice gear meshing unit of the left and right meshing pairs, respectively. is the stiffness of the i-th contact point.

[0041] The generalized coordinates of the i-th thin-slice gear meshing unit can be defined as

[0042]

[0043] In the formula and is the lateral displacement around each axis, and is the rotation angle around each axis.

[0044] The relative displacement of the i-th thin-slice gear meshing unit along the normal meshing line is:

[0045]

[0046] The expression of the projection vector T can be written as

[0047]

[0048] In the formula The "+" and the symbol "±" The “-” in the symbol “±” means that the driving wheel rotates counterclockwise. The “+” in the figure indicates that the driving wheel rotates clockwise.

[0049] (3) Use three-dimensional finite element method and substructure technology to consider the flexibility of the box, such as Figure 3As shown. For each bolt hole, the six degrees of freedom of all nodes on its inner surface are constrained. For each bearing seat, all nodes on its inner surface are coupled, and then the sixth-order stiffness matrix of the four bearing holes is compressed using the substructure method. The stiffness matrix obtained in this way fully reflects the flexibility of the box. After the models of the shaft system unit, bearing unit, thin-film gear meshing unit and box unit are established, the stiffness matrix of the system can be assembled according to the finite element method. The matrix form of the static equilibrium equation of the system can be written as

[0050] K GS (t)X GS (t) = P G

[0051] Where K GS (t) is the system stiffness matrix; X GS (t) is the static displacement vector of the system; P G is the external load vector.

[0052] (4) Solving the static equilibrium equation of the system can obtain the relative displacement and meshing misalignment of the left and right meshing pairs along the normal meshing line, as follows: Figure 4 The calculation formula for the meshing misalignment of the left and right meshing pairs is as follows:

[0053]

[0054] In the formula is the meshing misalignment of the i-th thin-slice gear meshing unit on the left, is the meshing misalignment of the i-th gear meshing unit on the right.

[0055] In summary, the present invention calculates the meshing misalignment of the herringbone gear pair taking into account the flexibility of the system, and has higher calculation efficiency and more precise calculation accuracy.

Claims

1. A method for calculating the meshing misalignment of a herringbone gear pair considering system flexibility, characterized by: (1) Discretize each axis into axis system units, and use Euler-Bernoull beam theory to establish the stiffness matrix of each axis system unit; (2) The left and right meshing pairs of the herringbone gears are discretized into thin-slice gear meshing units along the axial direction, and the stiffness matrix of the thin-slice gear meshing unit is established according to the basic parameters of the gears and the meshing principle; (3) Discretize the four bearings along the axial direction, establish the bearing-housing unit, and use a set of parallel springs to simulate their stiffness; (4) The box is regarded as a basic unit, a three-dimensional finite element model of the box is established, the inner holes of the four bearing seats are coupled as four concentrated mass points, and the finite element substructure technology is used to extract its stiffness matrix; (5) Based on the structural finite element method, the system stiffness matrix is ​​obtained by assembling the unit stiffness matrix, and then the system static equilibrium equations are established; (6) By solving the static equilibrium equations of the system, the meshing misalignment of the herringbone gear pair considering the flexibility of the system is obtained according to the relative displacement of each slice gear meshing unit; The flexibility matrix method is used to calculate the stiffness of the thin-plate gear meshing unit. The calculation formula of the flexibility matrix method is: Where [λ] is the elastic deformation compliance matrix of the contact point, {F} is the load borne by each contact point, C is the deformation of the contact point, and P is the total load; The static equilibrium equation of the bearing-housing unit is In the formula, K b is the bearing stiffness matrix, q bi is the displacement vector of the bearing node, q hi is the displacement vector of the equivalent node of the box; The static equilibrium equation of the box unit is K h q h =0 In the formula, K h is the box stiffness matrix obtained by the finite element substructure method, q h is the displacement vector of the equivalent node of the box; The meshing misalignment of the herringbone gear pair is: In the formula, It represents the meshing misalignment of the i-th thin-plate gear unit in the left meshing pair, represents the meshing misalignment of the i-th thin-plate gear unit in the right meshing pair, It represents the relative displacement of the i-th thin-slice gear pair in the left meshing pair along the normal meshing line direction, It represents the relative displacement of the i-th thin-plate gear pair of the right meshing pair along the normal meshing line; The stiffness calculation formula of the thin-film meshing element is written as: Wherein, N is the number of contact points of the sheet meshing unit.

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