A method for solving interdental load distribution coefficient based on finite element method

Abstract

The application provides a finite element method-based method for solving intertooth load distribution coefficient, which first needs to establish a finite element model of gear pair with regular tooth surface nodes, and solve the model to obtain the contact pressure value of any node on the tooth surface; the obtained tooth surface node pressure is used to solve the total pressure of each tooth surface participating in meshing by numerical integral division method, and the total pressure of each tooth surface participating in meshing is divided by the total pressure of all tooth surfaces participating in meshing to obtain the intertooth load distribution coefficient of the gear pair. The advantage of the method is that the solving of the load distribution coefficient of any non-standard gear pair can be completed while the mechanical analysis of the gear pair is performed, so that the purpose of reducing complex theoretical calculation is achieved, and the solving error caused by the traditional approximate solution method is reduced.

Description

Technical Field

[0001] This invention relates to the field of gear strength design technology, and in particular to a method for solving the inter-tooth load distribution coefficient based on the finite element method. Background Technology

[0002] During gear meshing, the contact between gear teeth occurs alternately, sometimes as a single tooth and sometimes as a double tooth. In gear transmissions with a high degree of overlap, double and triple tooth contacts may also alternate. Taking single and double tooth contact as examples, this paper analyzes the evolution of the normal force Fn on the tooth surface of a specific gear tooth during the meshing cycle; the inter-tooth load distribution can lay the foundation for gear strength design and elastohydrodynamic lubrication analysis.

[0003] Figure 1 These are three special meshing states during the gear meshing process. Figure 1 (a) shows the state when the gears first engage, with teeth 1 and 2 in contact simultaneously, and the contact point of tooth 2 located near the tooth root. As engagement progresses, the contact point on the gear teeth changes to the middle position of tooth 2, becoming a single-tooth contact state, as shown below. Figure 1 As shown in (b), when tooth number 2 is about to disengage, the contact point is located at the tip of tooth number 2, at which point teeth number 2 and 3 are in double-tooth contact. For tooth number 2, the contact point on the tooth surface changes from the tooth root to the tooth tip. Under the same torque T, the tooth surface curvature and load at the contact point also change with the change in meshing state. Under the action of torque T, the ratio of the normal load Fn on one tooth to the sum of the normal loads on every tooth of the entire gear is called the load-sharing factor (LSF).

[0004] During one meshing cycle, the tooth surface contact imprint of tooth #2 and the corresponding transmission error are as follows: Figure 2 As shown, Figure 2 In this configuration, single-tooth contact occurs only in the middle region of the gear teeth, while double-tooth contact occurs in the engagement and disengagement regions. Figure 3 In the diagram, at a certain moment, the contact position of tooth 1 is P1, and the contact position of tooth 2 is P2. The transmission error of tooth 1 is greater than that of tooth 2. As meshing progresses, the transmission error of tooth 1, which is about to disengage, increases, while the transmission error of tooth 2, which is entering meshing, decreases, until a single-tooth meshing state is reached. This change in transmission error can be considered to be caused by tooth surface contact deformation. If the initial angle of tooth 2 entering meshing is θ... START The termination angle when disengaging is θ END The angle at which the teeth disengage from the double teeth and engage with the single tooth is θ. LPSTC The angle at which the tooth disengages from a single tooth and enters double-tooth engagement is θ. HPSTCAccording to the American AGMA gear standard, the load distribution factor of tooth #2 can be calculated using the following formula;

[0005]

[0006] Because gear teeth undergo both bending deformation and contact deformation during meshing, and errors also exist during gear machining and installation, the load on the contact surface is nonlinear and non-uniform. Therefore, solving the inter-tooth load distribution using traditional potential energy methods and analytical methods is quite complex. Furthermore, traditional inter-tooth load distribution theories are only suitable for standard gear pairs and are ineffective for gear pairs with tooth surface modification and assembly errors. Consequently, the obtained load distribution coefficients have significant errors compared to the actual situation. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide a method for solving the load distribution coefficient between teeth based on the finite element method. By performing mechanical analysis on the gear pair, the load distribution can be solved at the same time, so as to reduce the error of the load distribution calculation results of non-standard gears.

[0008] A method for solving the inter-tooth load distribution coefficient based on the finite element method includes the following steps:

[0009] S1. Create a finite element model of the gear pair. The tooth surface nodes of the created finite element model must be regularly distributed, that is, the number of tooth surface nodes is Nl rows and Np columns. Divide one meshing cycle of the gear (Q = 360 / Z degrees, Z is the number of teeth) into m contact moments. That is, complete one static analysis for every Q / m degree rotation of the gear. At this time, the stress distribution data of the tooth surface at each contact moment can be obtained until the static analysis of m contact moments is completed.

[0010] S2. When a gear pair meshes, contact occurs between the two paired tooth surfaces, causing contact pressure and deformation on the tooth surfaces; if the normal load of one tooth on the other tooth is F... n ; under normal load F n Under the action of the force, the contact pressure at any point on the tooth surface can be represented by the function P(u,v); in the coordinate system established in the tooth direction, the contact pressure distribution P(v) in the tooth direction is represented in the coordinate system. Similarly, the pressure distribution in the tooth profile direction is represented by P(u). Based on the solution results of the finite element model of the gear pair (step S1), the contact pressure values ​​P(u,v) on all nodes of the tooth surface can be obtained.

[0011] S3. The formula for calculating the contact pressure distribution function P(u) along the tooth profile direction of the tooth surface is as follows:

[0012]

[0013] In Equation 1, u0 is the lower limit of integration along the tooth profile direction; uf This represents the upper limit of the integral along the tooth profile direction;

[0014] S4. The pressure values ​​at the nodes obtained after finite element analysis are discrete quantities and cannot be directly integrated. Therefore, the trapezoidal rule in numerical integration is used to replace formula (I), and the solution is as follows:

[0015]

[0016] In Equation 2, r n r n+1 is the position vector of two adjacent nodes on the tooth surface; the position vector data of all nodes on each tooth surface can be exported from the finite element model; Pn and Pn+1 are the contact pressure values ​​corresponding to these nodes obtained from the analysis of the gear finite element model, and n and n+1 represent the numbers of two adjacent nodes in the tooth profile direction;

[0017] S5. The equivalent normal load Fni on the tooth surface at the i-th (i = 1, 2, ..., m-1, m) contact instant is solved by integrating the unit load over the tooth width direction, as shown in the following formula:

[0018]

[0019] S6. The total normal load of all meshing teeth on the entire gear is calculated as follows:

[0020]

[0021] S7. Based on the definition of the load distribution coefficient, the load distribution coefficient of a certain gear tooth at the i-th contact instant is calculated as follows:

[0022]

[0023] In step S2, the variable u is the coordinate value in the tooth direction, and the variable v is the coordinate value in the tooth profile direction.

[0024] The unit of load on the tooth surface is N / mm.

[0025] The beneficial effects of this invention are:

[0026] 1. This method first requires establishing a finite element model of a gear pair with regular tooth surface nodes and solving it to obtain the contact pressure value at any node on the tooth surface. Using the obtained tooth surface node pressures, the total pressure of each meshing tooth surface is solved by numerical integration. Dividing the total pressure of a single tooth surface by the total pressure of all meshing tooth surfaces yields the inter-tooth load distribution coefficient of the gear pair. The advantage of this method is that it can simultaneously perform mechanical analysis of the gear pair and solve for the load distribution coefficient of any non-standard gear pair, thereby reducing complex theoretical calculations and minimizing the solution errors caused by traditional approximate methods. By obtaining the contact pressure values ​​of all nodes on the tooth surface based on the finite element analysis of the gear pair, the total normal load of the meshing teeth can be solved, further reducing complex theoretical calculations.

[0027] 2. This method can calculate the load distribution coefficient between teeth of any non-standard gear, thereby reducing the error caused by approximate solution using the American AGMA standard formula. Attached Figure Description

[0028] Figure 1 This is a structural diagram of the gear meshing process;

[0029] Figure 2 This is a schematic diagram of the contact marks on the tooth surfaces during gear meshing;

[0030] Figure 3 This is a schematic diagram of transmission error during gear meshing;

[0031] Figure 4 This is a schematic diagram of the contact pressure distribution on the tooth surface;

[0032] Figure 5 A schematic diagram illustrating the parameters of a single-tooth finite element model;

[0033] Figure 6 A schematic diagram comparing the load distribution factors of FEM and AGMA; Detailed Implementation

[0034] Example 1

[0035] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0036] like Figure 4 As shown, a method for solving the inter-tooth load distribution coefficient based on the finite element method includes the following steps:

[0037] S1. Create a finite element model of the gear pair. The tooth surface nodes of the created finite element model must be regularly distributed, that is, the number of tooth surface nodes is Nl rows and Np columns. Divide one meshing cycle of the gear (Q = 360 / Z degrees, Z is the number of teeth) into m contact moments. That is, complete one static analysis for every Q / m degree rotation of the gear. At this time, the stress distribution data of the tooth surface at each contact moment can be obtained until the static analysis of m contact moments is completed.

[0038] S2. When a gear pair meshes, contact occurs between the two paired tooth surfaces, causing contact pressure and deformation on the tooth surfaces; if the normal load of one tooth on the other tooth is F... n ; under normal load F n Under the action of the force, the contact pressure at any point on the tooth surface can be represented by the function P(u,v); in the coordinate system established in the tooth direction, the contact pressure distribution P(v) in the tooth direction is represented in the coordinate system. Similarly, the pressure distribution in the tooth profile direction is represented by P(u). Based on the solution results of the finite element model of the gear pair (step S1), the contact pressure values ​​P(u,v) on all nodes of the tooth surface can be obtained.

[0039] S3. The formula for calculating the contact pressure distribution function P(u) along the tooth profile direction of the tooth surface is as follows:

[0040]

[0041] In Equation 1, u0 is the lower limit of integration along the tooth profile direction; u f This represents the upper limit of the integral along the tooth profile direction;

[0042] S4. The pressure values ​​at the nodes obtained after finite element analysis are discrete quantities and cannot be directly integrated. Therefore, the trapezoidal rule in numerical integration is used to replace formula (I), and the solution is as follows:

[0043]

[0044] In Equation 2, r n r n+1 is the position vector of two adjacent nodes on the tooth surface; the position vector data of all nodes on each tooth surface can be exported from the finite element model; Pn and Pn+1 are the contact pressure values ​​corresponding to these nodes obtained from the analysis of the gear finite element model, and n and n+1 represent the numbers of two adjacent nodes in the tooth profile direction;

[0045] S5. The equivalent normal load Fni on the tooth surface at the i-th (i = 1, 2, ..., m-1, m) contact instant is solved by integrating the unit load over the tooth width direction, as shown in the following formula:

[0046]

[0047] S6. The total normal load of all meshing teeth on the entire gear is calculated as follows:

[0048]

[0049] S7. Based on the definition of the load distribution coefficient, the load distribution coefficient of a certain gear tooth at the i-th contact instant is calculated as follows:

[0050]

[0051] In step S2, the variable u is the coordinate value in the tooth direction, and the variable v is the coordinate value in the tooth profile direction.

[0052] The unit of load on the tooth surface is N / mm.

[0053] The beneficial effects of this invention are:

[0054] Based on the finite element analysis of the gear pair, the contact pressure values ​​of all nodes on the tooth surface are obtained, thereby solving the total normal load of the meshing gear teeth, which can reduce complex theoretical calculations.

[0055] This method can calculate the load distribution coefficient between teeth of any non-standard gear, thereby reducing the error caused by approximate solutions using the American AGMA standard formula.

[0056] Example of inter-tooth load distribution calculation:

[0057] The following analysis uses a cylindrical gear pair with certain parameters as the object of study. The gear pair parameters are: 40 teeth on the large gear, 31 teeth on the small gear, a module of 4 mm, a cutter head radius of 60 mm for machining both gears, a material elastic modulus of 208000 MPa, a Poisson's ratio of 0.298, a coefficient of friction of 0.25, and a driven gear resistance torque of 150 N·m. The analysis examines the load distribution between the teeth within a complete meshing cycle (ensuring the engagement and disengagement of a single tooth). A complete meshing cycle is evenly divided into 21 contact positions, and the load distribution coefficient (LSFi, i = 1, 2, ..., 21) for each contact position is calculated.

[0058] Furthermore, the finite element model of the gear pair in this example adopts a three-tooth model. Figure 5 For the single-tooth model, the number of elements on the tooth surface is Np×Nl=50×40, the number of elements in the tooth thickness direction is Nc=6, the mesh density near the tooth surface is appropriately increased, the number of elements on the tooth transition surface without contact behavior is Nf=6, and the number of elements on the tooth ring is Nr=5.

[0059] The fundamental angle for calculating the AGMA load distribution coefficient in this example can be solved using the TCA (tooth contact analysis) method. The results are as follows: θSTART = -9.64°, θ END = 9.95°, θ LPSTC = -1.66°, θ HPSTC =1.97°. Substituting these angular parameters into equation (a), the AGMA load distribution coefficient can be solved. The tooth surface load distribution coefficient solved by the finite element method can be automatically extracted through programming. The results obtained by the two methods are as follows: Figure 6 As shown, from Figure 6 It can be seen that the single-tooth meshing zone obtained by the two solution methods is the same, while the load distribution coefficient of the double-tooth meshing zone has a certain deviation. This is mainly because the tooth root undergoes bending deformation under the action of torque, as well as the tooth tip edge contact. The AGMA method does not consider the tooth deformation caused by the load, which is basically consistent with the deviation law caused by the experimental results obtained by Spitas et al.

[0060] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for solving the inter-tooth load distribution coefficient based on the finite element method, characterized in that; Includes the following steps: S1. Create a finite element model of the gear pair. The finite element model should have regularly distributed tooth surface nodes, i.e., the number of tooth surface nodes is Nl rows and Np columns. Divide one meshing cycle of the gear into m contact moments. The meshing cycle is Q = 360 / Z degrees, where Z is the number of teeth. That is, a static analysis is completed every time the gear rotates Q / m degrees. At this time, the stress distribution data of the tooth surface at each contact moment is obtained until the static analysis of m contact moments is completed. S2. When a gear pair meshes, contact occurs between the two paired tooth surfaces, causing contact pressure and deformation on the tooth surfaces; if the normal load of one tooth on the other tooth is F... n ; under normal load F n Under the action of the force, the contact pressure at any point on the tooth surface is represented by the function P(u,v); in the coordinate system established in the tooth direction, the contact pressure distribution P(v) in the tooth direction is represented in the coordinate system. Similarly, the pressure distribution in the tooth profile direction is represented by P(u). Based on the solution results of the finite element model of the gear pair, the contact pressure values ​​P(u,v) on all nodes of the tooth surface are obtained. S3. The formula for calculating the contact pressure distribution function P(u) along the tooth profile direction of the tooth surface is as follows: In Equation 1, u0 is the lower limit of integration along the tooth profile direction; u f This represents the upper limit of the integral along the tooth profile direction; S4. The pressure values ​​at the nodes obtained after finite element analysis are discrete quantities and cannot be directly integrated. Therefore, the trapezoidal rule in numerical integration is used instead of Equation 1, and the solution is as follows: In Equation 2, r n r n+1 is the position vector of two adjacent nodes on the tooth surface; the position vector data of all nodes on each tooth surface are derived from the finite element model; Pn and Pn+1 are the contact pressure values ​​corresponding to these nodes obtained from the analysis of the gear finite element model, and n and n+1 represent the numbers of two adjacent nodes in the tooth profile direction; S5. The equivalent normal load Fni on the tooth surface at the i-th (i = 1, 2, ..., m-1, m) contact instant is solved by integrating the unit load over the tooth width direction, as shown in the following formula: S6. The total normal load of all meshing teeth on the entire gear is calculated as follows: S7. Based on the definition of the load distribution coefficient, the load distribution coefficient of a certain gear tooth at the i-th contact instant is calculated as follows:

2. The method for solving the inter-tooth load distribution coefficient based on the finite element method as described in claim 1, characterized in that: In step S2, the variable u is the coordinate value in the tooth direction, and the variable v is the coordinate value in the tooth profile direction.

3. The method for solving the inter-tooth load distribution coefficient based on the finite element method as described in claim 1, characterized in that: The unit of load on the tooth surface is N / mm.

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