Newton-Raphson-based angular contact bearing pre-tightening load calculation method

Abstract

The invention discloses a method for calculating a pre-tightening load of an angular contact bearing based on Newton-Raphson, and belongs to the technical field of angular contact bearing pre-tightening load calculation, and the method comprises the following steps: 1, inputting a bearing structure parameter and a maximum axial load Famax; 2, calculating an initial contact angle alpha degree; 3, the initial value Fp0 of the pre-tightening load is determined to be equal to 0.2 * Famax; 4, calculating the pre-deformation delta P by using the current pre-tightening load Fp; 5, calculating the contact angle alpha 2 of the bearing 2 by using the following formula; sixthly, whether the deformation delta 2max of the bearing 2 is larger than 0 or not is judged; 7, calculating a new iteration step length by using a formula 11; 8, updating the pre-tightening load value by using the new iteration step length; ninthly, the pre-tightening load in the current step is the pre-tightening load needed by the paired bearing. According to the method, a Newton-Raphson-based pre-tightening load solving equation set is deduced by constructing a bearing pre-tightening load mathematical model, and the pre-tightening load calculation efficiency is greatly improved on the premise that the precision is met by adopting a self-adaptive step length adjustment method.

Description

Technical Field

[0001] The invention relates to the technical field of angular contact bearing preload calculation, and in particular to a method for calculating the angular contact bearing preload based on Newton-Raphson. Background Art

[0002] Angular contact bearings have good structural performance and can withstand axial and radial loads. They are widely used in high-precision rotating structures. Angular contact bearings are generally used in pairs. In order to improve the rotation accuracy, support stiffness and load-bearing capacity of the bearings, the angular contact bearings need to be preloaded. Whether the calculation and control of the bearing preload are reasonable will directly determine the rotation accuracy, structural stiffness, and service life of the rotating shaft. If the preload force is too large, the bearing friction torque will increase, the bearing wear will accelerate, and the bearing service life will be shortened; if the preload force is too small, the bearing support stiffness will deteriorate, the bearing runout will increase, and the rotation accuracy and running stability of the rotating shaft will deteriorate, affecting the performance indicators.

[0003] The existing bearing preload calculation method uses a simplified formula for bearing displacement preload, assuming that the bearing stiffness remains unchanged, and calculates the bearing preload, which is approximately 35% of the maximum axial load. Some methods recommend different preloads based on experience according to different load ranges.

[0004] Defects of existing technical solutions:

[0005] 1) The existing technical solutions do not consider the influence of bearing load on bearing stiffness. For bearings working under heavy load conditions, the calculation error of preload is relatively large.

[0006] 2) The existing technical solutions lack accurate theoretical derivation and mathematical calculation, and cannot accurately calculate the bearing preload. The preload calculation is not universal and accurate. Therefore, we propose a method based on Newton-Raphson to calculate the preload of angular contact bearings to solve this problem. Summary of the invention

[0007] The object of the present invention is to provide a method for calculating the preload of an angular contact bearing based on Newton-Raphson to solve the problems raised in the above background technology.

[0008] In order to achieve the above object, the present invention adopts the following technical solutions:

[0009] A method for calculating the preload of an angular contact bearing based on Newton-Raphson includes the following steps:

[0010] Step 1: Input bearing structural parameters and maximum axial load F amax ;

[0011] Step 2: Calculate the initial contact angle α ° , the calculation formula is:

[0012]

[0013] Calculate the displacement constant K using the formula:

[0014]

[0015] In the above formula:

[0016] Fa is the axial load borne by a single angular contact bearing;

[0017] K is the axial displacement constant;

[0018] The initial contact angle of the bearing changes from α° to α;

[0019] Step 3: Determine the initial value of the preload F p0 =0.2*F amax , initial iteration step length step =

[0020] 0.001*F amax ;

[0021] Step 4: Use the current preload F p Calculate the pre-deformation δ P ;

[0022] Step 5: Use the following formula to calculate the contact angle α2 of bearing 2 and the deformation δ of bearing 2 when the axial load is 0 2max , the axial load is F amax Deformation of bearing 2 δ 2min :

[0023]

[0024] Calculate the Jacobi matrix J(α) based on f1(a1,α2), f2(α1,α2);

[0025]

[0026] In the above formula:

[0027]

[0028] Step 6: Determine the deformation of bearing 2 δ 2min Is it greater than 0? If it is satisfied, jump to step 9; if it is not satisfied, jump to step 7;

[0029] Step 7: Use formula 11 to calculate the new iteration step size:

[0030] step=0.001*Famax*(1+abs(δ2min / δ2max)*100) Formula 11;

[0031] Step 8: Use the new iteration step to update the preload value and jump to step 5;

[0032] Step 9: The preload in the current step is the preload required by the paired bearing;

[0033] Preferably, in step 2, the parameter K is calculated using the following formula:

[0034] K=-26451B 3 +36402B 2 +19704B-16.827 Formula 3

[0035] In the above formula, B is the total curvature of the bearing.

[0036] Preferably, in step 2, the Newton-Raphson method is used to numerically solve formula 3, which is converted into formula 4. The bearing contact angle α of the load Fa can be obtained by using a linear approximation numerical calculation method. The calculation formula is as follows:

[0037]

[0038] According to the relationship between the force and deformation of the paired bearings, the following formula can be obtained:

[0039]

[0040] In the above formula:

[0041] α1 is the contact angle of bearing 1;

[0042] α2 is the contact angle of bearing 2;

[0043] Use the following formula to calculate the pre-deformation δp of the paired bearing under the preload Fp:

[0044]

[0045] In the above formula, Fp is the preload borne by the paired bearings, and δp is the predeformation of the two bearings.

[0046] Preferably, in step 5, the indeterminate equations are changed into definite equations by assuming different preloads Fp; the preloads required under different axial load conditions need to be solved by the above calculation formula.

[0047] Preferably, in step 2, when the paired bearings are subjected to an axial load F a When the load on bearing 1 increases, the deformation increases; for bearing 2, the axial load Fa With preload F P The displacement produced is opposite. As the axial load increases, the axial displacement decreases. When the displacement of bearing 2 is zero, bearing 2 is unloaded and no longer bears the load. The design constraint condition of the bearing preload is that under the maximum load, the preload of bearing 2 is not completely unloaded.

[0048] Preferably, under given axial load conditions, the preload is determined by a numerical iterative approximation method, and the specific method is:

[0049] First, determine the initial value of the preload and the iteration step according to the maximum axial load. Use Formula 8, Formula 9, and Formula 10 to calculate the contact angle α2 of bearing 2 according to the current preload, and calculate the deformation δ of bearing 2 when the axial load is 0. 2min , the axial load is F amax The deformation of bearing 2 is δ 2min .

[0050] Preferably, the minimum displacement δ of the bearing 2 is determined 2min is greater than zero. If the minimum displacement is greater than zero, the current preload is the required preload. The iterative calculation is completed. If the minimum displacement δ 2min If it is less than zero, the iteration step size is recalculated using formula 11. The step size can be adaptively adjusted according to the current calculation result, and a larger iteration step size is used at the beginning of the iteration.

[0051] The beneficial effects of the present invention are:

[0052] 1. In the present invention, the method for calculating the preload of an angular contact bearing based on Newton-Raphson is to derive a set of equations for solving the preload based on Newton-Raphson by constructing a mathematical model of the bearing preload;

[0053] 2. In the present invention, the method for calculating the preload of an angular contact bearing based on Newton-Raphson innovatively adopts a linear approximation method for the preload, converts the solution of an indeterminate set of equations into a solution of a definite equation, and uses Newton-Raphson to solve the precise value of the preload;

[0054] 3. In the present invention, the method for calculating the preload of angular contact bearings based on Newton-Raphson is more versatile than the traditional method in that it uses a numerical calculation method to solve the preload, and can accurately calculate the preload of the paired bearings according to different loads.

[0055] 4. In the present invention, the method for calculating the preload of an angular contact bearing based on Newton-Raphson adopts an adaptive step adjustment method to greatly improve the efficiency of preload calculation while meeting the accuracy requirement. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is a schematic diagram of the axial load and axial displacement structure;

[0057] Figure 2 It is a schematic diagram of the paired bearing structure;

[0058] Figure 3 Flow chart for bearing preload calculation. DETAILED DESCRIPTION

[0059] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0060] Reference Figure 1 - Figure 3 , a method for calculating the preload of an angular contact bearing based on Newton-Raphson, comprising the following steps:

[0061] Step 1: Input bearing structural parameters and maximum axial load F amax ;

[0062] Step 2: Calculate the initial contact angle α°, the calculation formula is:

[0063]

[0064] Calculate the displacement constant K using the formula:

[0065]

[0066] In the above formula:

[0067] Fa is the axial load borne by a single angular contact bearing;

[0068] K is the axial displacement constant;

[0069] The initial contact angle of the bearing changes from α° to α;

[0070] Step 3: Determine the initial value of the preload F p0 =0.2*F amax , initial iteration step length step =

[0071] 0.001*F amax ;

[0072] Step 4: Use the current preload F p Calculate the pre-deformation δ P ;

[0073] Step 5: Use the following formula to calculate the contact angle α2 of bearing 2 and the deformation δ of bearing 2 when the axial load is 0 2max , the axial load is F amax Deformation of bearing 2 δ 2min :

[0074]

[0075] Calculate the Jacobi matrix J(α) based on f1(α1,α2), f2(α1,α2);

[0076]

[0077] In the above formula:

[0078]

[0079] Step 6: Determine the deformation of bearing 2 δ 2min Is it greater than 0? If it is satisfied, jump to step 9; if it is not satisfied, jump to step 7;

[0080] Step 7: Use formula 11 to calculate the new iteration step size:

[0081] step=0.001*Famax*1+absδ2min / δ2max*100 formula 11;

[0082] Step 8: Use the new iteration step to update the preload value and jump to step 5;

[0083] Step 9: The preload in the current step is the preload required by the paired bearing;

[0084] In this embodiment, in step 2, the parameter K is calculated using the following formula:

[0085] K=-26451B 3 +36402B 2 +19704B-16.827 Formula 3

[0086] In the above formula, B is the total curvature of the bearing.

[0087] In this embodiment, in step 2, the Newton-Raphson method is used to numerically solve formula 3, which is converted into formula 4. The bearing contact angle α of the load Fa can be calculated using a linear approximation numerical calculation method. The calculation formula is as follows:

[0088]

[0089] According to the relationship between the force and deformation of the paired bearings, the following formula can be obtained:

[0090]

[0091] In the above formula:

[0092] α1 is the contact angle of bearing 1;

[0093] α2 is the contact angle of bearing 2;

[0094] Use the following formula to calculate the pre-deformation δp of the paired bearing under the preload Fp:

[0095]

[0096] In the above formula, Fp is the preload borne by the paired bearings, and δp is the predeformation of the two bearings.

[0097] In this embodiment, in step 5, the indeterminate equation group is changed into a definite equation by assuming different preloads Fp; the preload required under different axial load conditions needs to be solved by the above calculation formula.

[0098] In this embodiment, in step 2, when the paired bearings are subjected to the axial load F a When the load on bearing 1 increases, the deformation increases; for bearing 2, the axial load F a With preload F P The displacement produced is opposite. As the axial load increases, the axial displacement decreases. When the displacement of bearing 2 is zero, bearing 2 is unloaded and no longer bears the load. The design constraint condition of the bearing preload is that under the maximum load, the preload of bearing 2 is not completely unloaded.

[0099] In this embodiment, under given axial load conditions, the preload is determined by a numerical iterative approximation method, and the specific method is:

[0100] First, determine the initial value of the preload and the iteration step according to the maximum axial load. Use Formula 8, Formula 9, and Formula 10 to calculate the contact angle α2 of bearing 2 according to the current preload, and calculate the deformation δ of bearing 2 when the axial load is 0. 2min , the axial load is F amax The deformation of bearing 2 is δ 2min .

[0101] In this embodiment, the minimum displacement δ of the bearing 2 is determined 2min is greater than zero. If the minimum displacement is greater than zero, the current preload is the required preload. The iterative calculation is completed. If the minimum displacement δ 2minIf it is less than zero, the iteration step size is recalculated using formula 11. The step size can be adaptively adjusted according to the current calculation result, and a larger iteration step size is used at the beginning of the iteration.

[0102] The specific embodiments of the present invention are as follows:

[0103] Take a pair of angular contact bearings 71821AC as an example to calculate the preload of back-to-back installation. The parameters of bearing 718212AC are as follows:

[0104] Inner ring raceway diameter

[0105] Outer ring raceway diameter

[0106] Inner ring raceway curvature r i =3.27mm

[0107] Outer ring raceway curvature r o =3.33mm

[0108] Ball diameter D = 6.35 mm

[0109] Number of balls: 45

[0110] Radial dynamic load rating 22.7KN

[0111] Radial static load rating 38.3KN

[0112] According to the light load condition, the maximum axial load is F amax =0.07C r =1600N. Formula 1 is used to calculate the initial contact angle α°=25°, and Formula 2 is used to calculate the displacement constant K=813.8MPa. The initial value of the preload is 320N, and the initial step is 1.6N. After 16 iterations, the minimum deformation of bearing 2 is greater than 0, and the preload at this time is 531.82N, that is, when the preload is 531.82N, the maximum axial load is 1600N and bearing 2 will not be completely unloaded. The iterative calculation results are shown in Table 1.

[0113] According to different load conditions of the bearing, this method is used to calculate the precise preload, and the calculation results are shown in Table 2. The existing bearing preload generally adopts a simplified calculation method, that is, the preload is 0.35 times the maximum axial load. When the bearing works in a light load state, the calculation results of the two methods are similar. As the maximum axial load increases, the ratio of the preload to the maximum axial load decreases. There is a large error in calculating the preload according to 0.35 times the maximum axial load. This method has better versatility and calculation accuracy.

[0114] Table 1 Preload iteration calculation results

[0115]

[0116]

[0117] Table 2 Calculation results of preload for different axial loads

[0118]

[0119] The above is a detailed introduction to a method for calculating the preload of an angular contact bearing based on Newton-Raphson provided by the present invention. This article uses specific embodiments to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea. It should be pointed out that for ordinary technicians in this technical field, without departing from the principle of the present invention, the present invention can also be improved and modified, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.

Claims

1. A method for calculating the preload of an angular contact bearing based on Newton-Raphson, characterized in that: The steps include: Step 1: Input bearing structural parameters and maximum axial load F amax ; Step 2: Calculate the initial contact angle α°, the calculation formula is: Calculate the displacement constant K using the formula: In the above formula: F a A single angular contact bearing bears axial loads; K is the axial displacement constant; The initial contact angle of the bearing changes from α° to α; Step 3: Determine the initial value of the preload F p0 =0.2*F amax , initial iteration step step=0.001*F amax ; Step 4: Use the current preload F p Calculate the pre-deformation δ P ; Step 5: Use the following formula to calculate the contact angle α2 of bearing 2 and the deformation δ of bearing 2 when the axial load is 0 2max , the axial load is F amax Deformation of bearing 2 δ 2min : Calculate the Jacobi matrix J(α) based on f1(α1, α2), f2(α1, α2); In the above formula: Step 6: Determine the deformation of bearing 2 δ 2min Is it greater than 0? If it is satisfied, jump to step 9; if it is not satisfied, jump to step 7; Step 7: Use formula 11 to calculate the new iteration step size: step=0.001*Famax*(1+abs(δ2min / δ2max)*100) Formula 11; Step 8: Use the new iteration step to update the preload value and jump to step 5; Step 9: The preload in this step is the preload required for the paired bearings.

2. A method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: In step 2, the parameter K is calculated using the following formula: K=-26451B 3 +36402B 2 +19704B-16.827 Formula 3 In the above formula, B is the total curvature of the bearing.

3. The method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: In the step 2, the Newton-Raphson method is used to numerically solve Formula 3 and convert it into Formula 4. The bearing contact angle α of the load Fa can be calculated using the linear approximation numerical calculation method. The calculation formula is as follows: According to the relationship between the force and deformation of the paired bearings, the following formula can be obtained: In the above formula: α1 is the contact angle of bearing 1; α2 is the contact angle of bearing 2; Use the following formula to calculate the pre-deformation δp of the paired bearing under the preload Fp: In the above formula, Fp is the preload borne by the paired bearings, and δp is the predeformation of the two bearings.

4. The method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: In step 5, by assuming different preloads Fp, the indeterminate equations are changed into definite equations; the preloads required under different axial load conditions need to be solved by the above calculation formula.

5. The method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: In step 2, when the paired bearings are subjected to the axial load F a When the load on bearing 1 increases, the deformation increases; for bearing 2, the axial load F a With preload F P The displacement produced is opposite. As the axial load increases, the axial displacement decreases. When the displacement of bearing 2 is zero, bearing 2 is unloaded and no longer bears the load. The design constraint condition of the bearing preload is that under the maximum load, the preload of bearing 2 is not completely unloaded.

6. The method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: Under given axial load conditions, the preload is determined by numerical iterative approximation. The specific method is: First, determine the initial value of the preload and the iteration step according to the maximum axial load. Use Formula 8, Formula 9, and Formula 10 to calculate the contact angle α2 of bearing 2 according to the current preload, and calculate the deformation δ of bearing 2 when the axial load is 0. 2min , the axial load is F amax The deformation of bearing 2 is δ 2min .

7. The method for calculating the preload of an angular contact bearing based on Newton-Raphson according to claim 1, characterized in that: Determine the minimum displacement δ of bearing 2 2min is greater than zero. If the minimum displacement is greater than zero, the current preload is the required preload. The iterative calculation is completed. If the minimum displacement δ 2min If it is less than zero, the iteration step size is recalculated using formula 11. The step size can be adaptively adjusted according to the current calculation result, and a larger iteration step size is used at the beginning of the iteration.

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