Random rotating speed-oriented gear meshing stiffness calculation method

Abstract

The invention provides a random rotation speed-oriented gear meshing stiffness calculation method, which belongs to the technical field of gear performance evaluation, and comprises the following steps: describing the actual operation state of a gear system by using a random rotation speed; determining the total potential energy of the gear according to the acting force of the meshing points; solving the total potential energy of the gear to obtain a potential energy calculation function about a meshing rotation angle; determining a meshing stiffness calculation function about the meshing rotation angle by using the potential energy calculation function; solving an average value of a single-tooth meshing area and a double-tooth meshing area of the meshing stiffness calculation function; and the meshing rigidity is simplified into a square wave function through the average value, and the meshing rigidity is changed into a time function from a corner function through the generated random rotating speed. According to the method, the running state of the gear system is described by using the random rotating speed, the influence caused by random fluctuation of the rotating speed is considered, the meshing rigidity of a wind power gear box and the like in the actual working environment can be reflected more truly, and the accuracy and applicability of meshing rigidity calculation are remarkably improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of gear performance evaluation, and more specifically, it particularly relates to a calculation method for gear meshing stiffness facing random rotational speeds. Background Art

[0002] Gear systems have significant advantages such as high transmission efficiency, high transmission accuracy, and high reliability. Therefore, gear transmission has been widely applied in mechanical equipment. Gear transmission is a typical non-linear vibration system, including non-linear factors such as time-varying meshing stiffness and backlash. Phenomena such as jump discontinuity, subharmonic resonance, and chaotic motion are common in gear transmission systems. Therefore, the vibration response of gear systems is extremely complex, directly affecting the smoothness and reliability of gear operation. Time-varying meshing stiffness is the main excitation source for gear systems to generate vibration. Accurately characterizing the time-varying meshing stiffness of gears is a prerequisite for accurately predicting the dynamic response of the system. Currently, there are many research methods for gear system meshing stiffness, but most models assume that gears operate at a constant rotational speed for analysis, ignoring the influence of rotational speed changes under actual working conditions on gear meshing stiffness. Summary of the Invention

[0003] In order to overcome the deficiencies of the prior art, the purpose of the present invention is to provide a calculation method for gear meshing stiffness facing random rotational speeds.

[0004] To achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0005] A calculation method for gear meshing stiffness facing random rotational speeds, comprising:

[0006] Step 1: Use KL expansion to generate random rotational speeds and describe the operating state of the gear system using the random rotational speeds;

[0007] Step 2: Determine the total potential energy of the gear according to the acting force at the meshing point;

[0008] Step 3: Solve the total potential energy of the gear to obtain a potential energy calculation function with respect to the meshing rotation angle;

[0009] Step 4: Use the potential energy calculation function to determine a meshing stiffness calculation function with respect to the meshing rotation angle;

[0010] Step 5: Solve the average values of the single-tooth and double-tooth meshing regions of the meshing stiffness calculation function, and use the average values to obtain a simplified meshing stiffness;

[0011] Step 6: Substitute the random rotational speeds, convert the simplified meshing stiffness into a time function, and change the meshing period of the tooth pair, thereby completing the solution of the gear meshing stiffness at random rotational speeds.

[0012] Preferably, step 2: determining the total potential energy of the gear according to the acting force at the meshing point, includes:

[0013] Using the formula:

[0014]

[0015] determining the total potential energy of the gear; where U represents the total potential energy, F d represents the acting force at the meshing point, k represents the total stiffness, U h represents the contact potential energy, U b1 represents the bending potential energy of the driving gear, U b2 represents the bending potential energy of the driven gear, U s1 represents the shear potential energy of the driving gear, U s2 represents the shear potential energy of the driven gear, U a1 represents the compression potential energy of the driving gear, U a2 represents the compression potential energy of the driven gear, U f1 represents the tooth base potential energy of the driving gear, U f2 represents the tooth base potential energy of the driven gear.

[0016] Preferably, step 3: solving the total potential energy of the gear to obtain a potential energy calculation function with respect to the meshing rotation angle, includes:

[0017] Step 3.1: calculating the Hertz contact stiffness and the bending stiffness based on the tooth width;

[0018] Step 3.2: solving the bending potential energy, the shear potential energy and the compression potential energy by using the theory of elasticity and the mechanics of materials;

[0019] Step 3.3: calculating the bending stiffness, the shear stiffness and the compression stiffness according to the bending potential energy, the shear potential energy and the compression potential energy;

[0020] Step 3.4: obtaining a potential energy calculation function with respect to the meshing rotation angle based on the Hertz contact stiffness, the bending stiffness, the shear stiffness and the compression stiffness.

[0021] Preferably, step 3.1: calculating the Hertz contact stiffness and the bending stiffness based on the tooth width, includes:

[0022] Using the formula:

[0023]

[0024]

[0025] calculating the Hertz contact stiffness and the bending stiffness; where k h represents the Hertz contact stiffness, k f represents the bending stiffness, E represents the modulus of elasticity, L BLet \(b\) denote the tooth width, \(\nu\) denote the Poisson's ratio, \(\theta\) denote the angle between the force acting on the action point and the horizontal; \(u\) f is the root displacement; \(S\) f is the root thickness; \(L\) * , \(M\) * , \(P\) * , \(Q\) * are functions of \(R\) a / \(R\) f and \(\theta\) f for the gear body deformation coefficient, \(R\) a is the gear shaft radius; \(R\) f is the root radius; \(\theta\) f is the angle corresponding to half a tooth.

[0026] Preferably, in the step 3.2, the expressions of the bending potential energy, shear potential energy and compression potential energy are:

[0027]

[0028]

[0029]

[0030] wherein, \(F\) b = \(F\) d \(\cos\theta\); \(F\) a = \(F\) d \(\sin\theta\); \(x\) b is the distance between the meshing point and the gear center line; \(y\) θ is the horizontal distance between the meshing point and the origin; is the shear modulus; \(y_1\), \(y_2\) are the horizontal coordinates of the starting point and the ending point of the transition curve; \(I\) y1 , \(I\) y2 , \(S\) y1 , \(S\) y2 are the sectional moment of inertia and cross-sectional area at any position on the transition curve and involute; \(M_1 = F\) b (\(y\) θ -\(y_1\)) - \(F\) a \(x\) θ and \(M_2 = F\) b (\(y\) θ -\(y_2\)) - \(F\) a \(x\) θ are the torques generated by the meshing force on any point on the transition curve and involute respectively.

[0031] Preferably, in the step 3.3, the bending stiffness is obtained by integrating with the angular displacement:

[0032]

[0033] Shear stiffness:

[0034]

[0035] Compression stiffness:

[0036]

[0037] where x θ , y θ , y1, y2, I y1 , I y2 , S y1 , S y2 are all expressed as functions of the angular displacements γ or τ;

[0038] From the properties of the involute:

[0039] x θ = r b [(θ + θ b ) cos θ + sin θ], y θ = r b [(θ + θ b ) sin θ + cos θ], where θ b represents half of the base tooth angle, y c is the horizontal distance between the starting point of the involute and the origin, and can be expressed as y C = r b [(τ C + θ b ) sin τ C + cos τ C ;

[0040]

[0041] y1 = r cos φ - (a d / sin γ + r ρ ) sin(γ - φ), x2 = r b [(τ + θ b ) cos τ - sin τ];

[0042] y2 = r b [(τ + θ b ) sin τ - cos τ];

[0043]

[0044]

[0045] α C is the pressure angle corresponding to point c; r C is the radius of point c; is the addendum coefficient; m is the module of the gear; θC is the angle between point c and the x-axis; x1 and x2 are the distances from any point on the transition curve and involute to the vertical coordinate; γ represents the variable with respect to angular displacement; φ is the radian value corresponding to different positions; a d is the distance from the center of the tool tip rounding to the midline; r ρ is the tool tip rounding radius; b1 is the distance from the center of the tool tip rounding to the center line of the tool tooth groove.

[0046] Preferably, in the step 5, the meshing stiffness calculation function with respect to the meshing rotation angle is:

[0047]

[0048] where is the potential energy calculation function with respect to the meshing rotation angle, is the meshing stiffness calculation function with respect to the meshing rotation angle;

[0049] The average values of the meshing stiffness in the single-tooth region and double-tooth region are solved by integration:

[0050]

[0051] In the formula, K d is the average value of the meshing stiffness in the double-tooth region, K s is the average value of the meshing stiffness in the single-tooth region, is the meshing angular displacement in the double-tooth region, is the meshing angular displacement in the single-tooth region, and they can be solved by the following formula:

[0052]

[0053]

[0054] In the formula, ε is the contact ratio.

[0055] Preferably, in the step 6: Substitute the random rotational speed, convert the simplified meshing stiffness into a time function, and change the meshing period of the tooth pair, so as to complete the solution of the gear meshing stiffness at the random rotational speed, including:

[0056] Step 6.1: Based on the integral random rotational speed sample function, obtain the expression function of the meshing rotation angle with respect to time;

[0057] Step 6.2: Divide the rotational meshing angle into zones to obtain the angular displacement expressions in the double-meshing zone and single-meshing zone;

[0058] Step 6.3: Obtain the times corresponding to single-tooth meshing and double-tooth meshing from the time function of the meshing rotation angle;

[0059] Step 6.4: The stiffness in the time period of single-tooth meshing takes Ks , the meshing stiffness during the double-tooth meshing period is taken as K d , and thus the time-varying meshing stiffness of the gear system at random rotational speeds is obtained.

[0060] Preferably, in the step 6.1, the expression function of the meshing angle with respect to time is:

[0061]

[0062] where represents the mean value of the random rotational speed at a point, N is the number of items of the intercepted sample, ξ i (t) is a set of uncorrelated standard normal distribution random variables with a mean of 0 and a standard variance of 1, λ i and f i (t) are the eigenvalue and eigenfunction respectively, and t represents time.

[0063] Preferably, in the step 6.4, the meshing stiffness of the gear system at random rotational speeds is:

[0064]

[0065] where K d is the average value of the double-tooth meshing area of the meshing stiffness, K s is the average value of the single-tooth meshing area of the meshing stiffness, and i is the number of meshing teeth.

[0066] The beneficial effect of a method for calculating the meshing stiffness of gears facing random rotational speeds provided by the present invention is that: compared with the prior art, by using the random rotational speed to describe the operating state of the gear system, the present invention takes into account the influence brought by the rotational speed fluctuation. This method can more truly reflect the dynamic characteristics under the actual working environment, and significantly improves the accuracy and applicability of the meshing stiffness calculation. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below only represent some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained based on these drawings without creative efforts.

[0068] Figure 1 is a flowchart of a method for calculating the meshing stiffness of gears facing random rotational speeds provided by an embodiment of the present invention;

[0069] Figure 2 is a schematic diagram of the solution principle of the meshing stiffness of gears provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0070] In order to make the technical problems, technical solutions and beneficial effects to be solved by the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0071] Please refer to Figure 1 , to achieve the above object, the technical solution adopted by the present invention is: a method for calculating the meshing stiffness of gears facing random rotational speeds, including:

[0072] Step 1: Use random rotational speed to describe the operating state of the gear system;

[0073] The present invention mainly considers the randomness of the rotational speed. For example, when the load components of the reducer are affected by random wind, waves or other random external loads, the rotational speed of the reducer is a random process. Therefore, due to the action of the random rotational speed, the stiffness of the internal gears of the reducer also shows a certain degree of randomness. Usually, the calculated meshing stiffness of gears is periodically changing and cannot accurately express the meshing stiffness under the actual working conditions. Therefore, the present invention introduces a method for calculating the meshing stiffness of the gear system under random rotational speed. Let the random rotational speed be a one-dimensional Gaussian random field with respect to time:

[0074]

[0075] In the formula: represents the mean value of the random rotational speed at a point, N is the number of items in the intercepted sample, ξ i (t) is a set of uncorrelated standard normal distribution random variables with a mean of 0 and a standard deviation of 1, λ i and f i (t) are the eigenvalue and eigenfunction respectively, and t represents time.

[0076] is the random rotational speed part, which is expressed by the product of a set of uncorrelated standard normal distribution random variables and the eigenvalue and eigenfunction.

[0077] Step 2: Determine the total potential energy of the gear according to the acting force at the meshing point;

[0078] The meshing stiffness is solved by the improved potential energy method. The total potential energy U of a pair of gears can be expressed as:

[0079]

[0080] In the formula, U h is the contact potential energy, U b is the bending potential energy, U s is the shear potential energy, U a is the compression potential energy, U f is the tooth base potential energy, and k is the total stiffness.

[0081] The contact potential energy is expressed as:

[0082]

[0083] In the formula, F d represents the acting force at the meshing point, and k h is the Hertz contact stiffness.

[0084] The bending potential energy is expressed as:

[0085]

[0086] In the formula, F d represents the acting force at the meshing point, and k b is the bending stiffness.

[0087] The shear potential energy is expressed as:

[0088]

[0089] In the formula, F d represents the acting force at the meshing point, and k s is the shear stiffness.

[0090] The compression potential energy is expressed as:

[0091]

[0092] In the formula, F d represents the acting force at the meshing point, and k a is the axial compression stiffness.

[0093] The tooth base potential energy is expressed as:

[0094]

[0095] In the formula, F d represents the acting force at the meshing point, and k f is the tooth base stiffness.

[0096] Furthermore, the total potential energy can be expressed as:

[0097]

[0098] The subscripts 1 and 2 represent the driving and driven wheels respectively. The total stiffness of a pair of meshing teeth can be expressed as:

[0099]

[0100] The total stiffness of two pairs of meshing teeth can be expressed as:

[0101]

[0102] Step 3: Solve the total potential energy of the gear to obtain the potential energy calculation function with respect to the meshing rotation angle;

[0103] Step 4: Use the potential energy calculation function to determine the meshing stiffness calculation function with respect to the meshing rotation angle;

[0104] In the above Steps 3 - 4, solve the stiffness of each part:

[0105]

[0106] In the formula: E is the modulus of elasticity; L B is the tooth width; v is the Poisson's ratio.

[0107]

[0108] In the formula: θ is the angle between the force acting on the point and the horizontal; u f is the tooth root displacement; S f is the tooth root thickness; L * , M * , P * , Q * are functions of R a / R f and θ f , R a is the gear shaft radius; R f is the tooth root radius; θ f is the angle corresponding to half a tooth. L * , M * , P * , Q * are the gear body deformation coefficients. (In general calculations, take values according to the preset standards)

[0109]

[0110]

[0111]

[0112]

[0113] In the formula, χ = 3 - 4v;

[0114] γ k,h , λ k,h , δ k,h are the stress distribution coefficients of k.

[0115] Therefore, the bending, shear, and axial compression stiffness of the gear part are obtained through two representations of the strain energy in elastic mechanics and material mechanics:

[0116]

[0117]

[0118]

[0119] y C represents the coordinate of point c on the y-axis, y D represents the abscissa of point D;

[0120] where F b = F d cosθ; F a = F d sinθ; x b is the distance between the meshing point and the center line of the gear; y θ is the horizontal distance between the meshing point and the origin; is the shear modulus; y1, y2 are the horizontal coordinates of the starting and ending points of the transition curve; I y1 , I y2 , S y1 , S y2 are the moment of inertia of the cross-section and the cross-sectional area at any position on the transition curve and the involute; M1 = F b (y θ - y1) - F a x θ and M2 = F b (y θ - y2) - F a x θ are the torques generated by the meshing force on any point on the transition curve and the involute respectively.

[0121] As Figure 2 shown, when angular displacement is used for integration, the bending stiffness of the tooth part is:

[0122]

[0123] The shear stiffness of the gear part is:

[0124]

[0125] The axial compression stiffness is:

[0126]

[0127] α0 is the pressure angle; τ is the angle between the meshing force and the vertical direction; τ c is the angle between the tooth root point c and the vertical direction;

[0128] where x θ , y θ , y1, y2, Iy1 , I y2 , S y1 , S y2 are all expressed as functions of the angular displacements γ or τ.

[0129] From the properties of the involute:

[0130] x θ = r b [(θ + θ b ) cosθ + sinθ], y θ = r b [(θ + θ b ) sinθ + cosθ], where θ b represents half of the base tooth angle, y c is the horizontal distance between the starting point of the involute and the origin, and can be expressed as y C = r b [(γ C + θ b ) sinτ c + cosτ c .

[0131] r b is the base circle radius; N is the number of teeth, τ c is the angle between the root c point and the vertical direction;

[0132] In the formula:

[0133]

[0134] y1 = r cosφ - (a d / sinγ + r ρ ) sin(γ - φ), x2 = r b [(τ + θ b ) cosτ - sinτ];

[0135] y2 = r b [(τ + θ b ) sinτ - cosτ];

[0136]

[0137] α C is the pressure angle corresponding to the c point; r C is the radius of the c point; is the addendum coefficient; m is the module of the gear; θ C is the angle between the c point and the x-axis; x1 and x2 are the distances from any point on the transition curve and the involute to the vertical coordinate; γ represents a variable related to the angular displacement; φ is the radian value corresponding to different positions; a dis the distance from the center of the tool tip fillet to the center line; r ρ is the tool tip fillet radius; b1 is the distance from the center of the tool tip fillet to the center line of the tool tooth groove.

[0138] In summary, the calculation of potential energy is a function of θ:

[0139]

[0140] Therefore, its meshing stiffness is also a function of the rotation angle θ:

[0141]

[0142] Step 5: Solve the average values of the single-tooth and double-tooth meshing regions of the meshing stiffness calculation function, and use the average values to obtain the simplified meshing stiffness;

[0143] Step 6: Substitute the random rotational speed, convert the simplified meshing stiffness into a time function, and change the meshing period of the tooth pair, so as to complete the solution of the gear meshing stiffness at the random rotational speed.

[0144] Specifically, solve the average values of the meshing stiffness in the single-tooth region and the double-tooth region by integration:

[0145]

[0146] In the formula, K d is the meshing stiffness of the double-tooth region, K s is the meshing stiffness of the single-tooth region, is the meshing angular displacement of the double-tooth region, is the meshing angular displacement of the single-tooth region, and they can be solved by the following formula:

[0147]

[0148]

[0149] In the formula, ε is the contact ratio.

[0150] Based on the above, the meshing rotation angle can be obtained for the random rotational speed:

[0151]

[0152] In the formula: t represents the time for a pair of meshing gears to rotate; represents the random rotational speed at any time.

[0153] The meshing rotation angle can be transformed into:

[0154]

[0155] During the meshing process of the gear, the angles of the single and double meshing zones are fixed, but the rotational speed is random, resulting in different rotational times for each zone.

[0156] Use to represent the angular displacement of the double meshing zone, and use to represent the angular displacement of the single meshing zone:

[0157]

[0158]

[0159] By inversely solving the above numbers (21) and (22), the time of the single and double tooth intervals during one meshing can be solved.

[0160] Therefore, the meshing stiffness can be expressed as:

[0161]

[0162] After calculating the meshing stiffness of the gear system, the dynamic response and vibration characteristics of the gear under random working conditions can be predicted, which can help engineers understand its working performance and reasonably design the gear to reduce vibration and improve the reliability of the gear system.

[0163] The present invention describes the actual operating state of the gear system by using random rotational speed, considering the influence brought by the random fluctuation of the rotational speed. This method can more realistically reflect the dynamic characteristics of actual working environments of equipment such as wind turbines and hydro-generator sets, and significantly improve the accuracy and applicability of meshing stiffness calculation.

[0164] The present invention also provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the steps in the above-mentioned method for calculating gear meshing stiffness for random rotational speed are implemented.

[0165] Compared with the prior art, the beneficial effects of the computer-readable storage medium provided by the present invention are the same as those of the above-mentioned method for calculating gear meshing stiffness for random rotational speed, and will not be elaborated here.

[0166] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention shall be included in the protection scope of the present invention.

Claims

1. A calculation method for gear meshing stiffness facing random rotational speeds, characterized in that Including: Step 1: Generate random rotational speeds using KL expansion and describe the operating state of the gear system with the random rotational speeds; Step 2: Determine the total potential energy of the gear according to the acting force at the meshing point; Step 3: Solve the total potential energy of the gear to obtain a potential energy calculation function with respect to the meshing rotation angle; Step 4: Use the potential energy calculation function to determine a meshing stiffness calculation function with respect to the meshing rotation angle; Step 5: Solve the average values of the single-tooth and double-tooth meshing regions of the meshing stiffness calculation function and obtain a simplified meshing stiffness using the average values; Step 6: Substitute the random rotational speed, convert the simplified meshing stiffness into a time function, change the meshing period of the tooth pair, and thus complete the solution of the gear meshing stiffness at random rotational speeds.

2. The gear meshing stiffness calculation method for random rotational speeds according to claim 1, characterized in that, The said Step 2: Determining the total potential energy of the gear according to the acting force at the meshing point includes: Adopting the formula: Determine the total potential energy of the gear; where, U represents the total potential energy, F d represents the acting force at the meshing point, k represents the total stiffness, U h represents the contact potential energy, U b1 represents the bending potential energy of the driving gear, U b2 represents the bending potential energy of the driven gear, U s1 represents the shear potential energy of the driving gear, U s2 represents the shear potential energy of the driven gear, U a1 represents the compression potential energy of the driving gear, U a2 represents the compression potential energy of the driven gear, U f1 represents the tooth base potential energy of the driving gear, U f2 represents the tooth base potential energy of the driven gear.

3. The gear meshing stiffness calculation method for random rotational speeds according to claim 2, wherein The said Step 3: Solving the total potential energy of the gear to obtain a potential energy calculation function with respect to the meshing rotation angle includes: Step 3.1: Calculate the Hertz contact stiffness and bending stiffness based on the tooth width; Step 3.2: Solve the bending potential energy, shear potential energy, and compression potential energy using elastic mechanics and material mechanics; Step 3.3: Calculate the bending stiffness, shear stiffness, and compression stiffness according to the bending potential energy, shear potential energy, and compression potential energy; Step 3.4: Obtain a potential energy calculation function with respect to the meshing rotation angle based on the Hertz contact stiffness, bending stiffness, shear stiffness, and compression stiffness.

4. A method for calculating the gear meshing stiffness for random rotational speeds as claimed in claim 3, wherein, The said Step 3.1: Calculating the Hertz contact stiffness and bending stiffness based on the tooth width includes: Adopting the formula: Calculate the Hertz contact stiffness and the bending stiffness; where, k h represents the Hertz contact stiffness, k f represents the bending stiffness, E represents the modulus of elasticity, L B represents the tooth width, v represents the Poisson's ratio, θ represents the angle between the force at the action point and the horizontal; u f is the tooth root displacement; S f is the tooth root thickness; L * , M * , P * , Q * are the gear body deformation coefficients.

5. The gear meshing stiffness calculation method for random rotational speeds according to claim 4, characterized in that, In the said Step 3.2, the expressions of the bending potential energy, shear potential energy, and compression potential energy are: Among them, U b is the bending potential energy expression, U s is the shear potential energy expression, U a is the compression potential energy expression, k b is the bending stiffness, k s is the shear stiffness, k a is the axial compression stiffness, F b = F d cosθ; F a = F d sinθ; x b is the distance between the meshing point and the gear center line, θ represents the angle between the force at the action point and the horizontal, y θ is the horizontal distance between the meshing point and the origin; is the shear modulus; y1 and y2 are the horizontal coordinates of the starting and ending points of the transition curve; I y1 , I y2 are the sectional moments of inertia at any position on the transition curve and the involute, S y1 , S y2 are the cross-sectional areas at any position on the transition curve and the involute; M1 is the moment generated by the meshing force on any point on the transition curve, M2 is the moment generated by the meshing force on any point on the involute, y C represents the coordinate of point c on the y-axis, y D represents the abscissa of point D.

6. The gear meshing stiffness calculation method for random rotational speeds according to claim 5, wherein In the said Step 3.3, the bending stiffness is obtained by integrating using the angular displacement: Shear stiffness: Compression stiffness: where x θ , y θ , y1, y2, I y1 , I y2 , S y1 , S y2 are all functions of the angular displacements γ or τ; From the properties of the involute: x θ = r b [(θ + θ b ) cosθ + sinθ], y θ = r b [(θ + θ b ) sinθ + cosθ], where θ b represents half of the base tooth angle, y c is the horizontal distance from the starting point of the involute to the origin and can be expressed as y C = r b [(τ C + θ b ) sinτ C + cosτ C ; r b represents the base circle radius; θ C represents the angle between point c and the y-axis; r represents the pitch circle radius, a d is the distance from the center of the tool tip fillet to the center line, α0 represents the pressure angle of the pitch circle, τ c represents the angle between the tooth root point c and the vertical direction, and τ represents the angle between the tangent line passing through a certain point and the pitch circle and the y-axis; y1 = r×cosφ - (a d / sinγ + r ρ )×sin(γ - φ), x2 = r b [(τ + θ b )cosτ - sinτ]; y2 = r b [(τ + θ b )sinτ - cosτ]; α C is the pressure angle corresponding to point c; r C is the radius of point c; is the addendum coefficient; m is the module of the gear; θ C is the angle between point c and the x-axis; x1 and x2 are the distances from any point on the transition curve and the involute to the vertical coordinate; γ represents the variable with respect to the angular displacement; φ is the radian value corresponding to different positions; a d is the distance from the center of the tool tip fillet to the center line; r ρ is the tool tip fillet radius; b1 is the distance from the center of the tool tip fillet to the center line of the tool tooth groove.

7. The method for calculating gear meshing stiffness for random rotational speeds according to claim 6, characterized in that In the said Step 5, the meshing stiffness calculation function with respect to the meshing rotation angle is: Among them, is the potential energy calculation function with respect to the meshing rotation angle, is the meshing stiffness calculation function with respect to the meshing rotation angle; is the derivative function of displacement with respect to the rotation angle, is the derivative function of the meshing force with respect to the rotation angle; Solve the average values of the meshing stiffness in the single-tooth region and the double-tooth region by integration: where K d is the average meshing stiffness of the double-tooth region, and K s is the average meshing stiffness of the single-tooth region, is the angular displacement of meshing in the double-tooth region, is the angular displacement of meshing in the single-tooth region, and they can be solved by the following formula: where ε is the contact ratio, is the angular displacement of single-tooth engagement.

8. A method for calculating the gear meshing stiffness for random rotational speeds according to claim 7, characterized in that, The said Step 6: Substitute the random rotational speed, convert the simplified meshing stiffness into a time function, change the meshing period of the tooth pair, and thus complete the solution of the gear meshing stiffness at random rotational speeds, including: Step 6.1: Based on the integral random rotational speed sample function, obtain an expression function of the meshing rotation angle with respect to time; Step 6.2: Divide the rotational meshing angle into zones to obtain the angular displacement expressions in the double-meshing zone and the single-meshing zone; Step 6.3: Obtain the times corresponding to single-tooth meshing and double-tooth meshing from the time function of the meshing rotation angle; Step 6.4: The stiffness during the single-tooth meshing period is taken as K s , and the meshing stiffness during the double-tooth meshing period is taken as K d , thereby obtaining the time-varying meshing stiffness of the gear system at random rotational speeds.

9. A method for calculating the gear meshing stiffness for random rotational speeds according to claim 8, characterized in that In the said Step 6.1, the expression function of the meshing rotation angle with respect to time is: Among them, represents the magnitude of the meshing rotation angle in one cycle, represents the magnitude of the angular velocity, represents the mean value of the random rotational speed at a point, N is the number of terms of the intercepted sample, and ξ i (t) is a set of uncorrelated standard normal distribution random variables with a mean of 0 and a standard deviation of 1, and λ i and f i (t) are the eigenvalue and the eigenfunction respectively, and t represents time.

10. A method for calculating gear meshing stiffness for random rotational speeds according to claim 9, characterized in that, In Step 6.4, the meshing stiffness of the gear system at random rotational speeds is: Among them, K d is the average value of the meshing stiffness in the double-tooth meshing zone, and K s is the average value of the meshing stiffness in the single-tooth meshing zone, and i is the number of meshing teeth.

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